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Review of Industrial Organization

Erschienen in:

16.02.2023

Uncertainty of Outcome Hypothesis: Theoretical Development and Empirical Evaluation

verfasst von: Hayley Jang, Doyoung Kim, Young Hoon Lee

Erschienen in: Review of Industrial Organization | Ausgabe 3/2023

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Abstract

This paper studies the effect of outcome uncertainty (OU) on fan attendance in sports. First, it develops a theoretical model of fan attendance in which fans derive utilities from match quality and team quality. The theoretical results suggest that empirical models should control for the effect of team quality (“standard model”) to identify the effect of OU or include a cubic term of win probability (“alternative model”). Second, the paper evaluates previous empirical studies. It finds that they adopted the standard model, but often failed to control for the effect of team quality. Third, the paper applies the two empirical models to samples that are drawn from the English Premier League and from Major League Baseball. The empirical results from the standard model are sensitive to model specifications, whereas those from the alternative model are consistent across different specifications.

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See Humphreys and Miceli (2019) for the explanation of how outcome uncertainty improves consumer welfare.

We coin the term “loss-enduring” here to contrast with the well-known concept of “loss-aversion.” Loss-aversion is the case where a loss has a larger utility consequence than a gain of the same size, whereas loss-enduring is opposite. An importance caveat for Coates, Humphreys, and Zhou (2014) is that loss-aversion has been accepted as the standard preference in economics since well before Kahneman and Tversky (1979), whereas loss-enduring is not (see O’Donoghue & Sprenger, 2018). If fans have preferences for loss-aversion instead, the fan attendance exhibits a U shape, which contradicts the UOH.

As the previous studies did, we use win percentage as a proxy for team quality in our empirical specifications. However, unlike the previous studies, we specify team quality \({p}_{i}\) as a function of individual win probability \({p}_{ij}\). It allows us to show that individual win probability and its squared and cubic terms can account for both match quality (or OU) and team quality.

A simple example is the case of balanced schedules in which \({w}_{ij}=\frac{1}{k-1}\), where \(k\) is the number of teams in the league where team \(i\) plays.

We assume this independency for simplicity. When \({w}_{ij}\) depends on \({p}_{ij}\), we can show that the fan’s attendance can decrease with \({p}_{ij}\) for all \({p}_{ij}\) if \(\frac{\partial {w}_{ij}}{\partial {p}_{ij}}<0\). It may not be consistent with the fact that fan attendance increases with win percentage at least for some range.

Another potential explanation would be price discrimination for differentiated products combined with strong preference for high-quality wines by very rich people.

For the additive utility function, we assume that the effects of match and team qualities on the fan’s utility are independent because match quality depends on who the opponent team of the match is, whereas team quality does not. If match and team qualities interact with each other in the fan’s utility, we can think of a multiplicative form of the fan’s utility function.

Note that \({u}_{m}^{{\prime}}\left({p}_{ij}\right)=0\) for \({p}_{ij}=0.5\) and \({u}_{t}^{{\prime}}\left({p}_{ij}\right)>0\) for \({p}_{ij}=0.5\). Thus, the marginal utility from team quality can never be smaller than the marginal disutility from match quality at \({p}_{ij}=0.5\). It can become smaller for \({p}_{ij}\) above 0.5. Accordingly, the peak point in case (ii) occurs at \({p}_{ij}={p}_{0}>0.5\).

If the marginal utility from team quality measures the importance of team quality to fans, the positive third derivative means that the importance of team quality increases at an increasing rate as the probability of winning increases.

\(\varepsilon_{ijs}\) is assumed to be censored when there are many cases of sell-out matches.

Instead of absolute deviation, an analysis can use a squared deviation: \((p_{ijs} - 0.5)^{2}\). According to our empirical exercises in the next section, the absolute and squared deviation produce qualitatively similar results. Note also that the absolute deviation of the win probability of home team from 0.5 (\(\left| {p_{ijs} - 0.5} \right|\)) is the same as that from the win probability of visiting team (\(\left| {p_{ijs} - p_{jis} } \right|\)) because \(p_{jis} = 1 - p_{ijs}\).

See Szymanski (2003) for details.

They also used spline regressions to examine the non-linear relationship between fan attendance and win probability.

In their comprehensive review on match outcome uncertainty, Johnson and Fort (2022) suggested that there is evidence failing to reject the UOH that was previously acknowledged.

For example, a bookmaker quotes "2 to 1 for a home team win," "4 to 1 for a home team loss," and "2.5 to 1 for a draw." The provisional probabilities are then (1/2), (1/4), and (1/2.5). The sum of the three probabilities is always greater than 1 because of the bookmaker's earning spread. Therefore, probability estimates \((p_{1ij} ,\,\,p_{2ij} ,\,\,p_{3ij} )\) are obtained by imposing the restriction that the sum of the three probabilities is 1.

The bottom three teams in EPL are relegated to its second division (English Football League Championship, EFLC), while the top two teams in EFLC are promoted automatically and the next four teams compete in playoffs for the third promotion position.

Our sample excludes the first six matches of each team in every season as we use the average goals scored and allowed in the last six matches for TQ variables.

Their signs and statistical significance are reasonable, and they are available upon request to us.

They are available upon request to us.

Meanwhile, the effect of draw probability is statistically significant and negative for most models, which is consistent with the findings by Alavy et al. (2010).

The total effects from Models 1 and 3 are calculated by \(-0.23\mathrm{x}\left|p-0.5\right|+0.105p\) and \(0.48p-0.839{p}^{2}+0.59{p}^{3}\), respectively.

To obtain reliable win percentage to date, our sample excludes the first 10 games of each team in every season. The sample also excludes Washington Nationals, Toronto Blue Jays, and the 2017 season because of data availability limitations.

The R²s are smaller than those of the EPL attendance regressions in Table 2. This difference is consistent with previous empirical studies. A main cause of this difference is the explanatory power of the individual effects. According to our additional regression, the addition of home team effects increases R² by 0.38 in MLB, but by 0.55 in EPL.

Note that levels of outcome uncertainty are not equal to levels of competitive balance in the context of unbalanced schedules. See Fort and Lee (2020) for details.

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Alavy, K., Gaskell, A., Leach, S., & Szymanski, S. (2010). On the edge of your seat: Demand for football on television and the uncertainty of outcome hypothesis. International Journal of Sport Finance, 5(2), 75.

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Titel: Uncertainty of Outcome Hypothesis: Theoretical Development and Empirical Evaluation
verfasst von: Hayley Jang
Doyoung Kim
Young Hoon Lee
Publikationsdatum: 16.02.2023
Verlag: Springer US
Erschienen in: Review of Industrial Organization / Ausgabe 3/2023
Print ISSN: 0889-938X
Elektronische ISSN: 1573-7160
DOI: https://doi.org/10.1007/s11151-023-09899-w

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