First server's effect on the expected number of games in tennis

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 $0.4$ games at both the set and match level. This translates, for example, to roughly a $9$ percent shift in the probability that a match exceeds $19.5$ games when the players' serve point win probabilities differ by $10$ percent. 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.