Disequilibrium Play in Tennis

Are the serves of the world’s best tennis pros consistent with the theoretical predictions of Nash equilibrium in mixed strategies? We analyze their serve direction choices (to the receiver’s left, right or body) with data from an online database called the Match Charting Project. Using a new methodology, we test and decisively reject a key implication of a mixed strategy Nash equilibrium, namely, that the probability of winning a service game is the same for all serve directions. We also use dynamic programming (DP) to numerically solve for the best-response serve strategies in models of the service game of tennis estimated for individual server-receiver pairs, such as Novak Djokovic serving to Rafael Nadal. We show that for most elite professional servers, the DP serve strategy significantly increases their service game win probability compared to the mixed strategies they actually use, which we estimate using flexible reduced-form logit models. Stochastic simulations verify that our results are robust to estimation error.