Abstract. We study the complexity of computing Nash equilibria in games where players arranged as the vertices of a graph play a symmetric 2-player game against their neighbours. We call this a pairwiseinteraction game. We analyse this game for n players with a fixed number of actions and show that (1) a mixed Nash equilibrium can be computed in constant time for any game, (2) a pure Nash equilibrium can be computed through Nash dynamics in polynomial time for games with a symmetrisable payoff matrix, (3) determining whether a pure Nash equilibrium exists for zero-sum games is NP-complete, and (4) counting pure Nash equilibria is #P-complete even for 2-strategy games. In proving (3), we define a new defective graph colouring problem called Nash colouring, which is of independent interest, and prove that its decision version is NP-complete. Finally, we show that pairwise-interaction games form a proper subclass of the usual graphical games.

Pavlov, a well-known strategy in game theory, has been shown to have some advantages in the Iterated Prisoner's Dilemma (IPD) game. However, this strategy can be exploited by inveterate defectors. We modify this strategy to mitigate the exploitation. We call the resulting strategy Rational Pavlov. This has a parameter p which measures the "degree of forgiveness" of the players. We study the evolution of cooperation in the IPD game, when n players are arranged in a cycle, and all play this strategy. We examine the effect of varying p on the convergence rate and prove that the convergence rate is fast, O(n log n) time, for high values of p. We also prove that the convergence rate is exponentially slow in n for small enough p. Our analysis leaves a gap in the range of p, but simulations suggest that there is, in fact, a sharp phase transition.

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