Institutional Design of Cooperation: Incentive and ...jiac/pdfs/incent_screen.pdf · mechanisms:...
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Institutional Design of Cooperation: Incentiveand Screening under Uncertainty
Jia Chen∗
Department of Political ScienceUniversity of Colorado, Boulder
Abstract
Most of the existing theories of international cooperation implicitlyadopted the assumption that defection or cheating are easily detectable,either because the behavior of other actors is directly observable, or thepayoffs received by the relevant actors are perfect indicators of past be-havior. These assumptions are not tenable in many contexts of interna-tional cooperation where the observability of behavior is low and payoffsare volatile. This paper is a theoretical examination of how such “objec-tive uncertainty” interacts with the strategic incentives of actors in in-ternational cooperation. The model developed in the paper shows therandomness of payoffs have a major impact on the cooperative behaviorof actors. Actors adopt very different strategies given different structuresof payoff uncertainty. In particular, the presence of observable behaviorand payoff uncertainty induce a bifurcation in cooperation objectives andhence strategies. In such a context, the objectives of inducing cooperativebehavior from the opponent, which begets moderate strategies of selection,is now incompatible with the objectives of screening different types of theopponent which begets a more unusual and counterintutive strategies.
Preliminary draft. Please do not cite without permission.
∗333 UCB, Boulder, CO 80309. E-mail: [email protected]
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1 Introduction
Information problems constitute one of the most salient types of obstacle to inter-
national cooperation. Actors involved in cooperation oftentimes maintain private
information regarding their preferences, capabilities, and behavior, all of which
could make cooperation difficult to achieve by spawning opportunistic tendencies
in strategy and behavior. While information asymmetries on the characterisics
of actors in cooperation have been thoroughtly studied in the literature, few has
examined the impact of “objective uncertainty” on the structure and outcome of
cooperation in the rantionalist framework. Objective uncertainty could be con-
ceptualized as the volatility and randomness in the “natural” invironment that
does not have behavioral or strategic origins but neverless affect the outcomes of
behavioral interactions in some stochastic manner. The presence of objective un-
certainty in international cooperation breaks the deterministic linkages between
profiles of behavior and the ensuing outcomes. The action with benevolent in-
tentions may not always return welcoming outcomes and ill-intentioned behavior
could occasionally lead to good results. Such phenomenon is prevalent in inter-
national econonimc cooperation. For example, the performance of a country in
international trade is shaped by the policy of its trade partners as well as other
stochastic factors in the complex systems of global economy. A bad performance
in the country’s exports could not always be blamed on protectionism, as sys-
temic factors could lead to slumps in trade even when good policies are in place.
The problem of objective uncertainty is further complicated by the unobserv-
ability of behavior in cooperation. If the actors are able to perfectly observe each
other’s past behavior, the stochastic factors in the environment would only inflate
the volatility in the realized payoffs instead of substantially altering the strategic
incentives in cooperation. When there is no other way of obtaining authentic
information regarding the opponent’s past behavior, strategic actors will handle
the realized payoffs under objective uncertainty with greater attention and care,
as these realized outcomes are complex compounds of strategic incentives of the
opponents intertwined with stochastic aspects of the environment. Returning to
the example of trade liberalization, the low transparency of many trade policy
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tools has made it difficult to assoicate international trade performance with the
policy behavior of the trade partner. Imagine a country, who is not sure if its
trade partner is also a believer of free trade, realizes the recent slump in exports
could be the innocuous consequence of random shock in the international trade
system, or the aftermath of a new protectionist but non-tarrif measure covered
up by policies indirectly related to trade. Given the complexity in the signals car-
ried in the realized outcome, how should this country respond to the trade loss?
Problems as such are very common not just in the area of international trade but
also in international financial regulation and global environmental governance.
The rest of this paper seeks to develop a theoretical approach to understand
how actors strategize their behavior in an environment featuring randomly dis-
tributed payoffs. In devising strategic responses to realized outcome in such
a context of objective uncertainty, actors will be particularly careful with two
mechanisms: incentivizing and screening. The opponent could have multiple
types: some find mutual cooperation most desirable whileas some are prone to
take advantage of cheating in a noisy informational environment. The unin-
formed actors is able to strategize their response in a way to induce maximal
cooperation from the “opportunistic” type of opponent. The key of this type
of strategies is incentivizing the “opportunistic” type to pool his behavior along
with the “cooperative” type such that the instantaneous reward of cooperation
for the uninformed actor is maximized. This strategy is yet incompatible with
the intrinsic information need of the uninformed actors as it creats minimal in-
centive for the opponent with different types to self-separate from each other.
On the other hand, the uninformed actor could use strategies as a screening tool
to reveal the true type of the opponent by prompting the “opportunistic” type
actor to cheat. Contrary to the “incentivizing” mechanism, this strategy would
bring long term informational gains at the cost of the instantaneous payoffs of
cooperation. While some of the existing studies have examined the differentials in
incentivizing and screening in international cooperation, this paper is the first to
characterize the intrinsic linkage between these mechanisms and the phenemenon
of randomly distributed payoffs and objective uncertainty.
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The theoretical findings in this paper yield implications for a number prob-
lems in the study of international cooperation. Most importantly, it reveals the
strategic nature of compliance systems in international cooperation. As Chayes
and Chayes (1993) and Downs, Rocke and Barsoom (1996) suggested, the stan-
dard and definition of compliance could be subjective. This paper further shows
the compliance systems in international institutions is strategically structured in
the context of objective uncertainty. Depending on the structure of responses
to non-compliance in the compliance system, complying with existing rules and
laws may be profitable for players who are not cooperative in nature. For those
who design and administer the compliance system, the system reflect distinct
calculations of gains through structuring interactions of the strategic incentives
in a noisy informational environment.
2 Literature Review and Theoretical Framework
Information problem in various contexts remains a central strategic issue in in-
ternational cooperation. Early scholarship on the topic called attentions to the
importance of the information and monitoring regime supporting cooperation ef-
fort. Many of them argued that transparent and well-functioning information
exchange systems enable states to better understand each other’s interests and
preferences, making cooperation less vulnerable to opportunistic and strategic
motivations. Scholars also emphasize the features of different issue areas as key
determinants of the information regime adopted in institutionalized cooperation
(Dai, 2002; Mitchell, 1998). In particular, uncertainty in the objective environ-
ment is regarded as a chief factor affecting the prospect of cooperation and com-
pliance. For example, some of them argue that institutionalization of cooperation
is intrinsically difficult because ambiguity in both bargaining and enforcement of
cooperation is impossible to be eliminated, making cooperation difficult to be
monitored.
As Chayes and Chayes (1993) suggested, non-compliance is frequently identi-
fied in cooperation not because the participating states deliberately violate treaty
stipulations driven by strategic incentives, but because the objective uncertainty
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and unpredictability of factors in the complex system of cooperation made the
convergence of expectations difficult. Most relevantly, they emphasize the impact
of the uncertainty about state’s capability in fulfilling obligations stipulated in
the agreement on compliance. It is illustrated in their discussion that when en-
gaged in the bargaining phase of cooperation, states have only limited knowledge
on the randomness in the objective environment in fulfilling the cooperation ar-
rangements and the national representatives can be considerably ignorant of the
domestic or international consequences of policy change required by cooperation
arrangements. Also, in the long run the path of the evolution of the structure
of state preference and capability is difficult to be predicted. State’s capacity
and willingness to fulfill obligations stipulated in treaty provisions in the long
term remain a stochastic factor that could potentially jeopardize international
cooperation.
While Chayes and Chayes (1993) pointed out the existence of uncertainties
in the objective environment that are not subject to strategic manipulations but
have impact on the process and outcome of international cooperation, they do
not examine how such uncertainty reshape strategic incentives of actors partic-
ipating in cooperation. As a reponse to Chayes and Chayes (1993)’s argument,
Downs and Rocke (1995) more explicitly consider the randomness in the objective
environment and its impact on the strategic interactions in cooperation. They
specifically address how the rational actors design strategies in a way to offset
the impact of the stochastic shocks on utility realization. They pointed out that
objective uncertainty existing at the domestic level results in more lenient pun-
ishment strategy following defections. The main implication, therefore, is that
uncertainties compromise the prospect of cooperation as the compliance system
has to be relaxed accommodate the informational environment. Taking on a
very similar problem regading domestic level uncertainty, Rosendorff and Milner
(2001) suggested in their model that instead of letting the objective uncertainty
reduce the durability of cooperation, international institutions could incorporate
escape clause in the compliance system to allow members adjust their policies in
times of domestic political difficulty. They shows this institutional component
incorporating escape clause actually enables state to reach agreement faster than
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otherwise. A similar argument is seen in Carrubba (2005) in an examination of
the role of European Court of Justice in response to stochastic shocks at the do-
mestic level that affect the capability of member states to comply with EU law.
Kucik and Reinhardt (2008) also find support for the cooperation promotion ef-
fect of flexible compliance system of GATT/WTO. Koremenos (2005) further
considered the cost associated with flexible institutions in the presence of uncer-
tainty and explore its impact on institutional design of international cooperation.
One missing mechanism in the theoretical framework in the studies cited
above, however, is how the uncertainty in the objective environment could lead
to a bifurcation of the strategic responses of rational actors. A key concern is the
tradeoff between long turn informational gains and instant reward from present
behavior. In an environment filled with random stochastic factors affecting real-
izations of payoffs from cooperation, the actors now face more complex calculus
regarding the optimal strategy to implement in an scenarios where the key mech-
anisms in the system are probabilistic instead of deterministic. While strategies
could be designed to tackle the noisy signals carried in the realized payoffs such
that the informational gains in the long run is maximized, actors may also be
interested in just optimize the strategy in the current period if the discount factor
is very big. In such a scenario, the problem becomes how the structure and mag-
nitude of uncertainty shapes the actors’ strategies in maximizing the cooperation
gain in the current period or the informational gains in the future.
Given this implication generated with regard to screening and incentivization
in international cooperation, the paper is also related to recent debate regarding
the role of international institutions in cooperation as seen in Simmons (2000a)
and Von Stein (2005). The key question been controverted, which is closedly
related with former discussion, is whether international institutions such as the
International Monetary Fund have imposed substantial constraints to change the
incentive of participating states. The findings in Simmons (2000a) support the
incentive-changing effect of IMF whereas Von Stein (2005) disputes the conclu-
sion by suggesting the institutional constraints imposed by IMF only induce states
with different incentives to self-select into accepting different arrangements. An
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important questions derived from this debate regarding screening function of in-
stitutions is how to design mechanisms that deter defection without deterring
participation. While a rigorous compliance system with stringent and binding
commitments are oftentimes appreciated as the key in promoting cooperation,
the potential negative effect of too stringent cooperation arrangement on cooper-
ation should be recognized, particularly in an complex informational environment.
One of the studies closely related to this theme is the study of the pattern of rat-
ification of the global environmental legislation by Von Stein (2008). Von Stein
adopted a similar theoretical framework to that of the aforementioned works on
the “flexibility theory” such Rosendorff (2005) and Kucik and Reinhardt (2008)
in the emphasis of the cooperation promotion effect of the incorporation of a
flexible complicance system. Her main contribution, however, is a more explicit
characterization of a tradeoff between the incentive changing function and the
screening function of international institutions in her theoretical framework. She
particularly mentioned that stringent treaty provision and compliance system is
a double-edged sword in that “when governments are likely to be held to their
international legal commitments, they will particularly concerned, when consid-
ering ratification, about their subsequent ability to comply”. A key missing part
of the argument in Von Stein (2008), however, is the strategic incentive under-
lying the observed pattern of compliance and more fundamentally the design of
compliance system of the respective institution.
Another key aspect of flexible compliance systems is the moral hazard asso-
ciated with escape clause with safeguard measure. While institutional flexibility
allows participating members to adjust policy without breaking the rules, it is
possible that strategic incentives will motivate states to lie about the domestic
imperatives that justify a shift away from complying with the rules. Bagwell
and Staiger (2005) examined the strategic incentive underlying the design of es-
cape clause of GATT/WTO when the cost of compliance is private information.
They explicitly derived the conditions under which flexible compliance system
could promote cooperation instead of harming it. Taking the problem to a more
politically structured contect, Svolik (2006) explored the problem of lying in
cooperation institutions with a flexible compliance system in a contrast between
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democratic and autocratic regimes. Recognizing that states have incentive to mis-
represent the domestic political circumstance to take advantage of the flexibility
in the compliance system, Svolik specifically examines the role that democratic
institutions plays in counteracting the negative incentive of lying in a flexible
compliance system.
This paper extends this problem of flexibility vs rigidity tradeoff theoretically
by looking at one specific factor, objective uncertainty, that leads to bifurcating
incentives in cooperation underlying different institutional design of compliance
system. Some of the institutions for cooperation have greater effect of altering
the incentives of actors whereas others institutions are designed deliberately to
bar non-cooperative or opportunistic actors from joining the party. In the bigger
context of objective uncertainty, institutions fulfilling different functions incor-
porate different systems of compliance correspondingly. The model presented in
the following section characterize the objectives of cooperation in this specific set-
ting of informational environment and seeks to unpack the structure of strategic
incentives underlying different designs of compliance system.
3 A Simple Model of Cooperation
I start the analysis with a simple model of collaboration with no systemic un-
certainty which provide a benchmark for contrasting the incentives and strategic
outcomes of cooperation under different informational environment. To give a
brief overview, the benchmark model represents a scenario that combines the
strategic elements of the Prisoner’s Dilemma and Stag Hunt. The uninformed
actor is cooperative in nature in that he is willing to reciprocate cooperation if
the other party is expected to cooperate as well. The other actor in the game
has two types. His preference could be identical to the uninformed player. He
could also be “opportunistic” who has the preference ordering as the players in
the Prisoner’s Dilemma. The game is set to consists of two periods: in the first
period both actors take actions simultaneously. The second period only allow the
uninformed player to take action, which is to infer the true type of the opponent
based on the realized payoffs from the first period of the game. The actor with
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private information, regardless of his type, always prefer to be identify as the
“cooperative” type. The uninformed player, understanding this incentive, faces
a strategic decision: should the strategy in period prioritize incentivizing coop-
eration or enhancing the accuracy in the inference of the type of the opponent.
The analysis in this part is centered around this issue with details explicated as
follows.
3.1 Basic Settings
There are two actors in the model, denoted 1 and 2. The one-shot game model
has two periods of play. The first peroid is a 2× 2 simultaneous move game with
the following payoff structure.
Player 2Cooperate Defect
Player 1Cooperate r, r s, v2
Defect v1, s p, p
Table 3.1: Period One Payoffs
The payoffs to the players in the first period can be denoted ut=1i (). The pref-
erence orderings are as follows: s < p < v1 < r. The defection payoff to player
2 given player 1 cooperated follows a Bernoulli distribution B(α). v2 ∈ {v, v̄},and Pr(v2 = v) = α, Pr(v2 = v̄) = 1 − α. Let p < v < r < v̄. Obviouslly the
incentive of the players is structured by the realized value of v2. If v2 = v̄, the
game is one-sided prisoner’s dilemma whereas the game is coordination if v2 = v.
v2 = v v2 = v̄
Player 1B bv1, b2 bv̄1, b2
E ev1, e2 ev̄1, e2
Table 3.2: Period Two Payoffs
The second period of the game only envolves the behavior of the uninformed
player, player 1. He is trying to identify the real type of player 2 given the payoffs
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from the first period of the game. Player 1 will obtain higher payoff at the end
the period if he correctly infers the type of player 2. Both types of player 2 prefers
to be identified as the cooperative type with v2 = v. The payoffs to the players
in the second period can be denoted ut=2i ().1 The ordering of the payoffs are as
follows: e2 > b2, ev1 > bv1, ev̄1 < bv̄1.2 The total payoff from the two periods of play
is given by
Vi(·) = ut=1i + δ · ut=2
i
where δ is the common discount factor for all players. Multiple equilibria exist
in this game. The following names a few.
3.1.1 Total Cooperation
One cooperative equilibrium could exists where player 1 and both types of player
2 cooperate, player 1 choose “E” if and only if payoff r is received at the end
of period one. This equilibrium hinges on the willingness of the non-cooperative
type player 2 to cooperate in the first period, which formally provides that
r + δe2 > v̄ + δb2 (3.1)
⇔ δ >v̄ − re2 − b2
≡ δk (3.2)
Conceptuallizing the payoff for player 2 in the second period as the future re-
ward for reputation, this condition indicates the long horizon of the future benefit
from being identified as cooperative could shift the behavior of non-cooperative
type player.
3.1.2 Partial Cooperation
The other equilibrium is that where only player 1 and cooperative-type player 2
cooperate, and player 1 plays “E” if and only if r is received at the end of period
one. Obviously, δ < δk such that non-cooperative player 2 would not want to
1This can be thought as a scenario in international cooperation where the uninformed playeris deciding on a long term strategy towards his partner depending on the types.
2Substantively, it means that if player 2 is believed to be cooperative type, a favorabletreatment will be granted which benefits both types of player 2 but only benefit player 1 if theinference is correct.
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cooperate. Furthermore, it must be verified that player 1 still find it preferable
to cooperate in period one even though only the cooperative type player 2 is
expected to do the same. Formally this requires
rα + s(1− α) > v1α + p(1− α) (3.3)
⇔ α >p− s
p− s+ r − v1
≡ αk (3.4)
Thus the probability that player 2 is cooperative type must be high enough to
outweigh the loss from sucking upon the non-cooperative type player 2.
3.1.3 Rewarded Forebearance
Another equilibrium related to the former is one where player 1 and non-cooperative
type player 2 defect and cooperative type player 2 cooperate. Player 1 plays “E”
if and only if r is received at the end of period one. This happens first due to
that α < αk so cooperation is not good for player 1. Secondly the cooperative
type player 2 find it favarable to cooperate despite the defection from player 1 be-
cause of the future benefit from being recognized as cooperative type. Formally,
forebearance works if
s+ δe2 > p+ δb2 (3.5)
⇔ δ >p− se2 − b2
≡ δf (3.6)
Together with the condition for non-cooperation from non-cooperative type player
2, δ has to be bound in the interval (δf , δk), which also requires δf < δk or
v̄ − r < p − s. It is easy to see that an equilibrium of total defection exists if
δ ≤ min{δf , δk} and α < αk.
3.1.4 Randomized Mimicing
One obvious semi-pooling equilibrium with mixed strategy exists where player 1
and the cooperative type player 2 cooperate whereas the non-cooperative type
player 2 cooperate with probability q, and player 1 always chooses “B” if s is the
payoff from period one and chooses “E” with probability η if r is the payoff from
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period one.
This equilibrium obviously involves Bayesian updating. Player 1 evaluates
the strategy in period two based on the posterior belief about the type of player
2. Given the strategy combination, the posterior belief is
Pr(v̄|r) =α
α + q(1− α)(3.7)
The expected utilities for player 1 in period two given r are
E[u1(E|r)] = Pr(v̄|r)ev1 + [1− Pr(v̄|r)]ev̄1 (3.8)
E[u1(B|r)] = Pr(v̄|r)bv1 + [1− Pr(v̄|r)]bv̄1 (3.9)
Setting u1(E|r) = u1(B|r) and plugging in Pr(v̄|r) from (3.7), q = qm that
makes player 1 indifferent in period two strategy can be obtained.
αev1 + q(1− α)ev̄1 = αbv1 + q(1− α)bv̄1 (3.10)
⇔ q =α
1− α· e
v1 − b
v1
bv̄1 − ev̄1≡ qm (3.11)
If the surplus from correct identification of player 2’s type is the same, i.e.
bv̄1 − ev̄1 = ev1 − bv1, then the proportion of cooperative type player 2 (α) must
not be greater than 50% in order to support player 1’s randomization.
Now to make mixing strategies rational for non-cooperative player 2, η should
make E[uv̄2(Cooperate) = E[uv̄2(Defect)], which provides
r + δ[ηe2 + (1− η)b2] = v̄ + δb2 (3.12)
⇔ η =v̄ − r
δ(e2 − b2)≡ ηm (3.13)
It also needs to be verified in the equilibrium that player 1 and cooperative
type player 2 find it profitable to cooperate in the first period given qm and ηm.
That is
E[u1(Cooperate|qm)] > E[u1(Defect|qm)]
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and
E[uv2(Cooperate|ηm)] > E[uv2(Defect|ηm)]
One may question the effectiveness of the strategy of the non-cooperative
player 2 in this semi-pooling equilibrium: since the cooperative player 2 never
defects, why cannot player 1 tell the player is non-cooperative type upon observ-
ing defection if the randomization in one period comes to play defection? The
problem can be rounded with the idea of purification of Bayesian equilibrium un-
der payoff uncertainty elaborated in Harsanyi (1993). The randomized strategy is
simply the convergence of the players’ response in the Bayesian equilibrium when
the payoffs are randomly distributed. This equilibrium basically captures the
impact of payoff uncertainty on the player’s behavior in the general framework
of the game.
4 A Model of Cooperation with Payoff Uncer-
tainty
The model in this section introduces another layer of uncertainty into the anal-
ysis. Keeping the assumption that behaviors are non-observable, let me now
assume that the payoffs to the players in the first period of the model are ran-
domly distributed. For example the payoff to both players when both cooperated,
is now r · σθ. σ could be any positive real number. and θ is a random variable
following any symmetric and unimodal distribution with continuous and differ-
entiable distribution function F (·) bounded between −1 and 1. Intuitively while
the presence of the random variable θ introduce uncertainty to the payoffs in a
standard way, σ indicates the variability of the random payoffs. The payoffs in
the first period is represented in the following table.
The payoff is of uncertainty structured in this way as long as there is at least
one player cooperated. If both player defected, the payoff is doomed to be un-
desirable. Even though the result of the model is actually insensitive to this
specification, such assumptions are reasonable to impose in the practical context
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Player 2Cooperate Defect
Player 1Cooperate r + σθ, r + σθ s+ σθ, v2 + σθ
Defect v1 + σθ, s+ σθ p, p
Table 4.1: Randomly Distributed Payoffs
of the study. This specification is particulaly relevant in the context of interna-
tional cooperation where mutual defection is destructive for sure while the payoff
is more uncertain and subject to greater variablility when at least there is some
cooperation in the interaction. Particularly, if only one of the players cooperated,
he could still harvest some of the reward from his only cooperation behavior by
chance even such cooperation is not reciprocated. And it could be difficulty to
tell the gain from cooperation is coming from mutual cooperation or not when
the behavior of the other party is unobservable. The payoffs from the second
period of the game where the uninformed player made inference about the true
type of the other party remain identity to the earlier specification.
Since now the payoffs are randomly distributed in the extended model, anal-
ysis of the game starts with the uninformed player, Player 1, whose decision in
the second period should be one of Bayesian optimality. The only observable
outcome that player 1 could use to structure his response is the realization of
the randomly distributed payoffs from period one. Using x to denote the payoff
receive in period one, the updating of player 1’s belief regarding the type of player
2 is thus characterizes in the following expanded Bayes’s formula:
Pr(v2 = v̄|x) =Pr(v2 = v̄) · Pr(x|v2 = v̄)
Pr(v2 = v̄) · Pr(x|v2 = v̄) + Pr(v2 = v̄) · Pr(x|v2 = v)(4.1)
where
Pr(x|v2) = f( xrσ
)· Pr(Cooperate|v2) + f
( xsσ
)· Pr(Defect|v2). (4.2)
f( xrσ
) is the density function of θ and is used to approximate Pr(rσθ = x).
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Player 1 thus uses this updated belief after observing x to make inference regard-
ing the type of player 2. The behavior of player 2 is most likely to be shaped by
how the inference will be made by player 1 regarding his type. I let player 1 to
implement a cut-point strategy where E will be played iff the observed payoff is
greater than or equal to certain value, i.e. x ≥ x̂.
4.1 Strategic Response under Uncertainty
The “cooperative” type Player 2’s response is straightforward that he always
reciprocate cooperation as it provides strictly better payoff than defection even
after payoff uncertainty is introduced into the story.
x′′x′ s+r2
x̂
F ( x̂sσ
)
F ( x̂rσ
)
Figure 4.1: F (x̂/sσ) and F (x̂/rσ)
Given the cut-point strategy, x̂, the expected payoff from cooperating and
defecting for player 2 are given by:
E[u2(Cooperate)] = r + δ
{[1− F
(x̂
rσ
)]· e2 + F
(x̂
rσ
)· b2
}(4.3)
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and
E[u2(Defect|v2)] = v2 + δ
{[1− F
(x̂
sσ
)]· e2 + F
(x̂
sσ
)· b2
}. (4.4)
If player 1 plays cooperate in the first period, what would the strategic type
player 2 do? To address this question, the following inequality is obtained.
E[u2(Cooperate)] ≥ E[u2(Defect|v2 = v̄)] (4.5)
⇔ F (x̂
sσ)− F (
x̂
rσ) ≥ v̄ − r
δ(e2 − b2)(4.6)
Define ϕ(x̂, σ) ≡ F ( x̂sσ
) − F ( x̂rσ
) and k(δ) ≡ v̄−rδ(e2−b2)
. It can be verified that
under the assumption that F (·) is continous and strictly unimodal, ϕ(x̂, σ) is
also continuous and unimodal in x. There exists xE ≡ arg maxx ϕ(x|σ). And
xE = (s+ r)/2. Intuitively, the cut-point x̂ = xE induces the maximum incentive
to cooperate from Player 2.
Lemma 4.1. Under the cut-point strategy, the incentive of the “strategic” type
player 2 to reciprocate cooperation in period one, indicated by ϕ(x̂, σ), reaches the
maximum under the cut-point x̂ = xE ≡ (s + r)/2. The incentive to cooperate,
ϕ(x, σ), decreases in |x̂− xE|.
The other feature of ϕ(x̂, σ) is regarding σ. It is found that ϕ(x̂, σ) increases in
σ if only if the cut-point x̂ is more around xE which is considered the “reasonable”
range of x̂. ϕ(x̂, σ) decreases in σ if x̂ is more on the far left or right tail of ϕ(x̂|σ).
Proposition 4.1. Greater variance of the random payoff reduces the incentive
of the “strategic” type player to cooperate in the first period if the cut-point x̂
is around the “reasonable” range. Formally, let xL(σ′, σ′′) and xR(σ′, σ′′) be the
value of x̂ such that ϕ(x̂, σ′) = ϕ(x̂, σ′′).ϕσ(x̂, σ) < 0 if xL < x < xR
ϕσ(x̂, σ) ≥ 0 otherwise
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xRxL s+r2
x̂
k(δ′)
k(δ′′)
ϕ(x̂, σ = 1)
ϕ(x̂, σ = 1.5)
Figure 4.2: ϕ(x̂, σ) and k(δ)
Intuitively, if Player 1 sets x̂ = xE to induce maximum cooperation from
the strategic opponent, greater variance of the random payoff, σ, will muffle the
cooperation incentive. But if for some reason inducing cooperation is not the
sole purpose of the cut-point strategy and Player 1 sets x̂ more on the tails of
ϕ(), meaning Player 1 is extremely tolerant or stringent, greater variance of the
random payoff actually strengthens the strategic type player 2’s willingness to
cooperate. Particular, if the present value of being recognized as “cooperative”
type is high, which is indicated by a small RHS of the inequality in 4.6, greater
σ can result in cooperation from the “strategic” type Player 2 who would defect
otherwise.
Corollary 4.1. Greater variance of the random payoffs boosts the “stragetic” type
Player 2’s incentive to cooperate in period one if 1) the future gain from mimick-
ing is high and 2) Player 1 sets a extreme cut-point x̂ either on the “tolerant” or
the “stringent” side.
Corollary 4.2. Greater variance of the random payoffs reduces the “stragetic”
type Player 2’s incentive to cooperate in period one if 1) the future gain from
mimicking is low and 2) Player 1 sets the x̂ around the “reasonable” cut-point
xE = (s+ r)/2.
17
4.2 Baysian Updating based on Randomly Distributed
Payoffs
Give Player 2’s response to the cut-point strategy, Player 1 can update his belief
about the type of Player 2 using the equation formulated in (4.1). For example,
the Bayesian updating by Player 1 given that the realized payoff is smaller than
the cut-point x̂ is characterized by the following formula:
Pr(v̄|x ≤ x̂) =Pr(v̄) · Pr(x ≤ x̂|v̄)
Pr(v̄) · Pr(x ≤ x̂|v̄) + Pr(v̄) · Pr(x ≤ x̂|v)(4.7)
where
Pr(x ≤ x̂|v̄) = F
(x̂
rσ
)· Pr(Cooperate|v̄) + F
(x̂
sσ
)· Pr(Defect|v̄). (4.8)
To make the process of updating more comprehensible, I now discuss a couple
of scenarios with the application of the Bayes’s formula. First suppose that only
the cooperative-type of Player 2 prefers to cooperate given cut-point x̂. The
posterior belief that Player 2 is strategic given the realized payoff to Player 1 is
smaller than x̂:
Pr(v̄|x ≤ x̂) =Pr(v̄) · F (x̂/sσ)
Pr(v̄) · F (x̂/sσ) + Pr(v) · F (x̂/rσ))≡ πD2|x̂(v̄|x ≤ x̂) (4.9)
πD2|x̂(v̄|x ≤ x̂) denotes the posterior probability that Player 2 is strategic
type given 1) the strategic type Player 2 is expected to defect under x̂ and 2) the
observed payoff is smaller than x̂. It can be shown that πD2|x̂(v̄|x ≤ x̂) > Pr(v̄)
given F (x̂/sσ) > F (x̂/rσ). It can also be verified following the same procedure
that πD2|x̂(v̄|x > x̂) < Pr(v̄).
In the second scenario where both types of Player 2 cooperate, the belief on v2
cannot be updated. If the strategic-type Player 2 prefers to cooperate given the
cutpoint x̂, the posterior belief would be identical to the prior belief regardless
18
of the realized payoff to Player 1. That is,
πC2|x̂(v̄|x > x̂) = πC2|x̂(v̄|x ≤ x̂) = Pr(v = v̄). (4.10)
As has been shown earlier, Player 1 has the ability to shape the behavior of
the strategic-type Player 2 through setting the cut-point x̂ at various values. If
inducing cooperation from the strategic type player is the only objective, player 1
should set x̂ = (s+r)/2. The next subsection explores the optimal x̂ to maximize
the accuracy of the posterior inference on Player 2’s types.
4.3 The Optimal Cut-point x̂ for Posterior Inference
Player 1 can manipulate the cut-point x̂ to affect the response from the strategic
type Player 2, which in turn shapes the updating of the prior belief through equa-
tion (4.7). Based on Lemma 4.1 and 4.1, there could exist xl and xr (xl < xr)
such that the strategic type Player 2 prefers to defect in period one if x̂ < xl or
x̂ > xr. Thus Player 1 can determine if there is pooling or separating behavior
among the two types of Player 2 in equilibrium.
As is being shown below, if making an accurate posterior inference is the
sole objective, Player 1 always prefer inducing a separating equilibrium which
provides a strictly higher expected payoff in the second period of the game. I list
the expected utilty when pooling or separating dominate the equilibrium. Since
pooling makes updating irrelevant, the expected utility of infering Player 2 as
strategic given the realized payoff smaller than the cut-point x̂ is provided by:
E[uC2|x̂1 (B|x ≤ x̂)] = Bv
1 + (Bv̄1 −B
v1) · Pr(v2 = v̄) (4.11)
The expected utility where separating dominates the equilibrium is
E[uD2|x̂1 (B|x ≤ x̂)] = Bv
1 + (Bv̄1 −B
v1) · πD2|x̂(v̄|x < x̂) (4.12)
Given Bv̄1 > Bv
1 and πD2|x̂(v̄|x ≤ x̂) > Pr(v̄), E[uD2|x̂1 (B|x ≤ x̂)] is strictly
19
greater than E[uC2|x̂1 (B|x ≤ x̂)]. Simularly, E[u
D2|x̂1 (E|x > x̂)] is strictly greater
than E[uC2|x̂1 (E|x > x̂)]. To see this, check the expected payoffs listed below.
E[uC2|x̂1 (E|x > x̂)] = Ev
1 − (Ev1 − E v̄
1 ) · Pr(v2 = v̄) (4.13)
E[uD2|x̂1 (E|x > x̂)] = Ev
1 − (Ev1 − E v̄
1 ) · πD2|x̂(v̄|x > x̂) (4.14)
Given Ev1 > E v̄
1 and πD2|x̂(v̄|x > x̂) < Pr(v2 = v̄), it is easily seen that
E[uD2|x̂1 (E|x > x̂)] is greater than E[u
C2|x̂1 (E|x > x̂)]. Along with the results
established earlier, these inequalities establish Proposition 4.2.
Proposition 4.2. (Optimal Posterior Inference) Given that Player 1 has the
ability to determine the cut-point x̂, Player 1 always prefers setting the x̂ regard-
less of the realized payoff such that a seperating profile of behavior is induced in
the first period of the game where only the “cooperative” type Player 2 cooperates,
i.e.
E[uD2|x̂1 (B|x)] > E[u
C2|x̂1 (B|x)],∀x ∈ X (4.15)
Together with Lemma 4.1, Proposition 4.2 derives the following corollary re-
garding the cut-points that maximizes the accuracy of the posterior inference.
Corollary 4.3. (Optimal Cut-points for Posterior Inference) As the in-
centive of mimicking grows, indicated by k(δ), more extreme cut-points either on
the “tolerant” or “stringent” side are required to induce a separating profile of
behavior.
5 Implications and Concluding Remarks
The paper studies how uncertainty in objective environment, state’s strategic
motivation to conceal policy from being directly observed, and the enforceabil-
ity of institutionalized agreement interact with one another in determining the
prospect of cooperation and compliance. In the model I construct it has been
20
δS(σ)δP (0)
δP (α)
Shadow of the Future
Screening
Cooperation
Mim
ickin
g(A
bor
ted
Scr
een
ing)
α
Pri
orP
r.of
Coo
pera
tive
Pla
yer
Figure 4.3: Equilibrium Space in δ and α
shown that, contrary to the claims made in previous studies that prospect of co-
operation hinges on the level of uncertainty in objective environment, states can
be very proactive in designing strategies that manipulate “strategic uncertainty”
in cooperation to maximize the benefit. Also, counter-intuitively, low level of
uncertainty in objective environment does not necessarily make cooperation and
compliance more likely, high level of uncertainty in objective environment does
not necessarily make cooperation less likely. Instead, the comparative statics
shows that the complex interaction between uncertainty in objective environ-
ment and actor’s strategic motivation determines the prospect of cooperation.
Furthermore, in the presence of third party adjudication as institutional guar-
antee of cooperation, enforceability of treaty as well as judiciary power of the
adjudication can be only substitutes for universal transparency of state policy
could in maintaining equilibrium behavior on the cooperation and compliance
path. Treaty enforceability, as shown in the model, may not a necessary condi-
tion for cooperation and compliance.
There are a couple of implications of these conjectures generated. Firstly, the
conjectures generated above provide some explanation for the observation that
21
in some specific areas of cooperation featured by high objective uncertainty in
the environment states are willing to make policy voluntarily more transparent to
other cooperating states. If we can observe in reality that despite the high level of
uncertainty in objective environment, states actually enhance policy transparency
to each other and maintain cooperation and compliance, then such observation
is likely to substantiate the conjecture that state will choose cooperation and
maintain high level of transparency even when the level of objective uncertainty
is very high. And conversely, in the circumstance where the objective uncertainty
is low, state will still have the incentive to manipulated strategic uncertainty to
block outsiders from directly observing the policy choices made. Such strategic
uncertainty will unfortunately lead to greater risk of collapse of cooperation due
to actor’s worry about the potential vulnerability resulted from zero environment
uncertainty and more importantly other actor’s strong opportunistic motivation
unleashed by low uncertainty in objective environment. Also the logic underlying
the finding that transparency to international adjudication body is substitute for
universal transparency to state actors can also be found in cases in reality. For
example, Mitchell (Mitchell 1998) discusses the fact that reporting rate under hu-
man right treaties is generally high. Drawing the insight from the second model,
this regularity can be explained that because the level of policy transparency is
low (partly because the cost enhancing policy transparency is too high), coop-
erating states will voluntarily entitle the international adjudication substantial
access to the situation of domestic policy implementation. State is willing to do
so because that is the one way in which other potential cooperating states can
be assured of the intention and willingness to cooperate. Empirical analysis in
greater depth is to be carried out to further elaborate the mechanism underlying
the regularity with the insights from the conjectures.
22
References
Alt, James E., Randall L. Calvert and Brian D. Humes. 1988. “Reputation andHegemonic Stability: A Game-theoretic Analysis.” American Political ScienceReview 82(02):445–466.
Bagwell, Kyle and Robert W Staiger. 2005. “Enforcement, Private PoliticalPressure, and the General Agreement on Tariffs and Trade/World Trade Or-ganization Escape Clause.” The Journal of Legal Studies 34(2):471–513.
Carrubba, Clifford J. 2005. “Courts and Compliance in International RegulatoryRegimes.” Journal of Politics 67(3):669–689.
Chayes, Abram and Antonia Handler Chayes. 1993. “On Compliance.” Interna-tional Organization 47(02):175–205.
Dai, Xinyuan. 2002. “Information Systems in Treaty Regimes.” World Politics54(04):405–436.
Dai, Xinyuan. 2005. “Why Comply? The Domestic Constituency Mechanism.”International Organization 59(02):363–398.
Downs, George W. and David M. Rocke. 1995. Optimal Imperfection?: DomesticUncertainty and Institutions in International Relations. Princeton UniversityPress.
Downs, George W., David M. Rocke and Peter N. Barsoom. 1996. “Is the GoodNews about Compliance Good News about Cooperation?” International Or-ganization 50(03):379–406.
Fearon, James D. 1995. “Rationalist Explanations for War.” International Orga-nization 49(03):379–414.
Fearon, James D. 1998. “Bargaining, Enforcement, and International Coopera-tion.” International Organization 52(02):269–305.
Guzman, Andrew T. 2008. How International Law Works: A Rational ChoiceTheory. Oxford University Press.
Koremenos, Barbara. 2005. “Contracting around International Uncertainty.”American Political Science Review null:549–565.
Kucik, Jeffrey and Eric Reinhardt. 2008. “Does Flexibility Promote Coopera-tion? An Application to the Global Trade Regime.” International Organization62(03):477–505.
23
Meirowitz, Adam and Anne E Sartori. 2008. “Strategic Uncertainty as a Causeof War.” Quarterly Journal of Political Science 3(4):327–352.
Mitchell, Ronald B. 1994. “Regime Design Matters: Intentional Oil Pollutionand Treaty Compliance.” International Organization 48(03):425–458.
Mitchell, Ronald B. 1998. “Sources of Transparency: Information Systems inInternational Regimes.” International Studies Quarterly 42(1):109–130.
Rosendorff, B Peter. 2005. “Stability and Rigidity: Politics and Design of theWTO’s Dispute Settlement Procedure.” American Political Science Review99(03):389–400.
Rosendorff, B. Peter and Helen V. Milner. 2001. “The Optimal Design of Interna-tional Trade Institutions: Uncertainty and Escape.” International Organization55:829–857.
Simmons, Beth A. 2000a. “International Law and State Behavior: Commitmentand Compliance in International Monetary Affairs.” American Political ScienceReview pp. 819–835.
Simmons, Beth A. 2000b. “The Legalization of International Monetary Affairs.”International Organization 54(03):573–602.
Simmons, Beth A. 2001. “The International Politics of Harmonization: The Caseof Capital Market Regulation.” International Organization 55:589–620.
Svolik, Milan. 2006. “Lies, Defection, and the Pattern of International Coopera-tion.” American Journal of Political Science 50(4):909–925.
Thompson, Alexander. 2010. “Rational Design in Motion: Uncertainty and Flex-ibility in the Global Climate Regime.” European Journal of International Re-lations 16(2):269–296.
Von Stein, Jana. 2005. “Do Treaties Constrain or Screen? Selection Bias andTreaty Compliance.” American Political Science Review 99(04):611–622.
Von Stein, Jana. 2008. “The International Law and Politics of Climate ChangeRatification of the United Nations Framework Convention and the Kyoto Pro-tocol.” Journal of Conflict Resolution 52(2):243–268.
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