Evan Osborne Wright State University
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- Introduction
- 1. The classic tree problem
- 2. Upsets as beyond the trees
- 3. Model 1 – harvest time as the decision variable
- 4. Model 2 – Uncertainty
- 5. Optimization under uncertainty
- 6. Implications
- 8. Conclusion
| Upsets
Evan Osborne Wright State University Department of Economics 3640 Col. Glenn Hwy. Dayton, OH 45435 evan.osborne@wright.edu
June 30, 2010 Portland, Oregon
This paper explores one such aspect – the upset. Without question, fans often take great pleasure when highly favored teams lose games, especially important games. This is true even when the loss does not affect fans' own teams' place in the standings. Instead, the low-probability outcome seems to generate excitement in its own right. From the point of view of a sports league trying to decide how frequently upsets should happen, this problem bear some resemblance to optimal asset harvesting – the decision of when to end of the growth of an asset that is more productive the longer the owner can wait, even while waiting requires that income be deferred and other options be foregone. Here two models of the upset are developed, with an eye to comparing optimal talent dispersal and competitive balance with standard models. Section 1 lays out the classic tree model that is the basis for the analysis, Section 2 extends this basic model to make it relevant for the upset problem, sections 3-5 present two resulting models of the upset, section 6 derives their empirical implications, and section 7 places the results within the space of existing models of fan demand. 1. The classic tree problem The model developed here is an extension of a classic result first achieved by Hirshleifer (1970) and confirmed by Samuelson (1976). The model of upsets is demand-driven, with a league facing imperfectly elastic demand curves optimizes against the constraint of specific consumer preferences. It is a model of profit rather than win maximization. The tree-harvesting model on which it is based investigates the optimal lifespan of a harvested asset, known as a tree. There is a function g(T) that relates output from the tree, which can be thought of as lumber, to its (continuous-time) harvest date T. g is characterized by positive but diminishing returns. Each period of not harvesting a tree incurs a maintenance cost of c. The standard statement of the farm owner’s maximization problem is  , (1)
The tree farmer must in other words choose a rotation period, a length T, at which to cut each tree. Solving (1) leads to the first-order condition  (2)
The farmer will extend the harvest period until the discounted marginal revenue equals the discounted marginal cost. The tradeoff occurs because on the one hand allowing the tree to grow further creates more lumber to harvest, albeit subject to diminishing returns. On the other, income from selling the lumber must be deferred, and leaving the tree uncut requires continued expenditure on tree care. The tree-farm owner selects a single optimal age at which to harvest. In equilibrium a tree farmer thus will have a field full of trees, each of a distinct age, but all of them harvested at the same age. 2. Upsets as beyond the trees A sports league can be thought of as facing an analogous but more complex problem. Teams compete continuously, and for any pair of teams one is more likely to win than the other in each moment of the competition series. The time at which the underdog wins can be likened to the time at which the tree is harvested. But one difference in an upset setting is that while (like a tree) an upset is the end stage of a series of competitions between teams, revenue is earned throughout instead of being deferred until the end, because there are fans willing to pay for the favorite to win as well. In addition, a sports league is not managing a farm consisting exclusively of one kind of tree but a portfolio of assets involving competition between each pair of teams in its league. Given that its teams compete against one another, that each team has its own distinct set of fans, and that one of the primary sources of value for fans of a particular team is its success on the field, each competition between each pair of teams is in a sense a separate asset. Refer to the fans of a pair of teams as a “market pair," and to fans of one side of that pair as a "market side." The league's assets differ in terms of the willingness to pay by each market side for a particular outcome (either a victory or loss). In much literature, this willingness to pay is assumed to be a function of a market side’s population size, wealth, intensity of devotion to its team, and so on. 3. Model 1 – harvest time as the decision variable In the first model I adopt an extremely simplified albeit illustrative approach, in which leagues can control, perhaps through differential allocation of talent, game outcomes, although the decision rule, procedure and results are unknown to fans. Let i and j denote two teams of an n-team league. The model contains a series of competitions, indexed by continuous time, between every pair of teams i and j. Let  be the combined net revenue at time t from markets i and j if team i defeats team j. Implicit within it is the idea that market i has a higher willingness to pay for victory at any moment in time than market j, so that the revenue from market j if j wins is the opportunity cost of i winning, and vice versa. Thus,  . Note that thus incorporates many of the standard elements of competitive-balance models, including different willingness to pay for each market side in a market pair, which should yield differences in talent and hence the likelihood of victory. will depend on population, wealth and individual fans’ availability of substitutes for the competition series. Assuming away ties, it will be true for any market pair i, j that the net revenue is greater for either or  . By the structure of the model in which team j is the team for whom victory is the upset that ends the series of victories by team i, team i can be thought of as the team that is expected to win. Since for all t there is one competition for each i, j pair, it is only necessary to specify one of either or , for a total of  games. As for the upset, people outside of markets i and j have a willingness to pay to see one, which is dependent on time until it occurs. Let  be the revenue gained at the moment when team j defeats team i.
Like the tree problem, and unlike the asset-pricing models descended from Lucas (1978), which require adjusting asset holdings so as to equate the expected intertemporal marginal utility of consumption in consecutive periods, the league must think in terms of optimal asset lifespan. Unlike the Hirshleifer/Samuelson model but like the portfolio-balancing model of Markowitz (1952), the league faces a problem of balancing a series of choice variables, in this case the timing of maturity of an entire portfolio of assets. Note also that the opportunity cost of failing to cut the tree is different from the classic model. There, the cost involves some type of expense on labor and other resources needed to keep the tree growing. Here, the cost at any moment in time is foregoing the one-time revenue from the upset. Refer to an uninterrupted series of victories by team i as continuity. To further specify the problem, consistent with the stylized facts about upsets, assume first that from the league’s point of view each market-pair competition series is an independent asset. The team’s problem is to optimize the “harvest date” of an upset across each competition series. This is a series of  (1)
The first-order condition is  (2)
(2) has a very sensible interpretation. The optimal harvesting date for a particular market pair equates the marginal revenue from extending series continuity (the left-hand side) to the marginal cost, which is the foregone upset income (right-hand term 1) minus the marginal increase in foregone w from extending continuity. As Figure 1 suggests, market pairs for which willingness to pay is high will defer upsets. If this means that market side i is a large market (depicted in the figure as R1), the marginal revenue from i for continuity is higher, and at equilibrium the marginal willingness to pay for an upset is correspondingly higher. The reason is that 1’s willingness to pay for deferring the upset is higher. The rising marginal cost of extending continuity mentioned above, in other words, is more tolerable when the favored team is willing to pay more to do so. This suggests that wealthier market halves (or market halves with greater fan loyalty because of history, fan intensity, etc.) should win more. So far this is a standard result, but comparison will be made later to more conventional models of competitive balance. In addition, in this model we speak not of the likelihood of winning a particular game, but of the length of success by one market side against the other. Powerful market halves will dominate weaker ones for longer periods of time. Note the partial overlap with the Osborne (2008) model of rivalry, which argues that large markets ignore the effects of defeating specific smaller markets, while smaller-market fans emphasize competition against specific larger ones. Here, larger markets will be able to defer the upset longer. 4. Model 2 – Uncertainty Now assume that game outcomes are not certain for the league. Thus, it no longer makes sense to speak of optimal harvest time. Instead, the decision variable is the distribution of talent, which probabilistically affects the likelihood of continuity or an upset. In particular, assume that the probability of an upset is a function of the talent differences between teams. A game between two teams that are equal in every respect would, at least on a neutral field, presumably be one in which each team has a 50-percent chance of winning. Si is the talent level of team i. Define also  as the talent difference between i and j. The league must allocate talent among the n teams in such a way that total talent is equal to some constrained amount S. The set of these variables, i.e. the differential of skill levels (and hence expected competitive balance) among teams, is the choice vector for the league. Since i is the favorite, it will presumably generally be true (although need not be) that  .
Assume that there is a function Not all upsets are equal. In particular, their appeal depends on both the seeming talent gap between the teams and the length of time since the last upset in the series. The benefit to an upset in period t is, extending the specification in Model 1, A league‘s optimization problem here for each competition series ij is  (1)
This maximand is simply the net present value of the competition series. If it has a unique maximum, then in this version of the model the upset is not a single, rigidly timed event. Instead, the upset is a partly random event, which occurs from time to time as a function of optimized competitive imbalance in the league. In addition, because time is continuous, there is no way to speak in terms of the probability of an upset at a particular tij. However, for any given length of time Tij we may refer to the probability that an upset has not happened by that time, since Tij is an implicit function of Sij. If Rij lies above Rkj for any i, j, k then as before this raises revenue from continuity at any moment in time, and this will increase the optimal amount of skill differential Sij compared to Skj. F(T; Sij) will be correspondingly higher for any T. 6. Implications Note that other models in the literature on competitive balance can be reduced to special cases of this problem. If demand for upsets per se does not exist, only demand in i and j for victory by their market sides, then wij(.) = 0. Upsets are then undefined as a distinct event, but are a mere byproduct of a standard allocation of talent. Optimal Sij is then simply proportional to Rij – higher-willingness-to-pay market sides should win more often, the more so the relatively higher is demand in market side i. Such willingness to pay determines the optimal distribution of talent, and the probability of winning in any interval T. This is the classic competitive-balance framework of Quirk and El Hodiri (1974). The Quirk and Fort (1992) argument that ideal competitive balance requires that each game have a 50-percent probability for each outcome means that in addition to the previous condition, Rij = 0, so that each market side is equally willing to pay for a victory. The league is then indifferent as to game outcome (although not to skill distribution, which should be uniform, i.e. Sij = 0 for all i, j.) But the existence of independent revenue from an upset whose marginal value is increasing in both skill differential and time to occurrence means that competitive balance is altered. In fact, the classic statements of competitive balance overstate the optimal level. The extra element of demand for upsets, which at least for some interval Tij is positive, means that the league should make upsets more attractive by decreasing their frequency, and in the meantime earn income from continuity, by increasing skill differential. As before, if a hypothetical 7. Context The literature on the determinants of demand for professional sports leagues is vast. The unit of analysis can be an individual game, or a team or a league over the entire season. It can be without too much violence to reality be divided into four categories. Sports demand as a standard microeconomic category. Sports teams or leagues are simply ordinary goods, characterized by a certain market structure on the supply and standard consumer preferences on the demand side. Most commonly this structure is termed "monopoly," despite the term's imprecision and its camouflaging of the complexity of actual firm decisions (Shmanske, 2006). Since monopolists with linear demand curves are supposed to price in the elastic portion of the demand curve, that they are so often found not to is a continuing puzzle, although Coates and Humphreys (2007) solve this problem by taking attendance at a live sports event as a consumption basket that includes concessions and other goods. Sports is also sometimes modeled as one good that enters with others into consumer preferences. Provocatively, Sandercock and Turner (1981, p. 91), perhaps depicting sport as Marx and Engels once depicted religion, describe fan loyalty for Australian football as “the main escape from the unfair play of the capitalist system.” Good weather on game day can be a substitute for game attendance by making other activities more attractive (Garcia and Rodriguez, 2002). Alternatively, good weather can be a complement for outdoor events by making attendance itself more enjoyable (Bruggink and Eaton, 1996; Welki and Zlatoper, 1999; Butler, 2002). And if the time costs of attendance decline because a game is played on a holiday (Carmichael et al., 1999) or on weekends (Knowles et al., 1992), attendance increases. High ticket prices across a variety of sports (Garcia and Rodriguez, 2002; Simmons, 1996; Whitney, 1988; Borland, 1987) and higher time costs (Falter and Perignon, 2000; Carmichael et al., 1999) lower attendance. Subsidizing attendance through the provision of promotional materials (Bruggink and Eaton, 1996) or, in the case of U.S. college football, the homecoming-weekend experierence (Wells et al., 2000) also increases it. Demand for success. Fans want to see their team win. A corollary is that if a particular team has more or wealthier fans, total market demand for its success is correspondingly higher. Rothenberg (1956) even argues that wealthy teams themselves, in the interest of maintaining the overall quality of the product, will wish to restrain their margin of success. Falter, Perignon and Vercruysse (2008) find that success in and even close-up exposure to the World Cup as the host nation increases future attendance. Schmidt and Berri (2006) claim that winning loomed ever larger in the demand for American baseball over the course of the 20th century. Demand for uncertainty/competitive balance. Highly predictable games are unattractive games, perhaps excepting fans of teams who are heavily favored. But competitive imbalance trades off against the demand for success in bigger markets (Quirk and El Hodiri, 1974, Quirk and Fort, 1992). Whitney (2005) poses an option problem sharing some common features with this one, in the sense that teams’ decisions have third-party effects. However, his is a supply-side argument -- teams selling talent benefit not just the team buying it but also third-party clubs, who improve relative to the seller. Chan, Courty and Li (2009) draw a link between a taste for uncertainty of outcome and optimal compensation for players. The evidence in Lee and Fort (2008) for the value of uncertainty in attendance is positive but modest, but considerably stronger in Forrest, Simmons and Buraimo (2005) for demand for televised games. Demand for quality. Fans are also interested in the general quality of play. There is undoubtedly a huge supply of players at local recreational facilities who would be willing to play for pay, but for which fans' willingness to pay is essentially zero. Rather, higher-quality play is play in higher demand (Scully,1989). Hausman and Leonard (2005) merge this quality argument with competitive balance as joint objects of fan utility. The quality of players in particular may enter fan demand. Berri, Schmidt and Brook (2004) show evidence that demand to watch star players may offset lack of competitive balance in the NBA. Brandes, Franck and Nuesch (2008) find that in German soccer both great players from other teams and the good but not great players from the local team increase attendance. Most of this work is good news, in the sense that sports demand seems to follow standard microeconomic principles, and that demand for live and broadcast consumption seems to have identifiable characteristics. But this literature collectively is too simple a framework for thinking about sports demand. Watching a sports event – a game or a season, in person or on television – is a complex experience, with nonrivalrous consumption and even positive network consumption externalities (from, e.g., the psychological satisfaction of sharing an experience with other fans, either of the own or other team, or, mundanely, the concessions or party atmosphere generally accompanying game consumption) and (relevant here) complex tastes over game outcomes. The early generations of models of sports demand, while essential to making progress in understanding the market for sports, are inadequate for many phenomena. In a survey of the literature on sports demand, Borland and Macdonald (2003, p. 479) give a fairly cramped definition of what demand is about: “The league product includes a series of individual contests, but also has additional saleable and non-saleable elements such as licensed reproduction of the seasonal fixture list and non-saleable externalities such as the ‘league standing’ effect. The essence of demand for the game or sporting contest is ‘fan interest.’ This interest is manifested in watching or listening to a description of the contest (live or on TV/radio), buying products associated with the contest (for example, team merchandise, products of team sponsors, or gambling), or ‘following’ the contest (for example, reading newspaper reports). Even as a broad framework this leaves something to be desired, because of the complex ways fans interact with and judge the competitions they witness. Mason (1999, p.405) argues that the primary feature of fan demand is that it is “enhanced by association to a specific team or its competitors.” But almost the entirety of the nature of fan preferences over their teams’ competitors remains unexplored territory. The analysis here concerns the quality of the game (or at least a series of games) in a sense, but not that of the sheer athletic level of the play. It concerns uncertainty, but not in the simplistic sense of merely valuing more. It concerns success, but of other teams. There are in sports demand undoubtedly many of these kinds of cross-terrain stories to be told. 8. Conclusion The objective function of sports fans is thus still significantly unexplored. The model developed here can perhaps be thought of as an example of a more general theory of the utility of witnessing the improbable. Such a theory has many sports manifestations. It is said that the number of people who claim to have witnessed Wilt Chamberlain’s 100-point game vastly exceeds the capacity of the arena in Hershey, Pennsylvania where it happened. Resale prices of ticket stubs for games where a no-hitter is thrown or another similarly improbable event happens are much higher than those for ordinary games of the same vintage. Such an adornment model of utility, in which the consumer seeks to maximize his collection of experienced rare events, can also be applied to such experiences as travel to exotic destinations or the race to be first to experience a trendy new restaurant. This kind of model goes beyond the mere taste-for-variety utility function of the Krugman (1980) type. The appeal of rarity, demonstrated by the appeal of the upset, is an area worth further exploration both inside sports and out. References Berri, David J., Schmidt, Martin B., and Brook, Stacey L. “Stars at the Gate: the Impact of Star Power on NBA Gate Revenues." Journal of Sports Economics, February 2004, 5 (1), 33-50. Borland, Jeffery. “The Demand for Australian Rules Football.” Economics Record 63, 1987, 220-230. Borland, Jeffery and MacDonald, Robert. “Demand for Sport.” Oxford Review of Economic Policy 19 (4), 2003, 478-502. Brandes, Leif, Franck, Egon, and Nuesch, Stephan. “Local Heroes and Superstars: An Empirical Analysis of Star Attraction in German Soccer." Journal of Sports Economics, June 2008, 9 (3), 266-286. Bruggink, E.H. and Eaton, J.W. “Rebuilding Attendance in Major League Baseball: The Demand for Individual Games.” In J. Fizel, E. Gustafson and L. Hadley (eds.), Baseball Economics: Current Research (Westport, CT: Praeger, 1996, 9-31). Butler, M. “Interleague Play and Baseball Attendance.” Journal of Sports Economics 3 (4), 2002, 320-334. Carmichael, F., Millington, J., and Simmons, R. “Elasticity of Demand for Rugby League Attendance and the Impact of BSkyB.” Applied Economics Letters 6 (12), 1999, 797-800. Chan, William, Courty, Pascal, and Li, Hao. “Suspense: Dynamic Incentives in Sports Contests.” Economic Journal, January 2009, 119 (534), 24-46. Dennis Coates and Brad R. Humphreys. "Ticket Prices, Concessions and Attendance at Professional Sports Events." International Journal of Sport Finance, August 2007, 2 (3), 161-170. Falter, J.M. and Perignon, C. “Demand for Football and Intramatch Winning Probability: AN Essay on the Glorious Uncertainty of Sports.” Applied Economicsi 32, 2000, 1757-1765. The case of a soccer World Cup victory Falter, Jean-Marc, Perignon, Christophe, and Vercruysse, Olivier. “Impact of Overwhelming Joy on Consumer Demand: The Case of a Soccer World Cup Victory." Journal of Sports Economics 9 (1), February 2008, 20-42. Garcia, J. and Rodriguez, P. “The Determinants of Football Match Attendance Revisited: Empirical Attendance from the Spanish Football League.” Journal of Sports Economics 3 (1), 1999. 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"Asset Prices in an Exchange Economy." Econometrica 46, 1978, 1429-1445. Osborne, Evan. “Rivalries.” Presented at the annual meetings of the Western Economic Association, 2007. Quirk, J. and El Hodiri, M. "The Economic Theory of a Professional Sports League." In R. Noll (ed.), Government and the Sports Business (Washington, DC: Brookings Institution, 1974) Quirk, J. and Fort, Rodney D. Pay Dirt: the Business of Professional Team Sports (Princeton: Princeton University Press, 1992) Samuelson, Paul A. "Economics of Forestry in an Evolving Society." Economic Inquiry 14 (4), December 1976, 466-492. Sandercock, L. and Turner, I. Up Where Czaly? The Great Australian Game. Sydney: Granada, 1981. Scully, Gerald W. The Business of Major League Baseball (Chicago: University of Chicago Press, 1989) Schmidt, M., and Berri, David. “Competitive Balance and Attendance: The Case of Major League Baseball.” Journal of Sports Economics 2 (2), 2001, 145-167. Simmons, R. “The Demand for English League Football: A Club Level Analysis.” Applied Economics 28, 139-155. Shmanske, Stephen. “The Monopoly Nonproblem: Taking Price Discrimination Seriously." Independent Review, 10 (3), Winter 2006, 337-350. Welki, A.M. and Zlatoper, T.J. “U.S. Professional Football Game-Day Attendance.” Atlantic Economic Journal 27 (3), 285-298. Wells, D.E., Southall, R.M. and Peng, H-H. “An Analysis of Factors Related to Attendance at Division II Football Games.” Sport Marketing Quarterly 9 (4), 2000, 203-210. Whitney, J. “Winning Games Versus Winning Championships: The Economics of Fan Interest and Team Performance.” Economic Inquiry 26, 1988, 703-724. Whitney, J.D. "The Peculiar Externalities of Professional Team Sports." Economic Inquiry 43 (2), 2005, 330-343. 
Figure 1 – Optimal time to upset. Download 44.93 Kb. Share with your friends:
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> 0. The effect is to decrease over time the marginal willingness to pay of market i for further victories. The assumption is analogous to Eckard’s (2001) study of free agency and competitive balance. Assume also that 
, which is a cdf indicating the probability that an upset does not occur in the interval T. Its range is [0, 1] and its domain is 
is a positive shift parameter, indicating that a greater talent advantage for i means a greater likelihood of no upset over any time interval. F is twice continuously differentiable, and f(.) is the associated pdf. 
is then the probability that no upset has occurred by time T. As the interval lengthens, the probability that an upset has happened asymptotically approaches one. So 
.
. So here skill differential, in addition to time, is now a determinant of willingness to pay for an upset. It is assumed, consistent with the discussion above, that 
. In addition, the greater the upset, the greater the valuation of it, so that 
, and 
. Talent differences play no role in the demand for continuity by market side i, so that 
, with 
, the optimal length of time until j defeats i is higher for 
. The reason is that 1’s willingness to pay for deferring the upset is once again higher at any moment in time. This suggests that wealthier market halves (or market halves with greater fan loyalty because of history, fan intensity, etc.) should win more. Fan bases able to extensively support their teams will succeed more, as has been suggested in the competitive-balance literature for decades. Powerful market halves will dominate weaker ones for longer periods of time.