Searching for the Effects of Momentum in Beach Volleyball
An Empirical Comparison of Different Models
Abstract
Abstract: The goal of the present study was to investigate whether in a beach volleyball game between teams of equally skilled players, a team is more likely to win a three-set match if it has won the second set than if it has won the first one. Furthermore, we analyzed how many games end after two or after three sets. We compared three models making different predictions: the psychological momentum model (PMM), the strategic effects model (SEM), and the independent probability model (IPM). In line with the PMM and the SEM, the results showed that regardless of how strictly we controlled for ability, there were always significantly more two-set than three-set matches (33 – 44 % three-set matches). In line with the SEM, but not the PMM, the analysis of almost 1,000 three-set matches between equally skilled players suggested that in a third set neither the winner of the first nor the winner of the second set has an advantage in the third set.
Zusammenfassung: Ziel der vorliegenden Studie ist zu untersuchen, ob bei einem Beachvolleyballmatch zwischen Teams gleicher Spielstärke dasjenige Team mit höherer Wahrscheinlichkeit den dritten Satz gewinnt, das den zweiten Satz gewonnen hat als das Team, welches den ersten Satz gewonnen hat. Zusätzlich analysieren wir wie viele Matches nach zwei und nach drei Sätzen enden. Wir vergleichen drei unterschiedliche Modelle, die unterschiedliche Vorhersagen machen: Das Psychological Momentum Model (PMM), das Strategic Effects Model (SEM) und das Independent Probability Model (IPM). In Einklang mit dem PMM und dem SEM legen die Ergebnisse nahe, dass es, ungeachtet dessen, wie stark wir die Leistungsstärke berücksichtigen, signifikant mehr Zweisatz- als Dreisatzspiele gibt (33 – 44 % Dreisatzspiele). In Einklang mit dem PMM, aber nicht dem SEM legt die Auswertung von beinahe 1000 Dreisatzmatches zwischen Teams gleicher Spielstärke nahe, dass im dritten Satz weder der Gewinner des ersten noch des zweiten Satzes einen Vorteil haben.
A central tenet of sports competitions is the idea that the better athlete is supposed to win (Appleton, 1995). Athletes may differ regarding more or less stable variables such as physiological factors, tactical skills, and psychological skills (e. g., self-regulation and coping). Thus, we usually assume that athletes who possess more of these skills than their opponents will be more successful, both during the course of a competition (e. g., points scored during a match) and regarding the outcome of a competition (e. g., winning a match). However, is it possible that the experience of success may itself be a factor that influences future success? Does “success breed success”?
For decades, sport psychologists have investigated the effects of athletic success both on athletes’ psychological experience and on their future performance (Morgulev & Avugos, 2020). Various theories and findings regarding the experience and the consequences of athletic success have been subsumed under the term psychological momentum (PM). Recently, there has been a surge in scholarly as well as public interest in momentum in sports (for a comprehensive review of the current literature and a novel, integrative theoretical conceptualization of momentum, see Morgulev & Avugos, 2020). However, the phenomenon remains somewhat elusive, both on a theoretical level and on an empirical level. Thus, it is unclear what exactly (psychological) momentum is and whether it really exists.
(Psychological) Momentum – A Theoretical Background
In the early 1980s, Adler introduced his concept of momentum as a “state of dynamic intensity marked by an elevated or depressed rate of motion, grace, and success” (Adler, 1981, p. 29), which is aroused through (unusual) successful performances, so-called momentum starters. Although this concept was criticized for lacking theoretical and empirical background (Silva III et al., 1992), it still served as a reference point for subsequent theoretical approaches. Silva III et al. (1988) widened Adler’s theory by observing the different combinations of success and failure. They investigated the data of sets in intercollegiate tennis between similarly skilled players (controlling for the tournament’s stage) and found that the outcome of the first set was a good predictor of the outcome of the second set and the whole match. They argued that if positive PM was operating, players who had won the second set would have higher chances of winning the third set, but the results did not show evidence for this hypothesis.
Some years earlier, Iso-Ahola and Mobily (1980) had defined PM in a similar way, focusing on changes in the (self–)perception caused by initial success and its influence. They tested their approach with data from a racquetball tournament and found that the outcome of the first set was useful for predicting the outcome of the second set and the whole match. They hypothesized that in the case of a third set, both players would experience the same amount of PM and consequently have the same winning probabilities (the fact of a three-set match was used to control for ability). The studies by Iso-Ahola and Blanchard (1986) and Richardson et al. (1988) confirmed these results with data from racquetball and tennis matches. They additionally found that the players perceived a psychological advantage. Likewise, the PM definition given by Vallerand et al. (1988) in their antecedents–consequences model refers to the phenomenon as a perception toward a goal caused by an antecedent event, and they also consider its effect on (among other variables) performance.
While the approaches described up to this point focus on changes in behavior and performance through psychological mediators, Taylor and Derrick (1994) added the aspect of physical arousal in their multidimensional model of momentum in sports. According to them, changes in cognition, affect, and physiology, produced by an initial event, influence players’ performance. Markman and Guenther (2007), in turn, based their model on Newtonian physics and describe PM as an external force given by the product of mass and velocity. They tested their approach in a non-sport setting with psychology students and found that the more PM is generated during one task, the more PM can be transferred to the next task.1
There are also more recent momentum models in the psychological domain, for example, the mediational model of Iso-Ahola and Dotson (2014), where the central idea of success breeding success stays the same, but the mediating variables of intensity, frequency, and duration are added. Morgulev and Avugos (2020) have analyzed the research done to date on PM. They stress the physiological component and establish the concept of psychophysiological momentum as a combination of psychological and physiological variables.
The concept of PM is also strongly related to other phenomena such as the hot hand, flow, or the so-called back-to-the-wall effect. The back-to-the-wall effect states that players will exert more effort in a “do or die” situation when they are facing elimination from the competition. Since some studies (Pope & Schweitzer, 2011; Scarf & Shi, 2008; Szymanski, 2003) support the hypothesis and others do not (Apesteguia & Palacios-Huerta, 2010; Cao et al., 2011; Dohmen, 2008; Morgulev & Galily, 2018;), research remains inconclusive about the question of whether the back-to-the-wall effect is only assumed or whether it truly exists.
Taken together, a core prediction of most models of PM is that an initial success increases the probability of further success (Morgulev & Avugos, 2020). In other words, positive momentum is present when the probability of success is higher after a previous success than after a previous failure. Conversely, negative momentum is present when the probability of failure is higher after a previous failure than after a previous success. This understanding of momentum is sometimes summarized as “success breeds success,” and it is this understanding (and not the individual emotional and cognitive experience) that we refer to when we use the term “momentum” during the remainder of the present manuscript. Morgulev and Avugos (2020) argue that both momentum itself and its perceptions and consequences can be explained by the evolved psychophysiological responses to success in competitive situations. They also emphasize that the definitions of the different phenomena mentioned earlier (PM, hot hand, back-to-the-wall effect, flow) are not strictly separated.
Whereas there is substantial evidence that an initial success indeed influences psychological states and behavior, Morgulev and Avugos (2020) summarize the current evidence on the existence of the effects of PM as follows: “It is not clear to what extent, where and when initial success actually breeds subsequent success” (p. 3). They go on to summarize that “there is a fair amount of evidence of positive momentum in some contexts, but also a fair amount of evidence that no such effect exists” (Morgulev & Avugos, 2020, p. 9). Accordingly, some studies report evidence for momentum (e. g., Cohen-Zada et al., 2017; Page & Coates, 2017), whereas others do not (e. g., Morgulev et al., 2019, 2020). Even among those studies that report finding momentum effects, some open questions and challenges remain.
(Psychological) Momentum – Open Questions and Challenges
First, in many analytical approaches, effects can only be interpreted as evidence for momentum after interindividual differences regarding ability have been controlled for (e. g., Malueg & Yates, 2010; Page & Coates, 2017). For example, one contestant scoring several times in a row does not constitute evidence of momentum when this contestant is simply a substantially better player than their opponent. Thus, evidence for momentum is strongest the more strictly interindividual differences in abilities have been controlled for. More generally, the predictions of various momentum models regarding competitors’ success only hold to the extent that there are no further differences between competitors that may influence their probabilities of winning. If such differences exist, (a) momentum models cannot arrive at meaningful predictions, (b) it becomes difficult to find evidence for the effects of momentum, and (c) different winning probabilities cannot be explained by momentum. Particularly in economic models, this restriction is referred to by the term “all else equal.”
Second, at least in some analytical approaches, it is difficult to distinguish PM and strategic momentum. In contrast to PM models, models of strategic momentum do not assume that athletes are influenced by their prior successes, but instead that athletes react to changing incentives during games (Gauriot & Page, 2019; Malueg & Yates, 2010). Very generally speaking, models of strategic momentum assume that competitors exert effort based on their perceived chances of winning. When both competitors perceive the same chances of winning, all else equal they should exert the same amount of effort. However, when one competitor starts to take the lead, again all else being equal, they should start to exert more effort, because their chances of winning have increased. Specifically, effort is determined by a cost–benefit analysis, where the main cost is exerted effort, and the main benefit is the prize associated with winning (Malueg & Yates, 2010). Meier et al. (2020) explain:
If the position of athletes varies relative to their competitors, athletes have asymmetric incentives, and “strategic momentum” arises. Because the losing athlete must put in extra effort to catch up, the winning athlete’s net prize, i. e., the final award minus the level of effort to win this award, is larger. Thus, the winning athlete will exert more effort than the losing athlete. (p. 3)
Models of strategic momentum are based on economic theorizing and models (e. g., Tullock, 1980). Thus, one key advantage of models of strategic momentum is that they can be formulated as formal models with a high degree of mathematical sophistication.
Not surprisingly, papers with a more psychological background tend to treat PM (i. e., prior successes influence internal states that in turn influence future successes) as the dominant explanatory process and strategic momentum as the alternative that needs to be ruled out, whereas papers with a more economics background tend to do the opposite. For example, in their authoritative review in the International Review of Sport and Exercise Psychology, Morgulev and Avugos only briefly refer to strategic momentum when they state that “economists sometimes attempt to distinguish between psychological and strategic momentum” (Morgulev & Avugos, 2020, p. 2), and they never come back to the topic during the remainder of their text. On the contrary, Gauriot and Page remark in The Economic Journal that “[it] is also sometimes suggested that a momentum can arise for psychological reasons” (Gauriot & Page, 2019, p. 3130). Interestingly, both groups of authors refer to Mago et al. (2013) for their observations. As Gauriot and Page observe, “[in] many situations, the pure effect of winning from the psychological momentum is not distinguishable from the predictions from the strategic momentum” (Gauriot & Page, 2019, p. 3130). Thus, evidence both for psychological and for strategic momentum is strongest when it is supported by a design and an analytical strategy that allows one to distinguish between both concepts (Meier et al., 2020). Distinguishing between psychological and strategic momentum is possible in a setting where the underlying theoretical models arrive at different predictions.
For example, Malueg and Yates (2010) present the strategic effects model, which is a formal model of best-of-three contests (see also Malueg & Yates, 2006). In best-of-three contests (e. g., games where competitors have to win two out of three sets in order to win the game), the strategic effects model predicts that (all else equal) both competitors exert the same amount of effort in the first set, because both perceive the same chance to win. In the second set, the winner of the first set is supposed to exert more effort, because their chances to win the entire game have increased. If the game is tied after the second set and a third set is played, both competitors are again assumed to exert the same amount of effort, because both have the same chance to win. Thus, the strategic effects model predicts that the winner of the first set is more likely to win the second set. Consequentially, according to the model’s predictions, more games are supposed to end in two than in three sets. When it comes to a third set, both competitors are equally likely to win it. While the strategic effects model and PM models arrive at the same prediction for winning the second set (i. e., winners of the first set are more likely to win the second set), they arrive at different predictions for winning the third set: Whereas the strategic effects model predicts that both competitors are equally likely to win the third set, most models based on PM predict that the winner of the second set is more likely to win the third set, because they perceive momentum and thus benefit from heightened self-efficacy or the like.2
Malueg and Yates (2010) tested the strategic effects model against a PM model in a dataset of 351 tennis matches with the same betting odds, thus attempting to control for interindividual differences regarding abilities. They report support for the strategic effects model, but not for the PM model. Mago et al. (2013) confirmed these results for best-of-three contests, nevertheless with a study in a non-sports setting (i. e., a virtual game). Cohen-Zada et al. (2017) tested strategic momentum against PM in judo. Unlike in other disciplines, in judo there are two bronze medal fights reached by former winners and losers, respectively. The results showed evidence for PM, as the former winners had higher winning probabilities than the former losers. It is worth noting that the effects were found for men, but not for women. Meier et al. (2020) found evidence for PM in men’s professional tennis data. They tested PM against strategic momentum using exogenous interruptions. They argue that such interruptions leave strategic momentum unaffected and since they found a negative effect between the interruption and the result, they determined PM to be the trigger. Interestingly, both studies report evidence for psychophysiological momentum from research on 1×1 competitions. For future research it seems to be fruitful for psychologists and economists alike to be aware of both approaches and to try distinguish between them (as in the studies reported above), or even try to synthesize them.
Additionally, a third model exists that the PM model and the strategic effects model are often compared against, namely, the independent probability model. In the independent probability model, all sets of a match are considered to be independent of each other (Ferrall & Smith Jr., 1999; Jackson & Mosurski, 1997). Thus, two equally skilled teams would have the same probability of winning in every set, regardless of how many sets they have won or lost. In the case of equally skilled players, this would result in the same number of two- and three-set games. On the one hand, the independent probability model can be considered to be rather simple, as it does not make any other assumptions than the independency of sets. On the other hand, this might appear to be a reasonable assumption given equal skills of competitors.
Third, different authors have speculated that both psychological and strategic momentum effects – even if there is evidence for them in individual sports – should be less pronounced in team sports than in individual sports. For example, Malueg and Yates (2010, p. 692) observe that “studies with data from individuals usually find evidence of strategic effects, while studies with data from teams usually do not.” One reason why PM might be more pronounced in individual sports could be that psychophysiological changes occurring in individual athletes do not spread to the rest of the team. However, one might also argue that momentum should be more pronounced in team sports than in individual sports because momentum-related changes spread from team member to team member in a process resembling emotional contagion (Hatfield et al., 1993; Moll et al., 2010). Regarding the question of whether there is a difference in PM between individual and team sports, Morgulev and Avugos note that “the literature shows no consistency in the results” (2020, p. 7). Thus, we consider it an open question whether data from team sports are in line with predictions from models both on psychological and on strategic momentum regarding the association between prior and subsequent success.
The Present Research
The goal of the present study was to investigate whether data from real-world competitions are in line with core predictions of the PM model regarding the association of previous success and future success. We aimed to address the aforementioned challenges by (a) controlling for stable interindividual differences regarding ability, (b) choosing a performance situation where the PM model and the strategic effects model arrive at different predictions, and (c) extending research on momentum to teams instead of individuals. We realize that it is impossible to answer the questions of whether (if at all), where, and when momentum effects exist with a single study. Instead, we aim to contribute one piece of information, with the hope that many such studies may together form a comprehensive picture on the existence, extent, and boundary conditions of both psychological and strategic momentum.
Specifically, we aimed to investigate whether in the case of a third set in a beach volleyball game between equally skilled teams, the team who won the second set is more likely to win the third set as a consequence of an advantage caused by momentum. This is a prediction that can be derived from PM models. We chose beach volleyball for several reasons. First, beach volleyball is played in a best-of-three mode and thus allows us to differentiate between the PM model and the strategic effects model. Both models predict that a match in beach volleyball is more likely to end in two than in three sets. However, crucially, the models are in disagreement as to whether the winner of the first or the second set has higher chances of winning the third set. Whereas the PM model predicts that the winner of the second set is more likely to win the third set, the strategic effects model predicts that the winner of the first and the winner of the second set are equally likely to win the third set.
Second, beach volleyball is a team sport. Thus, we can extend research on PM from individuals to (dyadic) teams.
Furthermore, a recent study in judo conducted by Cohen-Zada et al. (2017) suggests that men are affected by the impact of momentum whereas women are not. For this reason, we analyzed the results of men and women separately, but only in an exploratory manner.
Method
We considered the following models and approaches. Although our main focus was on comparing the PM model with the strategic effects model, for sake of completion we also considered the independent probability model.
- 1.PM model: A best-of-three contest between equally skilled teams is more likely to end in two than in three sets. In the case of a third set, the team who won the second set is more likely to win the final set.
- 2.Strategic effects model: A best-of-three contest between equally skilled teams is more likely to end in two than in three sets. If it comes to a third set, both teams have the same probability of winning.
- 3.Independent probability model: In a best-of-three contest between equally skilled teams, both teams have the same probability of winning, regardless of the number of the set. Consequently, a game is as likely to end in two than to end in three sets.
Data Analysis
We used results of tournaments at international level held by the Fédération Internationale de Volleyball (FIVB). Data from the period 2010 – 2019 (excluding 2017) were considered.3 Until 2017 the tournaments were divided into Open and higher awarded Grand Slams. Since 2017 there have been five classes (one star to five stars) of tournaments with different performance levels and prize money. In total, over 27,300 games were taken into consideration in our dataset.
Procedure
First of all, games that were not completed regularly, for example, due to injuries, were excluded from the dataset. Over 25,500 games remained. To pursue the main questions of the present research, we gathered the following information: games in total, two-set games, three-set games, first-set winners among three-set game winners, second-set winners among three-set game winners. The corresponding information can be found in Table 1 and Figure 1. We divided the total sample of matches into three categories with regard to their competitive balance (operationalization of equally skilled teams). In each subsequent category, we employed a stricter operationalization of equally skilled than in the previous category. Thus, matches in the latter categories are subsets of all matches in the previous categories. In the remainder of the manuscript, we refer to analyses pertaining to the different categories as “stages” of the analysis.
- 1.Third set: The fact that a third set is required is seen as an indicator of the similar performance levels of two teams.
- 2.Small betting odds difference: Two teams are considered equally skilled if their betting odds do not differ more than 0.3.
- 3.Small betting odds difference and close first set: Two teams are considered equally skilled if their betting odds do not differ more than 0.3 and the first set ended with a difference of three points or less.
The categorization described above is based on the following studies: Category 1 is based on Silva III et al. (1988) and Iso-Ahola and Blanchard (1986). Category 2, the main category, follows the procedure of Malueg and Yates (2010). Their approach also informed the third category.
The betting odds were available on the website oddsportal“.4 The analysis comprises two steps. First, we analyzed the probability that a game would end in two or three sets and the probability that the team who won the first or the second set would win the third set (Table 2 and Figure 2). Second, a hypothesis test (two-tailed binomial test) was used to test whether the percentage of two-set or three-set games and first-set or second-set winners differed significantly from 50 % at a 5 % significance level. Thereby, the null hypothesis H0 states that the probability for a two-set or three-set game and for the first-set or second-set winners is 50 %.
Results
Stage 1: All Games
Considering all games without restrictions (Stage 1), we noted that about two thirds of all games (65.7 %) ended after two sets (see Table 2 for percentages and p values). The percentage was slightly higher among the female players (67 %) and slightly lower among the male players (64.5 %). In other words, the number of three-set matches was slightly higher among men. The results of the binomial tests were significant in all the three cases, that is, the probability for a two- or three-set game differed significantly from 50 %. The probability for a two-set game was about 15 % higher than expected in H0.
Regarding the three-set games without restrictions (Stage 1), we found that in all groups (total, women, men) the winners of the second set won the third set with a probability of about 51 %. Thus, all effects were very small. Only in the group including the matches of both genders (total) did the result differ significantly from 50 %.
Stage 2: All Games With Small Odds Difference
With respect to the second stage of analysis, that is, all games with a small odds difference, 58.9 % of all matches (2,293 games) ended in two sets. In the group of women, the percentage was somewhat higher (60.3 %) and in the group of the men somewhat lower (57.7 %). In other words, as in the first analysis stage, the number of three-set matches among men was somewhat higher. The results of the binomial tests were significant in all the three cases, that is, the probability for a two- or three-set game differed significantly from 50 %. The probability for a two-set game was about 9 % higher than expected in H0 in the group that covers both genders, about 10 % higher than expected in the women’s group and about 7 % higher than expected in the men’s group.
Regarding the three-set games (943 games) of the second analysis stage, we did not find significant results with regard to whether the winner of the first or the second set won the third set. In the group of both genders and in the men’s group, the second-set winners more often won the third set in absolute numbers.
Stage 3: All Games With Small Odds Difference and Close First Set
With respect to the third stage of analysis, that is, all games with a small odds difference and close first set, 57.1 % of all matches (1,106 games) ended in two sets. In the group of women, the percentage was slightly higher (58.4 %) and in the group of men it was slightly lower (56.1 %). As in the other analysis stages, we found that there was a slightly higher number of three-set matches among men. The results of the binomial tests were significant in all three cases, that is, the probability for a two- or three-set game differed significantly from 50 %. The probability for a two-set game was about 7 % higher than expected in H0 in the group with both genders, about 8 % higher than expected in the women’s group and about 6 % higher than expected in the men’s group.
In all groups, the chances for winning the third set, in a three-set game in this third analysis stage (436 games), did not differ significantly from 50 %, neither for the first-set winners nor for the second-set winners. In absolute numbers, the teams who won the second set more often won the third set in all groups.
All Stages
In all three stages of analysis, there were significantly more two-set games than three-set games, both for men and for women. The better we controlled the relative strength of the teams, the more three-set games occurred. At the same time, there were always more three-set games among men than among women.
Regarding the question of whether the winner of the second set is more likely to win the third set, we found that only in the first stage of analysis, that is, taking into account all three-set games without restrictions, and only in the group that included both genders (total), did the percentage of second-set winners among the three-set game winners (51.4 %) differ significantly from 50 %. Among men, there were always more second-set winners who won the whole match. Nevertheless, for both genders none of the results were statistically significant.
Discussion
In all three stages of the analysis, we found that significantly more matches ended in two sets than in three. This finding is both in line with the predictions of the PM model and the strategic effects model; however, it is not in line with the predictions of the independent probability model. The proportion of three-set matches increased with the more restrictions we made concerning the equality of the skill level. This is because, all else equal, the more the skills of two teams are equal, the more likely it is for a three-set match to occur. Still, even when we restricted our sample to matches with an odds difference equal to or smaller than 0.3 and a close first set, significantly more matches ended in two than in three sets.
Furthermore, (with the exception of one sample) we found that the team that won the second set was not significantly more likely to win the third set. Although in all stages of the analysis, the team that won the second set was descriptively more likely to also win the third set, this advantage only reached significance for the sample with the least restrictions regarding equal skill. However, the effect here was small (51.4 %), while at the same time the sample size was the largest of all sample sizes in the different stages of the analysis (as the sample size got smaller, the more restrictions we employed regarding equal skill). The finding that the second-set winner was not more likely to win the third set is in line with both the predictions of the strategic effects model and the independent probability model. The reported patterns do not differ for female and male teams, with the exception of the second stage of analysis. Here, at least descriptively, there seems to be some support for the assumption of Cohen-Zada et al. (2017) that the effects of PM are stronger for men than for women. However, as there was no significant overall effect regarding the likelihood for the second-set winners to win the third set, we refrain from interpreting this difference any further.
Taken together (i. e., when combining the results regarding the overall proportion of two-set matches and the results regarding the likelihood of the second-set winner to win the third set), these findings most strongly support the strategic effects model. On a descriptive level, there seems to be some support for the PM model in the third stage of analysis (most restrictions regarding equal skill); however, it failed to reach statistical significance.
Strengths and Weaknesses
We consider it a strength that the sample size of the present study was large. In the first stage of analysis, we analyzed 25,532 matches overall. Of these matches, 8,757 were three-set matches. In the third stage of analysis (small odds difference and close first set), we still analyzed 1,016 matches overall, out of which 436 were three-set matches. For comparison, Malueg and Yates (2010) analyzed 351 matches with the same odds, out of which 38 were three-set matches with the same odds and a close first set (note that Malueg and Yates did indeed analyze same-odds matches, and not matches with a small odds difference). The samples analyzed by Page and Coates (2017) and Cohen-Zada et al. (2017) were of a comparable size (N = 375 and N = 217, respectively). The decreasing sample sizes from the first stage of analysis to the last one means that it is difficult to compare significance levels between the different stages, as power decreases with decreasing sample sizes. Thus, the subsample of the third stage (with the most restrictions regarding the skill level) that is the most promising for comparing the three models (because it is the one where competitors are most likely to be equally skilled) also has the smallest power. This is particularly problematic since for the three-set matches the strategic effects model constitutes the null hypothesis (i. e., no difference) that the PM model is tested against (i. e., there is a difference). Thus, the chosen approach favors the strategic effects model insofar as decreasing power leads to decreasing chances of rejecting the null hypothesis. Still, with 436 three-set matches in the third stage of analysis, the respective analyses were reasonably powered. Given the sizes of the total samples in all three stages of analysis (i. e., 8,757, 943, and 436 games, respectively), the respective power of the one-tailed binomial test to detect an effect size of g = 0.05 (i. e., 55 % instead of 50 % of three-set games are won by the winner of the second set) at a significance level of 5 % was 1 in the first stage, .92 in the second stage, and .66 in the third stage.
It is noteworthy that when looking only at the data of the first 3 years in the sample (2016, 2018, 2019), we found significant differences for most of the (sub–) samples. It cannot be finally clarified in this study whether the difference between the results for the subsample and the total sample are random or systematic. To answer this question, further research would be necessary. Speculatively, if it were systematic, the difference might have something to do with the level of the tournaments included. The data for the period 2010 – 2016 only included tournaments of the highest level, whereas for 2018 and 2019 also lower classified tournaments were considered.5 Hence, the performance level in general could be related to the occurrence of PM. We are aware of the fact that not controlling for the tournament level can be seen as a limitation of this study.
We also consider it a strength that we collected real-world data from tournaments where athletes with varying cultural, national, and ethnic backgrounds participated, thus making our results less dependent on the specific characteristics of athletes. With few exceptions, results were robust across all three stages of analyses.
The most severe challenge to our interpretation of the present data is whether we succeeded in controlling for the competitors’ skill level. The present data only allow for conclusions regarding the three different models when competitors are indeed equally skilled. For example, when competitors are not equally skilled, the finding that there are more two-set matches than three-set matches can easily be explained because the better of both teams is both more likely to win the first and to win the second set.
Furthermore, as a dyadic team sport, beach volleyball lies on the separation line between individual and team sports. Thus, it remains subject to further research to what extent the present results can be applied to sports with larger teams.
Thus, our results, just like the results from Malueg and Yates (2010), are in line with the notion that athletes (or teams) strategically allocate effort in such a way that the winner of the first set exerts more effort in the second set. When it comes to a third set, both competitors appear to exert the same amount of effort. This is true for both men and women. Interestingly, and contrary to earlier speculation (see Malueg & Yates, 2010), our results are in line with the notion that not only individual athletes but also teams allocate their efforts strategically. We would like to emphasize that this interpretation is only valid when teams are indeed equally skilled. Otherwise, the reported patterns may be a function of skill differences instead of the allocation of effort.
We caution against interpreting the present results as strong evidence against the existence of PM. First, we did not investigate athletes’ experiences of momentum, but only the behavioral consequences of momentum according to the idea that success breeds success. Thus, our results can only be informative regarding the consequences of momentum, and not regarding the experience of momentum. Second, a substantial body of existing research reports evidence for the existence of PM (Morgulev & Avugos, 2020). Third, recently authors have suggested that PM exerts its strongest influence when there are no breaks between single performances (or points; Den Hartigh & Gernigon, 2018). As there are breaks between sets in beach volleyball, these breaks might have hindered the development of PM. Thus, it might be more promising to look for PM not between sets, but on a point-by-point basis. Still, the present results add to our understanding of best-of-three contests and PM: In these contests we find evidence for strategic effects, but not for the effects of PM. Thus, the present research may contribute to understanding the boundary conditions for PM to develop.
Conclusion
We replicate and extend the results reported by Malueg and Yates (2010). We replicate them insofar as our results, just like the results presented by Malueg and Yates (2010), offer most support for the strategic effects model and less support for the PM model and the independent probability model, when analyzing best-of-three contests. We extend them by following the call from Malueg and Yates to extend research on best-of-n contests from individual to team contests. We caution against overinterpreting the results as strong evidence against PM in general. However, as far as best-of-three contests are concerned, it seems safe to conclude that so far there is no strong evidence for the effects of PM.
Literatur
(1981). Momentum: A theory of social action. Sage Publications.
(1995). May the best man win? The Statistician, 44 (4), 529 – 538. https://doi.org/10.2307/2348901
(2010). Psychological pressure in competitive environments: Evidence from a randomized natural experiment. American Economic Review, 100 (5), 2548 – 2564. https://doi.org/10.1257/aer.100.5.2548
(2011). Performance under pressure in the NBA. Journal of Sports Economics, 12 (3), 231 – 252. https://doi.org/10.1177/1527002511404785
(1996). Momentum strategies. The Journal of Finance, 51 (5), 1681 – 1713. https://doi.org/10.1111/j.1540-6261.1996.tb05222.x
(2017). Psychological momentum and gender. Journal of Economic Behavior & Organization, 135, 66 – 81. https://doi.org/10.1016/j.jebo.2017.01.009
(2018). Time-out! How psychological momentum builds up and breaks down in table tennis. Journal of Sports Sciences, 36 (23), 2732 – 2737. https://doi.org/10.1080/02640414.2018.1477419
(2008). Do professionals choke under pressure? Journal of Economic Behavior & Organization, 65(3-4), 636 – 653. https://doi.org/10.1016/j.jebo.2005.12.004
(1999). A sequential game model of sports championship series: Theory and estimation. Review of Economics and Statistics, 81 (4), 704 – 719. https://doi.org/10.1162/003465399558427
(2019). Does success breed success? A quasi-experiment on strategic momentum in dynamic contests. The Economic Journal, 129 (624), 3107 – 3136. https://doi.org/10.1093/ej/uez040
(1993). Emotional contagion. Current Directions in Psychological Science, 2 (3), 96 – 100. https://doi.org/10.1111/1467-8721.ep10770953
(1986). Psychological momentum and competitive sport performance: A field study. Perceptual and Motor Skills, 62 (3), 763 – 768. https://doi.org/10.2466/pms.1986.62.3.7
(2014). Psychological momentum: Why success breeds success. Review of General Psychology, 18 (1), 19 – 33. https://doi.org/10.1037/a0036406
(1980). “Psychological momentum”: A phenomenon and an empirical (unobtrusive) validation of its influence in a competitive sport tournament. Psychological Reports, 46 (2), 391 – 401. https://doi.org/10.2466/pr0.1980.46.2.391
(1993). Independent trials are a model for disaster. Journal of the Royal Statistical Society: Series C (Applied Statistics), 42 (1), 211 – 220. https://doi.org/10.2307/2347421
(1997). Heavy defeats in tennis: Psychological momentum or random effect? Chance, 10 (2), 27 – 34. https://doi.org/10.1080/09332480.1997.10542019
(2013). Best-of-three contest experiments: Strategic versus psychological momentum. International Journal of Industrial Organization, 31 (3), 287 – 296. https://doi.org/10.1016/j.ijindorg.2012.11.006
(2006). Best-of-three contests between equally-skilled players. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.900001
(2010). Testing contest theory: Evidence from best-of-three tennis matches. The Review of Economics and Statistics, 92 (3), 689 – 692. https://doi.org/10.1162/REST_a_00021
(2007). Psychological momentum: Intuitive physics and naive beliefs. Personality and Social Psychology Bulletin, 33 (6), 800 – 812. https://doi.org/10.1177/014616720730102
(2020). Separating psychological momentum from strategic momentum: Evidence from men’s professional tennis. Journal of Economic Psychology, 78, 102269. https://doi.org/10.1016/j.joep.2020.102269
(2010). Emotional contagion in soccer penalty shootouts: Celebration of individual success is associated with ultimate team success. Journal of Sport Sciences, 28 (9), 983 – 992. https://doi.org/10.10 80/02640414.2010.484068
(2020). Beyond heuristics, biases and misperceptions: the biological foundations of momentum (hot hand). International Review of Sport and Exercise Psychology, 1 – 21. https://doi.org/10.1080/1750984X.2020.1830426
(2019). Does a “comeback” create momentum in overtime? Analysis of NBA tied games. Journal of Economic Psychology, 75, 102126. https://doi.org/10.1016/j.joep.2018.11.005
(2020). Searching for momentum in NBA triplets of free throws. Journal of Sports Sciences, 38 (4), 390 – 398. https://doi.org/10.1080/02640414.2019.1702776
(2018). Choking or delivering under pressure? The case of elimination games in NBA playoffs. Frontiers in Psychology, 9, 979. https://doi.org/10.3389/fpsyg.2018.00979
(2017). Winner and loser effects in human competitions. Evidence from equally matched tennis players. Evolution and Human Behavior, 38 (4), 530 – 535. https://doi.org/10.1016/j.evolhumbehav.2017.02.003
(2013). Rising stars and sinking ships: Consequences of status momentum. Psychological Science, 24 (8), 1579 – 1584. https://doi.org/10.1177/0956797612473120
(2011). Is Tiger Woods loss averse? Persistent bias in the face of experience, competition, and high stakes. American Economic Review, 101 (1), 129 – 157. https://doi.org/10.1257/aer.101.1.129
(1988). Game, set, match: Psychological momentum in tennis. The Sport Psychologist, 2 (1), 69 – 76. https://doi.org/10.1123/tsp.2.1.69
(2008). The importance of a match in a tournament. Computers and Operations Research, 35 (7), 2406 – 2418. https://doi.org/10.1016/j.cor.2006.11.005
(1992). Psychological momentum and skill performance: A laboratory study. Journal of Sport & Exercise Psychology, 14 (2), 119 – 133. https://doi.org/10.1123/jsep.14.2.119
(1988). Analysis of psychological momentum in intercollegiate tennis. Journal of Sport & Exercise Psychology, 10 (3), 346 – 354. https://doi.org/10.1123/jsep.10.3.346
(2003). The economic design of sporting contests. Journal of Economic Literature, 41 (4), 1137 – 1187. https://doi.org/10.1257/jel.41.4.1137
(1994). A multidimensional model of momentum in sports. Journal of Applied Sport Psychology, 6 (1), 51 – 70. https://doi.org/10.1080/10413209408406465
(1980).
Efficient rent seeking . In J. M. BuchananR. D. TollisonG. Tullock (Eds.), Toward a theory of the rent-seeking society (pp. 97 – 112). Texas A&M University Press.(1988). Psychological momentum and performance inferences: A preliminary test of the antecedents-consequences psychological momentum model. Journal of Sport and Exercise Psychology, 10 (1), 92 – 108. https://doi.org/10.1123/jsep.10.1.92
1Momentum research has reached other domains such as economy (e. g., companies’ performances at the stock market, Chan et al., 1996) politics (e. g., candidates’ chances to win the election, Iso-Ahola & Dotson, 2014) and others (e. g., social status, Pettit et al., 2013).
2Although this is true for the majority of approaches to psychological momentum, there are few conceptualizations of psychological momentum that predict the same winning chances for the winners of the first and the second set or no influence of psychological momentum in a third set (Jackson, 1993). In consequence, these approaches cannot be distinguished from the Strategic Effects Model and make it impossible to test the two possible models against each other.
3The data is available on the official website of the federation (https://www.fivb.com/en/beachvolleyball/competitions). At the time the data was collected, there were no results available for 2017. Hence, this year’s data could not be included.
4The address of the website is https://www.oddsportal.com/.
5For every year, we considered all games available on the official FIVB-website. Since the tournament system changed in 2017 from Opens and Grand Slams to 1- to 5-stars tournaments, there is a wider range of skill level for 2018 and 2019.
