ECONOMIC STUDIES OF WAGERING ON SPORTS

I. The Efficient Markets Hypothesis

A. What characterizes equilibrium prices in financial markets?

1. Equal expected rates of return across similar prospects

2. Efficient prices incorporate all available information

3. Absence of profit opportunities

B. Are markets efficient?

1. Why is this question important?

2. What is the evidence?

3. What can wagering on sports tell us?

II. Implications of Efficient Pricing for Racetrack Betting

A. The Constant Expected Returns Model

1. Parimutuel Odds (betting to win)

Notation:

W_i - amount be on horse i to win the race

W - amount bet on all horses

w_i = W_i/W = % of wagering pool bet on horse i

Sum(w_i) = 1

p_i = probability that horse i wins the race

Sum(p_i) = 1

Parimutuel Rules:

The pool is divided up among the winning bettors in proportion to the amount bet.

Assume zero takeout (W split among winning bettors)

Example: only two people bet $1 on horse #4 & it wins:

they split W equally (each gets $W/2)

Add someone else who bets $2 on horse #4; she gets ½ W; 1^st 2 get W/4

General rule: return to $1 bet on the winning horse is W/W_i

R_i = W/W_i = 1/w_i is the return to betting $1 of horse i

2. Efficient Odds in the Betting Market

Mathematical Expected Return to betting $1 on horse i:

ER_i = p_i R_i = p_i / w_i

(i) Absence of profit opportunities:

all horses must have expected returns < = 1.0

(ii) Now suppose bettors care only about returns:

all horses must have the same expected returns

(iii) With zero takeout, these imply that ER_i = 1.0 for each horse

(example)

(iv) With positive takeout rate

t = takeout rate

Q = 1 - t = fraction of W returned to bettors

R_i = QW/W_i

Equilibrium betting: ER_i = Q for each horse

(v) Efficient (equilibrium) betting implies that

w_i = p_i

i.e. that horse i's share of the win pool is the probability it wins the race!

proof: (zero takeout) ER_i = p_i / w_i = 1; or p_i = w_i

w/ positive takeout: ER_i = p_i QW/W_i = Q; or p_i / w_i = 1 as before

Tables 2 & 3 Test This Major Implication of Efficient Pricing

Table 2: w & p are fairly close

Table 3: longshots are more likely to finish up the track, favorites to win

(See Sauer, Journal of Economic Literature, 1998, for all figures and tables)

III. The Favorite-Longshot Bias

A. Examine the tails in Hoerl & Fallin's Table 2

For the longest shots, w > p

i.e. w (from the betting market) is an upward biased estimate of p

B. More evidence of the favorite-longshot-bias

Ali (1977) and Asch, Malkiel & Quandt (1982) -- Table 4

Snyder (Figure 1)

C. Why does the favorite-longshot bias exist?

1. Bettors are risk-lovers

willing to accept lower returns on longshots

lower returns are required in equilibrium if all bettors like risk

Problem: some racetracks do not exhibit this bias (Busche & Hall, T. 4)

2. Heterogeneous agents

(i) some agents are uninformed, bet blindly

some agents know true probabilities; bet only on horses with

+ net expected returns (bias dissolves with informed bettors)

see Table 1: bias falls with # of informed bettors

(ii) Shin's model of optimal bookmaker prices (Great Britain):

bookmakers face an adverse selection problem

adjust by shortening the odds on longshots

V. The Racetrack Market as an Information Aggregation Device

A. Parimutuel Betting (Asch, Malkiel & Quandt, 1982)

Betting improves the forecast of the winner above the morning line

=> Late bettors have more knowledge that early bettors

B. Fixed Odds Betting (Crafts, 1985)

Betting coups in the U.K.

Horses whose odds fall during the betting were phenomenally profitable bets prior to the fall (Table 6B)

C. The Bottom Line on Racetrack Betting

Odds (w) provide good first approximations to p

In some cases we observe biased odds, particularly for very low p horses

Final odds are more accurate estimates of p than initial odds.

There is strong evidence of differentially informed agents.

The betting period creates more efficient prices - final odds provide better forecasts of winning than experts, computer programs, etc.

During the betting period information is aggregated from diverse sources, resulting in more efficient prices.