HELPING BEHAVIOR IN LARGE SOCIETIES

• Author: Francesc Dilmé
• Published date: 01 November 2016
• Date: 01 November 2016
• DOI: http://doi.org/10.1111/iere.12197

INTERNATIONAL ECONOMIC REVIEW

Vol. 57, No. 4, November 2016

HELPING BEHAVIOR IN LARGE SOCIETIES∗

BYFRANCESC DILM ´

E1

University of Bonn, Germany

This article investigates how helping behavior can be sustained in large societies in the presence of agents who never

help. I consider a game with many players who are anonymously and randomly matched every period in pairs. Within

each match, one player may provide socially optimal but individually costly help to the other player. I introduce and

characterize the class of “linear equilibria” in which, unlike equilibria used in the previous literature, there is help

even in the presence of behavioral players. Such equilibria are close to a tit-for-tat strategy and feature smooth help

dynamics when the society is large.

1. INTRODUCTION

Consider a finite but potentially large society. Each agent in such a society sometimes needs

help to avoid incurring a big cost. When this happens, another randomly chosen agent has the

opportunity to incur a small cost to help him, so helping is socially optimal but individually

suboptimal. Assume anonymity (i.e., agents do not recognize other agents) and that each

agent observes only the outcomes of her own matches. Assume also that the society contains

behavioral agents who never help. My goal is to find equilibrium mechanisms that promote

helping behavior in such a hostile environment.

Sustaining helping behavior in a society in the previous environment requires balancing the

incentive of helping and and the incentive not to help. If players find helping too attractive,

they “forgive” not being helped by other agents in the past. In this case, each agent would have

the incentive not to help, avoiding incurring an individual cost while not changing the others’

behavior. If, instead, the agents have a strong incentive not to help, there is no helping behavior

in equilibrium.

I identify a set of equilibria that balance the incentive to help and not to help after each

history. The incentive to help is provided to each agent through “worsening” the rest of the

society when he or she does not help; that is, not helping lowers the likelihood of being helped

in the future. Still, the incentive to help cannot be too strong, since nonhelping behavior has to

be persistent enough in the society that defectors are punished. In equilibrium, the increase in

the per-period payoff that not helping provides is compensated by the decrease in continuation

payoff that it also implies. This balance of incentives is achieved through equilibrium strategies

that generate linear dynamics of the average level of help in the society, so I call them “linear

equilibria.”

When the society is large, my equilibrium strategies are close to a tit-for-tat strategy; that

is, with a very high probability each agent reproduces in the current period the action that he

or she played or received in the previous period. Tit-for-tat-like strategies make each action’s

effects in the continuation play of the rest of the society small but persistent enough that each

agent’s incentives are balanced. As a consequence, a defection does not lead to an exponential

contagion of nonhelping behavior. Instead, it leads to a contagion that, in expectation, slowly

disappears, so the expected fraction of agents helping in each period evolves smoothly over time.

∗Manuscript received April 2015; revised September 2015.

1I am grateful to George Mailath and Andy Postlewaite for their comments. Please address correspondence to:

Francesc Dilm´

e, Institute for Microeconomics, University of Bonn, Adenauerallee 24−42, 53113 Germany. E-mail:

fdilme@uni-bonn.de.

1261

C

(2016) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1262 DILM´

E

I show that the existence of equilibria remains generic when behavioral types are present in

the society, there is entry/exit of agents, and agents make mistakes. Only the “total size” of the

perturbation (e.g., the number of behavioral types, but not their action) affects the existence of

equilibria with help. The reason is that my mechanism to promote help relies on the persistence

of the effects of individual actions on the continuation play of the society. Since the “total size”

of the perturbation is a measure of how fast information is lost, it determines the existence of

equilibria with help. In contrast, I find that the long-run maximum level of help depends also

on the direction of the perturbation, so full help in the long run is reachable only when, for

example, behavioral agents always help.

My construction uses an important feature of my stage game: Conditional on being chosen

the helper, the stage-game marginal gain from helping is independent of the history. This allows

me to generalize my results to games where the payoff of the stage game is linear in the mixing

probabilities of each of the players. I illustrate this by modeling bilateral trade in markets with a

lot of traders, where the separability of the utility function is a reasonable assumption given the

different timing between producing a good and enjoying the good obtained in the trade. I model

trade with asymmetric information through assuming that traders decide the (privately known)

quality of their goods and incur an extra cost when producing high-quality goods instead of

low-quality goods.

I find that equilibria with trust, where traders produce high-quality goods generically exist

and that the equilibrium mechanism to spread defections is transmissive instead of contagious.

After this introduction, there is a review of the literature related to my model. Section 2

introduces my unilateral help model, I introduce linear equilibria where helping behavior takes

place, and I analyze their properties when the society gets large. In Section 3, I analyze the

existence and efficiency of equilibria in the presence of behavioral agents, entry/exit of agents,

and mistakes. Section 4 extends my results to a bilateral-trade model, and Section 5 concludes.

The Appendix is used to provide the proofs of the results stated in the previous sections.

1.1. Literature Review. Kandori (1992), Ellison (1994), Harrington (1995), and Deb and

Gonz´

alez-D´

ıaz (2010) consider repeated games with many agents and random matching and

assume that there is no information about the opponent. These papers analyze grim-trigger-

like (contagious) equilibria in which each player cooperates until his opponent defects. After

a defection, there is an exponential contagion, leading to a fully nonhelping behavior. Using a

public randomization device, cooperation can be restored. These equilibria fail to exist when

the society contains a single nonhelping behavioral agent. Also, when mistakes are introduced,

the whole society oscillates between total cooperation or complete defection. My equilibrium

strategies, instead, are close to a tit-for-tat strategy, its existence survives to the presence of

behavioral types, and feature smooth evolution of the aggregate level of cooperation. They high-

light reciprocity as a mechanism through which cooperation can be sustained, which generates

significantly different cooperation dynamics.

Other models in the game theoretical literature on cooperation in large societies assume

some degree of knowledge about the opponent in order to sustain cooperation. For example,

Kandori (1992) considers the case where every agent knows everything that happens in the

game. Okuno-Fujiwara and Postlewaite (1995) assume that agents have a social status that is

updated over time according to some social norm. In Takahashi (2010), agents observe only

their opponent’s past play. Deb (2008) introduces cheap talk before each stage game takes

place. In most of these models, folk theorems can be proved; that is, keeping the size of the

society constant, if agents are patient enough, all rational payoffs can be achieved. My model

focuses on the potentially most hostile environment for cooperation, where agents have no

other information than their own past play. Still, equilibria in a model with less information

remain equilibria when more information is added into the model. So, even though additional

information can potentially improve the efficiency of equilibria with cooperation, it is important

to understand mechanisms that do not depend on the amount information structure.