We analyze the performance of the best-response dynamic across all normal-form games using a random games approach. The playing sequence-the order in which players update their actions-is essentially irrelevant in determining whether the dynamic converges to a Nash equilibrium in certain classes of games (e.g. in potential games) but, when evaluated across all possible games, convergence to equilibrium depends on the playing sequence in an extreme way. Our main asymptotic result shows that the best-response dynamic converges to a pure Nash equilibrium in a vanishingly small fraction of all (large) games when players take turns according to a fixed cyclic order. By contrast, when the playing sequence is random, the dynamic converges to a pure Nash equilibrium if one exists in almost all (large) games.

We present a test of the two most established reciprocity models, an intention factor model and a reference value model. We test characteristic elements of each model in a series of twelve mini-ultimatum games. Results from online experiments show major differences between actual behavior and predictions of both models: the distance of actual offers to the proposed reference value provides a poor measure for the kindness of offers, while a comparison of offers with extreme offers as suggested by the intention factor model makes offers indiscriminable in richer settings. We discuss possible combinations of both models better describing our observations.