When Nash meets Stackelberg

Margarida Carvalho1,2, Gabriele Dragotto2, Felipe Feijoo4, Andrea Lodi2.3, and SriramSankaranarayanan5

1 Universite de Montreal carvalho@iro.umontreal.ca2 CERC Data Science, Polytechnique Montrealgabriele.dragotto, andrea.lodi@polymtl.ca

3 Jacobs Technion-Cornell Institute, Cornell Tech, New York City4 Pontifica Universidad Catolica de Valparaıso felipe.feijoo@pucv.cl

5 Production and Quantitative Methods, Indian Institute of Management, Ahmedabad srirams@iima.ac.in

Abstract. Motivated by international energy trade between countries with profit-maximizing domesticproducers, we analyze Nash games played among Stackelberg games leaders (NASP). In particular, wefocus on NASPs where each leader program is a linear bilevel with quadratic convex followers, and weassume the standard optimistic version of such bilevels. We prove it is both Σp

2 -hard to decide if thegame has a pure-strategy (PNE) or a mixed-strategy Nash equilibrium (MNE). We provide a finitealgorithm that computes exact MNEs for NASPs when there is at least one or returns a certificate ifno MNE exists. To enhance computational speed, we introduce an inner approximation hierarchy thatincreasingly grows the description of each Stackelberg leader feasible region. Furthermore, we extend thealgorithmic framework to retrieve a PNE if one exists specifically. Finally, we provide computationaltests on a range of NASPs instances inspired by international energy trades.

Keywords: Game Theory, Algorithmic Game Theory, Stackelberg Game, Nash Game, EquilibriumProblems with Equilibrium Constraints, Mixed Integer Programming

1 Games, definitions, and overview

Game theoretical frameworks model complex interactions among agents and are widely employed for real-world applications. Their effectiveness relies on two key ingredients. First, their modeling capabilities forthe specific field of application and the ease of interpretability of such models. Secondly, the power andefficiency of the underlying algorithmic arsenal available to solve these models. In this paper, we study a classof non-cooperative, simultaneous games between the leaders (i.e., the first-level players) of bilevel programswith an optimistic followers’ response. In other words, Stackelberg games’ leaders are playing a Nash gameamong themselves with complete information. We call such problems Nash Games among Stackelberg Leaders( NASPs), schematically represented in Figure 1. NASPs are part of the well-known family of equilibriumproblems with equilibrium constraints ( EPECs), which has a wide variety of applications in energy markets. Aconcise representation of an elementary (or trivial, as more formally defined in Definition 7) NASP between aLatin Stackelberg game and a Greek Stackelberg game is given byLatin Leader

minx,y

: cTx+ dT y +

(G

(ξχ

))T (xy

)(1a)

subject to Ax+By ≤ b (1b)

y ∈ arg miny

fT y : Qy ≤ g − Px

(1c)

Greek Leader

minξ,χ

: αT ξ + βTχ+

(Γ

(xy

))T (ξχ

)(1d)

arX

iv:1

910.

0645

2v5

[cs

.GT

] 7

Sep

202

1

Leader1

F 11 F 1

f1

Leader`

F 1` F `

f`

Fig. 1: A schematic representation of a NASP . The vertical arrows are Stackelberg interactions (i.e, sequentialdecisions), while the horizontal ones are Nash interactions (i.e, simultaneous decisions).

subject to Φξ + Ψχ ≤ ρ (1e)

χ ∈ arg minχ

φTχ : Ωφ ≤ γ −Πξ

. (1f)

The interaction among the Stackelberg leaders happens within their objective function, as in (1a) and (1d).On the other hand, within each Stackelberg game, each leader anticipates the reaction of their distinctfollowers, each of which solves a lower-level parametric linear program. NASPs can be extended (i) tohandle more than two leaders, (ii) to handle more than one follower per leader, (iii) to have the follow-ers of each leader interact in a Nash game, and (iv) to enforce each follower to solve a convex quadraticprogram as opposed to linear programs in (1c) and (1f). A dominant solution concept is the one of Nashequilibrium — namely when each player cannot profitably and unilaterally deviate from the prescribed strategy.

Applications. NASPs, in their full generality, could solve a wide range of problems. We outline three differentpotential applications related to energy, vaccines, and insurances. In this work, we are primarily motivated tomodel international energy markets with climate change-aware regulatory authorities with profit-maximizingdomestic energy producers, and we provide a game-theoretic framework to analyze this problem. In thisgame, energy producers – namely the Stackelberg followers – compete in the domestic market and are usuallysubject to restrictions in the form of tax and caps from the regulatory authorities. The regulatory authorities– namely the Stackelberg leaders – negotiate environmental-conscious agreements for energy trade, thusengaging in a Nash game. The theoretical abstraction NASP models this problem and provides a generalframework to analyze games, in and outside the field of energy, when there are multiple Stackelberg leaders,each with their set of followers, playing a Nash game with each other.Similarly – yet in a different context – NASPs can model a complex drug trade system. For instance, at thetime of writing, the COVID-19 vaccine production and trade situation pose severe threats to the world’simmunization programs and may lessen inter-country cooperation with the so-called vaccine nationalism(Weintraub et al. 2020). In several cases, countries threatened and successfully blocked vaccine exports whilealso imposing strict regulations on indigenous producers (Fleming et al. 2021, Boffey 2021a,b). In this scenario,the homogeneous good would be the vaccine, and in analogy to emission factors, we would see efficacyproperties. Countries act as Stackelberg leaders, regulating vaccines’ trade and incentivizing indigenousproducers (followers). Further, the leaders’ objectives could model a wide variety of tactical requirements (e.g.,prioritize the production of some doses reserved for vulnerable classes, incentivize the exports of prioritizeddoses to neighboring countries, prioritize more effective vaccines).

2

Finally, as a third example, we draw attention to the insurance business. Users of a given good may be subjectto disruptions (e.g., cyberattacks for energy generators) and may need to contract insurance services (Gordonand Loeb 2002, Cashell et al. 2004). Insurers – namely Stackelberg leaders – provide such services at a cost totheir clients – or followers. The NASP framework extends this hierarchical model to a multi-insurer setting,introducing a mechanism of re-insurance. In plain English, the leaders mutually protect their insurances’portfolios, shielding them against large-scale disruptions (e.g., natural disasters). With these motivatingexamples, we now detail our primary contributions.

Primary Contributions

In the first place, we characterize the computational complexity of deciding if a given instance of NASP hasa pure-strategy Nash equilibrium (PNE ). Even with restrictive assumptions – such as single follower andbounded strategy sets for all players – we show it is Σp

2 -hard to decide if the instance has a PNE . Namely,even with oracle access to solve NP-hard problems instantaneously, there would be exponentially many callsto such an oracle to decide the existence of PNE for a given instance of NASP . In other words, withoutsubstantial consequences in complexity theory, this translates to a bound of Ω(22n

) elementary operationsrequired to solve the problem, where n is the size of the representation of the corresponding decision problem.This is quite surprising since, in most literature cases, one can either prove that all games in a consideredcategory have a PNE or prove sufficiently fast that a given instance has no PNE . Second, we consider thecomputational complexity of deciding the existence of mixed-strategy Nash equilibrium (MNE ) for NASPs.We demonstrate that with exactly one follower for each leader and boundedness in every player’s problem, anMNE always exists (Corollary 1). However, even if one of the leaders has an unbounded feasible set, it isagain Σp

2 -hard to decide the existence of an MNE .Third, given these lower bounds to computationally find PNE or MNE for NASPs, we provide a finite-time

algorithm to do so. It retrieves an MNE for an instance of NASP when it exists and provides a (doubleexponentially-large) proof of infeasibility when an MNE does not exist. To the best of our knowledge, this isthe first algorithm to identify MNE or PNE for a game of this type. Fourth, we provide an enhancementto the algorithm to exclusively seek PNEs, or provide proof of infeasibility. This is the case of interest ifmixed-strategies are not implementable in practice. Fifth, we provide another enhancement to the algorithmsto find MNEs and PNE s, with an iterative inner-approximation procedure that proves to be considerablyfaster in practice. We also remark that the negative results (Σp

2 -hard complexity) are for the easier versionof the problem (the latter defined trivial NASP), and our positive algorithmic results are, on the contrary,for the harder version of the problem with multiple followers. Besides, we also present several observations,for instance, Remarks 1 and 3, which enlight on equilibria for Nash games where players solve non-convexoptimization problems.

We believe that the above contributions, both from the complexity and algorithmic (computational) sidesestablish a solid benchmark against which future progress can be measured.

Literature review.

Nash games gained popularity with the Nobel awarded papers of Nash (1951, 1950). Nash proven the existenceof the so-called Nash equilibrium for games with a finite number of players and a finite number of strategies.By definition, these equilibrium strategies ask that no player has an incentive to unilaterally deviate fromthe prescribed strategy. Generally, we distinguish between the pure strategy Nash equilibrium (PNE ) andthe mixed strategy one (MNE ). The latter generalizes the pure one since each strategy in the support ofthe equilibrium has an associated probability of being played. The Nash equilibrium concept extends togames where players have an uncountable set of strategies. From an application perspective, Nash Gamesare extensively adopted for modeling interactions within economic markets. For instance, gas market bilevelformulations usually involve players solving convex optimization problems parametrized in other playersvariables (Egging et al. 2010, Feijoo et al. 2016, Sankaranarayanan et al. 2018, Feijoo et al. 2018, Holz et al.2008, Egging et al. 2008, Stein and Sudermann-Merx 2018). On the other side, the cross-border kidney

3

exchange model (Carvalho et al. 2017), competitive lot-sizing models (Li and Meissner 2011, Carvalho et al.2018b), and the fixed charge transportation model (Sagratella et al. 2019) feature players solving non-convexproblems.

In contrast with Nash games, sequential ones partition the players into different groups, and each groupdecides in a round – or level. If the rounds are two, then the game is known as Stackelberg game (Stackelberg1934, Candler and Norto 1977). Here, the agents playing in the first round are the leaders, while the onesplaying afterward are called followers. When each Stackelberg player solves an optimization problem, then wehave a bilevel program. In general, bilevel formulations can model interactions where leaders have specificadvantages over the followers, such as government taxing companies. Indeed, bilevel formulations allure anourished community of researchers. Bard et al. (1998, 2000) model tax credits strategies in the context ofbiofuel production, and Brotcorne et al. (2008), Labbe and Violin (2013) create bilevel pricing problems.Hobbs et al. (2000), Gabriel and Leuthold (2010), Feijoo and Das (2014) model pricing and environmentalpolicies for energy markets, where power generators are leaders, and network operators are followers.

When multiple leaders – each with possibly multiple followers – seek for an equilibrium between eachother, we fall into the category of EPEC s. Thereby leaders often have a common set of followers, andthe equilibrium of interest is PNE . Sherali (1984) introduced EPEC s where both leaders and followersproduce a homogeneous commodity, and followers adopt a reaction curve. Gabriel et al. (2012) providesa Gauss-Seidel iteration technique to find PNEs for a restricted class of EPEC s, where followers fromdistinct leaders can interact. Ralph and Smeers (2006), and Hu and Ralph (2007) extend the analysis on theexistence of a PNE to specialized classes of EPEC s arising in electricity markets. Leyffer and Munson (2010)introduces a weaker solution concept based on a nonlinear programming reformulation. DeMiguel and Xu(2009) craft the concept of stochastic multi-leader Stackelberg-Nash-Cournot equilibrium for a particularform of investment-production interaction between the players. More recently, Kulkarni and Shanbhag (2014,2015) considered EPEC s with shared constraints, presenting solution concepts and algorithms starting fromthe potentiality of players’ objectives.

Complexity of Equilibria. As previously mentioned, Nash (1950, 1951) proven that a Nash equilibrium forfinite games always exists, and thus the associated decision problem is trivial. However, since the proof isnon-constructive, it already unveils the difficulty of computing an equilibrium. Indeed, even for two-playersfinite games in strategic form, the problem of determining an equilibrium is PPAD-complete (Chen and Deng2006). Furthermore, even for games where equilibria are guaranteed to exist, many variations of associateddecision problems are known to be NP-complete (Gilboa and Zemel 1989). A few illustrative examples arethe existence of two equilibria or the existence of an equilibrium where a player’s payoff exceeds a giventhreshold. Besides, Carvalho et al. (2018a, 2020) proved the existence of PNE and MNE for games whereplayers solve parametrized non-convex problems to be Σp

2 -hard. Under this setting, if players’ strategies arebounded, then an MNE always exists. For congestion games, another widely studied class of Nash games,PNE s always exist due to their potential nature (Rosenthal 1973). Del Pia et al. (2017) focus on congestionsgames where totally unimodular matrices describe the players’ strategies. Within this context, the authorsprove that if players have the same feasible set of strategies, a PNE can be computed in polynomial time. Inany other case, the problem is PLS-complete. For what concerns Stackelberg games’ complexity, the seminalresult of Jeroslow (1985) enlightens the matter. It proves that sequential games’ computational complexityrises one layer up in the polynomial hierarchy for every additional round, even for linear problems. Thereupon,the classification of the computational complexity for NASPs becomes almost natural.

Paper Organization. We organize the manuscript as follows. Section 2 provides definitions and restates someknown results. Section 3 provides the complexity results regarding NASPs. Section 4 presents an algorithmto find MNE for NASP , proves its finiteness and correctness. Section 5 builds on top of the developedalgorithm by extending it with an inner approximation hierarchy and introduces a heuristic for computingPNE . Section 6 presents computational tests, and, finally, Section 7 draws conclusions.

4

2 Preliminaries

In this section, we provide definitions, notations and recall some known results in the context of polyhedraltheory, Nash games, and Stackelberg games.

2.1 Definitions

Nash Games. When players decide simultaneously, and with complete information, we have a Nash Game.As a standard notation in game theory, let the operator (·)−i denote (·) except i.

Definition 1 (Nash games). A Nash game P among n players is a finite tuple of optimization problemsP =

(P 1, . . . , Pn

), where each P i is the problem of the ithplayer. Simultaneously, each player i solves an

optimization problem of the form minxi∈Rni f i(xi;x−i) : xi ∈ Fi, where f i and Fi are their objectivefunction and the feasible set, respectively.

Moreover, we can further characterize a Nash game as (i) simple if, for every player i and for somepositive semidefinite matrix Qi, and ci, Ci of appropriate dimensions, the objective function is in the form

of f i(xi;x−i) = 12x

iTQixi +(ci + Cix−i

)xi, (ii) linear, if Qi = 0 for all i, namely each leader has a linear

objective function, (iii) facile, if the game is simple, and Fi is a polyhedron for all i ∈ [n].

Definition 2 (Simple parameterization). An optimization problem in y has a simple parameterizationwith respect to x ∈ Rn` if the problem is in the form of miny∈Rnf f(y) + (Cx)T y : y ∈ F , Ax + By ≤ b,where f : Rnf → R, and C, A, B, b are matrices and vectors of appropriate dimensions, and F ⊆ Rnf .

A Nash game P = (P 1, . . . , Pn) has a simple parameterization with respect to x ∈ Rn` if each optimizationproblem P 1(x), . . . , Pn(x) has a simple parameterization with respect to x.

Definition 3 (Mixed and Pure-strategy Nash equilibria). Let ν = (ν1, . . . , νn) where νi is a Borelprobability distribution on Fi with finite support. Then, ν is a MNE if – for all i ∈ [n] and xi ∈ Fi –E(f(νi, ν−i)) ≤ E(f(xi, ν−i)) holds. If all the distributions have a singleton support, then the set of strategiesis referred to as PNE.

PNE is a strong notion of equilibrium, and even relatively trivial games — for example, rock-paper-scissors —do not possess one. In contrast, an MNE always exists for finite games (Nash 1950, 1951).

Stackelberg Games. A Stackelberg game is a multi-level game with 2 rounds of decisions. First, the leaderdecides, optimizing their objective subject to some constraints. Subsequently, the followers decide, with theirobjective and constraints now depending upon the leader’s decision (Candler and Townsley 1982).

Definition 4 (Stackelberg game). Let P (x) be a Nash game with a simple parametrization with respectto x, SOL(P (x)) denotes its solution set, and f : Rn`+nf → R. Then, a Stackelberg game is an optimizationproblem of the form min

x∈Rn` ;y∈Rnff(x, y) : (x, y) ∈ F , y ∈ SOL(P (x)).

In a Stackelberg game, the set SOL(P (x)) is parametrized given the leader’s strategy x. Namely, given anupper-level strategy x, the followers should play optimally. Theoretically, each Stackelberg game’s solution isthen a subgame perfect Nash-Equilibrium (SPNE). For the purposes of this manuscript, we only considerSPNEs. The previous definition implies the Stackelberg game to be optimistic. Namely, if the game hasmultiple optimal solutions SOL(P (x)), then y takes the value among SOL(P (x)) benefitting the leader themost. If P (x) is an optimization problem (i.e., one follower), then the optimistic assumption is natural aswe could state that the leader incentivizes the follower by an arbitrarily small amount to choose the mostfavorable solution. However, this could be a strict restriction if there are multiple followers and the followers’equilibrium is not unique. As commented before, the optimistic assumption on the bilevel solution selection isnot restrictive here, because despite having multiple followers, their equilibrium is unique. Such uniquenessfollows from the fact that the followers play a Nash-Cournot game with strictly convex cost functions.

5

Definition 5 (Simple Stackelberg game). A Stackelberg game P is simple if P (x) is a facile Nash gamewith a simple parameterization with respect to the upper-level variables x, F is a polyhedron, and f(x, y) is alinear function.

Definition 6 (NASP). A NASP is a linear Nash game N = (P 1, . . . , P k) where for each i, P i(xi) is asimple Stackelberg game.

Combining Definitions 1, 2 and 4 to 6, a NASP refers to the following game. There is a set L of players calledthe leaders and another set F of players called the followers. Each follower f ∈ F has a unique leader `(f) ∈ L,such that the objective function and the feasible set of the follower f depends only upon the decision variablesof `(f) and other followers f ′ ∈ F such that `(f ′) = `(f). In other words, only the followers of the sameleader interact with each other, and they only interact with their leader. We assume all parameters of theplayers are common information. First, with complete anticipation of their followers’ behavior, simultaneously,every leader ` ∈ L chooses their decision to maximize their utility. Then, every follower f observes theirrespective leader’s (`(f)’s) decision and, simultaneously, every follower chooses their decision, maximizingtheir utility. We assume an optimistic behavior from the followers in the sense that – if there are multipleoptimal strategies for the followers over which they are indifferent – they will choose the strategy whichbenefits their leader the most.

Definition 7 (Trivial NASP). A trivial NASP is a NASP where k = 2, and P 1 and P 2 are simple bilevelgames whose lower levels are linear programs with a simple parameterization with respect to the upper-levelvariables.

The additional assumptions holding on a trivial NASP (as of Definition 7) compared to a general NASP (asof Definition 6) are seemingly strong. We require that each leader has precisely one follower – as opposed tofinitely many followers – and that each follower solves a linear program – as opposed to a quadratic program –with a simple parameterization with respect to the upper-level variables. The game between the Latin andGreek leaders presented in (1) is an example of trivial NASP . Finally, in NASPs with lower-level facile NashGames, the feasible region for the followers are convex, and the leaders’ objective functions are convex in xi.As a consequence, the existence of a PNE is guaranteed whenever the feasible regions are compact (Debreu1952, Glicksberg 1952, Fan 1952). Therefore, one can restrict the search of equilibria to PNEs among thefollowers despite looking for both MNE s and PNE s among the leaders.

Typically, within the optimization literature, Nash games are reformulated as linear complementarityproblems (LCPs). This reformulation leverages the complementarity conditions (KKT ) of all the players inthe game. LCPs have a rich theoretical basis (Facchinei and Pang 2015b,a, Cottle et al. 2009), and can beformulated as mixed-integer programs (MIPs). Following the usual notation, let operator x ⊥ y be equivalentto xT y = 0.

Definition 8 (Linear complementarity problem). Given M ∈ Rn×n, q ∈ Rn, the linear complemen-tarity problem ( LCP) asks to find a x ∈ Rn so that 0 ≤ x ⊥Mx+ q ≥ 0, or to show that no such x exists.We denote as feasible set induced by the LCP, the set of all x satisfying the condition of the LCP.

Simplifying assumptions and limitations. Here, we sum up the assumptions made for the entirity of thispaper. First, a NASP is a game among leaders of Stackelberg games, each of who have a personal set offollowers. In other words, actions of the followers of a certain leader do not affect another leader or followersof another leader. Second, we assume optimistic behavior by the leader. That is, given a leader’s decision,should there be multiple equilibria for the followers, the leader will choose the most favorable equilibriumfrom its standpoint. This also has consequences for the upcoming results about absence of equilibria. Whenwe state that a NASP does not have an equilibrium (PNE or MNE), our claim is as follows: should theleaders always pick the most favorable lower-level equilibrium, no equilibrium exists among the leaders. Thisindeed means, if the leader does not pick the most favorable equilibrium among the followers, an equilibriumamong the leaders might exist. In this sense, we consider non-optimistic equilibrium selection by the leadersto be beyond the scope of this work.

6

2.2 Known results.

Cottle et al. (2009) proven that a facile Nash game can be solved as an LCP .

Theorem 1 (Cottle et al. (2009)). Let P be a facile Nash game. Then, there exist M, q such that everysolution to the LCP defined by M, q is a PNE for P and every PNE of P solves the LCP.

Basu et al. (2019), with Theorem 2, provide an extended formulation for the feasible region of a simpleStackelberg game. This result is a critical ingredient of our contribution since it enables us to provide apolyhedral characterization NASPs.

Theorem 2 (Basu et al. (2019)). Let S be the feasible set of a simple Stackelberg game. Then, S isa finite union of polyhedra. Conversely, let S be a finite union of polyhedra. Then, there exists a simpleStackelberg game with P (x) containing exactly 1 player such that the feasible region of the simple Stackelberggame provides an extended formulation of S.

Finally, the celebrated Theorem 3 from Balas (1985) allows us to retrieve the closure of the convex hull forthe union of a finite set of polyhedra.

Theorem 3 (Balas (1985)). Given k polyhedra Si = x ∈ Rn : Aix ≤ bi for i = 1, . . . k, then

cl conv(⋃ki=1 Si) is given by the set x ∈ Rn : ∃(x1, . . . , xk, δ) ∈ (Rn)k × Rk : x ∈ Aixi ≤ δib

i,∑kw=1 x

w =

x,∑kw=1 δw = 1, δi ≥ 0,∀i ∈ [k]

3 Hardness of finding a Nash equilibrium

In what follows, we characterize the computational complexity of NASPs. We formalize the intuitionstemming from Jeroslow (1985) with a reduction from the SUBSET SUM INTERVAL problem. The main resultsare summarized below.

Theorem 4. It is Σp2 -hard to decide if a trivial NASP has a PNE.

Corollary 1. If each player’s feasible set in a trivial NASP is a bounded set, an MNE exists.

Theorem 5. It is Σp2 -hard to decide if a trivial NASP has an MNE.

In what follows, we will provide the proof of Theorems 4 and 5. First, we formally introduce the SUBSET

SUM INTERVAL.

Definition 9 (SUBSET SUM INTERVAL). Given q1, . . . , qk, p, t, k ∈ Z+, with none of them equal to zero, andlog2(t− p) ≤ k, does there exist a s ∈ Z : p ≤ s < t, so that for all I ⊆ 1, 2, . . . , k then

∑i∈I qi 6= s.

In other words, we seek – within an interval of integers – for a number s that cannot be expressed as a sumof a subset of q1, . . . , qk or alternatively show that no such s exists. Here, t− p can be chosen as a powerof 2. For instance, we may ask if there exist an r in ∈ Z+ such that 2r = t− p. Eggermont and Woeginger(2013) proven that, given r in Z+ such that t− p = 2r, the problem is Σp

2 -hard.

Theorem 6 (Eggermont and Woeginger (2013)). Given that there exists r ∈ Z+ such that t− p = 2r,SUBSET SUM INTERVAL is Σp

2 hard.

Proof. Proof of Theorem 4. To show the hardness of NASP, we will rewrite SUBSET SUM INTERVAL as atrivial NASP of comparable size. Then, appealing to Theorem 6, we establish the hardness of a trivial NASP .Finally, we claim that NASP is only a generalization of trivial NASP , which could not be any easier.

Consider a trivial NASP as of in Definition 7. For the sake of clarity, we call the two Stackelberg gamesassociated with the trivial NASP the Latin, and Greek game, respectively. The decision variables of the Latingame’s leader are x, and their follower controls y variables. Similarly, the decision variables of the Greekgame are ξ, and χ for their follower. As for the SUBSET SUM INTERVAL, we stick to the notation introducedin Definition 9.

Let b1, . . . , br ∈ 0, 1 as the unique r-bit binary representation of s− p: for instance, biri=1 satisfies

s− p =∑ri=1 bi2

i−1. Then, let P = k + 2r, Q =∑ki=1 qi, and T = t− 1 + rQ, where both can be computed

in polynomial time with respect to the data in SUBSET SUM INTERVAL.

7

The Latin Game.

maxx0,x1,...,x2P

∈Ry0,y1,...,y2P

∈R

: (T − 1)ξ0x0 +

k∑i=1

qiξixP+i +Q

P∑i=k+1

ξixP+i (2a)

subject to xi = 0 i = 1, . . . , k (2b)

yi ≥ 0 i = 1, . . . , 2P (2c)

xi ≥ 0 i = 1, . . . , 2P (2d)

P∑i=k+1

xi ≤ r (2e)

xi + xP+i ≤ 1 i = 1, . . . , P (2f)

x0 + xP+i ≤ 1 i = 1, . . . , P (2g)

(y0, . . . , y2P ) ∈ arg miny

2P∑i=0

yi :yi ≥ −xiyi ≥ x1 − 1

∀ i = 0, . . . , 2P

(2h)

The Greek Game.

maxξ0,ξ1,...,ξP∈R

χ0,...,ξP∈R

: (T − 1)ξ0 +

k∑i=1

qiξi(1− xP+i) +Q

P∑i=k+1

ξi(1− xi − xP+i)

+

k+r∑i=k+1

2i−k−1ξi(1− xi − xP+i)

−P∑

i=k+1

T (xiξi + (1− xi)(1− ξi − ξ0)) (2i)

subject to ξi ≥ 0 ∀ i = 0, . . . , P (2j)

ξi ≤ 1 ∀ i = 0, . . . , P (2k)

χi ≥ 0 ∀ i = 0, . . . , P (2l)

P∑i=k+1

ξi + rξ0 ≥ r (2m)

T ≥ Tξ0 +

k∑i=1

qiξi +Q

P∑i=k+1

ξi +

k+r∑i=k+1

2i−k−1ξi (2n)

(χ0, . . . , χP ) ∈ arg minχ

P∑i=0

χi :χi ≥ −ξiχi ≥ ξi − 1

∀ i = 0, . . . , 2P

(2o)

We claim the game in (2) has a PNE , if and only if the SUBSET SUM INTERVAL instance has a decision YES.

Claim 1 The game defined in (2) is a trivial NASP.

Claim 2 The region in the space of x defined by (2c) and (2h) is the Cartesian product of (xi : xi ≤ 0 ∪ xi : xi ≥ 0),for i = 0, . . . , 2P . Similarly, the region in the space of ξ defined by (2l) and (2o) is the Cartesian product of(ξi : ξi ≤ 0 ∪ ξi : ξi ≥ 0), for i = 0, . . . , P .

8

We refer the reader to the electronic companion for the proofs of Claims 1 and 2.

Claim 3 If((x, y), (ξ, χ)

)is a PNE for (2), then ξ0 6= 0.

Proof (Proof of Claim 3). First, observe that ξ 6= 0, since setting ξ0 = 1 is a feasible profitable deviation forthe Greek leader, regardless of the Latin leader’s decision. Suppose ξ0 = 0 and for some ∅ 6= L ⊆ 1, . . . , P,ξ` 6= 0. Note that the Latin leader has no incentive to keep x0 = 1, which forces an objective value of 0.Instead, it can choose x0 = 0, and xP+` = 1 for all ` ∈ L and any feasible value for xP+` for ` ∈ 1, . . . , P\L.One can check that this is feasible and optimal for the the Latin leader, given ξ0 = 0. This also means thatthe Greek leader’s objective is 0, as each of the summands in their objective vanishes, and ξ0 = 0 makes thefirst term vanish. Hence, this cannot be a Nash equilibrium since the Greek leader has a profitable deviationby setting ξ0 = 1 and ξi = 0 for all i 6= 0, which is feasible and yields an objective value of T − 1 > 0.

Claim 4 If SUBSET SUM INTERVAL has decision YES, then (2) has a PNE.

Proof (Proof of Claim 4). Suppose there exists s ∈ Z+ such that p ≤ s ≤ t− 1, and for all I ⊆ 1, . . . , k,∑i∈I qi 6= s. Also, recall the unique r-bit binary representation of s− p, namely b1, . . . , br ∈ 0, 1. Consider

the following strategy:

x0 = 1 (3a)

xi = 0 ∀ i = 1, . . . , k (3b)

xi = bi−k ∀ i = k + 1, . . . , k + r (3c)

xi = 1− bi−k−r ∀ i = k + r + 1, . . . , P = k + 2r (3d)

xi = 0 ∀ i = P + 1, . . . , 2P (3e)

yi = 0 ∀ i = 0, . . . , 2P (3f)

ξ0 = 1 (3g)

ξi = 0 ∀ i = 1, . . . , P (3h)

χi = 0 ∀ i = 1, . . . , P (3i)

It is easy to check that the strategy in (3) is feasible. Given ξ, observe that the strategy is optimal for theLatin leader as follows. Due to the choice ξi = 0 for i 6= 0, all but the first term of the Latin leader vanish.The largest value the first term can take corresponds to x0 = 1. The remaining terms do not affect the Latinleader’s objective, as long as they are feasible.

For what concerns the Greek leader, the current objective is T − 1. We show there exist no deviationwhich can improve their objective. With ξ0 = 1, clearly no other deviation is feasible. Consider the deviationξ0 = 0: with such strategy the first term in the objective vanishes. Let M = i ∈ k+ 1, . . . , k+ 2r : xi = 1.Observe that |M | = r, and let L = k+ 1, . . . , k+ 2r \M . Notice that we require ξ` = 1 for ` ∈ L, otherwisethe fifth term in the objective would be a large negative quantity. Hence, the objective would not exceedthe value of T − 1. With such a choice of ξ` for ` ∈ L, the fifth term in the objective evaluates to 0, and thefourth term evaluates to

∑`∈L 2`−k−i =

∑k+ri=k+1(1− bi−k)2i−k−1 = 2r − 1 + p− s = t− 1− s. Therefore, the

objective value is t−1 + rQ−s. However, since it is a YES instance of SUBSET SUM INTERVAL, the deficit s inthe objective value can never be made up by any choice of ξi for i = 1, . . . , k and by making the second termequal to s. If such ξi are chosen to exceed s, then (2n) is violated if it is strictly less than s, and the objectivecannot exceed T − 1. Hence, this is no longer a valid deviation. Thus (3) is indeed a Nash equilibrium. ut

Claim 5 If SUBSET SUM INTERVAL has decision NO, then (2) has no PNE.

Proof (Proof of Claim 5). We prove the result by contradiction. In orter to establish the latter, assume that theSUBSET SUM INTERVAL instance has an answer NO, and there exists a PNE

((x, y), (ξ, χ)

)for (2), with ξ0 = 1.

From Claims 2 and 3, any PNE necessarily has ξ0 = 1. From (2n), ξ0 = 1 enforces that ξi = 0 for i = 1, . . . , T ,and hence the Greek leader has an objective value of T − 1. Therefore, with ξ =

(1 0 . . . 0

), observe that

the Latin leader’s objective is (T − 1)x0. Thus, we necessarily have x0 = 1. From (2g), we deduce xP+i = 0

9

for i = 1, . . . , P , while from (2e) we obtain xi ≤ rr+1 for i = 1, . . . , k. The only value of xi that satisfies this

condition along with (2h) is xi = 0 for i = 1, . . . , k That only leaves xi for i = k + 1, . . . , k + 2r = P .We can now show that – for any value of xi – the Greek leader has a profitable deviation. Namely, it

can get an objective strictly greater than T − 1. Let M = i ∈ k + 1, . . . , k + 2r : xi = 0. From (2e), wehave |M | ≥ r. We choose some L ⊆ M such that |L| = r, and for i ∈ L, we set ξi = 1. Since |L| = r, andL ⊆M , the third term in the Greek leader’s objective evaluates to rQ. The fourth term is in between 0 and2r − 1, and the fifth term vanishes. Keeping in mind that ξ0 = 0, the objective now evaluates to a number

between∑ki=1 qiξi + rQ and

∑ki=1 qiξi + rQ+ 2r − 1. In other words, the objective is T − s+

∑ki=1 qiξi and

p ≤ s ≤ t− 1. Since this is a NO instance of SUBSET SUM INTERVAL, ∃I ⊆ 1, . . . , k such that∑i∈I qiξi = s.

Set ξi = 1 if i ∈ I, and ξi = 0 if i ∈ 1, . . . , k \ I. This is feasible, and makes the objective value equal to T ,which is a profitable deviation from T − 1. Therefore

((x, y), (ξ, χ)

)is not a Nash equilibrium. ut

From Theorem 4, we have a direct implication of Corollary 2.

Corollary 2. Consider a linear Nash Game N = (P 1, . . . , Pn) where each P i is an MIP . It is Σp2 -hard to

decide if N has a PNE.

Proof (Proof of Corollary 2). Bounded and continuous bilevel programs can be reformulated as boundedinteger programs of polynomial-size (Basu et al. 2019). The Greek and the Latin leaders’ problems defined in(2) are bounded bilevel programs, where each variable necessarily takes value in [0, 1]. ut

Furthermore – under an assumption of boundedness – we prove Corollary 1, showing that an MNE alwaysexists.

Proof (Proof of Corollary 1). Let Fi be the feasible region of the i-th player (leader), namely a bounded set.Given x−i, the objective of its optimization problem is linear. Hence, an optimal solution always exists, whichis an extreme point of conv(Fi). However, given that Fi are feasible sets of bilevel linear programs, we knowthat the feasible region of the leaders is a finite union of polyhedra from Theorem 2. It follows that conv(Fi)is a polyhedron. Since we also assume boundedness, conv(Fi) is indeed a polytope. Thus, the i-th player’sstrategy is the set of extreme points of this polytope, finite in number. Since the same reasoning holds foreach player, this is a Nash game with finitely many strategies. From Nash (1950, 1951), such a game has anMNE . ut

From Corollary 1, deciding on the existence of an MNE is trivial if each player has a bounded feasible set.We extend this result with Theorem 5, showing that even if one player’s feasible region is unbounded, thendeciding on the existence of an MNE is Σp

2 -hard.Before proving Theorem 5, we introduce the technical Lemma 1. While Theorem 2 shows that any finite

union of polyhedra can be written as a feasible region of a bilevel problem in a lifted space, Lemma 1 explicitlydescribes this set for a given union of two polyhedra.

Lemma 1. Consider the set S defined as the union of two polyhedra, namely

S =

(h, y, x) ∈ R3+ : h = x; y = 1

∪

(h, y, x) ∈ R3+ : h = 0; y = 0

(4)

S has an extended formulation as a feasible set of a simple bilevel program.

From Lemma 1 we can further derive Lemma 2.

Lemma 2. Suppose S ⊆ Rn1 and T ⊆ Rn2 have an extended formulation as bilevel programs. So does S × T .

Therefore, with Lemmata 1 and 2, we can then prove Theorem 5. Both the proofs for these two lemmas canbe found in the electronic companion.

Proof. Proof of Theorem 5. We reduce SUBSET SUM INTERVAL into a problem of deciding the existence of anMNE for a trivial NASP . Let Q =

∑ki=1 qi. Also, as of Theorem 4, let the Latin game and the Greek game

have Latin and Greek terms, respectively.

10

Latin Game. The Latin game is a Stackelberg game. The variables of the leader and the follower are denotedby Latin alphabets x and y, respectively.

maxx0,...,xk+3r+1

∈Ry0,...,yk∈R

:x0

2+

k∑i=1

qixi + 2(Q+ 1)ξr+1xk+3r+1

− (Q+ 1)

(r∑i=1

2i−1xk+i + pxk+3r+1

)(5a)

subject to xi ≥ 0 ∀ i = 0, . . . , k (5b)

yi ≥ 0 ∀ i = 0, . . . , k (5c)

xi ≥ 1 ∀ i = 0, . . . , k (5d)

xk+3r+1 = xk+2r+i ∀ i = 1, . . . , r (5e)

xk+3r+1 = p+

r∑i=1

2i−1xk+r+i (5f)

x0

2+

k∑i=1

qixi ≤ xk+3r+1 (5g)

(xk+i, xk+r+i, xk+2r+i) ∈ S (as in (4)) ∀ i = 1, . . . , r (5h)

(y0, . . . , yk) ∈ arg miny

k∑i=0

yi :yi ≥ −xiyi ≥ xi − 1

∀ i = 0, . . . , k

(5i)

Greek Game. Similarly, the Greek game is a Stackelberg game, where leader and the follower variables aredenoted by Greek alphabets ξ and χ, respectively.

maxξ0,...,ξr+1

∈Rχ1,...,χr∈R

: (1− x0)ξ0 (5j)

subject to ξi ≥ 0 ∀ i = 1, . . . , r (5k)

χi ≥ 0 ∀ i = 1, . . . , r (5l)

ξi ≤ 1 ∀ i = 1, . . . , r (5m)

p+

r∑i=1

2i−1ξi = ξr+1 (5n)

(χ1, . . . , χr) ∈ arg minχ

r∑i=1

χi :χi ≥ −ξiχi ≥ ξi − 1

∀ i = 0, . . . , r

(5o)

Claim 6 The game defined in (5) is a trivial NASP.

Claim 7 The region of space for x – defined by (5c) and (5i) – is the Cartesian product of (xi : xi ≤0 ∪ xi : xi ≥ 0) for i = 0, . . . , k. Similarly the region of the space for ξ – defined by (5l) and (5o) – is theCartesian product of (ξi : ξi ≤ 0 ∪ ξi : ξi ≥ 0) for i = 1, . . . , k.

The proof of this claim is analogous to the ones of Claims 1 and 2.

Claim 8 xk+3r+1 takes integer values only.

11

Proof (Proof of Claim 8). From (5h), each xk+r+i for i = 1, . . . , r can take a value of either 0 or 1, dependingupon which of the two polyhedra (in the definition of S) the variable falls in. Moreover, since in (5f) the RHSis a sum of integers, the LHS xk+3r+1 is also an integer.

Claim 9 (xk+3r+1)2 =∑ri=1 2i−1xk+i + pxk+3r+1 holds for the Latin game’s feasible set.

Proof (Proof of Claim 9). Consider the set S defined in (4). For a point h = x and y = 1 in the first polyhedra,one can write h = xy. Similarly, for a point h = 0 and y = 0 in the second polyhedron, then h = xy. Thus,the nonlinear equation h = xy is valid for the set S. By multiplying both sides of (5f) with xk+3r+1, one gets

(xk+3r+1)2 = pxk+2r+1 +

r∑i=1

2i−1xk+r+ixk+3r+1

= pxk+3r+1 +

r∑i=1

2i−1xk+r+ixk+2r+i

= pxk+3r+1 +

r∑i=1

2i−1xk+i

The second equality follows from (5e), and the third equality from the fact that h = xy is valid for S and(5h).

Claim 10 Given some ξr+1 ∈ Z between p and t− 1, the Latin player has a profitable unilateral deviationfor any feasible strategy with xk+3r+1 6= ξr+1.

Proof (Proof of Claim 10). Note that if ξr+1 is between p and t− 1, then xk+3r+1 = ξr+1 is feasible for theLatin game. Observe the last two terms of the objective function. From Claim 9, we can rewrite them as(Q+ 1)(2ξr+1xk+3r+1 − x2

k+3r+1). By focusing just on the last two terms, these reach a maximum value forthe feasible choice of xk+3r+1 = ξr+1. We can now argue that the player can never be optimal by choosingxk+3r+1 6= ξr+1. As established in Claim 8, xk+3r+1 is restricted to take integer values, and for any otherchoice xk+3r+1, the deficit in objective value is at least Q+ 1. However, even if each of the other terms taketheir maximum possible value, the largest value they can add to is 0.5 +Q < Q+ 1. the claim follows.

Claim 11 If SUBSET SUM INTERVAL has decision YES, then (2) has a PNE (and hence an MNE).

Proof (Proof of Claim 11). Let s be an integer such that p ≤ s < t and ∀I ⊆ 1, . . . , k,∑i∈I qi 6= s, and let

b1, . . . , br ∈ 0, 1 be the unique r-bit binary representation of s− p. Consider the following pure strategiesfor the players:

xk+3r+1 = s (6a)

xk+2r+i = s i = 1, . . . , r (6b)

xk+r+i = bi i = 1, . . . , r (6c)

xk+i = bis i = 1, . . . , r (6d)

x0 = 1 (6e)

ξ0 = 0 (6f)

ξi = bi i = 1, . . . , r (6g)

ξr+1 = s (6h)

Finally, choose xi ∈ 0, 1 for i = 1, . . . , k such that∑ki=1 qixi is the largest value not exceeding s. Since

it is a YES instance of SUBSET SUM INTERVAL,∑ki=1 qixi ≤ s− 1, and thus the strategy is indeed feasible

for both the players. The Latin player has no feasible profitable deviation. This follows from the fact thatxk+3r+1 cannot be chosen differently due to Claim 10. Moreover, the first two terms in the above strategyalready take the largest possible value not violating (5g). Thus the Latin player has no profitable deviation.Now for the Greek player, since x0 = 1, the objective value is always zero, and cannot be improved. Thus,the strategy in (6) is indeed a PNE .

12

Algorithm 1 Enumeration algorithm to obtain an MNE for a NASP

Input: A description of NASP N = (P 1, . . . , Pn).Output: For each i = 1, . . . , n, xij is a pure-strategy played with probability pij , presenting a mixed-strategy with

support size ki.1: for i = 1, ..., n do2: Enumerate the polyhedra whose union defines the feasible set Fi of P i.3: Fi ← cl conv Fi by applying Theorem 3.4: P i ← objective function of P i and a feasible set of Fi.5: end for6: Solve the facile Nash game N = (P 1, ..., Pn) to obtain either a PNE , (x1, . . . , xn) or show that no PNE exists.

7: if no PNE exists for N then8: There is no MNE for N ; exit returning failure.9: end if

10: for i = 1, ..., n do11: if xi ∈ Fi then12: xi1 ← xi; pi1 ← 1; ki ← 1.13: else14: xi =

∑ki

j=1 ηj xij for xi1, . . . , x

iki ∈ Fi with ηj ≥ 0 and

∑ki

j=1 ηj = 1.

15: pij ← ηj for j = 1, . . . , ki.16: end if17: end for18: return (xij , p

ij) for each i = 1, . . . , n and j = 1, ..., ki.

Claim 12 If SUBSET SUM INTERVAL has decision NO, then (2) has no MNE.

Proof (Proof of Claim 12). Recall xk+3r+1 is forced to be an integer between p and t − 1. For any choiceof xk+3r+1, x0 = 0 is selected and x1, . . . , xk are so that (5g) holds with equality. There is no incentive tochoose x0 = 1, which will contribute to only 0.5 in the objective. However, with x0 = 0, the Greek player canchoose arbitrarily large values of ξ0. Hence, there is always a larger choice of ξ0 which constitute a profitabledeviation. Thus, no equilibrium exists for the game. ut

4 An enumeration algorithm to find MNEs for NASPs

First, we introduce Algorithm 1, which enumerates the polyhedra whose union corresponds to each player’sfeasible region. Then, it finds a pure-strategy Nash equilibrium in the convex hull of each player’s feasibleregions. We prove the equivalence between finding a PNE over the convex hull and the original problem.

The feasible region. Consider the feasible region of a simple Stackelberg game, given by A′u+B′v ≤ b, v ∈SOL(P (u)). Using the KKT conditions of the players in P (u), we can rewrite the latter by an extendedformulation, as

S =

x :Ax ≤ b

z = Mx+ q0 ≤ xi ⊥ zi ≥ 0, ∀ i ∈ C

. (7)

Note that the set S is a union of polyhedra.

Preliminary Enumeration Algorithm. We present Algorithm 1, which exploits the knowledge that eachplayer’s feasible region is a finite union of polyhedra (Basu et al. 2019). Step 2 explicitly enumerates all suchpolyhedra, while Step 3 computes the closure of their convex hull using Theorem 3. Since this convex hull is

13

(a) The players’ feasible regions. From Theorem 2, these arefinite unions of polyhedra. Step 2 of Algorithm 1

(b) With Theorem 3, we compute the convex hull of eachplayer’s feasible region. Step 3 of Algorithm 1

(c) Given the convex hulls, the problem reduces to a MIP(LCP) as of Theorem 1. Step 6 of Algorithm 1

!

!

(d) The solution ? can be interpreted as a convexcombination of feasible strategies. Steps 12 and 14 of

Algorithm 1

Fig. 2: A pictorial reprsentation of Algorithm 1.

also a polyhedron, the game N (defined in Step 6) is a facile Nash game, and we can get a PNE for the gameusing Theorem 1.

Let x be a PNE of N and xi be the strategy of the i-th player. If xi belongs to Fi, then at equilibrium iplays xi in N . If xi does not belong to Fi, it is still contained in cl conv Fi. Thereby, xi can be expressed asa convex combination of points – or strategies – in Fi or a limit of such points. Player i would then playa mixed-strategy where each weight in the convex combination – or δ of Theorem 3 – is the probability ofplaying the corresponding pure-strategy, as in Step 14 of Algorithm 1. We remark that the LCP solved inStep 6 is implemented as a feasibility problem and solved as a MIP . If the user is interested in a specificequilibrium, an objective function can be added to this problem. This gives the user the ability to effectivelyperform equilibria selection, if more than one exists. A visualization of the rationale behind the algorithm isin Figure 2. We formalize the correctness and finite termination of the above procedure in Theorem 7.

Theorem 7. Algorithm 1 terminates finitely and (i) if it returns xij , pij for each i = 1, . . . , n, and j = 1, . . . , ki,

then the strategy profile is indeed an MNE for the NASP, (ii) if it returns failure, then N has no MNE.

Proof (Proof of Theorem 7). For the purpose of this proof, we adopt the same symbols introduced inAlgorithm 1. First, the algorithm terminates in a finite number of steps: all loops in Algorithm 1 are finiteloops, Step 2 ends finitely since there are only finitely many polyhedra (see Theorem 2), and Step 3 is also afinite procedure.

Proof of Statement (i) . Observe that if Algorithm 1 does not return failure, then Step 6 finds PNE x for N .Each player’s objective function is linear, and the distribution for the MNE has finite support. Therefore,one can observe that - for each player i - the following holds:

E((ci + Cix−i

)Txi)

=∑j′

ki∑j=1

p−ij′ pij

(ci + Cix−ij′

)Txij =

(ci + Cix−i

)Txi. (8)

Assume a generic player i has an unilateral profitable deviation †xij , and †pij for i = 1, . . . , `i from xi in

their P i problem. Such a deviation is also a mixed-strategy profile. Consider now the pure-strategy for N

given by∑`i

j=1(†pij † xij). It is feasible for the facile game P i. Therefore, leveraging on the linearity of each

player’s objective function, we can show that this is also a profitable deviation for P i in N , and hence find acontradiction.

14

(ci + Cix−i

)Txi =

∑j′

ki∑j=1

p−ij′ pij

(ci + Cix−ij′

)T (xij)

(9)

≥∑j′

`i∑j=1

p−ij′ † pij

(ci + Cix−ij′

)T (†xij)

(10)

=

ci + Ci

∑j′

p−ij′ x−ij′

T `i∑j=1

†pij † xij

(11)

=(ci + Cix−i

)T `i∑j=1

†pij † xij

(12)

Here (12) follows by plugging the profitable deviation into (8), and exploiting its linearity. Since we havea profitable deviation for the mixed strategy for N , there exists a unilateral deviation for N from x. Thiscontradicts the fact that x is a PNE for N . Therefore, such a deviation cannot exist.

Proof of Statement (ii) . To prove this statement, we prove its contrapositive. Namely, we show that if

N has an MNE , then Step 6 obtains a PNE for N and will not return failure. Therefore, it is sufficientto show that N has a PNE . Let the MNE of N be given by each player i ∈ [n] playing xi1, . . . , x

iki

with

probability pi1, . . . , piki

, respectively. Let xi =∑kij=1 p

ijxij be the a feasible pure-strategy for player i. It follows

that (x1, . . . , xn) is a feasible pure-strategy for N , and we now show it is indeed a PNE for N . Given theabove MNE for N , we know that

∑j′

ki∑j=1

p−ij′ pij(C

ix−ij′ + ci)Txij ≤∑j′

p−ij′ (Cix−ij′ + ci)Txi,∀ xi ∈ Fi.

Due to the linearity of the objective function, it follows that:(Cix−i + ci

)Txi ≤

(Cix−i + ci

)Txi ∀xi ∈ Fi. (13)

If (13) holds for all xi ∈ cl conv(Fi), for all i, then x is a PNE of N and the proof will be complete. First,

we show that (13) holds for xi ∈ conv(Fi). Let xi =∑`j=1 λjx

ij , where xij ∈ Fi and λj ≥ 0 and

∑`j=1 λj = 1.

Now consider the ` inequalities of (13), each one for xij for j = 1, . . . , l. Multiply these inequalities bynon-negative λj on both sides, and add to obtain

(Cix−i + ci

)Txi ≤

∑j=1

λj(Cix−i + ci

)Txij

=(Cix−i + ci

)Txi

In the second instance, to show the same holds for xi ∈ cl conv(Fi), consider a convergent sequence xi1, xi2, . . .

with each xij ∈ conv(Fi) and limj→∞ xij = xi.(Cix−i + ci

)Txi ≤

(Cix−i + ci

)Txij ∀ j = 1, 2, . . .

=⇒ limj→∞

(Cix−i + ci

)Txi ≤ lim

j→∞

(Cix−i + ci

)Txij

=⇒(Cix−i + ci

)Txi ≤

(Cix−i + ci

)T (limj→∞

xij

)

15

=(Cix−i + ci

)Txi

Thus, (13) holds for all xi ∈ cl conv(Fi), and x is indeed a PNE of N . ut

Remark 1. Within the proof of Theorem 7, we never exploit any specific properties of simple Stackelberggames. The only assumption we leverage is that the problem is a linear Nash game (i.e., the objective ofeach player is of the form (ci + Cix−i)Tx). In this case, it is sufficient to solve the problem for PNE in theconvex hull of each player’s feasible set to compute an MNE for the original problem. In this spirit, if one cancompute the convex hull of the player’s feasible region, and if objectives are linear, then every game is aconvex game.

5 Enhancing the algorithm

In this section, we present two enhancements of Algorithm 1. In Section 5.1 we introduce an iterative procedureto approximate the closure of the convex hull of each player feasible set. Thereby, we avoid the possibly costlyand unnecessary enumeration of all the polyhedra defining the feasible sets. In Section 5.2, we tailor thealgorithms to specifically retrieve PNE s as opposed to general MNE s.

5.1 Inner approximation algorithm

While Algorithm 1 is guaranteed to terminate and solve the problem, we introduce a procedure that canimprove computational tractability. The feasible region of a simple Stackelberg game is a finite union ofpolyhedra (see Theorem 2), and their convex hull can be computed using Theorem 3. However, since there maybe exponentially many polyhedra, the convex hull description could become untractably large. Algorithm 1intensively leverage on the complete enumeration of such polyhedra in Step 2. The central intuition is tolimit the enumeration by iteratively refining the convex hull’s description for each player. This procedure isalso valid for an individual Stackelberg game or a bilevel program. However, its importance is more relevantwhen dealing with NASPs, where the computation of this convex hull is essential. The key components ofthis approach are the polyhedral relaxation of the set S defined in (7), and the concept of selected polyhedron.

Definition 10 (Polyhedral relaxation). The polyhedral relaxation of the set S defined in (7) is given bythe set O0 = x : Ax ≤ b, z = Mx+ q, xi ≥ 0, zi ≥ 0 ∀ i ∈ C

Clearly, this set contains cl conv(S), and is hence a relaxation. Also, while S is generally not a polyhedron,its polyhedral relaxation is.

Definition 11 (Selected polyhedron). Let b ∈ 0, 1|C| and let C = c1, . . . , ck. Then, the selectedpolyhedron corresponding to b is P(b) = xci ≤ 0, ∀ i ∈ i : bi = 0

⋂[Mx+ q]ci ≤ 0, ∀ i ∈ i : bi = 1.

We can then formally define the concept of inner approximation.

Definition 12 (Inner Approximation). Let J = j1, . . . , j` ⊆ 0, 1mf . Then the inner approximationdefined by J is IJ = cl conv

(⋃b∈J P(b) ∩ O0

).

Remark 2. The size of the extended formulation of IJ is bounded by O(|J |). To ensure a perfect description,we need a choice of J = 0, 1|C|. However, |J | = 2|C| and a description of cl conv(S) will be exponentiallylarge. Unless P = NP, there cannot be any asymptotical improvements (Bard 1991).

Algorithm 2 presents the inner approximation algorithm – an enhancement to Algorithm 1– to retrievean MNE for NASPs. It iteratively constructs an increasingly accurate inner approximation for the players’feasible regions in the NASP , and seeks for a PNE for this restricted game N (Step 4).

Let F1, . . . ,Fn, be the inner approximations of the feasible sets of player 1, . . . , n. One can compute theconvex hull’s closure for each approximation and solve the associated facile Nash game N . If x is a Nashequilibrium of N , the algorithm checks if x – or the associated mixed-strategy implied by x (similarly to

16

Algorithm 2 Inner approximation to obtain an MNE for a NASP

Input: A description of NASP N = (P 1, . . . , Pn) and J = (J1, ..., Jn) where J i ⊆ 0, 1|Ci| where Ci is the set ofindices of complementarity (⊥) conditions for the i-th player.

Output: For each i = 1, . . . , n, xij is a pure-strategy played with probability pij , presenting a mixed-strategy withsupport size ki.

1: function IterInnerApproxNash(N, J)

2: Fi ←inner approximation defined by J i and Fi ← cl conv Fi .3: P i ← objective function of P i and a feasible set Fi.4: Solve the facile Nash game N = (P 1, ..., Pn) to obtain solution x. . Might fail5: x1, ..., xn ← getDeviation(P, x)6: if xi = NULL for all i = 1, ..., n then7: return x.8: end if9: for i = 1, ...n do

10: if xi 6= NULL then11: bi ←binary encoding of a polyhedron containing xi. J

i ← J i ∪ bi.12: end if13: end for14: return InnerApproxNash(N, J).15: end function

Step 14 of Algorithm 1) – is a Nash equilibrium for the original game N . If this is the case, then the algorithmterminates and returns the equilibrium. Conversely, if this mixed-strategy is not an MNE of N , there exists

a profitable deviation xi for some players such that xi 6∈ Fi. Thereby, we refine the inner approximation

of i-th player’s feasible set by adding a polyhedron containing xi. At each iteration of the algorithm, wekeep on adding polyhedra containing the profitable deviations. However, N may not have a PNE in a giveniteration (Step 4). In this case, we gain no additional knowledge about which polyhedra to add to the innerapproximation. Therefore, we arbitrarily add one or more polyhedra to the feasible region of each player inthe problem, keeping the algorithm running. We define as the extension strategy the criteria by which suchpolyhedra are selected.

Broadly speaking, in optimization problems, a point contained in an inner approximation of the feasibleset is feasible for the original problem and provides a primal bound for the original problem. However, this isnot true in the case of a Nash game. In Remark 3 below, we show that the inner approximation game mighthave a Nash equilibrium while the original game does not. Conversely, we also show that the original gamemight have a Nash equilibrium while an inner approximation does not.

Remark 3 (Inner approximation N might have an MNE but N might not). There might be cases where theinner approximation has no MNE , but the original NASP does. Consider the following players’ problems andtheir inner approximation.

Latin Player: minxξx : x ∈ R, x ≥ 0 (14a)

Greek Player: minξ,χxξ : ξ ∈ [−5, 5];χ ≥ 0;

χ ∈ arg minχχ : χ ≥ ξ − 1;χ ≥ −ξ − 1 (14b)

Using KKT conditions on the follower’s problem, the Greek ’s problem can be rewritten as

minξ,χ,µ

xξ : ξ ∈ [−5, 5];µ1 + µ2 = 1;χ ≥ 0;

0 ≤ µ1 ⊥ χ− ξ + 1 ≥ 00 ≤ µ2 ⊥ χ+ ξ + 1 ≥ 0

The polyhedra P (b) corresponding to b = (0, 0), and b = (1, 1) are empty. The remaining two polyhedra canbe projected to the ξ space as [−5,−1] ∪ [1, 5]. We claim that the problem in (14) has no Nash equilibrium.

17

This is because, irrespective of the Latin player’s decision, an optimal decision for the Greek player is ξ = −5.For such a value of ξ, the Latin player has an unbounded objective. Consider the inner approximation due tothe choice J = (0, 1). The equivalent feasible regions can be expressed as follow.

Latin Player: min ξx : x ∈ R, x ≥ 0 (15a)

Greek Player: min xξ : ξ ∈ R, ξ ∈ [1, 5] (15b)

In (15), the inner approximation is exact for the Latin player and is a strict inner approximation for theGreek player. However, (15) has a PNE (ξ, x) = (0, 1).

Conversely, it can also happen that the original NASP has no MNE , but the inner approximation does.For such an example, replace the objective of the Greek player in (14) with a minimization of −xξ, and thecorresponding inner approximation of the Greek player in (15) with ξ ∈ [−5,−1]. This inner approximationgame has no Nash equilibrium. However, the original game has a Nash equilibrium of (ξ, x) = (0, 5).

5.2 Enhancements for PNEs

In specific applications, deterministic strategies are preferred over randomized ones. Thus, one necessarilyrequires a PNE or show that no PNE exists. With this motivation, we alter Algorithm 1 to retrieve PNE sspecifically or prove no PNE exists.

Enumeration for PNE. This algorithm is similar to Algorithm 1, hence we assume the same notation. First,the procedure explicitly enumerates all the polyhedra in the feasible region of each player, and computes theirconvex hull. In addition, it introduces in N a set of binary variables forcing the equilibrium strategy, for eachplayer, to be strictly in the original feasible region rather than solely in the convex hull. From Theorem 2,the feasible region for each NASP ’s player is a finite union of polyhedra. Let the feasible region of the i-thleader be Fi =

⋃gij=1 P

ij , where P ij = Aijx ≤ bij is a polyhedron. Moreover, Theorem 3 gives cl conv(Fi)

as Aijxij ≤ bijδ

ij for j ∈ [gi], x

i =∑gij=1 x

ij , and

∑gij=1 δ

ij = 1. If for some j, δij = 1, then the projection x is

strictly in the polyhedron P ij . Since we can reformulate a NASP as a MIP feasibility problem, we enforce a

new set of constraints in N requiring each δij to be binary in N . Hence, each PNE for N is also a PNE for

N , and if N has no PNE , also N has no PNE . In addition, for the equivalence between PNE s in N and N ,the condition of N being a NASP can be relaxed. In particular, it is sufficient that leaders’ objectives in Nare convex – observe that under this case, the reasoning in the proof of statement (ii) for Theorem 7 directlyfollows.

We refer the reader to the electronic companion for the pseudocode of this procedure.

6 Computational Tests

We test our algorithms1 with the energy-trade model (18) and (19).

The Model. We consider different geographical regions, where Governments of such regions act as leaders.Governments determine the amount of energy export/import and the CO2 taxation scheme imposed on theirrespective followers (energy producers). Each country seeks to minimize the sum of three components: (i) theproduct between each follower’ production and the emission cost (e.g., the social cost of carbon, SCC), (ii) theproduct between import price and quantity to any other country, (iii) the negative product between importprice and quantity of any other country, namely the maximization of export revenues. Besides, countriesmay also include a negative (maximized) tax-revenue term in their objectives, namely the sum of all theirrespective followers’ taxes. We distinguish between three forms of taxation: (i) Standard-Taxation, whereeach follower has a possibly different tax per unit-energy produced, (ii) Single-Taxation, where every followerhas the same tax per unit-energy produced, (iii) Carbon-Taxation, where every follower has the same taxper unit-emission. The lower-level players are energy producers deciding the amount of production of theirplants based on their linear and quadratic unit costs and their leader’s taxation levels. In specific, followersare playing a Cournot game where the homogeneous good is the amount of energy produced.

18

Instances. We propose three sets of computational instances (InstanceSet A, B, and Insights), and a casestudy on a real-world inspired Chile-Argentina instance 2. The goal of our computational tests is twofold.On the one hand, we showcase our algorithms’ compelling computational capabilities and compare theirperformances. With this respect, we extensively test all our algorithms on the instance sets A and B. On theother hand, we derive managerial insights from our models’ solutions, focusing on the Chile-Argentina casestudy and the instance set Insights.

Data generation. We synthetically generate our instance sets as follow: (i) InstanceSet A contains 150instances with 3 to 5 countries and up to 3 followers per country, (ii) InstanceSet B contains 50 instanceswith 7 countries and up to 3 followers per country, (iii) InstanceSet Insights contains 50 instances with 2countries and 3 followers per country.

We randomly draw each of the followers from three classes of producers: highly-polluting (e.g., coal, oil),averagely-polluting (e.g., gas), and green (e.g., renewables such as solar, hydro). Their emission costs perunit-energy (e.g., GWh) takes an integer value in the range [300, 500], [100, 200], and [25, 50], respectively.These are USD values of emission assuming a social cost of carbon at USD 25 per tonne of CO2 equivalentand typical emission values in these technologies. We set linear and quadratic production costs – negativelycorrelated to the emission factors – in the respective ranges [150, 300], and [0, 0.6] for unit energy. Theproduction capacities are discrete unit-energies in the interval [50, 20000]. We refer the reader to Section 11for a more detailed review of the parameters.

6.1 Strategic insights

Starting from InstanceSet Insights, we solve each instance 4 times by testing a discrete grid of 2 parameters.The first one is the Carbon-Taxation revenue in every country’s objective, while the second dictates whethertrade among countries is allowed. Table 6 provides comprehensive results. We attempt to answer the followingstrategic questions:

(i) Tax policy. Are countries reducing further their emissions if they consider the carbon tax as a source ofincome?

(ii) Trade policy. How does competitive energy trade among countries affect global emission?

Tax policy. Some literature argues that carbon tax revenues can further help reduce carbon emission, spurgreener technologies (e.g., carbon sequestration, electric vehicles) or even to meet other governmental expenses(Olsen et al. 2018, Liu and Lu 2015, Amdur et al. 2014). One might instinctively think that an income-hungry(e.g., GDP) government could levy a more aggressive carbon tax policy if that could be a revenue sourceand help reduce emissions. However, we observe the opposite to be true. We consistently find that when thegovernment’s objective (b = 1 in (18)) model incomes through a carbon tax, the government is systematicallyincentivized to impose a smaller tax. This is because, with smaller tax rates, the production through coal andnatural gas is larger. Thus, this increases the governmental revenue, which is the product of production andtax per unit of emission. In summary, decreased carbon tax could give increased revenue for the government.However, emissions are decreased compared to the no-taxation scheme but increased compared to the casewhen the government does not look for revenue from these taxes.

In particular, in 40 out of the 50 test instances, both countries’ total emission was greater if the individualgovernments considered the objective’s tax revenue. On an absolute basis, emissions were about 13.5% moreon average when governments imposed taxes, keeping the revenue in their objective. A statistical t-test rejectsthe null hypothesis that the global emissions are equal with and without the countries considering carbon taxas a revenue source with a p-value of 0.00018.

In a similar vein, we observe that the trade is lesser in 30 out of 50 instances and, on average, about 7.8%lesser when the countries consider tax as a revenue source. However, a similar t-test does not suggest enoughevidence to reject the null hypothesis (p-value = 0.29) that the traded quantities in the two cases have thesame population mean.

19

Trade policy. Secondly, we observe that the tax rate is typically lesser when countries can trade. Quantitatively,we find that the average tax rate without trade is about 12.9% higher than the tax rate when trade existsbetween the countries. However, we also observe that in 63 of the 100 possible cases (50 instances in InstanceSetInsights with 2 countries each), the tax rate is higher if there is trade between them. In other words, tax isslightly higher in many instances when the trade is enabled. Nevertheless, in those instances where the taxrate is lower with trade enabled, the tax rate is significantly lower.

Next, one might wonder if increased emissions might accompany trade between countries. Since energytrade is an economic activity, one can think it could worsen the externality of emission. However, we observethat emissions are consistently less when countries can trade. This is primarily driven by the fact that cleanmeans of energy in a different country could fulfill the demand in a country without forcing domestic producersto produce using non-green means of production. Quantitatively, we compare the average emission by boththe countries when a trade happens between them instead of no-trade being allowed between them (seeTable 6). We observe that the average emission dropped by about 35.9% when trade was enabled. Further,never in those 50 test instances did the emission ever increase after trade was enabled. We also note thatwhen countries can trade, emissions could increase in one country, but the decrease in another country isalways significant enough to ensure that the total emission decreases while keeping the consumption in bothcountries roughly the same.

Final comments. Besides our consistent insights that (i) a tax revenue-hungry government might imposelesser carbon tax than a government inclined to reduce emissions (ii) enabling trade reduces total globalemissions, the answers to the more general questions were predominantly instance dependent. In particular,we observed that opening up trade increased domestic carbon taxes in some cases and decreased them inothers. We observed similar behaviors for trade with revenue-hungry governments. The answers to thesequestions were sensitive to the cost, capacity, and emission factors of production units and the domesticenergy demand of each country. These observations suggest that one has to solve a NASP (or even a morecomplex model) to identify the specific dynamics for a given situation. Furthermore, in NASPs one canalways perform equilibria selection – if multiple MNE s exist – by solving the problem with Algorithm 1 andenforcing the MIP ’s objective to optimize a given criterion.

6.2 Case study – Chile-Argentina energy markets

We implemented the model using actual data (2018-2019) of Chile’s and Argentina’s electricity markets.Electricity trade between these countries started in 2016, with Chile exporting to Argentina a small amount –close to 1558 MWh – of electricity. However, the transfers are expected to increase as both countries signedan energy cooperation agreement in 2019 (both electricity and gas). These efforts have created some debateregarding electricity prices, which may impact one of the Chilean government’s main goals: make electricitymore affordable. Furthermore, both Chile and Argentina have signed the Paris agreement and promised rapiddecarbonization of their energy systems. Chile was the first country in Latin America to implement a carbontax (USD 5/tCO2), followed by Argentina, which defined a carbon tax that became operational in 2019.Given this context, this analysis focuses on determining the impacts of an integrated market where electricitytrade is viable while each country’s government can define internal carbon tax policies.

We model different energy producers in each country. Electricity producers in Chile and Argentina havevarious technologies. We consider hydro, solar, wind, natural gas, and coal technologies in Chile’s case.Historical data shows that Argentina heavily relies on thermal plants fueled by natural gas and on hydroenergy. Technical data for different technologies, obtained from the Chilean Comision Nacional de Energia(National Energy Agency) and the US Energy Information Administration, include fuel consumption, capacityfactors, and variable costs. We model a stake of coal-based production technology only in Chile, and minimalto none in Argentina. We analyzed how the markets react under different renewable sources’ future levels andwith/without limits on energy trade imposed between these two countries.

If no trade is allowed (representing current operations), we calibrated the model to match historical data(2018/2019) for both countries. There is a significantly larger demand in Argentina (129 TWh/y) than inChile (60 TWh/y). Approximately 71% of the generation in Argentina roots in natural gas thermal power

20

plants. Hydro energy fulfills the remaining demand. In Chile’s case, coal and gas power plants have a marketshare of 42%, hydro accounts for 36%, and renewable sources (solar and wind) supply approximately 15% ofthe electric demand.

When trade is allowed among countries and install capacities are not varied (existing capacities in bothcountries), an interesting substitution effect is observed. Imports from Argentina replace conventional meansof production in Chile (coal and gas). The Chilean government curtails fossil-fueled electricity by increasingthe carbon tax faced by such technologies. The opposite effect shows in Argentina, where the governmentlowers the carbon tax to incentivize electricity generation from natural gas technologies. Such an exporthurts the local market. As expected, an increase in exports to Chile yields increased local electricity prices inArgentina, significantly lowering the indigenous consumption levels.

As observed above, with a possibility for energy trade between the countries, our model predicts thatwithout a significant increase in the renewable capacity on either country or without a significant decreasein carbon’s social cost, Argentina’s economy could be highly impacted. Therefore, unless cheap (near-zero)renewable sources produce energy in Chile or Argentina, it is expected that trade among countries will remainlow.

To assess the likelihood of future trade under large deployments of renewable energy, we consider twocases of increased wind and solar capacity in Chile. The two scenarios consider capacity additions of 20 GWand 40 GW, respectively (Amigo et al. 2021).

In these cases, we initially observe that Chile benefits from increased renewable capacity if energy trade isnot allowed. Electricity prices are reduced by 13% when there is an increase of 40 GW, while consumptiongrows by 20%. Interestingly, when energy trade is allowed, we notice that Argentina becomes a net importerof electricity as Chile increases its renewable energy capacity. When 40 GW of renewable capacity is installed,Argentina has net imports of 12 TWh/y. Therefore, Argentina goes from being a net exporter (withoutrenewable capacity installed) to a net importer of electricity. This import is a direct result of the availabilityof cheap energy, which increases the demand.

6.3 Speed analysis

In terms of performance analysis, we focus on InstanceSet A, and B. An instance is marked a solved if ithas an MNE , or an algorithm finds a certificate of inexistence, namely, no MNE exists. The time limit isTL = 1800 seconds. In our implementation, we introduce 3 extension strategies for Algorithm 2: given alexicographic order for each leader’s polyhedra, k of them are added sequentially, reverse-sequentially, orrandomly.

Table 1: Results summary of different algorithmic configurations for InstanceSetA.

Time (s) WinsAlgorithm ES k EQ NO All EQ NO Solved

FE - - 29.08 0.12 120.21 6 82 140/149

Seq 1 6.65 0.35 51.33 3 0 145/149Seq 3 17.76 0.18 55.82 5 0 145/149Seq 5 6.40 0.15 51.08 3 0 145/149

Rev.Seq 1 7.97 0.36 3.73 26 0 149/149Rev.Seq 3 11.29 0.18 53.12 4 0 145/149Rev.Seq 5 9.53 0.15 76.41 5 0 143/149Random 1 5.22 0.36 26.60 8 0 147/149Random 3 32.42 0.18 85.65 5 0 143/149

MNEInnerApp

Random 5 23.67 0.15 58.26 2 0 145/149

PNE FE-P - - 7.25 0.12 328.23 – – 122/149

Tables 1 and 2 summarize the computational results for InstanceSetA and InstanceSetB, respectively.The upper parts of the tables reports results for the full enumeration Algorithm 1 (FE ) and Algorithm 2

21

Table 2: Results summary of different algorithmic configurations for InstanceSetB.

Time (s) WinsAlgorithm ES k EQ NO All EQ NO Solved

FE - - 260.29 1.12 1174.32 0 2 20/50

Seq 1 39.26 9.64 672.24 1 0 32/50Seq 3 62.66 3.88 616.25 1 0 34/50Seq 5 24.03 2.83 733.97 1 0 30/50

Rev.Seq 1 171.47 9.66 262.74 27 0 47/50Rev.Seq 3 13.85 3.86 585.27 4 0 34/50Rev.Seq 5 78.57 2.83 798.90 6 0 29/50Random 1 34.65 9.65 497.06 0 0 37/50Random 3 123.02 3.87 588.03 2 0 36/50

MNEInnerApp

Random 5 39.18 2.86 711.77 4 0 41/50

PNE FE-P - - 7.36 1.12 1441.95 – – 10/50

(InnerApp), where an MNE solves the instances. In the bottom part, we specifically look for PNEs withthe enhanced algorithm presented in Section 5.2 (FE-P). In the third column, if the algorithm is the innerapproximation, we highlight the extension strategies, and the relative parameter k in the following column.Fifth, sixth, and seventh columns are, respectively, average time when: (i) an MNE is found (EQ), (ii) acertificate of non-existence is returned (NO) and (iii) for all instances. In the eighth and ninth column, wereport the number of times the row’s algorithm outperforms all the others, namely wins in terms of computingtimes. Finally, the tenth column reports how many instances do not trigger the time limit.

For MNE s, InnerApp achieves better performances than FE, being on average 2x faster on all instances,and up to 30x when an MNE exists (see InnerApp-RevSeq-1 in Table 1). Table 2 shows the full potentialof InnerApp, which remarkably reduces computational times compared to FE. Especially, InnerApp cansolve almost all the 50 hard instances compared to the 20 solved by FE. Besides, when no equilibrium exists,InnerApp will always terminate at its last iteration, namely the one corresponding to FE. It is not surprisingthat FE returns a non-existence certificate always faster than InnerApp. Both the algorithms InnerApp andFE – when asked to retrieve a generic MNE – may return a PNE . This happens 37.6%, and 30.4% withinInstanceSetA and InstanceSetB, respectively. Hence, there is a natural need for FE-P.

7 Concluding Remarks

Our theoretical and computational framework tackles NASPs, where players of a Nash game solve linearbilevel programs, and each leader can have several followers playing a simple Nash game among themselves.We have shown that deciding existence of PNE and MNE for NASPs is Σp

2 -hard, and we provided a familyof algorithms to find MNE s as well as PNE s for the problem. We shown it is sufficient to compute an MNEover the convex hull of each player’s feasible region to retrieve a MNE for the original problem.

This work expands our knowledge of algorithmic approaches to compute equilibria, in particular MNE s.In addition to a theoretical characterization of these algorithmic methods, it analyzes their practical efficiency,settles their limitations, and opens up new future directions by establishing a solid benchmark againstwhich future progress can be measured. From an application standpoint, we demonstrated how the NASPsframework could help unveil counterintuitive consequences of policymaking within the context of internationalenergy trade.

In terms of forthcoming work, the computation of multiple equilibria, or their selection according to somespecified criteria, is interesting. Furthermore, it may be worth developing procedures to prune parts of thefeasible regions (e.g., polyhedra) not in the support of any equilibrium. This last direction would remarkablyspeed up equilibria computation. Any advancements on these proposed research lines would enable us totackle more general cases, for instance, where interactions among followers of different leaders are allowed.

22

Endnotes

1. Full implementation with detailed documentation are available on https://github.com/ssriram1992/

EPECsolve.2. All instances are available on https://github.com/ds4dm/EPECInstances.

23

Bibliography

Amdur D, Rabe BG, Borick CP (2014) Public views on a carbon tax depend on the proposed use of revenue. Issues inEnergy and Environmental Policy -(13).

Amigo P, Cea-Echenique S, Feijoo F (2021) A two stage cap-and-trade model with allowance re-trading and capacityinvestment: The case of the chilean ndc targets. Energy 120129.

Balas E (1985) Disjunctive Programming and a Hierarchy of Relaxations for Discrete Optimization Problems. SIAMJournal on Algebraic Discrete Methods 6(3):466–486, ISSN 0196-5212, URL http://dx.doi.org/10.1137/

0606047.

Bard JF (1991) Some properties of the bilevel programming problem. Journal of Optimization Theory and Applications68(2):371–378, ISSN 1573-2878, URL http://dx.doi.org/10.1007/BF00941574.

Bard JF, Plummer J, Sourie JC (1998) Determining tax credits for converting nonfood crops to biofuels: An applicationof bilevel programming. Migdalas A, Pardalos PM, Varbrand P, eds., Multilevel Optimization: Algorithms andApplications, 23–50 (Boston, MA: Springer US), ISBN 978-1-4613-0307-7.

Bard JF, Plummer J, Sourie JC (2000) A bilevel programming approach to determining tax credits for biofuelproduction1this work was supported by a grant from institut national de la recherche agronomique and thetexas higher education coordinating board under the advanced research program – arp 003.1. European Journalof Operational Research 120(1):30 – 46, ISSN 0377-2217, URL http://dx.doi.org/10.1016/S0377-2217(98)

00373-7.

Basu A, Ryan CT, Sankaranarayanan S (2019) Mixed-integer bilevel representability. Mathematical ProgrammingISSN 1436-4646, URL http://dx.doi.org/10.1007/s10107-019-01424-w.

Boffey D (2021a) Eu threatens to block covid vaccine exports amid astrazeneca short-fall. The Financial Times URL https://www.theguardian.com/world/2021/jan/25/

eu-threatens-to-block-covid-vaccine-exports-amid-astrazeneca-shortfall, accessed on Marchthe 4th 2021.

Boffey D (2021b) Italy blocks export of 250,000 astrazeneca vaccine doses to aus-tralia. The Guardian URL https://www.theguardian.com/world/2021/mar/04/

italy-blocks-export-of-250000-astrazeneca-vaccine-doses-to-australia, accessed on March the4th 2021.

Brotcorne L, Labbe M, Marcotte P, Savard G (2008) Joint design and pricing on a network. Operations Research56(5):1104–1115, URL http://dx.doi.org/10.1287/opre.1080.0617.

Candler W, Norto R (1977) Multi-level programming and development policy. Working Paper 1–56.

Candler W, Townsley R (1982) A linear two-level programming problem. Computers & Operations Research 9(1):59–76.

Carvalho M, Lodi A, Pedroso JP (2018a) Existence of nash equilibria on integer programming games. Vaz AIF,Almeida JP, Oliveira JF, Pinto AA, eds., Operational Research, 11–23 (Cham: Springer International Publishing),ISBN 978-3-319-71583-4.

Carvalho M, Lodi A, Pedroso JP (2020) Computing Nash equilibria for integer programming games.

Carvalho M, Lodi A, Pedroso JP, Viana A (2017) Nash equilibria in the two-player kidney exchange game. MathematicalProgramming 161(1):389–417, ISSN 1436-4646, URL http://dx.doi.org/10.1007/s10107-016-1013-7.

Carvalho M, Pedroso JP, Telha C, Vyve MV (2018b) Competitive uncapacitated lot-sizing game. International Journalof Production Economics 204:148 – 159, ISSN 0925-5273, URL http://dx.doi.org/https://doi.org/10.1016/

j.ijpe.2018.07.026.

Cashell B, Jackson WD, Jickling M, Webel B (2004) The economic impact of cyber-attacks. Congressional ResearchService Documents, CRS RL32331 (Washington DC) .

Chen X, Deng X (2006) Settling the complexity of two-player nash equilibrium. 2006 47th Annual IEEE Symposiumon Foundations of Computer Science (FOCS’06), 261–272 (IEEE).

Cottle R, Pang JS, Stone RE (2009) The Linear Complementarity problem (Society for Industrial and AppliedMathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104)), ISBN 898719003.

Debreu G (1952) A Social Equilibrium Existence Theorem. Proceedings of the National Academy of Sciences 38(10):886–893, ISSN 0027-8424, 1091-6490, URL http://dx.doi.org/10.1073/pnas.38.10.886.

Del Pia A, Ferris M, Michini C (2017) Totally unimodular congestion games. Proceedings of the Twenty-Eighth AnnualACM-SIAM Symposium on Discrete Algorithms, 577–588, SODA ’17 (USA: Society for Industrial and AppliedMathematics).

DeMiguel V, Xu H (2009) A stochastic multiple-leader stackelberg model: analysis, computation, and application.Operations Research 57(5):1220–1235.

Eggermont CE, Woeginger GJ (2013) Motion planning with pulley, rope, and baskets. Theory of Computing Systems53(4):569–582.

Egging R, Gabriel SA, Holz F, Zhuang J (2008) A complementarity model for the european natural gas market.Energy policy 36(7):2385–2414.

Egging R, Holz F, Gabriel SA (2010) The world gas model: A multi-period mixed complementarity model for theglobal natural gas market. Energy 35(10):4016–4029.

Facchinei F, Pang JS (2015a) Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol 1,volume 1 (Springer-Verlag), ISBN 9788578110796, URL http://dx.doi.org/10.1017/CBO9781107415324.004.

Facchinei F, Pang JS (2015b) Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol 2,volume 2 (Springer-Verlag), ISBN 9788578110796, URL http://dx.doi.org/10.1017/CBO9781107415324.004.

Fan K (1952) Fixed-point and Minimax Theorems in Locally Convex Topological Linear Spaces. Proceedings of theNational Academy of Sciences 38(2):121–126, ISSN 0027-8424, 1091-6490, URL http://dx.doi.org/10.1073/

pnas.38.2.121.

Feijoo F, Das TK (2014) Design of pareto optimal co2 cap-and-trade policies for deregulated electricity networks.Applied energy 119:371–383.

Feijoo F, Huppmann D, Sakiyama L, Siddiqui S (2016) North american natural gas model: Impact of cross-bordertrade with mexico. Energy 112:1084–1095.

Feijoo F, Iyer GC, Avraam C, Siddiqui SA, Clarke LE, Sankaranarayanan S, Binsted MT, Patel PL, Prates NC,Torres-Alfaro E, et al. (2018) The future of natural gas infrastructure development in the united states. Appliedenergy 228:149–166.

Fleming S, Brunsden J, Miles (2021) Italy blocks shipment of oxford/astrazeneca vaccine to australia. The FinancialTimes URL https://www.ft.com/content/bed655ac-9285-486a-b5ad-b015284798c8, accessed on March the4th 2021.

Gabriel SA, Conejo AJ, Fuller JD, Hobbs BF, Ruiz C (2012) Complementarity Modeling in Energy Markets (Springer-Verlag), ISBN 9781441961235.

Gabriel SA, Leuthold FU (2010) Solving discretely-constrained mpec problems with applications in electric powermarkets. Energy Economics 32(1):3–14.

Gilboa I, Zemel E (1989) Nash and correlated equilibria: Some complexity considerations. Games and EconomicBehavior 1(1):80 – 93, ISSN 0899-8256, URL http://dx.doi.org/https://doi.org/10.1016/0899-8256(89)

90006-7.

Glicksberg IL (1952) A Further Generalization of the Kakutani Fixed Point Theorem, with Application to NashEquilibrium Points. Proceedings of the American Mathematical Society 3(1):170, ISSN 00029939, URL http:

//dx.doi.org/10.2307/2032478.

Gordon LA, Loeb MP (2002) The economics of information security investment. ACM Transactions on Informationand System Security (TISSEC) 5(4):438–457.

Hobbs BF, Metzler CB, Pang JS (2000) Strategic gaming analysis for electric power systems: An mpec approach.IEEE transactions on power systems 15(2):638–645.

Holz F, Von Hirschhausen C, Kemfert C (2008) A strategic model of european gas supply (gasmod). Energy Economics30(3):766–788.

Hu X, Ralph D (2007) Using EPECs to Model Bilevel Games in Restructured Electricity Markets with LocationalPrices. Operations Research 55(5), URL http://dx.doi.org/10.1287/opre.1070.0431.

Jeroslow RG (1985) The polynomial hierarchy and a simple model for competitive analysis. Mathematical programming32(2):146–164.

25

Kulkarni AA, Shanbhag UV (2014) A shared-constraint approach to multi-leader multi-follower games. Set-valued andvariational analysis 22(4):691–720.

Kulkarni AA, Shanbhag UV (2015) An existence result for hierarchical stackelberg v/s stackelberg games. IEEETransactions on Automatic Control 60(12):3379–3384.

Labbe M, Violin A (2013) Bilevel programming and price setting problems. 4OR 11(1):1–30, ISSN 1614-2411, URLhttp://dx.doi.org/10.1007/s10288-012-0213-0.

Leyffer S, Munson T (2010) Solving multi-leader–common-follower games. Optimisation Methods & Software 25(4):601–623.

Li H, Meissner J (2011) Competition under capacitated dynamic lot-sizing with capacity acquisition. InternationalJournal of Production Economics 131(2):535 – 544, ISSN 0925-5273, URL http://dx.doi.org/10.1016/j.ijpe.

2011.01.022.

Liu Y, Lu Y (2015) The economic impact of different carbon tax revenue recycling schemes in china: A model-basedscenario analysis. Applied Energy 141:96–105.

Nash J (1951) Non-Cooperative Games. Annals of Mathematics 54(2):286–295.

Nash JF (1950) Equilibrium Points in N-Person Games. Proceedings of the National Academy of Sciences of the UnitedStates of America 36(1):48–9, ISSN 0027-8424, URL http://dx.doi.org/10.1073/pnas.36.1.48.

Olsen DJ, Dvorkin Y, Fernandez-Blanco R, Ortega-Vazquez MA (2018) Optimal carbon taxes for emissions targets inthe electricity sector. IEEE Transactions on Power Systems 33(6):5892–5901.

Ralph D, Smeers Y (2006) EPECs as models for electricity markets. 2006 IEEE PES Power Systems Conference and Ex-position, 74—-80, URL http://www3.eng.cam.ac.uk/~dr241/Papers/Ralph-Smeers-EPEC-electricity.pdf.

Rosenthal RW (1973) A class of games possessing pure-strategy nash equilibria. International Journal of Game Theory2:65 – 67.

Sagratella S, Schmidt M, Sudermann-Merx N (2019) The noncooperative fixed charge transportation problem. EuropeanJournal of Operational Research ISSN 0377-2217, URL http://dx.doi.org/https://doi.org/10.1016/j.ejor.

2019.12.024.

Sankaranarayanan S, Feijoo F, Siddiqui S (2018) Sensitivity and covariance in stochastic complementarity problems withan application to North American natural gas markets. European Journal of Operational Research 268(1):25–36,ISSN 0377-2217, URL http://dx.doi.org/10.1016/J.EJOR.2017.11.003.

Sherali HD (1984) A multiple leader stackelberg model and analysis. Operations Research 32(2):390–404.

Stackelberg H (1934) Von, 1934, marktform und gleichgewicht. Vienna: Julius Springer English translation availableat https://doi.org/10.1007/978-3-642-12586-7.

Stein O, Sudermann-Merx N (2018) The noncooperative transportation problem and linear generalized nash games.European Journal of Operational Research 266(2):543 – 553, ISSN 0377-2217, URL http://dx.doi.org/10.

1016/j.ejor.2017.10.001.

Weintraub R, Bitton A, Rosenberg M (2020) The danger of vaccine nationalism. Harvard Business Review URLhttps://hbr.org/2020/05/the-danger-of-vaccine-nationalism, accessed on March the 4th 2021.

26

Supplementary Material

In this electronic companion, we complement the proofs of Section 3 in Section 8, the pseudo-code for thePNE s algorithm in Section 10, and an overview of computational instances in Section 11.

8 Extensions to proofs of hardness

Proof (Proof of Claim 1). All the constraints are linear, and if the variables of the other player are fixed, theobjectives are also linear. Also, the follower is simply parameterized in their leader’s variables. There areprecisely two leaders, and their interaction follows the definition of a simple Nash game. Hence – by definition– the game in (2) is a trivial NASP .

Proof (Proof of Claim 2). Notice that the constraints in (2h) enforce yi ≥ max(−xi, xi − 1), and since yi isminimized, it has necessarily to be equal to max(xi−1,−xi). However, if this quantity should be non-negative– as enforced in (2c) – then either xi ≤ 0 or 1− xi ≤ 0 should hold. The claim follows.

Proof (Proof of Lemma 1). The following bilevel problem gives the necessary extended formulation. Variablesz1, z2, . . . are the variables in the lifted space, which can be projected out.

x ≥ 0 (16a)

y ≥ 0 (16b)

h ≥ 0 (16c)

y ≤ 1 (16d)

h ≤ x (16e)

z1, . . . , z6 ≥ 0 (16f)

(z1, , . . . , z6) ∈ arg minz

6∑i=1

zi :

z1 ≥ h− x ; z1 ≥ −hz2 ≥ 1− y ; z2 ≥ −hz3 ≥ y − 1 ; z3 ≥ −hz4 ≥ x− h ; z4 ≥ −yz5 ≥ h− x ; z5 ≥ −yz6 ≥ y − 1 ; z6 ≥ −y

(16g)

Proof (Proof of Lemma 2). If S has an extended formulation given by (x, y) : ASx + BSy ≤ bS ; y ∈arg minfTS y : CSx + DSy ≤ gS, and if T has an extended formulation given by (x, y) : ATx + BT y ≤bT ; y ∈ arg minfTT y : CTx+DT y ≤ gT , then the following is an extended formulation of S × T :

(x, y, u, v) : ASx+BSy ≤ bS ;ATu+BT v ≤ bT ;

(y, v) ∈ arg minfTS y + fTT v :CSx+DSy ≤ gSCTu+DT y ≤ gT

Proof (Proof of Claim 6.). All constraints are linear, and if the variables of the other player are fixed, theobjectives are also linear. The constraints (5h) are valid due to Lemma 1. Also, for Lemma 2, we can havemultiple bilevel constraints in (5h) and (5i). Each follower is simply parameterized in their leader’s variables.There are precisely two leaders, and their interaction follows the definition of a simple Nash game.

27

9 NASP with no PNE but only an MNE

Example 1. Considering the following Latin-Greek trivial NASP .Latin Player

maxx,y

: x1ξ1 + x2ξ2 (17a)

x, y ≥ 0 (17b)

x ≤ 1 (17c)

x1 + x2 = 1 (17d)

y ∈ arg miny

y1 + y2 :

yi ≥ −xiyi ≥ xi − 1

for i = 1, 2

(17e)

Greek Player

maxξ,χ

: x2ξ1 + x1ξ2 (17f)

ξ, χ ≥ 0 (17g)

ξ ≤ 1 (17h)

ξ1 + ξ2 = 1 (17i)

χ ∈ arg minχ

χ1 + χ2 :

χi ≥ −ξiχi ≥ ξi − 1

for i = 1, 2

(17j)

The only feasible decisions for both the Latin and the Greek player in (17) are (1, 0, 0, 0), (0, 1, 0, 0). So thegame can be written as a normal form game. The payoffs for these finitely many strategies can be computedto be that if the Latin and the Greek player choose the same strategy, then the Latin player gets a payoff of 1and the Greek player gets a payoff of 0. If they choose different strategies, then the Latin player gets a payoffof 0, and the Greek player gets a payoff of 1. One can easily check that this game’s unique Nash equilibriumis an MNE and that no PNE exists.

10 Enumeration algorithm to obtain a PNE

Algorithm 3 reports the pseudo-code for the algorithm described in Section 5.2.

Algorithm 3 Enumeration algorithm to obtain a PNE for a NASP

Input: A description of NASP N = (P 1, . . . , Pn).Output: For each i = 1, . . . , n, a pure-strategy xi, such that the strategy profile is a PNE or a proof that no PNE

exists.1: for i = 1, ..., n do2: Enumerate the polyhedra whose union defines the feasible set Fi of P i.3: Fi ← cl conv Fi by applying Theorem 3.4: P i ← objective function of P i and a feasible set of Fi.5: end for6: N = (P 1, ..., Pn) the facile Nash game.

7: Enforce δij for i = 1, . . . , n, j = 1, . . . , gi in N to be binary.

8: if N is infeasible then9: return No PNE exists.

10: else11: return Project the solution of N to the space of the original variables of N .12: end if

28

11 Computations

Governments act as Stackelberg leaders by trading energy, intending to minimize their emissions, andeventually to maximize tax incomes. Within each country, energy producers act as Stackelberg followersand play a Nash game between themselves, aiming to maximize their profits. Each country is interestedin imposing a tax that is not preventing profitable domestic production, as it is constrained to keep thedomestic energy price less than a predetermined threshold. We present the optimization problems of theplayers formally below. For ease of understanding, the quantities in red are parameters, i.e., inputs to themodel. And the quantities in blue are decision variables, decided by the country or of the energy producersin the same country. Quantities in green are variables in a problem, but not decided by the country beingconsidered. Each country C solves the following problem.

minqp,tp,

qC′→Cimp ,qC

exp

:

∑p∈P

Cpemmisionqp − btpqp

+∑

C′∈C\C

πCqC′→C

imp − πCqCexp (18a)

subject to tp ≤ tp (18b)

αC − βC∑p∈P

qp + qCimp − qCexp

≥ πC (18c)

∑C′∈C

qC′→C

imp = qCimp (18d)

qp ∈ SOL(Lower Level Nash Game) (18e)

Cpemmision is the dollar value of the emission caused by producer p while producing a unit quantity of energy.

This number is the product of cost incurred due to the emission of one unit of greenhouse gases (GHG),some times referred to as the social cost of carbon and the quantity of GHG emitted for each unit of energyproduced by the producer p, called as the emission factor. b dictates whether the objective should include thetax revenue earned by the government or not. qp is the quantity of energy produced by the producer p ∈P,qCimp,q

Cexp are respectively import and export quantities, and αC , βC are the intercept and the slope of the

demand curve. The domestic price, for each country, is given by αC − βCQ, where Q is the quantity of energyavailable domestically. Finally, πC is the price at which the country can import energy from other countries,hence the variable linking the optimization problems of different countries. Thus, πC can be interpreted asthe shadow price to the market-clearing constraint∑

C′∈C

qC→C′

imp =∑C∈C

qCexp. (18f)

We note that including (18f) does not make the game into a generalized Nash game. This is because(18f) can be written as the KKT conditions of a fictitious optimization problem, generally referred to as theinvisible hand in the market. An alternative manner of looking at this is as if there is a perfect competitionin the international energy markets and the most efficient allocation of resources happens. This is again astandard simplifying assumption considered, for example, in Egging et al. (2010, 2008), Gabriel and Leuthold(2010), Sankaranarayanan et al. (2018), Feijoo et al. (2018).

Optionally for some countries, as a domestic policy, we introduce a carbon tax paradigm, where thetax imposed on the followers is proportional to the emissions they cause. i.e., there is a constraint tp =Cp

emmisiontGHG, where the government decides the tax payable per unit emission. Furthermore, note that if bis non-zero, the objective is no longer linear. In such a case, we replace the product term with a McCormickrelaxation. Finally, tp, and πC are the tax cap and price cap, respectively. The lower level problem that eachproducer p solves is formulated as follow:

minqp

: Cpqp +

1

2Dpq

p2 + tpqp −

αC − βC∑p′∈P

qp′+ qCimp − qCexp

qp (19a)

29

subject to qp ≥ 0 (19b)

qp ≤ qp (19c)

The first two terms in the objective correspond to the energy producer’s cost, while the third term is the taxexpense. The parenthesis results in the revenue of p, which is the product of domestic price and the quantityproduced. Further, the producer is constrained by their capacity limits. Note that the product of variables(tpqp) in the objective does not pose any additional difficulty to the problem. This is because the follower’sproblem is still convex quadratic for a fixed value of tp, and the KKT conditions give complementarityconstraints with only linear terms.

Further, we also note that the previously-mentioned assumption of optimistic equilibrium selection by theleaders and the limitations imposed as a result, are irrelevant here. This is because, for Dp > 0, which isalways the case, the test examples have a unique lower-level equilibrium. Thus any technique for equilibriumselection is not warrented.

11.1 Instance sets

We generated three instances sets for our computations. (i) InstanceSetA contains 149 instances where there are3 to 5 countries (ii) InstanceSetB contains 50 instances with strictly 7 countries. These instances were selectedif Algorithm 1 was not able to solve them within 10 second on a single core machine. (iii) InstanceSetInsightscontains 50 instances with 2 countries with 3 followers each. Such instances are useful to derive managerialinsights from our model. The specific parameters for all these instances are described in Table 3 and areavailable in our open-source GitHub repository. All our tests run on a 8-cores Intel(R) Xeon Gold 6142, with32GB of RAM and Gurobi 9.0.

Parameter Distribution Notes

Capacities 50, 100, 130, 170, 200, 1000, 1050, 20000 Each follower’s capacity is randomly drawn from these values.

Emission Costs 25, 50, 100, 200, 300, 500, 550, 600 The first two values are reserved for green producers. The following two foraveragely-polluting producers, while the remaining three for highly-pollutingones.

Linear Costs 150, 200, 220, 250, 275, 290, 300 Linear costs are generally inversely proportional to the emission cost. Forinstance, a producer with a 50 emission cost will generally have a linear costaround 290.

Quadratic Costs 0, 0.1, 0.2, 0.3, 0.5, 0.55, 0.6 Quadratic costs are generally inversely proportional to the emission cost.Same rationale as linear costs.

Tax Caps 0, 50, 100, 150, 200, 250, 275, 300 Tax caps are assigned following the same rational of emission costs. Thelower the emission cost of a given producer, the lower the maximum taxapplicable to it.

Demand Alpha 275, 300, 325, 350, 375, 450 Each country alpha is randomly drawn from this set.

Demand Beta 0.5, 0.6, 0.7, 0.75, 0.8, 0.9 Each country beta is randomly drawn from this set.

Price Cap 0.8, 0.85, 0.90, 0.95 Each country price-limit is randomly drawn from this set. The final price-limit is made of the product of this value and the country’s demand alphaparameter.

Tax Paradigm Standard, Single, Carbon A country tax scheme can be: (i.) Standard, where followers are taxed at dif-ferent levels per unit-energy, or (ii. ) Single, where all the followers are taxedwith the same level per unit-energy (iii.) Carbon, where all the followers aretaxed with the same level per unit-emission

Table 3: Description of the parameters for our instances.

.

11.2 Results tables

Tables 4 and 5 contains the full results for InstanceSetA and InstanceSetB, respectively. The first threecolumns are the instance number, the number of leaders, and – for each leader – their respective number

30

of followers in squared parenthesis. The MNE column is the status of the instance, namely if it has anequilibrium (YES ), if it does not (NO), or if the time limit was triggered (TL) for all the methods. In theremaining column, we report the clock time and the status for each algorithmic configuration. In particular,we have Algorithm 1 (FE ), and the inner approximations. We report three extension strategies, namelythe sequential (seq), the reverse sequential (rseq), and the random one (rand). They are followed by theirrespective parameter k, as reported in Section 6. The last two columns are related to PNE s.

Table 6 reports the results for InstanceSet Insights. The first column reports each instance’s number.The second and third column are boolean values reporting whether the tax (Ta) and the trade (Tr) areallowed (value of 1) or not. The following 16 columns are results for the first country (Country One), whilethe remaining 16 are for the second country (Country Two). Following the column order, for each country wereport: the unit-energy production level Prod, the domestic price per unit-energy $(E), the import Imp andexport Exp unit-energies, the export price $(E), and the tax per unit-emission Tax. Furthermore, for each ofthe the 3 followers of each country, we have the type Ty (C for coal, G for gas, or S for solar), the associatedemission cost per unit-energy E, and its production Prod.

31

Table

4:

MN

Eand

PN

Ere

sult

sfo

rIn

stan

ceS

etA

.C

olu

mns:

#-

Inst

ance

Num

ber

.L

-N

um

ber

of

leader

sin

the

inst

ance

.F

-N

um

ber

of

foll

ower

sea

chle

ader

has

.F

E-

Tim

eta

ken

for

full

enu

mer

ati

on

alg

ori

thm

.se

q1to

ran

d5

-T

ime

take

nfo

rin

ner

ap

pro

xim

ati

on

wit

hd

iffer

ent

exte

nsi

on

stra

tegie

s.M

NE

-ex

iste

nce

(or

tim

eli

mit

reach

edT

L).

FE

-P-

Tim

efo

rfu

llen

um

erati

on

tofi

nd

aP

NE

.P

NE

-ex

iste

nce

(or

tim

eli

mit

reac

hed

TL

).

#L

FFE

seq1

seq3

seq5

rseq1

rseq3

rseq5

rand1

rand3

rand5

MNE

FE-P

PNE

13

[1

22

]0.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

4NO

0.0

4NO

23

[2

23

]0.0

70.1

50.0

80.0

80.1

60.0

80.0

80.1

60.0

80.0

8NO

0.0

7YES

33

[2

22

]0.0

50.1

30.0

60.0

60.1

30.0

60.0

60.1

30.0

60.0

6NO

0.0

5NO

43

[1

21

]0.0

40.1

00.0

50.0

50.1

00.0

50.0

50.1

00.0

50.0

5NO

0.0

4TL

53

[2

13

]0.3

30.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

4YES

0.0

9YES

63

[2

21

]0.0

60.1

80.1

10.0

70.1

80.1

10.0

70.1

80.1

10.0

7NO

0.0

6YES

73

[2

22

]0.0

60.1

40.0

70.0

70.1

40.0

70.0

70.1

40.0

70.0

7NO

0.0

6YES

83

[1

21

]0.1

00.0

80.0

80.0

80.0

80.0

80.0

80.0

80.0

80.0

8YES

0.0

8NO

93

[1

22

]0.0

50.1

00.0

60.0

60.1

00.0

60.0

60.1

00.0

60.0

6NO

0.0

4NO

10

3[2

21

]0.1

60.2

40.1

70.1

70.1

30.1

90.2

00.2

40.2

00.2

0YES

0.1

6NO

11

3[3

22

]0.0

80.2

20.1

50.0

90.2

30.1

40.0

90.2

20.1

40.0

9NO

0.0

8NO

12

3[2

22

]0.7

61.5

01.4

11.0

70.3

80.2

80.7

91.7

21.7

70.5

5YES

2.0

7YES

13

3[2

12

]0.0

40.0

70.0

50.0

50.0

70.0

50.0

50.0

70.0

50.0

5NO

0.0

4NO

14

3[2

13

]0.0

60.1

30.0

70.0

70.1

30.0

70.0

70.1

30.0

70.0

7NO

0.0

6NO

15

3[2

12

]0.3

90.0

80.0

80.0

80.0

90.0

80.0

80.0

80.0

80.0

8YES

0.1

8NO

16

3[2

32

]5.7

35.0

77.6

84.8

30.3

71.4

812.1

913.4

91370.2

81.3

0YES

TL

YES

17

3[2

22

]0.0

60.1

30.0

70.0

70.1

30.0

70.0

70.1

30.0

70.0

7NO

0.0

6YES

18

3[2

22

]0.7

90.1

50.1

60.1

50.1

50.1

50.1

50.1

50.1

50.1

5YES

3.1

7NO

19

3[2

22

]0.0

70.1

60.0

80.0

80.1

60.0

80.0

80.1

60.0

80.0

8NO

0.0

7NO

20

3[1

32

]0.0

50.0

90.0

60.0

60.0

90.0

60.0

60.0

90.0

60.0

6NO

0.0

5NO

21

3[1

22

]0.0

40.1

10.0

60.0

60.1

20.0

60.0

60.1

10.0

60.0

6NO

0.0

5YES

22

3[2

11

]0.0

40.1

00.0

50.0

50.1

00.0

50.0

50.1

00.0

50.0

5NO

0.0

4NO

23

3[1

21

]0.0

90.1

30.1

10.1

10.1

00.1

10.1

10.1

30.1

10.1

1YES

0.0

7YES

24

3[2

21

]0.0

50.1

40.0

90.0

60.1

40.0

90.0

60.1

40.0

90.0

6NO

0.0

5YES

25

3[2

22

]0.1

40.2

40.1

70.1

70.1

40.1

60.1

70.1

20.1

70.1

7YES

0.1

2NO

26

3[2

22

]0.0

60.1

50.0

70.0

70.1

60.0

70.0

70.1

50.0

80.0

7NO

0.0

6NO

27

3[1

12

]0.1

00.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

4YES

0.1

2NO

28

3[3

13

]0.2

61.1

00.5

50.9

60.1

00.3

30.5

30.1

80.3

50.2

8YES

TL

NO

29

3[1

12

]0.0

50.1

50.0

90.0

60.1

50.0

90.0

60.1

50.0

90.0

6NO

0.0

5YES

30

3[1

11

]0.0

30.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

4NO

0.0

3YES

31

3[2

21

]0.5

37.0

40.5

80.5

50.5

80.3

70.5

70.5

80.4

80.6

0YES

TL

YES

32

3[1

13

]0.1

10.1

90.1

40.1

50.1

00.1

40.1

40.1

90.1

50.1

5YES

0.1

1YES

33

3[3

23

]0.0

70.1

50.0

80.0

80.1

50.0

80.0

80.1

50.0

80.0

8NO

0.0

7TL

34

3[1

22

]0.4

20.1

50.1

50.1

50.1

50.1

50.1

50.1

50.1

50.1

5YES

1.0

3TL

35

3[2

22

]0.0

60.1

40.0

70.0

70.1

40.0

70.0

70.1

40.0

70.0

7NO

0.0

6NO

36

3[1

13

]0.5

72.6

92.6

46.3

70.6

719.0

23.3

31.6

8316.9

40.6

8YES

TL

TL

37

3[2

13

]0.2

50.2

00.2

50.2

50.3

40.2

00.2

00.3

50.1

90.2

6YES

0.1

5NO

38

3[2

22

]0.0

60.2

00.1

00.0

70.2

00.1

00.0

70.2

00.1

00.0

7NO

0.0

6NO

39

3[3

31

]0.0

70.1

60.0

80.0

80.1

60.0

80.0

80.1

60.0

80.0

8NO

0.0

7TL

40

3[2

12

]0.1

30.2

40.1

80.1

80.1

50.1

70.1

70.1

90.1

70.1

8YES

0.1

4YES

41

3[2

23

]0.0

80.2

10.1

40.0

80.2

20.1

40.0

90.2

20.1

40.0

9NO

0.0

7NO

42

3[1

11

]0.0

30.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

40.0

4NO

0.0

3NO

43

3[1

32

]1.0

20.1

00.1

00.1

00.1

00.1

00.1

00.1

00.1

00.1

0YES

TL

YES

44

3[3

22

]0.1

00.4

20.1

80.2

00.4

20.1

80.2

00.4

20.1

80.1

9NO

0.1

0NO

45

3[2

21

]0.0

50.1

30.0

60.0

60.1

30.0

60.0

60.1

30.0

60.0

6NO

0.0

5TL

46

3[1

22

]0.0

50.1

20.0

60.0

60.1

20.0

60.0

60.1

20.0

60.0

6NO

0.0

5NO

47

3[1

32

]0.4

00.0

30.0

30.0

30.0

30.0

30.0

30.0

30.0

30.0

3YES

0.5

8NO

48

3[2

21

]0.0

40.0

50.0

50.0

50.0

50.0

50.0

50.0

50.0

50.0

5NO

0.0

4YES

49

3[2

11

]0.0

40.1

10.0

70.0

50.1

10.0

70.0

50.1

10.0

70.0

5NO

0.0

4NO

50

4[1

21

2]

0.1

80.0

60.0

60.0

60.0

60.0

60.0

60.0

60.0

60.0

6YES

0.1

7NO

51

4[1

11

1]

0.0

60.1

10.0

70.0

70.1

10.0

70.0

70.1

10.0

70.0

7NO

0.0

6NO

52

4[3

13

1]

0.1

00.1

70.1

10.1

10.1

70.1

10.1

10.1

80.1

10.1

1NO

0.1

0NO

53

4[3

12

1]

775.5

50.3

60.3

60.3

60.3

60.3

60.3

50.3

60.3

60.3

7YES

TL

TL

54

4[1

12

3]

0.0

90.2

20.1

00.1

00.2

20.1

00.1

00.2

20.1

00.1

0NO

0.0

9NO

55

4[1

21

2]

0.1

40.4

30.3

60.3

60.1

60.1

80.1

80.1

60.1

80.3

6YES

0.1

7NO

56

4[2

21

2]

1.6

90.2

00.2

00.2

00.2

00.2

10.2

00.2

00.2

00.2

0YES

TL

NO

57

4[1

22

2]

0.2

90.4

20.2

80.2

80.2

10.3

10.3

10.4

80.4

50.3

1YES

0.6

4YES

58

4[2

22

1]

0.0

90.2

40.1

10.1

10.2

40.1

10.1

10.2

40.1

10.1

1NO

0.0

9YES

59

4[1

22

1]

0.0

90.2

20.1

10.1

10.2

30.1

10.1

10.2

20.1

10.1

1NO

0.0

9NO

60

4[1

31

3]

0.0

90.1

60.1

10.1

10.1

60.1

10.1

10.1

60.1

10.1

1NO

0.1

0YES

61

4[3

13

2]

38.8

3TL

TL

17.7

00.3

12.6

454.9

994.3

81.3

6TL

YES

152.4

8NO

62

4[1

13

2]

24.3

00.8

00.7

90.7

90.7

90.8

00.8

00.7

90.8

00.8

0YES

TL

NO

63

4[2

32

3]

0.3

20.7

00.4

10.4

10.2

40.3

60.3

60.4

70.4

10.4

2YES

0.2

5NO

64

4[2

23

1]

0.1

80.7

60.3

20.3

40.7

70.3

20.3

40.7

60.3

20.3

4NO

0.1

8NO

65

4[2

13

2]

1.3

22.6

32.2

51.8

90.4

21.2

61.2

73.0

61.8

01.6

4YES

1.6

1NO

66

4[3

33

3]

0.5

81.3

40.7

70.8

20.3

70.8

80.6

00.6

60.5

70.5

0YES

0.4

4YES

32

67

4[3

22

1]

0.1

20.4

40.2

20.1

40.4

40.2

20.1

30.4

40.2

20.1

4NO

0.1

2NO

68

4[3

22

2]

1.9

00.2

00.2

00.2

00.2

00.2

00.2

00.2

00.2

00.2

0YES

TL

YES

69

4[2

13

3]

0.1

20.2

80.1

30.1

30.2

80.1

30.1

30.2

80.1

30.1

3NO

0.1

2NO

70

4[2

22

1]

0.0

80.1

60.1

00.1

00.1

60.1

00.1

00.1

60.1

00.1

0NO

0.0

9YES

71

4[1

22

2]

0.0

80.0

90.0

90.0

90.0

90.0

90.0

90.0

90.0

90.0

9NO

0.0

8YES

72

4[1

21

3]

0.1

00.3

20.2

00.1

20.3

30.2

00.1

20.3

30.2

00.1

2NO

0.1

0NO

73

4[2

22

2]

160.8

212.6

612.8

012.7

112.6

512.8

112.6

412.7

512.8

312.6

6YES

TL

NO

74

4[2

11

3]

0.1

60.8

50.3

70.2

80.8

50.3

70.2

80.8

50.3

90.2

8NO

0.1

6NO

75

4[1

21

3]

0.1

00.2

40.1

10.1

10.2

40.1

10.1

10.2

40.1

10.1

1NO

0.1

0YES

76

4[1

11

2]

0.0

60.1

10.0

70.0

70.1

10.0

70.0

70.1

10.0

70.0

7NO

0.0

6YES

77

4[3

12

2]

0.1

40.5

20.2

60.1

60.5

20.2

60.1

60.5

20.2

60.1

6NO

0.1

4NO

78

4[2

11

2]

0.2

40.3

40.2

80.2

80.2

10.2

80.2

80.2

10.2

80.2

8YES

0.2

5YES

79

4[3

21

3]

54.9

9297.6

5297.3

4296.3

74.6

8415.6

4422.4

235.0

457.3

261.4

7YES

TL

NO

80

4[2

11

2]

0.0

70.0

80.0

80.0

80.0

80.0

80.0

80.0

80.0

80.0

8NO

0.0

7YES

81

4[3

22

2]

0.1

90.8

00.3

30.3

60.8

10.3

30.3

70.8

10.3

30.3

7NO

0.1

9YES

82

4[2

31

1]

0.2

30.6

40.3

20.2

60.1

70.2

20.2

60.2

30.1

90.2

6YES

0.4

0NO

83

4[2

12

2]

TL

1.5

01.3

6TL

8.7

351.2

8TL

8.7

476.2

61209.7

6YES

0.8

8NO

84

4[3

31

2]

0.2

61.8

90.7

90.6

61.9

00.8

00.6

61.9

10.7

90.6

6NO

0.2

6NO

85

4[1

11

2]

0.0

70.1

20.0

80.0

80.1

20.0

80.0

80.1

20.0

80.0

8NO

0.0

7YES

86

4[1

11

3]

0.1

50.1

50.1

50.1

50.1

50.1

50.1

50.1

50.1

40.1

5YES

0.2

1NO

87

4[2

31

2]

0.4

234.4

434.3

10.4

60.7

40.4

20.4

40.5

8TL

0.4

9YES

TL

NO

88

4[1

22

1]

0.1

80.3

30.2

20.2

20.1

80.2

10.2

20.2

50.2

10.2

2YES

0.1

8YES

89

4[1

12

1]

0.0

50.0

60.0

60.0

60.0

60.0

60.0

60.0

60.0

60.0

6NO

0.0

5NO

90

4[3

32

1]

0.4

70.5

10.3

30.3

30.2

40.3

20.3

10.5

20.3

40.3

2YES

0.7

1NO

91

4[3

13

2]

TL

0.3

80.3

80.3

80.3

80.3

80.3

80.3

80.3

80.3

8YES

TL

NO

92

4[3

32

3]

0.4

60.9

20.7

30.8

60.2

30.5

70.4

80.4

60.3

60.4

0YES

1.0

0YES

93

4[2

22

2]

0.4

10.6

60.5

10.4

00.6

00.4

40.4

60.7

00.8

50.4

3YES

0.3

2NO

94

4[2

23

2]

1.0

03.2

82.0

32.8

60.4

41.7

41.2

33.3

157.7

22.6

5YES

15.2

4NO

95

4[1

12

2]

0.0

80.2

00.0

90.0

90.2

00.0

90.0

90.2

00.0

90.0

9NO

0.0

8YES

96

4[1

21

1]

0.0

60.1

10.0

70.0

70.1

10.0

70.0

70.1

10.0

70.0

7NO

0.0

6YES

97

4[2

22

2]

0.2

20.3

20.2

60.2

60.2

20.2

50.2

50.3

20.2

50.2

6YES

0.2

2NO

98

4[1

12

1]

0.2

70.1

80.1

80.1

80.1

80.1

80.1

80.1

80.1

80.1

8YES

98.1

4NO

99

4[2

22

2]

0.1

70.6

30.3

20.1

90.6

40.3

20.1

90.6

40.3

20.1

9NO

0.1

7YES

100

5[2

22

11

]0.1

50.3

80.1

70.1

60.3

90.1

60.1

70.3

80.1

70.1

7NO

0.1

5NO

101

5[2

33

22

]TL

TL

TL

TL

1.8

2TL

TL

TL

TL

TL

YES

TL

YES

102

5[2

22

33

]4.4

52.3

61.1

41.3

64.7

12.8

441.9

751.4

71.3

53.7

6YES

TL

YES

103

5[1

23

12

]1.2

90.2

80.2

80.2

80.2

80.2

80.2

80.2

80.2

80.2

8YES

TL

YES

104

5[1

31

11

]0.3

70.3

10.3

30.3

30.4

20.3

10.3

10.6

00.3

10.3

3YES

0.2

3NO

105

5[2

32

21

]0.3

21.7

60.7

70.5

81.7

80.7

70.5

81.7

80.7

70.5

8NO

0.3

2NO

106

5[2

21

22

]0.1

90.4

80.2

10.2

10.4

80.2

00.2

00.4

80.2

00.2

0NO

0.1

9NO

107

5[1

23

21

]0.2

10.8

10.3

80.2

30.8

30.3

80.2

30.8

30.3

80.2

3NO

0.2

1NO

108

5[3

22

11

]0.1

40.3

50.1

70.1

60.3

50.1

70.1

60.3

60.1

60.1

7NO

0.1

5YES

109

5[2

22

31

]582.9

42.1

22.0

92.1

12.1

12.1

22.1

12.1

02.1

12.1

0YES

TL

YES

110

5[3

22

33

]0.2

90.6

70.3

30.3

20.6

90.3

20.3

20.6

70.3

20.3

2NO

0.3

0YES

111

5[2

13

13

]0.2

10.6

60.4

10.2

30.6

60.4

10.2

30.6

60.4

10.2

3NO

0.2

1NO

112

5[1

21

32

]0.4

90.7

40.4

60.5

00.3

20.4

30.5

10.7

50.4

30.5

2YES

1.1

1YES

113

5[1

12

11

]0.3

40.7

60.3

80.3

80.2

50.3

80.3

80.5

10.3

80.3

8YES

0.5

3YES

114

5[1

33

11

]0.2

10.5

30.2

30.2

30.5

30.2

30.2

30.5

30.2

30.2

3NO

0.2

2NO

115

5[3

12

32

]0.2

40.5

90.2

60.2

60.5

90.2

60.2

60.5

90.2

60.2

7NO

0.2

4YES

116

5[1

22

32

]0.1

60.3

00.1

80.1

80.3

00.1

80.1

80.3

00.1

80.1

8NO

0.1

6NO

117

5[2

21

22

]6.0

720.3

65.1

85.0

11.0

91.7

92.8

10.3

427.8

13.0

9YES

4.2

0NO

118

5[2

22

22

]0.2

00.4

70.2

20.2

20.4

80.2

20.2

20.4

80.2

20.2

2NO

0.2

0NO

119

5[3

32

11

]1.6

12.0

325.4

96.8

80.6

81.8

61.3

412.0

03.3

547.5

3YES

TL

NO

120

5[3

21

22

]0.1

90.5

00.2

20.2

10.5

00.2

20.2

20.4

90.2

20.2

2NO

0.2

0NO

121

5[1

21

21

]0.1

40.3

50.1

60.1

50.3

60.1

50.1

60.3

60.1

60.1

5NO

0.1

3YES

122

5[1

22

23

]0.2

00.5

10.2

30.2

30.5

10.2

30.2

30.5

20.2

30.2

3NO

0.2

1NO

123

5[2

22

32

]0.2

10.5

30.2

30.2

30.5

30.2

30.2

30.5

30.2

30.2

4NO

0.2

1NO

124

5[1

12

13

]2.1

00.2

60.2

60.2

60.2

60.2

60.2

60.2

60.2

60.2

6YES

TL

NO

125

5[2

32

12

]2.1

20.5

00.5

10.5

00.4

90.5

00.5

00.5

00.5

10.5

1YES

TL

NO

126

5[2

22

22

]0.9

50.4

70.4

70.4

70.4

70.4

70.4

70.4

70.4

80.4

8YES

TL

YES

127

5[2

12

12

]0.3

00.5

20.3

30.3

30.2

20.3

00.3

00.5

10.2

90.3

3YES

0.2

3YES

128

5[2

12

23

]TL

0.4

80.4

80.4

80.4

80.4

80.4

80.4

80.4

80.4

8YES

TL

YES

129

5[2

23

33

]TL

TL

TL

TL

204.9

5TL

TL

TL

TL

83.4

6YES

TL

NO

130

5[3

21

22

]TL

TL

TL

TL

65.0

1TL

TL

74.4

2TL

TL

YES

TL

NO

131

5[2

12

21

]0.2

10.6

80.4

00.2

30.6

80.4

10.2

30.6

80.4

10.2

3NO

0.2

1YES

132

5[2

21

12

]0.1

90.4

80.2

10.2

10.4

80.2

10.2

10.4

80.2

10.2

1NO

0.1

9TL

133

5[2

22

23

]TL

0.4

50.4

50.4

50.4

50.4

50.4

50.4

50.4

50.4

5YES

TL

NO

134

5[2

21

22

]5.1

61.0

40.7

51.1

20.7

2170.7

3TL

0.7

2TL

TL

YES

0.8

7NO

135

5[2

21

21

]0.1

40.3

60.1

60.1

60.3

60.1

60.1

60.3

60.1

60.1

6NO

0.1

4YES

136

5[2

21

32

]0.2

40.9

40.4

50.2

60.9

40.4

50.2

60.9

40.4

50.2

6NO

0.2

4YES

137

5[2

22

21

]0.4

00.8

80.6

10.6

10.3

50.4

80.4

70.6

50.4

50.4

5YES

0.2

9YES

138

5[2

22

21

]0.2

61.3

70.6

90.4

91.3

80.6

90.4

91.3

70.6

90.4

9NO

0.2

7NO

139

5[3

21

23

]TL

0.5

80.5

80.5

80.5

80.5

80.5

80.5

80.5

80.5

8YES

TL

NO

140

5[2

21

12

]0.1

10.2

10.1

30.1

30.2

10.1

30.1

30.2

20.1

30.1

3NO

0.1

1YES

141

5[1

21

21

]0.6

20.9

41.1

30.6

70.3

30.5

80.7

10.7

20.9

50.9

9YES

1.1

5TL

142

5[2

12

21

]0.2

10.8

30.3

90.2

30.8

40.3

90.2

20.8

30.3

90.2

2NO

0.2

1YES

143

5[1

21

22

]0.1

90.4

60.2

00.2

10.4

60.2

00.2

00.4

60.2

00.2

0NO

0.1

9YES

33

144

5[1

31

22

]0.1

70.3

10.1

90.1

90.3

10.1

90.1

90.3

10.1

90.1

9NO

0.1

7YES

145

5[1

22

21

]0.1

60.4

20.1

80.1

80.4

20.1

80.1

80.4

20.1

80.1

8NO

0.1

7NO

146

5[3

22

12

]TL

3.1

7688.6

720.6

0203.0

7TL

TL

2.2

7TL

21.6

6YES

TL

NO

147

5[1

12

23

]0.1

60.2

90.1

80.1

80.3

00.1

80.1

80.3

00.1

80.1

8NO

0.1

6YES

148

5[2

22

23

]0.2

50.6

20.2

80.2

70.6

20.2

70.2

80.6

20.2

70.2

7NO

0.2

5YES

149

5[2

11

22

]0.1

30.3

50.1

50.1

50.3

50.1

50.1

50.3

50.1

50.1

5NO

0.1

3NO

Tab

le5:

MN

Ean

dP

NE

resu

lts

for

Inst

an

ceS

etB

.S

am

en

ota

tion

as

Tab

le4.

#L

FFE

seq

seq3

seq5

rseq1

rseq3

rseq5

rand1

rand3

rand5

MNE

FE-P

PNE

07

[1

12

22

22

]TL

TL

TL

TL

TL

8.8

6TL

240.9

9TL

TL

YES

TL

TL

17

[1

11

21

21

]62.1

22.8

21.9

3102.9

71.6

46.9

1279.0

61.8

75.9

883.8

4YES

TL

TL

27

[2

32

13

22

]TL

6.2

86.2

96.3

26.4

36.4

26.3

56.2

86.3

56.3

2YES

TL

TL

37

[1

22

11

31

]572.6

90.5

10.5

00.5

00.5

10.5

00.5

00.5

00.5

00.5

0YES

TL

TL

47

[1

32

21

21

]TL

1.3

41.3

31.3

31.3

31.3

51.3

21.3

31.3

51.3

2YES

TL

TL

57

[2

12

21

22

]192.3

70.2

80.2

70.2

80.2

80.2

80.2

70.2

70.2

70.2

7YES

2.4

0YES

67

[2

22

33

21

]TL

TL

TL

TL

20.0

2TL

TL

TL

TL

TL

YES

TL

TL

77

[2

21

13

23

]TL

TL

TL

TL

464.2

1TL

TL

TL

TL

TL

YES

TL

TL

87

[3

31

13

33

]TL

0.3

80.3

80.3

90.3

80.3

80.3

90.3

80.3

90.3

9YES

TL

TL

97

[2

12

22

12

]TL

2.4

52.4

62.4

52.4

32.4

72.4

62.4

72.4

62.4

6YES

TL

TL

10

7[2

32

21

22

]TL

TL

TL

TL

1704.4

2TL

TL

TL

TL

TL

YES

TL

TL

11

7[1

23

13

21

]TL

TL

418.9

6TL

2.8

18.9

1TL

4.1

7TL

TL

YES

TL

TL

12

7[3

22

12

11

]9.2

93.6

72.7

08.1

71.3

84.5

9TL

4.8

69.1

43.9

0YES

TL

TL

13

7[2

33

11

12

]TL

251.7

8250.4

864.7

9687.6

3TL

22.4

5TL

21.4

331.4

6YES

15.6

6YES

14

7[3

22

22

22

]TL

TL

TL

TL

234.3

3TL

TL

480.5

6TL

TL

YES

TL

TL

15

7[2

23

22

21

]10.6

52.0

91.1

216.3

30.6

919.9

14.1

35.1

15.2

518.6

8YES

2.2

4YES

16

7[2

22

21

22

]TL

162.8

9162.1

9TL

14.1

451.7

5TL

82.2

1975.4

9TL

YES

TL

TL

17

7[3

31

32

21

]635.2

7TL

TL

TL

1.8

1TL

TL

6.8

892.2

9TL

YES

TL

TL

18

7[1

12

22

22

]TL

TL

TL

TL

TL

8.7

9TL

TL

10.9

2269.5

7YES

TL

TL

19

7[2

11

32

32

]0.9

78.9

93.5

72.2

89.0

03.5

72.3

08.9

63.5

92.3

0NO

0.9

7NO

20

7[2

23

22

21

]10.7

72.1

21.1

216.3

20.7

020.1

54.0

82.5

64.7

25.8

2YES

2.2

4YES

21

7[3

22

22

22

]TL

TL

TL

TL

231.0

1TL

TL

TL

TL

TL

YES

TL

TL

22

7[2

11

23

13

]1.2

710.2

94.1

93.3

910.3

34.1

63.4

010.3

54.1

63.4

1NO

1.2

6NO

23

7[2

33

11

12

]TL

247.8

0248.0

164.2

8674.3

7TL

22.2

6TL

56.8

9TL

YES

15.7

3YES

24

7[1

12

22

22

]TL

TL

TL

TL

TL

8.8

5TL

40.1

4TL

72.5

2YES

TL

TL

25

7[3

31

32

21

]634.3

0TL

TL

TL

1.8

2TL

TL

21.9

247.9

7TL

YES

TL

TL

26

7[2

22

21

22

]TL

163.1

3162.7

7TL

14.1

351.8

4TL

4.8

249.2

3TL

YES

TL

TL

27

7[2

23

22

21

]10.8

02.1

01.1

216.5

00.7

020.2

04.1

11.3

23.9

629.0

7YES

2.2

5YES

28

7[3

22

22

22

]TL

TL

TL

TL

232.5

6TL

TL

TL

TL

TL

YES

TL

TL

29

7[2

33

11

12

]TL

251.4

1250.9

264.2

4687.3

8TL

22.5

2TL

26.5

52.1

1YES

15.8

3YES

30

7[3

22

12

11

]9.3

23.6

32.7

18.1

71.3

74.6

2TL

3.8

6TL

2.7

7YES

TL

TL

31

7[1

23

13

21

]TL

TL

419.4

8TL

2.8

08.8

8TL

3.5

51403.1

9TL

YES

TL

TL

32

7[2

32

21

22

]TL

TL

TL

TL

1687.5

3TL

TL

TL

TL

TL

YES

TL

TL

33

7[2

12

22

12

]TL

2.4

62.4

62.4

52.4

62.4

62.4

32.4

42.4

72.4

3YES

TL

TL

34

7[3

31

13

33

]TL

0.3

80.3

90.3

80.3

90.3

80.3

80.3

80.3

80.3

8YES

TL

TL

35

7[2

21

13

23

]TL

TL

TL

TL

458.7

2TL

TL

TL

TL

TL

YES

TL

TL

36

7[2

22

33

21

]TL

TL

TL

TL

20.1

5TL

TL

TL

TL

TL

YES

TL

TL

37

7[2

12

21

22

]194.2

80.2

70.2

80.2

80.2

90.2

80.2

70.2

80.2

80.2

8YES

2.4

9YES

38

7[1

32

21

21

]TL

1.3

21.3

21.3

31.3

21.3

31.3

31.3

21.3

31.3

3YES

TL

TL

39

7[1

22

11

31

]572.4

20.5

00.5

00.5

00.5

00.5

00.5

00.5

00.5

00.5

0YES

TL

TL

40

7[2

32

13

22

]TL

6.3

16.3

66.3

16.3

46.3

26.3

06.3

66.3

26.3

1YES

TL

TL

41

7[1

11

21

21

]62.5

52.7

91.9

2103.2

81.6

16.8

8281.1

88.2

923.7

8TL

YES

TL

TL

42

7[1

23

32

13

]TL

4.9

64.9

94.9

44.9

85.0

04.9

54.9

54.9

44.9

7YES

TL

TL

43

7[2

22

22

21

]TL

9.4

79.5

89.5

09.4

89.5

99.5

19.5

39.5

59.5

6YES

TL

TL

44

7[3

11

22

22

]328.7

223.7

322.8

0151.0

336.6

336.9

11190.5

511.5

9168.2

04.9

8YES

TL

TL

45

7[2

22

31

22

]62.3

90.1

20.1

30.1

20.1

20.1

20.1

20.1

30.1

30.1

2YES

TL

TL

46

7[2

12

22

32

]TL

TL

TL

TL

357.4

3TL

TL

TL

TL

TL

YES

TL

TL

47

7[2

22

11

32

]1131.2

51.6

71.6

91.6

61.6

71.6

61.6

61.6

91.6

61.6

6YES

TL

TL

48

7[1

22

22

32

]72.6

819.0

918.0

618.0

8130.4

7136.2

1136.3

747.9

7173.2

2113.8

7YES

TL

TL

49

7[2

22

12

33

]113.3

0TL

TL

TL

4.8

5TL

115.9

6413.0

11065.6

7689.1

0YES

TL

TL

34

Tab

le6:

Inst

ance

s’so

luti

ons

for

Inst

an

ceS

etIn

sigh

ts.

The

colu

mns

are,

inor

der

ofap

pea

rance

:th

ein

stan

ce’s

num

ber

,th

eb

ool

ean

tax

swit

ch(T

a)

an

dth

etr

ad

esw

itch

(Tr).

Th

en,

the

set

of

resu

lts

ass

oci

ate

dw

ith

each

of

the

two

cou

ntr

ies

(Cou

ntr

yO

ne,

an

dC

ou

ntr

yT

wo

).In

part

icu

lar:

the

un

it-e

ner

gypro

du

ctio

nle

vel

Pro

d,

the

dom

esti

cp

rice

per

un

it-e

ner

gy$(

E),

the

imp

ort

Imp

and

exp

ort

Exp

un

it-e

ner

gies

,th

eex

por

tp

rice

$(E

),an

dth

eta

xp

erunit

-em

issi

onT

ax.

Furt

her

mor

e,fo

rea

chof

the

the

3fo

llow

ers

ofea

chco

untr

y,w

ehav

eth

ety

pe

Ty

(Cfo

rco

al,G

for

gas,

orS

for

sola

r),

the

asso

ciat

edem

issi

onco

stp

eru

nit

-en

ergy

E,

an

dit

sp

rod

uct

ion

Pro

d.

Country

One

Country

Tw

oFollow

er

1Follow

er

2Follow

er

3Follow

er

1Follow

er

2Follow

er

3#

Ta

Tr

Prod

$(D

)Im

pExp

$(E)

Tax

Ty

EP

rod

TE

Prod

Ty

EP

rod

Prod

$(D

)Im

pExp

$(E)

Tax

Ty

EP

rod

TE

Prod

Ty

EP

rod

I0

00

83,3

3300,0

00,0

00,0

0-

80,0

0C

500

0,0

0G

200

41,6

7S

25

41,6

760,0

0255,0

00,0

00,0

0-

0,3

0C

500

27,7

6G

100

27,7

6S

50

4,4

8I0

01

98,8

1300,0

00,5

616,0

3279,4

380,0

0C

500

0,0

0G

200

57,1

4S

25

41,6

744,5

2255,0

016,0

30,5

6278,4

39,3

5C

500

20,5

2G

100

20,5

2S

50

3,4

9I0

10

83,3

3300,0

00,0

00,0

0-

0,1

4C

500

7,3

5G

200

37,2

2S

25

38,7

660,0

0255,0

00,0

00,0

0-

0,0

0C

500

27,4

2G

100

27,8

8S

50

4,6

9I0

11

119,3

2300,0

00,0

035,9

9103255,8

60,0

7C

500

32,0

2G

200

47,1

0S

25

40,2

024,0

1255,0

035,9

90,0

0103256,8

60,0

7C

500

0,1

4G

100

22,4

3S

50

1,4

5I1

00

37,5

0270,0

00,0

00,0

0-

11,4

1C

300

29,6

9G

100

7,8

1S

50

0,0

050,0

0315,0

00,0

00,0

0-

15,7

7G

100

26,9

2S

50

11,5

4S

25

11,5

4I1

01

0,0

0270,0

046,5

99,0

963,5

450,0

0C

300

0,0

0G

100

0,0

0S

50

0,0

087,5

0315,0

09,0

946,5

962,5

47,2

7G

100

36,3

6S

50

25,5

7S

25

25,5

7I1

10

37,5

0270,0

00,0

00,0

0-

0,0

6C

300

24,7

3G

100

12,7

7S

50

0,0

050,0

0315,0

00,0

00,0

0-

0,2

8G

100

13,5

6S

50

13,8

8S

25

22,5

6I1

11

3,8

3270,0

033,6

70,0

05498,0

30,1

6C

300

0,5

7G

100

3,2

6S

50

0,0

083,6

7315,0

00,0

033,6

75497,0

30,1

1G

100

31,8

2S

50

24,1

5S

25

27,7

0I2

00

30,5

6247,5

00,0

00,0

0-

15,5

6C

300

22,0

3G

100

8,5

3S

25

0,0

097,5

0276,2

50,0

00,0

0-

20,0

4C

500

53,5

3G

200

36,2

1S

50

7,7

6I2

01

52,4

0247,5

030,0

351,8

8304,1

00,0

0C

300

32,7

6G

100

19,6

4S

25

0,0

075,6

5276,2

551,8

830,0

3303,1

040,5

7C

500

33,9

8G

200

26,3

2S

50

15,3

5I2

10

30,5

6247,5

00,0

00,0

0-

0,0

8C

300

16,5

2G

100

14,0

4S

25

0,0

097,5

0276,2

50,0

00,0

0-

0,0

9C

500

31,2

4G

200

38,8

7S

50

27,3

8I2

11

52,4

0247,5

00,0

021,8

5161540,0

60,0

0C

300

32,7

6G

100

19,6

4S

25

0,0

075,6

5276,2

521,8

50,0

0161541,0

60,1

2C

500

17,1

6G

200

32,9

6S

50

25,5

3I3

00

84,3

7382,5

00,0

00,0

0-

116,8

8C

500

35,0

9G

200

35,0

9S

25

14,2

030,5

6247,5

00,0

00,0

0-

32,7

3S

25

10,1

9S

50

10,1

9S

50

10,1

9I3

01

16,6

5382,5

079,6

211,9

0178,8

5153,3

4C

500

2,8

2G

200

7,0

5S

25

6,7

998,2

8247,5

011,9

079,6

2179,8

50,0

0S

25

32,7

6S

50

32,7

6S

50

32,7

6I3

10

134,7

1342,2

30,0

00,0

0-

0,2

4C

500

0,0

0G

200

56,4

2S

25

78,2

954,2

2226,2

00,0

00,0

0-

0,0

0S

25

18,0

7S

50

18,0

7S

50

18,0

7I3

11

178,2

8374,0

414,9

698,3

064,8

70,3

1C

500

0,0

0G

200

71,1

0S

25

107,1

80,0

0200,0

098,3

014,9

665,8

70,0

0S

25

0,0

0S

50

0,0

0S

50

0,0

0I4

00

108,4

3277,4

10,0

00,0

0-

25,0

0S

25

36,1

4S

50

36,1

4S

50

36,1

493,7

5318,7

50,0

00,0

0-

0,2

9G

200

11,6

8S

50

36,4

9S

25

45,5

9I4

01

151,6

4298,2

90,0

066,4

127,1

125,0

0S

25

50,5

5S

50

50,5

5S

50

50,5

527,3

4318,7

566,4

10,0

028,1

10,8

8G

200

0,0

0S

50

0,0

0S

25

27,3

4I4

10

93,5

5290,8

00,0

00,0

0-

1,0

9S

25

43,7

6S

50

24,9

0S

50

24,9

093,7

5318,7

50,0

00,0

0-

0,2

9G

200

11,6

8S

50

36,4

9S

25

45,5

9I4

11

101,4

2298,4

40,0

016,3

6159,0

91,1

9S

25

47,4

4S

50

26,9

9S

50

26,9

977,3

9318,7

516,3

60,0

0160,0

90,3

4G

200

0,1

7S

50

33,2

5S

25

43,9

7I5

00

112,5

0318,7

50,0

00,0

0-

110,9

6C

300

52,5

4G

200

52,5

4S

25

7,4

280,0

0240,0

00,0

00,0

0-

300,0

0C

500

0,0

0G

200

49,2

3G

200

30,7

7I5

01

192,5

0318,7

560,9

3140,9

3188,8

691,3

9C

300

70,3

3G

200

70,3

3S

25

51,8

50,0

0240,0

0140,9

360,9

3189,8

690,0

0C

500

0,0

0G

200

0,0

0G

200

0,0

0I5

10

136,6

4306,6

80,0

00,0

0-

0,5

2C

300

0,0

0G

200

47,4

8S

25

89,1

680,0

0240,0

00,0

00,0

0-

0,1

3C

500

20,3

4G

200

48,1

4G

200

11,5

3I5

11

150,8

4318,7

50,0

038,3

4116,2

20,5

6C

300

0,0

0G

200

51,1

4S

25

99,7

041,6

6240,0

038,3

40,0

0117,2

20,1

8C

500

0,0

0G

200

39,3

1G

200

2,3

6I6

00

56,2

5255,0

00,0

00,0

0-

0,0

0C

500

25,3

0G

100

26,6

0S

50

4,3

581,2

5276,2

50,0

00,0

0-

9,0

1G

200

42,9

4S

50

19,1

5S

50

19,1

5I6

01

28,0

3255,0

028,2

20,0

0398,9

70,0

6C

500

3,8

9G

100

22,3

2S

50

1,8

2109,4

7276,2

50,0

028,2

2397,9

70,0

0G

200

51,1

4S

50

29,1

7S

50

29,1

7I6

10

56,2

5255,0

00,0

00,0

0-

0,0

0C

500

25,3

0G

100

26,6

0S

50

4,3

581,2

5276,2

50,0

00,0

0-

0,1

0G

200

33,6

2S

50

23,8

1S

50

23,8

1I6

11

28,0

3255,0

028,2

20,0

0190054,1

40,0

6C

500

3,8

9G

100

22,3

2S

50

1,8

2109,4

7276,2

50,0

028,2

2190053,1

40,0

0G

200

51,1

4S

50

29,1

7S

50

29,1

7I7

00

58,3

3297,5

00,0

00,0

0-

103,7

5C

300

29,1

7G

100

29,1

7S

25

0,0

056,5

2399,1

30,0

00,0

0-

199,1

3C

500

0,0

0G

100

0,0

0G

200

56,5

2I7

01

0,0

0297,5

058,3

30,0

099,0

0147,5

0C

300

0,0

0G

100

0,0

0S

25

0,0

0108,3

3405,0

00,0

058,3

3100,0

0205,0

0C

500

0,0

0G

100

47,6

2G

200

60,7

1I7

10

91,9

4267,2

50,0

00,0

0-

0,3

9C

300

0,1

3G

100

52,1

6S

25

39,6

5122,6

1339,6

50,0

00,0

0-

0,2

8C

500

0,0

0G

100

77,0

5G

200

45,5

6I7

11

124,5

6297,5

00,0

066,2

2114,1

80,4

9C

300

0,1

7G

100

65,6

1S

25

58,7

891,5

7307,9

866,2

20,0

0115,1

80,2

2C

500

0,0

0G

100

59,5

8G

200

31,9

9I8

00

150,0

0360,0

00,0

00,0

0-

2,4

6S

50

32,1

7S

25

85,6

5S

50

32,1

760,9

4276,2

50,0

00,0

0-

58,9

1C

500

48,1

0G

200

12,8

4S

50

0,0

0I8

01

210,9

4360,0

00,0

060,9

445,0

01,9

0S

50

56,5

5S

25

97,8

4S

50

56,5

50,0

0276,2

560,9

40,0

046,0

0126,2

5C

500

0,0

0G

200

0,0

0S

50

0,0

0I8

10

150,0

0360,0

00,0

00,0

0-

2,4

6S

50

32,1

7S

25

85,6

5S

50

32,1

760,9

4276,2

50,0

00,0

0-

0,2

4C

500

5,4

6G

200

21,3

4S

50

34,1

5I8

11

157,5

0360,0

00,0

07,5

01045,0

02,3

9S

50

35,1

8S

25

87,1

5S

50

35,1

853,4

3276,2

57,5

00,0

01046,0

00,2

5C

500

0,5

3G

200

19,2

9S

50

33,6

1I9

00

150,0

0360,0

00,0

00,0

0-

2,3

0C

300

0,0

0S

25

75,0

0S

25

75,0

093,3

3280,0

00,0

00,0

0-

0,4

7G

100

25,4

7S

25

38,6

2S

50

29,2

4I9

01

219,3

3360,0

00,0

069,3

325,0

00,7

7C

300

0,0

0S

25

109,6

7S

25

109,6

724,0

0280,0

069,3

30,0

026,0

01,2

0G

100

0,0

0S

25

24,0

0S

50

0,0

0I9

10

190,1

5335,9

10,0

00,0

0-

0,4

5C

300

0,0

0S

25

95,0

8S

25

95,0

8100,4

5274,6

60,0

00,0

0-

0,3

2G

100

32,4

6S

25

37,2

4S

50

30,7

5I9

11

230,4

7360,0

09,1

589,6

23239,8

10,5

3C

300

0,1

4S

25

115,1

6S

25

115,1

660,6

6244,1

589,6

29,1

53238,8

10,1

9G

100

19,2

0S

25

22,6

5S

50

18,8

1I10

00

69,5

7283,2

60,0

00,0

0-

83,2

6G

100

0,0

0S

25

34,7

8S

50

34,7

837,5

0270,0

00,0

00,0

0-

11,4

1C

300

29,6

9G

100

7,8

1S

50

0,0

0I10

01

91,6

7292,5

00,0

037,5

025,0

092,5

0G

100

0,0

0S

25

48,4

8S

50

43,1

8-0

,00

270,0

037,5

00,0

026,0

050,0

0C

300

-0,0

0G

100

0,0

0S

50

0,0

0I10

10

67,0

9284,7

50,0

00,0

0-

0,8

0G

100

4,0

1S

25

40,6

5S

50

22,4

337,5

0270,0

00,0

00,0

0-

0,0

6C

300

24,7

3G

100

12,7

7S

50

0,0

0I10

11

87,8

4292,5

00,0

033,6

711284,5

50,8

0G

100

10,7

0S

25

47,6

8S

50

29,4

63,8

3270,0

033,6

70,0

011285,5

50,1

6C

300

0,5

7G

100

3,2

6S

50

0,0

0I11

00

125,7

6286,9

70,0

00,0

0-

25,0

0S

50

34,9

7S

50

34,9

7S

25

55,8

175,0

0337,5

00,0

00,0

0-

0,5

8C

500

0,0

0G

200

1,7

9S

50

73,2

1I11

01

173,4

8306,0

60,0

075,0

026,5

225,0

0S

50

50,8

8S

50

50,8

8S

25

71,7

2-0

,00

337,5

075,0

00,0

027,5

21,7

5C

500

0,0

0G

200

0,0

0S

50

-0,0

0I11

10

138,7

4277,8

80,0

00,0

0-

0,0

6S

50

45,8

5S

50

45,8

5S

25

47,0

4123,4

2313,2

90,0

00,0

0-

0,1

9C

500

0,0

0G

200

55,9

7S

50

67,4

5I11

11

82,2

3254,3

190,1

90,0

0284,8

40,0

3S

50

27,1

7S

50

27,1

7S

25

27,8

8165,1

9337,5

00,0

090,1

9283,8

40,2

4C

500

0,0

0G

200

70,5

0S

50

94,6

9I12

00

97,5

0276,2

50,0

00,0

0-

0,0

6C

500

28,1

2G

100

50,6

3G

200

18,7

5150,0

0300,0

00,0

00,0

0-

8,5

7C

300

51,7

9G

100

62,5

0S

25

35,7

1I12

01

86,7

9276,2

510,7

10,0

0364,7

10,0

7C

500

21,8

2G

100

49,3

6G

200

15,6

0160,7

1300,0

00,0

010,7

1363,7

10,0

0C

300

62,5

0G

100

62,5

0S

25

35,7

1I12

10

97,5

0276,2

50,0

00,0

0-

0,0

6C

500

28,1

2G

100

50,6

3G

200

18,7

5150,0

0300,0

00,0

00,0

0-

0,0

2C

300

55,0

0G

100

60,0

0S

25

35,0

0I12

11

86,7

9276,2

510,7

10,0

088598,0

00,0

7C

500

21,8

2G

100

49,3

6G

200

15,6

0160,7

1300,0

00,0

010,7

188599,0

00,0

0C

300

62,5

0G

100

62,5

0S

25

35,7

1I13

00

135,0

0382,5

00,0

00,0

0-

0,7

2C

300

16,2

2G

200

81,2

7G

200

37,5

238,8

9315,0

00,0

00,0

0-

135,8

3C

500

19,4

4G

200

19,4

4G

100

0,0

0I13

01

173,8

9382,5

031,0

869,9

7231,7

40,6

3C

300

24,6

6G

200

95,0

3G

200

54,2

00,0

0315,0

069,9

731,0

8232,7

4165,0

0C

500

0,0

0G

200

0,0

0G

100

0,0

0I13

10

135,0

0382,5

00,0

00,0

0-

0,7

2C

300

16,2

2G

200

81,2

7G

200

37,5

283,9

5274,4

40,0

00,0

0-

0,2

5C

500

0,0

0G

200

49,7

8G

100

34,1

8I13

11

96,6

5382,5

071,2

732,9

2169,0

70,7

8C

300

0,0

0G

200

70,4

5G

200

26,1

9104,4

0290,5

632,9

271,2

7168,0

70,2

8C

500

0,0

0G

200

56,2

2G

100

48,1

8I14

00

90,0

0382,5

00,0

00,0

0-

2,8

7S

50

30,0

0S

50

30,0

0S

50

30,0

060,0

0255,0

00,0

00,0

0-

6,5

7C

500

37,2

5G

100

22,7

5S

50

0,0

0I14

01

150,0

0382,5

00,0

060,0

050,0

02,3

5S

50

50,0

0S

50

50,0

0S

50

50,0

00,0

0255,0

060,0

00,0

051,0

055,0

0C

500

0,0

0G

100

0,0

0S

50

0,0

0I14

10

207,6

1294,2

90,0

00,0

0-

0,0

9S

50

69,2

0S

50

69,2

0S

50

69,2

060,0

0255,0

00,0

00,0

0-

0,0

3C

500

30,9

9G

100

25,6

5S

50

3,3

6I14

11

233,0

2305,8

30,0

040,8

0290,6

60,1

0S

50

77,6

7S

50

77,6

7S

50

77,6

719,2

0255,0

040,8

00,0

0291,6

60,1

1C

500

0,0

0G

100

19,2

0S

50

0,0

0I15

00

60,0

0405,0

00,0

00,0

0-

118,1

6C

300

35,0

8S

50

12,4

6S

50

12,4

640,0

0268,0

00,0

00,0

0-

50,0

0S

50

13,3

3S

50

13,3

3S

50

13,3

3I15

01

0,0

0405,0

060,0

00,0

051,0

0155,0

0C

300

0,0

0S

50

0,0

0S

50

0,0

097,5

0270,0

00,0

060,0

050,0

026,1

3S

50

32,5

0S

50

32,5

0S

50

32,5

0

35

I15

10

128,7

9353,4

10,0

00,0

0-

0,3

4C

300

0,0

0S

50

64,3

9S

50

64,3

977,4

2238,0

60,0

00,0

0-

0,0

6S

50

25,8

1S

50

25,8

1S

50

25,8

1I15

11

201,6

4392,5

221,4

7146,4

760,8

80,4

8C

300

0,0

0S

50

100,8

2S

50

100,8

20,0

0200,0

0146,4

721,4

761,8

80,0

0S

50

0,0

0S

50

0,0

0S

50

0,0

0I16

00

60,0

0270,0

00,0

00,0

0-

0,1

0S

50

19,0

0S

50

19,0

0S

25

22,0

0128,5

7360,0

00,0

00,0

0-

110,0

0C

500

0,0

0G

100

34,1

3S

50

94,4

4I16

01

75,0

0270,0

00,0

015,0

099,0

00,0

0S

50

25,0

0S

50

25,0

0S

25

25,0

0113,5

7360,0

015,0

00,0

098,0

0110,0

0C

500

0,0

0G

100

19,1

3S

50

94,4

4I16

10

60,0

0270,0

00,0

00,0

0-

0,1

0S

50

19,0

0S

50

19,0

0S

25

22,0

0147,2

4346,9

30,0

00,0

0-

0,1

9C

500

0,4

1G

100

77,6

3S

50

69,2

0I16

11

17,7

5270,0

042,2

50,0

0543,0

00,3

7S

50

2,1

0S

50

2,1

0S

25

13,5

5170,8

3360,0

00,0

042,2

5544,0

00,2

2C

500

0,4

6G

100

88,0

9S

50

82,2

7I17

00

112,5

0318,7

50,0

00,0

0-

9,6

3C

500

48,7

5S

25

31,8

8S

25

31,8

843,7

5315,0

00,0

00,0

0-

0,3

8C

300

0,5

3G

200

28,7

5G

200

14,4

7I17

01

113,7

5318,7

531,6

832,9

3211,0

812,3

2C

500

44,9

0S

25

34,4

3S

25

34,4

342,5

0315,0

032,9

331,6

8212,0

80,3

8C

300

0,0

0G

200

28,4

0G

200

14,1

0I17

10

112,5

0318,7

50,0

00,0

0-

0,0

6C

500

21,4

6S

25

45,5

2S

25

45,5

243,7

5315,0

00,0

00,0

0-

0,3

8C

300

0,5

3G

200

28,7

5G

200

14,4

7I17

11

88,8

7318,7

523,6

30,0

038393,5

70,0

9C

500

0,2

9S

25

44,2

9S

25

44,2

967,3

8315,0

00,0

023,6

338394,5

70,3

4C

300

10,5

5G

200

35,4

3G

200

21,4

1I18

00

50,0

0337,5

00,0

00,0

0-

1,0

3G

100

0,0

0S

50

11,4

0S

25

38,6

0150,0

0360,0

00,0

00,0

0-

85,0

0C

300

0,0

0G

200

106,2

5G

200

43,7

5I18

01

200,0

0337,5

00,0

0150,0

073,5

60,0

9G

100

75,1

8S

50

61,2

8S

25

63,5

40,0

0360,0

0150,0

00,0

072,5

685,0

0C

300

0,0

0G

200

0,0

0G

200

0,0

0I18

10

84,1

0311,9

30,0

00,0

0-

0,3

0G

100

30,2

1S

50

22,9

7S

25

30,9

2150,0

0360,0

00,0

00,0

0-

0,1

8C

300

39,3

4G

200

61,6

4G

200

49,0

2I18

11

140,5

6337,5

00,0

090,5

66225,3

20,4

3G

100

42,6

8S

50

43,3

2S

25

54,5

679,9

1347,7

190,5

60,0

06226,3

20,2

1C

300

13,7

2G

200

39,4

4G

200

26,7

5I19

00

64,2

9405,0

00,0

00,0

0-

2,2

1S

50

21,4

3S

50

21,4

3S

50

21,4

360,0

0270,0

00,0

00,0

0-

0,2

6C

500

0,0

0G

200

16,5

7S

25

43,4

3I19

01

83,0

4405,0

00,0

018,7

550,0

02,1

0S

50

27,6

8S

50

27,6

8S

50

27,6

841,2

5270,0

018,7

50,0

051,0

00,3

5C

500

0,0

0G

200

0,0

0S

25

41,2

5I19

10

64,2

9405,0

00,0

00,0

0-

2,2

1S

50

21,4

3S

50

21,4

3S

50

21,4

378,7

4260,6

30,0

00,0

0-

0,1

1C

500

4,4

7G

200

36,4

3S

25

37,8

3I19

11

150,0

0405,0

016,6

1102,3

25570,4

81,7

0S

50

50,0

0S

50

50,0

0S

50

50,0

045,9

1234,1

9102,3

216,6

15569,4

80,0

6C

500

2,5

2G

200

20,5

4S

25

22,8

4I20

00

48,6

5331,2

20,0

00,0

0-

25,0

0S

25

16,2

2S

25

16,2

2S

50

16,2

2107,1

4300,0

00,0

00,0

0-

44,7

6C

500

29,3

7G

100

50,0

0S

50

27,7

8I20

01

142,5

0337,5

00,0

0100,8

349,0

00,0

0S

25

47,5

0S

25

47,5

0S

50

47,5

06,3

1300,0

0100,8

30,0

048,0

080,0

0C

500

0,0

0G

100

0,0

0S

50

6,3

1I20

10

68,1

4313,6

70,0

00,0

0-

0,0

3S

25

22,9

5S

25

22,9

5S

50

22,2

4107,1

4300,0

00,0

00,0

0-

0,0

7C

500

39,5

1G

100

43,4

8S

50

24,1

6I20

11

142,5

0337,5

00,0

0100,8

3499,2

40,0

0S

25

47,5

0S

25

47,5

0S

50

47,5

038,5

1277,4

6100,8

30,0

0500,2

40,1

1C

500

0,3

8G

100

24,5

1S

50

13,6

2I21

00

43,7

5315,0

00,0

00,0

0-

0,9

8S

50

14,5

8S

50

14,5

8S

50

14,5

878,5

7220,0

00,0

00,0

0-

3,0

7C

500

51,4

8S

50

13,5

4S

50

13,5

4I21

01

122,3

2315,0

00,0

078,5

750,0

00,4

0S

50

40,7

7S

50

40,7

7S

50

40,7

70,0

0220,0

078,5

70,0

051,0

070,0

0C

500

0,0

0S

50

0,0

0S

50

0,0

0I21

10

84,2

1282,6

30,0

00,0

0-

0,0

4S

50

28,0

7S

50

28,0

7S

50

28,0

778,5

7220,0

00,0

00,0

0-

0,0

2C

500

47,8

2S

50

15,3

7S

50

15,3

7I21

11

177,2

7315,0

00,0

0133,5

2583,6

30,0

0S

50

59,0

9S

50

59,0

9S

50

59,0

97,8

6176,0

3133,5

20,0

0584,6

30,0

5C

500

0,1

0S

50

3,8

8S

50

3,8

8I22

00

36,6

7247,5

00,0

00,0

0-

13,9

4C

300

25,8

2G

200

10,8

5S

50

0,0

062,5

0337,5

00,0

00,0

0-

2,0

8C

500

0,0

0S

50

12,5

0S

25

50,0

0I22

01

0,0

0247,5

036,6

70,0

051,0

047,5

0C

300

0,0

0G

200

0,0

0S

50

0,0

099,1

7337,5

00,0

036,6

750,0

01,2

7C

500

0,0

0S

50

49,1

7S

25

50,0

0I22

10

36,6

7247,5

00,0

00,0

0-

0,0

6C

300

23,6

2G

200

13,0

4S

50

0,0

0100,4

6314,7

30,0

00,0

0-

0,2

3C

500

0,4

6S

50

50,0

0S

25

50,0

0I22

11

0,3

2246,2

238,0

50,0

0777,9

10,1

5C

300

0,3

2G

200

0,0

0S

50

0,0

0100,5

5337,5

00,0

038,0

5776,9

10,2

7C

500

0,5

5S

50

50,0

0S

25

50,0

0I23

00

100,0

0300,0

00,0

00,0

0-

38,3

3S

25

33,3

3S

25

33,3

3S

25

33,3

370,0

0315,0

00,0

00,0

0-

19,0

0G

100

10,0

0S

50

30,0

0S

50

30,0

0I23

01

170,0

0300,0

00,0

070,0

025,0

09,1

7S

25

56,6

7S

25

56,6

7S

25

56,6

70,0

0315,0

070,0

00,0

024,0

025,0

0G

100

0,0

0S

50

0,0

0S

50

0,0

0I23

10

100,0

0300,0

00,0

00,0

0-

1,5

3S

25

33,3

3S

25

33,3

3S

25

33,3

370,0

0315,0

00,0

00,0

0-

0,0

9G

100

27,2

7S

50

21,3

6S

50

21,3

6I23

11

158,9

1300,0

00,0

058,9

1441,6

70,5

5S

25

52,9

7S

25

52,9

7S

25

52,9

711,0

9315,0

058,9

10,0

0442,6

70,2

5G

100

0,4

9S

50

5,3

0S

50

5,3

0I24

00

120,0

0360,0

00,0

00,0

0-

0,7

3C

500

0,0

0G

100

66,6

7G

100

53,3

370,0

0297,5

00,0

00,0

0-

38,8

8C

500

30,9

0G

100

30,9

0G

100

8,2

1I24

01

184,8

6360,0

02,8

567,7

1227,3

10,3

2C

500

0,0

0G

100

98,4

6G

100

86,4

05,1

4297,5

067,7

12,8

5226,3

174,9

7C

500

1,6

3G

100

2,0

3G

100

1,4

8I24

10

153,4

1334,9

40,0

00,0

0-

0,2

7C

500

0,0

0G

100

83,0

4G

100

70,3

674,9

1293,8

20,0

00,0

0-

0,1

5C

500

0,0

0G

100

47,2

4G

100

27,6

7I24

11

148,5

2331,0

510,0

80,0

085,1

20,2

6C

500

0,0

0G

100

80,6

5G

100

67,8

780,0

8297,5

00,0

010,0

886,1

20,1

6C

500

0,0

0G

100

49,6

0G

100

30,4

8I25

00

70,0

0315,0

00,0

00,0

0-

95,0

0C

500

0,0

0G

200

0,0

0S

50

70,0

042,8

6270,0

00,0

00,0

0-

39,1

1G

200

24,7

1S

25

9,0

7S

25

9,0

7I25

01

81,2

5315,0

021,4

532,7

090,9

895,0

0C

500

0,0

0G

200

0,0

0S

50

81,2

531,6

1270,0

032,7

021,4

589,9

853,4

1G

200

11,8

5S

25

9,8

8S

25

9,8

8I25

10

70,0

0315,0

00,0

00,0

0-

0,4

0C

500

0,0

0G

200

14,0

5S

50

55,9

555,6

7261,0

30,0

00,0

0-

0,3

1G

200

0,0

0S

25

27,8

4S

25

27,8

4I25

11

49,9

7312,0

725,8

90,0

057,8

30,4

6C

500

0,0

0G

200

0,0

0S

50

49,9

768,7

5270,0

00,0

025,8

958,8

30,3

5G

200

0,0

0S

25

34,3

8S

25

34,3

8I26

00

135,0

0382,5

00,0

00,0

0-

0,7

4C

500

0,0

0G

200

15,3

6S

50

119,6

490,0

0382,5

00,0

00,0

0-

70,7

1C

500

38,7

3S

25

25,6

4S

50

25,6

4I26

01

114,8

4382,5

073,1

152,9

6124,2

30,8

1C

500

0,0

0G

200

0,0

0S

50

114,8

4110,1

6382,5

052,9

673,1

1123,2

363,2

6C

500

20,5

8S

25

44,7

9S

50

44,7

9I26

10

192,8

9353,5

60,0

00,0

0-

0,2

7C

500

0,0

0G

200

80,1

3S

50

112,7

5131,2

8351,5

40,0

00,0

0-

0,1

5C

500

0,0

0S

25

67,8

9S

50

63,3

9I26

11

150,2

4320,5

4175,3

866,7

1102,7

60,2

0C

500

0,0

0G

200

60,3

3S

50

89,9

1198,6

8382,5

066,7

1175,3

8103,7

60,2

2C

500

0,0

0S

25

102,5

0S

50

96,1

8I27

00

115,2

2305,8

70,0

00,0

0-

50,0

0S

50

32,6

1S

50

32,6

1S

25

50,0

075,0

0337,5

00,0

00,0

0-

82,8

8C

300

34,6

2G

200

34,6

2G

200

5,7

7I27

01

142,8

6334,2

98,3

383,3

322,4

82,3

8S

50

47,6

2S

50

47,6

2S

25

47,6

20,0

0337,5

083,3

38,3

323,4

8117,5

0C

300

0,0

0G

200

0,0

0G

200

0,0

0I27

10

62,5

0337,5

00,0

00,0

0-

2,2

1S

50

6,2

5S

50

6,2

5S

25

50,0

093,1

2328,4

40,0

00,0

0-

0,3

0C

300

19,7

2G

200

49,2

9G

200

24,1

1I27

11

150,0

0337,5

03,5

391,0

35779,9

81,2

5S

50

50,0

0S

50

50,0

0S

25

50,0

067,1

0297,7

091,0

33,5

35778,9

80,2

1C

300

14,1

3G

200

35,3

2G

200

17,6

6I28

00

80,0

0240,0

00,0

00,0

0-

14,4

7C

300

55,9

4G

200

19,6

3S

25

4,4

2150,0

0360,0

00,0

00,0

0-

0,4

7G

200

0,0

0S

25

83,3

3S

50

66,6

7I28

01

0,0

0240,0

080,0

00,0

0153,5

090,0

0C

300

0,0

0G

200

0,0

0S

25

0,0

0230,0

0360,0

00,0

080,0

0152,5

00,2

1G

200

52,8

8S

25

92,3

8S

50

84,7

5I28

10

80,0

0240,0

00,0

00,0

0-

0,0

8C

300

47,9

1G

200

17,7

8S

25

14,3

1152,0

8358,7

50,0

00,0

0-

0,4

2G

200

0,3

5S

25

83,3

1S

50

68,4

1I28

11

26,7

7240,0

053,2

30,0

028153,5

00,2

3C

300

15,3

8G

200

0,0

0S

25

11,3

8203,2

3360,0

00,0

053,2

328152,5

00,2

9G

200

34,1

4S

25

89,7

0S

50

79,4

0I29

00

75,0

0382,5

00,0

00,0

0-

4,4

0S

25

37,5

0S

50

0,0

0S

25

37,5

040,6

2292,5

00,0

00,0

0-

0,1

6C

500

0,0

0G

200

31,1

2G

200

9,5

1I29

01

115,6

3382,5

00,0

040,6

225,0

03,2

6S

25

57,8

1S

50

0,0

0S

25

57,8

10,0

0292,5

040,6

20,0

026,0

00,3

6C

500

0,0

0G

200

0,0

0G

200

0,0

0I29

10

75,0

0382,5

00,0

00,0

0-

4,4

0S

25

37,5

0S

50

0,0

0S

25

37,5

040,6

2292,5

00,0

00,0

0-

0,1

6C

500

0,0

0G

200

31,1

2G

200

9,5

1I29

11

69,8

6382,5

033,8

828,7

5425,8

84,8

9S

25

28,8

3S

50

12,2

1S

25

28,8

345,7

6292,5

028,7

533,8

8426,8

80,1

4C

500

0,0

1G

200

33,4

7G

200

12,2

8I30

00

112,5

0360,0

00,0

00,0

0-

54,1

4C

300

50,7

8S

50

30,8

6S

50

30,8

636,1

1292,5

00,0

00,0

0-

0,0

0S

50

15,9

1S

50

4,2

9S

25

15,9

1I30

01

100,8

8360,0

070,4

758,8

5141,7

668,9

0C

300

37,3

6S

50

31,7

6S

50

31,7

647,7

3292,5

058,8

570,4

7142,7

60,0

0S

50

15,9

1S

50

15,9

1S

25

15,9

1I30

10

121,5

1352,7

90,0

00,0

0-

0,3

4C

300

0,1

4S

50

60,6

9S

50

60,6

936,1

1292,5

00,0

00,0

0-

0,1

0S

50

11,2

6S

50

11,2

6S

25

13,5

9I30

11

133,5

3360,0

00,0

021,0

3227,1

50,3

7C

300

0,1

5S

50

66,6

9S

50

66,6

920,0

9287,9

921,0

30,0

0226,1

50,1

4S

50

5,6

7S

50

5,6

7S

25

8,7

4I31

00

46,6

7315,0

00,0

00,0

0-

56,6

2C

300

30,7

0S

25

7,9

8S

25

7,9

875,0

0382,5

00,0

00,0

0-

162,5

0C

300

0,0

0G

100

0,0

0S

50

75,0

0I31

01

-0,0

0315,0

046,6

70,0

0101,0

095,0

0C

300

-0,0

0S

25

0,0

0S

25

0,0

0121,6

7382,5

00,0

046,6

7100,0

0162,5

0C

300

0,0

0G

100

23,9

4S

50

97,7

3I31

10

73,5

9294,8

10,0

00,0

0-

0,2

5C

300

0,1

0S

25

36,7

5S

25

36,7

5104,2

7356,1

50,0

00,0

0-

0,4

5C

300

0,3

0G

100

50,7

6S

50

53,2

1I31

11

108,8

8315,0

038,7

0100,9

2133,0

70,3

2C

300

0,1

3S

25

54,3

8S

25

54,3

884,0

9318,3

3100,9

238,7

0132,0

70,3

3C

300

0,2

2G

100

39,5

8S

50

44,2

8I32

00

86,6

7260,0

00,0

00,0

0-

0,0

8C

300

27,5

1S

50

28,7

7S

25

30,3

865,0

0276,2

50,0

00,0

0-

3,5

0S

25

21,6

7S

50

21,6

7S

50

21,6

7I32

01

76,6

7260,0

010,0

00,0

0246,6

90,1

2C

300

19,5

8S

50

27,3

9S

25

29,7

075,0

0276,2

50,0

010,0

0245,6

90,0

0S

25

25,0

0S

50

25,0

0S

50

25,0

0I32

10

86,6

7260,0

00,0

00,0

0-

0,0

8C

300

27,5

1S

50

28,7

7S

25

30,3

868,1

8273,8

60,0

00,0

0-

-0,0

0S

25

22,7

3S

50

22,7

3S

50

22,7

3I32

11

76,6

7260,0

010,0

00,0

071673,2

60,1

2C

300

19,5

8S

50

27,3

9S

25

29,7

075,0

0276,2

50,0

010,0

071674,2

60,0

0S

25

25,0

0S

50

25,0

0S

50

25,0

0I33

00

75,0

0318,7

50,0

00,0

0-

0,0

8C

300

21,0

8G

200

29,4

0G

100

24,5

254,1

7292,5

00,0

00,0

0-

0,0

9C

300

17,0

1G

100

37,1

5G

200

0,0

0I33

01

97,6

9318,7

50,0

022,6

9230,7

70,0

4C

300

32,2

0G

200

36,8

2G

100

28,6

631,4

8292,5

022,6

90,0

0231,7

70,1

4C

300

0,0

0G

100

31,4

8G

200

0,0

0I33

10

75,0

0318,7

50,0

00,0

0-

0,0

8C

300

21,0

8G

200

29,4

0G

100

24,5

254,1

7292,5

00,0

00,0

0-

0,0

9C

300

17,0

1G

100

37,1

5G

200

0,0

0I33

11

97,0

6318,7

50,0

022,0

644365,3

80,0

4C

300

31,9

0G

200

36,6

2G

100

28,5

532,1

0292,5

022,0

60,0

044366,3

80,1

4C

300

0,4

7G

100

31,6

4G

200

0,0

0I34

00

69,6

4276,2

50,0

00,0

0-

0,0

8C

300

26,4

6G

200

33,2

6G

200

9,9

297,5

0276,2

50,0

00,0

0-

0,2

5S

25

32,5

0S

50

32,5

0S

25

32,5

0I34

01

68,7

1276,2

526,5

525,6

1243,8

80,0

9C

300

23,8

4G

200

31,5

2G

200

13,3

598,4

4276,2

525,6

126,5

5244,8

80,0

0S

25

32,8

1S

50

32,8

1S

25

32,8

1I34

10

69,6

4276,2

50,0

00,0

0-

0,0

8C

300

26,4

6G

200

33,2

6G

200

9,9

297,5

0276,2

50,0

00,0

0-

0,0

1S

25

32,5

8S

50

32,3

4S

25

32,5

8

36

I34

11

68,7

1276,2

526,5

225,5

867586,1

00,0

9C

300

23,8

5G

200

31,5

2G

200

13,3

498,4

4276,2

525,5

826,5

267587,1

00,0

0S

25

32,8

1S

50

32,8

1S

25

32,8

1I35

00

107,1

4300,0

00,0

00,0

0-

0,0

3C

500

37,4

5G

100

47,4

9G

200

22,2

0100,0

0360,0

00,0

00,0

0-

0,6

6G

100

3,5

7S

25

48,2

1S

25

48,2

1I35

01

37,5

0300,0

069,6

40,0

0101,0

00,1

3C

500

0,0

0G

100

37,5

0G

200

0,0

0169,6

4360,0

00,0

069,6

4100,0

00,0

0G

100

69,6

4S

25

50,0

0S

25

50,0

0I35

10

107,1

4300,0

00,0

00,0

0-

0,0

3C

500

37,4

5G

100

47,4

9G

200

22,2

0119,5

3342,4

30,0

00,0

0-

0,1

7G

100

34,9

5S

25

42,2

9S

25

42,2

9I35

11

45,5

7300,0

061,5

70,0

010101,0

00,1

0C

500

0,0

1G

100

40,0

0G

200

5,5

6161,5

7360,0

00,0

061,5

710100,0

00,0

8G

100

61,5

7S

25

50,0

0S

25

50,0

0I36

00

75,0

0337,5

00,0

00,0

0-

59,3

2C

500

35,2

3G

100

35,2

3G

100

4,5

570,0

0315,0

00,0

00,0

0-

0,0

0S

25

41,6

7S

50

0,0

0S

50

28,3

3I36

01

20,0

0337,5

069,9

814,9

8233,4

381,6

4C

500

6,7

0G

100

7,3

2G

100

5,9

8125,0

0315,0

014,9

869,9

8234,4

30,0

0S

25

41,6

7S

50

41,6

7S

50

41,6

7I36

10

112,0

7318,9

70,0

00,0

0-

0,1

4C

500

0,0

0G

100

68,9

7G

100

43,1

085,7

1307,1

40,0

00,0

0-

0,0

0S

25

28,5

7S

50

28,5

7S

50

28,5

7I36

11

151,7

9337,5

00,0

076,7

9277,3

30,1

8C

500

0,0

0G

100

87,5

0G

100

64,2

99,4

3306,8

976,7

90,0

0278,3

30,4

5S

25

9,4

3S

50

0,0

0S

50

0,0

0I37

00

68,7

5220,0

00,0

00,0

0-

64,4

9C

300

3,9

4G

100

50,0

0G

200

14,8

1135,0

0382,5

00,0

00,0

0-

98,5

0G

100

50,0

0S

25

42,5

0S

25

42,5

0I37

01

0,0

0220,0

0131,6

062,8

535,1

370,0

0C

300

0,0

0G

100

0,0

0G

200

0,0

0203,7

5382,5

062,8

5131,6

034,1

375,6

6G

100

26,7

2S

25

88,5

1S

25

88,5

1I37

10

68,7

5220,0

00,0

00,0

0-

0,1

1C

300

26,5

6G

100

42,1

9G

200

0,0

0183,8

7358,0

60,0

00,0

0-

1,3

8G

100

0,0

0S

25

91,9

4S

25

91,9

4I37

11

16,8

0185,2

895,3

50,0

085,5

20,1

2C

300

0,0

0G

100

16,8

0G

200

0,0

0230,3

5382,5

00,0

095,3

584,5

21,6

2G

100

0,4

1S

25

114,9

7S

25

114,9

7I38

00

50,0

0337,5

00,0

00,0

0-

128,9

2C

300

43,4

0G

100

6,6

0S

50

0,0

050,0

0405,0

00,0

00,0

0-

99,2

8C

300

27,9

3G

200

15,7

2S

50

6,3

5I38

01

100,0

0337,5

034,8

484,8

4164,2

9130,6

4C

300

39,4

2G

100

32,0

7S

50

28,5

00,0

0405,0

084,8

434,8

4165,2

9130,0

0C

300

0,0

0G

200

0,0

0S

50

0,0

0I38

10

90,6

9306,9

80,0

00,0

0-

0,5

2C

300

0,0

0G

100

42,0

4S

50

48,6

571,7

9385,3

80,0

00,0

0-

0,3

7C

300

0,0

0G

200

21,7

9S

50

50,0

0I38

11

126,6

9337,5

00,0

076,6

988,8

40,6

3C

300

0,0

0G

100

57,6

9S

50

69,0

042,5

0342,7

276,6

90,0

089,8

40,2

3C

300

0,0

0G

200

7,5

7S

50

34,9

3I39

00

50,0

0337,5

00,0

00,0

0-

128,9

2C

300

43,4

0G

100

6,6

0S

50

0,0

069,6

4276,2

50,0

00,0

0-

47,2

3S

50

23,2

1S

50

23,2

1S

50

23,2

1I39

01

0,0

0337,5

050,0

00,0

051,0

0187,5

0C

300

0,0

0G

100

0,0

0S

50

0,0

0119,6

4276,2

50,0

050,0

050,0

026,4

0S

50

39,8

8S

50

39,8

8S

50

39,8

8I39

10

90,6

9306,9

80,0

00,0

0-

0,5

2C

300

0,0

0G

100

42,0

4S

50

48,6

569,6

4276,2

50,0

00,0

0-

0,9

4S

50

23,2

1S

50

23,2

1S

50

23,2

1I39

11

120,5

6328,4

314,5

172,9

887,5

90,5

9C

300

0,0

0G

100

54,9

0S

50

65,6

611,1

7276,2

572,9

814,5

188,5

91,4

3S

50

3,7

2S

50

3,7

2S

50

3,7

2I40

00

72,2

2260,0

00,0

00,0

0-

38,1

3G

200

15,0

8S

25

28,5

7S

50

28,5

760,0

0255,0

00,0

00,0

0-

0,8

7S

25

25,5

8S

25

25,5

8S

50

8,8

5I40

01

28,5

7260,0

043,6

50,0

036,5

060,0

0G

200

0,0

0S

25

28,5

7S

50

-0,0

0103,6

5255,0

00,0

043,6

537,5

00,3

0S

25

36,4

9S

25

36,4

9S

50

30,6

7I40

10

73,3

4259,0

00,0

00,0

0-

0,1

2G

200

24,0

8S

25

25,7

0S

50

23,5

560,0

0255,0

00,0

00,0

0-

0,8

7S

25

25,5

8S

25

25,5

8S

50

8,8

5I40

11

38,1

2230,4

668,3

71,4

53501,3

90,0

6G

200

12,4

3S

25

13,4

0S

50

12,2

9126,9

2255,0

01,4

568,3

73502,3

90,0

0S

25

42,3

1S

25

42,3

1S

50

42,3

1I41

00

112,5

0318,7

50,0

00,0

0-

0,5

5C

300

2,1

0G

100

102,9

7G

200

7,4

2120,0

0360,0

00,0

00,0

0-

85,0

0G

200

0,0

0S

50

82,3

5S

50

37,6

5I41

01

67,7

9318,7

544,7

10,0

098,0

00,9

4C

300

0,0

0G

100

67,7

9G

200

0,0

0164,7

1360,0

00,0

044,7

199,0

085,0

0G

200

0,0

0S

50

82,3

5S

50

82,3

5I41

10

112,5

0318,7

50,0

00,0

0-

0,5

5C

300

2,1

0G

100

102,9

7G

200

7,4

2120,4

3359,6

80,0

00,0

0-

0,4

0G

200

4,0

4S

50

58,2

0S

50

58,2

0I41

11

108,6

1318,7

53,8

90,0

012977,4

20,5

6C

300

0,1

9G

100

102,3

4G

200

6,0

8123,8

9360,0

00,0

03,8

912976,4

20,4

0G

200

5,8

9S

50

59,0

0S

50

59,0

0I42

00

40,0

0270,0

00,0

00,0

0-

23,8

1C

500

20,9

5G

100

19,0

5S

25

0,0

093,7

5318,7

50,0

00,0

0-

0,5

6S

50

19,5

3S

25

37,1

1S

25

37,1

1I42

01

0,0

0270,0

040,0

00,0

038,5

050,0

0C

500

0,0

0G

100

0,0

0S

25

0,0

0133,7

5318,7

50,0

040,0

037,5

00,2

4S

50

39,5

3S

25

47,1

1S

25

47,1

1I42

10

40,0

0270,0

00,0

00,0

0-

0,0

4C

500

24,6

2G

100

15,3

8S

25

0,0

093,7

5318,7

50,0

00,0

0-

0,5

6S

50

19,5

3S

25

37,1

1S

25

37,1

1I42

11

9,6

1270,0

030,3

90,0

01288,5

00,1

0C

500

0,0

7G

100

9,5

4S

25

0,0

0124,1

4318,7

50,0

030,3

91287,5

00,3

2S

50

34,7

3S

25

44,7

1S

25

44,7

1I43

00

55,0

0247,5

00,0

00,0

0-

0,0

0C

300

27,5

0G

200

27,5

0G

200

0,0

087,5

0297,5

00,0

00,0

0-

1,0

9C

300

51,5

7G

100

26,7

7S

25

9,1

6I43

01

50,8

8247,5

04,1

20,0

0258,0

00,0

1C

300

25,0

3G

200

25,8

5G

200

0,0

091,6

2297,5

00,0

04,1

2259,0

00,0

0C

300

52,7

8G

100

28,1

3S

25

10,7

1I43

10

55,0

0247,5

00,0

00,0

0-

0,0

0C

300

27,5

0G

200

27,5

0G

200

0,0

087,5

0297,5

00,0

00,0

0-

0,0

1C

300

50,0

0G

100

27,0

8S

25

10,4

2I43

11

50,8

8247,5

04,1

20,0

090258,0

00,0

1C

300

25,0

3G

200

25,8

5G

200

0,0

091,6

2297,5

00,0

04,1

290259,0

00,0

0C

300

52,7

8G

100

28,1

3S

25

10,7

1I44

00

125,0

0300,0

00,0

00,0

0-

64,7

2G

200

13,8

9S

25

55,5

6S

50

55,5

666,6

7240,0

00,0

00,0

0-

0,0

0S

25

27,5

9S

25

27,5

9S

25

11,4

9I44

01

108,9

1300,0

016,0

90,0

050,0

080,0

0G

200

0,0

0S

25

55,5

6S

50

53,3

582,7

6240,0

00,0

016,0

949,0

00,0

0S

25

27,5

9S

25

27,5

9S

25

27,5

9I44

10

125,0

0300,0

00,0

00,0

0-

0,2

2G

200

32,3

8S

25

49,3

9S

50

43,2

366,6

7240,0

00,0

00,0

0-

0,3

1S

25

22,2

2S

25

22,2

2S

25

22,2

2I44

11

108,9

1300,0

016,0

90,0

013864,5

20,2

8G

200

21,3

5S

25

47,7

1S

50

39,8

682,7

6240,0

00,0

016,0

913863,5

20,0

0S

25

27,5

9S

25

27,5

9S

25

27,5

9I45

00

128,5

7360,0

00,0

00,0

0-

0,4

0C

300

16,0

0G

100

83,1

1G

200

29,4

650,0

0315,0

00,0

00,0

0-

40,0

0C

300

0,0

0G

200

44,4

4G

200

5,5

6I45

01

102,8

8360,0

025,6

90,0

0229,2

50,4

5C

300

3,9

5G

100

79,1

0G

200

19,8

375,6

9315,0

00,0

025,6

9230,2

540,0

0C

300

-0,0

0G

200

44,4

4G

200

31,2

5I45

10

128,5

7360,0

00,0

00,0

0-

0,4

0C

300

16,0

0G

100

83,1

1G

200

29,4

650,0

0315,0

00,0

00,0

0-

0,0

9C

300

15,4

2G

200

25,1

0G

200

9,4

8I45

11

96,5

2360,0

044,7

712,7

138214,9

10,4

6C

300

0,9

7G

100

78,1

0G

200

17,4

482,0

5315,0

012,7

144,7

738213,9

10,0

5C

300

28,3

8G

200

33,7

3G

200

19,9

5I46

00

43,3

3292,5

00,0

00,0

0-

0,0

9C

300

15,6

4S

50

13,8

5S

50

13,8

568,7

5233,7

50,0

00,0

0-

0,9

2C

500

28,5

4G

200

28,5

4G

100

11,6

6I46

01

77,3

2292,5

01,1

935,1

8305,1

80,0

0C

300

40,4

8S

50

18,4

2S

50

18,4

234,7

7233,7

535,1

81,1

9304,1

817,2

7C

500

14,3

3G

200

14,3

3G

100

6,1

0I46

10

43,3

3292,5

00,0

00,0

0-

0,0

9C

300

15,6

4S

50

13,8

5S

50

13,8

568,7

5233,7

50,0

00,0

0-

0,0

0C

500

27,8

3G

200

28,7

4G

100

12,1

8I46

11

77,3

2292,5

00,0

033,9

8155739,6

80,0

0C

300

40,4

8S

50

18,4

2S

50

18,4

234,7

7233,7

533,9

80,0

0155740,6

80,0

5C

500

6,7

1G

200

20,2

9G

100

7,7

7I47

00

87,5

0297,5

00,0

00,0

0-

77,5

0C

500

0,0

0G

200

34,7

2G

200

52,7

875,0

0337,5

00,0

00,0

0-

0,7

8S

50

14,1

7S

25

46,6

7S

50

14,1

7I47

01

-0,0

0297,5

087,5

00,0

046,0

077,5

0C

500

0,0

0G

200

-0,0

0G

200

0,0

0162,5

0337,5

00,0

087,5

045,0

00,3

6S

50

49,1

7S

25

64,1

7S

50

49,1

7I47

10

87,5

0297,5

00,0

00,0

0-

0,1

2C

500

14,2

4G

200

47,9

7G

200

25,2

989,6

1330,1

90,0

00,0

0-

0,5

3S

50

22,4

5S

25

44,7

2S

50

22,4

5I47

11

19,2

8240,9

3162,5

00,0

07183,8

00,0

4C

500

1,5

4G

200

12,0

3G

200

5,7

0237,5

0337,5

00,0

0162,5

07182,8

00,0

0S

50

79,1

7S

25

79,1

7S

50

79,1

7I48

00

54,2

2276,2

00,0

00,0

0-

50,0

0S

25

18,0

7S

25

18,0

7S

50

18,0

739,2

9247,5

00,0

00,0

0-

24,5

1C

300

18,4

0G

100

18,4

0S

50

2,5

0I48

01

93,4

5276,2

50,0

039,2

925,0

043,2

4S

25

22,7

6S

25

52,5

9S

50

18,1

00,0

0247,5

039,2

90,0

026,0

047,5

0C

300

0,0

0G

100

0,0

0S

50

0,0

0I48

10

54,1

7276,2

50,0

00,0

0-

1,5

0S

25

26,6

9S

25

26,6

9S

50

0,7

940,6

4246,5

50,0

00,0

0-

0,1

5C

300

0,1

0G

100

24,8

6S

50

15,6

8I48

11

51,6

4276,2

531,2

528,7

2434,5

21,9

8S

25

18,5

3S

25

18,5

3S

50

14,5

841,8

1247,5

028,7

231,2

5435,5

20,1

6C

300

0,1

1G

100

25,3

7S

50

16,3

4I49

00

69,6

4276,2

50,0

00,0

0-

0,0

8C

300

26,4

6G

200

33,2

6G

200

9,9

270,3

1318,7

50,0

00,0

0-

0,6

0C

300

0,0

0S

50

35,1

6S

50

35,1

6I49

01

15,6

3276,2

554,0

20,0

0230,3

50,1

9C

300

0,0

0G

200

15,6

3G

200

0,0

0124,3

3318,7

50,0

054,0

2229,3

50,2

4C

300

20,9

9S

50

51,6

7S

50

51,6

7I49

10

69,6

4276,2

50,0

00,0

0-

0,0

8C

300

26,4

6G

200

33,2

6G

200

9,9

290,4

8302,6

10,0

00,0

0-

0,2

1C

300

14,2

4S

50

38,1

2S

50

38,1

2I49

11

15,7

3276,2

553,9

10,0

064793,5

70,1

9C

300

0,0

6G

200

15,6

7G

200

0,0

0124,2

2318,7

50,0

053,9

164794,5

70,2

4C

300

20,9

1S

50

51,6

6S

50

51,6

6

37