Win-stay lose-shift strategy in formation changes in football
- Kohei Tamura^{1, 2} and
- Naoki Masuda^{3}Email author
Received: 9 March 2015
Accepted: 15 June 2015
Published: 17 July 2015
Abstract
Managerial decision making is likely to be a dominant determinant of performance of teams in team sports. Here we use Japanese and German football data to investigate correlates between temporal patterns of formation changes across matches and match results. We found that individual teams and managers both showed win-stay lose-shift behavior, a type of reinforcement learning. In other words, they tended to stick to the current formation after a win and switch to a different formation after a loss. In addition, formation changes did not statistically improve the results of succeeding matches. The results indicate that a swift implementation of a new formation in the win-stay lose-shift manner may not be a successful managerial rule of thumb.
Keywords
reinforcement learning sports soccer1 Introduction
Exploring rules governing decision making has been fascinating various fields of research, and its domain of implication ranges from our daily lives to corporate and governmental scenes. In economic contexts in a widest sense, individuals often modify their behavior based on their past experiences, attempting to enhance the benefit received in the future. Such decision making strategies are generally called reinforcement learning. In reinforcement learning, behavior that has led to a large reward will be selected with a larger frequency, or the behavior will be incrementally modified toward the rewarded one. Reinforcement learning is common in humans [1, 2] and non-humans [3], is implemented with various algorithms [4], has theoretical underpinnings [1, 4], and has neural substrates [5, 6].
A simple version of reinforcement learning is the so-called win-stay lose-shift (WSLS) strategy [7, 8]. An agent adopting this strategy sticks to the current behavior if the agent is satisfied. The agent changes its behavior if unsatisfied. Experimental studies employing human participants have provided a line of evidence in favor of WSLS in situations such as repeated Prisoner’s Dilemma [9, 10], gambling tasks [11, 12], and tasks in which participants construct virtual stone tools [13–15]. It has also been suggested in nonscientific contexts that decisions by athletes and gamblers are often consistent with WSLS patterns even if the outcome of games seems to be independent of the decision [16].
Association football (also known as soccer; hereafter refer to it as football) is one of the most popular sports in the world and provides huge business opportunities. The television rights of the English Premier League yield over two billion euros per year [17]. Transfer fees of top players can be tens of millions of euros [18]. Various aspects of football, not only watching but also betting [19] and the history of tactics [20], enjoy popularity. Football and other team sports also provide data for leadership studies because a large amount of sports data is available and the performance of teams and players can be unambiguously measured by match results [21–23].
In the present study, using data obtained from football matches, we examine the possibility that managers of teams use the WSLS strategy. Managers can affect the performance of teams through selections of players, training of players, and implementation of tactics including formations [18]. In particular, a formation is a part of tactics to determine how players participate in offense and defense [24] and considered to affect match results [24, 25]. Managerial decision making in substituting players during a match may affect the probability of winning [25]. We hypothesize that a manager continues to use the same formation if he has won the previous match, whereas he experiments on another formation following a loss in the previous match.
The WSLS and more general reinforcement learning posit that unsuccessful individuals modify their behavior to increase the probability of winning. Therefore, we are interested in whether a formation change improves the performance of a team. To clarify this point, we also investigate effects of formation changes on the results of succeeding matches.
2 Materials and methods
2.1 Data set
Statistics of the J-League and Bundesliga data sets
Quantities | J-League | Bundesliga |
---|---|---|
Year | 1993-2014 (1999-2014) | 1963-2014 |
Number of seasons | 33 (22) | 51 |
Number of matches | 5,944 (4,318) | 15,548 |
Number of teams | 29 (28) | 54 |
Number of managers | 176 | 372 |
Number of wins (or losses) | 4,961 (3,435) | 11,543 |
Number of draws | 983 (883) | 4,005 |
Between 1993 and 2004, except for 1996, each season of J-League was divided into two half seasons. After the two half seasons had been completed, two champion teams, each representing a half season, played play-off matches. We regarded each half season as a season because intervals between two half seasons ranged from ten days to two months and therefore are longer than one week, which was a typical interval between two matches within a season. We also carried out the same analysis when we regarded one year, not one half season, as a season and verified that the main results were unaltered (Appendix A).
We also collected data on Bundesliga from another website, Fussballdaten [28]. We focused on the Kicker-online data rather than the Fullballdaten data because the definition of the position was coarser for the Fussballdaten data (i.e., a player was not assumed to change his position during a season) than the Kicker-online data. Nevertheless, to verify the robustness of the following results, we also analyzed the Fussballdaten data (Appendix B).
2.2 Definition of formation
The definition of formation was different between the two data sets. In the J-League data, each of the ten field players was assigned to either defender (DF), midfielder (MF), or forward (FW) in each match. We defined formation as a triplet of the numbers of DF, MF, and FW players, which sum up to ten. For example, a formation 4-4-2 implies four DFs, four MFs, and two FWs.
Among all matches in the Bundesliga data, the smallest nonzero distance between two players was equal to 31 px. Therefore, we did not have to worry about the possibility that players possessed almost the same distance values while being classified into distinct positions. For example, there was no case in which the distances of two field players from the goal line were equal to 113 px and 114 px.
For both data sets, a formation was defined as an ordered set of numbers, whereas the definition differs for the two data sets. For example, forward players possessing distance values 359 and 441 were classified into different positions in the Bundesliga data, whereas they belonged to the same position in the J-League data if they were both assigned to FW. In the following, we regarded that formation was changed when the ordered set of numbers differed between two consecutive matches.
The frequency of formation changes as a function of time is shown in Figures 3(e) and 3(f) for J-League and Bundesliga, respectively. The figures suggest that the frequency of formation change is stable over years in J-League, but not in Bundesliga. Finally, we measured burstiness and memory coefficient [29] for interevent times of formation changes to quantify temporal patterns of formation changes. The results are shown in Appendix C.
2.3 GLMM
To statistically examine whether patterns of formation changes were consistent with WSLS behavior, we investigated effects of previous matches and other factors on the likelihood of formation change for each team. If managers used the WSLS, the effect of the win and loss in the previous match on the likelihood of formation change should be significantly negative and positive, respectively. We used a generalized linear mixed model (GLMM) with binomial errors and a logit-link function.
The dependent variable was the occurrence or lack thereof of formation changes, which was binary. As independent variables, we included the binary variable representing whether or not the stadium was the home of the team (i.e., home or away) and the ternary result of the previous match (i.e., win, draw, or loss). We designated the draw as the reference category for the match result. Because the likelihood of formation changes may be affected by a streak of wins or losses, we also included the result of the second last match as an independent variable. The difference between the focal team’s strength and the opponent’s strength was also an independent variable. The strength of a team was defined by the probability of winning in the season. We estimated the strength of a team separately for each season because it can vary across seasons. The name of the manager was included as a random effect (random intercept).
In this and the following analysis, we excluded the first match in each season for each team because we considered that the result of the last match in the preceding season would not directly affect the first match in a new season. In addition, we excluded matches immediately after a change of manager because we were not interested in formation changes induced by a change of manager. We further excluded the second match in each season for each team from the GLMM analysis when we employed the result of the second last match as an independent variable. Because the J-League data set did not have the information on managers between 1993 and 1998, we only used data between 1999 and 2014 in the GLMM analysis. We performed the statistical analysis using R 3.1.2 [30] with lme4 package [31].
2.4 Ordered probit model
We also investigated the effects of formation changes on match results. We used the ordered probit model because a match result was ternary. Because the strength was considered to heavily depend on teams, we controlled for the strength of teams. The same model was used for fitting match results in football in the Netherlands [32] and the UK [18].
We excluded the matches that were the first game in a season at least for either team. We also excluded matches immediately after a change of manager in either team. Because the J-League data set did not contain the information on managers between 1993 and 1998, we only used data between 1999 and 2014 in this analysis. We performed the analysis using R 3.1.2 [30] and maxLik package [34].
2.5 Influence of individual manager’s behavior on match results
Different managers may show WSLS behavior to different extents to respectively affect match results. Therefore, we analyzed data separately for individual managers. For each manager i, we calculated the probability of winning under each of the following four conditions: (i) i’s team won the previous match, and i changed the formation, (ii) i’s team won the previous match, and i did not change the formation, (iii) i’s team lost the previous match, and i changed the formation, and (iv) i’s team lost the previous match, and i did not change the formation. We then compared the probability of winning between cases (i) and (ii), and between cases (iii) and (iv) using the paired t-test. In the t-test, we included the managers who directed at least ten pairs of consecutive matches in both of the two cases in comparison. In this and the next sections, we treated a manager as different data points when he directed different teams, as explained in Section 2.1. In addition, we excluded the pairs of consecutive matches when the managers changed the team between the two matches.
2.6 Degree of win-stay lose-shift
3 Results
3.1 Win-stay lose-shift behavior in formation changes
Results of the GLMM analysis when the results of the previous match were used as the sole independent variables
Data set | Variable | Coefficient | SE | p -value |
---|---|---|---|---|
J-League | \(\mathrm{Win}_{t-1}\) | −0.360 | 0.101 | <0.001 |
\(\mathrm{Loss}_{t-1}\) | 0.388 | 0.093 | <0.001 | |
Bundesliga | \(\mathrm{Win}_{t-1}\) | −0.179 | 0.032 | <0.001 |
\(\mathrm{Loss}_{t-1}\) | 0.164 | 0.032 | <0.001 |
Results of the GLMM analysis when all the independent variables were considered
Data set | Variable | Coefficient | SE | p -value |
---|---|---|---|---|
J-League | \(\mathrm{Win}_{t-1}\) | −0.299 | 0.124 | 0.016 |
\(\mathrm{Win}_{t-2}\) | −0.204 | 0.116 | 0.079 | |
\(\mathrm{Loss}_{t-1}\) | 0.387 | 0.120 | 0.001 | |
\(\mathrm{Loss}_{t-2}\) | 0.146 | 0.126 | 0.248 | |
\(\mathrm{Win}_{t-1} \times \mathrm{Win}_{t-2}\) | 0.005 | 0.182 | 0.979 | |
\(\mathrm{Loss}_{t-1} \times\mathrm{Loss}_{t-2}\) | 0.023 | 0.164 | 0.888 | |
Home | −0.062 | 0.072 | 0.392 | |
Strength | −0.192 | 0.202 | 0.343 | |
Bundesliga | \(\mathrm{Win}_{t-1}\) | −0.207 | 0.040 | <0.001 |
\(\mathrm{Win}_{t-2}\) | −0.117 | 0.039 | 0.003 | |
\(\mathrm{Loss}_{t-1}\) | 0.136 | 0.039 | <0.001 | |
\(\mathrm{Loss}_{t-2}\) | 0.007 | 0.040 | 0.867 | |
\(\mathrm{Win}_{t-1} \times\mathrm{Win}_{t-2}\) | 0.025 | 0.059 | 0.676 | |
\(\mathrm{Loss}_{t-1} \times\mathrm{ Loss}_{t-2}\) | 0.108 | 0.060 | 0.072 | |
Home | −0.118 | 0.027 | <0.001 | |
Strength | −0.530 | 0.082 | <0.001 |
For the J-League data, the effects of all the additional independent variables were insignificant. We analyzed the J-League data by regarding a pair of half seasons (i.e., an yearly season) as a season to confirm that the results remained qualitatively the same except that winning in the second last match also significantly decreased the probability of formation change (Appendix A). We also confirmed that matches played in the further past affect the probability of formation change to progressively small extents (Appendix E).
For the Bundesliga data, winning in the second last match also significantly decreased the probability of formation change in the extended GLMM model (Table 3). These results are consistent with WSLS behavior. We also found for the Bundesliga data that stronger teams less frequently changed the formation and that a team would not change the formation to fight home games. We also investigated the Fussballdaten data for Bundesliga, in which the definition of formation was different, and confirmed that managers tended to use the WSLS strategy (Appendix B).
3.2 Determinants of match results
Effects of variables on match results as obtained from the ordered probit model
Data set | Variable | Coefficient | SE | p -value |
---|---|---|---|---|
J-League | Formation change (\(\beta_{\mathrm{f}}\)) | 0.112 | 0.063 | 0.075 |
Home (\(\beta_{\mathrm{ h}}\)) | 0.087 | 0.033 | 0.009 | |
Win (\(\beta_{\mathrm{ w}}\)) | 0.082 | 0.099 | 0.405 | |
Loss (\(\beta_{\ell}\)) | 0.193 | 0.100 | 0.053 | |
Strength (\(\beta_{\mathrm{r}}\)) | 2.889 | 0.169 | <0.001 | |
Bundesliga | Formation change (\(\beta_{\mathrm{f}}\)) | −0.011 | 0.016 | 0.509 |
Home (\(\beta_{\mathrm{h}}\)) | 0.420 | 0.011 | <0.001 | |
Win (\(\beta_{\mathrm{w}}\)) | −0.060 | 0.018 | 0.001 | |
Loss (\(\beta_{\ell}\)) | −0.006 | 0.018 | 0.738 | |
Strength (\(\beta_{\mathrm{r}}\)) | 2.792 | 0.060 | <0.001 |
4 Discussion
We have provided evidence that football managers tend to stick to the current formation until the team loses, consistent with the WSLS strategy previously shown in laboratory experiments with social dilemma games [9, 10] and gambling tasks [11, 12]. Formation changes did not significantly affect (at least did not improve) a match result in most cases. This result seems to be odd because managers change formation to lead the team to a success. Generally speaking, when the environment in which an agent is located is fixed or exogenously changing, reinforcement learning usually improves the performance of the agent [4]. However, computational studies have suggested that it is not always the case when agents employing reinforcement learning are competing with each other, because the competing agents try to supersede each other [36–39]. The present finding that manager’s WSLS behavior does not improve team’s performance is consistent with these computational results. Empirical studies also suggest that humans obeying reinforcement learning does not improve the performance in complex environments. For example, players in the National Basketball Association were more likely to attempt 3 point shots after successful 3 point shots although their probability of success decreased for additional shots [40]. Also in nonscientific accounts, it has been suggested that humans engaged in sports and gambles often use the WSLS strategy even if outcome of games is determined merely at random [16]. We have provided quantitative evidence underlying these statements.
Many sports fans possess the hot hand belief in match results, i.e., belief that a win or good performance persists [41]. However, empirical evidence supports that streaks of wins and those of losses are less likely to occur than under the independence assumption [41]. By analyzing patterns of matches in the top division of football in England, Dobson and Goddard suggested the existence of negative persistence effects, i.e., a team with consecutive wins tended to perform poorly in the next match and vice versa [18]. Their results are consistent with the present results; we observed the negative persistence effects, i.e., anticorrelation between the results of the previous and present matches.
In the present study, we have neglected various factors that potentially affect the likelihood of formation change because our data sets did not contain the relevant information. For example, managers may change formations due to injuries, suspensions of players, and other strategic reasons including transfer of players. More detailed data will be able to provide further understanding of the relative importance of strategic versus accidental factors in formation changes.
An important limitation of the present study is that we have oversimplified the concept of formation. Effective formations dynamically change during a match owing to movements of players. Because of the availability of data and our interests in the manager’s long-term behavior rather than formation changes during a match [25], we used the formation data released in the beginning of the matches. Based on recent technological developments, formations can be extracted from tracking data on movement patterns of players [42, 43]. Investigations on manager’s decision making using such technologies warrant further research.
Declarations
Acknowledgements
This work is supported by JST, CREST.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Fudenberg D, Levine DK (1998) The theory of learning in games. MIT Press, Cambridge Google Scholar
- Camerer C (2003) Behavioral game theory: experiments in strategic interaction. Princeton University Press, Princeton Google Scholar
- Pearce JM (2013) Animal learning and cognition: an introduction. Psychology Press, Hove Google Scholar
- Sutton RS, Barto AG (1998) Reinforcement learning: an introduction. MIT Press, Cambridge Google Scholar
- Schultz W, Dayan P, Montague PR (1997) A neural substrate of prediction and reward. Science 275(5306):1593-1599 View ArticleGoogle Scholar
- Glimcher PW, Camerer C, Fehr E, Poldrack RA (2009) Neuroeconomics: decision making and the brain. Academic Press, New York Google Scholar
- Kraines D, Kraines V (1989) Pavlov and the prisoner’s dilemma. Theory Decis 26(1):47-79 MathSciNetView ArticleGoogle Scholar
- Nowak M, Sigmund K (1993) A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner’s Dilemma Game. Nature 364(6432):56-58 View ArticleGoogle Scholar
- Wedekind C, Milinski M (1996) Human cooperation in the simultaneous and the alternating Prisoner’s Dilemma: Pavlov versus Generous Tit-for-Tat. Proc Natl Acad Sci USA 93(7):2686-2689 View ArticleGoogle Scholar
- Milinski M, Wedekind C (1998) Working memory constrains human cooperation in the Prisoner’s Dilemma. Proc Natl Acad Sci 95(23):13755-13758 View ArticleGoogle Scholar
- Hayden BY, Platt ML (2009) Gambling for Gatorade: risk-sensitive decision making for fluid rewards in humans. Anim Cogn 12(1):201-207 View ArticleGoogle Scholar
- Scheibehenne B, Wilke A, Todd PM (2011) Expectations of clumpy resources influence predictions of sequential events. Evol Hum Behav 32(5):326-333 View ArticleGoogle Scholar
- Mesoudi A, O’Brien MJ (2008) The cultural transmission of Great Basin projectile-point technology I: an experimental simulation. Am Antiq 73(1):3-28 Google Scholar
- Mesoudi A, O’Brien MJ (2008) The cultural transmission of Great Basin projectile-point technology II: an agent-based computer simulation. Am Antiq 73(4):627-644 Google Scholar
- Mesoudi A (2014) Experimental studies of modern human social and individual learning in an archaeological context: people behave adaptively, but within limits. In: Akazawa T, Ogihara N, Tanabe HC, Terashima H (eds) Dynamics of learning in Neanderthals and modern humans, vol 2. Springer, Heidelberg, pp 65-76 View ArticleGoogle Scholar
- Vyse SA (2013) Believing in magic: the psychology of superstition, updated edn. Oxford University Press, Oxford Google Scholar
- Panja T (2013) Top soccer leagues get 25% rise in TV rights sales, report says. In: Bloomberg Business. http://www.bloomberg.com/news/articles/2013-11-11/top-soccer-leagues-get-25-rise-in-tv-rights-sales-report-says. Accessed 28 Feb 2015
- Dobson S, Goddard J (2001) The economics of football. Cambridge University Press, Cambridge View ArticleGoogle Scholar
- Dixon MJ, Coles SG (1997) Modelling association football scores and inefficiencies in the football betting market. J R Stat Soc, Ser C, Appl Stat 46(2):265-280 View ArticleGoogle Scholar
- Wilson J (2013) Inverting the pyramid: the history of soccer tactics. Nation Books, New York Google Scholar
- Audas R, Dobson S, Goddard J (1997) Team performance and managerial change in the English Football League. Econ Aff 17(6):30-36 View ArticleGoogle Scholar
- Dawson P, Dobson S, Gerrard B (2000) Estimating coaching efficiency in professional team sports: evidence from English association football. Scott J Polit Econ 47(4):399-421 View ArticleGoogle Scholar
- Audas R, Dobson S, Goddard J (2002) The impact of managerial change on team performance in professional sports. J Econ Bus 54(6):633-650 View ArticleGoogle Scholar
- Bangsbo J, Peitersen B (2000) Soccer systems & strategies. Human Kinetics, Champaign Google Scholar
- Hirotsu N, Wright MB (2006) Modeling tactical changes of formation in association football as a zero-sum game. J Quant Anal Sports 2(2):4 MathSciNetGoogle Scholar
- J-League Data site https://data.j-league.or.jp/SFTP01/. Accessed 4-6 June 2014
- Kicker-online http://www.kicker.de/. Accessed 14-16 Aug 2014
- Fussballdaten http://www.fussballdaten.de/. Accessed 28-31 July 2014
- Goh KI, Barabási AL (2008) Burstiness and memory in complex systems. EPL 81:48002 View ArticleGoogle Scholar
- R Core Team (2014) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. http://www.R-project.org/ Google Scholar
- Bates D, Maechler M, Bolker B, Walker S (2014) lme4: linear mixed-effects models using Eigen and S4. http://CRAN.R-project.org/package=lme4. R package version 1.1-7
- Koning RH (2000) Balance in competition in Dutch soccer. J R Stat Soc, Ser D, Statist 49:419-431 View ArticleGoogle Scholar
- Parent E, Rivot E (2012) Introduction to hierarchical Bayesian modeling for ecological data. CRC Press, Boca Raton Google Scholar
- Henningsen A, Toomet O (2011) maxLik: a package for maximum likelihood estimation in R. Comput Stat 26(3):443-458 MathSciNetView ArticleMATHGoogle Scholar
- Albert J, Koning RH (2010) Statistical thinking in sports. CRC Press, Boca Raton Google Scholar
- Taiji M, Ikegami T (1999) Dynamics of internal models in game players. Physica D 134(2):253-266 MathSciNetView ArticleMATHGoogle Scholar
- Macy MW, Flache A (2002) Learning dynamics in social dilemmas. Proc Natl Acad Sci USA 99(Suppl 3):7229-7236 View ArticleGoogle Scholar
- Masuda N, Ohtsuki H (2009) A theoretical analysis of temporal difference learning in the iterated prisoner’s dilemma game. Bull Math Biol 71(8):1818-1850 MathSciNetView ArticleMATHGoogle Scholar
- Masuda N, Nakamura M (2011) Numerical analysis of a reinforcement learning model with the dynamic aspiration level in the iterated prisoner’s dilemma. J Theor Biol 278(1):55-62 MathSciNetView ArticleMATHGoogle Scholar
- Neiman T, Loewenstein Y (2011) Reinforcement learning in professional basketball players. Nat Commun 2:569 View ArticleGoogle Scholar
- Bar-Eli M, Avugos S, Raab M (2006) Twenty years of “hot hand” research: review and critique. Psychol Sport Exerc 7(6):525-553 View ArticleGoogle Scholar
- Bialkowski A, Lucey P, Carr P, Yue Y, Matthews I (2014) “Win at home and draw away”: automatic formation analysis highlighting the differences in home and away team behaviors. In: MIT Sloan sports analytics conference (SSAC) Google Scholar
- Bialkowski A, Lucey P, Carr P, Yue Y, Sridharanand S et al. (2014) Large-scale analysis of soccer matches using spatiotemporal data. In: IEEE international conference on data mining (ICDM) Google Scholar
- Stan Development Team (2014) Rstan: the R interface to Stan, version 2.5.0. http://mc-stan.org/rstan.html