Social Learning in the Ultimatum Game

In the ultimatum game, two players divide a sum of money. The proposer suggests how to split and the responder can accept or reject. If the suggestion is rejected, both players get nothing. The rational solution is that the responder accepts even the smallest offer but humans prefer fair share. In this paper, we study the ultimatum game by a learning-mutation process based on quantal response equilibrium, where players are assumed boundedly rational and make mistakes when estimating the payoffs of strategies. Social learning is never stabilized at the fair outcome or the rational outcome, but leads to oscillations from offering 40 percent to 50 percent. To be precise, there is a clear tendency to increase the mean offer if it is lower than 40 percent, but will decrease when it reaches the fair offer. If mutations occur rarely, fair behavior is favored in the limit of local mutation. If mutation rate is sufficiently high, fairness can evolve for both local mutation and global mutation.