Strategies generalization and payoff fluctuation optimization in the iterated ultimatum game

An iterated version of ultimatum game, based on generalized probabilistic strategies, which are mathematically modeled by accepting proposal functions is presented. These strategies account for the behavior of the players by mixing levels of altruism and greed. We obtained analytically the moments of the payoff of the players under such a generalization. Our analysis is divided into two cases: (i) no memory players, where players do not remember previous decisions, and (ii) one-step memory players, where the offers depend on players' last decision. We start considering the former case. We show that when the combination of the proposer's altruism and responder's greed levels balances the proposer's greedy and responder's altruism levels, the average and variance of the payoff of both players are the same. Our analysis is carried out considering that the acceptance of an offer depends on: (a) a fixed probability p or (b) the value offered. The combination of cases (i) and (a) shows that there exists a p value that maximizes the cumulative gain after n iterations. Moreover, we show n×p diagrams with ïso-average” and ïso-variance” of the cumulative payoff. Our analytical results are validated by Monte Carlo simulations. For the latter case, we show that when players have no memory (i), there are cutoff values, which the variance of the proposer's cumulative payoff presents local maximum and minimum values, while for the responder, the same amount presents a global maximum. Case (b) combined with one-step memory players (ii), we verified, via MC simulations that, for the same number of iterations, the responder obtains different cumulative payoffs by setting different cutoff values. This result composes an interesting pattern of stripes in the cutoff per n diagrams. Simultaneously, looking at variance of this amount, for the responder player in a similar diagram, we observe regions of iso-variance in non trivial patterns which depend on initial value of the proposal. Our contributions detailed by analytical and MC simulations are useful to design new experiments in the ultimatum game in stochastic scenarios.