Abstract—This paper is concerned with the effect of nonmonetary incentives on behavior, in particular with the study of social pressure as a determinant of corruption. We offer empirical evidence that shows how professional soccer referees favor home teams in order to satisfy the crowds in the stadium. Referees have discretion over the addition of extra time at the end of a soccer game to compensate for lost time due to unusual stoppages. We find that referees systematically favor home teams by shortening close games where the home team is ahead, and lengthening close games where the home team is behind. They show no such bias for games that are not close. We further find that when the rewards for winning games increase, referees change their bias accordingly. Lastly, we identify that the mechanism through which bias operates is to satisfy the crowd, by documenting how the size and composition of the crowd affect referee favoritism. I.

The implications of the Minimax theorem are tested using natural data. The tests use a unique data set from penalty kicks in professional soccer games. In this natural setting experts play a one-shot two-person zero-sum game. The results of the tests are remarkably consistent with equilibrium play in every respect: (i) winning probabilities are statistically identical across strategies for players; (ii) players’ choices are serially independent. The tests have substantial power to distinguish equilibrium play from disequilibrium alternatives. These results represent the first time that both implications of von Neumann’s Minimax theorem are supported under natural conditions.

Unpredictable behavior is central for optimal play in many strategic situations because a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable behavior is presented in the context of repeated two-person zero-sum games in which the stage games have no pure strategy equilibrium. Computational complexity considerations are introduced to restrict players ’ strategy sets. The use of Kolmogorov complexity allows us to obtain a sufficient condition for equilibrium existence. The resulting theory has implications for the empirical literature that tests the equilibrium hypothesis in a similar context. In particular, the failure of some tests for randomness does not justify rejection of equilibrium play.

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In cricket, the right to make an important strategic decision is assigned via a coin toss. We utilize these “randomized trials ” to examine (a) the consistency of choices made by teams with strictly opposed preferences, and (b) the treatment effects of chosen actions. We find significant evidence of inconsistency, with teams often agreeing on who is to bat first. Estimated treatment effects show that choices are often poorly made since they reduce the probability of the team winning.

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