Exploiting prior knowledge in continuous decision-making under uncertainty: the case of tennis experts.

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Learn more: PMC Disclaimer | PMC Copyright Notice Exp Brain Res . 2026 Apr 15;244(5):92. doi: 10.1007/s00221-026-07288-9 Search in PMC Search in PubMed View in NLM Catalog Add to search Exploiting prior knowledge in continuous decision-making under uncertainty: the case of tennis experts Damian Beck Damian Beck 1 University of Bern, Bremgartenstrasse 145, CH-3012 Bern, Switzerland Find articles by Damian Beck 1, ✉ , Ernst-Joachim Hossner Ernst-Joachim Hossner 1 University of Bern, Bremgartenstrasse 145, CH-3012 Bern, Switzerland Find articles by Ernst-Joachim Hossner 1 , Stephan Zahno Stephan Zahno 1 University of Bern, Bremgartenstrasse 145, CH-3012 Bern, Switzerland Find articles by Stephan Zahno 1 Author information Article notes Copyright and License information 1 University of Bern, Bremgartenstrasse 145, CH-3012 Bern, Switzerland Communicated by Matthew Heath ✉ Corresponding author. Received 2025 Oct 29; Accepted 2026 Mar 25; Issue date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13083433  PMID: 41984228 Abstract Leading theories of sensorimotor control propose that humans navigate uncertainty by integrating prior and sensory information in a Bayesian manner. However, empirical evidence is largely limited to static and constrained lab tasks. How humans exploit prior knowledge during continuously unfolding decision-making in naturalistic behavior remains unclear. Here, we study a task that pushes the human sensorimotor system to its limits: returning fast tennis serves. This task is particularly informative for gaining insights into the dynamics of unfolding decisions in action, owing to a distinct feature of the return movement: the split step—a preparatory movement characterized by a small jump to increase initial speed in the desired direction. In the experiment, experienced tennis players returned serves in an immersive extended reality setup with unconstrained movements and task demands matching real tennis. We manipulated the distributions of the opponent’s preferred serve locations (80% to the right vs. 20% to the left of the service box and vice versa in a second session). As a continuous behavioral readout of the evolving decision-making process, we measured participants’ weight shift during the unfolding action. Results show that, over the experiment, participants increasingly rely on acquired prior knowledge of the more probable serve direction to improve performance. Participants exploit prior knowledge by shifting their weight toward the more probable serve direction—already before the serve—while continuously re-evaluating both options based on incoming sensory information and motor costs. Using tennis as an exemplary case, our findings provide evidence that humans use prior and sensory information to probabilistically optimize continuous decision-making in complex sensorimotor behavior. Keywords: Bayesian inference, Decision-making, Motor control, Affordance competition hypothesis, Anticipatory behavior, Extended reality Introduction As a fundamental challenge in human behavior, we continuously perceive the world and act upon it under uncertainty arising from noise and delays in neural signals (Faisal et al. 2008 ; van Beers et al. 2002 ), ambiguities in the environment (Kersten et al. 2004 ) and incomplete information (Weber 1987 ). How humans deal with uncertainty is a pivotal question in sensorimotor behavior research, central to leading theories in the field, such as the theory of internal models (Wolpert et al. 1995 ), optimal feedback control (Todorov and Jordan 2002 ), active inference (Friston 2010 ), and affordance competition (Cisek 2007 ). While differences between these theories are apparent (cf. Friston 2011 ), they converge on the core idea that humans continuously generate predictions about future states of the world to navigate uncertainty. These continuous predictions about expected changes in the environment, including the sensory consequences of one’s own actions (Wolpert and Flanagan 2001 ), are fundamental to evaluating and deciding between ongoing action options (Pezzulo and Cisek 2016 ) and controlling movements online (Desmurget and Grafton 2000 ). Recent theoretical developments suggest that decision-making and sensorimotor control are inherently intertwined (Cisek and Pastor-Bernier 2014 ; Gallivan et al. 2018 ; Kim et al. 2021 ; Shadmehr et al. 2016 ; Wolpert and Landy 2012 ). While traditional models typically separate perception, cognition, and motor control into sequential stages, positing that only after a decision is made, its outcome is passed to the motor system to execute the action (e.g., Sternberg 1969 ), this classic view has been challenged by mounting behavioral (Chapman et al. 2010 ; Cos et al. 2021 ; Gallivan et al. 2016 ; Lepora and Pezzulo 2015 ; Nashed et al. 2014 ; Selen et al. 2012 ) and neurophysiological data (Cisek and Kalaska 2005 ; Grent et al. 2015 ; Pastor-Bernier and Cisek 2011 ; Thura et al. 2022 ; Thura and Cisek 2014 ). From an evolutionary perspective, Cisek ( 2007 ) argues that a functional architecture in which action selection (what to do) and specification (how to do it) run in parallel, rather than in sequence, much better fits the demands that humans and other animals face in natural behavior. Throughout evolution, as well as in present everyday life, we are required to interact with our environment in real time, continuously adjusting to dynamically changing situations during ongoing movement—a challenge often labeled “embodied decisions” in the literature (e.g., Cisek and Pastor-Bernier 2014 ). The affordance competition hypothesis (Cisek 2007 ) proposes that, to address this challenge, multiple potential actions are specified in parallel and continuously compete against each other, biased by the desirability of their predicted outcomes (Pezzulo and Cisek 2016 ). To form predictions under uncertainty, extensive research suggests that humans combine prior knowledge based on experience with streams of incoming sensory inputs in a probabilistic (i.e., Bayesian) manner (Körding 2007 ). While Bayesian theory offers a powerful framework for understanding how humans reduce uncertainty in perception (Yuille and Kersten 2006 ), higher-level cognition (Tenenbaum et al. 2006 ), and motor control (Körding and Wolpert 2006 ), its empirical foundation is largely limited to constrained laboratory experiments (Beck et al. 2023 ). Accordingly, testing major theories in complex and naturalistic situations humans face in the real world has been increasingly emphasized as a key challenge in psychology and neuroscience (Cisek and Green 2024 ; Fooken et al. 2023 ; Ibanez 2022 ; Maselli et al. 2023 ; Tsay et al. 2024 ). In this endeavor, studying athletes solving complex, naturalistic (but structured) tasks has the potential to reveal novel insights into basic mechanisms of sensorimotor behavior (Gordon et al. 2021 ; Maselli et al. 2025 ; Yarrow et al. 2009 ). In recent years, an increasing number of sports-related studies have begun to test Bayesian predictions in such complex situations (Arthur and Harris 2021 ; Beck et al. 2025 ; Diaz et al. 2013 ; Harris et al. 2022 ; Helm et al. 2020 ; Loffing and Hagemann 2014 ; Zahno et al. 2026 ; for a review: Beck et al. 2023 ). Among other things, these studies have shown that in fast interception tasks, predictive gaze behavior (i.e., early estimates in action) is systematically biased toward learned priors in a Bayesian manner (Arthur and Harris 2021 ; Beck et al. 2025 ), whereas explicit judgments after the action are not (Beck et al. 2025 ). These findings support the idea that humans optimize performance by exploiting prior knowledge to put themselves in a (probably) advantageous state for the upcoming—uncertain—situation, while continuously updating their actions based on incoming sensory inputs. However, data on how prior knowledge is actually used during unfolding decisions in action remains to be provided. For the experimental study of the dynamics of unfolding decisions in natural (but standardized) actions, the task of returning a tennis serve offers a unique opportunity. Serves are most frequently directed either toward the sideline (wide serve) or toward the center service line (T-serve), which are approximately four meters apart (Tea and Swartz 2022 ). To return these serves, players must initiate a movement to the left or right under uncertainty—a real-world example of a “go-before-you-know” task (Chapman et al. 2010 ). During this decision-making process, experienced players perform a preparatory movement known as the “split step”—a small jump and landing to exploit neuromuscular mechanisms and biomechanical elastic energy, thereby increasing initial movement speed in the chosen direction (e.g., Nieminen et al. 2013 ; Uzu et al. 2009 ). We propose that the player’s weight shift during this unfolding movement provides a continuous readout of the ongoing decision-making process. We developed an immersive extended reality (XR) setup that allows us to study naturalistic, full-body behavior with spatial and temporal constraints matching real tennis, while maintaining full experimental control (Beck et al. 2025 ; Zahno et al. 2026 ). Using this setup, we can experimentally manipulate the distribution of serve locations (e.g., 80% to the right, 20% to the left) and capture players’ full-body movement during the task. Linking the affordance competition hypothesis and Bayesian integration, we expect players to incorporate prior knowledge by shifting their weight in a way that favors an upcoming movement in the more probable direction—gaining an early advantage in most cases while accepting a disadvantage in rare cases—while continuously re-evaluating both options based on incoming sensory information. Interestingly, how athletes use prior knowledge to improve performance is also a subject of current debate in sport science. While research on “anticipation” in sports has a long tradition (Williams and Jackson 2019 ), and experts’ superior capabilities to extract situational probabilities and predict opponents’ actions—such as the direction of a tennis serve—have been repeatedly demonstrated (Loffing and Cañal-Bruland 2017 ), a recent systematic review challenged the notion that tennis players actually use these predictions under naturalistic conditions (Avilés et al. 2019 ). The review questions the ecological validity of many anticipation studies in tennis, which often rely on experimental tasks where athletes are asked to predict the direction of the opponent’s serve in response to a video in which vision is temporally occluded. Under ecologically valid conditions (e.g., Triolet et al. 2013 ), the review finds no evidence that tennis players exhibit “anticipatory behavior”—defined as overt movements before the serve. Based on this finding, Avilés et al. ( 2019 ) conclude that players “do not try to spatially anticipate the first serve but rather regulate their actions upon the unfolding information regarding the kinematics of the server and the first moments of ball flight” (p. 24). We propose that re-examining this question through the lens of state-of-the-art models of motor behavior and by analyzing players’ weight shifts over time might thus yield valuable insights not only for theoretical reasons but also for this debate in applied sport science. Thus, the present study combines two goals. First, using tennis as an exemplary task, we aim to contribute to fundamental understanding of how humans use prior and sensory information in continuous decision-making during complex, full-body naturalistic behavior. Second, we respond to Avilés et al.’s ( 2019 ) call and investigate whether experienced tennis players actually exploit accumulated prior knowledge to improve performance under representative conditions. Specifically, we will take a close look at the temporal evolution of the weight shift over the split step, which makes a potential early use of prior knowledge observable, and examine whether this early use explains performance. To this end, we examine experienced tennis players returning fast serves in an immersive XR environment. In our experiment, we manipulate the distribution of the opponent’s preferred serve locations (80% to the right vs. 20% to the left, and vice versa in a second session). This design creates trials that either match players’ expectations (congruent, 80%) or violate them (incongruent, 20%), similar to cueing experiments (e.g., proportion valid cueing effect, Lanthier et al. 2015 ). We predict that with accumulating experience, players will increasingly rely on prior information and that this results in improved performance in congruent trials and diminished performance in incongruent trials (Mann et al. 2014 )—a functional (Bayesian) strategy to optimize overall success under uncertainty. Within single trials, we further hypothesize that participants continuously adjust their weight shift during the split step based on incoming sensory information, while a bias toward the expected direction can already be detected in the preparatory phase during the serve. In addition, we expect that the degree to which prior knowledge is used in these early phases predicts performance. Method Participants Fourteen right-handed male tennis players ( M age = 23.7 years, SD age = 4.7 years) participated in the experiment. Given the limited access to experts, the sample size is within the typical range for studies investigating tennis experts (typically N = 10–30, e.g., Cañal-Bruland et al. 2018 ; Farrow and Reid 2012 ; Loffing et al. 2014 ; Loffing et al. 2016 ; Murphy et al. 2016 ; Murphy et al. 2018 ; Triolet et al. 2013 ; Yamamoto et al. 2019 ). All players were licensed and actively competing with a minimum rating of R7 ( M = 5.5, SD = 1.3) on the Swiss Tennis regional scale. On average, they trained 3.3 h per week ( SD = 1.4 h) and had 9.5 years ( SD = 5.5 years) of competitive experience. Based on these characteristics, our participants can be classified as Tier 2 on the standardized scale proposed by McKay et al. ( 2021 ). The experiment was approved by the Ethics Committee of the Faculty of Human Science at the University of Bern (approval number: 2017-12-00003) and conducted in accordance with the Declaration of Helsinki. All players gave written informed consent to participate. In particular, the player shown in Figs. 1 and 2 , as well as in the video of the experimental task (10.5281/zenodo.15915771), provided written consent for publication. Fig. 1. Open in a new tab Participants were equipped with 3D glasses and six rigid marker bodies (left) as well as with a custom-made tennis racket with an integrated Wii controller and further markers (middle). Their task was to return virtual serves to either the forehand or the backhand, following one of two possible ball trajectories, with the aim of hitting the center of the target on the opponent’s side of the court (right) Fig. 2. Open in a new tab A Weight shifts over the split step in congruent versus incongruent trials with correct versus incorrect responses in the final blocks of the biased condition with serve direction probabilities of 80:20, plotted relative to the developed prior, in comparison to the weight-shift dynamics for the neutral condition with serve direction probabilities of 50:50, plotted relative to the correct movement direction. B The same measurement as in A, displayed with barplots in more detail at three timepoints (–100 ms, 200 ms, and 500 ms) with standard errors of the mean Extended reality tennis setup The tennis players performed returns in a custom-built life-size XR CAVE environment, which was described in detail by Beck et al. ( 2025 ) (Fig. 1 and video 10.5281/zenodo.15915771). The XR tennis environment is displayed in real time and rendered in high resolution from the player’s perspective on a 6.00 m × 3.75 m front wall, two 11.00 m × 3.75 m side walls, and a 6.00 m × 11.00 m floor. In a further improvement of the setup used by Beck et al. ( 2025 ), the players wore 3D glasses, which allowed display of the XR environment stereoscopically. To track their tennis strokes, the participants held a custom-made (physical) racket with a marker cluster and a Wii controller built into the handle (Fig. 1 , middle). A brief vibration of the Wii controller provided haptic feedback of the racket’s contact with the virtual ball. The players saw only their physical racket; the virtual racket (interacting with the virtual ball) was not displayed. The markers attached to the racket were tracked with the 3.1.4 Optitrack 3D motion capture system at a rate of 200 Hz. Six marker clusters were placed on the head, back, left and right hand, and left and right foot to calculate a full body skeleton in Motive software (Fig. 1 , left). The racket position data from the Optitrack system were streamed in real time to the virtual tennis scene using Unreal Engine 4.27 to dynamically calculate interactive elements. In the Motive application, the racket and head movements were smoothed and predicted forward to overcome a small delay optimized for the moment of ball contact with the racket. All the systems were coordinated simultaneously using Streamix, an in-house software development ( https://tpf.philhum.unibe.ch/portfolio/streamix ) based on the theoretical work of Maurer ( 2018 ). The return trajectory was calculated according to the impact of the racket swing based on simulated physics in the Unreal Engine. Gravity, friction (air resistance), and collision were turned on, but radial momentum was turned off to avoid spin effects. We also used continuous collision detection to prevent the ball from tunneling through the racket. Experimental task and design In a within-subjects design, the participants took part in two sessions, with a break of one week between them. The two sessions took place at the same time of day and each lasted one hour. After fitting the 3D glasses and markers, the players’ task was to repeatedly return serves from the right side of the court and try to hit the center of a target on the opponent’s side of the court. Players were free to position themselves to cover the entire field. Before each trial, a red dot was displayed at the position of the serving avatar, and a short acoustic signal indicated the start of the trial. After a random delay of 1–2 s, the simulation began. Importantly, the avatar performed exactly the same serving motion in each trial. The opponent’s serving motion was therefore uninformative and provided no cues regarding the upcoming trajectory of the ball. This design ensured that the participants could rely only on two sources of information: current sensory information during the ball’s flight and prior knowledge accumulated from previous trials. Despite the constant serving motion, two distinct ball trajectories were possible, namely, either to the left side for backhand returns or to the right side for forehand returns (Fig. 1 , right). All serves had identical characteristics in terms of flight distance and speed: Serves were hit at 144 km/h, and the ball bounced 0.446 s after the serve and reached the return area after exactly 1 s. Our goal was to select a serve speed that challenges participants and provokes a return rate comparable to ATP/WTA matches, where players achieve a return rate of 50–80% on first serves (Gillet et al. 2009 ; Mecheri et al. 2019 ). To do so, we tested different serve speeds with a pilot participant. The bounce location in the virtual space of the tennis court was always either 38 cm behind the service line and 61 cm right to the center line (red trajectory in Fig. 1 , right) or 80 cm behind the service line and 89 cm left to the sideline (blue trajectory in Fig. 1 , right), respectively. Participants were not informed about these regularities. Both sessions began with four blocks of 20 warm-up trials in which the ball was constantly played to the same side (either left or right), followed by a fifth warm-up block in which 10 serves to the left and 10 serves to the right side were presented in random order. Before the fifth block, the players were told that from then on, the ball could go to the right or to the left side; however, they were not provided with explicit information about the respective probabilities. Additionally, they were reminded to perform a split step as in a real tennis match. After the warm-up trials, players performed two blocks of 20 trials each, again with a 50% chance for each side (neutral condition). Without any further instructions, the probabilities then changed to 80% for one side and 20% for the other side for the final four blocks of 20 trials each (biased condition). In compliance with this specification, individual random sequences were created for each participant (for details, see protocols on GitHub https://github.com/ispw-unibe-ch/continuous_decision-making_under_uncertainty_in_tennis ). The experiment was counterbalanced; half of the players had an 80% chance of receiving serves to the left in the first session and an 80% chance of receiving serves to the right in the second session, and vice versa for the other half of the players. Consequently, in comparison to the first session, all trials were exactly mirrored in the second one. Measures and analyses For our investigation, we (1) checked whether a split step had actually been performed (split-step detection). As a continuous behavioral readout for the decision-making process, we (2) quantified the weight-shift dynamics over the return action. To obtain performance measures, we (3) detected the movement direction after the split step and calculated the percentage of trials in which players moved to the correct side (correct response rate) as well as (4) the percentage of trials in which they successfully hit the ball (hit rate). The weight-shift dynamics, correct response rate, and hit rates were analyzed as a function of accumulated prior knowledge (neutral vs. biased) and, in the biased case, as a function of confirmed or nonconfirmed expectations (congruent vs. incongruent). Finally, (5) we investigated the extent to which the weight-shift dynamics predict performance (correct response rate and hit rate). Split-step detection For determining whether and, if applicable, when a split step had actually been performed, the analyses started with the calculation of the participant’s center of mass (COM). To this end, a full-body model with 21 body segments (Fig. 4 in Appendix A ) was computed from the six rigid motion capture bodies (head, back, left hand, right hand, left foot, and right foot) and exported from Motive 3.1.4 (OptiTrack). Based on the anthropometric data from Shan and Bohn ( 2003 ), we assigned relative weights to the body segment positions to calculate a raw COM value. The data for each body segment were finally calculated using a Savitzky-Golay filter of a third-order polynomial with a window length of 99 to obtain COM data in 3D space for each time frame. Fig. 4. Open in a new tab Development of tennis players’ performance in congruent versus incongruent trials in the blocks of the biased condition with serve direction probabilities of 80:20 in terms of hit rate. The uncertainty area represents a 95% confidence interval of the regression lines. The subsequently conducted algorithmic detection of the split step involved several steps. First, we set the time in relation to the serve. Starting from the maximum COM height in the time interval of − 0.2 s to 1 s around the moment of serve, it was iteratively checked frame by frame in both directions whether the following frame had a lower COM height. The identified local minima were taken as the beginning and end of a potentially performed split step, respectively. A split step was confirmed if the difference between the maximum COM value and the COM value at the end of the calculated split step interval was greater than 5 cm. The median was used as a robust estimate to calculate the average time for initiation of the split step. Weight-shift dynamics The COM was also used to quantify the weight shift. Specifically, we compared the lateral distances between the COM and the left and right ankles, respectively, and defined the weight shift as the difference between these distances. In the biased condition, we considered the weight shift to be positive (i.e., in the direction of the prior) if the smaller distance was on the same side as the more probable serve side; otherwise, we classified it as negative. In the neutral condition, positive values were assigned to the weight-shift difference if the side of the smaller distance matched the side of the actually played serve. To analyze weight-shift dynamics, we calculated each participant’s mean values for the time interval from 0.1 s before to 1.0 s after the serve, that is, to the moment when the ball reached the return area. These curves were aggregated from the two blocks of the neutral condition as well as from the last two blocks of the biased condition per session, separately for correct and incorrect trials and for the biased condition, additionally for congruent and incongruent trials. To investigate whether the weight shift was drawn toward the prior in the biased condition, we conducted multilevel regression analyses at three timepoints relative to the serve onset that preceded the average initiation of the lateral movement, namely at − 0.1 s, 0.2 s, and 0.5 s. For these timepoints, we tested if the overall mean deviated from zero. On the basis of all individual measurements over the final two blocks of the biased condition, we had multiple data points for each participant, and the residuals were not independent. Therefore, we built and compared regression models in several steps in order to take the hierarchical structure of the data into account (Field et al. 2012 ). Descriptively, we compared models considering the Akaike information criterion (AIC), Bayesian information criterion (BIC), and log-likelihood (Tables 4 , 5 , 6 , 7 , 8 and 9 in Appendix B ). For the purpose of inferential model comparisons, we used the log-likelihood ratio χ 2 test. Only the best model according to the parsimony principle is reported. Further, if the model could not be estimated due to singularities, we took one step back in complexity and used the simpler model (Bates et al. 2015 ). The following procedures were used for missing values, outlier detection, and checking prerequisites to prepare the data for inferential statistical analyses: Under the assumption that missing values occur randomly, multilevel regression analyses can handle missing values (Field et al. 2012 ). We detected outliers by using Cook’s distance and excluded values that had more than three times the average influence on the intercept. We checked the homoscedasticity and normality of the residuals graphically. Further, we used the maximum likelihood for model estimations, and we tested intercept coefficients with the Wald test for significance. For all inferential tests, we chose an alpha level of 5%. For the required calculations, we applied the R package ‘nlme’ (Pinheiro 2009 ). Table 4. Weight shift directed to prior at 100 ms before serve Fixed effects B 95% CI SE B t (1037) p two-sided Intercept 2.49 [0.37, 4.61] 1.08 2.30 .022 Random effects Intercept variance (τ 00 ) 13.18 – – – – Level-1 residual (σ 2 ) 231.18 – – – – ICC 0.05 – – – – Open in a new tab B Unstandardized regression coefficient, CI Confidence intervals, SE Standard error, t (degrees of freedom), ICC Interclass correlation coefficient. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, N weight shifts = 1051 (outliers removed = 69), R 2 marginal < .001. Multilevel model comparison in Table 5 . Table 5. Model comparison for multilevel regression of weight shift in direction of prior at 100 ms before serve Model df AIC BIC logLik Comparison χ 2 p 1 Intercept 2 8766.824 8776.738 − 4381.412 2 Random intercepts 3 8732.574 8747.446 − 4363.287 1 versus 2 36.250 < .001 Open in a new tab df Degrees of freedom, AIC Akaike information criterion, BIC Bayesian information criterion, logLik Log-likelihood. Table 6. Weight shift directed to prior at 200 ms after serve Fixed effects B 95% CI SE B t (1044) p two-sided Intercept 3.35 [1.36, 5.35] 1.02 3.29 .001 Random effects Intercept variance (τ 00 ) 11.10 – – – – Level-1 residual (σ 2 ) 252.68 – – – – ICC 0.04 – – – – Open in a new tab B Unstandardized regression coefficient, CI Confidence intervals, SE Standard error, t (degrees of freedom), ICC Interclass correlation coefficient. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, N weight shifts = 1058 (outliers removed = 62), R 2 marginal < .001. Multilevel model comparison in Table 7 . Table 7. Model comparison for multilevel regression of weight shift in direction of prior at 200 ms after serve (biased condition) Model df AIC BIC logLik Comparison χ 2 p 1 Intercept 2 8905.439 8915.367 − 4450.719 2 Random intercepts 3 8881.836 8896.728 − 4437.918 1 versus 2 25.603 < .001 Open in a new tab df Degrees of freedom, AIC Akaike information criterion, BIC Bayesian information criterion, logLik Log-likelihood. Table 8. Weight shift directed to prior at 500 ms after serve Fixed effects B 95% CI SE B t (1026) p two-sided Intercept 5.02 [2.06, 7.97] 1.51 3.33 .001 Random effects Intercept variance (τ 00 ) 22.13 – – – – Level-1 residual (σ 2 ) 370.83 – – – – ICC 0.06 – – – – Open in a new tab B Unstandardized regression coefficient, CI Confidence intervals, SE Standard error, t (degrees of freedom), ICC Interclass correlation coefficient. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, N weight shifts = 1040 (outliers removed = 80), R 2 marginal < .001. Multilevel model comparison in Table 9 . Table 9. Model comparison for multilevel regression of weight shift in direction of prior at 200 ms after serve Model df AIC BIC logLik Comparison χ 2 p 1 Intercept 2 9033.488 9048.329 − 4513.744 2 Random intercepts 3 9024.690 9044.478 − 4508.345 1 versus 2 10.80 .001 Open in a new tab df Degrees of freedom, AIC Akaike information criterion, BIC Bayesian information criterion, logLik log-likelihood. Lateral movement direction Algorithmic detection of lateral movement initiation was also based on the COM and carried out in the following steps. First, the initial lateral COM position was defined as the average position over the time interval between 0.2 s and 0.1 s before the serve. Next, the algorithm searched for the first instance when the average lateral position of the COM over 10 consecutive frames deviated by at least 0.2 m from this initial position. The algorithm then iterated backward frame by frame until the lateral velocity of the COM fell below 0.4 m/s, and the identified frame was taken as the moment of lateral movement initiation. The median was used as a robust estimate to calculate the average time of lateral movement initiation. In addition to the derivation of the temporal marker’s position, the sign of the difference between the initial lateral COM position and the COM position at the moment of lateral movement initiation unambiguously allowed the movement direction either to the left or to the right side to be clearly determined. These directions were used to calculate the rate of correct responses; that is, the percentage of trials in which the players initiated their movement in the direction of the actual serve. This variable was computed separately for congruent and incongruent trials of the four blocks of the biased condition. To analyze the development of the correct response rate over the four blocks of the biased condition, we conducted a multiple regression analysis with the independent predictors: trial number and the dummy coded (in)congruency of the respective trial as well as their interaction, thereby, in accordance with the exponential law of practice (Heathcote and Brown 2000 ), taking the root of the trial number rather than its raw value. As we were able to calculate a correct response rate for each trial, we had no missing values in the analysis. For all inferential tests, we chose an alpha level of 5%. Hitting performance As an additional performance measure, the hit rate was calculated as the percentage of trials in which the players successfully hit the ball. Analogous to the correct response rate variable, the hit rate was analyzed for congruent and incongruent trials of the four blocks of the biased condition. To analyze its development over the four blocks of the biased condition, the same regression approach was chosen as for the correct response rate. Performance predictions To investigate the extent to which the weight-shift variable predicts later performance, we conducted multilevel logistic regression analyses, again at three timepoints relative to serve onset (–0.1 s, 0.2 s, 0.5 s). As described for weight-shift dynamics (Sect.  3.4.2), we considered the hierarchical structure of the data. Therefore, we built and compared models similarly (Tables 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 and 25 in Appendix B ). In contrast to the analyses described in Sect.  3.4.2, however, regression coefficients were tested for significance using the log-likelihood ratio χ 2 test. For all inferential tests, we chose again an alpha level of 5%. We applied the R package ‘lme4’ (Bates et al. 2015 ). Table 10. Logistic regression model of weight shift in direction of side played for predicting direction taken 100 ms before serve Fixed effects B SE B z (1067) p two-sided OR 95% CI Intercept − 0.05 0.06 − 0.81 0.417 0.95 [0.84, 1.07] Weight shift 0.02 0.00 4.97 < .001 1.02 [1.01, 1.03] Open in a new tab B Unstandardized regression coefficients, SE Standard error, z (degrees of freedom), OR Odds ratio, CI Confidence intervals. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, N weight shifts = 1069 (outliers removed = 51), R 2 Tjur = .024. No multilevel model comparison due to singularities in their model fits. Table 11. Logistic regression model of weight shift in direction of side played for predicting direction taken 200 ms after serve Fixed effects B SE B z (1086) p two-sided OR 95% CI Intercept − 0.13 0.06 − 1.98 .047 0.88 [0.78, 1.00] Weight shift 0.03 0.00 8.15 < .001 1.03 [1.02, 1.04] Open in a new tab B Unstandardized regression coefficients, SE Standard error, z (degrees of freedom), OR Odds ratio, CI Confidence intervals. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, N weight shifts = 1088 (outliers removed = 32), R 2 Tjur = .080. No multilevel model due to worse model fits; see Table 12 . Table 12. Model comparison for logistic regression of weight shift in direction of side played for predicting direction taken probability 200 ms after serve Model df AIC BIC logLik Comparison χ 2 p 1 Intercept 1 1510.108 1515.100 − 754.05 2 Base model 2 1415.037 1425.021 − 705.52 1 versus 2 97.072 < .001 3 Base model + RI 3 1414.927 1429.904 − 704.46 2 versus 3 2.11 .146 4 Base model + RS 5 – – – 3 versus 4 – – Open in a new tab df Degrees of freedom, AIC Akaike information criterion, BIC Bayesian information criterion, logLik Log-likelihood, RI Random intercepts, RS Random intercept and slopes. The base model includes the predictor weight shift directed to the side played. Table 13. Logistic regression model of weight shift in direction of side played for predicting direction taken 500 ms after serve Fixed effects B SE B z (1088) p two-sided OR 95% CI Intercept − 0.13 0.06 − 2.09 .037 0.88 [0.77, 0.99] Weight shift 0.02 0.00 7.92 < .001 1.02 [1.02, 1.03] Open in a new tab B Unstandardized regression coefficients, SE Standard error, z (degrees of freedom), OR Odds ratio, CI Confidence intervals. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, N weight shifts = 1090 (outliers removed = 30), R 2 Tjur = .068. No multilevel model comparison due to singularities in their model fits. Table 14. Multilevel logistic regression model of weight shift in direction of prior for predicting correct response probability 100 ms before serve Fixed effects B SE B z (1074) p two-sided OR 95% CI Intercept − 0.61 0.32 − 1.89 .058 0.54 [0.29, 1.02] Weight shift − 0.02 0.01 − 1.73 .084 0.98 [0.95, 1.00] Condition (0 = incongruent, 1 = congruent) 1.91 0.20 9.77 < .001 6.73 [4.59, 9.86] Weight shift × condition 0.03 0.01 2.64 .008 1.03 [1.01, 1.06] Random effects Intercept variance (τ 00 ) 1.11 – – – – – Slope variance (τ 11 ) 0.00 – – – – – Intercept-slope covariance (ρ 01 ) 0.20 – – – – – Level-1 residual (σ 2 ) 3.29 – – – – – ICC 0.33 – – – – – Open in a new tab B Unstandardized regression coefficients, SE Standard error, z (degrees of freedom), OR Odds ratio, CI Confidence intervals, ICC Interclass correlation coefficient. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, R 2 marginal = .154. Multilevel model comparison in Table 15 . Table 15. Model comparison for logistic regression of weight shift in direction of prior for predicting correct response probability 100 ms before serve Model df AIC BIC logLik Comparison χ 2 p 1 Intercept 1 1401.346 1406.331 − 699.67 2 Base model 4 1264.726 1284.668 − 628.36 1 versus 2 142.620 < .001 3 Base model + RI 5 1146.668 1171.596 − 568.33 2 versus 3 120.058 < .001 4 Base model + RS 7 1139.357 1174.257 − 562.68 3 versus 4 11.311 .003 Open in a new tab df Degrees of freedom, AIC Akaike information criterion, BIC Bayesian information criterion, logLik Log-likelihood, RI Random intercepts, RS Random intercept and slopes. The base model includes the predictors weight shift, condition, and weight shift × condition. Table 16. Multilevel logistic regression model of weight shift in direction of prior for predicting correct response probability 200 ms after serve Fixed effects B SE B z (1086) p two-sided OR 95% CI Intercept − 0.59 0.33 − 1.75 .080 0.56 [0.29, 1.07] Weight shift − 0.05 0.01 − 3.55 < .001 0.95 [0.93, 0.98] Condition (0 = incongruent, 1 = congruent) 1.77 0.20 8.92 < .001 5.87 [3.98, 8.66] Weight shift × condition 0.07 0.01 5.17 < .001 1.07 [1.04, 1.09] Random effects Intercept variance (τ 00 ) 1.20 – – – – – Slope variance (τ 11 ) 0.00 – – – – – Intercept-slope covariance (ρ 01 ) 0.02 – – – – – Level-1 residual (σ 2 ) 3.29 – – – – – ICC 0.33 – – – – – Open in a new tab B Unstandardized regression coefficients, SE Standard error, z (degrees of freedom), OR Odds ratio, CI Confidence intervals, ICC Interclass correlation coefficient. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, R 2 marginal = .190. Multilevel model comparison in Table 17 . Table 17. Model comparison for logistic regression of weight shift in direction of prior for predicting correct response probability 200 ms after serve Model df AIC BIC logLik Comparison χ 2 p 1 Intercept 1 1417.875 1420.771 − 707.94 2 Base model 4 1267.921 1290.993 − 629.96 1 versus 2 155.95 < .001 3 Base model + RI 5 1139.966 1172.613 − 564.98 2 versus 3 129.95 < .001 4 Base model + RS 7 1129.159 1171.764 − 557.58 3 versus 4 14.81 .001 Open in a new tab df Degrees of freedom, AIC Akaike information criterion, BIC Bayesian information criterion, logLik Log-likelihood, RI Random intercepts, RS Random intercept and slopes. The base model includes the predictors weight shift, condition, and weight shift × condition. Table 18. Multilevel logistic regression model of weight shift in direction of prior for predicting correct response probability 500 ms after serve Fixed effects B SE B z (1055) p two-sided OR 95% CI Intercept − 0.53 0.40 − 1.32 .188 0.59 [0.27, 1.29] Weight shift − 0.15 0.02 − 7.36 < .001 0.86 [0.82, 0.89] Condition (0 = incongruent, 1 = congruent) 1.73 0.27 6.46 < .001 5.65 [3.34, 9.56] Weight shift × condition 0.22 0.02 10.09 < .001 1.25 [1.20, 1.30] Random effects Intercept variance (τ 00 ) 1.55 – – – – – Slope variance (τ 11 ) 0.00 – – – – – Intercept-slope covariance (ρ 01 ) 0.01 – – – – – Level-1 residual (σ 2 ) 3.29 – – – – – ICC 0.42 – – – – – Open in a new tab B Unstandardized regression coefficients, SE Standard error, z (degrees of freedom), OR Odds ratio, CI Confidence intervals, ICC Interclass correlation coefficient. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, R 2 marginal = .682. Multilevel model comparison in Table 19 . Table 19. Model comparison for logistic regression of weight shift in direction of prior for predicting correct response probability 500 ms after serve Model df AIC BIC logLik Comparison χ 2 p 1 Intercept 1 1368.695 1549.624 − 683.35 2 Base model 4 946.972 1453.351 − 469.49 1 versus 2 427.724 < .001 3 Base model + RI 5 822.651 1280.728 − 406.33 2 versus 3 126.321 < .001 4 Base model + RS 7 818.754 1285.899 − 402.38 3 versus 4 7.897 .019 Open in a new tab df Degrees of freedom, AIC Akaike information criterion, BIC Bayesian information criterion, logLik Log-likelihood, RI Random intercepts, RS Random intercept and slopes. The base model includes the predictors weight shift, condition, and weight shift × condition. Table 20. Multilevel logistic regression model of weight shift in direction of prior for predicting hit probability 100 ms before serve Fixed effects B SE B z (1062) p two-sided OR 95% CI Intercept − 1.14 0.35 − 3.21 .001 0.32 [0.16, 0.64] Weight shift − 0.03 0.01 − 1.76 .079 0.97 [0.95, 1.00] Condition (0 = incongruent, 1 = congruent) 1.80 0.21 8.62 < .001 6.03 [4.01, 9.07] Weight shift × condition 0.05 0.01 3.77 < .001 1.05 [1.02, 1.08] Random effects Intercept variance (τ 00 ) 1.32 – – – – – Slope variance (τ 11 ) 0.00 – – – – – Intercept-slope covariance (ρ 01 ) − 0.08 – – – – – Level-1 residual (σ 2 ) 3.29 – – – – – ICC 0.36 – – – – – Open in a new tab B Unstandardized regression coefficients, SE Standard error, z (degrees of freedom), OR Odds ratio, CI Confidence intervals, ICC Interclass correlation coefficient. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, R 2 marginal = .168. Multilevel model comparison in Table 21 . Table 21. Model comparison for logistic regression of weight shift in direction of prior for predicting hit probability 100 ms before serve Model df AIC BIC logLik Comparison χ 2 p 1 Intercept 1 1475.848 1480.822 − 736.92 2 Base model 4 1363.833 1383.731 − 677.92 1 versus 2 118.015 < .001 3 Base model + RI 5 1194.925 1219.798 − 592.46 2 versus 3 170.908 < .001 4 Base model + RS 7 1184.875 1219.696 − 585.44 3 versus 4 14.051 < .001 Open in a new tab df Degrees of freedom, AIC Akaike information criterion, BIC Bayesian information criterion, logLik log-likelihood, RI random intercepts, RS Random intercept and slopes. The base model includes the predictors weight shift, condition, and weight shift × condition. Table 22. Multilevel logistic regression model of weight shift in direction of prior for predicting hit probability 200 ms after serve Fixed effects B SE B z (1061) p two-sided OR 95% CI Intercept − 1.27 0.36 − 3.54 < .001 0.28 [0.14, 0.57] Weight shift − 0.03 0.01 − 2.64 .008 0.97 [0.94, 0.99] Condition (0 = incongruent, 1 = congruent) 1.89 0.22 8.68 < .001 6.64 [4.33, 10.18] Weight shift × condition 0.04 0.01 3.44 .001 1.05 [1.02, 1.07] Random effects Intercept variance (τ 00 ) 1.31 – – – – – Slope variance (τ 11 ) 0.00 – – – – – Intercept-slope covariance (ρ 01 ) 0.28 – – – – – Level-1 residual (σ 2 ) 3.29 – – – – – ICC 0.34 – – – – – Open in a new tab B Unstandardized regression coefficients, SE Standard error, z (degrees of freedom), OR Odds ratio, CI Confidence intervals, ICC Interclass correlation coefficient. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, R 2 marginal = .189. Multilevel model comparison in Table 23 . Table 23. Model comparison for logistic regression of weight shift in direction of prior for predicting hit probability 200 ms after serve Model df AIC BIC logLik Comparison χ 2 p 1 Intercept 1 1479.854 1479.943 − 738.93 2 Base model 4 1369.016 1389.741 − 680.51 1 versus 2 116.839 < .001 3 Base model + RI 5 1197.478 1215.237 − 593.74 2 versus 3 173.538 < .001 4 Base model + RS 7 1194.064 1220.430 − 590.03 3 versus 4 7.414 .025 Open in a new tab df Degrees of freedom, AIC Akaike information criterion, BIC Bayesian information criterion, logLik Log-likelihood, RI Random intercepts, RS Random intercept and slopes. The base model includes the predictors weight shift, condition, and weight shift × condition. Table 24. Multilevel logistic regression model of weight shift in direction of prior for predicting hit probability 500 ms after serve Fixed effects B SE B z (1071) p two-sided OR 95% CI Intercept − 1.34 0.49 − 2.73 .006 0.26 [0.10, 0.69] Weight shift − 0.11 0.02 − 6.10 < .001 0.89 [0.86, 0.93] Condition (0 = incongruent, 1 = congruent) 1.84 0.25 7.28 < .001 6.27 [3.83, 10.28] Weight shift × condition 0.01 0.02 7.40 < .001 1.14 [1.10, 1.18] Random effects Intercept variance (τ 00 ) 2.68 – – – – – Slope variance (τ 11 ) 0.00 – – – – – Intercept-slope covariance (ρ 01 ) − 0.15 – – – – – Level-1 residual (σ 2 ) 3.29 – – – – – ICC 0.52 – – – – – Open in a new tab B Unstandardized regression coefficients, SE Standard error, z (degrees of freedom), OR Odds ratio, CI Confidence intervals, ICC Interclass correlation coefficient. Weight shift is positive in the direction of the prior. Model statistics: N players = 14, R 2 marginal = .410. Multilevel model comparison in Table 25 . Table 25. Model comparison for logistic regression of weight shift in direction of prior for predicting hit probability 500 ms after serve Model df AIC BIC logLik Comparison χ 2 p 1 Intercept 1 1483.919 1488.902 − 740.96 2 Base model 4 1296.260 1316.191 − 644.13 1 versus 2 193.659 < .001 3 Base model + RI 5 1107.124 1132.038 − 548.56 2 versus 3 191.136 < .001 4 Base model + RS 7 1088.760 1123.640 − 537.38 3 versus 4 22.364 < .001 Open in a new tab df Degrees of freedom, AIC Akaike information criterion, BIC Bayesian information criterion, logLik Log-likelihood, RI Random intercepts, RS Random intercept and slopes. The base model includes the predictors weight shift, condition, and weight shift × condition. Results Results were obtained for (1) the split-step detection, (2) the weight-shift dynamics over the split step, (3) the subsequent actual movement direction, (4) the hit rate, and (5) the prediction of the two performance variables (correct movement direction and hit rate) from the split step characteristics. Split step detection In 90.6% of the trials, a clear split step was detected based on the specified criterion of a definite raising and lowering of the COM around the opponent’s serve. Consequently, it can be inferred that the participants definitely performed a split step in the vast majority of cases. However, as weight-shift data can also be computed for vertical COM displacements of a lesser extent than 5 cm, all trials were included in the analysis. On average, the split step was initiated quite exactly at the moment of serve ( M = − 0.01 s) in a neutral forward direction. The resulting lateral movement initiation started on average about half a second later ( M = 0.62 s), and in successful trials, as prespecified by the experimentally manipulated serve speed, the ball was hit about one second after the serve ( M = 1.00 s). Weight-shift dynamics over the split step Figure 2 A illustrates the players’ average weight shifts during the return action after exposure to the opponent’s serve preferences (i.e., the final two blocks of the biased condition). Trials are categorized by whether the actual serve direction was congruent with the expected (prior) direction (blue lines) or incongruent with it (red lines). Additionally, trials are separated based on response correctness, defined as whether the movement was initiated in the serve direction (solid lines) or against it (dotted lines). Weight-shift data are plotted relative to the prior; thus, positive values reflect a behavior consistent with the opponent’s preferred serve direction. For comparison, Fig. 2 A also includes data for the neutral condition (green lines), corresponding to the first two blocks after the warm-up. Because the prior serve direction probability is 50:50 in the neutral condition, weight shifts are plotted relative to the correct movement direction. Figure 2 B zooms into players’ weight shifts at three different timepoints: 100 ms before the serve and 200 ms and 500 ms after the serve. Descriptively, Fig. 2 B reveals that, taking all four cases of the biased condition together (i.e., congruent/correct, incongruent/incorrect, incongruent/correct, incongruent/incorrect), a weight shift toward the more probable side is already evident 100 ms before the serve. This overall bias persists at 200 ms and 500 ms, as the increase in weight-shift magnitude in the congruent/correct and incongruent/incorrect trials exceeds the corresponding decrease in the congruent/incorrect and incongruent/correct trials. Inferential statistics confirm this overall bias, as the mean weight shift was significantly above zero at all three timepoints (at − 100 ms: N players = 14, B = 2.49, CI [0.37, 4.61], t (1037) = 2.30, two-sided p = .022; at 200 ms: N players = 14, B = 3.35, CI [1.36, 5.35], t (1044) = 3.29, two-sided p = .001; at 500 ms: N players = 14, B = 5.02, CI [2.06, 7.97], t (1026) = 3.33, two-sided p = .001; see full statistics in Appendix B , Tables 4 , 5 , 6 , 7 , 8 and 9 ). In contrast, the neutral cases (neutral/correct vs. neutral/incorrect) display a symmetrical, mirrored pattern without such a bias. While the weight-shift magnitudes at − 100 ms and 200 ms remain close to zero for both correct and incorrect responses, they tend toward the direction of the initiated movement at 500 ms, but with comparable absolute values. As a result, the overall mean of the weight shifts (neutral/correct and neutral/incorrect combined) was close to zero at all three timepoints and did not significantly deviate from this neutral value. Rate of correct movement initiation over the experiment To examine the effect of accumulated prior knowledge on performance over the experiment, we analyzed the response correctness (i.e., players’ movement initiation in the direction of the actual serve) over the experiment in congruent versus incongruent trials (Fig. 3 ). Overall, due to high serve speeds, the perceptual-motor demands were high, resulting in an overall response correctness of only 57.2% (56.8% for left/backhand, 57.6% for right/forehand). In the neutral condition after the warm-up trials, the overall correct response rate was 49.5%. However, in the biased condition with an 80% probability to one side and a 20% probability to the other, the players learned the probabilities quickly. They increased their correct response rate in the congruent trials to 79.1%, while the corresponding value for the incongruent trials fell to 36.8% (Fig. 3 ), resulting in an overall correct response rate of 70.6% at the end of the practice blocks in the biased condition. The regression model was significant with a large effect ( N players = 14, F (3,76) = 62.49, two-sided p < .001, R 2 adjusted = 0.700, f 2 = 2.33), and all coefficients were significant (Table 1 ). Fig. 3. Open in a new tab Development of tennis players’ performance in congruent versus incongruent trials in the blocks of the biased condition with serve direction probabilities of 80:20 in terms of correct response rate. The uncertainty area represents a 95% confidence interval of the regression lines. Table 1. Regression model for correct response rate Predictors B 95% CI t (76) p two-sided Intercept 58.17 [46.17, 70.17] 9.66 < .001 sqrt(trial number) − 2.39 [− 4.29, − 0.49] − 2.51 .014 Condition (0 = incongruent, 1 = congruent) − 13.84 [− 27.28, − 0.39] − 2.05 .044 sqrt(trial number) × condition 6.28 [4.16, 8.40] 5.89 < .001 Open in a new tab B Unstandardized regression coefficients, CI Confidence intervals, sqrt Square root function Model statistics N players = 14, N response rates = 80, F (3,76) = 62.49, two-sided p < .001, R 2 adjusted = .700, f 2 = 2.33. Hitting performance over the experiment Since hitting the ball is even harder compared to initiating a movement in the correct direction, the overall hit rate was no more than 35.5% in the neutral condition after the warm-up trials. However, the participants managed to increase this value to 67.7% in congruent trials at the end of the biased condition, with, at the same time, a decrease to 33.1% in incongruent trials (Fig. 4 ), which comes down to an overall final hit rate of 60.8%. The corresponding regression model reaches significance with a large effect ( N players = 14, F (3,76) = 39.04, two-sided p < .001, R 2 adjusted = 0.591, f 2 = 1.44), all coefficients again being significant (Table 2 ). Table 2. Regression model for hit rate Predictors B CI t (76) p two-sided Intercept 48.75 [36.04, 61.47] 7.64 < .001 sqrt(trial number) − 1.75 [− 3.77, 0.26] − 1.74 .086 Condition (0 = incongruent, 1 = congruent) − 13.09 [− 27.34, 1.17] − 1.83 .071 sqrt(trial number) × condition 5.33 [3.08, 7.58] 4.72 < .001 Open in a new tab B Unstandardized regression coefficients, CI Confidence intervals, sqrt Square root function Model statistics N players = 14, N hit rates = 80, F (3,76) = 39.04, two-sided p < .001, R 2 adjusted = .591, f 2 = 1.44. Performance prediction As shown in Fig. 2 , for the last two blocks of the biased condition, a weight shift in the direction of the learned prior can already be observed at rest before the start of the serve. Interestingly, it also appears that the weight shift tends toward the less likely side for correct responses in incongruent trials as well as for incorrect responses in congruent trials early in the split step. This suggests that the direction of the initial weight shift predicts the direction of the later initiated movement to a certain extent. Statistically, the odds ratio for initiating a movement in the same direction as the direction of the weight shift before the serve is significantly higher than one ( N players = 14, OR = 1.02, CI [1.01, 1.03], z (1067) = 4.97, two-sided p < .001), and this value increases further for the timepoints of 200 ms ( N players = 14, OR = 1.03, CI [1.02, 1.04], z (1085) = 8.22, two-sided p < .001) and 500 ms ( N players = 14, OR = 1.25, CI [1.20, 1.30], z (1055) = 10.09, two-sided p < .001) after the serve (see full statistics in Tables 10 , 11 , 12 and 13 in Appendix B ). Since players generally shift their weight toward the prior, and because this weight shift makes it more likely they will move in the same direction, there is an interaction between condition (incongruent/congruent) and weight shift toward the prior in increasing the odds of a correct response. These interaction effects are significant with increasing odds ratios (all higher than one) 100 ms before the serve ( N players = 14, OR = 1.03, CI [1.01, 1.06], z (1074) = 2.64, two-sided p = .008) as well as 300 ms after the serve ( N players = 14, OR = 1.07, CI [1.04, 1.09], z (1086) = 5.17, two-sided p < .001) and 500 ms after the serve ( N players = 14, OR = 1.25 CI [1.20, 1.30], z (1055) = 10.09, two-sided p < .001) (see full statistics in Tables 14 , 15 , 16 , 17 , 18 and 19 in Appendix B ). Furthermore, there is a significant interaction between condition (incongruent/congruent) and weight shift directed toward the prior in increasing the odds of hitting the ball. These interaction effects are significant with increasing odds ratios (all higher than one) 100 ms before the serve ( N players = 14, OR = 1.05, CI [1.02, 1.08], z (1062) = 3.77, two-sided p < .001) as well as 300 ms after the serve ( N players = 14, OR = 1.05, CI [1.02, 1.07], z (1065) = 3.44, two-sided p < .001) and 500 ms after the serve ( N players = 14, OR = 1.14 CI [1.10, 1.18], z (1071) = 7.40, two-sided p < .001) (see full statistics in Tables 20 , 21 , 22 , 23 , 24 and 25 in Appendix B ). Discussion This study pursued two aims. First, taking tennis as an exemplary case, we sought to advance fundamental understanding of how humans utilize prior and sensory information in the dynamics of continuously unfolding decisions in action. Second, we aimed to gain insights into whether and how experienced tennis players use prior knowledge to improve performance in tennis returns, thereby contributing to a current discussion in the applied field of sport science. To this end, we investigated a preparatory movement for the tennis return under ecologically valid task demands: the split step. We showed that experienced tennis players continuously adjusted their weight shifts over the split step based on an ongoing weighting of prior knowledge on serve direction probability and incoming sensory evidence for one proposition or the other (Fig. 2 ). Furthermore, over the course of the experiment, participants learned situational probabilities of the opponent’s serve directions and increasingly exploited this information to improve performance in terms of initiating a movement in the correct direction (Fig. 3 ) and successfully hitting the ball (Fig. 4 ). Consistent with Bayesian principles, participants increasingly relied on prior information, resulting in enhanced performance in congruent (thus frequent, 80%) trials while accepting lower performance in incongruent (thus rare, 20%) trials—a functional strategy that optimizes overall performance. Importantly, players’ performance could be explained by the extent to which prior knowledge is integrated in the early phase of the split step. Specifically, weight shifts were already biased toward the more probable side before the serve, which in turn increased the likelihood of a subsequent movement in the same direction and thus put them in a promising state for a successful return. With regard to applied considerations for sport science, we followed Avilés et al.‘s ( 2019 ) call to investigate the effect of prior knowledge on anticipatory behavior and performance in tennis returns under representative spatiotemporal task demands. To do so, we applied an XR-based experimental setup in which experienced tennis players could move freely and return virtual balls with a real racket. Ball–racket interactions were accompanied by synchronized visual and auditory feedback as well as haptic vibration. Although this vibration did not reproduce fully realistic haptic feedback—specifically, the force impulse associated with ball impact (for an experimental study on the role of haptic feedback in VR, Lavoie et al. 2025 )—the setup reflects key characteristics of naturalistic behavior, including actability and multisensory processing (Snow and Culham 2021 ). Importantly, task difficulty was representative of elite competition, as return success rates were comparable to those observed in ATP and WTA first-serve returns (approximately 50–80%, Gillet et al. 2009 ; Mecheri et al. 2019 ). Critically, the XR setup enabled a targeted manipulation of serve location probabilities while holding all other cues constant. The server’s body movements were uninformative about the serve direction (identical across all trials), ensuring that participants could rely only on two information sources: prior knowledge acquired from previous serves and sensory information of the ball trajectory on the current serve. In real tennis, players can additionally exploit kinematic cues from the opponent’s body movements, which feed into their probabilistic judgments (Loffing and Cañal-Bruland 2017 ). Thus, while examining the relative weighting of prior, kinematic, and ball trajectory information—and possible interactions (e.g., Navia et al. 2013 )—is a promising avenue for future research, intentionally holding kinematic information constant here was essential to examine the effect of increasing prior knowledge in an ecologically valid yet experimentally rigorous manner. Applying this representative experimental design, we obtained the following three main findings, which can claim practical relevance for the world of sports, particularly for sports situations that are classically studied in anticipation research (Loffing and Cañal-Bruland 2017 ). First, the results reconcile and put into context the seemingly incompatible positions that anticipation is, on the one hand, key to returning tennis serves (Williams and Jackson 2019 ) and, on the other hand, no overt anticipatory behavior can be observed when players actually return serves (Avilés et al. 2019 ). In this regard, we show that tennis players are not only able to extract situational probabilities (Farrow and Reid 2012 ; Loffing and Hagemann 2014 ; Loffing et al. 2016 ; Murphy et al. 2016 , 2018 ) but actively exploit predictive information by optimizing their weight shifts during and even before initiating the split step. As argued by Avilés et al. ( 2019 ), when interacting with a real opponent, players avoid overt anticipatory movements to conceal their intentions and thus do not display lateral displacement before racquet-ball contact during the serve and instead perform the split step in a natural forward direction. Our data confirms this argument and previous findings (e.g., Triolet et al. 2013 ). However, this does not mean that they do not use predictive information. Our data show that players do prepare to move in the expected direction, but in a more subtle way without revealing their intentions by optimizing their weight shift before the serve and throughout the movement. Second, confirming previous studies (Jackson et al. 2020 ; Mann et al. 2014 ), our results indicate that players increasingly rely on acquired prior information with exposure to an opponent’s action tendencies. This leads to performance increases in congruent trials and, conversely, performance decreases in incongruent trials. Importantly, however, this should not be taken as a “bug” of the system. On the contrary, according to Bayesian decision theory (Körding and Wolpert 2006 ), this behavior reflects the optimal strategy in the face of high uncertainty. Being systematically biased toward the more probable side comes with the cost of being wrong in rare cases (i.e., incongruent trials) but with the benefit of being right in most cases (i.e., congruent trials), which optimizes overall performance across multiple serves. Third, putting sports-related anticipation research in the context of state-of-the-art theories of sensorimotor behavior suggests that using predictions to guide action is fundamental to behavioral control in general and not a specific feature of actions that require “anticipation” due to high temporal demands. The difference between tasks with low and high time pressure then comes down to the fact that in the first case, sufficient time is available to wait for confirmation of one’s prediction by incoming sensory information, whereas in the second case of “anticipation,” one must rely on the currently usable predictions to initiate action. In both cases, however, behavioral control constitutes an ongoing process of continuous anticipatory decision-making. Beyond sports, the tennis task served as an informative exemplary case for gaining insights into fundamental mechanisms of prior knowledge integration in continuous sensorimotor decision-making. Players’ weight shifts throughout the return action offered a continuous behavioral readout of evolving decisions during ongoing movement (Fig. 2 A). In our data, we identified three clear patterns that fit well with the affordance competition hypothesis (Cisek 2007 ) and empirical findings on reach movements to competing targets (for a review, Gallivan et al. 2018 ). First, weight shifts in congruent/correct and incongruent/incorrect trials exhibit similar trajectories in the early phase of the split step. While an early bias toward the congruent side is observable in both conditions, these trajectories diverge when, in incongruent/incorrect cases, players appear to realize—based on incoming sensory evidence—that their initial movement direction is incorrect (see Fig. 2 A). A corresponding pattern of a late divergence is observed when comparing congruent/incorrect and incongruent/correct returns. This suggests that participants incorporate prior knowledge in their early weight shift to put themselves in a (probably) beneficial state, while continuously evaluating both response options during ongoing action and stop committing when they realize they are moving in the wrong direction. Second, in incongruent/correct and congruent/incorrect trials, the early weight shifts toward the prior are descriptively less pronounced (see Fig. 2 B). One potential explanation for this pattern could be the gambler’s fallacy: Participants may expect an incongruent serve from time to time after several congruent serves in a row (e.g., described for soccer goalkeepers by Misirlisoy and Haggard 2014 ). Notably, however, this explanation would predict negative values for the weight shift in these cases—which we did not find empirically. We therefore argue that this pattern is better explained by a continuous evaluation of the two response options given motor costs and noise. Weight shifts in early phases of the split step are, like all actions, subject to motor noise and distributed around a mean value. Consequently, it is unavoidable that players occasionally start with a less pronounced weight shift toward the congruent side, which temporarily enhances the attractiveness of the incongruent side. In turn, this random deviation increases the probability of initiating a movement in the right direction in incongruent trials, but also of moving incorrectly in response to congruent serves. Third, in congruent/correct and neutral/correct trials, weight shifts develop almost in parallel over the split step (Fig. 2 A). This pattern indicates that the advantage of integrating prior knowledge at an early stage of an action is preserved throughout the trial. Additionally, it suggests that if one is already inclined toward one option, this option gains relative attractiveness over time. Due to biomechanical costs (e.g., Griessbach et al. 2022 ), switching to the alternative option becomes progressively more difficult and thus unattractive. Taken together, these three observations in complex, full-body actions support the idea that humans specify competing affordances in parallel and continuously evaluate them in action (Cisek 2007 ), while exhibiting an early bias toward the more probable option. When both options were equally likely (50:50, neutral condition at the beginning of the experiment), participants initiated their split step with a neutral weight shift, suggesting that they hedged their bets between the two sides (e.g., Haith et al. 2015 ). After exposure to the opponent’s serve preferences (80:20, biased condition), an early weight shift was apparent (Fig. 2 B), suggesting that they incorporated prior probabilities to initiate action under uncertainty (Hudson et al. 2007 ; Chapman et al. 2010 ). This strategy enhanced performance in frequent (congruent) trials but impaired performance in rare (incongruent) trials, optimizing overall performance under uncertainty in line with Bayesian principles (Körding and Wolpert 2006 ). In conclusion, this study bridges fundamental research on human sensorimotor behavior and applied research in sport science, exemplifying how the investigation of complex, full-body sensorimotor behavior can advance both fields (Cisek and Green 2024 ; Maselli et al. 2023 ). By leveraging XR technologies, we examined naturalistic behavior under ecologically valid yet experimentally controlled conditions (Mangalam et al. 2023 ). Our findings provide evidence that experienced tennis players exploit prior knowledge in continuous anticipatory decision-making—an issue that remained debated in sport science (Avilés et al. 2019 ). Using tennis as an exemplary case, we further show that humans probabilistically integrate prior knowledge in continuous decision-making, consistent with leading approaches to sensorimotor behavior, such as the theories of internal models (Wolpert et al. 1995 ), optimal feedback control (Todorov and Jordan 2002 ), active inference (Friston 2010 ), and affordance competition (Cisek 2007 ). Acknowledgements The author would like to thank Dario Breitenmoser, Sascha Muhmenthaler, Janis Holzer, and Maurin Frank for recruiting players and collecting data. We would also like to thank Martin Widmer and Simon Maurer as well as the Technology Platform of the Faculty of Human Sciences at the University of Bern for their technical support. Finally, special thanks go to Ralf Kredel for his advice in implementing the experimental design and analyzing the raw data obtained. Appendix A See Fig. 5 ; Table 3 . Fig. 5. Open in a new tab Body segments of the full body model exported from Motive software 3.1.4 (OptiTrack) Table 3. Relative weights of body segments to calculate center of mass Body segment (Shan and Bohn 2003 ) Relative weight Motive body segment(s) Relative weight Head 0.07 Head 0.07 Upper torso 0.18 Neck 0.18/4 Left shoulder 0.18/4 Right shoulder 0.18/4 Upper torso 0.18/4 Middle torso 0.12 Middle torso 0.12 Lower torso 0.13 Lower torso 0.13 Thigh 0.14 Left thigh 0.14 Right thigh 0.14 Shank 0.05 Left shank 0.05 Right shank 0.05 Foot 0.01 Left ankle 0.01/2 Left foot 0.01/2 Right ankle 0.01/2 Right foot 0.01/2 Upper arm 0.03 Left upper arm 0.03 Right upper arm 0.03 Forearm 0.01 Left forearm 0.01 Right forearm 0.01 Hand 0.01 Left hand 0.01 Right hand 0.01 Open in a new tab Appendix B See Tables 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 and 25 Author contributions D.B. and S.Z. conceived and designed the research. D.B. performed the experiments. D.B. analyzed the data. D.B., S.Z., and E.-J.H. interpreted the results of experiments. D.B., S.Z., and E.-J.H. wrote the manuscript. All authors reviewed and approved the final version. Funding No funding was received for this work. Data availability All data and code used for statistical analysis and figure generation, a video of the experimental task, the experimental protocols, and the c3d files of the full body model of each trial are available at GitHub ( https://github.com/ispw-unibe-ch/continuous_decision-making_under_uncertainty_in_tennis ) or at Zenodo (10.5281/zenodo.15915771). Declarations Conflict of interest The authors report no conflict of interest. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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