First server effect on the expected number of games in tennis

TL;DR

This paper investigates how the initial server affects the expected total games and margin in tennis matches, providing theoretical bounds and empirical validation under constant serve point probabilities.

Contribution

It identifies the conditions under which the first server significantly impacts match outcomes and confirms the bounded nature of this effect through numerical and empirical analysis.

Findings

The first server influences the expected number of games in tennis.

The effect of the first server is bounded by at most one game.

Empirical data supports the constant-probability model for match analysis.

Abstract

We show that information on the first server influences the expected total number of games and margin in a tennis match under the standard assumption that each player's serve point win probability remains constant, and identify the exact regions, in terms of these probabilities, in which this effect is non-negligible. We confirm numerically that this effect is bounded by at most one game at both the set and match level. We complement the analysis with an empirical comparison on professional match data, illustrating the adequacy of the constant-probability assumption for modelling the total number of games.