Introduction

Humans make decisions and take actions based on their decisions, and these decisions and actions influence others, while being influenced by them in return. In his retrospective paper1, the famous game theorist Morgenstern used the example of Professor Moriarty’s battle with Sherlock Holmes to illustrate a strategy in game theory. He referred to this strategy as a circular argument, involving arbitrary decisions such as “I think he thinks that I think!!...” Interpersonal competition is similar to this circular argument; it is a problem, much like an ill-posed problem, for which a unique solution cannot be determined. However, game theory mathematically analyzes the interdependence of decision-making in economic activities and does not involve actions, such as those of Holmes as cited by Morgenstern1. Few studies had tried to apply a game theoretic framework to human motor interaction2,3,4, it has not yet fully addressed interpersonal competition involving individuals’ physical activities in their daily lives.

Interpersonal competition, as analyzed in game theory, is as fundamental to daily human life as interpersonal cooperation. The study of joint action suggests that, for successful cooperation, reducing variability in individuals’ actions and increasing predictability of these actions is important5,6. These are referred to as coordination smoothers7, which aim to enhance coordination by modifying actions, such as exaggerating movements8 or reducing action variability. Conversely, competitive situations increase unpredictability9. However, the competitive situation in this study involves an asymmetrical leader-follower relationship rather than a symmetrical reciprocal relationship, as is commonly observed in sports or everyday life. In a reciprocal relationship as Moriarty and Holmes, it is conceivable that both individuals increase the variability of their actions, making it more difficult for others to predict them.

However, from the perspective of the free energy principle or active inference, individuals are also expected to reduce uncertainty about others’ actions and minimize surprise10,11. However, it can be argued that if others become unpredictable owing to increased variability in their actions, predicting their actions corresponding to one’s own actions, especially in sequential and reciprocal interactions, will become challenging. In other words, increasing the variability of one’s behaviors will increase the uncertainty of others’ behaviors, making predictions more difficult.

Furthermore, the competitive situation in this study involves not only a symmetrical reciprocal relationship but also an asymmetrical leader-follower relationship or sequential relationship, as is commonly observed in sports or everyday life. Strategies employed by individuals in both sequential and reciprocal competitive situations remain largely unclear. Therefore, this study aims to elucidate how individuals attempt to control unpredictable others in competitive situations. To achieve this objective, this study focuses on court net sports, which involve players hitting a ball alternately rather than simultaneously, competing against each other while having opposing goals. Players’ decision-making can be inferred from the direction of the ball’s trajectory, while their actions or responses to others can be observed as movements on the court. Furthermore, quick decision-making and the execution of appropriate responses are required, and skilled players undergo extensive practice to improve, making them well-suited for observing refined strategies in sequential and reciprocal competitive situations.

Interpersonal competition in court net sports involves successive decision-making and abrupt switching of movement patterns in response to unpredictable environmental changes, such as forehand and backhand strokes triggered by opponents’ shots. In other words, to change movement patterns, players must perceive environmental changes, like their opponents’ shots, and make decisions based on their intentions and states. To describe these phenomena, one should consider a dynamical system model that explains the switching of movements in response to external temporal inputs and discrete decision-making within continuous movement. In this paper, we introduce a switching hybrid dynamical system (SHDS) model12,13 that satisfies these requirements.

The SHDS model consists of a discrete dynamical system with an external temporal input and a continuous dynamical system connected by a feedback loop. The discrete and continuous dynamical systems correspond to the brain and body, respectively14. In the discrete dynamical system, as a higher-level module, decisions are made in response to the external temporal input, and these decisions are then sent to the continuous dynamical system. In the lower-level module, the continuous dynamical system switches the movement pattern based on the decisions from the discrete dynamical system. The output pattern of the continuous dynamical system is fed back into the discrete dynamical system as a feedback signal. Consequently, in the discrete dynamical system, the next decision is made based on both the current external temporal input and its physical state. If the external temporal input remains constant and repetitive, the movement pattern converges to a stable state as an excited attractor. However, if the external temporal input switches at certain intervals, the movement pattern also switches in response to the external input. As a result, the movement pattern exhibits fractal transitions between excited attractors, and we can observe hysteresis15,16 (see details in Supplementary Text S1 and Supplementary Fig. S1 online).

This implies that the discrete dynamical system corresponds to decision-making that depends on opponents’ shots and players’ movements or preparatory stances, while the continuous dynamical system corresponds to the pattern of players’ hitting movements. A switching dynamics model with an external temporal input was examined using striking movements under well-controlled experimental conditions. These experiments introduced a pseudo-random sequence of temporal inputs, referred to as a third-order sequential effect17,18,19. Additionally, the fractal transitions between two excited attractors were confirmed, including the hysteresis observed in forehand and backhand strokes in tennis20 and the fractal dimensions in table tennis21. However, in these experiments, the temporal input was controlled experimentally, unlike in real games, and because only individual movements were observed, not interpersonal competition, individual decision-making was not required in the experiments that only responded to external input. In this study, the SHDS model was first applied to interpersonal competition in real matches as a two-coupled SHDS model14.

In this study, we initially confirmed that court net sports can be described as a two-coupled SHDS based on the SHDS model. Specifically, we represent the regularity of the sequences of shot angles for two players in an actual game as a discrete dynamical system, and we represent the corresponding movements on the court as a continuous dynamical system. We then tested the following hypotheses by examining the characteristics of the observed regularity in actual matches through numerical simulations of switching dynamics: Hypotheses: If individuals’ strategy is to increase the variability of their actions to make them less predictable to others, the sequence of shot angles will be closer to a more random sequence. Alternatively, if their strategy is to decrease the variability of their actions and increase the predictability of others, ultimately increasing their own predictability, the sequence of shot angles will become closer to a regular (alternating left-right) sequence.

Results

Analysis of real matches

Regularity in a shot angle sequence as discrete dynamics

Fig. 1
figure 1

Frequencies of significant fitting for match data and surrogate data. For a full definition of “international” and “collegiate” Y matches corresponding to expert and intermediate skill levels, respectively, please refer to the methods section. Panels (A) and (B) show the frequencies of the shot angle \(\Theta\) and the shot length \(\Delta\)L for the international matches, respectively. Panels (C) and (D) show the frequencies of the shot angle \(\Theta\) and the shot length \(\Delta\)L for the collegiate matches, respectively. Blue and green bars represent the frequencies of significant fitting for the original data, and yellow bars represent the frequencies of significant fitting for the surrogate data.

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We confirmed the skill difference between international and collegiate matches in rally length (see Supplementary Text S2 and Fig S2 online) and examined patterns in a sequence of shot angles. We focused only on long rallies with more than nine successive shot sequences. To analyze the regularity in the sequence of the shot angle \(\Theta\) and shot length \(\Delta L\), return map analysis was applied to the data at both levels.

Based on the coefficient of determination (\(R^2\)) and the significance of the regression model, we counted the number of sequences that were significantly fitted to the linear function \(X_{n+1} = a~X_n + b\). Figure 1 shows the frequencies of well-fitted results for the sequences including four to eight successive shots, which were calculated from the data of the actual match and artificial surrogate data.

In international matches, significant differences in the ratio of significantly fitted shot angle \(\Theta\) sequences between actual and artificial surrogate data were found in sequences including four to eight successive shots (4 shots: \(\chi ^2(1) = 5.95, p = 1.47 \times 10^{-2}\); 5 shots: \(\chi ^2(1) = 41.74, p = 1.04 \times 10^{-10}\); 6 shots: \(\chi ^2(1) = 52.86, p = 3.58 \times 10^{-13}\); 7 shots: \(\chi ^2(1) = 69.71, p < 2.20 \times 10^{-16}\); 8 shots: \(\chi ^2(1) = 150.18, p < 2.20 \times 10^{-16}\)). However, a significant difference between the actual and artificial data in the ratio of significantly fitted shot length \(\Delta L\) sequences was found only in rallies including four and six successive shots (four shots: \(\chi ^2(1) = 9.42, p = 2.15 \times 10^{-3}\); six shots: \(\chi ^2(1) = 6.64, p = 9.97 \times 10^{-3}\)).

On the other hand, in collegiate matches, significant differences in shot angles \(\Theta\) were found in four and five successive shots (four shots: \(\chi ^2(1) = 17.24, p = 3.29 \times 10^{-5}\); five shots: \(\chi ^2(1) = 6.98, p = 8.26 \times 10^{-3}\)), and for shot length \(\Delta L\) sequences, significance was found only in four successive shots (\(\chi ^2(1) = 5.83, p = 1.57 \times 10^{-2}\)).

Thus, in international matches, the significance of the shot sequences in fitting the linear functions was found in sequences with four to eight shots. Additionally, such significance in fitting was found only in the analysis of shot angle \(\Theta\) and not in the analysis of shot length \(\Delta L\). Moreover, the significance of the sequences of shot angles \(\Theta\) was found in the data measured in both international and collegiate matches.

Table 1 Frequencies of significant fitting in original data for each type of attractor and repeller.
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Therefore, shot angle \(\Theta\) is a more appropriate state variable that represents regularity in a rally for a real match. In particular, this regularity strongly constrains longer sequences but not shorter ones. Therefore, we focus on shot angle \(\Theta\) as the state variable of a discrete dynamical system and present the results of the analysis concerning shot angle \(\Theta\) only.

Table 1 displays the frequency ratios for a sequence of shot angles that were significantly fitted to four different functions: rotational repeller, rotational attractor, asymptotic repeller, and asymptotic attractor. The combined count of well-fitted data for both functions was 288 for international matches and 38 for collegiate matches. To assess the significance of the data ratios fitted to each of these four functions, we compared them with ratios calculated from surrogate data. The results indicated that the ratio calculated from the actual data was significantly higher for both the rotational repeller and rotational attractor functions. However, for the asymptotic repeller and asymptotic attractor functions, only 16 and 12 cases in international matches, respectively, were found to be significantly fitted. As a result, in both international and collegiate matches, approximately 90.3% and 86.8% of the significantly fitted sequences exhibited periodic patterns. This suggests that international players alternated their shot angles between the right and left sides while facing their opponents prior to taking a shot.

Fig. 2
figure 2

Typical examples of four types of attractors and repellers. Panels (A)–(D) show time series of the rotational repeller, rotational attractor, asymptotic repeller, and asymptotic attractor, respectively. Panels (E)–(H) show return maps corresponding to the aforementioned time series. The horizontal axis of panels (E)–(H) indicates the value of a given shot number shown in panels (A)–(D) (i.e., the Nth shot), and the vertical axis indicates the value of the subsequent shot (the N + 1st shot for the Nth shot). Only the first and last pairs of shots are indicated by bracketed labels in panels (E)–(H). For example, in panel A, the values for the 2nd-6th shots are plotted; therefore, the first points (2,3) and (5,6) are labeled on the side of the data on the return map in panel (E).

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Figure 2 panels A–D display typical examples of time series depicting five successive shot angles \(\Theta\) that exhibit significant and periodic fluctuations (panels A and B) or mild divergence or convergence (panels C and D). In each of Fig. 2E–H, a return map is constructed using the data presented in Fig. 2A–D. The time series depicted in Fig. 2A,C exhibit apparent differences in both amplitude and fluctuation pattern, i.e., periodic or asymptotic. However, as demonstrated in Fig. 2E,G, both dynamical patterns can be categorized similarly as repellers on the return map. Such a similarity in the dynamical order between different time series patterns can also be observed in Fig. 2F,H, both of which are categorized as attractors. The observed regularities for each rally per player at both levels are presented in Supplementary Figs. S3 and S4.

The rotational repeller (attractor) represents such dynamics in which the shot angle gradually moves away (towards) from the opponent’s previous shot, while alternating the shot’s direction to the right and left sides in response to the opponent’s preceding shot angle. In a relatively short sequence, which included five successive shots, the rotational repeller appeared more frequently than the rotational attractor. However, in longer sequences, including more than seven successive shots, attractors appeared more frequently than repellers. These results suggest that as the total number of shots increased to 10–12 (5–6 shots \(\times\) 2 players), the shot angles gradually increased, and then the shot angles gradually decreased as the rally continued. Both players continued to adjust their shot direction based on the opponent’s shot. This means that international players had a tendency to switch the shot direction between the right and left sides regularly during longer rallies.

Hysteresis on the continuous hitting movements

Fig. 3
figure 3

Switching hybrid dynamics in international matches. Panel (A) shows a switching hybrid system in which discrete transition probabilities, as a higher module, are connected to continuous trajectories in a cylindrical phase space as a lower module in all cases. Panels (B) and (C) show separately the case corresponding to the right side and left side input, respectively. Panel (D) shows that the mean trajectories in the cylindrical phase space corresponding to third-order sequence effects for switching input corresponding to panel (A). Panel (E) are expanded into the two-dimensional plane, \(\theta\)-(polar angle \(\phi\), tangential velocity \(v_\phi )\), corresponding to panel (D). Panel (F) shows five Poincaré sections \(\Sigma\) , i.e., \(\theta = 0, \pi /2, \pi , 3\pi /2,\) and \(2\pi\) corresponding to panel (E), with 0.8 equal probability ellipses. Four ellipses of different colours in panel (F) represent four different data sets with different histeresis: magenta, red, blue and cyan represent the data sets with histeresis of LR, RR, LL and RL, the respectively. These colors are corresponding with four data shown in panels (D) and (E).

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Second-order state transition probabilities, comprising the right or left sides, were calculated for a well-fitted sequence of shots as the external temporal input of discrete dynamics. This means that the two cases, including the preceding and current inputs, were the same-right-right (RR) or left-left (LL)-while the preceding and current inputs were different-left-right (LR) or right-left (RL). The output pattern corresponding to the external temporal input consisted of two state variables. In other words, the polar angle \(\phi\) and tangential velocity \(v_{\phi }\) were depicted in the hyper-cylindrical phase space \({\mathcal {M}}\) as trajectories starting at the moment the opponent hit a ball, i.e., the Poincaré section \(\Sigma (\theta = 0)\), until the opponent hit the next ball, i.e., \(\Sigma (\theta = 2\pi )\) as continuous dynamics in the lower module. The trajectory around the cylinder represented one cycle from the current opponent’s shot to the next opponent’s shot, including one shot in the cycle. The first column in Fig. 3 A shows the state transition probabilities as a discrete dynamical system, and the output pattern as a continuous dynamical system in all cases for international matches. The second and third columns show separate patterns for the right and left sides, respectively, as the current input. The trajectories corresponding to the right and left inputs were considered separately. Figure 3B shows the mean trajectories corresponding to the four kinds of external temporal input by the second-order sequence effect as discrete dynamics. To provide a more detailed observation, the distributions of the Poincaré maps on the five Poincaré sections \(\Sigma\), that is, \(\theta = 0, \pi /2, \pi , 3\pi /2,\) and \(2\pi\), are shown in Fig. 3C. Supplementary Fig. 5 shows the results of collegiate matches corresponding to Fig. 3.

One-way MANOVAs were applied to test the equality of the multivariate means of the second-order sequence effect of the right and left inputs on each of the five Poincaré sections. As a result, in two Poincaré sections of five Poincaré sections, i.e., \(\theta = 0\) and \(\theta = \pi /2\), there were significant differences between Poincaré maps for the second-order sequence effects for international matches only (\(\theta = 0\): Preceding, Wilks \(\Lambda = .921, F(2, 251) = 10.782, p = 3.22 \times 10^{-5}, \eta _p^2 = .079\); Current, Wilks \(\Lambda = .976, F(2, 251) = 3.147, p = .044, \eta _p^2 = .024\), \(\theta = \pi /2\): Preceding, Wilks \(\Lambda = .935, F(2, 251) = 8.659, p = 2.31 \times 10^{-4}, \eta _p^2 = .065\); Current, Wilks \(\Lambda = .883, F(2, 251) = 16.637, p = 1.64 \times 10^{-7}, \eta _p^2 = .117\)). This result suggests that the spreading distributions in the Poincaré section did not result from random errors but depended on the sequence of the external temporal input. Namely, the output movement patterns depend not only on the current input but also on the preceding input; thus, we observed four clusters in Poincaré sections. This indicates that a hierarchical input sequence structure is mapped onto the excited attractor, and that the third-order sequence effect can be observed as trajectories in a hypercylindrical phase space20.

We analyzed the characteristic configurations of the four clusters in Poincaré section \(\Sigma (\theta = \pi /2)\) in Fig. 3C. For example, we examined the set of iterative functions \(g_R\) and \(g_L\) that provide a simple fractal structure (i.e., the Cantor set) (see Supplementary Fig. S6 A). This simple example was introduced to demonstrate the behavior of dynamical systems excited by an external temporal input15. It shows how the clustered structure corresponding to an input sequence is constructed from a set of iterative functions, i.e., \(g_R\) and \(g_L\). The four clusters in the Poincaré section in the second stage \(x_2\) line up in the order of RR, LR, RL, and LL from the top (Supplementary Fig. S6 A). This differs from the results of LR, RR, LL, and RL in Poincaré section \(\Sigma\) (\(\theta = \pi /2\)) shown in Fig. 3B. The sets of iterative functions (\(g_R\) and \(g_L\)) in Supplementary Fig. S6 A were modified to provide those in Supplementary Fig. S6 C, with each iterative function having a rotatory attractive fixed point. Although they construct the same Cantor set, the sets in Supplementary Fig. S6 A and S6 C provide significantly different cluster distributions. To emphasize this difference, the case in Supplementary Fig. S6 C is called a Cantor set with rotation. It is conjectured that the Cantor set with rotation originates from Hamiltonian dynamics with complex eigenvalues. The results of this study correspond to the Cantor set with rotation. The locations of the four clusters in the Poincaré section \(\Sigma\) (\(\theta = \pi /2\)) (LR, RR, LL, and RL from the top), which are clearly second-order sequence effects, correspond to this theoretical prediction. Consequently, eight clusters of trajectories were observed in the hypercylindrical phase space, as shown schematically in Supplementary Fig. S7 In summary, four clusters could be distinguished in the Poincaré section \(\Sigma\), and second-order sequence effects were confirmed. Furthermore, there were eight trajectories between the excited attractors and third-order sequence effects in the hypercylindrical phase space. These sequence effects correspond to the hierarchical structure represented by the Cantor set with rotation.

These fractal transitions, corresponding to the switching input, provide strong evidence to support the SHDS. Furthermore, the switching input was not a well-controlled experimental condition20,21; it represented the sequence of the opponent’s shot angles. This result supports our hypothesis that the individual dynamics underlying court net sports can be understood as SHDS in a non-autonomous system.

Two-coupled switching hybrid dynamics

Fig. 4
figure 4

Two-coupled switching hybrid dynamics. The upper panel displays the sequences of shot angles for both players in video clips where the shot angles are superimposed. Panels (A) and (A’) depict the time series of shot angles \(\Theta\) for each player X and Y, respectively. The sequence is corresponding to the upper video clips. Panels (B) and (B’) show the return maps corresponding to the time series of panels (A) and (A’). Filled circles in panels (B) and (B’) show significant time series as a rotational repeller and a asymptotic attractor, respectively. Panels (C) and (C’) display the time series of positions (polar angle \(\phi\)) for both players in polar coordinates. Panels (D) and (D’) show the time series of tangential velocity for both players. The triangles represent the moment of the opponent’s shot, and the circles indicate the moment of the players’ own shot. The inverse triangles show the four kinds of temporal inputs (i.e., magenta: LR, red: RR, blue: LL and cyan: RL). Panels (E) and (E’) show the trajectories in a cylindrical phase space, with the line colors corresponding to the four types of input. Panels (F) and (F′) display the Poincaré sections and the sequences of Poincaré maps. \(\Sigma\) denotes the Poincaré section with \(\theta =0\) and each of four circular trajectories indicates the time series of polar angle \(\phi\) and tangential velocity \(v_{\phi }\) with each of four hysteresis. Panels (F) and (F’) display the Poincaré maps on the Poincaré section, with the numbers inside the figures indicating the shot number.

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We found that 91 sequences in 60 rallies and 27 sequences in 19 rallies for international and collegiate matches, respectively, exhibited regularities in successive shot angles as a discrete dynamical system (Supplementary Fig. S3 and S4). Finally, we present a two-coupled SHDS in Fig. 4 as an example of sequences in which regularities are found in both players in a rally. The upper panels Fig. 4A,A′ show the sequence of shot angles for both players as video clips and time series; Fig. 4 A shows player X, and Fig. 4A′ shows player Y. Figure 4B,B′ show the return map corresponding to the sequence of Fig. 4A,A′, respectively. These systems are regarded as discrete dynamical systems with higher modules. The sequence of shots for player X (Fig. 4A) showed alternating changes between the left and right sides from the second shot to eighth shot. This sequence of shots for Player X, from the second shot to seventh shot, fitted the rotational repeller, as shown in Fig. 4B. Conversely, the sequence of shots for player Y, from fourth shot to seventh shot, fitted the asymptotic repeller (Fig. 4B′). The sequence of shots for player Y, that is, the output of continuous dynamical systems for player Y (Fig. 4C′–F′), corresponded to the external temporal input \(I_{ext}(t)\) for player X (Fig. 4A′,B′) as the discrete dynamical system, and vice versa.

Figure 4C,C′ show the time series of the players’ position in polar coordinates represented by the angle \(\phi\), Fig. 4, while D and D′ display the time series of tangential velocity denoted as \(v_{\phi }\). The inverse triangles represent the moment of the opponent’s shot, with colors indicating the direction—red for the right side and blue for the left side. The triangles signify the moment when the player takes their own shot. The position of their own shot corresponds to the external temporal input; when the input is on the right side, the shot position is also on the right side (positive value), and vice versa. This implies that the continuous dynamical system exhibits movements in response to the external temporal input. When a right-side input is provided, the player moves to the right side to hit the ball and then returns to their next waiting position. Figure 4E,E′ illustrate the trajectories in a hyper-cylindrical phase space consisting of two state variables: the polar angle \(\phi\) and the tangential velocity \(v_{\phi }\). Additionally, Fig. 4F,F′ display the transitions of the Poincaré map on the Poincaré sections \(\Sigma (\theta = 2\pi )\) (see Supplementary Movie M1).

To examine the relationship between the opponent’s shot course as an external temporal input and movement pattern as an output, we calculated partial correlation coefficients for international matches. The partial correlation coefficients between the shot angle \(\Theta\), polar angle \(\phi\) and tangential velocity \(v_{\phi }\) are \(r_{\Theta \phi . v_{\phi }} = .408, p = 1.31 \times 10^{-8}, r_{\Theta v_{\phi } . \phi } = .488, p = 4.92\times 10^{-12}\), respectively (Supplementary Fig. S8). This suggests that the relationship between inputs and outputs is significant; namely, outputs strongly depend on inputs. The results show that the two systems are strongly and reciprocally connected, supporting our hypothesis that the dynamics of interpersonal competition underlying court net sports could be considered a two-coupled SHDS autonomous system.

Simulation experiment

Players’ movements on the court, in response to continuous switching of external inputs, were represented by a mass-spring-damper model. To investigate the regularity observed in a discrete dynamical system during real matches, the correlation dimensions on the Poincaré map were calculated by modifying the second-order transition probabilities as a sequence of external temporal inputs. Figure 5A shows the correlation dimension on the Poincaré map when the R–R and L–L state transition probabilities are independently modified. With perfect regularity, the R–R transition probability was 0, and the R–L transition probability was 1.0. Similarly, the L–L transition probability was 0, and the L–R transition probability was 1.0, resulting in an alternating repetition of different inputs. Complete regularity is located outside the lower-left quadrant of Fig. 5A, where the corresponding Poincaré map is Fig. 5E, and the correlation dimension is zero. However, in the case of complete randomness, the R–R transition probability was 0.5, the R–L transition probability was 0.5, the L–L transition probability was 0.5, and the L–R transition probability was 0.5. This configuration is located at the center of Fig. 5A, where the corresponding Poincaré map is shown in Fig. 5B. At least eight clusters are observed, indicating the presence of a fractal structure with a correlation dimension of 1.433.

Fig. 5
figure 5

Results of the simulation experiments. Panel (A) shows correlation dimensions on Poincaré maps which were obtained through simulation experiments, corresponding to various second-order state transition probabilities. Panels (B)–(E) show the Poincaré maps (top) and correlation dimensions (bottom) which were obtained for four representative state transition probabilities (see Supplementary Figure S11), respectively.

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The pseudo-international and pseudo-collegiate sequences, generated based on second-order state transition probabilities derived from both international and collegiate match sequences, were situated between complete regularity and complete randomness, as illustrated in Fig. 5A. The Poincaré maps for the college and international levels corresponded to Fig. 5C,D, respectively, with correlation dimensions of 1.192 and 0.989, respectively. This indicates that the pseudo-collegiate sequence is closer to a completely random sequence than the pseudo-international sequence. When compared with real matches, the correlation dimensions for international and collegiate matches were similar to the results obtained from the pseudo-sequences. The correlation dimension for international matches was smaller than that for collegiate matches, consistent with the simulation results (Supplementary Fig. S9).

Discussion

Interpersonal competition in court net sports requires successive responses to an opponent’s shot, quick decision-making, and appropriate execution within short time intervals. Because the intentions of others cannot be completely predicted, court net sports serve as an ideal platform for observing human characteristics. This includes examining how individuals attempt to control unpredictable opponents in sequential and reciprocal situations, aiming to minimize uncertainty in the environment. To achieve this goal, we applied the SHDS model12,13,14 to analyze shot angles and on-court movements in international and collegiate soft tennis matches. Furthermore, by incorporating the mass-spring-damper model into the SHDS framework, we conducted simulation experiments to investigate the fractal nature of on-court movements resulting from the switching of input sequences. This analysis helped clarify the regularity in the sequence of shot angles observed in actual matches.

One of the main findings of this study is that using regularity to reduce the uncertainty of unpredictable intentions of others is an essential interpersonal skill in controlling others in sequential and symmetric reciprocal competitive situations. This strategy is believed to involve skilled players capitalizing on their opponents’ intentions to predict or reduce uncertainty11,22. They create regular patterns so that the opponent will predict them, and then disrupt that regularity to confound the opponent’s predictions. This is evident from the observed skilled players’ regularity in the shot angle, which falls between a completely random series and a consistently alternating left-right series. This is also apparent in Supplementary Fig. S3 and Fig. S4, where regularity is not used in the final phase of the rally but predominantly before it. Conversely, collegiate players exhibit less regularity and tend to approach randomness in their actions, which may be attributed to their efforts to avoid predictability by their opponents9. These results suggest that skilled players possess interpersonal motor skills that involve initially behaving consistently to reduce others’ uncertainty, and then prompting others to predict the regularity of their actions as a means of deception. To achieve this, the motor skill is used to create regular patterns that opponents can predict. In other words, the players’ strategy is not to prevent the opponents from predicting their movements from the beginning, but to deceive the opponents by initially prompting them to predict the players’ behaviors5,6,7. This action decreases the unpredictability of the opponent and increases the predictability of the opponent’s behavior. Both individuals want to avoid surprising themselves with their actions but face a dilemma as they want to surprise others. This suggests sequential, symmetric, reciprocal, and competitive interactions that form a circular argument.

In actual matches, we also discovered that skilled players exhibited greater consistency in their shot angle sequences compared to collegiate players. This consistency would be a manifestation of the tit-for-tat strategy, a strategy that leads theoretically stable solution for both players such as Nash equilibrium.2,3,4. Additionally, we found a third-order sequence effect, that has been observed in the sequence of the return balls systematically designed to have random sequence20. This hysteresis would emerge depending on the course of the opponent’s hit (i.e., forehand side or backhand side) that is manifested by the temporal development of the Cantor set with rotation, which has been proposed as a fractal transition in the SHDS12,13,15. This hysteresis is believed to result from the inertia in the rotational motion during hitting, causing a time lag of the rotation of the trunk. This lag would emerge due to a delay in returning to the home position after making a shot on the court and also effects on opponent’s perceptual motor system. Therefore, each of two players tends to repeat same shot by using the strategy based on the game theory. Consequently, the rally tends to continue, however, the system of mutually connected two players would be gradually disrupted due to physical delay and be broken by either player’s point as a consequence. Thus, the game would evolve thorough the interaction between cognitive game strategy and physical constraint of body movement as hypothesized as SHDS.

Models of coupled oscillators, such as the HKB models23, have explained the synchronization between the two systems24,25,26,27. However, because these models represent autonomous systems, they do not consider the individual behaviors of the two systems. In contrast, SHDS is a model of a non-autonomous system that explicitly incorporates temporal external inputs, allowing us to analyze the individual behaviors of the two systems. Although previous applications of SHDS to human movements have experimentally controlled external temporal inputs20,21,28,29, the present results were obtained during an actual match between two players without controlling for external inputs. Thus, incorporating this external temporal input into SHDS may enable us to elucidate the underlying principles of seemingly complex behaviors in human interactions.

This study revealed that the behavior of two players in court-based net sports can be described as a two-coupled switching hybrid dynamical system (SHDS). Expert players employ interpersonal strategies that reduce the unpredictability of their opponents by maintaining regularity in their own actions. A two-coupled SHDS could apply other court-based net sports, such as badminton, table tennis including doubles matches, because in these sports players hit a ball or shuttle alternately. However, to identify the regularity using the return map analysis, we need to analyze the longer rallies, that is, more than nine successive shots. On the other hand, team sports like a football or handball and combat sports like a boxing or judo could be also regarded as zero-sum games, however we could not define the input pattern, because two teams or players attack simultaneously. As a result, we could not be applied a two-coupled SHDS to these types of sports14. This discovery concerning interpersonal strategies could assist policymakers in the realm of business and organizational behavior in formulating policies that enhance competitiveness.

Methods

Data acquisition in real matches

We recorded nine men’s singles matches during the 8th Asian Soft Tennis Championship in Chiba, Japan, in November 2016, and nine men’s singles matches in the second division of the Tokai collegiate league in Tsu, Japan, in September 2017 (see Supplementary Text S3 for “soft tennis”). We used a video camera (SONY, HDR-CX630, 30 fps). The international matches and collegiate matches scored 353 and 300 points, respectively, in all nine matches. However, 39 and 20 points from the international and collegiate matches, respectively, were excluded from the analysis due to recording problems.

The players in the international matches represented the top three countries in the world: Chinese Taipei, South Korea, and Japan. The players in the collegiate matches competed in the second division of the local (Tokai) regional collegiate league.

Each match featured two players, and the ground level was determined at the center of each player’s feet. This point was digitized at a sampling rate of 30 Hz from the service to the end of the rally, and two-dimensional Cartesian coordinates were reconstructed using the direct linear transformation method30. The origin was set at the center of the court, with the X-axis parallel to the baseline and the Y-axis parallel to the sideline. All positional data underwent low-pass filtering using a dual-pass zero-lag fourth-order Butterworth filter with a cut-off frequency of 1 Hz. Furthermore, the moments when the ball made contact were identified through frame-by-frame observation.

Rallies that spanned nine shots, including the service, were extracted from all data points because we analyzed more than four successive shots of each player to examine the regularity of the return map and the third-order sequence effect of hysteresis in continuous movements. As a result, 101 and 31 points were analyzed for the international and collegiate matches, respectively, with the analyzed data accounting for 28.6% and 10.3% of all points, respectively.

All participants provided written informed consent before recording. The procedures were approved by the Internal Review Board of the Research Center of Health, Fitness, and Sports at Nagoya University and conformed to the principles of the Declaration of Helsinki. Approval from the Japan Soft Tennis Association was obtained for the use of videos in this study.

Analysis of the shot sequences in a rally

State variables

To represent the sequential decision-making process using the return map as a discrete dynamical system, we defined the state variables, which were the shot courses, as state variables in decision-making. For the continuous dynamical system, continuous movements were defined as trajectories in the cylindrical phase space, with two state variables: the position and velocity of the movements.

The shot course consisted of the shot angle \(\Theta\) and shot length \(\Delta\)L as candidate state variables. The shot angle \(\Theta\) is calculated as the inner angle between each vector of the opponent’s shot trajectory and the subsequent shot trajectory. This variable can be calculated for shots, including the service return to the last shot in each rally. We defined the angular deviation of the shot trajectory to the right of the vector inverse to the preceding shot trajectory as positive, and that to the left as negative (Fig. 6A). Shot length \(\Delta\)L is defined as the depth of a player’s shot relative to the position at which the opponent released the preceding shot. If the opponent is forced to move backward by the player’s shot, \(\Delta\)L is calculated as positive (Fig. 6B).

Players’ positions and velocities were defined in polar coordinates with the origin at the center of the court. The position is represented by the polar angle \(\phi\), and the velocity is represented by the tangential velocity \(v_{\phi } = r \phi\) (Fig. 6C). The deuce court side is considered positive, while the advantage side is considered negative.

These four variables, \(\Theta\), \(\Delta\)L, \(\phi\) and \(v_{\phi }\) were calculated for sequences that included more than nine shots (the reason for this limitation on the rally’s length is explained in the following paragraph). Each of the four variables was calculated for 202 sequences (101 rallies \(\times\) two players) and 62 sequences (31 rallies \(\times\) two players) for international and collegiate matches, respectively.

Fig. 6
figure 6

Definition of state variables. Panels (A) and (B) show the definitions of shot angle (\(\Theta\)) and shot length (\(\Delta\)L) for the discrete dynamical system, respectively. In Panels (A) and (B), red circles represent the opponent’s hitting positions; red lines represent the preceding shot; blue circles represent the current hitting position; blue lines represent the current shot; and blue crosses represent the opponent’s next hitting position. Panel (C) shows position (polar angle \(\phi\)) and velocity (tangential velocity \(v_{\phi }\)) defined in polar coordinates with the origin at the center of the court.

Full size image

Return map analysis

The discrete dynamics of the two variables, \(\Theta\) and \(\Delta\)L, can be analyzed on a return map, similar to a Lorenz map31. The state of each data point for a given player \(\Theta _{n+1}\) can be visualized by plotting it as a vector relative to that of n th previous shot \(\Theta _{n}\). The transition pattern of the points represents the trajectory of the state in a discrete dynamical system.

The return map analysis can be applied to the sequential data \(\{\Theta \} = \Theta _1, \Theta _2...\Theta _n\) as shown in Fig.  2A. The data set of the first two points is represented by the point (\(\Theta _1\), \(\Theta _2\)), indicated by the filled marker, and moves to the next point (\(\Theta _2\), \(\Theta _3\)) through the point (\(\Theta _2\), \(\Theta _2\)) as shown in Fig. 2E. Therefore, the data set of the last two points can be represented by the point (\(\Theta _{n-1}\), \(\Theta _{n}\)) following the point (\(\Theta _{n-1}\),\(\Theta _{n-1}\)). As seen in Fig. 2A,B, if the sequence fluctuates periodically, the points on the map move along the rotated trajectory in Fig. 2E,F, whereas the sequence fluctuates and moves to the fixed point in Fig. 2C,D. The points move from the point asymptotically, as shown in Fig. 2G,H. Thus, we can determine the type of fluctuation in a given sequence from the pattern of the moving trajectory in the return map space.

We postulated that each of the two variables, \(\Theta\) and \(\Delta\)L, fluctuates in either of these two patterns periodically or asymptotically, and that the pattern of the given variable can be mathematically determined by analyzing the relationship between the geometrical distribution of \(\Theta\) and the linear function \(X_{n+1} = a~X_{n} + b\). For example, if the distribution pattern of \(\Theta\) on the map fits the linear regression function \(X_{n+1}=a~X_{n}+b\), this relationship can be easily determined by confirming the parameter a32,33. When \(a<0\), the point on the map moves along the rotational trajectory in Fig. 2E,F, whereas \(\Theta _n\) alternately decreases or increases, as shown in Fig. 2A,B, respectively. By contrast, in the case of \(0<a\), the trajectories are asymptotically close to, or away from, a certain point in Figure 2G,H, whereas \(\Theta\) decreases or increases gradually to that point, as shown in Fig. 2C,D, respectively. Additionally, if \(|a|<1\) the points are attracted to the fixed point (i.e., an “attractor”) as shown in Fig. 2F,H, whereas in the case of \(|a|>1\), the points move away from a given point (i.e., a “repeller”) as shown in Fig. 2E,G.

If the rally includes ten shots and starts with a service shot by player X and ends with a shot by player Y, players X and Y have four and five successive data points of \(\Theta\) and \(\Delta\)L, respectively, which were calculated from the shot angles excluding that of the service (his/her 1st shot) by player X. These data points, calculated from the shot angles of players X and Y, can be projected onto the corresponding three and four points on the return map, respectively. Then, for the data shot by player X, the linear fitting analysis can be applied to only one sequence of three successive points on the return map calculated from the second to the fifth shots by player X. On the other hand, for player Y, four successive plotted point fitting can be applied to two sequences of three successive points on the map representing the first to fourth shots and that representing the second to fifth shots, and the four-point fitting can be applied to successive data points from the first to fifth shots.

Thus, \(N+1\) successive shots are required to calculate N points on the return map. Therefore, a rally must include at least eight shots in total to project at least three points on the return map. In this case, the receiver returns at least four shots, which are sufficient to calculate the three points on the map. However, for the shot sequence of the server, only two points can be projected on the map calculated from three shots, excluding the service shot. Thus, to determine the linear function on the return map, the length of the shot sequence must be greater than four, that is, nine shots in total. Therefore, the range of lengths of the shot sequences was set to four, five, six, seven, and eight to calculate three, four, five, six, and seven points on the return map, respectively. Further, as explained above, we can extract \(i+1\) sequences of N successive shots from the rally, including \(N+i\) return shots, for each player. We independently analyzed each \(i+1\)-sequence on the return map extracted from the same rally.

Statistical procedure of linear fit analysis

To determine the function that best fits the given data significantly, a linear fitting analysis was applied to sets of points ranging from three to seven in each rally. The point on the return map was calculated for each data sequence using any of the five lengths (three, four, five, six, and seven points on the return map). A linear fit analysis was applied to all these sequences of points on the return map. The significance of fitting to the linear function \(X_{n+1}=a~X_{n}+b\) was assessed using the coefficient of determination \(R^2\), with a significance level set at \(R^2>0.8\) or \(p<.05\). This choice was made because the coefficient of determination, \(R^2\), tends to decrease as the number of fitting points increases, while the significance of the regression model tends to increase for longer fitting points. The ratio of significantly fitted sequences of a given length to the total number of sequences of the same length observed in the same match was calculated. This ratio was determined independently for each sequence with five different lengths for each of the nine matches played by international and collegiate players.

The significance of these ratios was tested by comparing them to the ratios calculated from 200 surrogate data points generated by randomly shuffling data sequences of the same length as the shot sequence34 (see Supplementary Fig. S10). These surrogate data sets were also fitted to a linear function using the same procedure and criteria that were applied to the data points calculated from the actual data. The ratio of significantly fitted data points to all 200 surrogate data points was calculated independently for each sequence. This analysis was conducted for five different sequence lengths for each of the nine matches involving international and collegiate players.

To estimate the significance of the difference in the ratios between the actual and surrogate data, we tested the hypothesis of equal proportions independently for each skill level and length of successive shots. The significance level was set at p < 0.05. A significant difference suggests that such regularity is present in the sequence as a rotational or asymptotic attractor or repeller.

Analysis of trajectories in cylindrical phase space

To examine a continuous dynamical system, the position (\(\phi\)) and velocity (\(v_{\phi }\)) are designated \(x_1\) and \(x_2\), respectively. The output pattern—the movements of players—was depicted as trajectories in the hyper-cylindrical phase space \({\mathcal {M}}\), which started at the moment of the opponent’s preceding shot (i.e., the Poincaré section \(\Sigma\)(\(\theta = 0\))), until the next opponent shot (\(\Sigma\)(\(\theta = 2\pi\))). The trajectory around the cylinder represents a single cycle.

The analyzed data were the output patterns corresponding to a well-fitted input pattern on the return map analysis, i.e., the sequence of the opponent’s shot angles. Further, to examine the hysteresis of the output pattern corresponding to the input sequence, we considered the third-order sequence effect of the input. In this study, two input patterns were used: right (R) and left (L). Thus, the third-order sequence effect was \(2^2 = 8\) pattern, that is, RRR, LRR, LLR, RLR, LLL, RLL, RRL, and LRL. The well-fitted input pattern consisted of four-to-eight sequences of shots; for the four-shot sequence, we could only observe two series of third-order sequence effects, i.e., from the first to the third shot and from the second to the fourth shot.

We also calculated the second-order transition probability for the well-fitted input sequences—RR, LR, LL, and RL—because if we observed the second-order sequence effect on the Poincaré section \(\Sigma\), it would indicate eight types of trajectories in the cylindrical phase space20. To examine the second-order sequence effect on the Poincaré section \(\Sigma\), a MANOVA using Wilks’ \(\Lambda\) of position \(\phi\) and velocity \(v_{\phi }\) as multivariate variables on five Poincaré sections \(\Sigma (\theta = 0, \pi /2, \pi , 3 \pi /2, 2\pi )\) and the effect size \(\eta _p^2\) were calculated.

Simulation experiment

Model for continuous movement on the court corresponding to external input

First, the damped mass-spring model, which was applied to discrete movements in the shepherding task27,35, was applied to the movements of players on the court.

$$\begin{aligned} m \ddot{x} = - b {\dot{x}} - k x \end{aligned}$$
(1)

where x represents the position of an object or body, \({\dot{x}}\) the velocity of x over time, and \(\ddot{x}\) the acceleration of x over time. Further, m is the object mass, b is a damping parameter that resists motion (for \(b > 0\)), and k is a spring force or stiffness parameter that induces motion (for \(k > 0\)). When \(b=0\), this mass-spring model is termed a simple harmonic oscillator. If we set \(m=1\) for normalization, Eq. (1) can be transformed into

$$\begin{aligned} \ddot{x} = - b {\dot{x}} - k x \end{aligned}$$
(2)

To apply this damped mass-spring model to the switching dynamics, this equation can be expressed as its modification into the first-order differential equation for a dynamical system. We substitute \((x_1(=x), x_2(={\dot{x}}))\) with \({\varvec{x}}\), which is generally expressed by the following equation15,36:

$$\begin{aligned} \dot{{\varvec{x}}} = A {\varvec{x}} + {\varvec{I}}_l(t) \end{aligned}$$
(3)

Equation (2) is expressed as follows:

$$\begin{aligned} A = \left( \begin{array}{cc} 0 & 1 \\ -b & -k \end{array} \right) , \qquad {\varvec{I}}_l(t) = \left( \begin{array}{c} 0 \\ F_l(t) \end{array} \right) \end{aligned}$$
(4)

where \(F_l(t)\) is a series of two types of external inputs, \([1, -1, -1, -1, 1,...]\), with constant values. \({\varvec{I}}_l(t)\) switches with time interval T.

Series of external temporal inputs

We defined the four types of 2,000 series of external inputs to examine the behavior of the system. First, pseudo-international and pseudo-collegiate series are generated based on the second-order state transition probabilities observed in international and collegiate matches (Fig. 3A, Supplementary Fig. S11 A and S11 B). Second, a periodic series is alternately generated on the right and left sides (Supplementary Fig. S11 C). Finally, a random series is generated based on a uniform distribution from 0 to 1 (Supplementary Fig. S11 D).

Parameter tuning

We chose b and k in Eq. (2) and the time interval T to switch to the external input so that the switching movement pattern could be observed (Supplementary Fig. S11 E–H). Thus, we set \(b=10, k=500\) in Eq. (2), with a time interval of 100 ms. However, it is important to note that this time interval was used solely for calculations and did not correspond to actual time.