Evolutionary Stability of Behavioural Rules in Bargaining

Abstract

I study the evolutionary stability of 'mildly responsive' behavioural rules in a bargaining game. Individuals in a population (that may be finite or be described by individuals distributed uniformly over a continuum of fixed mass) bargain with all other individuals in a pair-wise manner over a pie/surplus of fixed size by simultaneously demanding a share of the pie for themselves. Individuals respond to random samples drawn from demands made in the past by using a mildly responsive behavioural rule. The fitness of a behavioural rule is a function of the average share of the pie received by the individuals who follow that particular rule. Stability of the population requires the existence of a state (or configuration of demands made by the individuals of the population) that satisfies an internal stability condition, whereby all incumbent behavioural rules need to be equally fit, and an external stability condition that requires the incumbent behavioural rules to be fitter than a mutant behavioural rule. I show that internal stability implies a hardwired behaviour-responsive behaviour equivalence theorem: even though individuals actually respond to history of play, a necessary condition for stability is that the state should be such that it is as-if they are hardwired to make the same demand. I show that the state where all individuals choose to claim half of the pie is the unique neutrally stable state; all other states are unstable in the face of an invasion by a mutant behavioural rule.