We develop a consumer choice model of live attendance at a sporting event with reference-dependent preferences. The predictions of the model motivate the "uncertainty of outcome hypothesis" (UOH) as well as fans' desire to see upsets and to simply see the home team win games, depending on the importance of the reference-dependent preferences and loss aversion. A critical review of previous empirical tests of the UOH reveals significant support for models with reference-dependent preferences, but less support for the UOH. New empirical evidence from Major League Baseball supports the loss aversion version of the model. (JEL L83, D12)

I. INTRODUCTION

Recent events in economic and financial markets reemphasize the importance of understanding decision making under uncertainty. Individuals make decisions that involve uncertain outcomes in a variety of settings, including decisions about the purchase and holding of risky assets like stocks and bonds (Lintner 1965), human capital investment (Kodde 1986), labor supply (Camerer et al. 1997), gambling (Sauer 1998), entertainment (Post et al. 2008), and others. In the sports economics literature, a body of research focuses on the effect of uncertainty about game outcomes on consumer demand. The decision to attend a sporting event involves uncertainty, because the consumer does not know the outcome of the game at the time the ticket is purchased. Recent research points out the importance of reference points that reflect consumer's expectations when making decisions under uncertainty (Koszegi and Rabin 2006).

The idea that demand for sporting events depends on the uncertainty of game outcomes was first identified by Rottenberg (1956) in a seminal paper and has long been referred to as the "uncertainty of outcome hypothesis" (UOH). The UOH forms the basis of a large literature on competitive balance in sport (Fort and Quirk 1995). Rottenberg (1956) observed that attendance demand depends on "the dispersion of percentages of games won by the teams in the league" (p. 246) and further observed: "That is to say, the 'tighter' the competition, the larger the attendance. A pennant-winning team that wins 80 per cent of its games will attract fewer patrons than a pennant-winning team that wins 55 per cent of them" (p. 246, footnote 21). These observations form the basis of the UOH, which generated a large empirical literature. Neale (1964), in another early paper, also addressed the UOH. Neale (1964) observed that in order for fans to attend sporting events, listen to events on radio, or watch events on television, some uncertainty of outcome about the contest must exist: "Of itself there is excitement in the daily changes in the standings or the daily changes in possibilities of changes in standings. The closer the standings, and within any range of standings the more frequently the standings change, the larger will be the gate receipts" (p. 3).

Interestingly, neither Rottenberg (1956) and Neale (1964), nor any subsequent researcher, developed a model of consumer behavior to motivate this observation; the UOH has been accepted as an accurate description of the outcome of consumer choices with no theoretic basis for more than 50 years. Instead, research focused on developing models of team and league behavior that generated different levels of winning percentage dispersion, which reflects outcome uncertainty, depending on market and team characteristics.

In this article, we develop a consumer choice model of the decision to attend sporting events that include uncertainty and reference-dependent preferences to motivate the UOH. Our model adopts the basic framework of the model in Card and Dahl (2009), where consumers decide whether to watch a national football league (NFL) game involving their local team on television with rational anticipation that an unexpected loss may trigger family violence. We abstract from the emotional cues and triggers in Card and Dahl's (2009) model, and focus on the decision to attend a live game. The predictions of the model show that the existence of the UOH depends critically on the marginal utility of wins and losses; the UOH only emerges when the marginal utility generated by an unexpected win exceeds or equals the marginal utility generated by an unexpected loss. When the marginal utility of an unexpected loss exceeds the marginal utility of an unexpected win, a situation that can be motivated by prospect theory (Kahneman and Tversky 1979), the UOH does not emerge from the model, and demand increases when there is less uncertainty about game outcomes. Relatively little empirical evidence about reference-dependent preference models exists. A review of past research on the relationship between expected game outcomes and attendance reveals evidence supporting the reference-dependent preference model with loss aversion. Much of this evidence supports the presence of reference-dependent preferences and loss aversion in this setting. We also test the predictions of this model using data from major league baseball (MLB), where outcome uncertainty is proxied with a market-generated prediction based on betting odds data. The evidence from this empirical analysis supports the predictions that emerge from the prospect theory-based model with reference-dependent preferences; attendance is higher at games with less outcome uncertainty, other things equal. Taken together, the previous empirical research on the UOH, and the new evidence developed in this article provide a relatively large body of evidence supporting the predictions of reference-dependent preference models, which have only been empirically tested in a small number of papers. (1)

II. A MODEL OF THE ATTENDANCE DECISION UNDER UNCERTAINTY

We first develop a model of sports fans' behavior to motivate the decision to attend a live game, the outcome of which is uncertain. Our model is developed in the same reference-dependent preference framework as that of Card and Dahl's (2009) model, but strips away the family violence aspect that is much less relevant in the live game attendance decision. The model captures the idea that the outcome of the choice to attend a sports event depends on the actual result of the game relative to a reference point that reflects the consumer's expectation of the game outcome. These reference-dependent preferences allow us to model uncertainty directly as part of the consumer choice process. This model provides a theoretical basis for the UOH and extends existing models of reference-dependent preferences to a new setting. In this discrete choice model of consumer decision making under uncertainty, consumers receive two types of utility from attendance at sporting events: intrinsic "consumption utility" that corresponds to the standard utility from consumer theory and "gain-loss utility" that depends on what actually happens on the field or court compared to the consumer's reference point (Koszegi and Rabin 2006). The reference point explicitly brings expectations about game outcomes into the model and allows us to model uncertainty in a way consistent with the UOH. The consumer compares the expected utility from attending a game under these conditions to a reservation utility level and attends the game if the expected utility exceeds this reservation utility level.

The outcome of a sporting event can be represented by a binary indicator variable y, where y = 1 represents a win by the home team and y = 0 represents a loss by the home team. Individuals receive "gain-loss utility" from attending a game, based on the utility function developed by Koszegi and Rabin (2006), which assumes that individuals derive utility from the outcome of an uncertain event, determined by intrinsic taste for the outcome itself, and the deviation of the outcome from a reference point. Following Koszegi and Rabin (2006), we assume that an individual's reference point is his expectation about the outcome of a game E(y = 1) = [p.sup.r]. Attending a game that the home team wins (y = 1) generates both intrinsic "consumption utility" from the game [U.sup.W] and "gain-loss utility" from experiencing a win when y = 1 conditional on the reference point [p.sup.r]. Assume the marginal impact of a positive deviation from the reference point is [alpha] > 0. The utility from attending a game that the home team wins (y = 1) is

[U.sup.W] + [alpha](y - [p.sup.r]) = [U.sup.W] + [alpha](1 - [p.sup.r]).

A fan who attends a game where the outcome is a home loss (y = 0) also gets intrinsic "consumption utility" from the home loss [U.sup.L] and "gain-loss utility" from the sensation of a loss compared to the reference point [p.sup.r]. Assume the marginal effect of a negative deviation from the reference point is [beta] > 0. The utility from attending a game that the home team loses (y = 0) is

[U.sup.L] + [beta](y - [p.sup.r]) = [U.sup.L] + [beta] (0-[p.sup.r]).

Figure 1 illustrates the relationship between game outcomes, the reference point, and total utility from attending a game. (2) In Figure 1, total utility is graphed on the vertical axis and the reference point, the expectation of the game outcome, is graphed on the horizontal axis. The top line shows the utility generated by a home win and the bottom line shows the utility generated by a home loss. We assume that [U.sup.W] > [U.sup.L], implying that the consumption utility generated by a home win exceeds the consumption utility generated by a home loss.

From Figure 1, the maximum total utility from a home win comes when a fan expects that the home team has no chance to win the game, the reference point is [p.sup.r] = 0, and the home

team pulls off an epic upset and wins the game (y = 1). The leftmost point on the home win line represents this outcome, and a fan gets total utility of [U.sup.W] + [alpha]. Moving to the right along the home win line, total utility from a home win diminishes because the deviation of the outcome (y = 1) from the reference point ([p.sup.r]) declines. In other words, as the fan's expectation that the team will win a home game increases ([p.sup.r] increases), total utility declines because the thrill of experiencing an upset declines. At the right end of the win line, a fan fully expects that the home team will win ([p.sup.r] = 1) and when the home team wins, total utility is [U.sup.W].

Consider the total utility generated by a home loss (y = 0). At the left end of the loss line, the home team loses (y = 0) and the loss was fully expected by a fan ([p.sup.r] = 0). This outcome generates only the consumption utility from a home loss, [U.sup.L]. At the right end of the loss line lies a fan's worst possible outcome: the situation where the home team loses (y = 0), but a fan fully expected that the home team would win ([p.sup.r] = 1). This generates the smallest possible utility for a fan, [U.sup.L] = [beta], because the consumption utility from attending the game, UL, is reduced by the fact that the loss represents maximum deviation from the reference point. Moving from left to right along the loss line, the fan's consumption utility is reduced because the reference point increases; a fan has an increasing expectation that the home team will win and experiences additional lost utility because the outcome differs more and more from this expectation.

Game outcomes are uncertain. We assume that the reference point is equal to the objective probability that the home team wins a game, which is also equal to the expected outcome of the game: E(y) = [p.sup.*] 1 + [(1 - p).sup.*] 0 = p = [p.sup.r]. In other words, we assume that consumers who are trying to decide whether or not to attend a game form a reference point that is equal to the objective probability that the home team will win. Under this assumption, the expected utility from attending a game is the probability that the home team wins (p) times the total utility from a home win plus the probability that the home team loses (1 = p) times the total utility from a loss

(1) E[U] = p[[U.sup.W] + [alpha](1 - p)] + (1 - p)[[U.sup.L] + [beta](0-p)]

(2) E[U] = ([beta] - [alpha])[p.sup.2] + [([U.sup.W] - [U.sup.L]) -([beta]-[alpha])]p + [U.sup.L].

Equation (2) shows that, after some manipulation, the expected utility from attending a game is a quadratic function of the probability that the home team wins the game. This expected utility function incorporates game outcome uncertainty, and also takes into account the utility generated by watching the home team win in an upset and the disutility generated by watching the home team get upset when fans expected the team to win, important components of outcome uncertainty in sports.

The choice to attend a game is binary; the consumer either attends a game or does not attend a game. Assume that if an individual does not attend a game, she gets utility v, which can be interpreted as the reservation utility from not attending a game. We assume that v is uniformly distributed across the population over the support [[v.bar], [bar.v]]. Given this reservation utility, an individual will attend a live game if expected utility E[U] from attending the game is higher than the reservation utility v. If the expected utility of attending the game is lower than v, then the consumer does not attend. Fans of a team have a low reservation utility and will attend games regardless of the expected outcome of the game; other consumers have a higher reservation utility and will only attend games if the expected utility from attendance is high enough.

Consider the relationship between the probability that the home team wins a game and the expected utility generated from attendance when there is no reference-dependent utility. This corresponds to a standard Friedman-Savage utility function for decisions made by risk neutral consumers under uncertainty. Under this special case, [alpha] = [beta] = 0 and the expected utility function is

(3) E[U] = ([U.sup.W] - [U.sup.L]) p + [U.sup.L].

From Equation (3), in the absence of reference-dependent preferences, the expected utility of attending a game is an increasing function of the probability that the home team wins the game. Fans simply want to see the home team win in this case. Under this assumption, games that the home team is expected to win will have higher attendance, and games that the home team is not expected to win will have lower attendance. This prediction has considerable intuitive appeal, because it predicts that better teams will have higher attendance and worse teams will have lower attendance, other things equal. Assuming risk averse fans would simply add convexity to the function, increasing the utility from seeing a win. This case is not consistent with the UOH, since expected utility is an increasing function of the probability that the home team will win the game. In this case, the more certain is a home win, the greater the expected utility. Games with the most uncertain outcomes will have p near .5. From Equation (3), a game where p = .5 generates less expected utility than a game the home team is expected to win, which would have a higher p.

Another special case implied by the model is that of a pure fan of the game, an individual for whom the standard consumption utility of a win and a loss are equal ([U.sup.W] = [U.sup.L]) and gain-loss utility is not important ([beta] - [alpha]) = 0. For such an individual, utility from attending the game is simply [U.sup.W] = [U.sup.L], and both uncertainty of outcome and the probability of the home team winning the game play no role in determining attendance for such a fan. We assume this describes a trivial portion of potential attendees at sporting events.

A. Reference-Dependent Preferences and the UOH

We next motivate the UOH in the context of this model. In the competitive balance literature (e.g., El-Hodiri and Quirk 1971 and Fort and Quirk 1995), the UOH is modeled as a concave relationship between the gate revenue and the home win probability with a maximum achieved between (0.5, 1). In practice, the maximum has been interpreted as being located closer to 0.5 than to 1. For example, Rottenberg (1956), states that a " ... team that wins 80 per cent of its games will attract fewer patrons than a ... team that wins 55 per cent of them" (p. 246, footnote 21). We call this the "classic" UOH. In the context of this model, the classic UOH is consistent with an expected utility function that is concave in p and reaches a maximum at a value greater than or equal to 0.5 and substantially less than 1.

From Equation (2), the expected utility function will be concave if ([beta]--[alpha]] < 0. Under the assumption that the consumption utility from a win must be at least as large as the consumption utility from a loss ([U.sup.W] [greater than or equal to] [U.sup.L]), the term [([U.sup.W]--[U.sup.L])--([beta]--[alpha]]] must be positive if ([beta]--[alpha]] is negative. From Equation (3), the expected utility function will be linearly increasing in p if fans only have home win preference.

Figure 2 illustrates consumer decision making consistent with the classic UOH in this model. The expected utility function is concave in p and peaks at [p.sup.max]. If the reservation utility is v, this consumer will only attend a game if the probability that the home team wins is between [p.sub.0] and [p.sub.1]. These games have a relatively uncertain outcome. Since v is distributed over the support [[v.bar], [bar.v]], more people will have E[U] > v when games have a relatively uncertain outcome. Note that

[p.sup.max] = 1/2 - ([U.sup.W] - [U.sup.L])/2([beta] - [alpha]) [greater than or equal to] 1/2

in Figure 2 because [beta] < [alpha] by assumption. The classic UOH also requires

[p.sup.max] = 1/2 - ([U.sup.W] - [U.sup.L])/2([beta] - [alpha]) < 1,

which is equivalent to [U.sup.W] - [U.sup.L] < [alpha] - [beta], in which case fans' preferences for home wins are dominated by their preferences for tighter games. In the context of the reference-dependent preference model developed here, the classic UOH in the competitive balance literature is not just about whether fans have preferences for tighter games, but whether fans' preference for tighter games dominates their preference for home team wins.

A necessary, but not sufficient, condition for the classic UOH to emerge from our model is ([beta] - [alpha]] < 0, when the marginal utility generated by deviations of game outcomes from the reference point when the home team wins is greater than the marginal utility generated by deviations of game outcomes from the reference point when the home team loses.

B. Reference-Dependent Preferences and Loss Aversion

The UOH is not the only relationship between the probability that the home team wins a game and expected utility consistent with this model. For positive [alpha] and [beta], when [beta] > [alpha] the marginal utility from game outcomes that deviate from the reference point when the home team is expected to lose is larger than the marginal utility from game outcomes when the home team is expected to win. This outcome is known as loss aversion in the literature on decision making under uncertainty, and emerges from prospect theory (Kahneman and Tversky 1979). The UOH is not consistent with the presence of loss aversion in terms of home team losses, since the expected utility function is not concave when [beta] > [alpha].

Consumer decisions under loss aversion differ from those under the UOH. Figure 3 shows the expected utility function under the assumption of loss aversion and game attendance decisions made by consumers under this condition. Again, v is the reservation utility for game attendance. p is both the objective probability of a home win and the reference point of a consumer. An increase in p has two effects on the expected utility of a consumer with reference-dependent preferences and loss aversion. To make the point clearly, rearrange Equation (1) as follows

E[U] = [p[U.sup.W] + (1 - p)[U.sup.L]] + ([alpha]- [beta)p(1 - p).

The expected intrinsic "consumption utility" p[U.sup.W] + (1 - p)[U.sup.L] increases with p, the objective probability of a home win. The expected "gain-loss utility" (a[alpha] - [beta])p(1 - p) first decreases with p at a decreasing rate until p = 1 /2, then increases with p at an increasing rate. (3) When p is smaller than 1 /2 - ([U.sup.W] - [U.sup.L])/2([beta]--[alpha]], the negative impact of an increase in p on the expected "gain-loss utility" dominates. When p is larger than 1/2 - ([U.sup.W] - [U.sup.L])/ 2([beta]--[alpha]], the positive impact on the expected "consumption utility" dominates. The model developed by Card and Dahl (2011) features loss-aversion and reference-dependent preferences in a similar context. Kahneman, Knetsch, and Thaler (1991) review the literature on loss aversion.

The size of the reservation utility v, which likely varies from person to person and sport to sport, is theoretically and empirically important. In the general case shown in Figure 3, v is low enough that there is a range of declining attendance as p rises. If this occurs in practice, then data should enable identification of this manifestation of loss aversion. However, if the support for v is sufficiently large that it exceeds [U.sup.L], then the expected home win probability beyond which attendance rises with p, home win probability [p.sub.1] in Figure 3, could be relatively large before expected utility from attending a game exceeds the reservation utility. In such a case, the attendance-home win probability relationship may have a flat section where attendance is unresponsive to changes in the expected home win probability.

Under reference-dependent preferences and loss aversion, attendance at games with a relatively certain outcome, be it an expected loss or an expected win, generates higher expected utility than games with uncertain outcome. Again the presence of loss aversion makes fans less interested in seeing a game with an uncertain outcome, because the marginal utility generated from seeing an unexpected loss outweighs the marginal utility of seeing an unexpected win. Notice that under loss aversion, the expected utility of a relatively certain loss by the home team, where p is small and close to zero, generates more expected utility than games with relatively uncertain outcomes, where p is close to .5. This prediction motivates observed interest among casual sports fans in seeing upsets. Clearly, strong fans of a team, in this context consumers with a low reservation utility, will attend games with either certain or uncertain outcomes. But among casual fans, in this context consumers with a relatively high reservation utility, the possibility of watching an upset often holds some allure. In the context of this model, an upset takes place when the home team is expected to lose the game (p is small) but the home team actually wins the game. This outcome generates a relatively large amount of gain-loss utility, since [beta] > 0. The thrill of potentially seeing an upset explains the convexity of the expected utility function under loss aversion, in that the expected utility of seeing an upset when the home team is expected to lose outweighs the gain-loss utility of seeing a home team loss when the outcome of the game is relatively uncertain.

The UOH cannot explain fans' interest in upsets, since upsets, by definition, only occur in games with a relatively certain outcome (games with a strong favorite and large underdog). The presence of consumers with loss aversion and reference-dependent preferences can explain the frequently observed increase in fans' interest in upsets.

Note that this relationship between expected game outcomes and expected utility requires reference-dependent preferences and loss aversion, and not simply risk aversion. A consumer with risk-averse preferences over game outcomes would have a standard concave expected utility function and would always get more expected utility from a game with a p = .2 expected probability of a home team win relative to a p = .1 expected probability of a home team win. However, from Figure 3, a consumer with loss aversion might get more expected utility from a game with a p = .1 expected probability of a home win relative to the expected utility from a game with a p = .2 expected probability of a home team win. In other words, reference-dependent preferences and loss aversion can explain fans' interest in upsets, but not risk aversion.

Note that the difference between the consumption utility from a win ([U.sup.W]) and the consumption utility from a loss ([U.sup.L]) plays a key role in determining the relationship between the probability that the home team wins a game and expected utility. In Figures 2 and 3, the value of the expected utility function at [p.sup.r] = 0 and [p.sup.r] = 1 depends on [U.sup.L] and [U.sup.W], respectively. If [U.sup.W] - [U.sup.L] is sufficiently large and positive, then the expected utility function will be strictly increasing over the interval [0,1]. In this case, the expected utility function with reference-dependent preferences resembles the expected utility function without reference-dependent preferences and attendance will increase with a team's success. In this case, the model does not generate predictions consistent with the UOH or fans' interest in seeing upsets. The relationship between [U.sup.W] and [U.sup.L] is an empirical issue. [U.sup.W] [greater than or equal to] [U.sup.L] seems to be a reasonable assumption, since losses should not generate more consumption utility than wins. But the size of [U.sup.W] - [U.sup.L] cannot be easily determined. The model predicts that as [U.sup.W] - [U.sup.L] increases, the relationship between expected utility and the probability that the home team wins a game becomes strictly positive and increasing.

Despite the lack of a theoretical basis, the UOH has been extensively used to motivate decisions by consumers to attend live sporting events for more than 50 years. We develop a utility maximizing consumer choice model of decision making under uncertainty to motivate the UOH. This model features reference-dependent preferences where consumer choice depends on expectations of game outcomes. The prediction of the UOH emerges as one special case in this model, but the model is general enough to predict other outcomes. In particular, depending on the relative size of the marginal utility from wins when consumers expected wins and losses when consumers expected losses, the model can also explain why consumers would only prefer to watch winning teams, and why consumers might have an interest in watching upsets, two outcomes that cannot be explained by the UOH. We next turn to a critical examination of previous evidence about the relationship between expected game outcomes and attendance at sporting events, a topic that has received considerable attention over the past 30 years.

III. LINKING THEORY AND EVIDENCE: A STRUCTURAL ECONOMETRIC MODEL

A structural econometric model which links the behavioral model developed above to the existing empirical literature and motivates our empirical work can be derived from the model in the previous section.[4.sub. ]The derivation begins with the assumption that the observed attendance at games depend on the number of individuals in the area with expected utility of attending a game greater than their reservation utility, v [less than or equal to] E[U], and that v is uniformly distributed over [[bar.v], [v.bar]]. The structural econometric model is

(4) In [Attendance.sub.ijt] = [lambda] + [theta][p.sub.ijt], + [gamma][p.sup.2.sub.ijt] + [X.sub.ijt][mu], + [D.sub.i] + [D.sub.j] + [D.sub.t] + [[epsilon].sub.ijt]

where [X.sub.ijt], is a vector of home and visiting team characteristics, stadium and local market characteristics, and day of game and month of season indicator variables, [D.sub.i] is a local market fixed effect capturing any unobservable heterogeneity in the markets, [D.sub.j] is a visiting team fixed effect capturing unobservable heterogeneity in the visiting teams, D, is a vector of time-related factors that affect this group of consumers, and [[epsilon].sub.ijt] is a random error term clustered on i that captures all other factors that affect the size of the population of residents who would consider going to a game. The key parameters in this model, those on the variable reflecting the probability that the home team wins a game, are functions of the parameters in the behavioral model, [gamma] [equivalent to] ([beta] - [alpha]/([bar.v] -[v.bar]), [theta] [equivalent to] (([U.sup.W] - [U.sup.L]) - ([beta] - [alpha])/([bar.v] - [v.bar]), and [lambda] [equivalent to] ([U.sup.L] - [bar.v])/([bar.v] - [v.var]). This model shows that attendance depends on team and time effects, the probability that the home team will win the game, market characteristics, the minimum utility from attending a game, market characteristics, and random factors, y and 0 reflect the absence or presence of reference-dependent preferences and loss aversion, and the relationship between game uncertainty and attendance. Parameter estimates from this structural regression model can be used to test the following hypotheses about the relationship between expected game outcomes and attendance:

H1 [gamma] > 0 implies that [beta] > [alpha], supporting the hypothesis that the marginal consumer has loss aversion for home games.

(H1a). [gamma] > 0 and [theta] < 0 implies that 0 [less than or equal to] [U.sup.W] - [U.sup.L] < ([beta] - [alpha]], which means that the marginal consumer gets more consumption utility from a home win than from a home loss and has loss aversion for home games, and the marginal impact of loss aversion is bigger than the consumption utility difference between a home win and a home loss.

(H1b). [gamma] > 0 and [theta] > 0 implies that [U.sup.W] - [U.sup.L] > ([beta] - [alpha]] > 0, which means that the marginal consumer gets more consumption utility from a home win than from a home loss and has loss aversion for home games, and the marginal impact of loss aversion is smaller than the consumption utility difference between a home win and a home loss.

H2 [gamma] = 0 and [theta] > 0 implies that ([beta] - [alpha]] = 0 and [U.sup.W] - [U.sup.L] >0, suggesting that the marginal consumer does not have reference-dependent preferences for home games and gets more consumption utility from a home win than from a home loss.

H3 [gamma] < 0 and [theta] > 0 implies that ([beta] - [alpha]] < 0 [less than or equal to] [U.sup.W] - [U.sup.L], suggesting that the marginal consumer gets more utility from an unexpected win than an unexpected loss and has preferences for tighter games, a necessary condition for the classic UOH. (5)

This model, and the hypotheses listed above, can also motivate a critical review of the existing empirical literature on attendance and the probability that the home team wins a game or match. The results from the extensive literature on empirical tests of the UOH in sports economics can be interpreted as estimates of structural parameters, in terms of the relationship between the probability that a home team wins a game and game attendance. This observation provides a significant body of research supporting the importance of reference-dependent preferences in a setting where both demand and a market-based proxy for expected outcomes exists. We review this literature in the next section.

IV. EVIDENCE ABOUT OUTCOME UNCERTAINTY AND ATTENDANCE

A substantial empirical literature examining the relationship between expected game outcomes and attendance exists. These studies have been carried out in many settings, using data at the game or match and season level. Here, we summarize only the research that tests the UOH at the game level using a variable to proxy for the probability that the home team will win a given game, because the model developed in the previous section applies only to consumer decisions to attend a game. In future research, we plan to examine decisions to attend multiple games over the course of a season. In order to facilitate comparisons of the results in this large literature, consider a generic regression model based on Equation (4)

(5) A = [theta]p + [gamma][p.sup.2] + /(covariates) + [epsilon]

where A is game attendance or some transformed game attendance variable and p is some measure of the probability that the home team wins a specific game. In terms of the UOH, [theta] and [gamma] are the parameters of interest, as they capture the effect of the expected outcome of a specific game on attendance at that game as identified in the structural econometric model, Equation (4). We identified 24 studies that examined the relationship between expected game outcome and attendance that include a variable explicitly linked to the expected outcome of games. In some cases, this proxy was based on teams current winning percentage or position on the league table. In other cases, the proxy for expected game outcomes was based on betting odds or point spread data. Some of the studies do not include the quadratic term [p.sup.2], and two studies by Owen and Weatherston (2004a, 2004b), use [p.sup.4] instead of [p.sup.2]. Two studies by Benz et al. (2009) and Rascher (1999), use both functions of current team success or position in the league table or betting odds (home win probabilities) but not both in the same specification. Consequently, some of the 24 studies allow for better comparison to the theory developed above than do others.

Table 1 summarizes the context and results for these 24 papers. The studies are separated into groups based on their measure of uncertainty of outcome and whether the equation is specified as linear or quadratic in the uncertainty measure. Moving down the table, empirical specifications and results in the studies become more consistent with the theoretical model developed in Section II.

Note that the research surveyed in Table 1 was carried out in a wide variety of settings, including North American, European, and South American sports leagues, as well as leagues in Australia and New Zealand. Most of these papers, 16 of 24, use betting odds data or point spreads as a proxy for the expected outcome of a game. The remaining eight construct complicated functions of the success of the two teams involved as proxies for the expected outcomes of games. Half of the studies using functions of team success fail to find a statistically significant relationship between the expected game outcome and attendance. Of the 16 studies using betting odds or point spread data to capture game uncertainty, only the two papers using data from rugby in New Zealand, and the quartic rather than the squared value of p, find no relationship between uncertainty and attendance. This pattern suggests that it may be difficult to construct useful proxies for expected game outcome using only data on the success of the teams involved and that the specification may be best approximated by a quadratic function of the game uncertainty variable.

The important result, in the context of support for the predictions of the model developed in the previous section, is the shape of the relationship between expected game outcome and attendance. From Equation (5), results where [theta] > 0 and [gamma] < 0 are a necessary condition for the classic UOH, results where [theta] > 0 and [gamma] = 0 are consistent with an absence of reference-dependent preferences, and results where [theta] < 0 and [gamma] > 0 are consistent with the presence of reference-dependent preferences and loss aversion. Note that the papers that did not include a quadratic term for the variable representing the expected outcome of the game cannot distinguish between the loss aversion prediction and the no reference-dependent preference prediction, but they can reject the UOH.

No clear consensus about the relationship between expected game outcome and attendance emerges from Table 1. All three of the special cases of the model developed in the previous section have some empirical support in this literature. Four papers contain evidence consistent with H3, a necessary condition for the classic UOH. Seven studies contain evidence consistent only with the model based on reference dependent preferences and loss aversion, HI a. Seven studies contain evidence consistent with the reference-dependent preference model with no loss aversion, H2. A preponderance of the papers contain results supporting a reference dependent preference model, either with or without loss aversion. Despite the dominance of the UOH as a theoretical explanation for consumer decisions about attending games under uncertainty, the predictions of the UOH are not widely supported in the existing empirical evidence.

A natural question, and issue for further study, is why different studies generate different empirical implications. The explanation is unlikely to be related to sample size, which tends to be large in these studies, as the total number of games or matches played in a season in a sports league tends to be large. Moreover, many of these papers use data from multiple seasons. All of the papers summarized in Table 1 contain explanatory variables that reflect the quality and ability of the teams involved in the games, so the results hold these factors constant. Precise specifications of these other variables and the array of additional covariates vary among the studies, so differences in the results may be based on use of alternative proxies for team quality, local market conditions, ticket prices, and availability of substitutes. Perhaps, consistent specifications would produce greater consistency in the results regarding the UOH and reference-dependent preferences. Also, Equation (4) shows that the coefficients on the variables reflecting the probability that the home team will win a game are functions of both the consumption utility from wins and losses ([U.sup.W] and [U.sup.L]) and the marginal utility of wins and losses relative to the reference points (a and P). Significance tests on the structural parameters are effectively a test of the joint hypothesis that consumers have reference-dependent preferences and specific restrictions on the marginal utility of expected wins and losses. Some of the cross-study variation in Table 1 may be attributable to differences in [U.sup.W] and UL across sports or cultures. Regardless of what the explanation is for finding support in some studies for a concave expected utility function, as reflected in attendance being maximized at some home win probability between 0 and 1, and rejection of this in others, nearly all the studies produce support for a role for uncertainty in the outcome as a determinant of game day attendance.

V. A TEST OF THE MODEL USING MLB GAME DAY ATTENDANCE DATA

The existing literature on attendance and expected game outcomes reviewed above contains evidence supporting both the UOH and a model with reference-dependent preferences and loss-aversion. From Table 1, much of the evidence supporting the UOH comes from MLB using data from a single season. While MLB teams play a large number of games each season, the evidence from MLB supporting the UOH comes from the 1980s and 1990s. The paper by Lemke, Leonard, and Tlhokwane (2010) finds evidence consistent with loss-aversion using MLB data from the 2007 season. To reconcile these results, and to further assess the relationship between attendance and expected game outcomes in MLB, we estimate the parameters of the structural econometric model developed above, Equation(4), using data from MLB over six seasons.

To test our model, we collected data on attendance and other characteristics for all MLB games in the 2005 through 2010 regular seasons. Our dataset contains data from all home games of every MLB team except the Toronto Blue Jays over this period, over 13,300 games. The data come from a variety of sources. Game attendance data, and data on scoring in the games and the teams involved were collected from the MLB website (www.mlb.com). Average ticket price data come from the Fan Cost Index collected by Team Marketing Report (www.teammarketing.com).

The probability that the home team wins each game, the primary variable in our analysis, is derived from betting data that come from Sports Insights (www.sportsinsights.com), a sports gambling information site. The MLB money line data collected and distributed by Sports Insights is the average money line from three off-shore, online sports books: BetUS.com, FiveDimes.com, and Caribsports.com. The money line reported by Sports Insights for each game is the average money line across these three book makers. We converted the money line to odds, and then to the probability that the home team wins each game using the formula in Kuypers (2000). Our explanatory variable is a market-based measure of the probability that the home team will win a game. The empirical model also contains the home win probability squared as the theory suggests.

Descriptive statistics for key variables in the final dataset are reported in Table 2.

The dependent variable in our analysis is the natural logarithm of attendance. Of course, attendance is constrained by the seating capacity of stadiums, so we construct a dummy variable that identifies games that are sell outs. The capacity constraint differs for each stadium in the sample. The dataset contains 224 right-censored games. We use this variable to estimate the attendance equation using a maximum likelihood estimator for truncated dependent variables, a generalized form of the standard to bit estimator. Amemiya (1973) developed this estimator, which assumes that the unobservable error term in Equation (4) has a normal distribution with mean zero and constant variance [[sigma].sup.2]. We relax this assumption and assume that the variance of the equation error can have nonzero within-team correlation, and cluster correct the estimated standard errors at the team level. We also use the standard White-Huber "sandwich" correction for heteroskedasticity.

The model also includes home team, visiting team, season, month, and day of the week dummy variables as well as interactions between the home team and the season dummies to control for unobservable heterogeneity. The structural econometric model developed in the previous section motivates the inclusion of these variables. Added to these explanatory variables, we include several additional variables that have been shown to affect attendance. Rottenberg (1956) hypothesized that fans will be attracted to high-quality play. To address this we include several measures of team performance. First, we construct the winning percentage over all games played prior to the current game for both the home team and the visiting team. Teams that win a larger percentage of their games are higher quality. Similarly, we construct the average number of runs scored and the average number of runs allowed by both the home and visiting teams in all games prior to the current game.

Results are reported in Table 3 for three model specifications, Models I, II, and III. Model I is a basic model that contains the day, month, season, team, and home team-season effects variables and the probability that the home team will win each game, based on the betting odds on each game. Model I is Equation (4) when [X.sub.ijt], = 0.

The estimated parameters on the home win probability and home win probability squared variables in Model I are both statistically significant. The relationship between the expected home win probability and attendance is U-shaped. The signs support the reference-dependent preferences model with loss aversion developed above, and are not consistent with the standard prediction of the UOH. In terms of the hypotheses from Section IV, the data support both HI and H2; attendance decisions appear to be consistent with the presence of consumers with reference-dependent preferences and loss aversion. The coefficients indicate that the relationship between home win probability and attendance turns up at a win probability of .505. About 34% of the observations have a home winning percent below 0.505, so 66% of the observations are in the range where attendance is rising with home winning probability.

Models II and III introduce additional covariates that might be correlated with the expected home win probability and also explain observed variation in attendance. Model II adds variables that capture the quality of play of the teams involved, the average runs scored and allowed by each team over the course of the season prior to the current game. Evidence suggests that fans like to see well-played baseball games. Attendance is higher the more runs per game either team scores and the less runs per game either team allows. Interestingly, the impact of runs scored is similar for both home and visitor while the impact of runs allowed is about twice as large for home team as for visitor. This suggests that fans want to see their team hold the other team to few runs but the visitors' defense is less important.

Model III adds the cumulative winning percentage for both teams prior to the current game. Existing evidence supporting the UOH based on data from MLB come from the 1980s and 1990s, and many of these studies used functions of won-loss records to estimate the probability that the home team would win a game, so including winning percentages is an important robustness check. We estimate Model III using only observations from May 1 on to avoid spurious correlation between winning percentage and attendance in the early season, when a few consecutive wins or losses can have a large effect on winning percentage. The estimated parameters on the team winning percentage variables are positive and significant, supporting the idea that fans like to see high quality teams play. The sign and significance of the estimated parameters on the expected probability that the home team will win the game variables are unchanged by the inclusion of these variables in the regression model. The evidence from all three models suggests that the UOH does not describe attendance at MLB games over the period 2005-2010. Instead, reference-dependent preferences and loss aversion appear to characterize the decision made by MLB fans over this period.

We estimated several additional models with a wider variety of explanatory variables and alternative functional forms as a robustness check. The first included dummy variables identifying double headers, games between teams in the same division, games between teams in the same league, and teams playing in a stadium that opened at the start of the current season to control for possible novelty effects of a new stadium. The second included the average ticket price as captured by the Fan Cost Index to the model. The third added the log of average attendance from the previous season to the model. The estimated parameters on these additional variables were of the predicted sign, including a negative sign on the average ticket price variable, and were generally significant. The sign and significance of the estimated parameters on the key explanatory variables of interest, the expected probability that the home team will win the current game, were unchanged in these additional models. The results supporting the predictions of the reference-dependent preferences and loss aversion model are robust in this setting.

We also estimated a model that included a piecewise linear function of the probability that the home team will win the game instead of the linear-quadratic specification in Equation (4). A piecewise linear specification allows for more flexibility in the relationship between the probability that the home team wins a game and attendance since it does not force the relationship to fit a smooth quadratic curve. In these alternative models, the omitted category was games where the probability that the home team would win was less than .45; there were 2002 games in this category in the sample. In all the alternative specifications, the largest segment included games where the probability that the home team wins was greater than .75; there were only 38 games in this category in the sample. We used several alternative break points for the piecewise linear function between the two extreme categories. None of these alternative functional forms provided evidence supporting the UOH. Some of these alternative specifications provided weak support for the reference dependent preference with loss aversion model (a U-shaped stepwise relationship between the probability that the home team would win the game and attendance) and others provided support for the model without reference dependent preferences (a strictly increasing piecewise relationship). (6)

VI. CONCLUSIONS

A large empirical literature testing the UOH exists, based on empirical analysis of data at the game or match and season level. While the UOH has received considerable attention from researchers and posits a clear, testable hypothesis, it lacks a solid theoretical basis. We develop a model of consumer decision making about game attendance to motivate the UOH. The model includes reference-dependent preferences and uncertain game outcomes. In this context, the UOH emerges only from a model with reference-dependent preferences; game uncertainty alone, in the context of a standard Friedman-Savage model of attendance under uncertainty, cannot generate predictions consistent with the UOH. In addition, the UOH emerges only when the marginal utility generated by watching a win when the home team was expected to win the game exceeds the marginal utility from watching a loss when the home team is expected to lose. Under an alternative specification, based on loss aversion, when the marginal utility generated by watching an expected loss exceeds the marginal utility from watching an expected win, the UOH does not emerge from the model. This alternative, which can be motivated by prospect theory, is also consistent with consumer preferences for upsets, a key feature of sports markets.

The model incorporates reference-dependent preferences, like the model developed by Koszegi and Rabin (2006). Relatively few empirical tests of reference-dependent preference models exist. By linking a reference-dependent preference-based model to the UOH, we provide a significant new source of empirical evidence supporting the predictions of models with reference-dependent preferences. Moreover, this evidence comes from a setting where both consumer demand and a market-based proxy for the expected probability of a specific outcome can be readily observed. Economists have been testing the UOH for decades, and our survey of this literature reveals a significant amount of support for models with reference-dependent preferences. We also show that much of this evidence comes from regression models that can be interpreted as structural econometric models, further strengthening this evidence.

Mixed empirical support for the UOH in data on attendance at individual games exists in the surveyed literature. While some papers develop evidence of higher attendance at games with uncertain outcomes, others find attendance to be higher at games with more certain outcomes. Our model reconciles these contradictory results. We show that loss aversion by the marginal fan should result in higher attendance at games with certain outcomes. This result also motivates sports fans' interest in upsets, a largely ignored topic in the literature.

Our results suggest a number of interesting implications and extensions. First, the extensive theoretical and empirical literature on competitive balance in sports leagues assumes that the predictions of the UOH drive league objectives. The model of team and league behavior developed by El-Hodiri and Quirk (1971) and Fort and Quirk (1995) assumes that leagues attempt to stage games with uncertain outcomes in order to maximize attendance, fan's interest, and total profits. However, under loss aversion, the model developed here suggests that attendance and fan's interest could be higher when there is less outcome uncertainty. If this prediction holds in practice, then the widely used league models in the sports economics literature need to be reformulated to take this prediction into account.

Second, the model developed here applies to a consumer's decision to attend a single game. But team sports typically feature a regular season home schedule with between 10 and 80 home games, plus additional post-season games between the best teams in the regular season. While loss aversion may play a role in the decision to attend a single game, a model with reference-dependent preferences applied to the decision to attend one or more games over the course of a season may generate different predictions. Future research should apply this model to decisions to attend multiple games to determine the conditions under which the UOH emerge in this setting, which also closely matches the original description offered by Rottenberg (1956). Also, the model applies to a setting where the majority of consumers attending games are fans of the home team. In other settings, for example football in Europe, baseball in Japan, or Australian Rules Football in Australia, a significant number of attendees are fans of the visiting team. These consumers will have different reference points, and the probability that the home team wins a game will have a different impact on their expected utility. This could also explain differences in the empirical estimates from different sports in the literature.

Finally, this model applies to live attendance at sporting events. In addition to live attendance, mediated observation of sporting events, either on television, radio, or streamed on the web, represents an equally important form of consumer interest, and revenues, for sporting events. Consumers watching or listening to sporting events may behave differently than consumers attending a game. The costs of attending a game are larger, and the consequences of attending a game with an outcome that differs from the reference point are very different from the consequences of watching a game on television that turns out to have an outcome different from the reference point. Future research should assess how live attendance differs from mediated observation of sporting events, in the context of a model with reference-dependent preferences like this one, and how the mode of observation affects predictions about the relationship between outcome uncertainty and game observation.

ABBREVIATIONS

MLB: Major League Baseball

NFL: National Football League

UOH: Uncertainty of Outcome Hypothesis

doi: 10.1111/ecin.12061

REFERENCES

Abeler, J., A. Falk, L. Gotte, and D. Huffman. "Reference Points and Effort Provision." American Economic Review, 101(2), 2011, 470-92.

Amemiya, T. "Regression Analysis When the Dependent Variable Is Truncated Normal." Econometrica, 41, 1973, 997-1016.

Beckman, E. M., W. Cai, R. M. Esrock, and R. J. Lemke. "Explaining Game-to-Game Ticket Sales for Major League Baseball Games Over Time." Journal of Sports Economics, 13(5), 2011, 536-53.

Benz, M.-A., Brandes, L., and Franck, E. "Do Soccer Associations Really Spend on a Good Thing? Empirical Evidence on Heterogeneity in the Consumer Response to Match Uncertainty of Outcome." Contemporary Economic Policy, 27(2), 2009, 216-35.

Borland, J. "The Demand for Australian Rules Football." Economic Record, 63(3), 1987, 220-30.

Borland, J., and J. Lye. "Attendance at Australian Rules Football: A Panel Study." Applied Economics, 24(9), 1992, 1053-58.

Camerer, C., L. Babcock, G. Loewenstein, and R. Thaler. "Labor Supply of New York City Cabdrivers: One Day at a Time." Quarterly Journal of Economics, 112(2):, 1997, 407-41.

Card. D., and G. Dahl. "Family Violence and Football: The Effect of Unexpected Emotional Cues on Violent Behavior." Technical report. National Bureau of Economic Research, 2009.

--. "Family Violence and Football: The Effect of Unexpected Emotional Cues on Violent Behavior." Quarterly Journal of Economics, 126(1), 2011, 103.

Coates, D., and B. R Humphreys. "Week to Week Attendance and Competitive Balance in the National Football League." International Journal of Sport Finance, 5(4), 2010, 239-52.

--. "Game Attendance and Competitive Balance in the National Hockey League." Journal of Sports Economics, 13(4), 2012, 364-77.

Crawford, V. P., and J. Meng. "New York City Cab Drivers' Labor Supply Revisited: Reference-Dependent Preferences with Rational Expectations Targets for Hours and Income." American Economic Review, 101(5), 2011, 1912-32.

El-Hodiri, M., and J. Quirk. "An Economic Model of a Professional Sports League." Journal of Political Economy, 79(6), 1971, 1302-19.

Ericson, K. M., and A. Fuster. "Expectations as Endowments: Evidence on Reference-Dependent Preferences from Exchange and Valuation Experiments." Quarterly Journal of Economics, 126(4), 2011, 1879-907.

Falter, J., C. Perignon, and O. Vercruysse. "Impact of Overwhelming Joy on Consumer Demand." Journal of Sports Economics, 9(1), 2008, 20-.

Forrest, D., and R. Simmons. "Outcome Uncertainty and Attendance Demand in Sport: The Case of English Soccer." Journal of the Royal Statistical Society: Series D (The Statistician), 51(2), 2002, 229-41.

Forrest, D., J. Beaumont, J. Goddard, and R. Simmons. "Home Advantage and the Debate About Competitive Balance in Professional Sports Leagues." Journal of Sports Sciences, 23(4), 2005, 439-45.

Fort, R., and J. Quirk. "Cross-Subsidization, Incentives, and Outcomes in Professional Team Sports Leagues." Journal of Economic Literature, 33(3), 1995, 1265-99.

Gill, D., and V. Prowse. "A Structural Analysis of Disappointment Aversion in a Real Effort Competition." American Economic Review, 102(1), 2012, 469-503.

Kahneman, D., J. L. Knetsch, and R. H. Thaler. "Anomalies: The Endowment Effect, Loss Aversion, and Status Quo Bias." Journal of Economic Perspectives, 5(1), 1991, 193-206.

Kahneman, D., and A. Tversky. "Prospect Theory: An Analysis of Decision Under Risk." Econometrica, 47(2), 1979, 263-92.

Knowles, G., K. Sherony, and M. Haupert. "The Demand for Major League Baseball: A Test of the Uncertainty of Outcome Hypothesis.". American Economist, 36, 1992, 72-80.

Kodde, D. A. "Uncertainty and the Demand for Education." Review of Economics and Statistics, 68(3), 1986, 460-67.

Koszegi, B., and M. Rabin. "A Model of Reference-Dependent Preferences." Quarterly Journal of Economics, 121(4), 2006, 1133-65.

Kuypers, T. "Information and Efficiency: An Empirical Study of a Fixed Odds Betting Market." Applied Economics, 32(11), 2000 1353-63.

Lemke, R., M. Leonard, and K. Tlhokwane. "Estimating Attendance at Major League Baseball Games for the 2007 Season." Journal of Sports Economics, 11(3), 2010, 316.

Lintner, J. "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets." Review of Economics and Statistics, 47(1), 1965, 13-37.

Madalozzo, R., and R. Berber Villar. Brazilian Football: What Brings Fans to the Game? Journal of Sports Economics, 10(6). 2009, 639-50.

Meehan, J. W., R. A. Nelson, and T. V. Richardson. "Competitive Balance and Game Attendance in Major League Baseball." Journal of Sports Economics, 8(6), 2007, 563-80.

Neale, W. C. "The Peculiar Economics of Professional Sports." Quarterly Journal of Economics, 78(1), 1964, 1-14.

Owen, P. D., and C. R. Weatherston. "Uncertainty of Outcome and Super 12 Rugby Union Attendance." Journal of Sports Economics, 5(4), 2004a, 347-70.

--. "Uncertainty of Outcome, Player Quality and Attendance at National Provincial Championship Rugby Union Matches: An Evaluation in Light of the Competitions Review." Economic Papers: A Journal of Applied Economics and Policy, 23(4), 2004b, 301-24.

Peel, D., and D. A. Thomas. "Outcome Uncertainty and the Demand for Football: An Analysis of Match Attendances in the English Football League." Scottish Journal of Political Economy, 35(3), 1988, 242-49.

--. "The Demand for Football: Some Evidence on Outcome Uncertainty." Empirical Economics, 17(2), 1992, 323-31.

--. "Handicaps, Outcome Uncertainty and Attendance Demand." Applied Economics Letters, 4(9), 1997, 567-70.

Pope, D. G., and M. E. Schweitzer. "Is Tiger Woods Loss Averse? Persistent Bias in the Face of Experience, Competition, and High Stakes." American Economic Review, 101(1), 2011, 129-57.

Post, T., M. J. v. d. Assem, G. Baltussen, and R. H. Thaler. "Deal or No Deal? Decision Making Under Risk in a Large-Payoff Game Show." American Economic Review, 98(1), 2008, 38-71.

Rascher, D. A. "A Test of the Optimal Positive Production Network Externality in Major League Baseball," in Sports Economics: Current Research, edited by J. Fizel, E. Gustafson, and L. Hadley. Westport, CT: Praeger, 1999, 27-45.

Rascher, D. A., and J. P. Solmes. "Do Fans Want Close Contests? A Test of the Uncertainty of Outcome Hypothesis in the National Basketball Association." International Journal of Sport Finance, 2(3), 2007, 130-41.

Rottenberg, S. "The Baseball Players' Labor Market." Journal of Political Economy, 64(3), 1956, 242-58.

Sauer, R. D. "The Economics of Wagering Markets." Journal of Economic Literature, 36(4), 1998, 2021-64.

Stadtmann, G., and D. Czamitzki. "Uncertainty of Outcome Versus Reputation: Empirical Evidence for the First German Football Division." Empirical Economics, 27(1), 2002, 101-12.

Tainsky, S., and J. Winfree. "Short-Run Demand and Uncertainty of Outcome in Major League Baseball." Review of Industrial Organization, 37(3), 2010, 1-18.

Welki, A. M., and T. J. Zlatoper. "US Professional Football: The Demand for Game-Day Attendance in 1991." Managerial and Decision Economics, 15(5), 1994, 489-95.

Whitney, J. D. "Winning Games Versus Winning Championships: The Economics of Fan Interest and Team Performance." Economic Inquiry, 26(4), 1988, 703-24.

(1.) Card and Dahl (2011) show the importance of reference-dependent preferences for explaining observed incidents of family violence following NFL football games; Crawford and Meng (2011) test the predictions of a reference-dependent preference model using data on the behavior of cab drivers in New York City; Post et al. (2008) show that the behavior of contestants on a large-payoff game show make decisions consistent with the predictions of reference-dependent preference models; Pope and Schweitzer (2011) develop evidence that professional golfers are loss averse, based on putting performance. Ericson and Fuster (2011), Abeler et al. (2011), and Gill and Prowse (2012) test the predictions of reference-dependent preference models in experimental settings.

(2.) Card and Dahl (2009) contains a similar figure, although an application there is the risk of marital violence following the outcome of an NFL game.

(3.) Because ([partial derivative]([alpha] - [beta])p (1 - p))/[partial derivative]p = ([alpha] - [beta])(1 - 2p) and ([[partial derivative].sup.2]([alpha] - [beta])p (1 - p))/[partial derivative][p.sup.2] = 2([alpha] - [beta]). ([partial derivative]([alpha] - [beta]) p(1-p))/[partial derivative]p < 0 if p < 1/2 and ([partial derivative]([alpha] - [beta])p(1 - p))/[partial derivative]p [greater than or equal to] 0 if p [partial derivative] 1/2.

(4.) The full derivation can be found in the working paper version of this article http://ideas.repec.Org/p/ ris/albaec/2012_007 .html.

(5.) Note that the expected utility function could reach a maximum at 1 and still be concave. The sufficient condition for the classic UOH to hold is 0 + 2[gamma] < 0. Future research should report an estimate of this linear combination of parameters, and the estimated standard error, to establish conclusive evidence supporting the classic UOH.

(6.) The two models may perform differently because of specification error. The piecewise model forces quasilinearity onto the quadratic structural econometric model estimated above.

DENNIS COATES, BRAD R. HUMPHREYS and LI ZHOU *

* Humphreys thanks the Alberta Gaming Research Institute for financial support for this research.

Coates: Department of Economics, UMBC, Baltimore, MD 21250. Phone 410-455-2160, Fax 410-455-1054, E-mail [email protected]

Humphreys: Department of Economics, West Virginia University, Morgantown WV, 26506-6025. Phone 304-2937871, Fax 304-293-5652, E-mail brhumphreys@mail. wvu.edu

Zhou: Department of Economics, University of Alberta, Edmonton AB T6G 2H4, Canada. Phone 780-492-4133, Fax 780-492-3300, E-mail [email protected]

TABLE 1 Game Level Evidence on Expected Game Outcome and Attendance

Author(s)                           Setting

Borland (1987)                      Australia football 1950-1986
Borland and Lye (1992)              Australia football 1981-1986
Falter, Perignon, and Vercruysse    France soccer 1996-2000
  (2008)
Madalozzo and Berber Villar         Brazil soccer 2003-2006
  (2009)
Meehan, Nelson, and Richardson      MLB 2000-2002
  (2007)
Tainsky and Winfree (2010)          MLB 1996-2009
Whitney (1988)                      MLB 1970-1984
Rascher and Solmes (2007)           NBA 2001-2002
Peel and Thomas (1988)              UK soccer 1981-1982
Peel and Thomas (1997)              UK rugby 1994-1995
Welki and Zlatoper (1994)           NFL 1986-1987
Coates and Humphreys (2010)         NFL 1985-2010
Benz et al. (2009)                  Germany soccer 2001 -2004
Rascher (1999)                      MLB 1996
Owen and Weatherston (2004a)        New Zealand rugby 2000-2002
Owen and Weatherston (2004b)        New Zealand rugby 1999-2001
Coates and Humphreys (2012)         NHL 2005-2010
Stadtmann and Czarnitzki (2002)     Germany soccer 1996-1997
Forrest and Simmons (2002)          UK soccer 1997-1998
Forrest et al. (2005)               UK soccer 1997-1998
Lemke, Leonard, and Tlhokwane       MLB 2007
  (2010)
Peel and Thomas (1992)              UK soccer 1986-1987
Knowles, Sherony, and Haupert       MLB 1988
  (1992)
Beckman et al. (2011)               MLB 1985-2009

Author(s)                           Uncertainty Measure

Borland (1987)                      f (win%)
Borland and Lye (1992)              f (win%)
Falter, Perignon, and Vercruysse    f (points)
  (2008)
Madalozzo and Berber Villar         f (win%)
  (2009)
Meehan, Nelson, and Richardson      f (win%)
  (2007)
Tainsky and Winfree (2010)          f (win%)
Whitney (1988)                      f (win%)
Rascher and Solmes (2007)           f (win%)
Peel and Thomas (1988)              Betting Odds
Peel and Thomas (1997)              Betting Odds
Welki and Zlatoper (1994)           Point Spreads
Coates and Humphreys (2010)         Point Spreads
Benz et al. (2009)                  Betting Odds, f (win%)
Rascher (1999)                      Betting Odds, f (win%)
Owen and Weatherston (2004a)        Betting Odds
Owen and Weatherston (2004b)        Betting Odds
Coates and Humphreys (2012)         Betting Odds
Stadtmann and Czarnitzki (2002)     Betting Odds
Forrest and Simmons (2002)          Betting Odds
Forrest et al. (2005)               Betting Odds
Lemke, Leonard, and Tlhokwane       Betting Odds
  (2010)
Peel and Thomas (1992)              Betting Odds
Knowles, Sherony, and Haupert       Betting Odds
  (1992)
Beckman et al. (2011)               Betting Odds

Author(s)                           Results                    Support

Borland (1987)                      [theta] = 0                --
Borland and Lye (1992)              [theta] > 0                H2
Falter, Perignon, and Vercruysse    [theta] = 0                --
  (2008)
Madalozzo and Berber Villar         [theta] = 0                --
  (2009)
Meehan, Nelson, and Richardson      [theta] > 0                H2
  (2007)
Tainsky and Winfree (2010)          [theta] = 0                --
Whitney (1988)                      [theta] > 0                H2
Rascher and Solmes (2007)           [theta] > 0, [gamma] < 0   H3
Peel and Thomas (1988)              [theta] > 0                H2
Peel and Thomas (1997)              [theta] > 0                H2
Welki and Zlatoper (1994)           [theta] > 0, [gamma] = 0   H2
Coates and Humphreys (2010)         [theta] < 0, [gamma] > 0   H1a
Benz et al. (2009)                  [theta] > 0, [gamma] < 0   H3
Rascher (1999)                      [theta] > 0, [gamma] < 0   H3
Owen and Weatherston (2004a)        [theta] = 0, [gamma] = 0   --
Owen and Weatherston (2004b)        [theta] = 0, [gamma] = 0   --
Coates and Humphreys (2012)         [theta] > 0, [gamma] = 0   H2
Stadtmann and Czarnitzki (2002)     [theta] < 0, [gamma] > 0   H1a
Forrest and Simmons (2002)          [theta] < 0, [gamma] > 0   H1a
Forrest et al. (2005)               [theta] < 0, [gamma] > 0   H1a
Lemke, Leonard, and Tlhokwane       [theta] < 0, [gamma] > 0   H1a
  (2010)
Peel and Thomas (1992)              [theta] < 0, [gamma] > 0   H1a
Knowles, Sherony, and Haupert       [theta] > 0, [gamma] < 0   H3
  (1992)
Beckman et al. (2011)               [theta] < 0, [gamma] > 0   H1a

TABLE 2 Descriptive Statistics

Variable                              M        SD

Attendance                          31,442   10,937
Home team win probability            0.54     0.08
Home team win probability squared    0.30     0.09
Visiting team winning percent        0.50     0.10
Home team winning percent            0.50     0.11
Home team runs scored                4.65     0.70
Visiting team runs scored            4.65     0.71
Home team runs allowed               4.66     0.75
Visiting team runs allowed           4.66     0.74
Observations                        13,298

TABLE 3 Censored Attendance Regression Results

Dependent Variable: log(Game Attendance)

                            Model I

                            Coefficient   p Value

Home win probability          -1.093       0.002
Home win probability (2)       1.081       0.001
Home avg. runs scored          --            --
Visitor avg. runs scored       --            --
Home avg. runs allowed         --            --
Visitor avg. runs allowed      --            --
Home team winning %            --            --
Visiting team winning %        --            --
Month, day indicators          Yes
Team, season, indicators       Yes
Team-season interactions       Yes
Observations                  13,298

Dependent Variable: log(Game Attendance)

                            Model II

                            Coefficient   p Value

Home win probability          -1.134       0.002
Home win probability (2)       1.139      <0.001
Home avg. runs scored          0.017       0.027
Visitor avg. runs scored       0.023      <0.001
Home avg. runs allowed        -0.037      <0.001
Visitor avg. runs allowed     -0.009      <0.001
Home team winning %            --          --
Visiting team winning %        --          --
Month, day indicators          Yes
Team, season, indicators       Yes
Team-season interactions       Yes
Observations                  13,298

Dependent Variable: log(Game Attendance)

                            Model III

                            Coefficient   p Value

Home win probability          -1.104       0.004
Home win probability (2)       1.090       0.002
Home avg. runs scored          0.009       0.424
Visitor avg. runs scored       0.013       0.026
Home avg. runs allowed        -0.024       0.016
Visitor avg. runs allowed      0.016       0.109
Home team winning %            0.556       0.001
Visiting team winning %        0.235      <0.001
Month, day indicators          Yes
Team, season, indicators       Yes
Team-season interactions       Yes
Observations                  11,961