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Process Utility in High-Stakes Competition.
\noindentJEL Classification: D91, D81, D01, C57.
Keywords: Process utility; Intrinsic motivation; Salience weight; Strategic behavior; Nonparametric; Structural estimation.
A central question in economics is the extent to which individuals are willing to trade off extrinsic rewards against the intrinsic enjoyment of an activity (e.g., frey02; benabou03).\footnote{ See also deci99, Bandiera05, falk06, bryan11, larkin12, falk13, kube13, corgnet16 among others. } In many instances, this trade-off can be framed as one between an outcome (what) and the process (how) by which it is obtained. For instance, Adam Smith's discussion of compensating wage differentials and its formalization in modern labor economics (e.g., rosen86) acknowledges that job attributes such as effort, risk, or enjoyment enter workers' utility alongside wages. Empirically, however, quantifying this trade-off remains challenging. Observational data typically raises two important issues: preferences are confounded with constraints, selection, and unobserved heterogeneity, and identification relies on between-person comparisons, as indeed the frequency of such decisions is generally limited.\footnote{ For instance, in our labor market example, workers who accept lower wages for more enjoyable jobs may differ systematically in ability or outside options (e.g., mas25; lavetti23) and only change jobs a limited number of times in the course of their career.} Experimental evidence circumvents these issues and has provided important insights (e.g., gneezy00a,gneezy00b; ariely08), but often relies on low-stakes laboratory environments with limited external validity. As a result, there is relatively little direct field evidence at the individual level on whether people are willing to sacrifice measurable economic outcomes in exchange for intrinsically rewarding processes in high-stakes, competitive settings.
Professional tennis provides a uniquely well-suited setting to study this trade-off. Players operate in a high-stakes environment with strong extrinsic motivation (monetary and reputational incentives) to maximize performance. Yet, they face repeated, strategically rich decisions---most notably in serving with the second serve rule---where they can choose between higher-risk strategies that increase the probability of winning the point immediately in one shot (unreturned serve), but also of making a mistake, and more conservative strategies that lead to longer rallies. Crucially, because points are won either in a single shot or through multi-shot exchanges, tennis offers a setting to separately identify outcome and process motivations. Finally, being one of the most broadcast sports on the planet, a wealth of point-level data is recorded for many matches, allowing a rich analysis of tennis rallies for a large number of players. This combination of high incentives, clear strategic margins, and granular data makes tennis an ideal setting to test whether individuals deviate from outcome-maximizing behavior to engage in more intrinsically rewarding forms of play.
We propose a simple framework in which players derive utility from winning points -outcome utility-- but also from the way in which they win points -process utility--. In particular, we allow process utility to depend on whether a point is won in one shot, immediately after the serve, or through multi-shot rallies, and model serving decisions as the outcome of utility maximization under this extended preference structure. We call the relative importance of process utility in total utility the salience weight. Our framework yields a natural interpretation of observed deviations from outcome-maximizing strategies\footnote{ See borghans95, klaassen09 for instance.} as reflecting heterogeneous salience weights, which we recover for each player from point-level data.
Using optimal serving decisions in professional tennis, we develop a nonparametric identification strategy that exploits optimality conditions and the second-serve rule to derive a sufficient condition under which player-specific bounds on the salience weight are positive. Confronting this condition to detailed point-level data from the Sackmann Charting Project, we show that, under mild shape restrictions, a large majority of players likely have a positive salience weight. Further exploiting these optimality conditions, we adopt a parametric approach and propose an algorithm to structurally recover player-specific salience weights on process utility. We then estimate player-specific skills and salience weights using this algorithm in a maximum likelihood procedure. We find that $79\%$ of professional players have a positive salience weight---of which $64\%$ are statistically significant at the $5\%$ level---and therefore place greater weight on winning multi-shot rallies, resulting in systematically more conservative (second-)serve strategies than predicted by outcome-maximizing models.\footnote{ For 119 out of 151 players in the data, the estimated salience weight for winning multi-shot rallies is larger than 0. For 76 of them, this weight is significantly different from 0 at the $5\%$ level. For only 3 players, this weight is significantly lower than 0 at the $5\%$ level.} Counterfactual exercises show that eliminating process utility would increase point-winning probability on serve by about 0.4 percentage points, translating into a 2.4 percentage point increase in match-winning probability and an increase of \$33,000 (13.5%) in expected prize money at a Grand Slam tournament (US Open 2025). These results provide field evidence that individuals are willing to forgo measurable extrinsic rewards to engage in intrinsically rewarding activities, and illustrate how small deviations from outcome-maximizing behavior can have economically meaningful consequences in high-stakes competitive settings.
Methodologically, our model fits naturally within the canonical framework of trade-offs between extrinsic and intrinsic motivation. To fix ideas, consider an agent choosing among alternatives leading to an outcome. Each alternative $x$ is associated with outcome utility $p(x)$ and process utility $k(x)$. In its simplest form, utility is a weighted sum of the two, where ${\Greekmath 010E} \geq 0$ denotes the relative weight on process utility. When $ {\Greekmath 010E} = 0$, the agent is purely outcome-maximizing, while larger values of $ {\Greekmath 010E}$ reflect increasing salience of process utility.
This formulation is consistent with leading models of motivation, where process utility arises from civic duty (frey97), identity (loewenstein99, akerlof00), implicit contracts (gneezy00b,gneezy00a), beliefs (benabou03), procedures (frey04 ), meaning (ariely08, norton12), gambling (LeMenestrel01 ), or the act of choice itself (sen97). In our setting, a player chooses a serve strategy $x$ and trades off the probability of winning a point on serve $p(x)$ against the probability of winning the point through multiple shots $k(x)$. By structurally recovering player-specific salience weights from optimal choices, we quantify the extent to which individuals deviate from outcome maximization to engage in intrinsically rewarding play.
Our analysis builds on the premise that winning multi-shot rallies is intrinsically more rewarding than winning one-shot rallies. We motivate this in three steps. First, from the psychological literature, we learn that enjoyment and self-determination are important drivers of intrinsic motivation. Research first formalized by csik75, csik90 indicates that flow is a state of complete absorption in an activity, often described as being \textquotedblleft in the zone,\textquotedblright\ that arises when the challenge of the task aligns with the performer's skill level, and therefore \textquotedblleft \lbrack e]njoyment appears at the boundary between boredom and anxiety, when the challenges are just balanced with the person's capacity to act.\textquotedblright\ csik90, pp. 52--53.\footnote{ Csikszentmihalyi illustrates this with tennis: \textquotedblleft One cannot enjoy doing the same thing [at tennis] at the same level for long. We grow either bored or frustrated, and then the desire to enjoy ourselves again pushes us to stretch our skills, or to discover new opportunities for using them.\textquotedblright\ csik90, p.75.} Flow theory connects closely with Self-Determination Theory (SDT) deci00, which posits that intrinsic motivation is fostered when autonomy, competence, and relatedness are satisfied.\footnote{ In tennis, autonomy arises from controlling shot selection and tactics; competence from improving skills and executing difficult shots; and relatedness from interactions with coaches, opponents, and the public.} Flow states and Self-Determination are therefore more likely to arise during multi-shot rallies than through the execution of a single shot, the serve.
Second, from players' testimonies, we learn that enjoyment is indeed an important component of their motivation. Professional players frequently highlight enjoyment as a central goal: for instance, Carlos Alcaraz stated, \textquotedblleft I just want to step on court \ldots and try to enjoy as much as I can.\textquotedblright \footnote{ \url{https://www.atptour.com/en/news/alcaraz-lehecka-us-open-2025-qf}. A broader set of players' quotes is collected in Online Appendix ((ref)).}
Third, in tennis, although the server wins roughly 45% of his points on unreturned serves (one-shot), top players spend most of their training time (about $90\%$) on baseline rallies as shown in oshannessy19 and fitzpatrick24a. This apparent paradox in the behavior of players during practice supports the idea that players attach more importance to multi-shot rallies, likely so because they enjoy playing such rallies more. For these reasons, we expect process utility to arise from winning multi-shot rallies, and player-specific salience weights for these rallies capture the preference for winning points through multi-shot rallies relative to one-shot points.
This paper contributes in several important ways. First, it extends the analysis of tennis serving strategies of klaassen09 to account for process utility and distinguish between one-shot and multi-shot points. It proposes a novel method to compute player-specific salience weights from point-level data and quantifies the economic consequences of process-driven strategy, demonstrating that small deviations from outcome maximization can have substantial effects.
Second, more broadly, our results speak to a general class of preferences in which individuals derive utility not only from outcomes but also from the process by which these outcomes are achieved. This idea is closely related to the compensating differentials literature in labor economics, which documents that workers are willing to accept lower wages in exchange for non-pecuniary job attributes such as meaningful or intrinsically rewarding tasks, with estimated trade-offs on the order of 8--20% depending on the job attributes considered (e.g., Stern04,Mas17). Similarly, in consumer markets, a large body of evidence on fair trade, ethical consumption, and product provenance shows that individuals are willing to pay substantial price premia for goods produced under socially or environmentally desirable conditions, at comparable product quality. Experimental and field evidence typically finds willingness-to-pay premia in the range of approximately 5% to 30% for fair trade or ethically certified products, depending on product category (e.g., see Maertens09,Dragusanu14). Across these settings, individuals appear willing to trade off outcome against process, be it about having autonomy, providing meaning, or being ethical. Our contribution is to show that this same trade-off can be identified from high-frequency, within-individual, continuous choices in a high-stakes, competitive strategic environment, in contrast to labor and consumer studies, which typically rely on low-frequency or discrete-choice settings.
The remainder of the paper is structured as follows. Section 2 presents the model, discusses non-parametric bounds and introduces a parametric approach together with an algorithm to recover player-specific salience weights from the data. Section 3 describes the data and the estimation method, while Section 4 presents the results. Section 5 provides robustness checks, and Section 6 concludes.
In tennis, a central strategic decision for the server is how much risk to take. A “safe” serve increases the probability that the ball lands in the service box (i.e., a higher serve percentage) but reduces the likelihood of winning the point outright through an ace or unreturned serve. Conversely, a “risky” serve raises the probability of winning the point immediately, at the cost of a higher probability of a fault.
While this resembles a standard high-risk--high-reward trade-off, the tennis setting involves an additional margin: not only whether the point is won, but how it is won. A safe serve increases the likelihood that the point evolves into a multi-shot rally, forcing the player to win through multiple shots, whereas a risky serve increases the likelihood of a one-shot win.
The server's decision, therefore, determines both the overall probability of winning the point and the distribution of this probability over one-shot and multi-shot points. As a result, the serve strategy reflects a trade-off not only between high-risk--high-reward outcomes, but also between one-shot wins and multi-shot wins, which may be intrinsically valued differently.
In support of this distinction, we propose the following defintions.
Definition (Outcome Utility): Outcome utility captures preferences over the results of an action, independent of how they are achieved. In our setting, it corresponds to the probability of winning a point, regardless of whether it is won through one-shot or multi-shot rallies.
Definition (Process Utility): Process utility captures preferences over how outcomes are achieved. In our setting, it is represented by a preference for winning points through multi-shot rallies rather than one-shot points.
Let the probability of a serve being in be denoted by $x\in \left[ 0,1\right] $. This probability reflects the choice of the server. As depicted in the above observations, if the server wants to take more risk, he will choose a lower value of $x$, hence a lower serve percentage. In contrast, if he wants to take fewer risks, he will choose a higher value of $x$, a higher serve percentage. Since in tennis, players can serve a second serve if the first is out, the server's strategy consists, in fact, of two numbers $x_{1}$ and $ x_{2}$ reflecting respectively the probability of the first and second serve to be in. A player's probability to win a point on his serve, denoted $ p\left( x_{1},x_{2}\right) $, depends on his strategy $\left( x_{1},x_{2}\right) $. Denote $y\left( x\right) $ the probability of winning a point conditional on the serve being in as a function of the serve probability, $x$. $y\left( x\right) $ reflects the skills of the player, encompassing both his serving and rally skills (relative to his opponent). In condition ((ref)) below we assume that $y\left( x\right) $ is twice differentiable on $x\in \left[ 0,1\right] $ and in particular, strictly decreasing, i.e., $y^{\prime }\left( x\right) <0$, so that the safer the serve, that is, the higher the probability that it is in, the lower the probability of winning the point, conditional on the serve being in, and strictly concave, $y^{\prime \prime }\left( x\right) <0$.
With these definitions, the probability of winning a point on one's own serve, the outcome utility, reads as,
where $w\left( x\right) :=xy\left( x\right) $ is the unconditional probability of winning a point, and a server aiming at maximizing his probability of winning a point on his serve then does $\max_{x_{1},x_{2}}p \left( x_{1},x_{2}\right)$.
This setting corresponds to the basis of the model presented in klaassen09 and discussed in more detail in Online Appendix ((ref) ). In this paper, we depart from klaassen09 by assuming that players are perfect maximizers and allowing them to care about process utility, and hence possibly put different weights to the various possible ways of winning a point. This requires distinguishing between 4 possible ways to win a point: with one shot on the first or second serve, i.e., an ace or an unreturned serve, or with multiple shots on the first or second serve, i.e., a rally of more than 2 shots. We hence decompose the conditional probability to win a point into a conditional probability to win in one shot, say $ f\left( x\right) $, and in multiple shots, say $k\left( x\right) $. By definition, one has $y\left( x\right) =f\left( x\right) +k\left( x\right) $. It seems natural to expect that $f^{\prime }\left( x\right) <0$ and $ f^{\prime \prime }\left( x\right) <0$ so $f$ is strictly concave, meaning that the conditional probability of winning a one-shot point is decreasing with the probability of the serve to be in (increasing with risk), more so as the level of risks decreases (concave).\footnote{ This assumption is supported in the data. Indeed, for all professional players in the data, the percentage of first serves in is lower than the percentage of second serves in ($x_{1}<x_{2}$), but only for two players, the share of one-shot points won on first serve ($f\left( x_{1}\right) $) is lower than that on the second serve ($f\left( x_{2}\right) $).} In contrast, the conditional probability of winning a multi-shot point may be increasing or decreasing with the probability that the serve is in, depending on the skills of the player. Hence, we impose that, if it is decreasing, it is also concave, i.e., $k^{\prime }\left( x\right) <0$ and $k^{\prime \prime }\left( x\right) \leq 0$, whereas, if it is increasing, it is convex, i.e., $ k^{\prime }\left( x\right) >0$ and $k^{\prime \prime }\left( x\right) >0$.
To summarize, we assume that the following standing assumptions hold.
The second departure from the model in klaassen09, is that we consider the case where a player may attach more or less, but not necessarily the same, importance to winning a one-shot point rather than a multi-shot point. In the model, there are four possible (winning) outcomes for the server, listed below with their specific probability to occur and specific utility:
Hence, we assume that the player maximizes, not his probability to win a point $p\left( x_{1},x_{2}\right) $, but rather, $\tilde{p}\left( x_{1},x_{2}\right) $ defined as the weighted average of the probability to win a one-shot point and the probability to win a multi-shot point, which reads as
where $\tilde{w}\left( x\right) =x\tilde{y}\left( x\right) $, $\tilde{y} \left( x\right) ={\Greekmath 010B} f\left( x\right) +{\Greekmath 010C} k\left( x\right) $, ${\Greekmath 010B} $ is the utility attached to winning a one-shot point, and ${\Greekmath 010C} $ is the utility attached to winning a multi-shot point.
A first important remark is that normalizing the utility of winning one-shot rallies to ${\Greekmath 010B} =1$ is without loss of generality, as it does not affect the optimal solution of a player. From now on, we therefore set ${\Greekmath 010B} =1$ and interpret ${\Greekmath 010C} $ as the relative preference parameter for multi-shot rallies. Moreover, in the case ${\Greekmath 010C} =1$, distinguishing between $f$ and $k$ is irrelevant since all that matters is the conditional probability of winning a point $y\left( x\right) =f\left( x\right) +k\left( x\right) $ and not how. In terms of notation, we call $\tilde{y}$ the perceived conditional probability of winning a point because of the presence of preference parameter ${\Greekmath 010C} $ in the expression. It only coincides with the true probability when ${\Greekmath 010C} =1$. A similar interpretation and notation is used for $w(x)$ and $p\left( x_{1},x_{2}\right) $.
A second important remark is that this utility rewrites as
clearly distinguishing the outcome utility, i.e., the probability of winning a point on one's own serve, and process utility, i.e., the probability of winning a multi-shot point on one's own serve. Presented this way, it is clear that our model relates to the simple model of outcome and process utility presented in the introduction, where the salience weight ${\Greekmath 010E} $ obtains as ${\Greekmath 010E} ={\Greekmath 010C} -1$.
To summarize, our setting distinguishes between one-shot and multi-shot rallies and allows preference weights to be different for these two types of rallies, replacing $w\left( x\right) $ by $\tilde{w}\left( x\right) $ and decomposing $y\left( x\right) $ into the constituants $f\left( x\right) $ and $k\left( x\right) $ to compute $\tilde{y}\left( x\right) $.
The FOCs to this problem are obtained as
The second order conditions require that the expected utility is concave. For this, the Hessian of the expected utility needs to be semi-definite negative and since at optimum, the Hessian is diagonal (see Online Appendix ( (ref))), the SOCs therefore are
Interestingly, the SOCs provide restrictions on the curvature of the perceived conditional probability of winning a point. Indeed, rearranging both SOCs, one obtains $x_{j}^{\ast }\frac{\tilde{y}^{\prime \prime }\left( x_{j}^{\ast }\right) }{\tilde{y}^{\prime }\left( x_{j}^{\ast }\right) }\geq -2$, provided that $\tilde{y}^{\prime }\left( x_{j}^{\ast }\right) <0$, $ \forall j=1,2$.\footnote{ Note that, while it is possible that ${\Greekmath 010C} $ is so that $\tilde{y}^{\prime }\left( x\right) >0$, even if $y^{\prime }\left( x\right) <0$, in that case, the optimal second serve strategy would be $x_{2}^{\ast }=1$ since $\tilde{w} ^{\prime }\left( x\right) >0$. For all players in the data, the observed second serve percentage is strictly lower than $1$, and hence this situation never occurs.} Note that $x\frac{\tilde{y}^{\prime \prime }\left( x\right) }{ \tilde{y}^{\prime }\left( x\right) }$ is the elasticity of the perceived marginal probability $\tilde{y}^{\prime }\left( x\right) $ and the SOCs, in fact, indicate that this elasticity should be larger than $-2$ on the interval $x\in \left[ x_{1}^{\ast },x_{2}^{\ast }\right] $. Moreover, the FOCs reveal important information about the shape of the perceived conditional probability $\tilde{y}$ at the optimum. Indeed, consider the FOC associated with the optimal second serve strategy. Rearranging, one has $ \tilde{y}^{\prime }\left( x_{2}^{\ast }\right) =\frac{\tilde{y}\left( x_{2}^{\ast }\right) }{x_{2}^{\ast }}$. By a similar procedure, the FOC for the first serve strategy obtains as $\tilde{y}^{\prime }\left( x_{1}^{\ast }\right) =\frac{x_{2}^{\ast }\tilde{y}\left( x_{2}^{\ast }\right) -\tilde{y} \left( x_{1}^{\ast }\right) }{x_{1}^{\ast }}$. It follows that the optimal serve strategy $\left( x_{1}^{\ast },x_{2}^{\ast }\right) $ pins down the slope of the perceived conditional probability of winning a point at both $ x_{2}^{\ast }$ and $x_{1}^{\ast }$ and hence the average curvature of $ \tilde{y}\left( x\right) $ in the interval $\left[ x_{1}^{\ast },x_{2}^{\ast }\right] $ as
Figure ((ref)) shows how the optimal service strategy is determined given the skills parameters of the player (the shapes of $f$ and $ k$) and his relative preference for winning multi-shot rallies (${\Greekmath 010C} $). \footnote{ The additional source of identification provided by distinguishing between one-shot and multi-shot rallies is illustrated in Figure ((ref) ) of Online Appendix ((ref)).} We herewith use the example of Roger Federer.
First, the second serve strategy is found by looking at the value of $x$ for which $\tilde{w}^{\prime }\left( x\right) =0$, say $x_{2}^{\ast }$. Then, the first serve strategy is derived by looking at the value $x$ for which $ \tilde{w}^{\prime }\left( x\right) =\tilde{w}\left( x_{2}^{\ast }\right) $, say $x_{1}^{\ast }$. The conditional probabilities of winning a point given that the first (second) serve is in are indicated on the curve $y\left( x\right) $, for the respective values of $x_{1}^{\ast }$ and $x_{2}^{\ast }$ . This delivers two observed points $\left( x_{1}^{\ast },y_{1}^{\ast }\right) $ and $\left( x_{2}^{\ast },y_{2}^{\ast }\right) $.
Our aim is to nonparametrically bound the salience weight ${\Greekmath 010E} $. The key idea is that optimal serve choices impose inequality restrictions on $\tilde{ p}(.,.)$, which translate into player-specific bounds on ${\Greekmath 010E} $. To show this, we first need to briefly introduce the data. Suppose that, for each player $i=1,...,N$, we observe the probabilities $\left( x_{1i},x_{2i},f_{1i},f_{2i},k_{1i},k_{2i}\right) $ where $x_{1i}$ and $ x_{2i} $ are the probabilities of first and second serves in, $ f_{1i}=f\left( x_{1i}\right) $ and $f_{2i}=f\left( x_{2i}\right) $ are the conditional probabilities of winning a point with one shot on first and second serves, respectively, and $k_{1i}=k\left( x_{1i}\right) $ and $ k_{2i}=k\left( x_{2i}\right) $ are the conditional probabilities of winning a point with multiple shots on first and second serves, respectively. \footnote{ In the data section, we discuss how to estimate these probabilities $\left( x_{1i},x_{2i},f_{1i},f_{2i},k_{1i},k_{2i}\right) $ given data on rallies played, in possibly multiple matches, on the serve of each player $i$.}
We assume that each player $i$ is a perfect maximizer of his perceived probability of winning a point $\tilde{p}\left( x_{1},x_{2}\right) $ so that the observed data reflect the optimum of each player, i.e., $\left( x_{1i},x_{2i}\right) =\left( x_{1i}^{\ast },x_{2i}^{\ast }\right) $. Dropping the index $i$ for notational simplicity, optimality implies for each player
The first inequality compares the observed (optimal) strategy to a deviation in which the player uses the first-serve strategy in both serves; the second compares it to always using the second-serve strategy.\footnote{ Note that inequality ((ref)) implies $\tilde{p}\left( x,x_{2}^{\ast }\right) \geq \tilde{p}\left( x,x_{1}^{\ast }\right) $ for all $x$, so that the third inequality derived from optimality, i.e., $\tilde{p}(x_{1}^{\ast },x_{2}^{\ast })\geq \tilde{p}(x_{2}^{\ast },x_{1}^{\ast })$, is in fact implied by inequalities ((ref)-(ref)). Indeed, note that, as long as one maintains the first serve strategy constant, the first serve strategy contributes the same term $\tilde{w}\left( x\right) $ and a slope $ \left( 1-x\right) $ to $\tilde{p}\left( x,x_{2}\right) $ and $\tilde{p} \left( x,x_{1}\right) $ so that if the inequality $\tilde{p}\left( x,x_{2}\right) >\tilde{p}\left( x,x_{1}\right) $ holds for $x$, it holds for all $x^{\prime }\neq x$.} Each inequality above can be written in linear form $A+{\Greekmath 010E} B\geq 0$ implying ${\Greekmath 010E} \geq -\frac{A}{B}$ if $B>0$ and $ {\Greekmath 010E} \leq -\frac{A}{B}$ if $B<0$. The parameter $A$ is in fact the difference in the probability of winning a point ($p\left( .,.\right) $ not $ \tilde{p}\left( .,.\right) $) between the two strategies being compared in the inequality, whereas the parameter $B$ is a similar difference, but for the probability of winning a multi-shot point. By construction, $A-B:=C$ is the same difference, but for the probability of winning a one-shot point.
The optimality conditions can be used to define a lower bound $L$ (when $B>0$ ) and an upper bound $U$ (when $B<0$) on the value of the salience weight ${\Greekmath 010E} \in \left[ L,U\right] $ of each player. For all 151 players in our data, we note that the lower bound $L$ is determined by inequality ((ref)) whereas the upper bound is either unrestricted ($+\infty $) by the optimality conditions, which is the case for 23 players, or determined by inequality ((ref)). Regarding the sign of the bounds, the data actually show that for 8 players the lower bound is positive ($B>0$ and $A<0$ ) so that we can conclude that the salience weight of these players must be positive. For the remaining players, the lower bound is negative ($B>0$ and $ A>0$), and the upper bound positive ($B<0$ and $A>0$).
The optimality conditions alone are only enough to determine the sign of the salience weight ${\Greekmath 010E} $ of 8 players. However, these conditions highlight a strategy to derive a sufficient condition for a positive lower bound that can be checked against the data for the remaining players. We note that the lower bound $L$ is determined by an upper bound of $\tilde{p}\left( x_{1}^{\ast },x_{1}^{\ast }\right) $, i.e., inequality ((ref)). Building on this observation, we therefore investigate whether there exists a $x_{0}\in \left[ x_{1}^{\ast },x_{2}^{\ast }\right] $ so that the inequality
is satisfied, where the associated values of $A$ and $B$ for this inequality are
If there exists a $x_{0}\in \left[ x_{1}^{\ast },x_{2}^{\ast }\right] $ so that $A\left( x_{0}\right) \leq 0$ and $B\left( x_{0}\right) >0$ then this inequality delivers a positive lower bound on the salience weight.\footnote{ We already know that for 8 players in the data when $x_{0}=x_{1}^{\ast }$, which corresponds to Inequality ((ref)), $A\left( x_{1}^{\ast }\right) <0$ and $B\left( x_{1}^{\ast }\right) >0$.}
We proceed in two steps. First, we show in Lemma ((ref)), conditions under which there exists a unique $x_{0}\in \left[ x_{1}^{\ast },x_{2}^{\ast }\right] $ so that $A\left( x_{0}\right) =0$. Then, we show in Lemma ((ref)) conditions under which $B\left( x_{0}\right) $ is positive.
Conditions (b)--(c) of Lemma ((ref)) are sign restrictions directly verifiable in the data. As it turns out, we find that, for all but one player, these conditions are satisfied and hence Lemma ((ref)) applies.
Next, we want to show conditions under which $B\left( x_{0}\right) >0$. We first note that, since $A\left( x_{0}\right) =0$,
Inequality ((ref)) cannot directly be tested in the data since we do not observe $x_{0}$. However, a sufficient condition for this inequality to hold is
and Lemma ((ref)) below shows that one can quantify the extent to which Inequality ((ref)) restricts the graph of the function $ m\left( x\right) :=xf\left( x\right) $ using observable data for each player and hence quantify how likely this inequality is met.
Note that, condition (b) and quantity $A_{2}/A_{1}$ of Lemma ((ref)) can be checked/computed for each player in the data. We find that condition (b) is satisfied for all but one player (Pedro Martinez) so that Lemma ((ref)) applies for these players. Regarding the share $A_{2}/A_{1}$, we find the following interesting results. First, for the 8 players whose optimality conditions already guarantee a positive lower bound on the salience weight, we find that $A_{2}/A_{1}=0$ for 7 of them, and $A_{2}/A_{1}=11.4\%$ for the remaining one (Mats Wilander). This confirms that the metric $A_{2}/A_{1}$ is informative about the likelihood that the sufficient condition for having a positive lower bound for ${\Greekmath 010E} $ is met. We further find that the sufficient condition for a positive salience weight is met with certainty for 15 players ($A_{2}/A_{1}=0$) and restricts the feasible area by $5\%$ or less for 69 players. For 101 players, the restriction represents less than $10\%$; it is more than $20\%$ for only 20 players. Note that the median value of $A_{2}/A_{1}$ is $5.8\%$. Finally, as a last means of comparison, we note that there are 110 players with a value of $A_{2}/A_{1}<11.4\%$, i.e., the value obtained for Mats Wilander, the player known to have a positive lower bound for the salience weight from the optimality conditions but a non $0$ ratio for $A_{2}/A_{1}$. We conclude that the sufficient condition for the salience weight to be positive is likely satisfied for a large majority of players.
Our objective is to estimate the salience weight ${\Greekmath 010E}$ (or equivalently, the relative preference for winning a multi-shot point ${\Greekmath 010C}$) to conduct counterfactual analyses. This requires imposing a parametric structure on the model, in particular on the functions $f(x)$ and $k(x)$. Recall that the SOCs of the optimization problem give a restriction on the elasticity of $ \tilde{y}^{\prime }\left( x\right) $, the perceived marginal probability of winning a point. Since $\tilde{y}^{\prime }\left( x\right) =f^{\prime }\left( x\right) +{\Greekmath 010C} k^{\prime }\left( x\right) $, we propose to parametrize the elasticity of the marginal probabilities $f^{\prime }\left( x\right) $ and $k^{\prime }\left( x\right) $ as in the following condition.
Condition ((ref)) imposes that the elasticities of the marginal probability of winning a point with one-shot and with multi-shot, conditional on the serve being in, are equal to each other and to a strictly positive constant ${\Greekmath 0115} -1$.
This condition has three main implications. First, the elasticity of both the perceived and true marginal probability of winning a point is also equal to ${\Greekmath 0115} -1$, as under condition ((ref)) one has $x \frac{\tilde{y}^{\prime \prime }\left( x\right) }{\tilde{y}^{\prime }\left( x\right) }=x\frac{y^{\prime \prime }\left( x\right) }{y^{\prime }\left( x\right) }={\Greekmath 0115} -1$.\footnote{ Indeed, since one has $x\frac{f^{\prime \prime }\left( x\right) }{f^{\prime }\left( x\right) }=x\frac{k^{\prime \prime }\left( x\right) }{k^{\prime }\left( x\right) }={\Greekmath 0115} -1$, it follows that $xk^{\prime \prime }\left( x\right) =\left( {\Greekmath 0115} -1\right) k^{\prime }\left( x\right) $ and $ xf^{\prime \prime }\left( x\right) =\left( {\Greekmath 0115} -1\right) f^{\prime }\left( x\right) $ and since $xk^{\prime \prime }\left( x\right) +xf^{\prime \prime }\left( x\right) =xy^{\prime \prime }\left( x\right) $ and $ xk^{\prime \prime }\left( x\right) +x{\Greekmath 010C} f^{\prime \prime }\left( x\right) =x\tilde{y}^{\prime \prime }\left( x\right) $, one has
which for $\tilde{y}^{\prime }\left( x\right) ,y^{\prime }\left( x\right) >0$ $\ $yields the result in the text.} Furthermore, since, by assumption $ y^{\prime }\left( x\right) <0$, this means that $y\left( x\right) $ follows the law of diminishing marginal returns (read conditional probability of winning a point).
Second, it means that $f\left( x\right) $ and $k\left( x\right) $ are power functions of the form $f\left( x\right) =\frac{a_{f}-x^{{\Greekmath 0115} }}{{\Greekmath 011C} _{f}} $ and $k\left( x\right) =\frac{a_{k}-x^{{\Greekmath 0115} }}{{\Greekmath 011C} _{k}}$, offering great flexibility with only 5 unknown parameters.
Third, as a by product of the two preceeding remarks, the conditional probability of winning a point $y\left( x\right) $ is itself a power function with power ${\Greekmath 0115} $, as indeed $y\left( x\right) =f\left( x\right) +k\left( x\right) =\frac{a-x^{{\Greekmath 0115} }}{{\Greekmath 011C} }$, where ${\Greekmath 011C} = \frac{{\Greekmath 011C} _{f}{\Greekmath 011C} _{k}}{{\Greekmath 011C} _{f}+{\Greekmath 011C} _{k}}$ and $a=\frac{a_{f}{\Greekmath 011C} _{k}+a_{k}{\Greekmath 011C} _{f}}{{\Greekmath 011C} _{f}+{\Greekmath 011C} _{k}}$, see Online Appendix ((ref)).\footnote{ This corresponds to the functional shape assumed in Klaassen and Magnus (2009) for $y\left( x\right) $.}
Associated with this parametric choice, Conditions ((ref)) are met with the following restrictions on the parameters of $f\left( x\right) $ and $k\left( x\right) $.
Note that Condition ((ref).ii) garantees that $y\left( x\right) =f\left( x\right) +k\left( x\right) $ is decreasing, as indeed it leads to ${\Greekmath 011C} =\frac{{\Greekmath 011C} _{f}{\Greekmath 011C} _{k}}{{\Greekmath 011C} _{f}+{\Greekmath 011C} _{k}}>0$, and $ f^{\prime }\left( x\right) <0$ from ${\Greekmath 011C} _{f}>0$.\footnote{ In our data, these conditions are met for all players except for the condition $-{\Greekmath 011C} _{k}>{\Greekmath 010C} {\Greekmath 011C} _{f}$, which is not met for 3 of them.}
With these parametric choices, the optimal first and second serve strategies can be derived in closed form from the first-order conditions (see Online Appendix ((ref))). One obtains for the second serve strategy
yielding an interior solution, i.e., $0<x_{2}^{\ast }<1$, if and only if $ {\Greekmath 0115} +1>\frac{a_{f}+a_{k}{\Greekmath 010C} \frac{{\Greekmath 011C} _{f}}{{\Greekmath 011C} _{k}}}{1+{\Greekmath 010C} \frac{ {\Greekmath 011C} _{f}}{{\Greekmath 011C} _{k}}}>0.$
And, the first serve strategy, therefore obtains, after simple substitution, as
This is a remarkable result\footnote{ This result arises not only with power functions but also with softmax functions, see Online Appendix ((ref)).} indicating that, when players are optimizers, data on first and second serve percentages uniquely identify the curvature parameter ${\Greekmath 0115} $ as shown in section ( (ref)).
To summarize, under our standing assumptions, at optimality, the server adopts the following service strategy
and enjoys the following conditional probabilities of winning a point in one and multiple shots on first and second serve:
An important remark is that although the preference parameter ${\Greekmath 010C} $ and the relative skills $\frac{{\Greekmath 011C} _{f}}{{\Greekmath 011C} _{k}}$ only enter the expressions of the optimal serve strategy on first and second serve, through the term $ {\Greekmath 010C} \frac{{\Greekmath 011C} _{f}}{{\Greekmath 011C} _{k}}$, the expressions for the conditional probabilities of winning a one-shot or multi-shot point at the optimum depend respectively, only (directly) on the skills parameters ${\Greekmath 011C} _{f}$ and ${\Greekmath 011C} _{k}$. At same value of ${\Greekmath 010C} \frac{{\Greekmath 011C} _{f}}{{\Greekmath 011C} _{k}}$, i.e., at same optimal serve strategy, players with different skills ${\Greekmath 011C} _{f}$ and ${\Greekmath 011C} _{k}$ have different optimal conditional probabilities of winning a one-shot or multi-shot point. This is the source for the separate identification of the preference parameter and the skills parameters to exploit in the data.
$\left( x_{1}^{\ast },x_{2}^{\ast }\right) $ is an optimum if the expected utility is concave at $\left( x_{1}^{\ast },x_{2}^{\ast }\right) $. With the parametric shapes assumed above, the conditions for the expected utility to be concave are given as
Since ${\Greekmath 011C} _{f}>0$, ${\Greekmath 0115} >0$, $x_{1}^{\ast }>0$, and from Condition ( (ref).ii) one has either ${\Greekmath 011C} _{k}>0$ or $-{\Greekmath 011C} _{k}>{\Greekmath 010C} {\Greekmath 011C} _{f}$ so that $1+{\Greekmath 010C} \frac{{\Greekmath 011C} _{f}}{{\Greekmath 011C} _{k}}>0$, we conclude that the expected utility is concave.
It is easy to show by simple substitution that when ${\Greekmath 010C} =1$, the optimal strategy is
and the SOC reads as ${\Greekmath 011C} \geq 0$.\footnote{ See Online Appendix (ref) for comparative statics.}
For each player $i=1,...,N$, we observe the probabilities $\left( x_{1i},x_{2i},f_{1i},f_{2i},k_{1i},k_{2i}\right) $ and have 5 unknown skill parameters $\left( {\Greekmath 0115} _{i},a_{fi},{\Greekmath 011C} _{fi},a_{ki},{\Greekmath 011C} _{ki}\right) $ and 1 unknown preference parameter ${\Greekmath 010C} _{i}$. The two points $\left( x_{1i},f_{1i}\right) $ and $\left( x_{2i},f_{2i}\right) $ can be used to identify 2 of the 3 parameters of the function $f\left( x\right) $, whereas the two points $\left( x_{1i},k_{1i}\right) $ and $\left( x_{2i},k_{2i}\right) $ can be used to identify 2 of the 3 parameters of the function $k\left( x\right) $. Since $f\left( x\right) $ and $k\left( x\right) $ have one parameter in common, i.e., ${\Greekmath 0115} _{i}$, this means that the four points $\left( x_{1i},f_{1i}\right) $, $\left( x_{1i},k_{1i}\right) $, $\left( x_{2i},f_{2i}\right) $ and $\left( x_{2i},k_{2i}\right) $ together only identify 4 of the 5 skills parameters. Assuming that these players are perfect optimizers, the optimality conditions imply that $\left( x_{1i},x_{2i}\right) =\left( x_{1i}^{\ast },x_{2i}^{\ast }\right) $ which provides 2 restrictions to the system of equations. Hence, we have 6 parameters to be identified by 4 data points and 2 optimality restrictions.
To show the identification of parameters $\left( {\Greekmath 0115} _{i},a_{fi},{\Greekmath 011C} _{fi},a_{ki},{\Greekmath 011C} _{ki},{\Greekmath 010C} _{i}\right) $ given data $\left( x_{1i},x_{2i},f_{1i},f_{2i},k_{1i},k_{2i}\right) $, we first show identification of parameters $\left( a_{fi},{\Greekmath 011C} _{fi},a_{ki},{\Greekmath 011C} _{ki},{\Greekmath 010C} _{i}\right) $ given data $\left( x_{1i},x_{2i},f_{1i},f_{2i},k_{1i},k_{2i}\right) $ conditional on ${\Greekmath 0115} _{i}$, and propose a bisection algorithm that searches for the curvature parameter ${\Greekmath 0115} $ given the data $\left( x_{1i},x_{2i}\right) $.
First, the slope and constant terms of the functions $f\left( x\right) $ and $k\left( x\right) $ are identified given ${\Greekmath 0115} _{i}={\Greekmath 0115} $. Indeed, as soon as the value of ${\Greekmath 0115} $ is known, data $\left( x_{1i},x_{2i}\right) $ can be used to compute $\left( z_{1i},z_{2i}\right) =\left( x_{1i}^{{\Greekmath 0115} },x_{2i}^{{\Greekmath 0115} }\right) $. It follows that, using the functional form for $f\left( x\right) $ and $k\left( x\right) $, from the points $\left( z_{1i},f_{1i}\right) $ and $\left( z_{2i},f_{2i}\right) $, by simply rearranging terms, one can deduce the slopes
where $\Delta l_{i}=\frac{l_{1i}-l_{2i}}{l_{2i}}$ $\forall l=z,f,k$, and then the constants
Second, one can uncover the preference parameter ${\Greekmath 010C} _{i}$ once the slope and constant parameters of $f\left( x\right) $ and $k\left( x\right) $ are known. This is done by using the previous results together with the equation for the optimal second serve strategy to isolate
provided $z_{2i}\left( 1+{\Greekmath 0115} \right) -a_{ki}\neq 0$. Using the expressions previously obtained, for $x_{2i}>0$, $f_{2i}>0$ and $k_{2i}>0$, it can be shown that in fact
and it follows that one must have $\Delta z_{i}+{\Greekmath 0115} \Delta k_{i}\neq 0$.
Note that after following these steps, the curvature parameter is the only remaining unknown parameter. We can then use the last remaining condition, i.e., the FOC for the optimal first serve strategy, to implicitly solve for the curvature parameter and obtain
since $z_{1i}>0$.
Importantly, as shown in Online Appendix ((ref)), this expression simplifies considerably to read as
This expression actually shows that the curvature parameter only depends on data $x_{1i}$ and $x_{2i}$ through $z_{1i}$ and $z_{2i}$ and the initial value of ${\Greekmath 0115} $ selected. Of course, the value ${\Greekmath 0115} _{i}$ herewith obtained might be different than the ${\Greekmath 0115} $ used to compute $z_{1i}$ and $z_{2i}$. However, as it turns out, this equation together with the structure of the problem, provides a fixed-point, so that there exists, for each player $i$, a ${\Greekmath 0115} $ such that ${\Greekmath 0115} _{i}={\Greekmath 0115} $. We shall see below that a relatively simple algorithm allows us to recover this value, for all the players in the data, and that this value is larger than unity for all players.
We propose the following algorithm to compute the parameter ${\Greekmath 0115} _{i}$ of each player $i$, from the associated data $\left( x_{1i},x_{2i}\right) $.
Note that the iteration step of Algorithm ((ref)) defines a map $ \Lambda _{i}: \mathbb{R} _{0}^{+}\rightarrow \mathbb{R} _{0}^{+}$, associating for each ${\Greekmath 0115} ^{(t)}$ a new value ${\Greekmath 0115} ^{(t+1)}$ for the curvature parameter given the data $\left( x_{1i},x_{2i}\right) $. If it exists, a solution is therefore a fixed-point $ \Lambda _{i}\left( {\Greekmath 0115} _{i}^{\ast }\right) ={\Greekmath 0115} _{i}^{\ast }$. The following theorem shows that such a solution exists and is unique under mild conditions for the structure of the data $\left( x_{1i},x_{2i}\right) $.
We illustrate the theorem using the example of Roger Federer. Figure ((ref)) plots the map $\Lambda _{i}({\Greekmath 0115} )$ for Roger Federer. We see that the properties of the map exploited in the proof, are so that the map crosses the 45 degree line only ones at ${\Greekmath 0115} ^{\ast }$, the point $\left( {\Greekmath 0115} ^{\ast },\Lambda _{i}({\Greekmath 0115} ^{\ast })\right) $ forming the unique (non zero) fixed-point of the map. Suppose that the algorithm starts with lower bound $l^{\left( 0\right) }=1$ and upper bound $ u^{\left( 0\right) }=4$. The mid point value is 2.5. At ${\Greekmath 0115} =2.5$, the value of the map is below the 45 degree line and the lower bound is updated to 2.5. The termination condition is not satisfied (at conventional levels of precision) and the algortihm goes back to the iteration step with lower bound 2.5 and upper bound 4. The mid point is now 3.75 so that the value of the map is higher than the 45 degree line. The upper bound is updated to 3.75, etc.. The algorithm converges very fast to the value ${\Greekmath 0115} ^{\ast }=2.81$.
Once the curvature parameter ${\Greekmath 0115} _{i}$ is obtained from the algorithm, one can compute the remaining parameters using the steps outlined in the Section ((ref)).
We use the Match Charting Project by Jeff Sackmann,\footnote{ https://github.com/JeffSackmann/tennis_MatchChartingProject} which collects information about professional tennis matches, encoded by dozens of contributors. In particular, we use the point-by-point data for men's matches with information on more than 1,200,000 rallies of over 7,100 matches by the end of January 2026.
The unit of observation is a rally in a match. For each rally, we know who is serving, whether it is a rally on first or second serve, and the length of the rally, i.e., the number of shots that were recorded “in" the court. A rally of length 1 is necessarily ending with either an ace or an unreturned serve. The server wins all rallies of odd length, whereas the returner wins rallies of even length. For our analysis, for each player, we need to observe a large number of rallies on their own serve. For this reason, we only select those players with at least 20 matches charted in the data. 151 players satisfy this criterion.
Table ((ref)) presents service statistics for selected players, along with sample descriptive statistics. The selected players include the four players with the most Grand Slam titles\footnote{ The four most prestigious tournaments of the year are the Australian Open, Roland Garros, Wimbledon, and the US Open. These tournaments also have the most generous prize money distributions.}, i.e., the Greatest Of All Times (GOATs): Novak Djokovic, Rafael Nadal, Roger Federer, and Pete Sampras; Boris Becker; the best two players of the current generation, Carlos Alcaraz and Jannik Sinner; three players known for their big serves, John Isner, Reilly Opelka, and Ivo Karlovic; and two players known for their baseline game, David Ferrer and Diego Schwartzman.
The data include between $1{,}300$ and $58{,}000$ rallies per player. On average, the first serve percentage is about $61\%$, while the second serve percentage is roughly $91\%$. Players win approximately $72\%$ of points played on their first serve, with an equal distribution between one-shot and multi-shot rallies, whereas $63\%$ of points on the first serve are multi-shot rallies. On the second serve, being more conservative, players win fewer one-shot points ($17\%$) and about $37\%$ of multi-shot rallies. The percentage of multi-shot rallies on second serves is $83\%$, roughly $20$ percentage points higher than on the first serve.
There are also notable disparities across players. Big servers, such as John Isner, win about twice as many one-shot points as multi-shot points on their first serve, whereas the reverse is nearly true for baseline specialists like David Ferrer. Comparing newer top players to earlier ones, Carlos Alcaraz's statistics resemble those of Rafael Nadal, and Jannik Sinner's resemble those of Roger Federer. Finally, the percentage of points won on one-shot rallies ranges from $10\%$ to $41\%$ on the second serve and from $ 20\%$ to $61\%$ on the first serve, while for multi-shot rallies, the corresponding ranges are $27\%$ to $44\%$ on the second serve and $21\%$ to $ 43\%$ on the first serve.
Let ${\Greekmath 0112} _{i}=\left( x_{1i},x_{2i},f_{1i},f_{2i},k_{1i},k_{2i}\right) $ be the unknown probabilities of interest. From the data, for each player $ i=1,...,N$, we observe $N_{i}$ points played on his serve. We can compute the number of points played on the first serve, i.e., $n_{x_{1}i}$ and on the second serve $n_{x_{2}i}=N_{i}-n_{x_{1}i}$ and since the data identifies for each point, the length of the rally, i.e., the number of shots played into the court during the point, we can also compute the number of rallies of length 1 on first and second serves, i.e., $n_{f_{1}i}$ and $n_{f_{2}i}$ and the number of other rallies of odd length on the first and second serves as well, i.e., $n_{k_{1}i}$ and $n_{k_{2}i}$.
Note that klaassen01a showed that even though points in Tennis are not i.i.d., the deviations from the i.i.d. hypothesis are small. As a result, the i.i.d. hypothesis can still be used as a good approximation when aggregating over a large number of points as we do in this paper, where we use \textquotedblleft averages\textquotedblright\ over points by players. We therefore maintain the assumption that points are i.i.d. so that each of the aforementioned variables follows a binomial distribution.\footnote{ See Online Appendix ((ref)) for a tree representation of the statistics for each player.} For instance, on the first serve, one has:
It implies, for instance, that the log-likelihood of observing data $\left( n_{f_{1}i},n_{x_{1}i}\right) $ given probability $f_{1i}$ is
Applying the same logic to all data and collecting the associated terms of the log-likelihood, obtains
The first order condition of the maximum likelihood with respect to for instance $f_{1i}$ requires that
Hence, the frequency $\frac{n_{f_{1}i}}{n_{x_{1}i}}$ is the maximum likelihood estimate of probability $f_{1i}$. A similar argument holds for all other probabilities. Let $\hat{f}_{1i}=\frac{n_{f_{1}i}}{n_{x_{1}i}}$ denote the empirical of frequency of $f_{1i}$ with a similar notation for the other terms. Then, by maximum likelihood, the empirical frequencies $ \left( \hat{x}_{1i},\hat{x}_{2i},\hat{f}_{1i},\hat{f}_{2i},\hat{k}_{1i},\hat{ k}_{2i}\right) $ are the estimates of ${\Greekmath 0112} _{i}=\left( x_{1i},x_{2i},f_{1i},f_{2i},k_{1i},k_{2i}\right) $.
Theorem ((ref)) applies on data satisfying conditions (A1)-(A3), which garantees the convergence of Algorithm ((ref)) to a solution ${\Greekmath 0115} $ larger than unity. As indicated in Table ((ref)), for all professional tennis players in our data, the first serve percentage ($x_{1}$) ranges between $0.51$ and $0.72$ while the second serve percentage ranges between\footnote{ In fact, Maxime Cressy and Alexandre Bublik are the only two players with second serve percentages lower than $84\%$.} $0.73$ and $0.97$. This trivially shows that condition (A1) is always met. Since $x_{1i}>0.5$, one has
which also garantees that Condition (A2) is satisfied for the ranges of values for $x_{1}$ and $x_{2}$ observed in the data. It follows that, for all players in the data, there exists a unique solution ${\Greekmath 0115} _{i}^{\ast } $. Last, note that Condition (A3) rewrites as $x_{2i}\left( 2-x_{2i}\right) <2x_{1i}$ and in the data,
so that this condition is also satisfied for all players in the data. Hence, the unique solution ${\Greekmath 0115} _{i}^{\ast }$ is strictly greater than unity for all $i=1,..,N$.
Table ((ref)) presents the estimates of the parameters for our selected players as well as the summary statistics for all players (bottom rows). The mean and median salience weights in our sample are both $1/3$, indicating a preference for winning multi-shot rallies. In fact, $79\%$ (119/151) of the players have a positive salience weight, and for about $ 64\% $ (76/119) of them, that coefficient is statistically significant at $ 5\%$. In contrast, there are 32 players with negative salience weight, and only 3 of them for whom that estimate is statistically significant at $5\%$. The first four listed players in the table are the GOATs. We see that for all of them, the salience weight is positive and significant. The salience weight is also positive and significant for big servers like John Isner, Ralley Opelka, and Ivo Karlovic. Interestingly, the salience weight for the new top players, Carlos Alcaraz and Jannik Sinner, is positive (0.05) but not significant. In particular, although the statistics in Table ((ref)) for Jannik Sinner were quite close to those of Roger Federer and those of Carlos Alcaraz to those of Rafael Nadal, their salience weights are quite different. This reflects the fact that the identification of the salience weight shown in Section ((ref)) is non-trivial.
The curvature parameter ${\Greekmath 0115}$ is on average 2.9 and ranges from 1.3 to 6.3. It is significantly different from unity for all players in the data, so that the conditional probability of winning points of all players abbeys the law of diminishing marginal returns. The slope and constant parameters of its constituents ($f(x)$ and $k(x)$) are also mostly significantly different from 0. They can be best interpreted using a graphical representation, as in Figure ((ref)). This figure shows the skills parameters of the four GOATs through the plot of $f$ (top panel) and $ k$ (bottom panel) of these players. The figure clearly indicates that $f$ is decreasing and concave for these players (true for all players) while $k$ is increasing and convex (for a few players, i.e., David Ferrer and Diego Schwartzman, for instance, $k$ is slightly decreasing and concave). Interestingly, we clearly see that Pete Sampras has the most efficient serve (measured as the probability to win one-shot rallies) for serve percentages between 0 and about $80\%$ where Roger Federer's serve becomes more efficient. However, in terms of winning-point percentages on multi-shot rallies, Figure ((ref)) shows that Rafael Nadal is the dominating player at all serve percentages, with a relatively flat profile, and the profile of Pete Sampras lies at the other extreme.
Having estimated the salience weight for each player, we can now ask the natural question: what would have been the optimal service strategy if players were mere outcome-maximizers, that is, if their salience weight were zero and they paid no attention to process utility? In other words, this exercise quantifies how much players are willing to sacrifice in point-winning probability to enjoy a more appealing style of play.
The model outlined in this paper allows us to answer this question through a simple counterfactual exercise. Using the estimated parameters for each player, we set the salience weight to zero and compute the optimal serve strategy, along with the corresponding probabilities of winning points on first and second serves. Table ((ref)) presents the differences in optimal serve strategies between the observed and counterfactual scenarios (${\Greekmath 010E}=0$) in columns 3 and 4. As expected, the change in serve percentage is negative for almost all players, meaning their strategies would be more aggressive under the counterfactual, which is especially true on the second serve. Setting the salience weight to zero increases the point-winning probability on a player's own serve by approximately 0.39\,%-pt. While this change appears small, the cumulative nature of tennis scoring amplifies its impact.
To illustrate, we compute the probability of winning a set and a match (best-of-five). For this exercise, we assume that the opponent's probability of winning their own serve equals the player's observed probability under the estimated ${\Greekmath 010E} $.\footnote{ Hence, the player has a $50\%$ chance of winning a set and a match under the estimated ${\Greekmath 010E} $.} Columns 7 and 8 show the corresponding changes in set- and match-winning probabilities. For instance, although a player increases the probability of winning a point on their serve by only 0.39\thinspace %-pt on average, their probability of winning a best-of-five match increases by 2.42\thinspace %-pt.
Finally, we can estimate the probability of reaching each round of a Grand Slam and the expected prize money under both observed and counterfactual strategies. Using the 2025 US Open prize distribution, Table ((ref)) shows that, on average, players forego approximately $\$33{,} 000$, i.e., $13.5\%$, in expected price money per Grand Slam by optimizing for both outcome and process utility, with $50\%$ of the players forgoing more than $\$12{,}880$, i.e., $5.5\%$. When interpreting these results, one should also bear in mind that there are four Grand Slam tournaments per year, and while smaller tournaments offer lower prize money, this amount accumulates across tournaments over a professional career.
Our parametric results rely crucially on Condition ((ref)), which has two components: first, the elasticity of the marginal probability of winning one-shot points equals that of winning multi-shot points, and second, this elasticity is constant. We conduct two robustness checks by relaxing these assumptions one at a time.
In the first robustness check, we allow the elasticity of the marginal conditional probabilities to vary linearly with $x$, i.e., proportional to $ x{\Greekmath 0115}$ with ${\Greekmath 0115}>0$ (Condition ((ref)), Online Appendix ((ref))) rather than being constant. Under this specification of the model, we find that the estimated salience weights are slightly larger than in the baseline, indicating that our earlier computations of trade-offs are robust and, if anything, conservative (see Table ((ref)), Online Appendix ((ref))).
The second robustness check relaxes the equality of elasticities across one-shot and multi-shot points. Specifically, we let the former be ${\Greekmath 0115} -1$ and the latter $t{\Greekmath 0115} -1$ with $t>0$ (Condition ((ref)), Online Appendix ((ref))). Note that, for most players, the conditional probabilities of winning multi-shot rallies on first and second serve, $k(x_1)$ and $k(x_2)$, are closer to each other than for one-shot points, so that $k(x)$ is likely to be “flatter” than $f(x)$, which would obtain for $t<1$. Nevertheless, we evaluate the model over a grid of $t $ values from 0.5 to 2, so that the curvature of $k$ ranges from half to twice that of $f$. This grid approach is necessary because the data do not provide sufficient degrees of freedom to estimate $t$ jointly with the other parameters. Across this range, the estimated salience weights are remarkably stable as shown in Table ((ref)) of Online Appendix ((ref)).
We conclude from these results that the estimates of salience weights presented above are very robust to departure from Condition ((ref)).
To further challenge our findings, we consider two alternative explanations for the apparent preference for winning multi-shot rallies and resulting conservative serve strategies adopted by tennis players. First, one could argue that players value one-shot and multi-shot rallies differently because these rallies require different levels of effort. One-shot rallies by nature are less demanding and more energy-efficient, so if effort considerations were driving deviations from outcome-maximization, players would be expected to put more importance on winning one-shot rallies, conserving energy for later points on their opponents' serve. We actually find the opposite: players put more weight on winning multi-shot rallies on their own serve, suggesting that enjoyment, rather than energy considerations, drives these choices.
Second, one might argue that conservative second-serve strategies arise due to risk aversion and, in particular, aversion to double faults, rather than process utility. To examine this, we develop and estimate a model (in Online Appendix ((ref))) where players have a disutility for double faults, rather than a utility for process, and derive the associated optimal serve strategies. Importantly, this alternative model does not rely on the distinction between the conditional probabilities of winning one-shot and multi-shot rallies, only on the sum of the two. Therefore, if double-fault aversion was the true mechanism, the estimated disutility parameter should be unrelated to the probability of winning one-shot rallies conditional on serve strategy and point-winning probabilities. However, our analysis shows that $\mathbb{E}\left[ {\Greekmath 010D} \times f_{j}|x_{1},x_{2},y_{1},y_{2}\right] $ for $j=1,2$, is systematically positive and significant at $5\%$. This indicates that there is additional information in the conditional probability of winning one-shot rallies that is being forced into the parameter of double-fault aversion. This, in turns, supports the relevance of process utility in explaining strategic deviations from outcome-maximizing behavior.
We examine how individuals trade off outcome (\textquotedblleft what\textquotedblright ) and process (\textquotedblleft how\textquotedblright ) utility in high-stakes strategic decisions, namely in professional tennis. We first develop a nonparametric identification strategy based solely on optimality conditions and the second-serve rule that delivers a sufficient condition under which the nonparametric bounds on process utility are positive. Under mild shape restrictions, we show that this sufficient condition is likely met for a large majority of players. We then propose a parametric approach to estimate each player's salience weight and to conduct counterfactual analyses. We show that professional tennis players in our sample are willing to trade off a lower point-winning probability on their serve (outcome utility), on average $0.4\%$-pt, in exchange for a higher probability of winning multi-shot rallies (process utility). Although these differences are small in probability terms, due to the rules of tennis, they translate into a substantial forgone expected prize money. Using the 2025 US Open prize distribution as an example, we find that, on average, players forgo $\$33,000$, i.e., $13.5\%$, in prize money per tournament (Grand Slam), with $50\%$ of the players sacrificing $ \$13,000$ or more, i.e., $5.5\%$, per Grand Slam. These results demonstrate that process utility---i.e., the enjoyment of winning multi-shot rallies---is a significant component of tennis players' utility. That players are willing to sacrifice substantial expected prize money aligns well with the psychological literature on flow and intrinsic motivation. Indeed, according to csik90, when individuals enter a state of flow, they become fully absorbed in the activity itself, such that the experience is intrinsically rewarding even at a great material cost. This interpretation is further consistent with Self-Determination Theory, which emphasizes that intrinsic motivation is fostered when individuals experience autonomy and competence in the activity itself (e.g., RyanDeci24). In this light, multi-shot rallies may provide a richer environment for such experiences than one-shot outcomes, helping to rationalize the observed willingness to trade off performance for process enjoyment.
As discussed in the introduction, the tennis setting offers several advantages for identifying and estimating individual-specific salience weights while capturing a mechanism likely to be present in many other economic contexts, particularly in labor markets. A growing body of evidence on compensating differentials suggests that individuals are willing to make substantial monetary sacrifices to access intrinsically rewarding job attributes. Existing evidence, however, is typically based on either revealed preferences over discrete job choices or on experimentally elicited stated or incentivized choices. For instance, Stern04 uses multiple job offers received by PhD students to measure the preference for independent research and shows that scientists are willing to forgo approximately 19$\%$ of their wages to engage in independent research, while Mas17 estimates willingness to pay for non-wage job amenities using incentivized discrete-choice experiments over alternative work arrangements, finding wage trade-offs of approximately 8$\%$ to 20$\%$ for working from home and having regular time schedules respectively. More broadly, similar trade-offs between material outcomes and process-related attributes are also well documented in consumer markets, suggesting that the type of preferences identified in this paper may represent a general feature of economic behavior across domains. Consistent with these findings, this paper shows that such trade-offs can be identified from revealed repeated, high-frequency, continuous choices in a competitive field setting, thereby providing a novel approach to measuring process utility from observed behavior.
Our results are also important for policy interventions. For instance, in the tennis context, one might be tempted to conclude that, from a coaching perspective, a possible policy intervention would be to encourage players to set aside their desire for enjoyment and adopt strategies more closely aligned with outcome maximization. However, while such an approach may be effective in certain points of a match, the long-run implications of systematically neglecting process utility may be detrimental to performance. As suggested by flow and self-determination theories, suppressing the need for enjoyment and undermining autonomy and competence during play can lead to boredom, anxiety, stress, or “controled motivation” which in turn may reduce performance and contribute to adverse long-term outcomes, including burnout, disengagement and withdrawal (e.g., RyanDeci24).