Mathematical Social Sciences 37 1999 265–280
On the stability of monotone discrete selection dynamics with inertia
Herbert Dawid
¨ Department of Management Science
, University of Vienna, Brunnerstraße 72, A-1210 Vienna, Austria Received 10 February 1997; received in revised form 1 March 1998; accepted 1 May 1998
Abstract
In this paper we study the learning behavior of a population of boundedly rational players who interact by repeatedly playing an evolutionary game. A simple imitation type learning rule for the
agents is suggested and it is shown that the evolution of the population strategy is described by the discrete time replicator dynamics with inertia. Conditions are derived which guarantee the local
stability of a Nash-equilibrium with respect to these dynamics and for quasi-strict ESS with support of three pure strategies the crucial level of inertia is derived which ensures stability and
avoids overshooting. These results are illustrated by an example and generalized to the class of monotone selection dynamics.
1999 Elsevier Science B.V. All rights reserved.
Keywords : Stability; Monotone discrete selection dynamics; Inertia
JEL classification : C72
1. Introduction
Adaptive learning in games has been a major topic in recent economic research. In particular the behavior of a whole population playing an evolutionary game has been
studied using several different stochastic and deterministic models see e.g. Blume, 1997; Kandori and Rob, 1995; Friedman, 1991; Ellison and Fudenberg, 1995. One of
the most popular approaches to modelling learning in an evolutionary context is replicator dynamics Taylor and Jonker, 1976 which was first introduced as a model of
biological evolution but may also be interpreted in economic applications as a highly stylized description of the effects of imitation in a population of agents. It is well known
Corresponding author. Tel.: 143 1 29128 513; fax: 143 1 29128 504. E-mail address
: herbert.dawidunivie.ac.at H. Dawid 0165-4896 99 – see front matter
1999 Elsevier Science B.V. All rights reserved.
P I I : S 0 1 6 5 - 4 8 9 6 9 8 0 0 0 3 1 - 6
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. Dawid Mathematical Social Sciences 37 1999 265 –280
that there is a considerable difference in behavior between the continuous time and discrete time formulations of the replicator dynamics. Whereas any evolutionary stable
strategy is stable with respect to the continuous time replicator dynamics see e.g. van Damme, 1991 this does not necessarilly hold true for the discrete time version.
Moreover there is no comparable criterion to ESS which would guarantee the stability of the discrete time replicator dynamics. Furthermore, the property that only rationalizable
strategies Bernheim, 1984 can survive in the long run holds true for the continuous time dynamics Samuelson and Zhang, 1992 but fails to do so for the discrete time
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analogon Deckel and Scotchmer, 1992. Both of these discrepancies are due to the same phenomenon called overshooting which may of course also be observed for
different learning dynamics e.g. Dawid, 1999. In this paper we will propose a very simple learning model describing the boundedly
rational choice of strategy by agents playing an iterated game. We show that the evolution of the population strategy in this model is determined by a replicator dynamics
with inertia where some fraction of the population sticks to its old strategy whereas the choice of the rest of the population is determined by the discrete time replicator
dynamics. It is quite intuitive that the properties of this dynamics lie somewhere in-between the properties of the continous and discrete time replicator dynamics. It is the
aim of this paper to exactify this intuition and provide some lower bounds of inertia which guarantee the stability of the learning dynamics near equilibria which are
asymptotically stable with respect to the continuous time replicator dynamics. We also extend these results by considering a more general class of imitation models with inertia
and showing that the same criterion as in the case of the replicator dynamics assures the existence of a crucial level of inertia which avoids instability due to overshooting.
The existence of inertia in a population of boundedly rational agents seems to be a very plausible assumption and has been used in several game theoretic learning models
e.g. Samuelson, 1994; Ellison and Fudenberg, 1995. Confronted with a situation where gathering of information which is needed for sensible decision making might be costly
and on the other hand by no means is a guarantee of higher payoffs an agent might decide just to stick to his previous actions. Such behavior could also be interpreted as
‘acting out of habit’ see also Day 1984 for a discussion of imitational and habitual behavior, a mode of boudedly rational behavior which has also been detected in
experiments Day and Pingle, 1996. The results which are derived in this paper suggest that a high propensity to show this kind of habitual behavior might allow a population to
settle down at an equilibrium where a population of ‘more rational’ agents showing a higher propensity to adapt their strategy does not converge. Observations of this kind do
not crucially depend on the actual structure of the learning rule but hold for different type of models and simply restate the fact that inertia can avoid overshooting.
The paper is organized as follows. In Section 2 we describe our model and point out the relation to the continuous and discrete time replicator dynamics. A local stability
analysis is carried out in Section 3 and conditions ensuring stability near equilibria with support of three or less pure strategies are derived. We illustrate these results using the
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Deckel and Scotchmer 1992 show however that this property holds also for the discrete time dynamics if mixed rather than pure strategies are inherited.
H . Dawid Mathematical Social Sciences 37 1999 265 –280
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example of a generalized rock-scissors paper game in Section 4. In Section 5 we shortly deal with the more general case of monotone imitation dynamics and conclude with
some remarks in Section 6.
2. Replicator dynamics with inertia
