Monte Carlo Tennis

Abstract

The probability of winning a game, set, match, or single elimination tournament in tennis is computed using Monte Carlo simulations based on each player’s probability of winning a point on serve, which can be held constant or varied from point to point, game to game, or match to match. The theory, described in Newton and Keller [Stud. Appl. Math., 114 (2005), pp. 241–269], is based on the assumption that points in tennis are independent, identically distributed (i.i.d.) random variables. This is used as a baseline to compare with the simulations, which under similar circumstances are shown to converge quickly to the analytical curves in accordance with the weak law of large numbers. The concept of the importance of a point, game, and set to winning a match is described based on conditional probabilities and is used as a starting point to model non‐i.i.d.effects, allowing each player to vary, from point to point, his or her probability of winning on serve. Several non‐i.i.d.models are investigated, including the “hot‐hand‐effect,” in which we increase each player’s probability of winning a point on serve on the next point after a point is won. The “back‐to‐the‐wall” effect is modeled by increasing each player’s probability of winning a point on serve on the next point after a point is lost. In all cases, we find that the results provided by the theoretical curves based on the i.i.d.assumption are remarkably robust and accurate, even when relatively strong non‐i.i.d.effects are introduced. We end by showing examples of tournament predictions from the 2002 men’s and women’s U.S. Open draws based on the Monte Carlo simulations. We also describe Arrow’s impossibility theorem and discuss its relevance with regard to sports ranking systems, and we argue for the development of probability‐based ranking systems as a way to soften its consequences.