Bargaining by Heterogeneously Responsive Populations

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I study the process of bargaining over a pie of fixed size. Motivated by the argument that the manner in which bargaining unfolds often depends on antecedents, I embed the bargaining process in an evolutionary framework. First, I posit that there exist two separate populations, and in each period, one individual is randomly drawn from each population to bargain. The bargaining protocol used is the Nash demand game, and the demand that each individual makes is guided by the demands made by the other population in the recent past. I show that under very general conditions, the bargaining process reaches a convention, where each population settles on demanding a fixed share of the pie, and the demands of each population are both compatible with each other, and cumulatively exhaust the pie. Next, I identify the most advantageous behavioural trait in the long-run: in the convention that is stable in the long-run, a population of 'wildly optimistic' individuals obtains almost the entire pie against a population comprised of 'almost any other' behavioural type. Secondly, since this two-population bargaining game does not allow for evolutionary selection, I analyse the stability of a behavioural trait in a playing-the-field model of bargaining by examining the relative performance of an incumbent population described by a particular behavioural trait against a mutant of another behavioural trait. I show that all behavioural traits are unstable as they are susceptible to invasion by any mutant trait. However, the only state where any behavioural trait can co-exist with any other mutant trait is when the pie is shared equally. This demonstrates the importance of the equal-splitting norm for co-existence of various behavioural traits, and hence, for sustainable population diversity.