perspec-108 A copula-based multivariate hidden Markov model for modelling momentum in football
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Figure 7.5: Transition probabilities as functions of the covariate minute. The dashed lines indicate confidence intervals (obtained based on Monte Carlo simulation). The values of the score difference and the market value of the opponent are set to 0 and 200, respectively. Table A4 in Appendix C displays the coefficients of the multinomial logistic regression underlying this figure.
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shotsball touches
decoded state: ● 1 (low control and counter attacks) ● 2 (balanced) ● 3 (high control)
Figure 7.6: Decoded most likely state sequence of the match Borussia Dortmund against Schalke 04 according to the three-state Clayton-copula HMM including covariates. The vertical dashed lines denote goals scored by Borussia Dortmund (yellow lines) and Schalke 04 (blue lines).
tive, it is important to understand the causes of such shifts, and hence also how to potentially exert an influence on the match outcome. With data sets on in-game
sum-7.5 Discussion 109
Table 7.4: Stationary distributions when fixing the score difference at certain levels. Probabilities were calculated for each value of the score difference, with the market value of the opponent and the minute of the match fixed at 200 and 80, respectively, corresponding to situations in the final stage of a match against an opponent team of average strength.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
state 1 0.073 0.100 0.134 0.175 0.222 0.280 0.523 0.732 0.705 0.642 0.560 0.475 state 2 0.391 0.364 0.334 0.301 0.267 0.234 0.206 0.175 0.147 0.122 0.098 0.076 state 3 0.535 0.535 0.532 0.524 0.511 0.486 0.271 0.094 0.148 0.236 0.342 0.450
mary statistics becoming freely available, we now have the opportunity to statistically investigate the corresponding processes. To that end, here we provide a modelling framework — copula-based multivariate HMMs — which naturally accommodates po-tential changes in the dynamics of a match by relating the observed in-game match statistics to latent states. A key strength of the proposed approach is that we not only partition a given match into different phases but also allow for the investigation into drivers of how a match unfolds dynamically over time.
In our proof-of-concept case study, we tested the feasibility of our approach by analysing minute-by-minute data on matches of one particular team, namely Borussia Dortmund. The underlying states of the fitted model correspond to match phases where Borussia Dortmund exhibits a low level of control with counter attacks, to phases where the match is balanced, and to those with high level of control, respectively. In addition, the estimated effects of the covariates shed some light on what kind of events may lead to switches between those states. Specifically, we found that Borussia Dortmund has the highest probability of being in the high-control state when having a clear lead or when trailing.
Although the states of the fitted models are tied to different levels of control, it remains unclear whether these are clearly attributed to shifts in the underlying momen-tum. Specifically, some of the reported effects may arise due to tactical considerations rather than momentum shifts. For example, for one-goal leads, being in the low con-trol and counter attacks state may be a tactical consideration rather than a shift in the underlying momentum. The data considered here does not allow us to disentan-gle these two possible causes, rendering a definitive conclusion whether the switches between the states are momentum shifts or tactical considerations impossible. How-ever, with the states and effects of the covariates considered (cf. Figure 7.5 and Table 7.4) being easy to interpret, they still provide interesting insights to dynamics of foot-ball matches. In addition, using copula-based HMMs as presented in this chapter
110 A copula-based multivariate hidden Markov model for modelling momentum in football
may be helpful for bookmakers to obtain more precise estimations of betting odds.
For instance, when modelling the time until the next goal during a football match, bookmakers could take into account the latent dynamics of a match as modelled here.
A clear limitation of the approach as presented here is that we focus on the in-game dynamics of only one of the two teams involved in a match, when in fact it is clear that the dynamics of a match result from the combination of both teams’ actions. It seems conceptually desirable to extend the approach to allow for the joint modelling of both teams’ in-game statistics. This could be achieved using a bivariate Markov chain to represent both teams’ underlying states, resulting in N2 combinations of states (see, e.g., Sherlock et al., 2013). To further improve the realism of these models, it would be beneficial to also include tracking data, e.g. by considering the distances run per minute as covariate information.
The modelling framework used in this chapter, i.e. copula-based HMMs for mod-elling football minute-by-minute data, can easily be transferred to other sports for further investigations and possible characteristics of momentum shifts. These sports include, e.g., basketball, where the variables to be modelled comprise, for example, the number of points/shots, the number of rebounds, and the number of blocks/steals.
More general, sports with two individuals or teams competing against each other and multiple variables measured on a fine-grained scale are best suitable for analysing mo-mentum shifts using the modelling framework provided here.
8 Performance under pressure in skill tasks:
An analysis of professional darts
8.1 Introduction
The effect of pressure on human performance is relevant in various areas of the society, including sports competitions (Hill et al., 2010), political crises (Boin et al., 2016), and performance-based payment in workplaces (Ariely et al., 2009), to name but a few. A broad distinction differentiates between effort and skill tasks. Success in effort tasks is dependent on motivation to perform while skill task outcomes underlie precision of (often automatic) execution. For effort tasks, such as counting digits (Konow, 2000) or filling envelopes (Abeler et al., 2011), individuals will typically respond to increased pressure (e.g. resulting from performance-related payment schemes) by investing more effort, which given the nature of such tasks will improve their performance (Lazear, 2000; Paarsch and Shearer, 1999, 2000; Prendergast, 1999). However, the literature on the impact of pressure on performance in skill tasks, e.g. juggling a football (Ali, 2011), is inconsistent and effectively divided into two different strands of research.
On the one hand, the existing literature related to potential “choking under pres-sure” indicates broad agreement that performance in skill tasks declines in high-pressure or decisive situations. An individual is said to be choking under pressure when their performance is worse than expected given their capabilities and past performances (Beilock and Gray, 2007). While there may also be random fluctuations in skill levels, choking under pressure refers to systematic suboptimal performance in high-pressure situations. The associated empirical findings — both such that are based on experi-mental data but also those using field data — consistently confirm a negative impact of pressure on skill tasks. On the other hand, and to some extent in contrast to the literature related to choking under pressure, the literature related to the concept of
“social facilitation” refers to potential negative but also potential positive effects of (social) pressure on performance — depending on circumstances associated with the
112 Performance under pressure in skill tasks: An analysis of professional darts
performance. The social facilitation literature explicitly incorporates characteristics of the task and individuals’ level of expertise into their analyses, and generally states that the circumstances surrounding performance play an important role regarding the im-pact of pressure on performance. Existing contributions focusing on potential choking have largely neglected the corresponding more comprehensive picture drawn by the social facilitation literature, by simply relating performance decrements to changes in the execution of actions, or simply distraction, generated either by rewards in case of success (Ariely et al., 2009; Baumeister, 1984) or potential penalties in case of failure (Kleine et al., 1988).
Our empirical investigation of individual’s performance in pressure situations is based on a large data set from a skill task, namely professional darts, comprising 32,274 individual dart throws, for a comprehensive empirical test of performance un-der pressure. For the professional darts players analysed in this study, playing darts is a full time job. The top players regularly earn prize money exceeding one million euro per year. In professional darts, highly skilled players repeatedly throw at the dartboard from the exact same position effectively without any interaction between competitors, making the task highly standardised. The amount of data available on throwing performances not only allows for comprehensive inference on the existence and the magnitude of any potential effect of pressure on performance, but also enables to track the variability of the effect across players. The literature on choking would suggest that performance of professional darts players declines in high-pressure situa-tions. However, when considering the highly standardised task to be performed and players’ high level of expertise, we do not expect dart players to choke under pressure.
The chapter is structured as follows. Section 8.2 reviews the literature on perfor-mance under pressure, and in particular details what we consider to be advantages of the darts setting with respect to investigating performance under pressure. In Section 8.3, we explain the rules of darts and define what constitutes pressure situations in darts. Section 8.4 presents the empirical approach and results.