UpsetsEvan Osborne

Wright State UniversityDepartment of Economics

3640 Col. Glenn Hwy.Dayton, OH 45435

evan.osborne@wright.eduFor presentation at the meetings of the Western Economic Association

June 30, 2010Portland, 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.

We like…

• Our team to win;• Sustained irresolution;• To watch the best talent;• To be part of a collective experience;• To consume concessions and parking;• The other team to lose;• The wildly improbable?

The basic approach

• Assume that a sports league manages a portfolio of competition series, each one an asset.

• The question is the timing of asset liquidation, given that during the interim it produces income.

The Hirshleifer/Samuelson tree model

• In continuous time T, tree produces g(T) in lumber income if it is harvested at T.

• Cost of waiting until T to harvest is cT.• And so the optimization problem is

• The FOC is

Interpretation

• Extend harvest time until discounted, diminishing marginal return to extending it equals discounted marginal cost of tending to the tree.

• The farmer thus harvests all trees at same date, although he will have a farm full of trees of varying ages.

Upsets as an extension of the tree model - differences• An asset – a “tree” – is a competition series

between two teams i, j out of n. • Revenue can be earned before the asset is

harvested, from fans of better team.

Model 1 – certainty, with harvest time as the decision variable.• i denotes favored team, j team not favored.• Rij(T) is the combined net revenue at time t from

markets i and j if team i defeats team j. Revenue from market j if j wins is the opportunity cost of i winning, and vice versa. Thus, Rij(T) > 0 > Rji(T).

• wij(T) is revenue in T from outside markets i, j if j defeats i. It is instantaneous, convex, and wij(0) = 0. It is willingness to pay for the upset for its own sake, as distinct from the willingness of fans in market j to pay for a victory against i.

• Each Rij(T) and wij(T) is assumed to be independent of the others.

Example of an asset portfolio: 4-team leagueR12(T) R13(T) R14(T)

R23(T) R24(T)

R34(T)

• For each market pair the objective is to

• The FOC is

• Interpretation: extend harvest time until marginal revenue from extending continuity (Rij(T)) plus the marginal increase in the upset revenue when it is deferred (wij‘(T)) equals the marginal cost – discounted income from the upset (rwij(T)).

• Implication: victories by small-market teams against big ones have independent value (contrary to conventional competitive-balance models), but only if they are aged properly.

Model 2 - uncertainty• Instead of optimal harvest, model optimal

distribution of talent, which determines probability of upset.

• Sij = Si – Sj is the talent difference between i, j.• F(T; Sij) is a cdf indicating probability that upset

never occurs by time T.• Specifications: F’T > 0, F’’T = f’T < 0, F’S > 0.• w is now wij(T, Sij); w’T > 0, w’’T > 0, w’S > 0.• R is as before; fans in i,j care only about winning,

not talent per se.

• Optimization problem is now

Implications, model 2• If Rij(T) > Rkj(T), the talent differential, and hence expected

time to upset, should be greater for i against j than k against j.

• Standard competitive-balance models with only market based demand for success are special cases. If there is no demand for upsets, w = 0, and optimal talent distribution is driven by market WTP, as usual. If there is no difference among markets, Rij = o for all i, j, and then it is Quirk and Fort (1992) 50-percent model.

• But in this model competitive balance based purely on these considerations, which ignores w, equilibrium competitive balance is too high, once again because of gains to aging.

Where does this model belong?• Existing models of demand: • (1) Demand as standard microeconomic good, with

quantity dependent on price, and demand on available complements and substitutes.

• (2) Demand for success• (3) Demand for suspense• (4) Demand for quality performance• But what fans like to see is an area that could be expanded

further – this is a model about quality, but of the game experience rather than the players; of uncertainty, but the utility of the improbable rather than suspense; of success, but of some other generally ignored team.

Things still to be worked through

• Why do we like upsets so much? An adornment model of utility?

• Deriving an explicit equilibrium out of the talent-allocation model.