The hope statistic as an alternative measure of competitive balance.

Kaplan, Alan ; Nadeau, John ; O'Reilly, Norm 等

Introduction

Considerable research has been undertaken to address questions relating to competitive balance (CB) in professional team sports. More specifically, research has been aimed at:

1. Defining CB

2. Measuring CB at different points in time in different leagues

3. Investigating whether CB is a desirable characteristic of a professional sports league

4. Offering suggestions as to ways to promote or discourage CB.

We believe that these questions have been addressed due to a widely held belief that CB may have a meaningful impact on team revenues and profits, as well as having an influence on fan interest.

Fan Welfare and the Need for Competitive Balance

It is not axiomatic that CB promotes fan welfare; having said that, Rottenberg (1956; 2000) argued that competitors must be of approximately equal "size" (i.e., ability) to be successful. Contests between poorly matched teams would eventually lose fan interest. Zimbalist (2002) similarly suggests that the notion of CB being important derives from an assumption that fans have a strong preference for uncertainty of outcomes. The argument is that an increase in CB will increase the uncertainty of outcomes and stimulate fan interest. Many of the theoretical advances in CB theory have stemmed from this uncertainty of outcomes hypothesis (UOH), which often assumes that equal weight is placed on each game--from an expansion franchise's first game in the league to an established and competitive team's final game before postseason play.

A more recent definition proposed by the authors of The Commissioner's Blue Ribbon Report on Baseball Economics (Levin et. al., 2000) states that "proper CB will not exist until every well-run club has a regularly recurring hope of reaching postseason play" (p. 5). We argue that this definition of CB is not entirely consistent with the UOH; this difference is important when it comes to making prescriptive suggestions to enhance CB down the line. Lee and Fort (2008) make a similar statement.

Based on the statistics constructed to measure it, the UOH is often thought of as a continuous variable occupying a spectrum from low to high. In contrast, we think of the hope of reaching postseason play as a binary variable. Although baseball fans are not presumed to be homogeneous in their beliefs and expectations, we believe that, if a team falls out of contention, its fans will lose interest. Similarly, if a team is in contention, its fans will maintain their interest and their welfare will be increased. We argue that hope can be effectively modeled as a binary, rather than a continuous variable. In this respect, our work is distinct from other work in the field. While recognizing the value of the UOH definition, we believe that the hope of postseason play definition, at least as stated here, is reflective of a fan's perception of CB. O'Reilly et al. (2008) present empirical support for this argument. With that in mind, we developed a new metric of CB, one that captures the hope of postseason play.

Competitive Balance: Theory and Alternative Measures

Competitive Balance Theory

Previous empirical research has studied the level of CB in different leagues at different times, along with the relationship between these measures and some proxy for consumer welfare (usually attendance). The results are inconsistent and the metrics are many. These differences of opinion abound both due to a lack of a clear definition of CB and an inability to completely measure whatever we decide CB is meant to measure. Humphreys (2002), and more recently Fort (2006), spend some time defining and justifying a variety of measures of CB. These measures include an adjusted or relative standard deviation of wins percentage, the HHI, Humphrey's CBR, Lorenz curves, Gini coefficients, analysis of variance (ANOVA) statistics, and the use of the number of championship seasons. These common metrics of CB (and others) have recently been categorized into one or more of three groupings (Lee & Fort, 2008): game performance, single-season performance, and multi-season performance.

Some of the above measures look at game performance, others at full-season performance and still others at multi-season performance. None of them look directly at single or multi-season performance in terms of whether or not fans think that their team has the potential for postseason play. In contrast, Whitney (1988) and Lee and Fort (2008) do try to reflect this consideration in their definitions for CB. In both cases, the CB definition is based on how far out the best non-playoff team was from a playoff spot at season's end. Lee and Fort (2008) compared their measure--that is, to other measures of CB that reflect single game and multi-season measures--in terms of which measure is best related to league attendance. They find that their measure is better related to league attendance than any of the others.

Developing the Hope Statistic

We hypothesize that the hope of postseason play drives fan welfare and that fan welfare can be maintained or increased, even if the fan's team does not make the playoffs and as long as it is relatively close. Unlike with the UOH, fan welfare is not necessarily negatively affected by a dominant team as long as each fan's team has a chance of postseason play. Accordingly, the availability of multiple playoff spots--compared with just one spot--is reflected in our measure of CB, unlike other measures. Finally, while we think that multi-season performance is important to hope, we believe within-season performance is important as well. More specifically, if a club is in the hunt until late in the season, fans will have continued hope--at least until the club falls out of contention; this is compared with a club that falls out of contention early on and never returns to being a contender.

On this basis, we chose to measure hope for the fan by how "far out" a team was from a playoff position. Instead of using a statistic based on wins and losses at the season's end, or the number of championships, we have chosen games out of a playoff spot as our indicator of hope, and we chose to measure this statistic as of both the middle of and the end of the regular season (MOS and EOS). While Whitney (1988) and Lee and Fort (2008) also use games out of a playoff spot, or games behind lead (GBL), they only measure the GBL at the EOS and for the best team to not make the playoffs rather than all the teams that did not make the playoffs. Yet, in any particular season, there may be one team or several that were close to making the playoffs. We feel that a CB measure reflecting this distinction will more effectively capture the essence of CB. Unlike prior efforts in the literature that use GBL, our measure is intended to capture this information.

Based on O'Reilly and colleagues (2008), we chose 5.5 games behind a playoff position as the point at which fans turn from having hope to not having hope of postseason play in MLB. Briefly, O'Reilly and colleagues (2008) reviewed a listing of trades made by MLB teams at or around the trading deadline over a period of years; they determined how far out of a playoff position a team had to be before team management traded their present-value talent away. Teams were identified as trading away present value if the salaries of the players they traded away were higher than the salaries of the players that they received in return. This methodology was validated by comparing the current level of ability of the players traded away in comparison to the current level of ability of players received in the trade.

The authors then argued that, if management lost hope (i.e., traded away their present value), there were several reasons to assume that fans would lose hope at the same time. First, the act of trading away present-value talent could serve as a signal to fans that management did not believe that the team would compete that season. Second, since management and fans could equally see the performance of the team up until that point in time, it is reasonable to assume that if one group (management) decided that the team was not going to compete for a playoff spot, the other group would come to a similar conclusion. Finally, the trading away of better current talent for noncurrent talent would further reduce the chance of competing for postseason play and, consequently, further reduce expectations on the part of fans.

While the trading deadline is usually around the middle of a season, we chose to measure our statistic as both the middle of the year and the end of the regular season. We chose the middle of the season since our metric is based on decisions made by management around the midseason point. In addition, fan expectations/hope for the second half of the season, and the potential for post season play, will depend upon what has occurred through the first half of the season. Alternatively, the final team standings are intuitively important to most fans when it comes time to renew or otherwise buy their season tickets--by far the largest source of gate revenue for most teams. Whether fans gave up hope prior to the end of the season and/or have an absence of hope going into the following season, the point is that fans whose teams are sufficiently far removed from contention at some point will not have a reasonable hope of postseason play--be it for that year or the next or beyond.

We then accorded a value of 1 to teams that finished 5.5 games or less out of a playoff spot, and a value of 0 to teams that finished more than 5.5 games out of a playoff spot. We label this binary variable GBL for games behind lead. We argue that it makes little difference to most fans if their team is 10, versus 20, games out of contention. Still, it is reasonable to suppose that some fans will maintain interest even if the team's management has traded away present value, as long as the team is not too far out of contention and assuming that some fans will maintain hope when their team is 10 games out, but not when the team is 20 games out, of contention.

An alternative to a binary statistic is a continuous variable--perhaps based on a logarithmic scale in which the most important changes in fan welfare occur when a team's performance falls from just in contention to just out of contention, and in which changes in fan welfare are minimal when a team falls from 15 games out of a playoff spot to 20 games out. While recognizing the potential value to such a statistic, we see no way to model it so as to ensure that the true value associated with the dispersion of wins and losses is properly captured.

While recognizing the contributions of others to this literature, we do feel strongly that a standard deviation statistic, or any statistic based on an exponential scale, will give undue weight to observations that do not deserve much weight-that is, teams that are 20 games out of the playoffs compared with those that finish 15 games out.

Arguably, our decision to use a binary instead of a continuous variable was the most contentious aspect of this paper, based on comments that we have received. It was suggested that by not using a continuous variable we ignore/throw out valuable information. We certainly agree that a continuous variable can take on a broader spectrum of values than a binary variable, but if we cannot accurately value that information, we may do more harm than good.

The binary statistic developed by O'Reilly and colleagues (2008) did reflect the hope of postseason play based on a measurable indicator (i.e., management's decision to trade away present value). While conceding that this is not a perfect indicator of fan hope, we argue that this binary statistic more effectively measures fan welfare because we can be more confident that it better gauges changes in fan hope for postseason play.

Our statistic incorporates many of the same features present in other measures. Like Eckard (2001, 2003) and Humphreys (2002), we recognize the impact of both single-season and multi-year performance on CB. Like Hadley and colleagues (2005) and some of those who rely on Herfindal indexes (e.g., Gerrard, 2004), we use a binary construct. However, our work is distinguished by our efforts to model the idea of hope. Based on the above discussion, we have defined our statistic for CB as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As stated earlier, GBL takes on a value of 1 if team i finishes 5.5 games or less out of a playoff position; otherwise it takes on a value of 0. [GBL.sub.t,i] is the average of GBL for a particular team i over t years. We use it to state a particular team's average CB over time. [GBL.sub.t,i] is the average of GBL for all teams i over t years. [GBL.sub.N,i] is the average of GBL for N teams in a league in a particular season t. [GBL.sub.N] is the average [GBL.sub.N,i] for the league over t years, and [GBL.sub.t] is a measure of the dispersion of hope among different teams over time.

For further clarity, we have simulated results for a four-team league over a 3-year period, using 8.0 GBL as our inflexion point. We could have just as easily used the 5.5 GBL provided in this paper for our empirical study, but we chose 8.0 GBL as this was the point used in our sensitivity analysis (discussed later in the paper).

We ran four simulations. Each simulation corresponded to a different level of CB. In our first simulation, fans of two out of four teams had hope in each of the 4 years, but it was the same fans (i.e., teams). Similarly, in our second simulation, fans of two out of four teams had hope in each of the 4 years, but the fans/teams that had hope were mixed up from year to year. In our third simulation, only four teams had hope in all 3 years combined--an average of 1.33 per year--and it was the same team almost all the time. This simulation corresponded to the lowest level of hope for fans. Finally, in our fourth simulation, three out of four teams had hope every year, and the fans/teams that had hope were mixed up from year to year. This simulation corresponded to the highest level of hope for fans. Our results follow:

Simulation 1 2 out of 4 teams in each Hope Statistic: 1.34
 year, but the same teams
Simulation 2 2 out of 4 teams in each Hope Statistic: 3.00
 year, and all mixed up
Simulation 3 1.33 out of 4 teams in each Hope Statistic: 0.82
 year, but the same teams
Simulation 4 3 out of 4 teams in each Hope Statistic: 5.20
 year, and all mixed up

These results, ranging from a low of 0.82 to a high of 5.20, give the reader a feel for the meaning of the different values of the Hope Statistic, which will be used as a guide for interpreting the results found later in the paper. We have presented details for Simulations 1 and 2 in Tables 1 and 2 of the paper for the purposes of further exposition.

The theoretical range of the Hope Statistic ranges from a low of {(1/N)/highest possible standard deviation}, which occurs when no team other than the winner has hope of making the playoffs, to a high of {(N/N)/0 = infinity}, when all teams have hope of making the playoffs all the time. While the Hope Statistic does not have a neat range from 0-1, as do some of the other statistics current in the literature--and we cannot figure out how to adjust it to have such a range--the Hope Statistic does have the benefit of a scale (presented in our simulations from 0.82 to 5.20) that is easy to understand in real life terms.

In addition, it was important to allow for meaningful comparisons between values of the Hope Statistic and values of other statistics current in the literature. Accordingly, when we present our results later in the paper, we normalize the results for each of the statistics measured in such a way as to allow for easy comparisons.

The Hope Statistic inherently accounts for both the number of teams in a league and the number of playoff positions available. Leagues with fewer teams and more playoff positions, relative to other leagues, will and should have lower values of the Hope Statistic (i.e., fans of more teams will have hope that their team will make the postseason). We want to compare leagues on this basis. Adjusting the statistic depending on the number of teams per league and/or the number of playoff spots per league would result in a less meaningful statistic.

Hope in Major League Baseball: An Empirical Look

We measure CB over time, using both of our Hope Statistics, as well as others current in the literature. To do so, we have used 108 years worth of data for MLB through the 2008 season. This data set was chosen for its completeness, its availability, and for its depth (i.e., large population size). In addition, since this data set has been regularly studied by others (see our literature review) comparisons are easier and arguably more meaningful. (1)

Measuring CB Over Time

We first calculated the value of the Hope Statistic and three other statistics at EOS and presented results in 10-year bands starting with the 1901 season, and ending with the 2008 season, where the last band was limited to 8 years. (2) For example, the 1910s refers to the years from 1911 through 1920, inclusively. The alternative statistics chosen for comparison purposes are a relative standard deviation--used, for example, by Scully (1989) and Quirk and Fort (1997)--the CB Ratio statistic developed by Humphreys (2002) and used in subsequent studies (Lee & Fort, 2008), and an HHI of League Championships (used by Gerrard, 2004, and others). The results for the MOS Hope Statistic values are presented in the next subsection of the paper. We felt that it made more sense to compare the EOS Hope Statistic to other statistics current in the literature since all of the other statistics are measured as of the EOS.

These three alternatives were chosen to reflect recent standards in the literature. More specifically, the relative standard deviation statistic reflects an attempt to recognize CB for game-by-game, and possibly for in-season, performance as a function of the UOH. While Humphreys' (2002) CBR statistic also reflects in-season performance as a function of the UOH, it additionally incorporates a multiyear component. Finally, the HHI metric is a binary (not continuous) statistic that measures CB based on number of championships per team per period. Notably, we did not choose to work with the GBL measures developed by Whitney (1988) or Lee and Fort (2008). Results are presented in Table 3. In this paper, we decided to compare our metric to those that reflect a substantially different definition for CB. Future research should compare our metric to those of Whitney (1988) or Lee and Fort (2008).

Values for the Hope Statistic ranged from a low of 0.64 to a high of 1.94. Put in the context of the simulations that we ran with four teams and three seasons, a value of 0.64 would correspond to approximately one third of the teams having hope in any one season, and it would be mostly the same teams from year to year. Intuitively, this seems to be a very low level of CB. Alternatively, a value of 1.94 might be akin to about half of the teams having hope in any one season with some of the teams being competitive (i.e., having hope) year in and year out, and a fair mixture of other teams being competitive on a semi-regular basis. Even this latter result falls far short of the Commissioner's Blue Ribbon Panel on Baseball Economics declaration that every team should have hope every year.

We want to briefly note the decades during which the Hope Statistic reaches its maximum and, intuitively, the possible reasons for this.

The first decade of the 20th Century saw a league structure and hierarchy in the AL that was not to be repeated at any other time in the 20th Century for either league. The AL was founded in 1900 and, while some of the franchises were new, several were well and, possibly to some extent, equally financed. Some baseball historians feel that the AL Commissioner during this period reigned as the most powerful individual in baseball history, ruling on ownership changes, imposing standards on owners, and controlling both the flow of free agents in particular, and player's rights in general (see Thorn & Palmer, 1989, p. 16). The absence of a franchise in New York during the 1900s might also have contributed to the general CB of the league. While the above story does not prove that CB would have been unusually high, it is certainly a viable argument. Having said this, of the four statistics measured, only the results for the Hope Statistic show a particularly high level of CB in the AL during the first decade of the 20th Century.

In the National League, the value of the Hope Statistic is generally highest in the 1980s and, to a lesser extent, in the 1990s and 2000s. No such pattern exists with respect to the other measures--the minor exception being the CBR in the National League during this period. This is highly intuitive in that from 1969 onwards, and increasingly so, the number of teams eligible to make the postseason increased from 2 teams out of 16 total in the 1968 season to 8 teams out of 30 total at the current time. In addition to more teams actually making the playoffs, there were more teams that were close to making the playoffs. The other measures of CB failed to recognize this change in the game--the number of teams allowed into the postseason--not because the data used in the analysis was flawed, or because the chosen statistic did not accurately measure what it was intended to measure; rather, the other measures of CB failed to recognize this change in the game because they were not supposed to recognize it. Hope of postseason play was simply not the definition of CB used to guide the development of the other measures.

In Table 4, we clarify this observation by normalizing the results to allow for easier comparisons. More specifically, the decade during which CB was best achieved was given a value of 1.0 for each statistic, and the other decades were given a value as a percentage based on that decade's result in comparison to the best decade. For example, the Hope Statistic achieved its peak in the 1980s in the National League with a value of 1.94. In the 1950s, the Hope Statistic took on a value of 0.88 in the National League. The normalized value in Table 4 for the statistic in the 1950s for the National League was 0.45 (0.88/1.94).

At first glance, it appears that the CBR statistic, especially for the National League, rates the 1980s through the 2000s just as highly (if not more highly) in terms of CB as does the Hope Statistic. However, when you compare the 28-year period from 1981 through 2008 to the 80-year period before it, the Hope Statistic showed the higher jump in value.

Those papers that have studied this subject have often tried to link other changes in the game, such as the advent of television or free agency as seminal moments in the move towards or away from CB. While these factors have undoubtedly had a meaningful part to play in the piece, we would suggest that the value of the Hope Statistic, normalized or otherwise, supports the contention that the single greatest contributor to CB over the years has been the decision to increase the number of playoff positions relative to the number of teams in the league.

Measuring Hope at MOS vs. EOS

The data for the MOS Hope Statistics were gathered from http://www.baseball-reference.com/teams/. Won-Loss records were gathered as of July 1st of each year, with the exception of the 1981 strike-shortened season in which case records were taken as of June 11th.

Using the same time period and the same 10 or multi-year bands as we did in the prior subsection of the paper, we calculated the value for the MOS Hope Statistics. Results are presented in Table 5 for both the EOS and MOS Hope Statistics.

We thought that the values for the Hope Statistic would be higher for the MOS than for the EOS stats. Teams that are several GBL, but less than 5.5 GBL at the MOS, might double their GBL by season's end. This argument assumes that teams will repeat their first half performance in the second half of the season, everything else being equal. In reality, this assumption is unproven. Some form of mean reversion might occur. Alternatively, and perhaps more likely, it is the possible that good teams will get better in the second half of the season due to mid-season trade acquisitions. Meanwhile, poorer teams will get worse as they trade away good present value at mid-season and replace that talent with less costly, perhaps younger and likely less-able, talent. In any case, our results showed quite clearly that CB was greater at the MOS than at the EOS for each and every 10-year band in both leagues.

Our primary purpose in calculating the MOS values for the Hope Statistic was to compare them to the EOS values over the time bands that we had defined. We ranked the EOS stats from high to low for all of the 10 year bands and for each league, and then we similarly ranked the MOS stats from high to low for the same time periods. So, for example, the 1980s saw the second highest value for the EOS Hope Statistic in the AL amongst all the time bands. Comparatively, the 1980s saw the highest value for the MOS Hope Statistic in the AL amongst all the time bands. At the other end, the 1950s saw the lowest value for the EOS Hope Statistic in the AL amongst all the time bands and the second lowest value for the MOS Hope Statistic amongst all the time bands. From these and the other ordinal rankings, derived from the data in Table 5, we argue that, during periods when the EOS Hope Statistic was relatively high or low in comparison to other times, the MOS Hope Statistic was similarly high or low. All in all, measuring the Hope Statistic at MOS in comparison to EOS did not greatly affect our conclusions regarding when CB has been higher and when it has been lower.

Discussion

Like other papers before it, this paper is limited in that we have not proven that CB, or even fan welfare itself, is valued by professional team sports leagues. We also have not provided a conclusive definition of CB, and our definition for CB may not be accurately reflected in the model that we have developed. However, in response to the last of these concerns, while O'Reilly and colleagues (2008) suggested that fans would lose Hope if their team was 5.5 or more games behind a playoff position, that research also suggested that fans were not necessarily a homogeneous group maintaining identical expectations. Consequently, we also calculated the EOS Hope Statistic based on an 8.0 game break point--at or above which fans would lose hope, and below which fans would have hope. In Table 6 we present comparative findings for both the AL and NL over the 108 year period for these two Hope Statistics. Our purpose was to determine if the different cut-off point (i.e., 8 games instead of 5.5 games) resulted in a different trend for CB. Based on O'Reilly and colleagues (2008), we chose 8.0 games as an alternative cut-off point because it was meaningfully different from the optimal point (5.5 games) but not so different as to be irrelevant. By observation, while the Hope Statistic using the 8 game cut-off was higher in all decades than the corresponding Hope Statistic using the 5.5 game cut-off, the overall trend of CB, compared across decades, was similar using both cut-offs. Our conclusion is that, while the overall level of CB will rise if the cut-off is increased, we have no reason to believe that the decades during which CB is highest or lowest will change.

More generally, we see no way to determine a theoretical "natural" cut-off point where most fans will lose hope. Rather, we think that this is an empirical question and needs to be addressed based on evidence as presented by, for example, O'Reilly and colleagues (2008).

In this paper, we have extended the literature on CB by suggesting a goal of hope of postseason play. Determining an effective measure of fan welfare has, up to this time in the literature, been both difficult and divisive. And while this definition does not end the debate, we hope that it furthers the discussion. We then developed a measure of CB that intuitively matches up with the goal that we think drives fan interest. We think that the values associated with this measure are very intuitive--based in part on the examples in our simulations--and that this applicability may allow us to better understand what is and what is not CB.

We have also introduced the idea that CB might be best measured as of different points in the season, not just at season's end. The question, still unanswered, is when a fan is sensitive to changes in CB. Arguably, a fan is sensitive at various points in time, including MOS and EOS. While we have proven little with our decision to measure CB as of MOS, we have at least created a useful database for future use and reference and possibly given others something to think about.

There are several directions for further research that come to mind. While survey methodology has its attendant shortcomings, there may be value in attempting to survey professional sports team owners or governors as to their perceptions regarding whether CB is desired--if so, to what extent it is desired--and lastly, which of the measures used in the literature best reflects their perception of CB. Given the small number of owners and governors, a face-to-face interview process might be a better idea, yielding richer and more credible results than a survey while still being a tractable and time-efficient methodology.

Other studies have commonly tried to link a definition of CB with league attendance or other proxies of fan welfare or shareholder wealth. That can likewise be done with this measure. However, unlike most of these other studies, the Hope Statistic can be adapted to reflect CB on a team-by-team basis, and an effort can be made to link CB with individual team attendance over time. It may be that league wide conclusions related to the value of CB do not hold true at the team level.

Having said this, even in the absence of further study, we feel that there are practical implications to our current research.

Assuming that increasing CB does further the goal(s) of the firm, we argue that prescriptions for increasing CB should be based on whether they increase the hope of postseason play. Increasing the number of playoff teams as a percentage of the total number of teams is a very important policy position that comes out of our work; that is, compared to being just one of many suggestions that others have made. This difference in emphasis is important. More generally, if our measure of CB better reflects sensitivity in fan interest, then we can use this measure as a guideline when prescribing changes in the structure of the game, the contracting process, and property rights issues, among others, so as to truly maximize fan interest.

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Endnotes

(1) Comparisons with results of other studies, even those which use the same dataset, are still limited due to differences in the data after cleaning. Several adjustments were made to reflect imperfect data. Data was collected in 10-year bands, a process followed by several other researchers (e.g., Humphreys, 2002). Franchises were used instead of teams. Franchises that were not in place for all 10 years of a band were excluded from the data for the analysis of that band. Consequently, any calculations involving variance or standard deviation are defined using a sample statistic, rather than a population statistic. Data from years with work stoppages was included. Data was organized by league to reflect the method used in other studies. Of note, it is not clear why data was separated by league in other studies. It seems reasonable to presume that CB is not meaningfully different by league. This assertion rests on the assumption that the market for labor talent and franchise property rights is not meaningfully different by league. In fact, the three largest markets in the U.S. are allocated one National League and one American League franchise each, generally supporting our assertion.

(2) The time span of each band was chosen to be comparable to the bands used by prior researchers. A shorter time span might have been chosen to reflect turnover in Collective Bargaining Agreements (CBAs), but CBAs have only been in place for about 40% of our overall time span. We chose a 7-year band at the end to make the results of our other 10 bands of 10 years, each comparable to the results of Humphreys (2002).

Authors' Notes

We wish to recognize the contributions of several anonymous referees whose suggestions were incorporated into this version of the paper, along with research assistants Michael Emrich and Alexander Scott.

Alan Kaplan [1], John Nadeau [2], and Norm O'Reilly [3]

[1] Ryerson University

[2] Nipissing University

[3] University of Ottawa

Alan Kaplan is an associate professor of finance at the Ted Rogers School of Management. His areas of research interest include ethical issues in finance, corporate finance, and sport finance.

John Nadeau is an associate professor of marketing at the School of Business. His research interests include consumer behavior, the application of images, tourism marketing, sport marketing, and sport finance.

Norm O'Reilly is an associate professor of sport business in the School of Human Kinetics. His research interests include both sport finance and sport marketing.

Table 1: Hope Statistic for a 4-team league over 3 years;
same order of finish each year

Team Year 1

 Wins Losses GBL [GBL.sub.i]

A 91 71 0 1
B 84 78 7 1
C 77 85 14 0
D 72 90 19 0
Totals 324 324

[GBL.sub.N,i] 0.5

 [GBL.sub.t,i]

Team Year 2

 Wins Losses GBL [GBL.sub.i]

A 91 71 0 1
B 86 76 5 1
C 83 79 8 1
D 64 98 27 0
Totals 324 324

[GBL.sub.N,i] 0.75

 .37

Team Year 3

 Wins Losses GBL [GBL.sub.i] [GBL.sub.t,i]

A 91 71 0 1 1.00
B 82 80 9 0 0.67
C 78 84 13 0 0.33
D 73 89 18 0 0.00
Totals 324 324 [GBL.sub.i,j]
 = .5
[GBL.sub.N,i] 0.25 [GBL.sub.i,j]
 = .5
 Hope 1.34

Table 2: Hope Statistic for a 4-team league over 3 years;
different order of finish each year

Team Year 1

 Wins Losses GBL [GBL.sub.i]

A 91 71 0 1
B 84 78 7 1
C 77 85 14 0
D 72 90 19 0
Totals 324 324

[GBL.sub.N,i] 0.5

 [GBL.sub.t,i]

Team Year 2

 Wins Losses GBL [GBL.sub.i]

A 64 98 27 0
B 86 76 5 1
C 83 79 8 1
D 91 71 0 1
Totals 324 324

[GBL.sub.N,i] 0.75

 .17

Team Year 3

 Wins Losses GBL [GBL.sub.i] [GBL.sub.t,i]

A 78 84 13 0 0.33
B 82 80 9 0 0.67
C 91 71 0 1 0.67
D 73 89 18 0 0.33
Totals 324 324 [GBL.sub.t,i]
 = .5
[GBL.sub.N,i] 0.25 [GBL.sub.t,i]
 = .5
 Hope 3.00

Table 3. Statistics for CB: MLB from 1901 to 2008

Years 1900s 1910s 1920s 1930s 1940s 1950s

 AL

[Rel..sub.L] 1.75 1.86 1.61 1.88 1.65 1.66
CBR 0.63 0.84 0.71 0.43 0.71 0.59
HHI 0.26 0.30 0.44 0.36 0.40 0.66
Hope 1.86 1.62 0.85 0.73 0.94 0.64

 NL

[Rel..sub.L] 2.18 1.57 1.63 1.63 1.70 1.47
CBR 0.66 0.47 0.62 0.70 0.75 0.73
HHI 0.36 0.24 0.30 0.26 0.28 0.34
Hope 0.75 1.22 1.18 1.15 0.8 0.88

Years 1960s 1970s 1980s 1990s 2000s

 AL

[Rel..sub.L] 1.61 1.67 1.51 1.61 1.52
CBR 0.73 0.67 0.86 0.82 0.63
HHI 0.27 0.38 0.25 0.43 0.55
Hope 0.98 1.11 1.82 1.58 1.52

 NL

[Rel..sub.L] 1.65 1.54 1.43 1.66 1.29
CBR 0.66 0.72 0.99 0.82 0.82
HHI 0.22 0.39 0.23 0.47 0.40
Hope 1.23 1.08 1.94 1.18 1.19

Notes. AL = American League; NL = National League; Higher numbers
indicate increasing CB for all measures except that there is an
inverse relationship between CB and HHI.

Table 4. Normalized statistics for competitive balance: MLB
from 1901 to 2008

Years 1900s 1910s 1920s 1930s 1940s 1950s

 AL

Rel. [sigma]L 0.93 0.99 0.86 1.00 0.88 0.89
CBR 0.73 0.98 0.82 0.50 0.82 0.68
HHI 0.94 0.82 0.56 0.68 0.61 0.37
Hope 1.00 0.87 0.46 0.39 0.51 0.34

 NL

Rel.[sigma]L 1.00 0.72 0.75 0.74 0.77 0.67
CBR 0.66 0.48 0.62 0.70 0.76 0.73
HHI 0.63 0.95 0.76 0.87 0.81 0.67
Hope 0.39 0.63 0.61 0.59 0.41 0.45

Years 1960s 1970s 1980s 1990s 2000s

 AL

Rel.. [sigma]L 0.86 0.89 0.81 0.86 0.81
CBR 0.85 0.78 1.00 0.95 0.73
HHI 0.92 0.65 1.00 0.57 0.44
Hope 0.52 0.60 0.98 0.85 0.82

 NL

Rel. [sigma]L 0.70 0.70 0.65 0.76 0.59
CBR 0.67 0.72 1.00 0.82 0.82
HHI 1.04 0.58 1.00 0.49 0.57
Hope 0.64 0.56 1.00 0.61 0.62

Notes. AL = American League; NL = National League;
Higher numbers indicate increasing CB for all measures
except that there is an inverse relationship between
CB and HHI.

Table 5. Hope Statistics for end of season and mid-season;
MLB from 1901-2008

Years 1900s 1910s 1920s 1930s 1940s 1950s

 AL

Hope EOS 1.86 1.62 0.85 0.73 0.94 0.64
Hope MOS 1.93 2.54 1.32 1.18 1.14 1.17

 NL

Hope EOS 0.75 1.22 1.18 1.15 0.8 0.88
Hope MOS 0.97 1.62 1.46 1.29 0.9 1.71

Years 1960s 1970s 1980s 1990s 2000s

 AL

Hope EOS 0.98 1.11 1.82 1.58 1.52
Hope MOS 1.48 1.48 3.33 2.28 1.79

 NL

Hope EOS 1.23 1.08 1.94 1.18 1.19
Hope MOS 1.75 1.62 2.88 2.02 2.11

Notes. AL = American League; NL=National League; EOS = End
of Season, MOS = Middle of Season

Table 6. Hope Statistics for two different cut-off points;
MLB from 1901-2008

Years 1900s 1910s 1920s 1930s 1940s 1950s

 AL

Hope 5.5g 1.86 1.62 0.85 0.73 0.94 0.64
Hope 8g 1.99 2.15 1.04 0.78 1.01 0.73

 NL

Hope 5.5g 0.75 1.22 1.18 1.15 0.80 0.88
Hope 8g 0.78 1.31 1.37 1.18 0.86 1.03

Years 1960s 1970s 1980s 1990s 2000s

 AL

Hope 5.5g 0.98 1.11 1.82 1.58 1.52
Hope 8g 1.12 1.31 2.37 1.95 1.62

 NL

Hope 5.5g 1.23 1.08 1.94 1.18 1.19
Hope 8g 1.49 1.22 2.72 1.37 1.59

Notes. AL = American League; NL = National League