No cheering in the background? Individual performance in professional darts during COVID-19

3.1 Three-dart average and checkout performance

Players aim for high numbers to rapidly reduce their score, with the maximum possible score in a single turn, consisting of three darts, being 180 points. Therefore, a natural strategy for each player is to maximise points per turn. A player’s mean score per turn is called the ’three-dart average’, providing the most recognised performance measure in darts which is also used by television broadcasts. In contrast to other sports, it is solely attributable to a player as it is neither part of a team outcome nor can it be directly affected by the referee or other players. Thus, the response variable average enables a holistic analysis of the performance as it is based on the entire match and is not limited to certain parts of a match. As a rule of thumb, professionals routinely achieve averages of over 90, while averages over 100 are considered particularly outstanding. In our data, the tournament with the lowest average was the ‘European Darts Matchplay 2018’ (91.65), and the highest was ‘The Masters 2018’ (99.11).

To ensure reliable inference on player-specific effects, we reduced the data set to only those players who played at least ten matches pre- and post-COVID. The final data set for the analysis of the averages comprises 60 players. Table 1 displays summary statistics on the three-dart average for all matches as well as for those played pre- and post-COVID, respectively. We observe a lower mean for the three-dart average during COVID matches (93.97) than before COVID (95.25).

The second performance measure our analysis concerns is the checkout performance of each player. As introduced above, at the end of a leg, players must hit a double, a fairly small segment of the dart board. To evaluate the checkout performance, the commonly used performance indicator in darts is the so-called checkout percentage, given by the ratio of doubles hit divided by all attempts on double. However, as our data contain double attempts and the hits, we use the variable double hits to analyse the checkout performance.

For reliable inference on player-specific effects, we reduced the data set to only those players with at least five attempts at a double. Consequently, the data set for the analysis of the checkout performance comprises n = 6, 323 observations from 60 players, which is thus only slightly smaller than the data set considered for the three-dart averages. Table 1 indicates only minor differences in the checkout percentage pre- and post-COVID.

3.2 COVID-19 matches

Darts fosters a unique fan-player connection, distinct from other sports. In PDC tournaments, fans stand closely to players, shaking hands as they enter the stage and singing chants during the game. This intimate atmosphere contrasts with sports in large stadiums, like football or basketball, where athletes are more isolated from fans due to venue design. Enclosed halls hosting darts tournaments amplify the crowd noise, creating an environment characterised by partying, disguised, and singing spectators (Davis 2018, 2022b).

The COVID-19 pandemic ended such festival atmosphere in darts. Matches in late 2020 had only a few spectators, adhering to strict rules, including a ban on chants and costumes.^[1] Other tournaments were held behind closed doors, resulting in a subdued atmosphere compared to pre-COVID matches with reduced noise, shouting, and chants (Davis 2022a,b). The Guardian highlighted such novel game-play circumstances and the significance of spectators in darts, emphasising their close connection to players and their impact on the sport.^[2] Consequently, we anticipate a diminished social pressure on players due to the altered atmosphere.

To capture the effect of the social pressure exerted by the fans, we include the dummy variable COVID. Table 1 indicates that almost 70 % of the observations belong to matches played pre-COVID, which are thus labelled as regular, meaning that they took place under standard conditions, i.e., with packed arenas and no restrictions. Thus, about 30 % of our observations belong to matches that took place after the pandemic outbreak. To explore performance variations pre- and post-COVID among players, Figure 1 displays the performance for the 25 players with the highest number of observations in our data, revealing substantial differences. For some players, such as Michael van Gerwen and Gary Anderson, the three-dart average declines post-COVID, while it increases for other players, such as Dimitri van den Bergh and Luke Humphries. The benefitting players are relatively inexperienced in most cases. For instance, Dimitri van den Bergh and Luke Humphries have post-COVID experience levels of six and three years. In contrast, the experience levels of Michael van Gerwen and Gary Anderson are very high at 15 and 20 years, and their averages decrease. Similarly, the checkout percentages (right panel of Figure 1) show a decline in performance for some players (such as Gary Anderson and Kim Huybrechts), while others increased it during spectator deviations (such as James Wade or Chris Dobey).

Figure 1:

Changes in performance pre- and post-COVID as indicated by the three-dart average (left panel) and checkout percentage (right panel). Shown are the 25 players with the most observations in our data set.

To assess performance differences before and during COVID-19 between experienced and inexperienced players, we classify those with three or fewer years of experience as inexperienced (corresponding to the 20 % percentile). Prior to the pandemic, inexperienced players had a lower mean three-dart average (93.7) compared to experienced players (95.7). During the COVID-19 pandemic, inexperienced players had, on average, a higher three-dart average (94.4) than experienced players (93.9), although this difference is not statistically significant. Thus, both decreasing performances of experienced players (95.7 vs. 93.9) and, albeit smaller, increasing performances of inexperienced players (93.7 vs. 94.4) could be observed during the pandemic.

3.3 Players’ experience

Drawing on social facilitation theory, findings by Beilock et al. (2002), Otten (2009) or Jetter and Walker (2015) and Figure 1, we anticipate that coping with spectator-induced social pressure improves with increased experience. Hence, we expect experienced darts players to perform better under social pressure than inexperienced players. In contrast, Beilock et al. (2002) suggest that more experienced players struggle with new situations compared to inexperienced players. Therefore, matches played behind closed doors may have adversely affected the performance of experienced darts players.

We measure a player’s experience by the number of years he is considered a professional player. To participate in PDC tournaments regularly, players need a so-called tour card, allowing them to participate in the PDC Pro Tour. Founded in 1992 as an independent darts association, the PDC has become a global hub for top darts players, hosting prestigious tournaments. The key to PDC’s success lies in its elevated professionalism compared to national associations. PDC players consistently achieve higher three-dart averages, earn substantial prize money, and, unlike players from other associations, can make a living from their sport, highlighting the significance of the tour card. We thus measure a player’s experience by the number of years since he received the tour card for the first time.

A total of 128 tour cards are available, with 64 awarded to the top 64 players on the world ranking list, known as the Order of Merit. The remaining 64 tour cards can be obtained through participation in the PDC Qualifying Schools.^[3] The PDC introduced the tour card first in 2011; however, a few players in our data can also be considered professionals before 2011. To obtain an adequate experience level for these players, we surveyed the first season in their career in which they belonged to the top 64 of the Order of Merit, which would, by today’s rules, allow them to obtain a tour card.

The experience levels of the players considered in our data range from 0 to 20, with a mean of 7.72 years (cf. Table 4 in the Appendix). Our sample thus contains substantial variation in players’ experience levels, which enables us to reliably investigate the effect of experience on the ability to cope with social pressure.

3.4 Further variables

Our data further contain covariates on match characteristics that potentially affect a player’s performance, which were also considered in previous studies (e.g., Ötting et al. 2020; Teeselink et al. 2020). In later rounds of a tournament, players usually have a higher incentive to perform well, which is captured by a set of dummy variables containing the information on the tournament round (ranging from the first round or group stage to the final; 12 categories in total). To account for the heterogeneity of the two players, we include the difference in their betting odds of winning the match. The betting odds were taken from www.oddsportal.com. Two further covariates cover the score difference and the match length. If the score difference is large, the match was most likely decided early, leading to a reduced incentive for the players to perform well at the end of the match. By including the match length, we account for potential fatigue effects. In the results below, we refer to the set of variables introduced in this subsection as match controls. Table 4 in the Appendix displays descriptive statistics for these covariates.

4.1 Three-dart average

In our analysis, the response variable average _ij denotes the three-dart average of player i (i = 1, …, 60) in match j (j = 1, …, n _i ). Our first model can be considered as a simple benchmark model which includes only the COVID dummy variable:

where α is the intercept and β captures the change in performance post-COVID across all players. The second model for investigating the performance as measured by the three-dart average is given by

where the vector x _ij contains further variables which are likely to affect the performance of darts players as introduced above, i.e. the length and the score of the match, a dummy variable for each tournament as well as for the round of the tournament, the difference in the betting odds, and the experience.

As Figure 1 indicates substantial heterogeneity in performance across players, the next model includes varying intercepts:

Here, player-specific deviations from the average intercept α are given by α _player[i] to account for the different levels of players’ quality.

Figure 1 further suggests that the effect of an absent crowd on performance differs across players. Our next model for the average includes varying slopes to investigate such pattern:

where β denotes the average change in performance post-COVID across all players and β _player[i] are the corresponding player-specific deviations from the average effect.

Finally, to investigate the potential moderating effect of players’ experience as introduced above, we consider the following model:

where β again denotes the average change in performance post-COVID across all players and f is a smooth function to account for potential nonlinear effects.

4.2 Checkout performance

The second part of our analysis focuses on modelling the checkout performance, i.e., the situation at the end of a leg where darts players have to hit a small segment to reduce their score to 0. While the checkout percentage constitutes a proportion, we do not model it directly but rather consider the number of successes (double hits), i.e., the number of darts that hit a double field, together with the total number of darts thrown at a double field. For that purpose, we use aggregated binomial regression models since it is most natural to explicitly consider the underlying sample size instead of using only the resulting proportion.

As for the analysis of the averages, we start with a simple benchmark model that includes only the COVID dummy:

where n _ij and p _ij denote the number of throws at a double segment of player i in match j and the corresponding probability to hit a double segment, respectively. Similar to the model for the averages (A.0) α is the intercept, and β is the average effect of the change in checkout performance post-COVID (across all players).

The second model for the checkout performance is given by

where the vector x _ij contains the match control variables that are likely to affect players’ performance, as introduced above.

To account for heterogeneity in players’ performance as indicated by Figure 1, the next model for the checkout performance includes varying intercepts:

The next model is the most flexible one as it further includes varying slopes for the effect of post-COVID matches on checkout performance to fully capture any potential heterogeneity across players:

As for the three-dart average, we also investigate the potential moderating effect of players’ experience for the checkout performance, with f again denoting a smooth function:

4.3 Model fitting

To fit the multilevel models introduced above, we choose a Bayesian approach to obtain posterior distributions of the change in performance due to the absence of fans. Moreover, due to the hierarchical structure of our data, a Bayesian approach allows to obtain posterior distributions of the change in performance for each player. In addition, we provide a comprehensive analysis of player heterogeneity by evaluating the corresponding credible intervals for each player. In particular, we use the R package brms (Bürkner 2017), which provides an interface for the programming language Stan (Carpenter et al. 2017). We fitted each model using four Markov chains with 10,000 iterations and discarded the first 5000 of each chain as warm-up, thus resulting in 20,000 posterior samples.

To check for convergence of the chains, we consider the Gelman-Rubin R ̂ statistic for each parameter, which indicates convergence if the value is close to one (Brooks and Gelman 1998; Gelman and Rubin 1992). We also report the effective sample size.

As we use a Bayesian approach to obtain posterior distributions of the change in performance, the use of prior distributions is not motivated to incorporate domain expertise. We thus use weakly informative prior distributions. In particular, we choose a N ( 0,10 ) prior for the population-level parameters, i.e., for β and for all entries of γ throughout all model formulations. For models (A.2) and (C.2), we select a normal prior for both player-level effects α _player and β _player, each with mean zero and a half-Student t(3, 0, 6.3) hyperprior for the standard deviation. For models (A.4) and (C.4), we consider the mixed model representation of penalised splines (Wood 2017) and select a half-Student t(3, 0, 6.3) hyperprior for the wiggly components. In the most flexible model formulations (A.3) and (C.3), we use a bivariate normal prior for the player-level effects α _player and β _player, respectively, with a half-Student t(3, 0, 6.3) prior for each standard deviation. For the associated correlation matrix, we choose an LKJ-correlation prior with parameter η = 1, leading to a uniform density over the correlation matrix (Bürkner 2017). The data set and the R code used for model fitting are available in the online supplementary of this article.

5.1 Three-dart average

To get an overview of the impact of COVID restrictions on the performance of darts players, this section presents the fitted models for the average. Table 2 displays posterior summaries for models (A.0)A.4)–(A.4). Starting with model (A.0), which includes only the post-COVID dummy, we observe a negative effect, and the corresponding credible interval does not include zero. For model (A.1), which includes match control variables such as dummy variables for each tournament, the COVID effect becomes positive, and the corresponding credible interval includes zero. The main driver behind this changed effect is the PDC Home Tour: In 2020, the PDC established a new tournament format where matches were played from home. For this tournament, we observe a three-dart average of 91.84, which is lower compared to other tournaments held both before (mean: 95.25) and after COVID (mean: 95.86, calculated without Home Tour matches; see Table 5 in the Appendix). Thus, the negative effect of post-COVID matches on performance vanishes once dummy variables for the different tournaments are included in the model.

For the post-COVID dummy variable, we observe a similar effect for model (A.2), which further includes varying intercepts for each player. This model is favoured over the previous model formulations by the Watanabe–Akaike information criterion (WAIC; Watanabe and Opper 2010). When considering varying slopes as in model (A.3), we observe a further drop in the WAIC, thus suggesting player-specific effects of COVID restrictions on the three-dart average. Since the direction of causality of the two match control covariates match length and legs difference could also go the other way – higher averages can also lead to shorter games – Table 6 in the Appendix shows the results when omitting these two covariates.

Figure 2 shows the posterior distributions of the player-specific post-COVID throwing performances as considered for model (A.3). While some players increased their performance during the pandemic, our results also indicate a drop in the performance of other players. The posterior mean for the player-level correlation between the varying intercepts and the slopes as included in model (A.3) is estimated as −0.31 (95 % CI: [−0.58; − 0.005]), thus indicating that players of higher quality tend to perform worse after the COVID break. High-quality players with a drop in post-COVID performances include, for example, the former world champions Gary Anderson and Michael van Gerwen. According to the social facilitation theory, players’ expertise can be considered as an individual-specific variable that can affect the response to pressure, which will be further investigated below.

Figure 2:

Posterior distributions of the individual COVID effects on the three-dart average (model (A.3)). Shown are the 25 players with the most observations sorted by their posterior means. Figure 5 in the Appendix shows the posterior distributions for all players in our sample.

5.2 Checkout performance

For the analysis of the checkout performance, Table 3 displays posterior summaries for the fitted models (C.0)–(C.3). While for the most simple model formulation (i.e., model (C.0)), the post-COVID effect is slightly negative, the effect becomes positive for the more complex models (C.1)–(C.4). However, the corresponding 95 % CIs include zero for each model. Moreover, the WAIC does not indicate any improvement in the model fit for the more complex models. In particular, when considering varying intercepts and slopes in models (C.2) and (C.3), respectively, we only observe a minor drop in the WAIC – in other words, our results do not indicate any player-specific effects in the checkout performance. As for the three dart average, Table 6 in the Appendix displays the results when omitting the two match controls match length and legs difference.

Figure 3 shows the corresponding posterior distributions for the 25 players with the most observations in our sample. It confirms that there are only minor variations in the post-COVID checkout performance across the players, thus showing a less clear picture than for the three-dart averages. This is further underlined by the correlation between the varying intercepts and slopes, for which the posterior mean is estimated as 0.33 with a 95 % CI of [−0.76; 0.97].

Figure 3:

Posterior distributions of the individual COVID effects on the checkout performance as considered for model (C.3). Shown are the 25 players with the most observations sorted by their posterior means. Figure 6 in the Appendix shows the posterior distributions for all players in our sample.

5.3 COVID matches and experience

Our results indicate that the effect of social pressure on the three-dart average varies across players, while we do not find any evidence for differences across players concerning the checkout performance. Moreover, for the three-dart averages, our results suggest that more experienced players are more likely to perform better when they are exposed to social pressure, which is in line with the social facilitation theory.

To further investigate the potential moderating effect of experience, Figure 4 shows the estimated effect of players’ experience under models (A.4) and (C.4). For the three-dart average, our results suggest that more experienced players are more likely to perform better under social pressure, with a peak at about ten years of experience. Although the effect turns negative for players with more than ten years of experience, the corresponding CIs include 0. For post-COVID matches, the estimated smooth functional effect indicates a positive effect for inexperienced players; however, the corresponding CIs include 0.

Figure 4:

Left: Estimated smooth-functional effects of the experience on the three-dart average (model (A.4)) for periods before and during the COVID-19 pandemic. Right: Estimated smooth-functional effects of the experience on the checkout performance (model (C.4)) for periods before and during the COVID-19 pandemic.

The effect of experience on the checkout performance shows a less clear picture than the three-dart average, as the 95 % credible intervals include 0 (cf. Figure 4). However, the shape of the estimated smooth functions is similar to those obtained for the three-dart average. Hence, our results slightly indicate that also, for the checkout performance, more experienced players are more likely to perform better under social pressure, but the corresponding evidence is inconclusive. This inconclusive evidence is further underlined by the posterior summaries shown in Table 3. While the posterior means obtained from model (C.4) are similar to those of the other models, the overall model fit indicated by the WAIC values cannot be improved by adding smooth-functional effects.