The Shape of Payoff Functions in N-person Games

We report computer simulation experiments based on our agent-based simulation tool to model an N-person game for the case when the agents are greedy simpletons who imitate the action of that of their neighbors who received the highest payoff for its previous action. The individual agents may cooperate with each other for the collective interest or may defect, i.e., pursue their selfish interests only. After a certain number of iterations the proportion of cooperators stabilizes to either a constant value or oscillates around such a value. The payoff (reward/penalty) functions are usually given as two straight lines: one for the cooperators and another for the defectors. The payoff curves are functions of the ratio of cooperators to the total number of agents. Even for linear payoff functions we have four free parameters that determine the payoff functions. In real-life situations, however, the payoff functions are seldom linear. In this short note we will show the influence of their shape on the results of the simulations. The defectors’ payoff function will still be linear but parabolas will represent that of the cooperators.We investigated the influence of the shape of the cooperation payoff function by running a large number of simulations. The results show that the solutions of N-person games with quadratic cooperation functions have some predictable tendencies but they are non-trivial and quite irregular with sharp fluctuations in certain regions. The solutions show drastic changes in some very narrow parameter ranges. The shapes of the payoff functions have a profound influence on the outcome of the games.