Cooperation in the repeated prisoner's dilemma game with local interaction and local communication

• Author: Myeonghwan Cho
• Date: 01 September 2014
• Published date: 01 September 2014
• DOI: http://doi.org/10.1111/ijet.12039

doi: 10.1111/ijet.12039

Cooperation in the repeated prisoner’s dilemma game with

local interaction and local communication

Myeonghwan Cho∗

This paper considers a repeated prisoner’s dilemma game in a network in which each agent

interacts with his neighbors and cannot observe the actions of other agents who are not directly

connected to him. If there is global information processing through public randomization and

global communication, it is not difficult to construct a sequential equilibrium which supports

cooperation and satisfies a property, called stability, which requires that cooperation resumes

after any history. Inthis paper, we allow agents to locally communicate with their neighbors and

show that it is possible to construct such an equilibrium without global information processing.

Here, the role of local communicationis to resolve the discrepancy of agents’ expectations about

their neighbor’s future actions.

Key wor ds repeated prisoner’s dilemma game, local interaction, local communication, stable

equilibrium, network

JEL classification C72, C73

Accepted 12 August2013

1 Introduction

From the results called the folk theorem, it is well known that any outcome that Pareto dominates

a Nash equilibrium (or an individually rational outcome) in a stage game can be supported as

an equilibrium in infinitely repeated games for sufficiently patient agents. The earliest works on

this theorem, such as Friedman (1971) and Fudenberg and Maskin (1986), assume that perfect

monitoring is possible. After those works, many studies focused on situations in which monitoring

is not perfect. In particular, Kandori (1992) and Ellison (1994) study repeated games with a random

matching process, where agents are matched randomly in each period and each agent observes only

the outcome of his own match.

In this paper, we are interested in another kind of situation in which monitoring is not perfect.

More precisely, agents are located in a network and each agent observes only the actions chosen by

his neighbors. In each period, each agent plays prisoner’s dilemma games against his neighbors and

his stage game payoff is the sum of payoffs fromthe games against all of his neig hbors. Furthermore,

each agent has to choose the same action against his neighbors, and so he cannot play cooperation

with one of his neighbors while playing defection against another neighbor.

*Department of Economics, Universityof Seoul, Dongdaemun-gu, Seoul, Republic of Korea. Email: chomhmh@uos.ac.kr

This work is based on Chapter 1 of my PhD dissertation at PennsylvaniaState University. I am deeply indebted to Kalyan

Chatterjee for his invaluable guidance and encouragement. I would liketo thank Drew Fudenberg, Edward Green, James

Jordan, Vijay Krishna, Anthony Kwasnica, Marek Pycia, Neil Wallace, and participants at the 18th Stony Brook Game

Theory Conference and the 3rd WorldCongress of the Game Theory Society for their helpful comments. The comments

from the anonymous referees are also appreciated.

International Journal of Economic Theory 10 (2014) 235–262 © IAET 235

International Journal of Economic Theory

Repeated prisoner’s dilemma game Myeonghwan Cho

An example of this situation is local competition and collusion between adjacent car dealers. Car

dealers are located along the road and two neighboring car dealers compete on price for consumers

located between them. They can cooperate with each other by setting a monopoly price, or defect by

setting a competitive price. Since car dealers cannot tell consumers apart, each dealer has to choose

the same price against his neighbors. Since the price is a private offer to consumers, a dealer may not

figure out other dealers’ prices if they do not affect his profit directly. In addition, if there is a cost

involved in seeing the prices of others, a dealer maynot pay the cost to see prices which are irrelevant

to his profit.1

For this environment, the goal of this paper is to construct a sequential equilibrium for suffi-

ciently patient agents, which supports cooperation and satisfies a property called stability. Stability

requires that cooperation resumes and so the continuation payoffeventually goes back to the original

payoff after any history. Indeed, it is not difficult to construct an equilibrium supporting cooper-

ation. The trigger strategy that observing a deviation causes a permanent punishment can be a

sequential equilibrium for agents who are not sufficiently patient, and Ellison (1994) provides the

idea of constructing a sequential equilibrium supporting cooperation for sufficiently patient agents

by modifying the trigger strategy. The trigger strategy and its modification are sequential equilibria

supporting cooperation, but they are not stable because cooperation never resumes after a mistake

of playing defection.

A typical way to obtain an equilibrium in which cooperation resumes after any history is to havea

punishment of fixed finite length. That is, if an agent observes his neighbor playing defection, he plays

defection in order to punish the defector for a finite number of periods. However, local observability

may cause a discrepancy in his neighbors’ expectations about when he resumes cooperation. If this

happens, the agent whose neighbors have different expectations about his future actions may not

be able to satisfy his neighbors’ expectations in the future. This causes a difficulty in constructing a

sequential equilibrium in which cooperation resumes after any history.

If agents share common information through perfect monitoring, public randomization, or

global communication, they can reach an agreement on when they resume cooperation after it breaks

down. So it is not difficult to construct a stable sequential equilibrium.2In this paper, we introduce

local communication and showthe possibilit y of constructing a stable sequential equilibrium without

global information processing. Local communication means that agents can send a message only to

their neighbors, and messages do not affect their payoffs directly. The message can be interpretedas

a car dealer’s “on sale”billboard.

In the equilibrium we construct, an agent starts a finite-period defection phase if there was a

surprise in the previous period. A surprise means a deviation from expectation, not from strategy.

Because an agent cannot figure out whether his neighbor’s defection is a deviation from strategy or

a consequence of punishing other neighbors, it is not possible to construct an equilibrium in which

punishment occurs based on deviation. Thus, we first construct the expectations of agents about

their neighbors’ actions, and then construct an equilibrium in which each agent chooses his action

based on whether or not his neighbors follow his expectations. The role of local communication in

1Another example for our model is one in which there is production and sharing of local public goods. Each agent decides

whether or not to produce his public good, the benefit of which is shared with his neighbors. The individual benefit from

the public good is smaller than its cost, even though the social benefit from the public good is greater than its cost. Thus,

a dominant strategy for each agent is not to produce, but it does not yield an efficient outcome.

2In Ellison (1994) and Cho (2011), a public randomization is used as a coordination device to resume cooperation. Global

communication is discussed in Ben-Porathand Kahneman (1996), Kandori and Matsushima (1998), Compte (1998), and

Obara (2007).

236 International Journal of Economic Theory 10 (2014) 235–262 © IAET