A Short Note on the Solution of the Prisoner's Dilemma

Here we reconsider the solution of the well-known Prisoner's Dilemma from a binary relation point of view. We identify the Nash equilibrium as single maximum element of a relation (we call it coordination relation) between the cells of a multidimensional payoff array. The comparison between reward vectors of cells is according to a preference relation of each player. This approach allows for an easier extension to cases of n players and m strategies, but also cases of varying preference relations among the players. This way we can judge on negotiable situations by analyzing maximum set sizes of the coordination relation. As an example, we study the prospect of players focusing on maximal total or least rewards (the latter one being a model for fair decision making). It appears as an apparent counter-intuitive result that such player preferences do not lead to Nash equilibria at all since they result in a strongly increasing number of maximal elements, thus hardening a joint decision making.