Upsets
Evan Osborne
Wright State University
Department of Economics
3640 Col. Glenn Hwy.
Dayton, OH 45435
evan.osborne@wright.edu
For presentation at the meetings of the Western Economic Association
June 30, 2010
Portland, Oregon
Abstract: The paper models the upset – a low-probability outcome of a team competition. It is
assumed that the upset is an independent component of consumer preferences, whose marginal
willingness to pay grows with time. The decision rule for a league on upset timing is a competitive
balance problem, but is unlike standard models of competitive balance. Upset timing is likened to the
optimal harvest time of a growing asset, and implications for competitive balance in this environment
are derived.
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Introduction
What exactly is it that makes for-profit sports attractive to those who pay to consume it? As
with any emerging field, sports economics began by making very simple assumptions – demand is about
competition between teams of equal strength, about market size, or about fan association with a
successful team. But when one looks at what is written popularly about sports, this seems to be a very
narrow conception of what makes them appealing. In fact, much of the drama of sports involves events
before and after the competition, as well as enjoyment (or distress) from results that have little to do
with a direct rooting interest. While in recent years there has been progress in analyzing these other,
perhaps once-removed aspects of demand, much remains to be done.
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.
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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
max
𝑇
𝑔(𝑇)𝑒 −𝑟𝑇 −𝑐𝑇
,
1−𝑒 −𝑟𝑇
(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
[𝑔′(𝑇)𝑒 −𝑟𝑇 − 𝑟𝑒 −𝑟𝑇 − 𝑐](1 − 𝑒 −𝑟𝑇 ) = 𝑟𝑒 −𝑟𝑇 [𝑔(𝑇)𝑒 −𝑟𝑇 − 𝑐𝑇]
(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
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
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j. Let 𝑅𝑖𝑗 (𝑡) 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, 𝑅𝑖𝑗 (𝑡) > 0 > 𝑅𝑗𝑖 (𝑡). Note that 𝑅𝑖𝑗 (𝑡) 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. 𝑅𝑖𝑗 (𝑡) 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 𝑅𝑖𝑗 (𝑡)
or 𝑅𝑗𝑖 (𝑡). 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 𝑅𝑖𝑗 (𝑡)
or 𝑅𝑗𝑖 (𝑡), for a total of ∑𝑛−1
𝑖=1 𝑖 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 𝑤𝑖𝑗 (𝑡) 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.
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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. 𝑅𝑖𝑗 (𝑡) is
assumed to be concave for all i, j, with 𝑅𝑖𝑗 (0) > 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 𝑤𝑖𝑗 (𝑡) = 0, and that 𝑤𝑖𝑗 (𝑡) is convex for all i, j.
The upset is thus wine and not beer – its marginal value is increasing in time. This feature of the model
is critical, because it provides the rising marginal cost of upset deferral. Note critically that its value
does not come merely from market-side j. Rather, fans in the rest of the league may take an active
rooting interest in an upset merely for its own sake, and the longer they have gone without one, the
higher their willingness to pay for one becomes.
The team’s problem is to optimize the “harvest date” of an upset across each competition series.
This is a series of ∑𝑛−1
𝑖=1 𝑖 problems, each of which is to
𝑇
max ∫0 𝑖𝑗 𝑅𝑖𝑗 (𝑡𝑖𝑗 )𝑒 −𝑟𝑡𝑖𝑗 𝑑𝑡 + 𝑤𝑖𝑗 (𝑇𝑖𝑗 )𝑒 𝑇𝑖𝑗
(1)
𝑇𝑖𝑗
The first-order condition is
𝑅𝑖𝑗 (𝑇𝑖𝑗 ) + 𝑤′(𝑇𝑖𝑗 ) = 𝑟𝑤𝑖𝑗 (𝑇𝑖𝑗 )
(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
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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 𝑆𝑖𝑗 ≡ 𝑆𝑖 − 𝑆𝑗 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
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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 𝑆𝑖𝑗 > 0.
Assume that there is a function 𝐹(𝑇; 𝑆𝑖𝑗 ), 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 [0, ∞). 𝑆𝑖𝑗 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. 1 − 𝐹(𝑇; 𝑆𝑖𝑗 ) is
then the probability that no upset has occurred by time T. As the interval lengthens, the probability that
𝜕𝐹
𝜕2 𝑓
𝜕𝑓
an upset has happened asymptotically approaches one. So 𝜕𝑇 > 0, 𝜕𝑇 2 = 𝜕𝑇 < 0.
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, 𝑤𝑖𝑗 (𝑇, 𝑆𝑖𝑗 ). 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
𝜕𝑤
𝜕𝑇
> 0,
𝜕𝑤 2
𝜕𝑇 2
> 0. In addition, the greater the upset, the greater the valuation of
𝜕𝑤
it, so that 𝑤𝑖𝑗 (𝑇, 0) = 0 ∀ 𝑇, and 𝜕𝑆 > 0. Talent differences play no role in the demand for continuity
𝑖𝑗
by market side i, so that 𝑆𝑖𝑗 does not enter into R, which is thus as before only a function of time.
5. Optimization under uncertainty
A league‘s optimization problem here for each competition series ij is
∞
max ∫𝑜 𝑓(𝑡𝑖𝑗 ; 𝑆𝑖𝑗 )[𝑅𝑖𝑗 (𝑡𝑖𝑗 ) − 𝑤𝑖𝑗 (𝑡𝑖𝑗 ; 𝑆𝑖𝑗 )]𝑒 −𝑟𝑡𝑖𝑗 𝑑𝑡𝑖𝑗
𝑆𝑖𝑗
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(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.)
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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
1
2
(. ) > 𝑅𝑖𝑗
(. ), with 𝑅1 , 𝑅 2 > 0, the optimal
increasing skill differential. As before, if a hypothetical 𝑅𝑖𝑗
1
(. ). The reason is that 1’s willingness to pay for deferring
length of time until j defeats i is higher for 𝑅𝑖𝑗
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.
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)
9
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
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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
11
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
12
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.
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Figure 1 – Optimal time to upset.
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