Evolution of fairness in the divide-a-lottery game

In this paper, we show that fairness can evolve in the divide-a-lottery game which is more general than the divide-a-dollar game by using an indirect evolutionary approach. In the divide-a-lottery game, the size of a pie is uncertain. Two players sequentially bid for a share and they get their bid if the allocation based on the bids turns out to be feasible and otherwise neither gets anything. In this game, rational players over-compete for a higher share, resulting in a high probability of failure in agreement, whereas fair players who dislike the disparity between shares lower their bids thereby reducing the failure probability and thus increasing the expected payoff. As a result, fairness strictly dominates rationality. This is the mechanism through which fairness evolves. However, this result is not robust against even a slight uncertainty about the opponent's type. Surprisingly, we show a contrasted simulation result that only rational players who are strictly dominated by fair players survive evolutionarily for most of the parameter values if players have even a slight chance of not knowing the opponent's type. Our simulation results in a local interaction model in which players only know the type of closer neighbors capture both insights and demonstrate that moderate proportions of both types coexist evolutionarily over time, and that the population average fitness of this polymorphic population is higher than monomorphic population consisting only of fair types or rational types.