Background: The Prisoner's Dilemma (PD) is a widely used paradigm to study cooperation in evolutionary biology, as well as in fields as diverse as moral philosophy, sociology, economics and politics. Players are typically assumed to have fixed payoffs for adopting certain strategies, which depend only on the strategy played by the opponent. However, fixed payoffs are not realistic in nature. Utility functions and the associated payoffs from pursuing certain strategies vary among members of a population with numerous factors. In biology such factors include size, age, social status and expected life span; in economics they include socio-economic status, personal preference and past experience; and in politics they include ideology, political interests and public support. Thus, no outcome is identical for any two different players.
Results: We show that relaxing the assumption of fixed payoffs leads to frequent violations of the payoff structure required for a Prisoner's Dilemma. With variance twice the payoff interval in a linear PD matrix, for example, only 16% of matrices are valid.
Conclusions: A single player lacking a valid PD matrix destroys the conditions for a Prisoner's Dilemma, so between any two players, PD games themselves are fewer still (3% in this case). This may explain why the Prisoner's Dilemma has hardly been found in nature, despite the fact that it has served as a ubiquitous (and still instructive) model in studies of the evolution of cooperation.