Introduction
The aim of this essay is to identify a descriptive theory of choice under uncertainty.Sporting matches have an uncertain outcome as there is doubt in what the result will be. Live sporting events provide a real-world experiment with large sample sizes of decision makers, as tens of thousands of individuals attend each game. We can assume that sports clubs are profit maximizing companies, where demand is measured by the willingness to attend a game. This investigation will look for the best choice set to maximize attendance by looking at the level of uncertainty of game outcomes. A secondary investigation will try to confirm the strength of different predictors on the home team attendance.
There are two main discourses present in the literature (1) fiercer competition and therefore higher uncertainty in the outcome of the game drives attendance and (2) higher expected win/loss and higher certainty in the outcome of the game drives attendance. Three types of uncertainty have been identified in the literature, (1) match uncertainty, which will be covered in this paper, (2) seasonal uncertainty and (3) the absence of long-run domination by a specific team (Borland & MacDonald, 2003).
Uncertainty of Outcome Hypothesis (UOH)
The Uncertainty of Outcome Hypothesis introduces the concept that attendance demand is dependent on the competitive balance within a sports league (Rottenberg, 1956). Neale (1964) calls this the ‘Louis-Schmelling Paradox’, highlighting the ‘peculiar economics’ of sports. He assumes that the ideal position for any firm is that of monopoly, however, for a fighter like Joe Louis, to adopt such strategy and not having competitors to fight with would bring him no income. The competition is what drives interest and the harder the fight the higher the profit. A comparable league-wide effect, also coined in the same paper by Neale, is the ‘League Standing Effect’, where the closer the standings of two teams within a team, the higher the gate receipts. This is different to the ‘Louis-Schemelling Paradox’ as it eliminates the effects of the advertising feedback loop present in singular matches like boxing. The NBA has followed Neale’s advice, by introducing Collective Bargaining Agreements where salaries are capped in order to improve competitive balance and suppress wages (Johnson, 2021).
Further research into this topic has corroborated this theory in English soccer (Forrest and Simmons, 2002), in baseball (Kochman & Badarinathi 1995) and in many other studies (Knowles et al., 1992). Cairns (1986) argues that many of these studies ignore the playing at home advantage, current performance or improvement potential within the league. This is where betting information becomes useful, as it can provide one of the most complete sources of information of a team and can capture outside factors such as injuries, fatigue, current form, the advantage of playing at home and weather.
Loss Aversion and Reference-Dependent Preferences
The theory of loss aversion and reference-dependent decision making states that consumer utility is higher when attending a game with an expected win or loss versus, an unexpected win or loss (Johnson, 2021). This stems from the utility function given by prospect theory, where loss aversion emerges from decisions under uncertainty (Kahneman & Tversky, 1979). Studies have found loss aversion affecting attendance in both the MLB (Coates et al., 2014) and the NFL (Coates & Humphreys, 2010). Humphreys and Zhou (2015) highlight the asymmetry present in the loss felt from an unexpected loss than an unexpected win (in line with prospect theory), and therefore attendance is driven by the avoidance of such losses. They also find that the number one predictor of attendance is a highly expected win.
Other studies have found that US consumers do not value outcome at all (Nalbantis & Pawlowski, 2019) and that different teams can have different effects (positive or negative) of uncertainty on attendance (Mills & Fort, 2018).
The aim of this study is to see if the win probability of a game has any predictive ability of stadium attendance turnout for the St. Louis Cardinals. The first hypothesis is that the higher the win probability the higher the attendance will be. The second hypothesis is in regards to other metrics that may influence attendance such as the present rank within the conference and the importance of a game to win the world series.
Methodology
The data used in this paper contains a sample of 160 games from the 2019 Major League Baseball Season for the St. Louis Cardinals from the website “baseball-reference.com”. Two games had to be omitted because betting data was not available. The betting data was taken from “sportsbookreview.com“, using 888sport betting odds. Win percentages were calculated using the formula present in the book The Logic of Sports Betting (Davidow, 2019). Home team win probability shows the likelihood that the home team will win the game.Win Probability = 100 / ( 100 + positive odds )
Win Probability = ( |negative odds| ) / ( 100 + |negative odds| )
Rank is the St. Louis Cardinals’ rank in the conference at the time the game was played, attendance is the number of people who attended the game, cLI (Championship Leverage Index) is the importance of this game on this team’s probability of winning the World Series, odds are the sport book odds and winp is the win probability given by the formulas above. A cLI of less than one is below average importance, one is average importance and above one is higher importance. cLI is calculated by using the current probability of winning the world series and subtracting the same probability assuming that the team wins the game. Such difference is the possible impact of this game.
RESULTS
The first model specification is a linear regression given by:Model 1: attendance= 13574 + 2152*rank + 7146*cLI – 0.7193*odds + 14881*winp Using this model the only significant variables are rank at the 0.05 alpha level and cLI at the 0.001 level. This model achieves a p-value of 0.0047 and a low R2 of 0.09181. We need to remove the odds variable, because it creates issues for multicollinearity given that odds are completely correlated to the win probability. After doing this, win probabilities become significant at the 0.1 level.
Model 2: log(attendance) = 9.79 + 0.06*rank + 0.24*cLI + 0.35*winP
Model 3: log(attendance) = 10.27 + 0.31*winP
When modeling win probability on attendance alone (model 3) we do not get any significant predictive ability. It is a positive relationship so we can infer that the higher the
probability of winning the higher the turn out.
Further visual analysis was done to look at trends over the season as time series data. Graphs 1, 2 and 3 (in the appendix) map attendance, win probability and Championship Leverage Index across the season. Attendance has some cyclical seasonality present which may be due to natural weekly cycles (more people present during the weekend). There seems to be no overall trend and the points do not look sticky. Win probability has an increasing variance as time progresses as well as a slight positive upward trend over time. Graph 1 and 2 do not seem to follow the same pattern. Graph 3 is the most interesting as the points are naturally sticky. This is due to the nature of the cLI as each subsequent game is highly dependent on the previous. There is an upward linear trend and an interesting dip towards the end of the series followed by a sharp rise for the last four data points.
Discussion
The analysis has revealed that while winning probabilities do have a positive impact on attendance they are not significant enough to predict attendance. This may be due to multiple reasons. Firstly, the sample size was arguably small with only 160 games. Another limitation is the data set being limited to only one team, suggesting that there may be team differences present. For example, demand for games may have a variable elasticity depending on the city and nature of the fan base of a team. Say that a team has just won the World Series the previous year, the fan base may be more inelastic in watching games due to the excitement and expectations of the previous year. A team that did not have a very good previous season may have a more elastic fan base where people have less anticipation and therefore a wider margin of expectations for their team.Another question arises when looking at the seasonality present in the data series, as the World Series comes closer the more attendance is present. The very sticky nature of the Championship League Index, as well as the upward positive trend should capture these trends in the model. This shows the importance of not only using win probabilities of a single event but their relationship to the rest of the season and it’s outcome. Further analysis could include looking at the uncertainty of the outcome of the season as a whole rather than a single game. Summing all attendance of a season and comparing it to the initial likelihood of winning the world series may prove to be significant across seasons and different teams.
While these findings were insignificant in regards to win probabilities the rank and CLI proved to be significant predictors. No studies have previously been interested in using these metrics or found them to be significant in predicting stadium attendance. The rank metric is in line with the loss-aversion and reference-dependent preferences theory, and the Championship League Index is closer to the UOH theory. When a team is ranked higher, we assume that the fan base will be more hopeful in a win and thus engage in loss aversion. The League Index, where the anticipation of a game having a higher importance on the outcome of the overall season predicts higher attendance shows us the attraction that people have for competition. While the competition does not have to be very close the sole expectation of being able to compete in the finals with other successful teams drives attendance. My final interpretation of uncertainty in baseball as a driver of attendance is that there is an interplay of both loss aversion, shown by ranking (on the level of a singular game), and competitive balance, displayed by championship league index (on a seasonal level).
Appendix
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