Microeconometric essays in health, sports and education economics Bojke, Christopher

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1 University of Groningen Microeconometric essays in health, sports and education economics Bojke, Christopher IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2010 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Bojke, C. (2010). Microeconometric essays in health, sports and education economics. Delft: Eburon. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:

2 3 3.1 The Economics of the Design of Post-Season Play- Offs and the Impact on Regular Season Game Attendance. Introduction Popular team sports such as association football (soccer), baseball, basketball, rugby and American football are contested in league and cup competitions as a means of giving matches between two teams some contextual meaning; a match can be seen as a contest which will go some way to resolving whether a team will achieve some desirable end-of-season goal such as winning a cup, a league championship, gaining promotion or avoiding relegation. Identifying and understanding the relationship between a sporting competition s characteristics and the demand for that product in the form of attendance is an important component in the design of league and cup competition formats. One such common policy-amenable element added to league structures is the addition of a post-season play-off system, whereby the allocation of end of season outcomes such as winning the overall championship or promotion/relegation to different divisions are finally determined. Such play-off systems are common and feature in many diverse sports, from determining champions in North American sports such as Major League Baseball and American Football to partly determining promotion and relegation issues in open league formats such as the European soccer leagues. One such motivation for the presence of play-offs is that they are argued to influence regular season attendance by increasing the proportion of regular season games for which a team is still in contention for the end of season outcome. However although a common feature of many professional sporting competitions, there is little consistency in the design and size of play-off structure both within and across different sports, indicating that the size and nature of the impact of play-off designs on attendance at regular season matches is broadly unknown. Although there exists an extensive literature on the determinants of demand within support in general and on the impact of league design in particular, limitations in the statistical techniques and the lack of a model which relates play-off design to demand has not reduced this uncertainty (Borland and Macdonald, 2003; Kuypers, 1995; Noll, 2003; Cairns, 1987). 63

3 This chapter therefore outlines an approach that may be used to address this important research gap and is illustrated with an empirical investigation of the incremental impact of a promotional play-off system on the attendance at regular season league matches in the English professional soccer league immediately below the top-tier Premiership division. The process of conducting this research is conceptually simple and falls into three distinct steps: Identification of the theoretical means by which play-off design may influence regular season games; Estimation of the relevant parameters using empirical data; and finally Prediction of attendances under other hypothetical play-off designs to identify the effects of different designs. Although conceptually simple, all these steps have proved difficult in practice and methods for conducting each step are thus covered to some degree within this text, which is structured as follows: first I outline a simple model of the determinants of the demand for attendance which identifies the theoretical framework by which the introduction of a post-season play-off system may influence demand in regular season games. Secondly I describe the method by which the play-off relevant variables are derived before describing the data and discussing the statistical issues which arise in estimating the parameters of the model given skewed, heteroscedastic data generated by heterogeneous teams. I then present the results of the statistical estimation and discuss the policy implications. 3.2 The Demand for Attendance and the Theoretical Impact of Play-Off Designs The framework in which to assess this research question is provided by a conceptual microeconomic model of demand which argues that the demand for attendance,, of a match t between opponents i and j is a function of the characteristics of that match such as: the teams/individuals competing, the cost of attendance, whether the match is televised and, importantly, the context or significance of the match in 64

4 resolving who gets what end of season outcome at the end of the overall competition e.g. promotion, relegation or the championship itself. Where: the potential significance of the match in resolving end of season outcomes, etc. at the time of match t time of match t. = home team i = (,, ) characteristics such as the quality of the home team, whether the team is still in contention for a desirable end of season outcome, = an analogous set of team characteristics applicable to away team j at the = a set of characteristics applicable to both teams such as ticket price, whether the match is televised live, the match uncertainty, etc. (3.1) As attendance at matches tend to be dominated by home-team supporters one would a priori hypothesize that the factors contained within those contained in. have a larger influence than The nature of the demand for football is characterised by a high and notable degree of brand loyalty: supporters affiliated to one team are unlikely to derive the same degree of utility from attending matches not involving their team, even if the other characteristics of the game are relatively attractive. Furthermore such loyalty appears highly heterogeneous, with clubs from prima facia similar areas attracting different loyalty. Borland and Macdonald (op cit) assert that this identification with a team in a sporting contest is in fact the major constituent of demand and other quality measures are secondary in nature. From an analysis perspective, strong team-specific loyalties imply a panel data approach, in which team specific fixed effects account for underlying levels of demand given the differences in observable and variable measures of quality of individual games. It may also imply the use of rational addiction models if individual level data were available. However, even if individual football clubs can be regarded as monopoly suppliers of the product and supporters do not regard matches played on the same day involving different teams as substitutes for a match involving their own team, marginal consumers will always have the null 65

5 option of choosing not to attend. Thus, for a given team, variations in the characteristics of individual games imply variations in quality or cost and will be reflected in variations in attendance as marginal consumers decide to attend or not attend particular games. The notion of a strong component of utility being attached to affiliation is supported by the observed phenomenon of season ticket holders; consumers who purchase tickets for all their team s home fixtures in a block booking prior to a new season starting. For these consumers it is possible that utility derived from team affiliation is so strong that anticipated variations in the quality of the games over the season are not anticipated to alter the decision to attend. League organisers, and indeed individual clubs, may thus affect demand via affecting certain characteristics of the product, such as promotional significance, but the extent to which they may practically affect attendance will be limited by the volume of marginal consumers. The scope of the empirical literature has largely been to identify and quantify these demand and cost factors, and many have been specifically geared towards testing the Uncertainty of Outcome Hypothesis (UOH) in the demand for football which has particular relevance for this paper. Occasionally, and also relevant, papers have examined the marginal effects of league structure changes Uncertainty of Outcome Hypothesis The Uncertainty of Outcome Hypothesis stems from the Louis-Schmeling paradox which states that a sporting competition is more profitable than a sporting monopoly, because although supporters are attracted to winning teams, the certainty of victory kills interest. The paradox is named after the Louis versus Schmeling boxing re- match of 1938 where, in a match with high political tension, black American Louis defeated German Schmeling in just 2 minutes 8 seconds in boxing, the length of the competition as well as the quality is a function of the evenness between contestants. In the context of football, three broad uncertainties have been identified: uncertainty of individual match outcome; uncertainty of current season outcome; and uncertainty of long run seasonal outcomes. Long run seasonal outcomes refer to the extent to 66

6 which championships are dominated by certain teams and the long run distribution of competitive balance. For example, the declining aggregate attendances in Scottish top division league football is anecdotally attributed to the increased long run domination of the championship by the Glasgow giants Rangers and Celtic (Jennett, 1984). However such long run uncertainty measures are not directly relevant to this research and are not discussed further. Uncertainty of seasonal outcome is the most relevant for discussing play-off designs, but some detailed coverage of match uncertainty is also desirable: not only will this enter as an explanatory variable in the econometric specification, but observing the evolution of the definition of this measure over time is useful for the discussion involving a new measure of seasonal outcome uncertainty Match Uncertainty Simple and mostly early measures of match uncertainty were essentially ad hoc and arbitrary. With no obvious direct measure of the uncertainty regarding a match s outcome or an obvious available proxy measure, researchers tended to identify those observable variables that they a priori expected to influence match outcome uncertainty. Unsurprisingly, the most common example was a simple measure constructed from the difference in league ranking: with higher teams expected to beat lower ranked teams, uncertainty was assumed decreasing with larger differences (Baimbridge, Cameron and Dawson, 1996; Borland and Lye, 1992; Garcia and Rodriguez, 2002). Other attempts are broadly similar and include difference in league points obtained (Wilson and Sim, 1995); difference in average goals scored (Falter and Perignon, 2000); and runs of recent form (Price and Sen, 2003). One common problem with these measures is that they may use the same variables that are used to construct proxy measures of other unobserved explanatory variables, such as league rankings measuring the quality of the team, thus making it difficult to isolate the effects of quality and uncertainty. However, a potentially bigger problem is that it is difficult for any ad-hoc measure to not only fully capture the full myriad of factors affecting the uncertainty of outcome, but also to specify the relationships between these factors. For example, difference in league positions will not fully capture other perceived significant factors such as recent runs of form; injuries or suspensions to key players; change of management; etc. Furthermore, even if these 67

7 additional variables were observable and could be included in an econometric specification, it is not clear how they should enter the specification. One example of an attempt to link factors together is provided by Whitney (1988), who specifies a linear additive relationship between a team s current game winning percentage, last season s performance and current short term performance (measured by the winning percentages of the previous two months). The relationship is estimated using OLS on available data and an expected winning percentage is predicted and used as a proxy measure of uncertainty of future match outcomes. Note that the measure assumes fans use adaptive expectations in updating their perspectives of outcomes and ignores all elements other than measures of long term and shorter term form. Efforts may have continued further down the path of creating ever more elaborate and ad-hoc measures had it not been for the emergence of an alternative proxy measure, which not only may be assumed to capture all relevant information but also compresses it to a meaningful measurement of the uncertainty of outcomes: posted fixed bookmaker betting odds. Attributed to Peel and Thomas (1988), betting odds, where available, have since become the dominant form of measuring match uncertainty. It is argued that, less the bookmaker s mark-up, these odds represent the direct market valuation of the likelihood of each possible match outcome. If markets are assumed to be efficient and use all available information in a rational manner, then betting odds provide an unbiased ex-ante measurement of the uncertainty of each outcome without having to construct the mechanism by which these expectations are formed and obtain measurements of all the identified variables. The existence of a market determined measure is naturally quite appealing to economists and the increasing availability of betting odds has made this the generally preferred measure of match uncertainty. However, criticisms of this method remain and have generally focussed on two areas: whether it is reasonable to assume betting markets are efficient and the form by which the betting odds enter the econometric specification. As regards the efficiency of the market, there is mixed, though mostly negative, evidence on whether betting odds represent unbiased predictions of the true odds of certain match outcomes. Early research found little evidence of bias (Peel and Thomas 1997; Pope and Peel 1989) whilst others, mostly more recent studies, find evidence of various forms of systematic deviations from efficiency (Peel and Thomas 68

8 1997; Pope and Peel 1989; Goddard and Asimakopoulos 2004; Cain, Law, and Peel 2000; Kuypers 2000). Notably, Forrest and Simmons (2002) and Peel and Thomas (1992) not only provide evidence of systematic deviations, noting that commercial odds setters are primarily engaged in setting odds to maximise profit, but usefully propose a means of correcting for the bias they detect. However, if the fixed posted odds are unbiased representations of biased consumer expectations of the probability of outcomes, then a case may be made for including the posted uncorrected odds in the specification. As regards the econometric specification of the betting odds, Peel and Thomas in their initial research simply included the odds of a home win. As pointed out by Forrest and Simmons, this measures the likelihood of a home win and ignores the split of the remaining probability between the other two outcomes: a draw or an away win. Incorporating this valid criticism, Peel and Thomas (1996) used a Theil measure of uncertainty in their further research. THEIL =.log (3.2) Where = reported betting probabilities of home win, draw and away win respectively and = is a scale factor accounting for bookmakers mark-up. From this measure, maximum uncertainty is generated when all outcomes have the same probability, in this case the (re-scaled) probability of a home win, a draw, and an away win all equalling one third. Minimum uncertainty occurs when one outcome has a probability of one and the other outcomes, by definition, will have probabilities of zero. Thus uncertainty is measured by deviations away from all outcomes having a probability of a third. In practice Peel and Thomas discovered that this Theil measure was essentially, a quadratic function of the odds of a home win. and left their empirical specification as home odds and home odds squared for ease of interpretation. However, the mathematical constraints and limited range of odds over which observations were made has left analysts unable to make the fine distinction 69

9 between genuine match uncertainty and a diminishing marginal effect of increasing odds of a home win. Results have been mixed. In football, Peel and Thomas (1992) and Peel and Thomas, (1996) find a counter-intuitive U -shaped relationship, attendances go up when home teams are overwhelming favourites. Whereas Forrest and Simmons (2002) using ratios of bias-corrected odds of a home win against away win find evidence that fans respond negatively to an increasing likelihood of a home win. In other sports Carmichael, Millington, and Simmons (1999) with rugby and Knowles, Sherony, and Haupert (1992) with baseball find a diminishing effect of the odds of a home win, maximising at approximately 0.6. Using points-spread, another form of betting data, Welki and Zlatoper (1999) find that in US football there is statistically significant evidence indicating fans prefer closer games. Overall, although mixed, the evidence does suggest an element of the Louis- Schmelling paradox exists in sports (at least in sports other than football): home fans do like their team to be favourites, fairly strong favourites, but not overwhelmingly so. Regardless of the empirical results, there appears to a broad consensus amongst economists of a preference for using betting odds as a means of measuring fan expectation of match uncertainty over a serious of ad-hoc proxy measures (although concerns do remain about specification and assumptions about whether they are 1 unbiased or not. ) Notably the literature on seasonal uncertainty has not yet established an equivalent measure which has obtained anything near the same degree of acceptance Seasonal Uncertainty Outcome of season or seasonal uncertainty shares several common elements with match uncertainty and could potentially be treated in the same manner. For example, in match, uncertainty the outcomes were defined as a home win, draw or away win; 1 And this point was re-emphasised by David Forrest at the recent first European workshop on sports economics at the University of Groningen, Netherlands

10 with seasonal uncertainty the outcomes are typically defined as some end of season goal such as winning the championship, obtaining promotion, avoiding relegation or gaining qualification to the lucrative European competitions. Similarly the uncertainty surrounding the outcome at any point during the season may be defined as the probability of obtaining the stated goal at that particular point during that season. Outcomes will be certain when the probability converges to 1 or 0 and for a league of size n teams and if m teams can obtain the end-of-season goal (e.g. three teams being promoted), Theil measured uncertainty will be maximised across the league where each team has a probability of achieving that goal and for individual teams, where the probability of obtaining that goal themselves is 0.5. Note that over the course of a season, this probability is also liable to vary from match to match for teams and its value and the extent to which it may vary may be influenced by league design such as the imposition of a play-off structure. The Louis-Schmeling paradox could then be potentially tested by examining the effect on attendances of variations away from this probability. However, as Cairns, Jennett, and Sloane (1986) point out, empirical attention has focussed not on this uncertainty per se, but more on the effect of the probability of success. The hypothesis thus becomes that fans are motivated by an increased probability of obtaining an end-of-season goal rather than more uncertainty surrounding the likelihood of obtaining it. This provides an intuitive motivation for why play-offs may increase match attendance. Relative to strict meritocratic means of achieving end-of-season objectives, such as the three highest league finishers gaining automatic promotion, play-off systems typically increase the number of games for which the probability of obtaining the desired end-of-season outcome is non-zero. Note that if probability of success is the key measure of interest, then we could invoke Peel and Thomas s insight on match uncertainty and explore the use of betting data on seasonal outcomes in an analogous manner. Further reflection indicates that there must be something more to this concept than simply increased non-zero probabilities of achieving end-of-season outcomes. Otherwise, by continuing this line of thought to its logical conclusion, it would appear that an optimal play-off design would be one that simply maximised the number of play-off qualifying spots i.e. every team in that league/division qualifying for the 71

11 play-offs. At the time of writing this is almost the case in the US, where eight out of ten Major League Soccer (MLS) teams qualify for the end of season championship play-offs. A situation that has prompted US national coach Bruce Arena to comment that most of the MLS regular season games mean nothing (Gardner 2005). Thus the measure of seasonal uncertainty appears inherently more complex than that of match uncertainty and perhaps this is one explanation as to why the literature has struggled to find an accepted single measure Current Measures of Seasonal Uncertainty In looking at seasonal uncertainty regarding the Scottish league, Jennett (1984) provides the basis for much of the current research, introducing the term significance, a subtle deviation from uncertainty in outcome. Jennett conceptually defines significance as the influence that a game has in deciding the end-of-season outcomes and, importantly, providing a much discussed original measure of that significance. In defining this measure Jennett notes (pg 179) Games have relatively little significance for the overall outcome of the championship in the first weeks of the season, but have considerably more in the season s closing stages for clubs located at the top of the table. Thus Jennett identifies a time element as well as a league position dimension to the significance. Again, this seems intuitively correct and again implies that there is something more to significance than just a non-zero probability of obtaining an end- of-season outcome. However, Jennett does not provide a formal explanation of why time should have this impact on the notional measure of significance, though he notes (pg 181) As the season progresses more clubs drop out of contention, but the games of those remaining take an even greater significance. Thus hinting that influence of time may be indirect and that significance may be a function of how many teams are left competing. 72

12 Having described the concept of significance and identified some factors that influence it (though not formally deriving how they influence it), Jennett constructs his measure of championship significance by initially assuming that the number of points needed to win the championship is known. He then calculates how many games are required to be won in order to achieve that points total. If there are too few games left for a team to mathematically achieve that number of points then the significance of all future games in that season for that club are set to zero. Otherwise the measure of significance is defined as the reciprocal of the number of games needed to be won in order to reach that points total. Thus if a team is required to win another 20 games and this is mathematically possible, the significance of the next game is 0.05, whereas if the team requires just 2 games, the significance is 0.5. The assumption that the points needed to win the championship is known is crucial. Firstly, Jennett avoids having to model the performance and league positions of all other teams at given points during the season, as the important final positions of all other teams are effectively summarised in the points needed to win the championship measure. Furthermore, this assumption also means the definition coincides with Jennett s observation about significance increasing over time if the objective is still mathematically possible: assuming teams will not be deducted points, any game that is won will reduce the number of games needed to achieve the final points total and hence increase the significance of the next game as measured by the reciprocal. However this assumption has been criticised in the literature for assuming that consumers are using information they do not have at that point in time (Cairns, Jennett, and Sloane 1986; Cairns 1990; Peel and Thomas 1988). The assumption may not be untenable if one considers rational expectations of supporters and that the number of points required to win the championship remains fairly stable over time. However, even if one is able to justify an a priori known points total, the measure still has properties that seem counterintuitive. Consider the following examples where in both cases a team requires just two more wins to secure the championship 1) 2) the team has just two games remaining; or the team has ten games remaining. 73

13 In case 1, the next game is critical, in that if it is not won, the championship cannot be won. In case 2, the next game is clearly not so critical, if the game is lost then there will be another nine opportunities to gain the required two wins. Yet Jennett s measure would rate each of these matches as equally significant with a score of 0.5, though intuitively it feels as if the next game in case 1 would appear to be of far greater significance. Also note the respective probabilities of that team winning the championship: given games of comparable match uncertainty, a team will have a higher probability of winning the championship in case 2 than in case 1. This example unusually suggests a higher probability being inversely related to significance: again implying a more complex relationship between significance and the probability of obtaining the seasonal objective. Despite the objections, Jennett s method remains the most common starting point in the current literature and indeed is also occasionally used in its original form (Borland and Lye 1992; Wilson and Sim 1995). Dobson, Goddard, and Wilson (2001) define an adapted Jennett measure. The major contribution Dobson et al make, is that they relax the assumption that the winning total is known a priori and allow adaptive expectations of the winning total based on the current leaders actual accrued points and the expected amount of points the leader will gain (based on the historical average point per game of previous league winners, 1.68, and the remaining number of 2 games.) In their paper, the measure of match championship significance ranges between 1 and 2 (though it is not mathematically bound at the upper level to be no greater than 2). For teams that remain in contention to finish at the top of the division, the championship significance increases towards two as the season nears its conclusion and the importance of each individual match increases; for those that drop the time if it is mathematically possible for that team to achieve the expected end-ofout of contention, the value becomes one. A team is assumed to be in contention at season divisional points winning points total. So if in an analysis of rugby league attendance, is the expected winning total at 2 In rugby 2 points rather than 3 are awarded for a win. 74

14 the time of fixture i; is the number of games played by the home team by fixture i; and n is the total number of games a team will play in a season and gained of the team at the time of fixture i. Then is the points if + 2 if + 2 < then then = 1 = (3.3) Where is the measure of significance. This measure removes the fixed known winning total points assumption and it also marks a shift of emphasis from, what contribution the next game could make to a required points total, to measurements of distance from the seasonal outcome at that point in time. Note that, beyond determining whether the significance is 1 or not, the element of games remaining in the measure occurs indirectly via a diminishing distance between, an expected end of season final winning points total and current points total. In a similar approach in considering an adaptive winning number of total points, Cairns (1987), in looking at Scottish league football, identifies two main factors: points won relative to other teams and the number of matches played. He also observes that championship contention is largely relevant to the second half of the season. Acknowledging that any definition is unavoidably arbitrary, Cairns defines a match as significant for a team if it can win the division by winning 80% of their remaining games and the leading team(s) only win 50% (so only 3 points behind with 5 games left; 6 points if 10 games left; etc.) and then only if the match occurs in the second half of the season (which is how the games remaining element is incorporated). Kuypers (1997) suggests three potential measures for championship significance in football, all of which are a function of games left to be played and the number of points behind the leader. Again the basic idea being that the fewer games left and the fewer points behind the leader the more significant is the game. 75

15 () ((3. )). ( + 2. ) (3.4) Where PB = points behind leader and GL = games left. Where a team is the leader the points behind is set to 1 and if a team cannot mathematically win the championship the measure is 0. A curious (but uncommented on) artefact of all of Kuyper s measures is that significance is decreasing in this measure (smaller numbers indicate higher significance and maximum significance is given by a score of 1), except that absolute minimum match significance is measured by a score of 0. The second measure of significance is used by Garcia and Rodriguez (2002) in their analysis of Spanish league football. Owen and Weatherston (2004) in super 12 rugby union attendance, use points behind th the 4 th qualifying place (points above 4 measure of significance. They ignore the issue of games left but expand the measure of significance from championship winning to qualification for end of season tournament. The issue of how to deal with points above the winning threshold is dealt with in a less arbitrary fashion than by Kuypers. place are measured as negative) as the Other measures include: Baimbridge, Cameron, and Dawson (1996) who use a dummy variable approach, indicating whether both teams are in top 4 positions, and hence in direct competition for the championship. He also constructs an analogous measure for relegation and dummy variables to capture whether championship has been won by that team (glory effect) or already relegated. Carmichael, Millington, and Simmons (1999) in analysing attendance at rugby league matches use pre-season betting odds to capture probability of a team winning the championship. Note the return of significance as being measured by probabilities, though in this case whilst the actual probabilities of obtaining the end of season outcome will change over time, these measures remain fixed for the entire season. In an international football tournament, Baimbridge (1997) uses a trend variable (game number) to capture proximity to winning championship. 76

16 Avoiding a measure of significance altogether, Szymanski (2001) uses a natural experiment using FA Cup and leagues fixtures between the same two teams. Arguing that since the FA Cup has become less competitive over time, importance of seasonal outcome significance should be reflected in the league fixtures becoming relatively more attractive over time. This is empirically verified though it may be the case that these results are driven by possibility that winning the FA Cup has become devalued over time. In aggregate data studies: Borland (1987) in Australian rules football takes measures of dispersion of points (between first and last team) in division at four points during the season. Double weighting is given to the final observations as a given amount of uncertainty of outcome becomes more significant the closer the end of the season, when there is less time left to alter the gap between any two teams. Borland tries a second measure using only those teams still in contention, defined as within two games of the top five. Finally, Burkitt and Cameron (1992) using season averaged attendance data on rugby league teams define promotional significance as being captured by a dummy variable indicating whether the team finished in a promotion challenging position (either promoted or within two spaces). The multitude of measures used indicate that, unlike match uncertainty, the issue of measuring seasonal outcome significance is still largely unresolved. However there are common elements within many of the post-jennett attempts. Firstly a team s distance from the desired outcome, whether measured by distance from current leaders or distance from expected final points total required (based on current leader points plus expected points from remaining games or an assumed known winning total). It is arguable that distance from the target (as measured) is the key component in the probability of achieving that end of season outcome. A further common element is the inclusion of a time component increasing significance as the end of season draws closer, conditional on the team still being able to mathematically achieve the objective. The time element is commonly used to weight the other measure of significance, with larger weights given to games near the end of the season. However, notice that there is no common means of including this time element with Cairns arbitrarily choosing only second half of the season games, Borland doubling the weighting of latter season games and Kuypers offering three 77

17 variations. The variation perhaps indicative of the lack of a clear theoretical rationale for inclusion of time in a significance measure. There are also common omissions amongst measures. In computing the significance of a match for a team, virtually all of the measures systematically ignore whether the match is against a team effectively competing for the same seasonal outcome. Such games are colloquially known as six-pointers in football as they not only provide an opportunity to gain three points for a team s cause, but also the opportunity to deny a rival team three points. All other things being equal, such games will carry more significance than a game against an opponent not in contention for the same seasonal outcome. Furthermore, the term is in such common usage amongst fans, it is hard to imagine it does not represent something considered important to football fans. The exception within the literature is Baimbridge, Cameron, and Dawson (1996) who arbitrarily decided that teams within the top 4 are challenging for the championship. Measures also tend to calculate the points, or the expected points of the current leading team, but (at least directly) ignore the distribution of points of other teams. This may not suffice when there is more than one team which can achieve the seasonal outcome (e.g. promotion). Furthermore it is unable to distinguish between situations where there are many teams in contention and when there is only one or two. The probability of obtaining a seasonal outcome is decreasing in number of effective competitors and thus significance is also likely affected in some similar manner. Perhaps the time variables indirectly capture this element as when the season is drawing to a close there are likely to be fewer and fewer teams close to the leader. Current measures of expected end of season points are adaptive, not rational, and do not take in to account the degree of expected difficulty of the remaining schedules of both the current leaders and teams playing in the next match. This problem also occurs where current points are used, in that a current point deficit may be expected to vary if there are systematic differences in the remaining schedules. For example if the top two teams are due to meet, it is not mathematically possible for them both to average over half the available points for that game. 78

18 In conclusion, there has yet to emerge a clear commonly accepted definition of end of season outcome match significance, indeed significance may not be captured by a single variable. There are, however, a number of similar looking measures, which not only systematically incorporate a number of common elements, but also systematically omit others and have a number of objectionable qualities. Even with a number of common elements, there is no consensus on how they should be included. The clearest examples of which are the numerous (and inconsistent) methods of incorporating the time element. It is thus arguable that the state of measurement in match significance is similar to that which was in the measurement of match uncertainty prior to Peel and Thomas s suggestion of using betting odds. This lack of consensus hinders any analysis of the effects of league structure A New Measure of Seasonal Uncertainty, Match Significance and the Relationship with Competition Design A Theoretical Construct This research proposes a new measure of seasonal uncertainty/match significance which is more akin to Jennet s original proposal but follows a similar route in the way that match uncertainty has been resolved a measurement of the probability of achieving the end of season outcome and/or the influence a game may have via betting odds. However, and unlike match uncertainty, the seasonal uncertainty measure may not be collapsible to a single dimension. Based on the preceding discussion I propose three potential constituent parts: 1. The probability of obtaining the outcome (promotion in this case) for team i at match t, 2. A simple dummy variable indicating whether this probability is non-zero or not, 79

19 3. And a significance variable, in this case defined as the difference in the probabilities of promotion given a win for team i in the match t would make relative to a defeat, In principle these measures avoid the ad hoc nature of the previous attempts and the perfect foresight assumption of the original Jennett measure. In addition time does not feature directly but indirectly : one might envisage that over the course of time as teams drop out of promotion contention then the probabilities rise for the other teams and indeed the significance of matches increases. However, identification (and measurement) of these variables is insufficient to estimate the impact of different play-off designs, in addition one must identify the relationship between these variables and play-off design. In order to do this it is useful to consider the probability of promotion as the product of two different probabilities: the probability of team i finishing in league position m, () and the probability of a team finishing in position m being promoted, ( ), where the lack of subscripts on this latter term indicate that it is constant across all teams and invariant to the stage of the season. Thus, in terms of our demand variables: = () (prom ) 1 if = 0 if > 0 = 0 = ( wins ) ( loses ) (3.5) Post-season play-off systems may enter the demand function by the potential impact they have on the match significance and/or the probability of obtaining the end-ofseason outcome. This may be illustrated by the example of the English soccer leagues where prior to the introduction of play-off systems, the second highest division operated a strictly automatic promotion scheme whereby the teams which finished in the top three positions at the end of the season were automatically promoted to the higher division. In contrast, the current play-off design (as of 2010) is one in which the top two highest teams automatically get promoted to the higher division and the 80

20 following four teams (positions three through six) play in a cup-style knock-out competition in which the winner joins the two automatically promoted teams. Given those definitions, imagine a hypothetical two-thirds played season where a mid-table team no longer has any reasonable chance of obtaining third position but a distinct possibility of obtaining sixth or slightly higher, under the automatic promotion regime, this team has a zero probability of obtaining promotion, whereas under the stated play-off system, there would be a non-zero probability thus illustrating the potential impact of a post-season play-off function on the determinants of demand during regular season games. In order to assess the impact of play-off design on the distribution of the three variables across matches it is useful to start with a simple fact: at time t the sum across all i teams of the probabilities of the individual teams being promoted is necessarily the number of teams that will be promoted, i.e (3.6) And for any team, the maximum is obviously 1. Thus the probability of promotion, at any time point, is distributed across all teams and sums to 3. It may be the case that three teams have a probability of 1 and all others 0, or at the other extreme all teams have a probability of 3/24 = Now unless a play-off system changes, or potentially changes, the number of teams being promoted from the division, then this fact will still remain. Different play-off designs do not create additional probability they simply redistribute it! If we can assume that play-off design does not influence the final league positions of teams i.e. teams always try and finish as high up the league table as possible, then we may isolate the impact of play-off design on the probability of promotion via its impact on (prom ) which is then filtered through to other probability variables, i.e. in the automatic promotion scheme it would be: 81

21 1 (prom ) = 0 if = 1,2,3 otherwise (3.7) Whereas in the current play-off scheme it would appear to be more like: 1 (prom ) = In the automatic scheme these probabilities are known with certainty, whereas in the current play-off scheme they are estimated with some uncertainty. The assumption of equal probabilities of promotion from each of the four qualifying positions is supported by results from the second tier of English soccer but may not apply in other circumstances and so further research in this area may be required. Nevertheless equation 3.7 and equation 3.8 identify the potential impact play-off systems may have on attendance on regular league matches and can be used to evaluate claims of the impact of play-off design on attendance. promotion having finished fourth, fifth or sixth (each increasing by 0.25) have been created by the probability of promotion having finished third decreasing by an amount that is redistributed to the other positions i.e and even if these amounts are perhaps uncertain the balancing mechanism is not. if = 1,2 if = 3,4,5,6 otherwise (3.8) As can be seen the increased probabilities of Thus different play-off designs make changes to the probability of promotion via the second part of the probability equation. And in terms of the actual difference they make to individual games, these conditional probabilities are weighted by the probability of a team finishing in that position, this will become very clear when we look at the empirical data. This leads to the most obvious and stated impact of the play-off system i.e. that it creates more games where there is some non-zero probability. If auto stands for those results achieved under the automatic promotion scheme and po stands for that achieved under the play-off scheme (as described), then > (3.9) 82

22 This is the result that drives the conventional wisdom regarding the impact of play- offs it creates more games with some significance and since that is thought to be a positive driver of demand, it increases attendance. one that is thought to maximise attendance? This leads to the logical conundrum of why then does a play-off system which maximises nzp, a system which makes all games between teams have some non-zero probability of promotion, not be Potentially the answer to this may lie with what happens to the third measure of seasonal uncertainty the significance of individual matches in determining the promotional outcomes of the competing teams. We can analyse this by defining in an analogous fashion to i.e.: = [ (). (prom )] (3.10) The difference in the probabilities of promotion a match can make operates through the difference an individual match may make in the probabilities of achieving a league position () league/play-off design. weighted by (prom ) which is a function of the Thus although () is assumed independent of the play- off design, as (prom ) is a function of play-off design, then so is. It should be noted that although constant identities, 3 (prom ) [ (prom ) ] (). is not. and () 0 are This is most obviously seen if the league were organised such that every team qualifies for an after season play-off system where each team has an equal chance of promotion i.e. (prom ) = 3. Then = 3 = 3 () () = 0 (3.11) 83

23 I.e. if it doesn t matter where you finish in the league as regards the probability of promotion, then no game which influences final league position will have any significance. This is the situation Bruce Arena alludes to when he described the US MLS system in negative terms. For any other distribution of (prom ) this relationship will not hold unless () = 0 (3.12) Thus under the limiting circumstances that every single team qualifies for an end-of- season play-off and/or there is no possibility of any single game ever changing league positions, then the match significance of every single game is zero. Under an automatic promotion scheme the first limiting circumstance is ruled out, though the highly unlikely second condition may still theoretically occur. i.e. there is no possibility of a team changing final league positions. If a play-off system can avoid either of these two limiting cases, then as with promotion probability, match significance is a fixed stock which is redistributed across games. This is perhaps less intuitive than with promotion probability, but can be partly motivated by the following example. The automatic promotion scheme awards no significance to matches that only affect the probability of say finishing second or third i.e. if a team is guaranteed second or third spot, then any remaining matches have no significance they will be promoted with a certainty of 1 in each case. Similarly, matches which influence the probability of a team finishing sixth or seventh only has no significance they will have no probability of being promoted. On the other hand, matches which heavily influence the probability of finishing third or fourth have quite a lot of significance. With the current English play-off system, matches which influence third or fourth spot only now have no significance (given the assumption that the conditional probabilities of promotion from finishing in that position are the same). However matches that influence the probability of finishing second relative to third and sixth relative to seventh now have some significance that they did not have in the original automatic scheme. Thus the significance involved in 84

24 finishing third and fourth has now been redistributed to finishing second and third and sixth and seventh. So there does appear to be a conundrum; play-off systems create more games with a non-zero probability of promotion without changing the overall stock of promotion probability and (unless under extreme circumstances) the overall stock of match significance. Since it increases the number of one positive demand driver and only redistributes the other two demand drivers, are extended play-offs always going to create more demand? There are two (related) reasons why this logic may fail. Firstly, the demand/attendance production function may be non-linear. In such circumstances the marginal impact of the demand drivers may not be constant across games and redistribution may well cause overall attendance to change (potentially of course in either direction). Secondly, the league is played between heterogeneous football clubs with very different sized markets, redistributing promotion probability and or match significance, particularly from bigger clubs to smaller clubs as is likely in a play-off system, may potentially reduce attendance. Thus the theoretical model has identified a potential likely trade-off: marginal fans of smaller clubs may be attracted to matches in which their team still has a non-zero probability of obtaining a desirable end of season outcome however larger clubs will see a reallocation of demand drivers away from games in which they feature, to games featuring smaller clubs. Therefore, the impact of a play-off system is largely an empirical question of which of these counteracting forces dominates given the play-off structure Measuring the Probability of End-of-Season Outcomes and Match Significance The play-off related variables may only be estimated in a production function regression model if there are measures of these probabilities readily available. Though there exists a betting market for end of season outcomes potentially providing 85