Synthese (2012) 187:73–103Knowledge, Rationality & Action 555–585DOI 10.1007/s11229-012-0130-y

Effectivity functions and efficient coalitionsin Boolean games

Elise Bonzon · Marie-Christine Lagasquie-Schiex ·Jérôme Lang

Received: 5 July 2011 / Accepted: 22 May 2012 / Published online: 21 June 2012© Springer Science+Business Media B.V. 2012

Abstract Boolean games are a logical setting for representing strategic games in asuccinct way, taking advantage of the expressive power and conciseness of proposi-tional logic. A Boolean game consists of a set of players, each of which controls a setof propositional variables and has a specific goal expressed by a propositional formula.We show here that Boolean games are a very simple setting, yet sophisticated enough,for analysing the formation of coalitions. Due to the fact that players have dichotomouspreferences, the following notion emerges naturally: a coalition in a Boolean game isefficient if it has the power to guarantee that all goals of the members of the coalitionare satisfied. We study the properties of efficient coalitions.

Keywords Game theory · Propositional logic · Coalitions

1 Introduction

Boolean games (Harrenstein et al. 2001; Harrenstein 2004; Dunne and van der Hoek2004; Bonzon et al. 2009) are a logical setting for representing strategic games in a suc-cinct way, taking advantage of the expressive power and conciseness of propositional

E. Bonzon (B)Université Paris Descartes, LIPADE, 75270 Paris Cédex 06, Francee-mail: elise.bonzon@parisdescartes.fr

M.-C. Lagasquie-SchiexUniversité de Toulouse, UPS, IRIT, 31062 Toulouse Cédex 9, Francee-mail: lagasq@irit.fr

J. LangUniversité Paris Dauphine, LAMSADE, 75775 Paris Cédex 16, Francee-mail: lang@lamsade.dauphine.fr

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logic. Informally, a Boolean game consists of a set of players, each of which controlsa set of propositional variables and has a goal expressed by a propositional formula.

Boolean games are games with both a structural specificity and a preferential spec-ificity. The structural specificity expresses a restriction on strategy profiles: a player’s(pure) strategy is a truth assignment of the variables she controls. The preferentialspecificity expresses a restriction on the player’s preferences: a player in a Booleangame has a dichotomous preference relation, that is, either her goal is satisfied or it isnot, and this goal is represented succinctly by a propositional formula. The preferen-tial specificity can be easily relaxed, and there are a number of extensions to Booleangames that allow players to have nondichotomous preferences: in Harrenstein (2004,last chapter), each agent has a set of goals; in Bonzon et al. (2006, 2009), an agent’spreferences is described by a CP-net or more generally a specification in some compactpreference representation language; in Dunne et al. (2008), each agent has a quasi-dichotomous utility function, namely, a dichotomous utility function induced by hergoal, plus a negligible cost associated to her possible variable assignments (with thespecific choice that making a variable false is costless). The structural specificity, onthe other hand, is central to the framework, and relaxing it would probably make itdepart largely from Boolean games.

Previous work on Boolean games has focused on representational issues, by givinglogical characterizations of several solution concepts such as Nash equilibria, and byinvestigating the computational issues related to these solution concepts. Studying thepower of coalitions, as well as the formation of coalitions, in Boolean games, hasrarely been addressed (with the exception of Dunne et al. 2008). The goal of this paperis to address both issues. As these issues are, to a large extent, independent, this paperis composed of two almost independent parts.

The first part of the paper focuses on the power of coalitions in Boolean games.Equivalently, it amounts at studying the meaning of the structural specificity: howrestrictive is it, and how can it be characterized? A natural way of answering thisquestion is to study Boolean games from the point of view of effectivity functions,which model the power of coalitions of agents. More precisely, we would like tocharacterize the properties of effectivity functions that are implied by the structuralspecificity of Boolean games. Almost ten years ago, Pauly showed a correspondencebetween stategic games and a particular class of effectivity functions he named play-able effectivity functions (it has been shown recently that this correspondence is notcompletely exact, but this has no impact on our work; see footnote 1). Now, BooleanGames do not cover all strategic games: their structural specificity is a true restric-tion, and therefore we expect that Boolean games correspond to a strict subset ofplayable effectivity functions. The contribution of this first part of the paper consistsin characterizing this subset. Note that in this first part, the agents’ preferences areirrelevant, and therefore the results are totally independent from the question whetherthe preferential specificity is assumed or not.

The second part of this paper also focuses on the power of coalitions, but from adifferent perspective, taking agents’ preferences into account, and taking the prefer-ential specificity for granted (even if some of our results still hold under the weakerassumption that preferences are quasi-dichotomous). Due to the dichotomous natureof agents’ preferences, the following simple notion emerges naturally: a coalition in a

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Boolean game is efficient if it has the power to guarantee that all goals of the members ofthe coalition are satisfied. This notion is of primary importance, because it is expectedthat agents in a Boolean game will join such coalitions. Similarly as for related notionsin cooperative game theory, the existence of an efficient coalition is not guaranteed,and deciding whether a Boolean game possesses an efficient coalition is an importantissue. Efficient coalitions enjoy interesting structural properties, and are not easy toidentify, especially because a subset or a superset of an efficient coalition may not beefficient, and likewise, the union or the intersection of efficient coalitions may not beefficient. We characterize efficient coalitions in terms of some topological propertiesand study the relation between efficiency and some solution concepts coming fromcooperative game theory: the weak core (the standard notion of core) and the strongcore (which is a sort of generalization of Pareto optimality). Then, the computationalcomplexity of the membership and non-emptiness problems is identified for the threenotions of efficiency, weak core and strong core. Finally, the last contribution regardsthe representation of a Boolean game in terms of dependency graphs: we show thatdependency graphs and the relative notion of stable coalitions can be used as a correct(but not complete) method to find efficient coalitions. Completeness is restored in thespecial case where goals requires only one player to be satisfied.

We recall the Boolean game framework in Sect. 2. In Sect. 3 we study the spec-ificity of the power of coalitions in Boolean games, as compared to static games ingeneral. For this we show that the effectivity function in a Boolean game satisfiessome specific properties, that fully characterize Boolean games. In Sect. 4, we defineefficient coalitions in Boolean games, and focus first on their structural properties. Wegive an exact characterization of sets of coalitions that can be obtained as the set ofefficient coalitions associated with a Boolean game, and we relate coalition efficiencyto the well-known notion of core. In Sect. 5 we study efficient coalitions from a com-putational point of view. In Sect. 6, we address the role of dependencies betweenagents in the computation of efficient coalitions. Sections 7 and 8 discuss respectivelyrelated work and further research issues.

2 Boolean games

For any finite set V = {a, b, . . .} of propositional variables, LV denotes the prop-ositional language built up from V , the Boolean constants � and ⊥, and the usualconnectives. Formulas of LV are denoted by ϕ,ψ etc. A literal is a variable x of V orthe negation of a variable. A term is a consistent conjunction of literals. A clause is adisjunction of literals. If α is a term, then Lit (α) is the set of literals appearing in α.If ϕ ∈ LV , then V ar(ϕ) denotes the set of propositional variables appearing in ϕ.

2V is the set of the interpretations for V , with the usual convention that for M ∈ 2V

and x ∈ V,M gives the value true to x if x ∈ M and false otherwise. |� denotes theconsequence relation of classical propositional logic. Let V ′ ⊆ V . A V ′-interpretationis a truth assignement to each variable of V ′, that is, an element of 2V ′

. V ′-interpre-tations are denoted by listing all variables of V ′, with a ¯ symbol when the variableis set to false: for instance, let V ′ = {a, b, d}, then the V ′-interpretation M = {a, d}

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assigning a and d to true and b to false is denoted by abd. If V ar(ϕ) ⊆ X , thenModX (ϕ) represents the set of X -interpretations satisfying ϕ.

If {V1, . . . , Vp} is a partition of V and {M1, . . . ,Mp} are partial interpretations,where Mi ∈ 2Vi , (M1, . . . ,Mp) denotes the interpretation M1 ∪ . . . ∪ Mp.

Let ψ be a propositional formula. A term α is an implicant of ψ if and only ifα |� ψ holds. α is a prime implicant of ψ if and only if α is an implicant of ψ and forevery implicant α ′ of ψ , if α |� α ′ holds, then α ′ |� α holds. P I (ψ) denotes the setof all the prime implicants of ψ .

Given a set of propositional variables V , a Boolean game on V is an n-player game,where the actions available to each player consist in assigning a truth value to eachvariable in a given subset of V . The preferences of each player i are represented by apropositional formula ϕi formed upon the variables in V .

Without loss of generality, we can assume that V is finite. Indeed, only a finite set ofvariables occurs in the goals ϕi and the constraints γi , and the variables not occurringin them do not play any role and can safely be forgotten.

Definition 1 An n-player Boolean game is a 5-tuple (N , V, π, �,�), where

– N = {1, 2, . . . , n} is a set of players (also called agents);– V is a set of propositional variables;– π : N → 2V is a control assignment function mapping each player to the set of

variables she controls;– � = {γ1, . . . , γn} is a set of constraints, where each γi is a satisfiable propositional

formula of Lπ(i).– � = {ϕ1, . . . , ϕn} is a set of goals, where each ϕi is a satisfiable formula of LV .

For ease of notation, the set of all the variables controlled by i is written πi insteadof π(i). Each variable is controlled by one and only one agent, that is, {π1, . . . , πn}forms a partition of V . The role of constraints is to restrict the set of feasible strategiesof each agent: agent i assigns each variable of πi to a truth value, in such a way thatthe resulting assignment satisfies γi .

Definition 2 Let G = (N , V, π, �,�) be a Boolean game. A (pure) strategy forPlayer i in G is a πi -interpretation satisfying γi . The set of strategies for Playeri in G is i = {σi ∈ 2πi | σi |� γi }. A strategy profile σ for G is an n-tupleσ = (σ1, σ2, . . . , σn) where for all i, σi ∈ i . = 1 × . . . × n is the set of allstrategy profiles.

For each i, γi is a constraint restricting the possible strategy profiles for Player i .Note that since {π1, . . . , πn} forms a partition of V , a strategy profile σ is an inter-

pretation for V , i.e., σ ∈ 2V . The following notations are usual in game theory. Letσ = (σ1, . . . , σn) be a strategy profile. For any nonempty set of players I ⊆ N , theprojection of σ on I is defined by σI = (σi )i∈I and σ−I = σN\I . If I = {i}, wedenote the projection of σ on {i} by σi instead of σ{i}; similarly, we note σ−i insteadof σ−{i}. πI denotes the set of the variables controlled by I , and π−I = πN\I . Theset of strategies for I ⊆ N is I = ×i∈Ii , and the joint goal of coalition I ⊆ N is�I = ∧

i∈I ϕi .

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If σ and σ ′ are two strategy profiles, (σ−I , σ′I ) denotes the strategy profile obtained

from σ by replacing σi with σ ′i for all i ∈ I . For the sake of notation, the set of allstrategy profiles constructed from σC will be written {σ |σ ⊇ σC } instead of {σ |σ =(σ−C , σC ),∀σ−C }.

The goalϕi of player i is a compact representation of a dichotomous preference rela-tion, or equivalently, of a binary utility function ui : → {0, 1} defined by ui (σ ) = 0if σ |� ¬ϕi and ui (σ ) = 1 if σ |� ϕi . σ is at least as good as σ ′ for i , denoted byσ i σ

′, if ui (σ ) ≥ ui (σ′), or equivalently, if σ |� ¬ϕi implies σ ′ |� ¬ϕi ; σ is

strictly better than σ ′ for i , denoted by σ �i σ′, if ui (σ ) > ui (σ

′), or, equivalently,σ |� ϕi and σ ′ |� ¬ϕi .

As we said in the introduction, for the results of the first part of the paper, prefer-ences do not play any role (and a fortiori, neither does their dichotomous nature). Forthis we introduce the notion of pre-Boolean games, which are preference-free Booleangames.

Definition 3 A pre-Boolean game is a 4-tuple (N , V, π, �), with N , V, π, � as inDefinition 1.

Thus, a Boolean game consists of a pre-Boolean game together with a description� of the player’s (dichotomous) utilities.

Boolean games can easily be extended so as to allow for non-dichotomous pref-erences, represented in some compact language for preference representation (see(Harrenstein 2004; Bonzon et al. 2006, 2009; Dunne et al. 2008)). Among thesegeneralized Boolean games, an interesting subclass consists of Boolean games inwhich the players’ utility functions are nearly dichotomous.

Definition 4 A quasi-dichotomous Boolean game is a 6-uple (N , V, π, �,�, 〈c1, . . . ,

cn〉) where (N , V, π, �,�) is a Boolean game and for each Player i, ci is a functionmapping each strategy profile σ = (σ1, . . . , σn) to a cost such that ci (σ ) < 1. Theutility function ui : → {0, 1} of player i defined by ui (σ ) = −ci (σ ) if σ |� ¬ϕi

and ui (σ ) = 1 − ci (σ ) if σ |� ϕi . Note that for any σ such that σ |� ϕi and any σ ′such that σ ′ |� ¬ϕi we have ui (σ ) > ui (σ

′): whatever the cost function, an agent isalways better off in a state that satisfies her goal than in a state that does not.If G is a quasi-dichotomous Boolean game, the standard Boolean game G∗ associatedwith G is obtained from G by simply ignoring the cost function c.

Obviously, any standard Boolean game corresponds to a quasi-dichotomous Bool-ean game, obtained by letting ci (σ ) = 0 for all i and for all σ .

Quasi-dichotomous Boolean games were introduced first in Dunne et al. (2008),with the difference that the cost function c in Dunne et al. (2008) depends only onthe player’s own action, that is, ci (σ ) = ci (σi ), plus the additional assumption thateach agent has a cost associated to each “positive” action (setting one of her controlledvariables to true), and ci (σi ) is the sum of the costs of all of her variables assigned totrue.

In the definition above we did not specify how the cost function ci is represented.Representing it explicitly, by listing all combinations of strategies together with theirutility for each agent, would not fit the spirit of Boolean games, and would render

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somehow useless the compact representation of the goals. It is thus natural to assumethat each ci will be represented in some compact representation language, possiblymaking some further restriction, such as in Dunne et al. (2008).

3 Coalitions and effectivity functions in Boolean games

Recall that the structural specificity of Boolean games is that individual strategies aretruth assignments to a given set of propositional variables. We might wonder howrestrictive this specificity is. In this section we study Boolean games from the pointof view of effectivity functions. Effectivity functions have been developed in socialchoice to model the power of coalitions (Moulin 1983; Abdou and Keiding 1991;Pauly 2001). Clearly, the definition of i as Modπi (γi ) induces some constraints onthe power of players and coalitions. Our aim is to give an exact characterization ofeffectivity functions induced by Boolean games.

Since in Boolean games the power of an agent i is independent from her goal ϕi ,it suffices to consider pre-Boolean games when dealing with effectivity functions. Asusual, N is the set of agents, a coalition C is a subset of N , and S is a generic set ofstates.

Definition 5 A coalitional effectivity function is a function Eff : 2N → 22Ssatisfying

monotonicity: for every coalition C ⊆ N , X ∈ Eff(C) implies Y ∈ Eff(C) wheneverX ⊆ Y ⊆ S.

The function Eff associates with every group of players the set of states, or out-comes, for which the group is effective. We usually interpret X ∈ Eff(C) as “theplayers in C have a joint strategy for bringing about an outcome in X”.

A strategic game is usually defined as a tuple 〈N , , S, o〉, where is the setof strategy profiles for players in N , and o : ×i∈N i → S is the outcome function.Pauly (2001) gives a more precise account for effectivity in strategic games by definingα-effectivity: a coalition C ⊆ N is α-effective for X ⊆ S if and only if the players inC have a joint strategy for bringing an outcome of X , whatever the strategies of theother players are.

Definition 6 A coalitionalα-effectivity function for a non-empty strategic game G is afunction EffαG : 2N → 22S

defined by: X ∈ EffαG(C) iff ∃σC ∀σ−C , o(σC , σ−C ) ∈ X .

In a Boolean game, outcomes are identified with strategy profiles: S = . A pre-Boolean game G then induces an α-effectivity function EffαG as follows:

Definition 7 Let G = (N , V, π, �) be a pre-Boolean game. The coalitional α-ef-fectivity function induced by G is the function EffαG : 2N → 22 defined by: for anyX ⊆ and any C ⊆ N , X ∈ EffαG(C) if there exists σC ∈ C such that for anyσ−C ∈ −C , (σC , σ−C ) ∈ X .

For the sake of notation, the α-effectivity function induced by a pre-Boolean gameG will be denoted by EffG instead of EffαG . Note that effectivity functions inducedby pre-Boolean games can be equivalently expressed as mappings EffG : 2N → 2LV

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from coalitions to sets of logical formulas: ϕ ∈ EffG(I ) if ModπI (ϕ) ∈ EffG(I ). Thisdefinition obviously implies syntax-independence, that is, if ϕ ≡ ψ then ϕ ∈ EffG(I )iff ψ ∈ EffG(I ).

This definition is a particular case of the α-effectivity function induced by a stra-tegic game (see Pauly 2001, chap. 2). Therefore, these functions satisfy the followingproperties (cf. Pauly 2001, Theorem 2.27):

1. ∀C ⊆ N ,∅ �∈ EffG(C);2. ∀C ⊆ N , ∈ EffG(C);3. for all X ⊆ , if X �∈ EffG(∅) then X ∈ EffG(N );4. EffG is superadditive, that is, if for all C,C ′ ⊆ N such that C ∩ C ′ = ∅, and

X,Y ⊆ , if X ∈ EffG(C) and Y ∈ EffG(C ′) then X ∩ Y ∈ EffG(C ∪ C ′).An effectivity function satisfying these four properties is called strongly playable.

Note that strong playability implies regularity and coalition-monotonicity (Pauly 2001,Lemma 2.26). Pauly (2001) proves a correspondence between strong playability of aneffectivity function Eff and the existence of a strategic game G such that EffG = Eff. 1

However, pre-Boolean games are a specific case of strategic game forms, therefore wewould like to have an exact characterization of those effectivity functions that corre-spond to a pre-Boolean game. We first have to define two additional properties. DefineAt (C) as the minimal sets in Eff(C), that is, At (C) = {X ∈ Eff(C)| there is no Y ∈Eff(C) such that Y ⊂ X}. At (C) is called the nonmonotonic core of C , and denotedby Effnc(C), in Goranko et al. (2011).

Atomicity: Eff satisfies atomicity if for every C ⊆ N , At (C) forms a partitionof S.

Decomposability: Eff satisfies decomposability if for every two disjoints subsets I, Jof N and for every X ⊆ S, X ∈ Eff(I ∪ J ) if and only if thereexist Y ∈ Eff(I ) and Z ∈ Eff(J ) such that X = Y ∩ Z .

Decomposability is a strong property that implies superadditivity. Note also thatdecomposability and atomicity are strongly related to the following properties inAgotnes and Alechina (2011):

(2) for any C �= ∅,Effnc(C) = {∩i∈C Xi : Xi ∈ Effnc(i)}(3) X,Y ∈ Effnc(i) and X �= Y implies X ∩ Y = ∅(4) X ∈ Effnc( j) and x ∈ X implies ∃Y ∈ Effnc(i), x ∈ Y

where Effnc(C) denotes the set of all inclusion-minimal sets in Eff(C). Decompos-ability is equivalent to (2) whereas in the presence of decomposability and Eff(N ) =2S \ {∅}, atomicity is equivalent to the conjunction of (3) and (4). We give a shortproof of these equivalences in Appendix.

In the rest of this section we prove the following characterization result: a coalitionalα-effectivity function Eff satisfies strong playability, atomicity, decomposability andEff(N ) = 2S \ ∅ if and only if there exists a pre-Boolean game G = (N , V, π, �)

1 This result was recently shown by (Goranko et al. 2011) to be wrong for infinite game models. As thegames we consider are finite, this has no impact on the rest of the paper.

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and a bijective function μ : S → Mod(�) such that for every C ⊆ N : EffG(C) ={μ(X)|X ∈ Eff(C)}.

The proof of these two results go along a series of lemmas, which we establishfirst.2 If G is a pre-Boolean game, the set of atoms for the effectivity functions EffG

will be denoted by AtG .

Lemma 1 For any pre-Boolean game G,EffG satisfies strong playability, atomicity,decomposability, and EffG(N ) = 2 \∅.

Proof EffG is a (specific) α-effectivity function, therefore by Theorem 2.27 in Pauly(2001), EffG satisfies strong playability (and, a fortiori, superaddivity).

As for atomicity, remark first that X ∈ AtG(C) if and only if X is the set of allπC -interpretations satisfying γC = ∧

i γi , which clearly implies that any two dis-tinct subsets in AtG(C) are disjoint. Then remark that

⋃σC∈C

{σ |σ ⊃ σC } = .Therefore, AtG(C) forms a partition of .

As for decomposability, from left to right: let X ∈ EffG(I ∪ J ). Then there existsa joint strategy σI∪J such that if W = {σ ∈ |σ ⊇ σI∪J }, then W ⊆ X . Considernow Y = {σ ∈ |σ ⊇ σI } and Z = {σ ∈ |σ ⊇ σJ }. We have Y ∈ EffG(I ), Z ∈EffG(J ) and X = Y ∩ Z . From right to left: let Y ∈ EffG(I ) and Z ∈ EffG(J ), thenby superaddivity, Y ∩ Z ∈ EffG(I ∪ J ).

Lastly, let σ = (σ1, . . . , σn) ∈ . If each player i plays σi then σ is obtained, there-fore {σ } ∈ EffG(N ). By monotonicity, every nonempty subset of is in EffG(N ) aswell, therefore EffG(N ) = 2 \∅. ��Lemma 2 If there exists a pre-Boolean game G = (N , V, π, �) and an bijective func-tion μ : S → Mod(�) such that for every C ⊆ N : EffG(C) = {μ(X)|X ∈ Eff(C)},then Eff satisfies strong playability, atomicity, decomposability and Eff(N ) = 2S \∅.

Proof EffG satisfies these properties and μ is a bijection between S and μ() =Mod(�), therefore these properties transfer to Eff. ��Lemma 3 Let G be a pre-Boolean game, its set of strategy profiles and Ti be aminimal subset of EffG(i). Then Ti = {σ |σ ⊇ σi } for all σi ∈ i .

Proof Player i can only enforce a subset of i , that is, X ∈ EffG(i)if X contains1× . . .×i−1×∗i ×i+1× . . .×n for some∗i ⊆ i . Therefore the minimalsubsets of EffG(i) are exactly those of the form1×. . .×i−1×{σi }×i+1×. . .×n ,that is, of the form {σ |σ ⊇ σi }. ��

From now on, let Eff be a coalitional effectivity function satisfying strong playabil-ity, atomicity, decomposability and Eff(N ) = 2S \∅.

Let At (C) be the set of atoms for C associated with Eff. Due to decomposability,Eff is entirely determined by {At (i), i ∈ N }.

2 The strong links between our properties and the properties in Agotnes and Alechina (2011) that char-acterize injective games should not be seen as a surprise: because σ = σ ′ if and only if σi = σ ′i forevery i ∈ N , Boolean games are injective. Our Lemmas 3 and 4 can actually be seen as a proof that ourproperties, which imply all of the properties (Agotnes and Alechina 2011) (note that their property (1) istrivially satisfied for finite games) imply injectivity.

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Lemma 4 For every s ∈ S there exists a unique (Z1, . . . , Zn) ∈ At (1)× . . .× At (n)such that Z1 ∩ . . . ∩ Zn = {s}.Proof Let s ∈ S. Because Eff(N ) = 2S \ {∅}, we have {s} ∈ Eff(N ), and by decom-posability, there exists (T1, . . . , Tn) such that for every i, Ti ∈ Eff(i) and T1∩. . .∩Tn ={s}. Let i ∈ N . By definition of At (i), there exists Zi ∈ At (i) such that s ∈ Zi andZi ⊆ Ti . Suppose there exists Z ′i ∈ At (i) such that s ∈ Z ′i and Z ′i ⊆ Ti . Zi ∩ Z ′i �= ∅,since s belongs to both Zi and Z ′i . Therefore, by atomicity, Zi = Z ′i , and this holdsfor every i . ��

Lemma 4 allows us to write Zi (s) for every s and i to be the unique subset in At (i)containing s. For any non-empty coalition C , let us write ZC (s) = ⋂

i∈C Zi (s).Let us now build the Boolean game G∗ = G(Eff) as follows. The intuition of the

construction is that by the atoms of player i correspond to her strategies, and in orderto ensure that the number of i’s strategies is equal to the number of atoms for i , weintroduce a suitable number of variables controlled by i , add a constraint that limitsthe number of i’s strategies to the number of atoms. (We also give a detailed exampleafter the formal construction.)

– for every i , consider the following numbering of At (i): let ri be a bijective mappingfrom At (i) to {0, 1, . . . , |At (i)| − 1}. Then create pi = �log2 |At (i)|� proposi-tional variables x1

i , . . . , x pii . Finally, let V = {x j

i |i ∈ N , 1 ≤ j ≤ pi };– for every i , let πi = {x1

i , . . . , x pii };

– for every i and every j ≤ pi , let εi, j be the j th digit in the binary representationof i . Note that εi,pi = 1 by definition of pi . If x is a propositional variable thenwe use the following notation: 0.x = ¬x and 1.x = x . Then define

γi =∧

j∈{2,...,pi },εi, j=0

⎛

⎝∧

1≤k≤ j−1

εi, j .xki → ¬x j

i

⎞

⎠

– finally, for each s ∈ S, let μ(s) ∈ 2V defined by: x ji ∈ μ(s) if and only if the j th

digit of the binary representation of ri (Zi (s)) is 1.

For every i ∈ N and every Z ∈ At (i), let k = ri (Z) and σi (Z) the strategy ofPlayer i in G∗ corresponding to the binary representation of k using {x1

i , . . . , x pii }, x1

ibeing the most significant bit. For instance, if pi = 3 and r(Zi ) = 6 then σi (Z) =(x1

i , x2i ,¬x3

i ).We denote byG∗ the set of strategy profiles (or, equivalently, states) of G∗. Strat-

egies of G∗ are denoted by σG∗ . The set of atoms of EffG∗ is denoted by AtG∗(i).Since the decomposition of states into atoms is unique (Lemma 4), two different

states s and s′ are mapped to two different valuations, i.e., s �= s′ impliesμ(s) �= μ(s′).Now, for any Z ∈ At (i), constraint γi ensures that σi (Z) |� γi ; this being true for alli , for any σ we have μ(σ) |� γ1 ∧ . . .∧ γn . Therefore, μ is a bijection between S andG∗ = Mod(γ1 ∧ . . . ∧ γn).

To understand it better, it is helpful to see how this construction works onan example. Let N = {1, 2, 3}, S = {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B,C}, At (1) =

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{1234, 5678, 9ABC}, At (2) = {13579B, 2468AC}, At (3) = {12569C, 3478AB}(curly brackets for subsets of are omitted – 1234 means {1, 2, 3, 4} and so on). Bydecomposability, we obtain:

– At (12) = {13, 24, 57, 68, 9B, AC},– At (13) = {12, 34, 56, 78, 9C, AB}, and– At (23) = {159, 37B, 26C, 48A}.|At (1)| = 3, therefore p1 = 2. |At (2)| = |At (3)| = 2, therefore p2 = p3 = 1. Thus,V = {x1

1 , x21 , x1

2 , x13 }. Let At (1) = {Z0, Z1, Z2}, that is, r1(1234) = 0, r1(5678) = 1

and r1(9ABC) = 2. Likewise, r2(13579B) = 0, r2(2468AC) = 1, r3(12569C) = 0and r3(3478AB) = 1. Consider s = 6. We have s = 5678 ∩ 2468AC ∩ 12569C ,therefore μ(s) = (x1

1 ,¬x21 , x1

2 ,¬x13). The constraints are γ1 = (x1

1 → ¬x21 ), γ2 =

γ3 = �. Thus, G∗ = (N , V, π, �) where N = {1, 2, 3}, V = {x11 , x2

1 , x12 , x1

3}, π1 ={x1

1 , x21 }, π2 = {x1

2}, π3 = {x13}, γ1 = (x1

1 → ¬x21 ) and γ2 = γ3 = �.

Lemma 5 For every i ∈ N and Z ∈ At (i) : μ(Z) = {σG∗ ∈ G∗ |σG∗ ⊇ σi (Z)}.Proof Let i ∈ N and Z ∈ At (i). Let σG∗ ∈ μ(Z); by definition of μ(Z), there existsan s ∈ S such that μ(s) = σG∗ . Consider the decomposition of s into atoms, thatis,{s} = Z1(s)∩ . . .∩ Zn(s) (cf. Lemma 4). By construction of μ, the projection of μ(s)on {x1

i , . . . , x pii } corresponds to the binary representation of ri (Zi (s)). Therefore,

μ(s) = σG∗ extends σi (Z).Conversely, let σG∗ such that σG∗ ⊇ σi (Z). For every j ≤ n, let k j be the number

whose binary representation in {x1j , . . . , x

p jj } is the projection ofσG∗ on {x1

j , . . . , xp jj }.

Let σ be defined by {σ } = Z1(k1) ∩ . . . ∩ Zn(kn). By construction of μ, we haveμ(σ) = σG∗ . Moreover, Zi (ki ) = Z by atomicity, that is, σ ∈ Z . Therefore σG∗ ∈μ(Z). ��

We are now ready for establishing the main result of this section.

Proposition 1 An effectivity function Eff satisfies

1. strong playability,2. atomicity,3. decomposability and4. Eff(N ) = 2 \∅

if and only if there exists a pre-Boolean game G = (N , V, π, �) and a bijective func-tionμ : → Mod(�) such that for every C ⊆ N : EffG(C) = {μ(X)|X ∈ Eff(C)}.Proof The right-to-left direction is Lemma 2. In order to prove the opposite direction,we show that for every C ⊆ N and every X ⊆ S,X ∈ Eff(C) holds if and only ifμ(X) ∈ EffG∗(C).

Decomposability of both Eff and EffG∗ implies that it is enough to show that forevery i and every X ⊆ S, X ∈ Eff(i) if and only if μ(X) ∈ EffG∗(i). Because bothEff and EffG∗ satisfy coalition monotonicity, it is enough to show that for every i, Zi ∈AtG∗(i) implies μ(Zi ) ∈ EffG∗(i) and Ti ∈ AtG∗(i) implies μ−1(Ti ) ∈ Eff(i).

Let Zi ∈ At (i). Because r(Zi ) ≤ pi , we have σi (Zi ) |� γi , therefore σi (Zi ) ∈EffG∗(i). By Lemma 5, μ(Zi ) = {σG∗ |σG∗ ⊇ σi (Zi )}. Therefore, μ(Zi ) ∈ EffG∗(i).

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Conversely, let Ti ∈ AtG∗(i). By Lemma 3, Ti = {σ |σ ⊇ σi } for some σi ∈ i .Let σi = (εi,1.x1

1 , . . . , εi,pi .xpii ) and q(σi ) = ∑pi

k=1 2pi−k .εi,k . Note that q(σi ) ≤ pi ,

because σi ∈ i implies σi |� γi . Now, let j = r−1i (q(σi )). Let Z j

i ∈ At (i) such

that ri (Zji ) = j . We have μ(Z j

i ) = {σ |σ ⊇ σi } = Ti . Now, Z ji ∈ Eff(i), because

Z ji ∈ At (i). Therefore, μ−1(Ti ) ∈ Eff(i).We have now proven that for C ⊆ N and every X ⊆ S, X ∈ Eff(C) holds

if and only if μ(X) ∈ EffG∗(C). We can now conclude that if Eff satisfies strongplayability, atomicity, decomposability, and Eff(N ) = 2 \ ∅, then there exists agame G(= G∗) and an bijective function μ : S → Mod(�) such that for everyC ⊆ N : EffG(C) = {μ(X)|X ∈ Eff(C)}. ��

A natural question is, can we obtain a similar result without the need of constraints?This is obviously not the case, because for a Boolean game G without constraints, thecardinality of G is 2m for some m. Therefore, requiring that the cardinality of S is apower of 2 is necessary. But it is not sufficient: because the cardinality of every i isalso be a power of 2, this must also be the case for At (i).

We say that Eff satisfies regular atomicity if it satisfies atomicity and for all i ∈N , |At ({i})| = 2mi for some positive integer mi . We note that regular atomicity anddecomposability implies that this cardinality property propagates to every coalition,i.e., for all C ⊆ N , |At (C)| = 2mC for some positive integer mC .

Then we have the following:

Proposition 2 A coalitional α-effectivity function Eff satisfies

1. strong playability,2. regular atomicity,3. decomposability and4. Eff(N ) = 2S \∅

if and only if there exists a constraint-free pre-Boolean game G = (N , V, π,�) andan bijective functionμ : S → 2V such that for every C ⊆ N : EffG(C) = {μ(X)|X ∈Eff(C)}.

Proof The proof is almost identical to the proof of Proposition 1. The only differenceis that in the construction of G∗ is unchanged except that we don’t need to define�. Since |At ({i})| is a power of 2, we have pi = log |At ({i})|, and μ is a bijectionbetween S and G∗ = Mod(γ1 ∧ . . . ∧ γn) = Mod(�) = 2V . ��

4 Efficient coalitions

We now consider Boolean games and define efficient coalitions. Informally, a coalitionis efficient in a Boolean game if and only if it has the ability to jointly satisfy the goalsof all members of the coalition. This notion of efficient coalition is not totally new,as is coincides with the notion of successful coalition in qualitative coalitional games(QCG) introduced in Wooldridge and Dunne (2004).

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4.1 Definition and characterization

Definition 8 Let G = (N , V, π, �,�) be a Boolean game. A coalition C ⊆ N is effi-cient if and only if there exists σC ∈ C such that for all σ−C , we have (σC , σ−C ) |�∧

i∈C ϕi . The set of all efficient coalitions of a game G is denoted by EC(G). C is aminimal efficient coalition if there is no efficient coalition B ⊂ C .

Note that this definition still makes sense for quasi-dichotomous Boolean games;in this case, the cost ci is irrelevant, and a coalition is efficient if it is able to jointlysatisfy the goals of its members, whatever the induced costs (which, we recall, arealways smaller than the utility gain resulting from goal satisfaction): formally, C isefficient for a quasi-dichotomous Boolean game G if and only if it is efficient forG∗. (However, it no longer makes sense for generalized Boolean games with arbitraryutility functions.)

Note that the empty coalition ∅ is efficient, because ϕ∅ = ∧i∈∅

ϕi ≡ � is alwayssatisfied.

Example 1 Let G = (N , V, �, π,�) where V = {a, b, c}, N = {1, 2, 3}, γi = �for every i, π1 = {a}, π2 = {b}, π3 = {c}, ϕ1 = (¬a ∧ b), ϕ2 = (¬a ∨ ¬c) andϕ3 = (¬b ∧ ¬c).

First note that ϕ1 ∧ ϕ3 is inconsistent, therefore no coalition containing {1, 3} canbe efficient. {1} is not efficient, because ϕ1 cannot be made true only by fixing thevalue of a; similarly, {2} and {3} are not efficient either. {1, 2} is efficient, because thejoint strategy σ{1,2} = ab is such that σ{1,2} |� ϕ1 ∧ ϕ2. {2, 3} is efficient, becauseσ{2,3} = bc |� ϕ2 ∧ ϕ3. Therefore, EC(G) = {∅, {1, 2}, {2, 3}}.

From this simple example we see already that EC is neither downward closed norupward closed, that is, if C is efficient, then a subset or a superset of C may not beefficient. We also see that EC is not closed under union or intersection: {1, 2} and{2, 3} are efficient, but neither {1, 2} ∩ {2, 3} nor {1, 2} ∪ {2, 3} is.

Example 2 (kidney exchange, after Abraham et al. 2007) .Consider n pairs of individuals, each consisting of a recipient Ri in urgent need ofa kidney transplant, and a donor Di who is ready to give one of her kidneys to saveRi . Because the kidney of donor Di is not necessarily compatible with recipient Ri ,a strategy for saving more people consists in considering the graph 〈{1, . . . , n}, E〉containing a node i ∈ 1, . . . , n for each pair (Di , Ri ) and containing the edge (i, j)whenever Di ’s kidney is compatible with R j . A solution is any set of nodes that can bepartitioned into disjoint cycles in the graph: in a solution, Donor Di gives a kidney ifand only if Ri gets one. An optimal solution (saving a maximum number of lifes) is asolution with a maximum number of nodes. The problem can be seen as the followingBoolean game G:

– N = {1, . . . , n};– V = {gi j |i, j ∈ {1, . . . , n}}; gi j being true means that Di gives a kidney to R j .– πi = {gi j ; 1 ≤ j ≤ n};– for every i, γi = ∧

j �=k ¬(gi j ∧ gik) expresses that a donor cannot give more thanone kidney.

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– for every i , ϕi = ∨( j,i)∈E g ji expresses that the goal of i is to get a kidney that is

compatible with Ri .

For example, take n = 5 and E = {(1, 1), (1, 2), (2, 3), (2, 4), (2, 5), (3, 1), (4, 2),(5, 4)}. Then G = (N , V, �, π,�), with

– N = {1, 2, 3, 4, 5}– V = {gi j | 1 ≤ i, j ≤ 5};– ∀i, γi = ∧

j �=k ¬(gi j ∧ gik)

– π1 = {g11, g12, g13, g14, g15}, and similarly for π2, etc.– ϕ1 = g11 ∨ g31;ϕ2 = g12 ∨ g42;ϕ3 = g23;ϕ4 = g24 ∨ g54;ϕ5 = g25.

The corresponding graph is depicted below.

1 2

3 4

5

Clearly enough, efficient coalitions correspond to solutions. In our example, theefficient coalitions are ∅, {1}, {2, 4}, {1, 2, 4}, {1, 2, 3}, {2, 4, 5} and {1, 2, 4, 5}.

We have seen that the set of efficient coalitions associated with a Boolean gamemay not be downward closed nor upward closed, nor closed under union or non-emptyintersection. However, it is possible to characterize the efficient coalitions of a Booleangame: we will show that a set of coalitions corresponds to the set of efficientcoalitionsfor some Boolean game if and only if (a) it contains the emptyset and (b) it is closedby union of disjoint coalitions.

We will prove this characterization result in the rest of this subsection. To do so,we will need Lemmas 6 to 9, which will we establish first.

Lemma 6 Let I, J be two coalitions of a Boolean game G. If I and J are efficientand I ∩ J = ∅, then I ∪ J is efficient.

Proof If I is efficient, then we know that ∃σI ∈ I such that σI |� ∧i∈I ϕi , and the

same for J : ∃σJ ∈ J such that σJ |� ∧j∈J ϕ j . Moreover, as I ∩ J = ∅, we have

(σI , σJ ) |� ∧i∈I∪J ϕi , so I ∪ J is an efficient coalition. ��

We now need to define the following construction. Let SC be a set of coalitions satis-fying the following conditions:

(1) ∅ ∈ SC.(2) for all I, J ∈ SC such that I ∩ J = ∅ then I ∪ J ∈ SC.

Define the following Boolean game G as follows:

– V = {connect (i, j)|i, j ∈ N } (all possible connections between players);– ∀i , γi = �;

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– πi = {connect (i, j)| j ∈ N } (all connections from Player i);– ϕi = ∨

I∈SC|i∈I FI , where

FI =⎛

⎝∧

j,k∈I

connect ( j, k)

⎞

⎠∧

⎛

⎝∧

j∈I,k �∈I

¬connect ( j, k)

⎞

⎠

(Player i wants that all the players of her coalition are interconnected and that thereis no connection from the coalition to the “outside” of the coalition)

We want to show that the set ECG = SC (where ECG is the set of efficient coalitionsfor G).

Before proving that ECG ⊆ SC, we establish the following lemmas:

Lemma 7 For any collection SC = {Ci , i = 1, . . . , q} ⊆ 22N,∧

1≤i≤q FCi is satis-fiable if and only if for any i, j ∈ {1, . . . , q}, either Ci = C j or Ci ∩ C j = ∅.

Proof 1. Assume that for any i, j ∈ {1, . . . , q}, either Ci = C j or Ci ∩ C j = ∅.Then

∧1≤i≤q FCi is equivalent to

⎛

⎝∧

1≤i≤q

∧

j,k∈Ci

connect ( j, k)

⎞

⎠∧

⎛

⎝∧

1≤i≤q

∧

j∈Ci ,k �∈Ci

¬connect ( j, k)

⎞

⎠

∧1≤i≤q FCi is satisfied by any interpretation assigning each connect ( j, k) such

that j, k belong to the same Ci to true, and each connect ( j, k) such that j ∈ Ci

for some i and k �∈ Ci to false. Hence∧

1≤i≤q FCi is satisfiable.2. Assume that for some i, j ∈ {1, . . . , q}, we have Ci ∩ C j �= ∅ and Ci �= C j . Let

k ∈ Ci∩C j and (without loss of generality) l ∈ Ci \C j . Then FCi |� connect (k, l)and FC j |� ¬connect (k, l), hence FCi ∧ FC j is unsatisfiable, and a fortiori, so is∧

1≤i≤q FCi . ��We now define a covering of a coalition I by disjoint subsets of SC as a tuple

!C = 〈Ci |i ∈ I 〉 of coalitions such that: (i) for every k ∈ I , Ck ∈ SC; (ii) for allC j ,Ck ∈ !C , either C j = Ck or C j ∩ Ck = ∅; (iii) for every i ∈ I, i ∈ Ci .

Let Cov(SC, I ) be the set of all covering of I by disjoint subsets of SC. For instance,if SC = {1, 24, 123, 124} then Cov(SC, 12) = {〈1, 24〉, 〈123, 123〉, 〈124, 124〉},3Cov(SC, 123) = {〈123, 123, 123〉},Cov(SC, 124) = {〈1, 24, 24〉, 〈124, 124, 124〉}and Cov(SC, 234) = Cov(SC, 1234) = ∅.

Lemma 8 For any I �= ∅,�I is equivalent to∨

!C∈Cov(SC,I )∧

i∈I FCi .

Proof

�I ≡ ∧i∈I ϕi

≡ ∧i∈I

∨J∈SC|i∈J FJ

≡ ∨〈Ci ,i∈I 〉 such that Ci∈SC and i∈Ci for every i∈I

∧i∈I FCi

3 There are 2 players in I = {1, 2}, therefore each !C in Cov(SC, 12) contains 2 coalitions, one for eachplayer, satisfying (i), (ii) and (iii).

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Now, by Lemma 7,∧

i∈I FCi is satisfiable if and only if for all i, j ∈ I , eitherCi = C j or Ci ∩ C j = ∅. Therefore, �I ≡ ∨

!C∈Cov(SC,I )∧

i∈I FCi . ��For instance, if SC = {1, 24, 123, 124} then�12 ≡ (F1∧ F24)∨ F123∨ F124;�123 ≡F123;�124 ≡ (F1 ∧ F24) ∨ F124;�234 ≡ ⊥.

Lemma 9 Let I ⊆ 2N . As ∀I, J ∈ SC, I ∩ J = ∅ ⇒ I ∪ J ∈ SC,�I is satisfiableif and only if there exists J ∈ SC such that I ⊆ J .

Proof The case I = ∅ is straightforward: � ≡ � is satisfiable, and ∅ ∈ SC byassumption, therefore there exists J ∈ SC(J = ∅) such that I ⊆ J .

Now, let I �= ∅.

⇒ Assume �I is satisfiable. By Lemma 8, �I is equivalent to

∨

!C∈Cov(SC,I )

∧

i∈I

FCi

therefore there exists a !C in Cov(SC, I ) such that∧

i∈I FCi is satisfiable, there-fore Cov(SC, I ) is not empty. Now, !C ∈ Cov(SC, I ) implies that:

(i) for every i ∈ I,Ci ∈ SC;(ii) for every i, j ∈ I , either Ci = C j or Ci ∩ C j = ∅.

(iii) I ⊆ ⋃i∈I Ci

Now, (i), (ii) and∀I, J ∈ SC, I∩J = ∅ ⇒ I∪J ∈ SC imply that⋃

i∈I Ci ∈ SC,which together with (iii) proves that there exists a J ∈ SC (namely J = ⋃

i∈I Ci )such that I ⊆ J .

⇐ Assume that there is a J ∈ SC such that I ⊆ J . Then �J |� �I , and �J

is consistent (consider the interpretation assigning each connect (i, j) such thati, j ∈ J to true). ��

We can now establish the characterization of the efficient coalitions of a Booleangame.

Proposition 3 Let N = {1, . . . , n} be a set of agents and SC ∈ 22Na set of coalitions.

There exists a Boolean game G over N such that the set of efficient coalitions for Gis SC (i.e. EC(G) = SC) if and only if SC satisfies these two properties:

(1) ∅ ∈ SC.(2) for all I, J ∈ SC such that I ∩ J = ∅ then I ∪ J ∈ SC.

Proof Lemma 6 proves the (⇒) direction of Proposition 3. For the (⇐) direction, wewant to show that SC ⊆ ECG .

We first show that SC ⊆ ECG . Let I ∈ SC. If every agent i ∈ I plays(∧j∈I connect (i, j)

) ∧ (∧k �∈I ¬connect (i, k)

), then ϕi is satisfied for every i ∈ I .

Hence, I is an efficient coalition for G and SC is included in EC(G).It remains to be shown that ECG ⊆ SC. Let I be a coalition such that I �∈ SC

(which implies I �= ∅, because of assumption ∅ ∈ SC).

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– If I = N then there is no J ∈ SC such that I ⊆ J (because I �∈ SC), and thenLemma 9 implies that �I is unsatisfiable, therefore I cannot be efficient for G.

– Assume now that I �= N and define the following I -strategy I ( I = N \ I ): forevery i ∈ I , σi is such that for all j ∈ I, σi contains ¬connect (i, j) (whetherconnect (i, j) is true or false for j �∈ I is irrelevant). Let !C = 〈Ci , i ∈ I 〉 ∈Cov(SC, I ).We first claim that there exists i∗ ∈ I such that Ci∗ is not contained in I . Indeed,suppose that for every i ∈ I , Ci ⊆ I . Then, because i ∈ Ci holds for every i , wehave

⋃i∈I Ci = I . Now, Ci ∈ SC for all i , and any two distinct Ci ,C j are disjoint,

therefore, by Property (2) of Proposition 3, we get I ∈ SC, which by assumptionis false.Now, let k ∈ Ci∗ \ I (such a k exists because Ci∗ is not contained in I ). As i and kare in Ci , connect (k, i∗) has to be true to satisfy FCi . Therefore σk |� ¬FCi , anda fortiori σ I |� ¬FCi , which entails σ I |� ¬∧

i∈I FCi .This being true for any !C ∈ Cov(SC, I ), we have

σ I |�∧

!C∈Cov(SC,I )¬

∧

i∈I

FCi

that is, σ I |� ¬∨!C∈Cov(SC,I )

∧i∈I FCi . Together with Lemma 8, this entails

σ I |� ¬�I . Hence, I does not control �I and I cannot be efficient for G. ��

4.2 Efficient coalitions and the core

We now relate the notion of efficient coalitions to the usual notion of core of a coali-tional game. In coalitional games with ordinal preferences, the core is usually definedas follows (see e.g. (Aumann 1967; Owen 1982; Myerson 1991)): a strategy profile σis in the core of a coalitional game if and only if there exists no coalition C with a jointstrategy σC that guarantees that all members of C are better off than with σ . Here weconsider also a stronger notion of core: a strategy profile σ is in the strong core of acoalitional game if and only if there exists no coalition C with a joint strategy σC thatguarantees that all members of C are at least as satisfied as with σ , and at least onemember of C is strictly better off than with σ .

Definition 9 Let G be a Boolean game. The (weak) core of G, denoted by WCore(G),is the set of strategy profiles σ = (σ1, . . . , σn) such that there exists no C ⊆ N andno σC ∈ C such that for every i ∈ C and every σ−C ∈ −C , (σC , σ−C ) �i σ .The strong core of a Boolean game G, denoted by SCore(G), is the set of strategyprofiles σ = (σ1, . . . , σn) such that there exists no C ⊆ N and no σC ∈ C such thatfor every i ∈ C and every σ−C ∈ −C , (σC , σ−C ) i σ and there is an i ∈ C suchthat for every σ−C ∈ −C , (σC , σ−C ) �i σ .

Obviously enough, this notion of weak core is equivalent to the notion of strongNash equilibrium (Aumann 1959), where coalitions form in order to correlate thestrategies of their members.

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The relationship between the (weak) core of a Boolean game and its set of efficientcoalitions is expressed by the following simple result.

Proposition 4 Let G = (N , V, �, π,�) be a Boolean game. σ ∈ WCore(G) if andonly if σ satisfies at least one member of every efficient coalition, that is, for everyC ∈ EC(G), σ |� ∨

i∈C ϕi .

Proof σ = (σ1, . . . , σn) �∈ WCore(G) if and only if there exist a coalition C ⊆ Nand a tuple σC ∈ C such that (1) for every i ∈ C and σ−C ∈ −C , we have(σC , σ−C ) �i σ . Using the specific form that utility functions have in Boolean games,(1) is equivalent to (2a) ∀σ−C , (σC , σ−C ) |� ∧

i∈C ϕi and (2b) σ |� ∧i∈C ¬ϕi . As

{π1, . . . , πn} forms a partition of V , (2a) can be written as σC |� ∧i∈C ϕi . Therefore,

σ ∈ WCore(G) if and only if (3) for every C ⊆ N , either σ |� ∨i∈C ϕi or for

every σC ∈ C , σC |� ∨i∈C ¬ϕi . (3) can be rewritten into (4): for every C ⊆ N ,

if there exists σC such that σC |� ∧i∈C ϕi then σ |� ∨

i∈C ϕi . Now, the existenceof σC ∈ C such that σC |� ∧

i∈C ϕi means that Coalition C is efficient. There-fore, σ ∈ WCore(G) if and only if for every C ⊂ N , if C ∈ EC(G) then σ |�∨

i∈C ϕi . ��In particular, when no coalition of a Boolean game G is efficient, then all strategyprofiles are in WCore(G).

Moreover, the weak core of a Boolean game cannot be empty:

Proposition 5 For any Boolean game G,WCore(G) �= ∅.

Proof We construct the following set of coalitions E as follows. First, initialize Eto ∅. Then, while there exists a coalition C in EC(G) such that C ∩ C ′ = ∅ holdsfor every C ′ ∈ E , pick such a C and add it to E . At the end of the algorithm,E is a set of disjoint efficient coalitions {CI , i ∈ I }, therefore, by Proposition 3,∪i∈I Ci is efficient. Therefore, there exists σE ∈ E such that σE |� ∧

i∈E ϕi , andE contains at least one element of every efficient coalition (if this were not the case,there would remain an efficient coalition C that intersects none of the Ci ’s, and thealgorithm would have continued and incorporated C into E). Let σ extending σE . σsatisfies at least one member of every efficient coalition, therefore, by Proposition 4,WCore(G) �= ∅. ��The strong core of a Boolean game is harder to characterize in terms of efficientcoalitions. We only have the following implication.

Proposition 6 Let G = (N , V, �, π,�) be a Boolean game, and σ be a strategyprofile. If σ ∈ SCore(G) then for every C ∈ EC(G) and every i ∈ C, σ |� ϕi .

Proof Let C ∈ EC(G) and assume there exists i ∈ C such that (1) σ |� ¬ϕi . Wewant to show that σ �∈ SCore(G). Since C ∈ EC(G), (2) there exists σC ∈ C suchthat σC |� ∧

j∈C ϕ j . Applying (1) and (2) to i leads to σC �i σ , while applying (1)and (2) to j ∈ C \ {i} leads to σC j σ . Therefore, σ �∈ SCore(G). ��Thus, a strategy in the strong core of G satisfies the goal of every member of everyefficient coalition. The following counterexample shows that the converse does nothold.

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Example 3 Let G=(N , V, �, π,�)be the following Boolean game: V={a, b, c, d, e},N ={1, 2, 3, 4, 5}, γi = � for every i, π1 = {a}, π2 = {b}, π3 = {c}, π4 = {d}, π5 ={e}, ϕ1 = ¬a ∧ b, ϕ2 = ¬a, ϕ3 = d, ϕ4 = c ∧ a and ϕ5 = c ∧ e.

This game has one efficient coalition: {1, 2}.Let σ = abcde. We have σ |� ϕ1 ∧ ϕ2 ∧ ¬ϕ3 ∧ ¬ϕ4 ∧ ¬ϕ5. Therefore, ∀C ∈EC(G),∀i ∈ C, σ |� ϕi .However, σ �∈ SCore(G) : ∃C ′ = {1, 2, 3, 4} ⊂ N such that ∃σC = abcd |�ϕ1 ∧ ϕ2 ∧ ϕ3 ∧ ¬ϕ4. So, ∀σ−C , (σC , σ−C ) 1 σ, (σC , σ−C ) 2 σ, (σC , σ−C ) 4 σ ,and (σC , σ−C ) �3 σ . σ �∈ SCore(G).

Note that the strong core of a Boolean game can be empty: in Example 1, the setof efficient coalitions is {∅, {1, 2}, {2, 3}}, therefore there is no σ ∈ such that forall C ∈ EC(G), for all i ∈ C , σ |� ϕi , therefore, by Proposition 6, SCore(G) = ∅.However, we can show that the non-emptiness of the strong core is equivalent to thefollowing simple condition on efficient coalitions.

Proposition 7 Let G = (N , V, �, π,�) be a Boolean game. We have the following:Score(G) �= ∅ if and only if

⋃{C ⊆ N |C ∈ EC(G)} ∈ EC(G) – that is, if andonly if the union of all efficient coalitions is efficient.

Proof Let M EC(G) = ⋃C⊆N {C ∈ EC(G)}.

⇐ Score(G) �= ∅. Let σ ∈ Score(G). From Proposition 6, we know that ∀C ∈EC(G),∀i ∈ C, σ |� ϕi . So, ∀i ∈ M EC(G), σ |� ϕi . So, M EC(G) ∈ EC(G).

⇒ M EC(G)∈EC(G). Let σM EC(G) ∈ M EC(G) such that ∀σ−M EC(G),

σM EC(G) |� �M EC(G). We are looking for σ such that σ ∈ Score(G).Let σ−M EC(G) ∈ −M EC(G) such that M AX = {i |σ = (σM EC(G), σ−M EC(G))

|� ϕi } be maximal for ⊆. σ−M EC(G) exists, in worst case σ |� �M EC(G). AsM AX is maximal, we cannot find C ⊆ N such that ∃σC ∈ C , such that∀σ−C ∈ −C ,∀i ∈ C, (σC , σ−C ) i σ , and ∃i ∈ C, (σC , σ−C ) �i σ . Indeed, ifwe assume that this coalition C exists, then ∀i ∈ N such that σ |� ϕi , we haveσC |� ϕi , and ∃i ∈ N such that σ �|� ϕi and σC |� ϕi . In this case, M AX is notmaximal for ⊆. ��

5 Computational complexity of reasoning about efficient coalitions

We start by identifying the complexity of some key decision problems related to effi-cient coalitions. The key questions are: is a given coalition efficient? does there exist anonempty efficient coalition? is a given agent member of some efficient coalition? ofall nonempty efficient coalitions? In addition to these problems that are directly relatedto efficient coalitions, similar problems arise for the notions of weak and strong core.

Proposition 8 Deciding whether a given coalition is efficient for a Boolean game is

p2 -complete, and is p

2 -hard even if n = 2 and the coalition is a singleton.

Proof Membership is straightforward and hardness is a straightforward consequenceof the facts that (1) the coalition reduced to the singleton {i} is efficient if and only if ihas a winning strategy and that (2) deciding whether an agent has a winning strategyin a Boolean game is p

2 -complete (see Bonzon et al. 2009). ��

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The next result addresses the problem whether there exists an efficient coalition ina Boolean game.

Proposition 9 Deciding whether there exists a non-empty efficient coalition in a Bool-ean game is p

2 -complete, and is p2 -hard even if n = 2.

Proof Membership to p2 is immediate.

To show that deciding whether there is a non-empty efficient coalition in a Booleangame is p

2 -hard (even with 2 agents), consider the following polynomial reductionfrom qbf2,∃. To each instance Q = ∃a1 . . . ap∀b1 . . . bqϕ of qbf2,∃, let us considerthe following Boolean game G Q = 〈N , V, π, �,�〉, where N = {1, 2}, γ1 = γ2 =�, V = {a1 . . . ap, b1 . . . bq , x}, π1 = {a1, . . . , ap, x}, π2 = {b1, . . . , bq}, ϕ1 = ϕ

and ϕ2 = ¬ϕ ∧ x . Neither {2} nor {1, 2} can be efficient; therefore, the only possiblenonempty efficient coalition is {1}. Now, it is easily seen that {1} is efficient if andonly if Q is a valid instance of qbf2,∃. ��

We now consider the problems of determining whether a given agent belongs tosome efficient coalition, and whether she belongs to all nonempty efficient coalitions.

Proposition 10

– deciding whether an agent i belongs to at least one efficient coalition of a Booleangame is p

2 -complete, and is p2 -hard even if n = 2.

– deciding whether an agent i belongs at all nonempty efficient coalitions of a Bool-ean game is �p

2 -complete, and is �p2 -hard even if n = 2.

Proof For both problems, the membership part of the proof is easy.To show that deciding whether an agent i belongs to at least one efficient coalition

for G is p2 -hard (even with 2 agents), consider the following polynomial reduction

from qbf2,∃. To each instance Q = ∃a1 . . . ap∀b1 . . . bqϕ of qbf2,∃, let us considerthe following Boolean game G Q = 〈N , V, π, �,�〉, where N = {1, 2}, γ1 = γ2 =�, V = {a1, . . . , ap, b1, . . . , bq}, π1 = {a1, . . . , ap}, π2 = {b1, . . . , bq}, ϕ1 = ϕ

and ϕ2 = ¬ϕ. {1} is efficient if and only if there exists a strategy σ1 such that σ1 |� ϕ,that is, if and only if I is valid. Now, {1, 2} cannot be efficient, because ϕ1 ∧ ϕ2 = �.therefore, 1 belongs to an efficient coalition if and only if {1} is efficient, that is, if andonly if Q is valid.

To show that deciding whether an agent i belongs to all nonempty efficientcoalitions for G is �p

2 -hard (even with 2 agents), consider the following poly-nomial reduction from qbf2,∀. To each instance Q = ∃a1 . . . ap∀b1 . . . bqϕ ofqbf2,∀, let us consider the following Boolean game G Q = 〈N , V, π, �,�〉, whereN = {1, 2}, γ1 = γ2 = �, V = {a1 . . . ap, b1 . . . bq , x}, π1 = {a1, . . . , ap, x}, π2 ={b1, . . . , bq}, ϕ1 = ¬ϕ and ϕ2 = ϕ ∧ x . Neither {2} nor {1, 2} can be efficient; there-fore, 2 belongs to all nonempty efficient coalitions if and only if {1} is not efficient, thatis, if ∃a1 . . . ap∀b1 . . . bq¬ϕ is not valid, or equivalently, if ∀a1 . . . ap∃b1 . . . bqϕ isvalid. ��

Although we have stated them and proven them for standard Boolean gameswith dichotomous utilities, Propositions 8, 9 and 10 hold also for quasi-dichotomous

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Boolean games. This is trivially obtained from the fact that a coalition is efficient fora quasi-Boolean game G if and only if it is efficient for G∗.4

Given the strong relationships between efficient coalitions and the notions of weakand strong core of a Boolean game, these results allow us to derive complexity resultsregarding these. Note however that unlike the previous three propositions, the follow-ing three hold for standard Boolean games (with dichotomous preferences) only.

First, Proposition 4 leads the following result:

Proposition 11 Deciding if a strategy profile σ is in the weak core of a Boolean gameG is �p

2 -complete.

Proof Recall that σ �∈ WCore(G) if and only if there exists a coalition C ⊆ Nsuch that (a) σ |� ∧

i∈C ¬ϕi and (b) there exists a strategy σC ∈ C such thatσC |� ϕC = ∧

i∈C ϕi .Membership to�p

2 is immediate, as the formulation above immediately shows thatthe problem to decide if a strategy profile σ is not in WCore(G) is in p

2 .Hardness is obtained by proving that the complementary problem is p

2 -complete,using a reduction of the problem of deciding the validity of a QBF2,∃.

Given Q = ∃A,∀B,�, where A and B are disjoint sets of variables and � is aformula of L A∪B , we define the following Boolean game G Q by

– V = A ∪ B ∪ {c}, where c is a fresh variable (c �∈ A ∪ B);– N = {pv | v ∈ V };– for every v ∈ V, π(pv) = {v};– for every ai ∈ A, ϕai = ϕ ∧ c;– for every b j ∈ B, ϕb j = �;– ϕc = ϕ ∧ c;– σ is any assignment satisfying ¬c.

Assume Q is a positive instance of QBF2,∃, that is, there exists a !a such that forevery !b we have (!a, !b) |� ϕ. Let C = A ∪ {c} and σC = (!a, c). We have σC |�∧v∈C ϕv = ϕ ∧ c and σ |� ∧

v∈C ¬ϕv . Therefore, σ �∈ WCore(G Q).Conversely, assume that there exists a coalition C ⊆ N and a strategyσC ∈ C such

that σC |� ϕC = ∧i∈C ϕi . Note that because σ |� ¬c, we have σ |� ∧

i∈C ¬ϕi , thuscondition (a) is satisfied. Because σ |� ¬ϕi for any i ∈ C , we must have C ⊆ A∪{c},which implies that σC = c ∧ ϕ. Define !a ∈ A such that !a and σC agree all ai ∈ C .Because σC |� ϕ we have !a |� ϕ, that is, for all !b we have (!a, !b) |� ϕ.

We have shown that σ ∈ WCore(G Q) if and only if Q is a positive instance ofQBF2,∃, hence the result. ��Proposition 12 Deciding if a strategy profile σ is in the strong core of a Booleangame G is �p

2 -complete.

Proof Recall that we have σ �∈ SCore(G) if and only if there exists a coalition C ⊆ Nand a strategy σC ∈ C such that

4 We must be careful however. if the cost functions in a quasi-dichotomous Boolean game were representedexplicitly, then all problems considered would be trivially polynomial. Our statement holds only if the rep-resentation is compact enough so that the representation of some cost function (such as, typically, the nullcost function) has a polynomial size.

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(c) for all i ∈ C and all σ−C ∈ −C we have (σC , σ−C ) i σ ;(d) there exist i ∈ C and σ−C ∈ −C such that (σC , σ−C ) �i σ ;

We take the same reduction as in Proposition 11.If Q is satisfiable, then σ �∈ WCore(G Q), and a fortiori σ �∈ SCore(G Q).If σ �∈ SCore(G Q) then there exist a coalition C and σC ∈ C such that (c) and

(d) hold. But (d) implies that σ |� ¬(ϕ ∧ c) and (σC , σ−C ) |� ϕ ∧ c, therefore,σ �∈ WCore(G Q), which implies that Q is a positive instance of QBF2,∃. ��Proposition 13 Deciding whether SCore(G) is nonempty is in p

2 .

Proof We recall that SCore(G) �= ∅ if and only if the union of all efficient coalitionsof G is efficient.

We start by noticing that C is the union of all efficient coalitions of G if and onlyif the following two conditions hold:

(A) for every i ∈ C , there exists an efficient coalition Ci containing i ;(B) for all i �∈ C , no efficient coalition contains i .

Therefore, the following nondeterministic algorithm shows that SCore(G) �= ∅:

1. C := ∅2. for every i ∈ N do

3. if there exists an efficient coalition of G containing ithen add i to C(else nothing)

4. check that C is efficient.

Consider the problem of Step 3, namely: given i ∈ N , check that there exists anefficient coalition of G containing i . The problem can be solved by the followingnondeterministic algorithm: guess a coalition C ′, a strategy profile σC ′ ∈ C ′ , andcheck that σC ′ |� ϕC ′ . Thus, checking that there exists an efficient coalition of Gcontaining i is in NP and Step 2 amounts to a linear number of NP-oracles, whereasStep 4 amounts to one more NP-oracle. Therefore, the algorithm is a deterministicalgorithm using a polynomial number of NP-oracles, which shows that the problemof checking that SCore(G) �= ∅ is in p

2 . ��So far we do not have a p

2 -hardness result.

6 Efficient coalitions and dependencies between agents

We now study how the computation of efficient coalitions can be made easier by tak-ing benefit from specific restrictions on the agents’ preferences. First, the syntacticalnature of goals may help us identifying efficient coalitions easily. Second, exploitingthe dependencies between agents (where a dependency between i and j means thatthe goal ϕi of i involves a variable controlled by j) can allow us, in some cases, todecompose the computation of efficient coalitions into independent subproblems.

We first note that when ϕi does not involve any variable controlled by j , the satis-faction of i does not depend directly on j . This is only a sufficient condition: it may

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be the case that the syntactical expression of ϕi mentions a variable controlled by j ,but that this variable plays no role whatsoever in the satisfaction of ϕi , as variabley in ϕi = x ∧ (y ∨ ¬y). We therefore use a stronger notion of formula-variableindependence (Lang et al. 2003).

Definition 10 A propositional formula ϕ is independent from a propositional variablex if there exists a formula ψ logically equivalent to ϕ and in which x does not appear.

Definition 11 Let G = (N , V, �, π,�) be a Boolean game. The set of relevant vari-ables for a player i , denoted by RVG(i), is the set of all variables v ∈ V such that ϕi

is not independent from v.

For the sake of notation, the set of relevant variables for a player i in a given Booleangame G will be denoted by RVi instead of RVG(i). We now easily define the relevantplayers for a given player i as the set of players controlling at least one variable ofRVi .

Definition 12 Let G = (N , V, �, π,�) be a Boolean game. The set of relevant play-ers for a player i , denoted by R Pi ,5 is the set of agents j ∈ N such that j controls atleast one relevant variable of i : R Pi = ⋃

v∈RViπ−1(v).

Example 4 Three friends (1, 2 and 3) are invited at a party. 1 wants to attend the party.2 wants to attend if and only if 1 does. 3 wants to attend, and would like 2 to attendas well and 1 not to. This situation can be modelled by the following Boolean gameG = (N , V, �, π,�), defined by

– V = {a, b, c}, with a (resp. b, c) means “1 (resp. 2, 3) attends the party”,– N = {1, 2, 3},∀i, γi = �,– π1 = {a}, π2 = {b}, π3 = {c},– ϕ1 = a, ϕ2 = a ↔ b and ϕ3 = ¬a ∧ b ∧ c.

We can see that 1’s satisfaction depends only on herself, 2’s depends on 1 and herself,whereas 3’s depends on 1, 2 and herself. Therefore, we have: RV1 = {a}, RV2 ={a, b}, RV3 = {a, b, c}, R P1 = {1}, R P2 = {1, 2}, R P3 = {1, 2, 3}.

This relation between players can be seen as a directed graph containing a vertexfor each player, and an edge from i to j whenever j ∈ R Pi , i.e. if j is a relevant playerof i .

Definition 13 Let G = (N , V, �, π,�) be a Boolean game. The dependency graphof a Boolean game G is the directed graph P = 〈N , R〉, with ∀i, j ∈ N , (i, j) ∈ R(denoted by R(i, j)) if j ∈ R Pi .

Thus, R(i) is the set of players from which i may need some action in order tobe satisfied: j ∈ R(i) if and only if j ∈ R Pi . Remark however that j ∈ R(i) doesnot imply that i needs some action by j to see her goal satisfied. For instance, if

5 Again, the set of relevant players for a Boolean game G should be denoted by R PG (i): for the ease ofnotation we simply write R Pi .

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π1 = {a}, π2 = {b} and ϕ1 = a ∨ b, then 2 ∈ R(1); however, 1 has a strategy forsatisfying her goal (setting a to true) and therefore does not need an action by 2. Notethat the dependency graph may have cycles.

We denote by R∗ the transitive closure of R. R∗(i) is the set of all players whohave a direct or indirect influence on i . For I ⊆ N , we let R(I ) = ⋃

i∈I R(i).Example 4, continued:The dependence graph P induced by G is depicted as follows:

1 2

3

We already know that if two disjoint coalitions I and J are efficient then their unionis efficient. The converse does not hold in the general case, that is, there may exist twodisjoint sets I and J such that I ∪ J is efficient and neither I nor J is. However, theconverse holds in the following specific case:

Proposition 14 Let G = (N , V, �, π,�) be a Boolean game. Let I and J be twocoalitions such that I ∩ J = ∅, I ∪ J is efficient, R(I )∩ J = ∅ and R(J )∩ I = ∅.Then I and J are both efficient.

Proof The efficiency of I ∪ J implies that there exists σI∪J ∈ I∪J such that σI∪J |�(∧

i∈I∪J ϕi ). Since I ∩ J = ∅, we have this following chain of equivalences:

∃σI ∈ I , ∃σJ ∈ J : (σI , σJ ) |�(

∧

i∈I∪J

ϕi

)

⇔ ∃σI ∈ I , ∃σJ ∈ J : (σI , σJ ) |�⎛

⎝∧

i∈I

ϕi ∧∧

j∈J

ϕ j

⎞

⎠

⇔ ∃σI ∈ I , ∃σJ ∈ J :(

(σI , σJ ) |�(

∧

i∈I

ϕi

))

∧⎛

⎝(σI , σJ ) |�⎛

⎝∧

j∈J

ϕ j

⎞

⎠

⎞

⎠

Moreover, we know that ∀i ∈ I, j ∈ J, j �∈ R Pi (resp. i �∈ R P j). So, ∀i ∈ I, j ∈J,∀v ∈ V ar(P I (ϕi )), v �∈ π j (resp. ∀w ∈ V ar(P I (ϕ j )), w �∈ πi ). We know that noplayer in J controls a variable of a goal of a player in I (and vice versa).As we have ∃σI∈I , ((σI , σJ ) |� (

∧i∈I ϕi )) and ∀i ∈ I, j ∈ J , ∀v ∈ V ar(P I (ϕi )),

v �∈ π j , we have: σI |� (∧

i∈I ϕi ) (resp. σJ |� (∧

j∈J ϕ j ).Therefore, both I and J are efficient. ��

We now introduce the notion of stable set for a Boolean game.6 A subset of agentsB is stable for G if none of the agents in B has a relevant player outside B.

6 Note that this notion has nothing to do with the classical notion of stable set in graph theory.

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Definition 14 Let G = (N , V, π, �,�) be a Boolean game. B ⊆ N is stable for Gif and only if R(B) ⊆ B.

The following proposition is straightforward but useful:

Proposition 15 Let G = (N , V, �, π,�) be a Boolean game. If B ⊆ N is stable forR, then B is an efficient coalition of G (B ⊆ EC(G)) if and only if ϕB = ∧

i∈B ϕi isconsistent.

Proof Let B a stable set for R. Then, we have:

∀i ∈ B,∀ j such that j ∈ R(i), j ∈ B⇔ ∀i ∈ B, R Pi ⊆ B⇒ ∀i ∈ B, ∃σB ∈ B such that σC |� ϕi

⇔ ∃σB ∈ B such that σB |� ∧i∈B ϕi if and only if

∧i∈B ϕi �|� ⊥

��The converse is not necessarily true, as we can see on the following example:

Example 5 Let G=(N , V, �, π,�) be the Boolean game defined by V={a, b},N={1, 2}, π1 = {a}, π2 = {b}, ϕ1 = a ∨ b and ϕ2 = �.

The coalition {1} is efficient, but is not stable for R : R({1}) = {1, 2} �⊆ {1}.However, the converse can be true under the very restrictive condition that the sat-

isfaction of the goal of a player depends only on the actions of one player, that is, ifR Pi is a singleton for every i ∈ B.

Proposition 16 Let G = (N , V, �, π,�) be a Boolean game. If B ⊆ N is an efficientcoalition of G(B ⊆ EC(G)) such that ∀i ∈ B, |R Pi | = 1, then B is stable for R. Inthis case, a coalition B such that ϕB = ∧

i∈B ϕi �|� ⊥ is efficient if and only if B isstable for R.

Proof B is an efficient coalition, so∧

i∈B ϕi is consistent, and ∃σB ∈ B such thatσB |� ∧

i∈B ϕi .We know that ∀i ∈ B, |R Pi | = 1. So, ∃ j ∈ N such that R Pi = { j}, i.e. ∀v ∈V ar(P I (ϕi )), v ∈ π j . As we have σB |� ϕi , with σB ∈ B , we know that B controlsat least one variable in ϕi . So, j ∈ B, and thus B is stable for R. ��

In this specific case where R Pi is a singleton for every i ∈ B, we have furthermorethis intuitive graph-theoretic characterization of efficient coalitions:

Proposition 17 Let G= (N , V, �, π,�) be a Boolean game such that ∀i ∈N , |R Pi | = 1. For any coalition C ⊆ N, C is minimal efficient if and only if Cforms a cycle in the dependence graph.

Proof ⇒ As ∀i , |R Pi | = 1, only one edge can go out for each player. So, if there is acycle between p players, and if we rename these players with respect to the topo-logical order, we have R P1 = {2}, R P2 = {3}, . . . , R Pp−1 = {p}, R Pp = {1}.

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Let C = {1, . . . , p}. As we obviously have R(C) = C(∀i ∈ C, R Pi = {(i +1)modp} ∈ C),C is stable for R.Moreover, we know that ∀i, j ∈ C, R Pi �= R Pj , so ∀i, j ∈ C, ϕi ∧ϕ j �|� ⊥. So,from Proposition 15, C is efficient.Assume that∃I ⊂ C efficient. So,∃σI such thatσI |� ∧

i∈I ϕi . So,∀i ∈ I, R Pi ∈I . As |R Pi | = 1, and as ∀i, j ∈ C , and thus ∀i, j ∈ I, R Pi �= i and R Pi �= R Pj ,agents in I form a cycle. So, I = C . C is minimal efficient.

⇐ If C is stable for R, then ∀i ∈ C , ∃ j ∈ C such that R Pi = { j}. So, if C ={1, . . . , p}, we can rename these players in order to have R P1 = {2}, R P2 ={3}, . . . , R Pp−1 = {p}, R Pp = {1}, and C forms a cycle in the dependancegraph. ��

Another interesting issue is the study of efficient coalitions in Boolean games wheregoals have a specific syntactical structure. The characterization of efficient coalitionswhen goals are literals is a straightforward consequence of Propositions 16 and 17:

Corollary 1 Let G = (N , V, �, π,�) be a Boolean game. If for every i ∈ N , ϕi is aliteral, then we have the following results:

1. A coalition B is efficient if and only if B is stable for R.2. For any coalition B ⊆ N , B is minimal efficient if and only if B forms a cycle in

the dependence graph.

We also have the following intuitive characterization of efficient coalitions whengoals are either clauses or terms:

Proposition 18 Let G = (N , V, �, π,�) be a Boolean game.

1. If for every i ∈ N , ϕi is a term then for any B ⊆ N such that∧

i∈B ϕi is consistent,B is efficient if and only if B is stable for R.

2. If for every i ∈ N , ϕi is a clause such that∧

i∈N ϕi is consistent, then for anyB ⊆ N , B is efficient if and only if there exists a set of cycles in the dependencegraph of G such that the nodes of the union of these cycles are exactly the membersof B.

Proof

1. ⇒ Let B be a stable set for R. As∧

i∈B ϕi is consistent, we know from Propo-sition 15 that B is an efficient coalition.

⇐ Let B be an efficient coalition. So ∃σB ∈B such that σB |� ∧i∈B ϕi . As

∀i, ϕi = ∧v∈Lit (ϕi )

v, σB |� ∧v∈⋃

i∈B Lit (ϕi )v. So, ∀v ∈ ⋃

i∈B Lit (ϕi ),

V ar(v) ∈ πB , and then ∀i ∈ B, R Pi ⊆ B. B is stable for R.2. ⇒ Let B be an efficient coalition. So ∃σB ∈ B such that σB |� ∧

i∈B ϕi .Let decompose B in p minimal efficient coalitions.7 We have ∀k ∈{1, . . . , p}, Bk is minimal efficient, B1 ∪ . . . ∪ Bp = B.Let Bk ∈ {B1, . . . , Bp}. As ∀i ∈ N , ϕi = ∨

v∈Lit (ϕi )v, we know that

∀i ∈ Bk, ∃ j ∈ Bk, ∃σ j ∈ j such that σ j |� ϕi . As Bk is minimal efficient,

7 If B is minimal efficient, then p = 1.

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we know that we cannot find C ⊂ Bk such that ∀i ∈ C, ∃ j ∈ C, ∃σ j ∈ j

such that σ j |� ϕi . So, if Bk = {1, . . . ,m} we can rename these players inthe following way: let us take a player and call her 1. Then, call 2 the playerin Bk such that σ2 |� ϕ1 (as Bk is minimal efficient, we know that 1 and 2are two different players). Then, call 3 the player in Bk such that σ3 |� ϕ3(as previously, as Bk is minimal efficient, we know that 1, 2 and 3 are threedifferent players). We can rename all players in Bk in the same way, untilσm |� ϕm−1. As Bk is minimal efficient, we know that σ1 |� ϕm . Thus Bk

forms a cycle in the dependency graph.As each Bk ∈ {B1, . . . , Bp} forms a cycle in the dependency graph, B is theunion of these cycles in the dependency graph.

⇐ Let B = {1, . . . ,m} be a cycle between m players. So, ∀i ∈ B, ∃ j ∈ B suchthat j ∈ R Pi . As ∀i ∈ N , ϕi = ∨

v∈Lit (ϕi )v, we know that ∀i ∈ B, ∃ j ∈

B, ∃σ j ∈ j such that σ j |� ϕi . Moreover, we know that∧

i∈N ϕi �|� ⊥.Then, ∃σB ∈ B such that σB |� ∧

i∈B ϕi . So, B is an efficient coalition.Let assume now that we have p cycles {B1, . . . Bp} in the dependency graph.As seen previously, each Bi is an efficient coalition. As

∧i∈N ϕi �|� ⊥, B1 ∪

. . . ∪ Bp |� ∧i∈{1,...p}

∧j∈Bi

ϕ j . So, B1 ∪ . . . ∪ Bp is an efficient coalition.

��Again, due to the fact that a coalition in a quasi-dichotomous Boolean game G

is efficient if and only if it is efficient in the associated standard Boolean game G∗,Propositions 14, 15, 16, 17 and 18 also hold for quasi-dichotomous Boolean games.

7 Related work

Introduced in Dunne et al. (2008), cooperative Boolean games (CBG) are a specificclass of quasi-dichotomous Boolean games. In a cooperative Boolean game, as in aclassical Boolean game, each agent has a goal represented by a propositional logicformula, and each agent has control over a set of Boolean variables. In addition to this,every propositional variable xi is associated with a positive number c(xi ) representingthe cost, incurring for the agent who controls xi , of making xi true. Costs are negligiblewith respect to the utility of having a goal satisfied (an agent always prefers a statesatisfying her goal to a state that does not), therefore cooperative Boolean games arequasi-dichotomous. Standard Boolean games are recovered by letting c(xi ) = 0 forall xi .

Now, Dunne et al. (2008) focuses on two stability concepts, one of which is highlyrelated to our Sect. 4.2. This concept is also called the core of a (cooperative) Booleangame, and is defined as follows:

Definition 15 Let G be a Boolean game.8 A strategy profile σ is blocked by a coalitionC ⊆ N through a strategy profile σ ′ if

8 Since we aim at comparing both notions in standard Boolean games, we give the definition for standardBoolean games only. The definition would be exactly the same in the general case with non-zero costs.

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1. σ and σ ′ coincide on all variables that are not controlled by any member of C ;2. coalition C strictly prefers σ ′ over σ : for all i ∈ C, σ ′ �i σ .

The DHKW-core of G, denoted by CoreDH K W (G), is the set of strategy profiles thatare not blocked by any coalition.

When restricted on standard (zero-cost) Boolean games, this definition differs fromboth our weak and strong core notions, for the following reason: σ is in the DHKW-core of a Boolean game if no coalition C has an interest to deviate from σ , the actionsof the other players being fixed, whereas σ is in the weak or strong core if no coalitionhas a joint strategy which makes it better off (where the meaning of “better off” differswhether we talk about the weak or the strong core) than σ , whatever the actions of theother players. As remarked in Sauro et al. (2009) and Sauro and Villata (2011), thisdefinition corresponds to the strong Nash equilibrium in noncooperative game theory.The following is immediate from the definition of the DHKW-core:

Proposition 19 For any Boolean game G, the DHKW-core of G is contained in theweak core of G.9

The converse inclusion does not hold, as witnessed by the following example.

Example 6 Let n = 3, V = {x1, x2, x3}, π(i) = {xi }, and γ1 = x1 ↔ (x2∧x3), γ2 =γ3 = ¬γ1. The DHKW-core of G is empty: for every σ satisfying γ2, 1 can switch x1to make γ1 true, and for every σ satisfying γ1, 2 and 3 can adjust the values of x2 andx3 to make γ2 true. However, no coalition is efficient in G, therefore, no coalition canfind another way to act (than in σ ) that ensures it to be better off, whatever the actionof the other player(s), and any strategy profile is in the weak core of G.

Note finally that deciding membership to the DHKW-core of a Boolean game iscoNP-complete (whereas deciding membership to the weak and to the strong coreis, in both cases, �p

2 -complete), and that deciding the nonemptiness of the DHKW-core is p

2 -complete (whereas the weak core is always nonempty, and deciding thenonemptiness of the strong core is in p

2 ).Cooperative games have been further investigated in Endriss et al. (2011), who

study the design of taxation functions so as to modify a cooperative Boolean gamein order to ensure that some Nash equilibrium (or all Nash equilibria) satisfies somedesirable property.

Sauro et al. (2009); Sauro and Villata (2011) address the actual computation of theDHKW-core of a cooperative Boolean game, using the dependencies between playersand variables and/or players, as we do for efficient coalitions in Sect. 6.

Another related line of work is the study of coalition formation among goal-directedagents by Boella et al. (2005, 2006), and Sauro (2006). One of the main differencesbetween their framework and ours is in the expression of the problem input. Whilewe specify agents’ abilities and goals separately (abilities by a control assignment

9 There is no similar relationship between the DHKW-core and the strong core. Whereas both the DHKW-core and the weak core consider that a coalition is strictly better off if all its members are strictly better off,the strong core uses a weaker definition that makes the concept incomparable with the DHKW-core.

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function and goals by propositional formulae), Sauro (2006) defines a power struc-ture consisting of a set of abstract goals Goal(i) for each agent i , a power relationpow expressing, for every subset Goal ′ of Goal = ∪i Goal(i), which coalitions canachieve Goal ′, and a compatibility relation comp expressing which goals are jointlyfeasible, i.e., non-conflictual. An agent is satisfied as soon as one of its goals is satisfied.Now, a pair 〈C, E〉, where C ⊆ N and E ⊆ Goal = ∪i Goal(i) is do-ut-des if everyagent i in C (a) has one of her goals satisfied: Goal(i)∩ E �= ∅, and (b) contributed tothe achievement of some of the others’ goals: there exists g j ∈ E ∩ Goal( j), j �= i ,such that C \ {i} cannot achieve g j .

A Boolean game can be translated into a power structure in the following way:every propositional goal ϕi is expressed as an abstract goal gi with Goal(i) = {gi },while pow and comp express respectively which coalitions can achieve which setsof goals and which sets of goals are jointly feasible. The converse translation (frompower structures to Boolean games) is simple only in the special case where eachagent i has a single goal gi (the details of the translation do not present any particularinterest and we omit them).

Now, in the specific case where each agent i has a single goal gi , 〈C, E〉 is do-ut-desif (a′) E ⊇ GoalC = {gi , i ∈ C} and (b’) for every i ∈ C there exists j ∈ C, j �= i ,such that C \ {i} cannot achieve g j . This leads to the following characterization ofdo-ut-des coalitions:

Observation 1 〈C,GoalC 〉 is do-ut-des if and only if C is efficient and for everyi ∈ C,C \ {i} is not efficient.

Although, clearly, any minimally efficient coalition is do ut des, the converse is nottrue: consider the Boolean game with n = 4, i controls xi , γ1 = x2, γ2 = x1, γ3 = x4and γ4 = x3; then C is do-ut-des, but not minimally efficient. Now, Boella et al.define a further refined notion : 〈C, E〉 is a i-dud coalition if it is do-ut-des and isnot decomposable into two smaller do-ut-des coalitions. In other terms, C is i-dud ifand only if C is efficient and cannot be decomposed into disjoint efficient subcoali-tions. This stronger notion is still not equivalent to being minimally efficient: in theBoolean game where n = 4, i controls xi , γ1 = x2, γ2 = x1, γ3 = x1 ∧ x2 ∧ x4 andγ4 = x1 ∧ x2 ∧ x3, the only efficient coalitions are {1, 2} and {1, 2, 3, 4}, therefore{1, 2, 3, 4} is i-dud but not minimally efficient.

8 Conclusion

The results we have obtained are twofold – this paper can actually be seen as twoindependent parts.

The first part (Sect. 3) gives a characterization of effectivity functions induced by(pre-)Boolean games, thus allowing us to understand better the structural assumptionshidden behind the control assignment functions that define the power of agents inBoolean games. The results of Sect. 3 apply to pre-Boolean games, where preferencesdo not play any role.

The second part (Sects. 4–6) shows that Boolean games can be used as a compactrepresentation setting for coalitional games where players have quasi-dichotomous

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preferences. This specificity has lead us to define an interesting notion of efficientcoalitions. We have given an exact characterization of sets of coalitions that correspondto the set of efficient coalitions for a Boolean game, and several results concerningthe computation of efficient coalitions. The results of Sects. 4, 5 and 6 can be parti-tioned into two classes: those who apply to Boolean games with quasi-dichotomouspreferences, and those who apply only to standard Boolean games, with dichotomouspreferences.

There are many practical situations where preferences are naturally quasi-dichoto-mous, or even dichotomous (cf. Example 2). However, it is natural to ask whether ourresults of Sects. 4–6 extend to generalized Boolean games, with arbitrary preferencesrepresented in some compact representation language (Bonzon et al. 2006, 2009). Thisis a challenging issue for further research. Unfortunately, this does not appear to beeasy, because the notion of efficient coalition, which is dichotomous in essence, onlymakes sense when each agent has a primary goal whose satisfaction overweighs allpossible action costs, or, in other terms, when preferences are quasi-dichotomous.

Acknowledgments We would like to thank the anonymous reviewers for their valuable comments andsuggestions to improve this paper. Jérôme Lang thanks the ANR project ComSoc (ANR-09-BLAN-0305).

Appendix

We show here that decomposability is equivalent to (2) whereas in the presence ofdecomposability and Eff(N ) = 2S \ {∅}, atomicity is equivalent to the conjunction of(3) and (4), with

(2) for any C �= ∅,Effnc(C) = {∩i∈C Xi : Xi ∈ Effnc(i)}(3) X,Y ∈ Effnc(i) and X �= Y implies X ∩ Y = ∅(4) X ∈ Effnc( j) and x ∈ X implies ∃Y ∈ Effnc(i), x ∈ Y

where Effnc(C) denotes the set of all inclusion-minimal sets in Eff(C).

1. Decomposability is equivalent to (2).First, we show that decomposability is equivalent to

(D′) : for every C ⊆ N , X ∈ Eff(C) iff for all i ∈ C there exists Xi such thatX = ∩i∈C Xi

This is easily shown by induction on the size of C : the base case is obvious, andfor any i ∈ C, X ∈ Eff(C) if and only if there exists Xi and X−i such thatX = Xi ∩ X−i .Now, because Eff(C) is upward closed (X ⊆ Y and X ∈ Eff(C) implies Y ∈Eff(C)), we have X ∈ Eff(C) iff Y ∈ Effnc(C) for some Y ⊆ X , therefore (2) isequivalent to: X ∈ Eff(C) iff there exists (Yi )i∈C such that for all i,Yi ∈ Effnc({i})and X ⊇ ∩i∈C Yi , which, again because Eff(C) is upward closed, is equivalentto: X ∈ Eff(C) iff there exists (Yi )i∈C such that for all i,Yi ∈ Eff({i}) andX = ∩i∈C Yi , which is (D′).

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2. In the presence of decomposability and Eff(N ) = 2S \{∅}, atomicity is equivalentto (3) and (4).Assume atomicity holds. Recall that At (i) and E f f nc(i) coincide, therefore thefact that any two elements of At (i) are disjoint implies (3). Now, let X ∈ At ( j)and x ∈ X . Then x ∈ S; since At (i) is a partition of S, there exists Y ∈ At ( j)such that x ∈ Y , which shows (4).Conversely, assume (3) and (4) hold, as well as decomposability and Eff(N ) =2S\{∅}. Let s ∈ S; then {s} ∈ Eff(N ). By decomposability applied to (N \{i}, {i}),there is Y ∈ Eff(i) and Z ∈ Eff(N \ {i} such that Y ∩ Z = {s}. Therefore,s ∈ ∪{X |X ∈ At (i)}. Lastly, (3) implies that any two elements of At (i) aredisjoint, which shows that At (i) is a partition of S.

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