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When to challenge a call in tennis: A Markov decision process approach

Vamsi K. Nadimpalli and John J. Hasenbein

Abstract

In this paper, we develop a Markov decision process (MDP) model to determine when a player should challenge a line call in a game of tennis, if the objective is to maximize the probability of winning the game. The parameters in the model include the relative strength of the players and the fallibility of the officials. The player’s decision depends on the the current score, the number of challenges remaining, the outcome of a successful challenge, and his confidence that the line call is incorrect. The model developed is a multi-chain MDP operating under the long-run average cost criterion. We also performed extensive numerical studies when the player has one challenge remaining, varying the player strengths and the fallibility levels. These studies imply some general intuitive challenge strategies but also exhibit unusual strategic behavior in some game states. For example, in some states it is not optimal for very weak or very strong players to challenge calls. Furthermore, we demonstrate that the challenge decision is not “unimodal” as a function of the player’s strength, i.e., there may be multiple decision thresholds with respect to this parameter.


Corresponding author: John J. Hasenbein, Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, University of Texas at Austin, Austin, Texas, 78712, USA

We would like to thank the anonymous review team for helpful comments that improved the paper greatly. Thanks also goes to Eric Bickel at the University of Texas at Austin for his insight and advice on this research.

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Published Online: 2013-07-23
Published in Print: 2013-09-01

©2013 by Walter de Gruyter Berlin Boston