Coordination and institutions: A review of game-theoretic contributions

Stéphane StraubUniversity of Edinburgh

Introduction

Institutions are key in enhancing the efficiency of economic interactions.

Huge variation. Both temporal and spatial.

While the role of institutions as protectors of property rights has been extensively studied, a more neglected aspect is that of institutions as coordination devices (Bardhan, 2005).

Institutions and coordination failures Institutions can help to correct the coordination

failure that plague basic economic interactions.

- Developing economies:Witnesses in commercial exchange (Attali, 2003), contract enforcement (Fafchamps, 2004), dispute prevention (McMillan & Woodruff, 99, 2000).

- Economies in transition to industrial/market stage:

Japan after WWII, East Asian countries (Aoki et al., 1997), transition countries (Johnson et al., 2002).

- Specific markets:US Cotton Market (Bernstein, 2001), Diamond (Bernstein, 1992, Richman, 2005).

Player

C

j

D

C x,x -l, g

Player i

D g, -l y,y

Institutional coordination needed because when individuals act opportunistically, pareto inferior outcomes may arise. Example: prisoner’s dilemna.

(D,D) is the only Nash equilibrium, and it is dominated by (C,C).

Assumptions:g > x > y > -l

(g - l > 2x)

Application 1: Social Capital (Durlauf & Fafchamps, 2005)

‘‘You should always go to other people’s funerals; otherwise, they won’t come to yours.’’ Yogi Bera.

Social Capital (SK) is “something” that generates positive externalities for members of a group, through shared norms, trust and values and their effects on expectations and behavior. These arise from informal forms of organizations based on social networks and associations.

So SK looks very much like “informal institutions”.

Social Capital

To matter, SK must compensate for some inefficiency, i.e. we must be in a 2nd best world, e.g. because of externalities, free-riding, imperfect information and enforcement, imperfect competition, etc.

Social capital can act by: Facilitating information sharing; Modify preferences, alter identification to

groups; Facilitate coordination, provide

leadership;

Player

C

j

D

C x,x -l, g

Player i

D g, -l 0,0

Example: modified preferences induce shift from (D,D) to (C,C)

Example: altruistic preferences (within group, kinship, etc.) Each player’s payoff is a weighted sum of

hers and her opponent’s payoff: Ui = (1-α) πi + α πj

Then we get (see next slide) that (C,C) is a Nash equilibrium whenever:

α > (g - x) / g + l

which can arise for α << ½.

Player

C

j

D

C x,x αg -(1-α)l,

(1-α)g-αl

Player i

D

(1-α)g-αl,

αg -(1-α)l 0,0

Players’ payoffs with altruistic preferences:

Example: social structure that facilitates cooperation (Routledge & Amsberg, JME 03) Community with N

players, randomly matched to play a repeated PD.

Games are private (no info on other players’ trade).

Agents play C if it is an equilibrium.

Proba of trade between 2 agents depends on N.

Player

C

j

D

C 2,2 0,3

Player i

D 3,0 1,1

Nmax trades per period. If Nmax > N-1, at most N-1 trades. Proba of trade between 2 agents (i and

j):πij = min (1, Nmax /(N-1))

Discount rate β. A strategy profile sc that supports (C,C)

is to use trigger strategies: play C if history of play with agent j contains only (C,C), otherwise play D.

SK exists if all players following sc is an equilibrium.

For trade between 2 agents (i and j), strategies sij

c and sjic are a SPE iff :

πij > πc = (1-β)/β Proof: no deviation if

3 + β [(πij .1)/(1-β)] < 2 + β [(πij .2)/(1-β)]

πij > (1-β)/ β Intuition: no deviation as long as agents

value future cooperative trade more than one-time deviation gain + unfriendly trade forever thereafter.

Nmax = 3, β = 0.55, πc = 0.818 In closed communities, probability of trade

π(3) = 1 > πc . Each agent trades twice, for a utility of 4: 2 trades times 2, since cooperation is sustained.

When communities are linked by the bridge, probability of trade π(6) = 0.6 < πc . Each agent trades 3 times, for a utility of 3: 3 trades times 1, since cooperation is not sustained.

More opportunities for trade but reduction in welfare because SK destroyed (migration parabola).

Repeated PD and cooperation. In 2 agents repeated PD, Folk Theorem known to

hold: cooperative outcome can be sustained as an equilibrium.

Can Folk Theorem-type results be obtained in social games with (possibly random) matching?

when players have limited information about others (past) behavior?

Answer is yes, under certain assumptions. Informational assumptions appear to be crucial. Refs: Greif, 93; Milgrom, North and Weingast,

90; Kandori, 92, Ellison, 94, etc.

Greif (1994) – Informal institution 11th century Maghribi traders used to employ

overseas agents, despite the obvious commitment problem.

Complete information about past behavior of agents in the community.

Cooperative relationships sustained by a multilateral punishment strategy: a merchant offers an agent a wage W, rehires the same agent if he has been honest (unless forced separation has occurred), fires the agent if he has cheated, never hires an agent who has ever cheated any merchant, and (randomly) chooses an agent from among the unemployed agents who never have cheated if forced separation has occurred.

Kandori (1992) – Informal institution With no information, a “contagious” punishment

strategy may sustain cooperation: when one player cheats in period t, his opponent cheats from t+1 onwards, infecting other players, etc.

For any N, there are payoffs which allow cooperation, but as N grows large, extreme values of the payoff are required (to avoid agents not punishing to slow down contagion and enjoy high payoffs in the future, the risk associated (getting -l) must be high).

Ellison (1994) provides several refinements.

Milgrom, North and Weingast (1990)Formal institution The law merchant enforcement system

and Champagne fairs in the 12th and 13th centuries.

There is a specialized agent (judge) serving both as repository of information and adjudicator of disputes (both at a cost to trading agents).

Under certain conditions, cooperation is sustained at a (transaction) cost for trading agents.

Osborne & Rubinstein (1994): “…in our opinion the main contribution of the theory is the discovery of interesting stable social norms (strategies) that support mutually desirable payoff profiles, and not simply the demonstration that equilibria exist that generate such profiles.”

Problem: How do agents come up with these norms in the first place? In particular, how do they structure their interaction and allocate roles when some form of formal enforcement is required?

Sanchez-Pages & Straub 2007

We model the process through which institutions such as these may arise.

We characterize: The factors that make possible or hinder the

formation of institutions. The level of efficiency at which they arise.

Their emergence is the equilibrium of a game that agents play in the state of nature. It has to be self-enforcing.

Otherwise, the economy remains in the status-quo.

Player

C

j

D

C x,x -l, g

Player i

D g, -l 0,0

Who induces shift from (D,D) to (C,C)?

The Model In the state of nature, N+1 agents,

endowed with ω, are randomly matched to play the PD without interference.

Expected unit payoff is then αx. The parameter α denotes the status-quo

level of coordination or trust (without institution, they play (C,C) with proba α).

The institution is able to ensure that the (C,C) profile is played with proba 1.

But someone has to run it (Pepe…).

The Model

One of the agents becomes the “centre”.

She must relinquish the ability to trade.

But is compensated in exchange. Agents must pay a fee a ≤ ω to interact

under the centre’s umbrella (trade certification, dispute prevention / resolution, reputation management…).

The procedure of institution formation Our procedure of institution formation starts

with a lottery over the set of agents who freely participate in it, to determine who will become the central agent.

Justification: All equally likely to be center. centre is randomly drawn each period.

the institution must emerge in the most decentralized way possible. No commitment is assumed.

The procedure of institution formation First, the fee is freely chosen by the

central agent: The institution is a revenue-maximizer.

Having observed a, agents must decide whether to become formal or not.

2 problems (IR constraints): Ex ante, agents may not want to participate. Ex post center may renege.

Two sources of inefficiency (1) Efficient institution may not arise

This is more likely for economies of intermediate size and high levels of trust α.

If N small: WF lower than informality payoff. If N large: incentive to become the central

agent increases (more revenue). High trust undermines the position of the

institution (reminiscent of identification problem in social capital literature. See Durlauf and Fafchamps, 2005).

Two sources of inefficiency (2) Institution may be sub-efficient (too

extractive). When status-quo trust is high, revenue and

welfare maximisation are aligned. Otherwise, with low status-quo trust, the

institution arises at a sub-optimal level of efficiency (that is when it is most needed).

The rent associated with being the centre are the key motivation for agents to participate. With a high extractive fee, all other agents are just indifferent.

Appendix: The formal model

One of the agents becomes the “centre”. She must relinquish the ability to trade. But is compensated in exchange. Agents must pay a fee a ≤ ω to interact under

the centre’s umbrella (trade certification, dispute prevention / resolution, reputation management…).

Participation decisions

Having observed a, agents must decide whether to become formal or not.

If they become formal, interacting with another formal agents yields per unit return

where xa> 0, xaa< 0 and x(0) > 1/α. The efficiency of interactions depends on the fee

paid to the institution.

)(axvF

Participation decisions

Interacting with an informal agent yields

regardless of your status. Expected payoffs when K formal agents:

)(axv I

)()(1

)()(1

1)( axa

N

KNaxa

N

KKV F

)()( axKV I

Participation decisions

Define

K formal agents can be supported in equilibrium if and only if

But a (K) is increasing in K, so only corner solutions prevail (full formality or full informality)

))(1

)1(1()(

KNK

NKa

)()1( KaaKa

Participation decisions

Proposition 1: For a given level of the fee a

(i) Informality can be supported in equilibrium only if a ≥ a(1)=0

(ii) Full formality can be supported in equilibrium only if a ≤ a(N)

0 a(N)

Only informalityMultiple equilibria: full formality or informality

Formality sustainable

Informality sustainable

First-best level of the fee aPlanner objective function:

max WF = N[(ω-a)x(a)+(a-c)]+ ωs.t. a < a(N)

This defines a*. The first best fee is then:aF = min{a*,a(N)}

and there is a threshold α* s.t a*>a(N) if α > α*, so in this case the revenue maximizing fee coincides with the first best level.

Finally, informality may dominate for N and ω small and α high.

The procedure of institution formation Our procedure of institution formation starts with a

lottery over the set of agents who freely participate in it, to determine who will become the central agent.

One justification: All equally likely to be center. game repeated infinitely and centre is randomly

drawn each period. See Morgan (2000) for an application of lotteries to

reduce free-riding on public goods financing.

Timing

t=1

Agents decide whether to participate or not in a

lottery that will determine who will run the

institution.

t=2

If the institution has emerged, the fee a to be paid by formal

agents is set. If not, the status-quo remains (informal

exchanges).

t=4

Agents are randomly matched and play G. Payoffs are realized.

time

t=3

Agents decide whether to become formal or not.

The fully decentralized procedure In this procedure, the institution must emerge

in the most decentralized way possible. No commitment is assumed.

First, the fee is freely chosen by the central agent: The institution is a revenue-maximizer.

So it will set the maximum fee compatible with formality, a(N).

The fully decentralized procedure Second, the agent that runs the institution can

renege ex-post. For the institution to arise, an ex-post participation

constraint must be satisfied:

That’s for the centre. It is trivially satisfied for other agents.

)0()( xcaN

The fully decentralized procedure Ex-ante participation constraint given the fee a

because either all agents or none participate in the lottery.

With a(N), the institution arises iff the (stronger) ex-ante constraint is met. It rewrites:

)()()(1

))((1

1axaxa

N

NcaN

N

))(())(( NaxcNaN

The fully decentralized procedureProposition 2: If the ex-ante constraint holds, there exists a SPE of

the fully decentralized procedure that implements formality under a(N).

Two sources of inefficiency

Corollary 1: There exists a range of parameters for which a potentially welfare enhancing institution does not arise.

This is the case when parameters are such that the level of individual welfare obtained under formality dominates the level of welfare under full informality but is not high enough to induce ex ante participation in the lottery:

))(()0( )( NaxWx FNa

)()(1

))((1

1)( axa

N

NcaN

NWwhere F

Na

Two sources of inefficiency

This is more likely for economies of intermediate size and high levels of trust α.

If N small: WF lower than informality payoff. If N large: incentive to become the central agent

increases (more revenue). High trust undermines the position of the institution

(reminiscent of identification problem in social capital literature. See Durlauf and Fafchamps, 2005).

Two sources of inefficiency

Corollary 2: The utilitarian first best fee can be implemented in a SPE of the fully decentralized procedure only for high enough of status-quo trust α.

When status-quo trust is high, revenue and welfare maximisation are aligned (see first best fee a*).

Otherwise, the institution arises at a sub-optimal level of efficiency. The rent associated with being the centre are the key motivation for agents to participate. With a = a(N), all other agents are just indifferent.

Two types of commitment

Now imagine that commitment can be imposed along two lines:

1. Individual: Agents cannot renege ex-post whatever their role.

2. Collective: The fee is chosen collectively before the lottery takes place.

Different procedures arise from different combinations of assumptions.

4. Agents commit ex ante to participate in the lottery and not to renege ex post if chosen as the center. Furthermore, the center has no freedom to set a ex post.

3. Agents commit ex ante to participate in the lottery. If chosen as the center, they may renege, but have no freedom to set a if they accept to fulfill their role.

Fee a set ex ante

2. Agents commit ex ante to participate in the lottery and not to renege ex post if chosen as the center.

1. Agents’ only commitment is to participate in the lottery ex ante. The center may refuse to cooperate ex post and is free to set a.

Center maximizes revenue (sets a) ex post

Strong commitment(ex ante participation constraint)

Limited commitment(ex post participation constraint)

Other procedures

One can consider alternative procedures by combining these 2 types of commitment.

Imposing individual commitment alone has no effect. Collective commitment alleviates the second type of

inefficiency. Only when commitment is imposed in both

dimensions, does the institution arise whenever it is welfare enhancing.

These two types of commitment rely on some exogenous enforcement mechanism.

We consider two ways to endogenize commitment:

1. Trigger-like strategies.

2. Threat of secession.

Endogenous commitment

Secession

When there is no commitment, secession is an issue.

No group in society should be able to improve its situation by withdrawing and forming its own mini-society.

We study when the institution will be secession-proof and the impact of this threat on welfare.

Secession

Definition: Denote by aN the fee set by the institution. A coalition of S interacting agents is said to be blocking if and only if

Note that when a group secedes, it sets a self-enforcing fee.

A fee is secession-proof (it is in the core of the procedure of institution formation) if it does not spawn any blocking coalition.

))(())((1

)))(((1

)()( NaxNaS

ScNaS

Saxa NN

Secession

Proposition 4 : The set of secession-proof fees is non-empty if and only if N is low enough.

The reason for blocking is the prospect of becoming the central agent in the new mini society.

When the level of status-quo trust is low enough, the threat of secession can tame the central agent.

Secession

N

Secession and efficiency

A natural question is whether secession is bad or good for efficiency.

Let us look at the eventual outcome of the secession process.

We say that a coalition structure is secession-proof if all coalitions in it can set a (possibly different) fee that does not spawn any blocking coalition.

Secession and efficiency

Proposition 5 : For high enough levels of status-quo trust, the total sum of payoffs under a secession-proof structure is never greater than under a single institution.

In our model, only the center gets positive rents. This creates strong incentives for secession. Proliferation of institutions is however socially

inefficient because of duplication of costs. A trade-off may arise if transaction costs of

institution are lower in small groups.

Conclusions

We have presented a model where an institution emerges as the equilibrium of a game played in the state of nature.

The institution may not emerge despite being welfare enhancing

This happens for intermediate population sizes and high levels of status quo coordination.

But even if it emerges it can do it at a suboptimal level. This is because the rent associated with being the centre are the key motivation for agents to participate.