Prisoner’s dilemma may or may not appear in large random games

Consider a two-person general-sum game on n × n payoff random matrices A and B with iid continuous entries, for large n. It is shown that the probability that there exists a pure strategy Nash equilibrium that is not pure Pareto optimal remains bounded away from 0 and 1 as n increases. We also consider the number of mixed strategy Nash equilibria: It is shown that for a mixed strategy Nash equilibrium the number of rows that are given nonzero probability by player I must equal the number of columns given nonzero probability by player II. We further investigate the expected number of k × k mixed strategy Nash equilibria when the entries are normally distributed and prove it to be of order (log n)k−1/(2k(k!)2). As a consequence we derive that with high probability no k × k mixed equlibria will exist when K > e2 and k ≥ K(log n)1/2.