Insight: This series provides a blueprint for analyzing serve strategies, covering both serve direction win probabilities and serial correlation across successive serve directions.
In the previous two series, I show that whenever I compare win probabilities across different tactics (such as approaching the net vs. staying at the baseline) to identify which ones are most effective, I must control for the incoming ball’s characteristics. For example, players often have high win probabilities at net, but that’s often due to their propensity to approach on incoming balls that are easy to begin with. In contrast, relative to other shots, the serve, which is arguably the most important shot, is unique because serves don’t have incoming balls. Therefore, when I compare win probabilities across serve directions (or across speeds or spins, though I leave those for future analyses), I don’t need to include any control variables.
However, there are still confounders I must consider. For instance, suppose a server often hits to her opponent’s backhand on second serves, where win probabilities are generally lower than on first serves. (To avoid ambiguity about whether I’m referring to the server or returner, I use female pronouns for a generic server and male pronouns for a generic returner.) Then, if I were to pool first and second serve data together, I would find that the server’s win probability when serving to the returner’s backhand is lower than her win probability when serving to his forehand. Yet if I were to analyze first and second serves separately, I could possibly find higher win probabilities when serving to his forehand for both types of serves.
So what is the best way to analyze serve strategies? The 2001 paper “Minimax Play at Wimbledon” by economists Mark Walker and John Wooders provides a valuable blueprint. In this paper, Walker and Wooders (WW) introduce the term “point game” (not a tennis term), which they define as all first or second serves hit in a match by one player to the deuce or ad court. For example, in the 2016 Wimbledon Men’s Final between Andy Murray and Milos Raonic, one of the point games in the match is all of Murray’s first serves to the deuce court. As such, every tennis match has eight point games, though since WW only analyze first serves, they only have four point games per match (Player 1 deuce, Player 1 ad, Player 2 deuce, and Player 2 ad).
In total, WW analyze 40 point games across 10 men’s matches at Grand Slams and the Masters Cup. They specifically check for unequal win probabilities across serve directions, as well as for serial correlation, which occurs when one or more of the server’s previous serve direction choices influence her current serve direction choice. Positive serial correlation occurs if she switches back and forth between serve directions less often than a random server would, whereas negative serial correlation occurs if she switches more often than a random server.
Why do WW check for these phenomena? It’s because they prove mathematically that at a static equilibrium, which occurs when the server and returner care only about maximizing their odds of winning the current point, there should be equal win probabilities and serial independence (that is, no serial correlation) across all observed serve directions. Intuitively, these results make sense. A server shouldn’t serve to a direction with a lower win probability, and if there’s serial correlation, then the returner will have an advantage because he’ll have an idea of where the next serve is likely to go. Therefore, the server is incentivized to avoid any serial correlation.
Ultimately, when omitting body serves and focusing exclusively on serves hit to the returner’s left or right, WW find that tennis pros are very good at maintaining equal win probabilities but have a slight bias toward negative serial correlation. The latter result agrees with other studies finding that when asked to randomize, people tend to switch up their choices too often (see WW). In this series, I replicate WW’s analysis on the 2016 Wimbledon Final to demonstrate how it works. While dynamic models, such as those my co-authors and I have devised in the paper “Disequilibrium Play in Tennis,” allow for equilibria with unequal win probabilities or serial correlation, I keep things relatively simple for now because I believe it’s essential to start with the foundations before delving into more complex models, which I will reserve for future series.
