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 Please do not cite without permission.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: jiac@colorado.edu

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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

2

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.

3

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

4

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

5

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).

14

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)

15

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

16

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

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