Beyond indifferent players: on the existence of prisoners dilemmas in games with amicable and adversarial preferences

Why don’t agents cooperate when they both stand to gain? This question ranks among the most fundamental in the social sciences. Explanations abound. Among the most compelling are various configurations of the prisoner’s dilemma (PD), or public goods problem. Payoffs in PD’s are specified in one of two ways: as primitive cardinal payoffs or as ordinal final utility. However, as final utility is objectively unobservable, only the primitive payoff games are ever observed. This paper explores mappings from primitive payoff to utility payoff games and demonstrates that though an observable game is a PD there are broad classes of utility functions for which there exists no associated utility PD. In particular we show that even small amounts of either altruism or enmity may disrupt the mapping from primitive payoff to utility PD. We then examine some implications of these results.