Fairness and mutual profitability in collective behavior of automata

A certain class of noncooperative games is considered with two pure strategies for each of the automata players. The nature of the game is such that the pay-off of each player increases if any other participant in the game chooses the same pure strategy as that player, and decreases otherwise. The pay-off from a joint strategy is generally different for each automaton so that a joint strategy which is mutually profitable is not generally a fair play. A learning automaton model is proposed so that, without any prior agreement, as time unfolds a group of automata achieve a situation in mixed strategies, which yields a pay-off for each automaton that is a trade-off between “fairness” and “profitability.” A formal mathematical proof of convergence of the collective situation in mixed strategies is given. An example is given to demonstrate the theoretical results.