IMITATION AND COORDINATION IN SMALL‐WORLD NETWORKS

• Date: 01 April 2014
• Author: Edward J. Cartwright
• Published date: 01 April 2014
• DOI: http://doi.org/10.1002/isaf.1351

IMITATION AND COORDINATION IN SMALL-WORLD

NETWORKS

EDWARD J. CARTWRIGHT*

Department of Economics, Keynes College, University of Kent, Canterbury, Kent,UK

SUMMARY

Westudy aggregate behaviour in a settingwhere individualsrepeatedly interactvia a network to playa minimum-effort

(stag hunt) game.Of interest is whether play converges on thePareto-optimal or risk-dominant outcome. We contrast

the best-reply dynamic with the imitate-the-best dynamic. We also contrast forms of lattice, small-world and random

networks.Our main finding is thatplay is far more likelyto convergeon the Pareto-optimaloutcome if individuals learn

by imitation.We find that play in small-world networks is similar to that in a regular network. Copyright© 2014 John

Wiley & Sons,Ltd.

Keywords: imitate the best; best reply; small world; stag hunt; minimum effort

1. INTRODUCTION

A large literature models myopic learning in coordination games, principally addressing the issue of

equilibrium selection (e.g. Kandori et al., 1993; Fudenberg & Levine, 1998; Young, 1993, 2001). The

main finding of this literature is that play will more likely converge on the risk-dominant equilibrium.

Given that the risk-dominant equilibrium need not be the Pareto-optimal outcome, this finding raises

fundamental questions about how a society or organization can maintain mutually beneficial behaviour.

To illustrate the issues, and its applied importance, consider the following stylized example.

Suppose that each week the employees of an organization work together in pairs on different pro-

jects. They interact via a network, an employee working on a project with all those to whom they are

linked in the network. Each employee can put in either high or low effort into the projects that they

are involved with. The outcome of a project is determined by employee effort, as summarized in TableI.

These payoffs give rise to what is called a minimum-effort game or stag hunt game. The optimal out-

come is for both employees to put in high effort. And high effort is individually rational: if Employee

B puts in high effort then it is in the interests of Employee A to also put in high effort. High effort is,

however, also relatively risky: if Employee B puts in low effort then the high effort of Employee A

would be wasted. These differences are captured bysaying that high effort is the Pareto-dominant Nash

equilibrium while low effort is the risk-dominant equilibrium.

As already mentioned, the literature on learning in coordination games suggests that low effort is

more likely to emerge over time. Moreover, this theoretical finding is backed up by the experimental

literature (Van Huyck et al., 1990; Devetag & Ortmann, 2007). Clearly, however, this is a ‘bad news’

story, because low effort is a bad outcome for the organization and its employees. The minimum-effort

game, therefore, stylized as it is, offers a crucial way to understand and model the difficulties of

* Correspondence to: Edward J. Cartwright, Department of Economics, Keynes College, University of Kent, Canterbury, Kent,

UK. E-mail: E.J.Cartwright@kent.ac.uk

Copyright © 2014 John Wiley & Sons, Ltd.

INTELLIGENT SYSTEMS IN ACCOUNTING, FINANCE AND MANAGEMENT

Intell. Sys. Acc. Fin. Mgmt. 21,71–90 (2014)

Published online 27 February 2014 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/isaf.1351

incentivizing effort in the workplace (Knez & Camerer, 1994; Camerer & Knez, 2002; Brandts &

Cooper, 2006). More generally, it allows us to address questions concerning the coordination of behav-

iour within society (Hirshleifer, 1983; Skyrms, 2004).

In a widely cited work, Skyrms (2004) argues that the prior literature overstates the difficulties of

obtaining the Pareto-dominant equilibrium. In particular, he argues that high effort and cooperative

behaviour will emerge under the ‘right conditions’. This is an exciting possibility, but one that needs

to be carefully investigated, not least to determine exactly what the ‘right conditions’are.

1

In this paper,

we explore in detail whether play will converge on the Pareto-dominant equilibrium if individuals (i)

learn by imitation and (ii) interact in a small-world network. To motivate this approach, we briefly

introduce each of the issues in turn.

The literature on learning in coordination games has primarily focused on best reply learning. This

assumes that an individual will choose the action in the current period that would have maximized their

payoff in the previous period (Fudenberg & Levine, 1998). Individuals, thus, base their choices on the

observed actions of others. By contrast, imitation assumes that individuals base their choices on the

observed payoffs of others (Alós-Ferrer & Schlag, 2009). If an individual imitates the best then they

choose the same action in the current period as the best individual they observed in the previous period

(Alós-Ferrer & Weidenholzer, 2008).

2

Both best reply and imitate the best have intuitive appeal as

stylized models of behaviour. And it is probably no surprise that imitate the best can lead to different

outcomes to best-reply learning. More interesting is that imitation may lead to Pareto-optimal outcomes

where best reply does not (Nowak & May, 1992; Robson & Vega-Redondo, 1996; Vega-Redondo,

1997; Alexander & Skyrms, 1999; Alós-Ferrer & Weidenholzer, 2008, 2010).

The outcome of any learning dynamic will depend crucially on the network through which individ-

uals interact. Most of the literature has focused on highly structured, regular networks such as a lattice

(e.g. Ellison, 1993; Blume, 1995). Some attention has been given to the opposite extreme of a random

network (Cartwright, 2004). Little attention, however, has been given to network structures that lie in

between these extremes—Morris (2000) being a notable exception.

3

This omission is worrying given

that most of the networks we observe, within society and organizations, have an intermediate structure,

such as that found in small-world networks (Watts, 2000). Moreover, experimental evidence suggests

that coordination on the Pareto-optimal outcome may be easier to obtain in small-world networks

(Cassar, 2007). In this paper we use the Watts and Strogatz model of network formation to study a

1

Skyrms (2004) was by no means the first to show that the Pareto-dominant equilibrium can emerge under specific conditions.

See, for example, Binmore et al. (1995) and Robson and Vega-Redondo (1996).

2

An alternative to imitate the best is imitate the average. In this case an individual imitates the strategy that yielded the highest

average payoff amongst their neighbours (Kandori et al., 1993).

3

Following Nowak and May (1992), a lot of attention has been given to intermediate networks in the study of the prisoner’s di-

lemma. The experimental literature has also consider non-lattice networks (e.g. Kovarik et al., 2010).

Table I. The payoff of Employee A, Employee B

Employee B

High effort Low effort

Employee A High effort 10, 10 0, 7

Low effort 7, 0 7, 7

72 E. J. CARTWRIGHT

Copyright © 2014 John Wiley & Sons, Ltd. Intell. Sys. Acc. Fin. Mgmt., 21,71–90 (2014)

DOI: 10.1002/isaf