The cybernetics of nonzero sum games – the prisoner's dilemma reinterpreted as a pure conflict game with nature, with empirical applications

Abstract

In this thesis a new solution concept is developed for n-player, nonzero sum games. The solution concept is based in reinterpreting the n-player nonzero sum game into 2-player zero sum games. The n-player nonzero sum game is first rewritten as an n + 1 player coalition game. The definition of zero sum payment is that one player pays the other what he gets in a given outcome (coalition of the n + 1 player game). Who pays whom depends on the coalition. More than one 2-player zero sum interpretation game always results from the procedure, and criteria are established to select one of the zero sum interpretation games. The solution concept defines results identical to the minimax concept when applied directly to zero sum 2-player games.

When applied to 2-player prisoner’s dilemma games, the solution procedure assigns mixed strategies to the prisoners, thereby “resolving” the dilemma. The mixed strategies vary with the payoffs (up to a linear transformation). For prisoner’s dilemma matrices which have been used in large numbers of gaming experiments, the solution concept predicts dynamically, i.e., by play number, the “fraction of cooperative choices” for (approximately) the first 30 plays. In addition, the mixed strategy appears in a game between each subject (prisoner) and the n + 1st player (district attorney), suggesting that the subjects have been playing against the experimenter. Empirical evidence for this conclusion is given. A theorem is proved for n-player prisoner’s dilemma games.

Game theory is reviewed to show the roots of this solution concept in the heuristic use of zero sum n-player games in the von Neumann and Morgenstern theory, and in rational decision making models, e.g., “games against Nature.” The empirical and formal difficulties of the equilibrium point solution concept for nonzero sum games are discussed. Detailed connections between game theory and cybernetics are discussed.