---long before this classic match race between Man O’War and Sir Barton—a small group of ---long before this classic match race between Man O’War and Sir Barton—a small group of aristocrats hit upon an ingenious way to finance their ruinously expensive hobby, racing horses. aristocrats hit upon an ingenious way to finance their ruinously expensive hobby, racing horses. They invited the general public to attend the races, encouraged them to wager by organizing They invited the general public to attend the races, encouraged them to wager by organizing the betting pools and then simply extracted a share of the betting pool.the betting pools and then simply extracted a share of the betting pool.Many years later a small group of economists figured out that this clever scheme—which had Many years later a small group of economists figured out that this clever scheme—which had since grown into a multi-billion dollar industry—might be a useful way to test their theories since grown into a multi-billion dollar industry—might be a useful way to test their theories about how markets work and about how people deal with uncertainty.about how markets work and about how people deal with uncertainty.

Many Years Ago …

Transactions Costs, Transactions Costs, Preferences and the Preferences and the

Favorite/Longshot BiasFavorite/Longshot BiasMichael L. DavisMichael L. Davis

Department of FinanceDepartment of Finance

Cox School of BusinessCox School of Business

Southern Methodist UniversitySouthern Methodist University

Dallas, TexasDallas, Texas

(mldavis@mail.cox.smu.edu(mldavis@mail.cox.smu.edu))

Questions That Studying Questions That Studying Horse Race Betting Might Horse Race Betting Might

Help AnswerHelp AnswerMarket Market

microstructuremicrostructure What makes a What makes a

market more market more efficient?efficient?

How do transactions How do transactions costs influence costs influence markets?markets?

How does the How does the distribution of distribution of information information influence markets?influence markets?

Psychology of risk-Psychology of risk-takingtaking

Are choices consistent Are choices consistent with expected utility?with expected utility?

Or are the choices seen Or are the choices seen in horse race betting in horse race betting best explained by best explained by preferences that are preferences that are non-linear in non-linear in probabilities?probabilities?

A Brief Caricature of Part A Brief Caricature of Part of the Literatureof the Literature

The early literature on horse racing asked The early literature on horse racing asked whether the market for these state-whether the market for these state-contingent claims was efficient. (Usually contingent claims was efficient. (Usually taken to mean that the odds were taken to mean that the odds were consistent with the liklihood of winning.)consistent with the liklihood of winning.)

The answer is NO!!The answer is NO!! Horse racing is characterized by the Horse racing is characterized by the

“favorite/longshot bias”. “favorite/longshot bias”. Longshots (horses that go off at high odds) have Longshots (horses that go off at high odds) have

significantly lower expected returns than significantly lower expected returns than favorites.favorites.

How To Explain the Bias: How To Explain the Bias: Part 1Part 1

(Gamblers Like to Gamble)(Gamblers Like to Gamble) The bias is consistent with a market The bias is consistent with a market

where the marginal bettor is “rational” where the marginal bettor is “rational” (in the sense that preferences are (in the sense that preferences are consistent with expected utility) but just consistent with expected utility) but just likes risk.likes risk. Ali (1977)Ali (1977) Quandt (1986)Quandt (1986)

Bettors are rational but enjoy playing Bettors are rational but enjoy playing the gamethe game ZiembaZiemba

How To Explain the Bias How To Explain the Bias Part 2:Part 2:

(The bias is evidence that (The bias is evidence that people don’t care about people don’t care about

expected utility.)expected utility.) The bias is consistent with a market The bias is consistent with a market

where the marginal bettor has where the marginal bettor has preferences that are described by preferences that are described by one of the many models that are one of the many models that are non-linear in probability (rank-non-linear in probability (rank-dependent utility, cumulative dependent utility, cumulative prospect theory …..prospect theory …..

How To Explain the Bias: How To Explain the Bias: Part 3Part 3

(Bettors are Rational But (Bettors are Rational But the Market is Crazy)the Market is Crazy) Key Insight: Parimutual betting is both Key Insight: Parimutual betting is both

a contest against “nature” (pick the a contest against “nature” (pick the winner) but also against the other winner) but also against the other bettors (pick the horse offering the bettors (pick the horse offering the highest expected payoff).highest expected payoff).

And so even if bettors were fully And so even if bettors were fully informed and risk neutral, the bias informed and risk neutral, the bias might arise as a consequence of this might arise as a consequence of this gamegame Potters and WittPotters and Witt Ottaviani and SorensenOttaviani and Sorensen

Problem: These Problem: These Explanations of the Bias are Explanations of the Bias are

all Terrificall Terrific That is, they are logical and seem to fit That is, they are logical and seem to fit

the data.the data. And so how do we use racetrack data to And so how do we use racetrack data to

really distinguish between models?really distinguish between models? Perhaps we should compare “goodness of Perhaps we should compare “goodness of

fit” (Julliene and Salanie).fit” (Julliene and Salanie). Or maybe compare different types of bets Or maybe compare different types of bets

contrasting, say, compound gambles with contrasting, say, compound gambles with single gambles (Snowberg and Wolfers).single gambles (Snowberg and Wolfers).

A Missing Link, A Missing Link, Transactions CostsTransactions Costs

Parimutual pools are heavily taxed Parimutual pools are heavily taxed (somewhere between 14% and 25% (somewhere between 14% and 25% at U.S. tracks).at U.S. tracks).

The tax rate varies between tracks, The tax rate varies between tracks, and types of bets. and types of bets.

Even more intriguing, because of Even more intriguing, because of “carryovers” and “guarantees” the “carryovers” and “guarantees” the tax rate varies randomly from day-tax rate varies randomly from day-to-day.to-day.

Can This Variance in Tax Can This Variance in Tax Rates Help Us Distinguish Rates Help Us Distinguish

Models?Models? First Step: Include tax rates in the First Step: Include tax rates in the

usual models.usual models. If it turns out that different models imply If it turns out that different models imply

different reactions to changes in the tax different reactions to changes in the tax rate, then maybe we’ve got a tool that rate, then maybe we’ve got a tool that could falsify one or more explanations. could falsify one or more explanations.

(Obvious) Second Step: If the models (Obvious) Second Step: If the models suggest differences, get the data and suggest differences, get the data and do some tests.do some tests.

Basics (Notation and Assumptions)Basics (Notation and Assumptions)

Two-horse race between the favorite (f) and the Two-horse race between the favorite (f) and the longshot (l). longshot (l).

p> 0.5 is the objective probability that the favorite will p> 0.5 is the objective probability that the favorite will win.win.

OOhh = profit from $1 winning bet on horse h (the odds). = profit from $1 winning bet on horse h (the odds).

In parimutal betting In parimutal betting

OOhh = (1-t)/W = (1-t)/Wh h -1 -1

(t= tax rate, W(t= tax rate, Wh h = % of pool bet on h)= % of pool bet on h)

Note to Eliminate Needless Note to Eliminate Needless ConfusionConfusion

I have been told that the British state the odds I have been told that the British state the odds differently than we do in the U.S. differently than we do in the U.S.

Throughout this presentation, I will follow the U.S. Throughout this presentation, I will follow the U.S. convention. For example, a winning horse that paid convention. For example, a winning horse that paid $3 on a $1 bet would go off at odds of 2.0, “2 to 1” in $3 on a $1 bet would go off at odds of 2.0, “2 to 1” in U.S. parlance.U.S. parlance.

The horse would (I think) be said to have odds of The horse would (I think) be said to have odds of “1 to 2 on” in Britain. “1 to 2 on” in Britain.

I have also been told that the British have a different I have also been told that the British have a different spelling of the word “favorite”.spelling of the word “favorite”.

Odds Are Constrained By Odds Are Constrained By the Tax on the Poolthe Tax on the Pool

Since WSince Wll=1-W=1-Wff

OOll = (1-t)(Of+1)/(Of+t)-1 = h(Of,t) = (1-t)(Of+1)/(Of+t)-1 = h(Of,t) ““Feasible Odds” satisfy this constraintFeasible Odds” satisfy this constraint Increasing the tax rate shifts the Increasing the tax rate shifts the

feasible odds downfeasible odds down

These odds are feasible These odds are feasible given the tax rategiven the tax rate

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

Odds on favorite

Odd

s on

long

shot

t=0

t=15%

Odds on this curve are feasible when the there are no taxes

Feasible odds when t=15%

A “Wager Indifference Curve” is one A “Wager Indifference Curve” is one where the bettor is indifferent between where the bettor is indifferent between

a bet on either horse.a bet on either horse. Suppose the bettor is a risk-lover who cares Suppose the bettor is a risk-lover who cares

about expected utilityabout expected utility A winning bet on h gives utility U(OA winning bet on h gives utility U(Ohh)) Normalize the utility to zero if the bet is lost utility Normalize the utility to zero if the bet is lost utility

U(-1)=0.U(-1)=0. In equilibrium, the bettor must be indifferent to a wager on In equilibrium, the bettor must be indifferent to a wager on

either horseeither horse pU(OpU(Off) = (1-p)U(O) = (1-p)U(Oll))

Wager indifference for a bettor whose Wager indifference for a bettor whose utility isutility is

U(x) = (e U(x) = (e22 -e- -e-2x2x)/2)/2 (A risk-lover with CARA, and U(-1) = 0) (A risk-lover with CARA, and U(-1) = 0)

P=.60P=.60

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5

Odds on favorite

Odds

on

long

shot

If odds are here, the bettor prefers the longshot

If odds are here, the bettor prefers the favorite

Wager Neutral Equilibrium Wager Neutral Equilibrium The intersection marks the only feasible odds The intersection marks the only feasible odds where the marginal bettor is just indifferent where the marginal bettor is just indifferent

between the two horsesbetween the two horses

00.5

11.5

22.5

33.5

0 0.5 1 1.5 2 2.5 3 3.5

Odds on favorite

Odd

s on

long

shot

Wager Indifference

Feasible odds

equilibrium

What happens when the tax rate What happens when the tax rate changes?changes?

00.5

11.5

22.5

33.5

0 0.5 1 1.5 2 2.5 3 3.5

Odds on favorite

Odds

on lo

ngsh

ot

Feasible odds (t=0)

Feasible odds (t=20%)

Raising the tax rate willRaising the tax rate will

Lower the equilibrium odds on both Lower the equilibrium odds on both horsehorse A sensible conclusion since a higher tax A sensible conclusion since a higher tax

rate means there is less in the pool to rate means there is less in the pool to pay out to the winnerspay out to the winners

Raise the odds on the longshot Raise the odds on the longshot relative to the longshotrelative to the longshot That is increase the favorite/longshot That is increase the favorite/longshot

biasbias

Concern: The wager neutral Concern: The wager neutral equilibrium might actually result in the equilibrium might actually result in the

favorite offering a positive expected favorite offering a positive expected value bet.value bet.

If this happens will there be enough If this happens will there be enough fully-informed, risk neutral bettors fully-informed, risk neutral bettors to take advantage of this and drive to take advantage of this and drive the odds back into the range where the odds back into the range where no bets offer a positive EV?no bets offer a positive EV?

Model 2: Preferences that Model 2: Preferences that are not linear in are not linear in

probabilitiesprobabilities Issue: There are lots of models to Issue: There are lots of models to

pick from, which should be tested?pick from, which should be tested? Obvious answer (especially for a Obvious answer (especially for a

summer-time conference in a great summer-time conference in a great city): the simplest one. city): the simplest one.

Risk-Neutral/Subjective Risk-Neutral/Subjective Probability Probability

ππ(p) is the bettors subjective belief of (p) is the bettors subjective belief of the probability that the favorite will the probability that the favorite will win.win.

Suppose equilibrium is the point Suppose equilibrium is the point where the subjective expected gain where the subjective expected gain from a bet on either horse is the same from a bet on either horse is the same (same definition as Snowberg and (same definition as Snowberg and Wolfers). Wolfers).

ππ(p)(O(p)(Off+1) = +1) = ππ(1-p)(O(1-p)(Oll+1)+1)

Same thing, expressed in Same thing, expressed in terms of proportion of pool. terms of proportion of pool.

Remember, if wRemember, if whh is the proportion of the total is the proportion of the total pool bet on horse h, thenpool bet on horse h, then

OOhh+1=(1-t)/w+1=(1-t)/whh

Thus, the equilibrium condition can be Thus, the equilibrium condition can be written as written as

ππ(p)(1-t)/w(p)(1-t)/wff= = ππ(1-p)(1-t)/(1-w(1-p)(1-t)/(1-wff), or simply), or simply (1-w(1-wff)/w)/wff= = ππ(p)/ (p)/ ππ(1-p) (1-p) If this model is the right one, then the tax If this model is the right one, then the tax

rate shouldn’t matter—the proportion of the rate shouldn’t matter—the proportion of the pool (and hence the odds) should depend only pool (and hence the odds) should depend only on subjective probabilities.on subjective probabilities.

So far I have outlined “median bettor” models. So far I have outlined “median bettor” models. That is, the observed odds are assumed to be That is, the observed odds are assumed to be

consistent with the preferences of some consistent with the preferences of some typical horse-player.typical horse-player.

If this “subjective probability/risk-If this “subjective probability/risk-neutral” model correctly describes how neutral” model correctly describes how the market works, then the relative the market works, then the relative proportions bet on either horse should proportions bet on either horse should not vary with the tax rate.not vary with the tax rate.

This is different than the risk-loving, This is different than the risk-loving, expected utility model, which implied expected utility model, which implied that as the tax rate changed, the bias in that as the tax rate changed, the bias in favor of the longshot should increase. favor of the longshot should increase.

So far I have outlined “median bettor” So far I have outlined “median bettor” models--that is, the observed odds are models--that is, the observed odds are

assumed to be consistent with the preferences assumed to be consistent with the preferences of some typical horse-player. Here is a of some typical horse-player. Here is a

summarysummaryExpected Utility Risk Expected Utility Risk

LoveLove Increasing the tax Increasing the tax

rate on the betting rate on the betting pool should pool should increase the bias increase the bias in favor of the in favor of the longshotlongshot

Biased Subjective Biased Subjective ProbabilityProbability

Increasing the tax Increasing the tax rate on the betting rate on the betting pool should not pool should not change the bias in change the bias in favor of the favor of the longshot.longshot.

Model 3: (Some) Rational Bettors in an Model 3: (Some) Rational Bettors in an Irrational MarketIrrational Market

Assume two types of bettorsAssume two types of bettors Informed (know p—as well as some Informed (know p—as well as some

other stuff).other stuff). I=the proportion of the potential total pool I=the proportion of the potential total pool

who are informedwho are informed uninformed (bets may not be consistent uninformed (bets may not be consistent

with p)with p) UUff = % of uninformed betting on favorite = % of uninformed betting on favorite

The The RightRight Number of Number of Informed Bettors Will Informed Bettors Will

Correct the MarketCorrect the Market Assume Assume

p=.75 (that is, odds of 1 to 3 would be a zero EV p=.75 (that is, odds of 1 to 3 would be a zero EV bet) bet)

UUff=.50 (half of the uninformed bet the favorite)=.50 (half of the uninformed bet the favorite)

If only the uninformed bet, there will be a If only the uninformed bet, there will be a biasbias The favorite goes off at odds of 1 to 1 (that is, The favorite goes off at odds of 1 to 1 (that is,

pays $2.00 on $1 bet).pays $2.00 on $1 bet). But the favorite wins 75% of the time But the favorite wins 75% of the time

(EV=.75x2=1.5)(EV=.75x2=1.5)

The The RightRight Number of Number of Informed Bettors Will Informed Bettors Will

Correct the MarketCorrect the Market But now suppose that the informed But now suppose that the informed

make up half of the pool (I=.50).make up half of the pool (I=.50). If the informed all bet the favorite, If the informed all bet the favorite,

thenthen The favorite goes off at odds of 1 to 3 The favorite goes off at odds of 1 to 3

[1/(.5x.5+.5)-1][1/(.5x.5+.5)-1] Thus, EV=1. Thus, EV=1.

But this argument depends on But this argument depends on their being the correct number of their being the correct number of

informed betters.informed betters. If there are less than the correct number of informed If there are less than the correct number of informed

bettors (equivalently, if the informed bettors face some bettors (equivalently, if the informed bettors face some sort of budget constraint), then the bias may still exist.sort of budget constraint), then the bias may still exist. To continue with the previous example if the informed To continue with the previous example if the informed

make up only 20% of the total pool, then the favorite make up only 20% of the total pool, then the favorite goes off at odds of 3 to 2 and still has a positive EV—the goes off at odds of 3 to 2 and still has a positive EV—the bias is reduced, but not eliminated.bias is reduced, but not eliminated.

Even more striking, Even more striking, too many informed bettors too many informed bettors can be a bad thing, can be a bad thing, in that if there are too many in that if there are too many informed bettors, the odds might be even more informed bettors, the odds might be even more distorted.distorted.

Why too many informed bettors might Why too many informed bettors might be badbe bad

Before making a bet, the informed know p, I and UBefore making a bet, the informed know p, I and U ff They assume that all the other informed bettors know They assume that all the other informed bettors know

this too and so they assume that all the other informed this too and so they assume that all the other informed will be doing whatever they do.will be doing whatever they do.

If there were a lot of informed bettors, all of whom If there were a lot of informed bettors, all of whom tried to take advantage of what appears to be a tried to take advantage of what appears to be a positive EV opportunity, they would actually lose positive EV opportunity, they would actually lose money. To avoid this, they only bet on those races money. To avoid this, they only bet on those races where the uninformed have gotten it so wrong that the where the uninformed have gotten it so wrong that the bet is still a good deal bet is still a good deal even when all the other even when all the other informed bettors recognize and act on the same thing. informed bettors recognize and act on the same thing.

This means that sometimes a horse will go off at odds This means that sometimes a horse will go off at odds implying a positive expected value, but the informed implying a positive expected value, but the informed won’t bet.won’t bet.

In game-speak, this is really just a kind of Cournot-In game-speak, this is really just a kind of Cournot-Nash equilibrium, where there is no collusion between Nash equilibrium, where there is no collusion between the players.the players.

Formalize the story: What the informed do Formalize the story: What the informed do depends on how the uninformed betdepends on how the uninformed bet

Bet on the favorite ifBet on the favorite if

U<UU<Uff=(P(1-t)-I )/(1-I)=(P(1-t)-I )/(1-I) Bet on the longshot ifBet on the longshot if

U>UU>Ull = = [t+p(1-t)]/(1-I)[t+p(1-t)]/(1-I)

Don’t bet if Don’t bet if

UUff≤U ≤U≤U ≤Ull

Results of Simulation Describing Relationship Results of Simulation Describing Relationship Between Odds and EV of Bet (The exercise Between Odds and EV of Bet (The exercise assumed several different types of races, assumed several different types of races,

where the uninformed under-bet the favorite.) where the uninformed under-bet the favorite.)

1

1.1

1.2

1.3

1.4

1.5

1.6

0.25 0.45 0.65 0.85 1.05 1.25 1.45

Odds

Expe

cted

Valu

e

Here, the informed make up 50% of the bettors. But they never

bet and so the horses with low odds offer very profitable bets.

Here, the informed make up 25% of the bettors. But they always bet and so the horses with low odds are less profitable

Things can seem even stranger if the informed bet Things can seem even stranger if the informed bet some races but not others. This is the previous some races but not others. This is the previous simulation exercise, except with a different proportion simulation exercise, except with a different proportion

of informedof informed. . When there are a greater proportion of When there are a greater proportion of informed bettors, the relationship between odds and EV informed bettors, the relationship between odds and EV appears almost random. This is because the informed appears almost random. This is because the informed are only betting races with a strong favorite (high p)are only betting races with a strong favorite (high p)

0.5

0.7

0.9

1.1

1.3

1.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Odds

Expe

cted

Val

ue 20% of bettors are informed. They all bet and so the bias is reduced.

40% of bettors are informed.

There are certainly many objections to There are certainly many objections to this model. But let’s ask how, if this this model. But let’s ask how, if this does describe what’s going on, changes does describe what’s going on, changes in the tax on the prize pool would in the tax on the prize pool would change the relationship between the change the relationship between the odds and the expected returns.odds and the expected returns. A tax discourages the informed bettors from playing A tax discourages the informed bettors from playing

the more marginal races (that is, those races the more marginal races (that is, those races offering an EV only slightly greater than one.)offering an EV only slightly greater than one.)

If the tax rate were very low, the informed players If the tax rate were very low, the informed players might be betting on all the races, thus doing their might be betting on all the races, thus doing their bit to reduce the bias created by the uninformed.bit to reduce the bias created by the uninformed.

But as the tax goes up, the informed could drop out But as the tax goes up, the informed could drop out of more and more races, leaving an outside of more and more races, leaving an outside observer with the impression that odds and returns observer with the impression that odds and returns are unrelated. are unrelated.

This simulation compares two racing seasons. The This simulation compares two racing seasons. The proportion of informed bettors is 20% in both cases. In proportion of informed bettors is 20% in both cases. In

the first scenario, the tax rate is zero, the informed the first scenario, the tax rate is zero, the informed bettors play every race and so help reduce the bias. In bettors play every race and so help reduce the bias. In the other, the tax rate is 15%, the informed bettors play the other, the tax rate is 15%, the informed bettors play

only some of the races and the odds/EV relationship only some of the races and the odds/EV relationship appears random. (Of course, the tax also shifts the appears random. (Of course, the tax also shifts the

entire odds curve down.) entire odds curve down.)

0.5

0.7

0.9

1.1

1.3

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Odds

Expe

cted V

alue

If t=0, all informed bettors bet all races and so help reduce the bias

If t=15%, informed bet only some races, meaning that there appears to be no relationship.

Let’s recap the predictions as to Let’s recap the predictions as to what would happen if the tax rate what would happen if the tax rate

goes up.goes up.1.1. If the odds are consistent with a representative bettor who If the odds are consistent with a representative bettor who

likes risk and expected utility, thenlikes risk and expected utility, then1.1. If taxes goes up, the favorite/longshot bias increases.If taxes goes up, the favorite/longshot bias increases.

2.2. If the odds are consistent with a representative bettor who is If the odds are consistent with a representative bettor who is risk neutral but with biased assessment of probability, then risk neutral but with biased assessment of probability, then

1.1. Changes in tax rates will not change favorite/longshot bias.Changes in tax rates will not change favorite/longshot bias.

3.3. If the odds are consistent with the Cournot-Nash equilibrium If the odds are consistent with the Cournot-Nash equilibrium of a game played among risk neutral, fully-informed players, of a game played among risk neutral, fully-informed players, thenthen

1.1. The bias may be present when tax rates are low, but as tax The bias may be present when tax rates are low, but as tax rates go up, the relationship between returns and odds may rates go up, the relationship between returns and odds may appear random.appear random.

Three types of data might Three types of data might be helpful in empirical testsbe helpful in empirical tests Comparisons across racetracks (Different racetracks have Comparisons across racetracks (Different racetracks have

different tax rates.)different tax rates.) Issue: tracks are different in other regards as well Issue: tracks are different in other regards as well

Comparisons across types of wagers at the same track. Comparisons across types of wagers at the same track. (Different types of wagers have different tax rates.)(Different types of wagers have different tax rates.) Issue: higher tax rates are imposed on very complex wagers Issue: higher tax rates are imposed on very complex wagers

that are difficult to handicap and may attract bettors with that are difficult to handicap and may attract bettors with different attitudes towards risk.different attitudes towards risk.

Comparisons across the same type of wager at the same Comparisons across the same type of wager at the same racetrack, where the tax rate varies randomly. (Some racetrack, where the tax rate varies randomly. (Some racetracks are allowed to encourage certain types of racetracks are allowed to encourage certain types of wagers by injecting money into the wagering poolwagers by injecting money into the wagering pool—”carryovers” and “guarantees”. )—”carryovers” and “guarantees”. ) Issue: the data is not easy to access and if I do get it, it will Issue: the data is not easy to access and if I do get it, it will

not be easy to analyze since there are often over 1 million not be easy to analyze since there are often over 1 million possible outcomes. possible outcomes.

At present I’ve only done some very At present I’ve only done some very preliminary comparisons between preliminary comparisons between two tracks.two tracks.

But let me show you the analysis But let me show you the analysis that I have so far. And even more that I have so far. And even more important!!important!!

Shamelessly beg for suggestions and Shamelessly beg for suggestions and advice on where to go nextadvice on where to go next mldavis@mail.cox.smu.edumldavis@mail.cox.smu.edu

A Tale of Two RacetracksA Tale of Two Racetracks

Belmont DownsBelmont Downs Location: New Location: New

YorkYork Racing Dates/year: Racing Dates/year:

9797 Approx. Average Approx. Average

Daily Total Wagers: Daily Total Wagers: $1.5 million$1.5 million

Tax rate on win Tax rate on win bet: bet: 14%-15%14%-15%

Suffolk DownsSuffolk Downs Location: BostonLocation: Boston Racing Dates/year: Racing Dates/year:

117117 Approx. Average Approx. Average

Daily Total Wagers: Daily Total Wagers: $1.15 million.$1.15 million.

Tax rate on win Tax rate on win bet: bet: 19%-20%19%-20%

In 2005 there were 35 days where Belmont In 2005 there were 35 days where Belmont and Suffolk both held races. My data includes and Suffolk both held races. My data includes information about every horse in entered in information about every horse in entered in every race on these days. Information every race on these days. Information includes. includes. Race information: Type of race (allowance, Race information: Type of race (allowance,

graded stakes, etc.), condition of race (maidens, graded stakes, etc.), condition of race (maidens, 3 year-olds, etc.) and purse.3 year-olds, etc.) and purse.

Horse information: Jockey/Trainer, weight Horse information: Jockey/Trainer, weight allowance, medication and equipment.allowance, medication and equipment.

Wager information: types of wagers and odds.Wager information: types of wagers and odds.

ResultsResults

Summary StatisticsSummary Statistics

  Combined Belmont Suffolk

  Mean Std Dev Mean Std Dev Mean

Std Dev

Race per Day 9.37 1.21 9.37 0.85 9.37 1.53

Horses Per Race 7.77 1.83 7.83 2.02 7.72 1.62

Purse * $25,927 $ 18,993 $40,861 $6,589 $11,135 $6,986

*In 2005 Belmont hosted several races with very high purses, including the Belmont Stakes and eight Breeder’s Cup races. The statistics in this row exclude those races.

First Impression (or Maybe First Impression (or Maybe Just Wishful Thinking)Just Wishful Thinking)

Except for the differences in purses, and tax rates, Except for the differences in purses, and tax rates, Belmont and Suffolk are very similar.Belmont and Suffolk are very similar. Remember, the purse goes to the owner of the winning Remember, the purse goes to the owner of the winning

horse, it has nothing to do with the return to the winning horse, it has nothing to do with the return to the winning bettor. This means that Belmont horses should be faster bettor. This means that Belmont horses should be faster than Suffolk horses (although it is not uncommon for than Suffolk horses (although it is not uncommon for horses to ship between tracks), But is there reason to horses to ship between tracks), But is there reason to think that horses at one track will be easier to handicap think that horses at one track will be easier to handicap than horses at another track?than horses at another track?

The bettors are likely to be very similar as well. The bettors are likely to be very similar as well. The vast majority of horse wagering in the U.S. does not The vast majority of horse wagering in the U.S. does not

come from those actually at the race. Most bets are made come from those actually at the race. Most bets are made from off-track betting and “account-wagering” (i.e., from off-track betting and “account-wagering” (i.e., internet). These are (usually) legal and the pools are internet). These are (usually) legal and the pools are combined. combined.

Casual empiricism (i.e., hanging out at the track) suggests Casual empiricism (i.e., hanging out at the track) suggests that bettors will play several different tracks on the same that bettors will play several different tracks on the same day. day.

Comparing OddsComparing Odds

BelmontBelmont SuffolkSuffolk

MeanMean Std DevStd Dev MeanMean Std DevStd Dev

14.5914.59 16.3816.38 13.8713.87 16.8016.80

The mean odds at Belmont are significantly greater than the odds at Suffolk (at the 5% level). This is consistent with the higher tax rate at Suffolk, which reduces the payout available from a given pool.

Comparing Distribution of Odds Between Comparing Distribution of Odds Between Tracks the Proportion of Longshots and Tracks the Proportion of Longshots and Favorites Appears to be About the Same.Favorites Appears to be About the Same.

QuantileQuantile Suffolk OddsSuffolk OddsBelmont Belmont

OddsOdds

100%100% 99.599.5 9595

99%99% 8282 73.573.5

95%95% 5252 52.252.2

90%90% 35.635.6 38.238.2

75%75% 16.916.9 18.918.9

50%50% 7.47.4 8.38.3

25%25% 3.23.2 3.73.7

10%10% 1.81.8 1.91.9

5%5% 1.21.2 1.31.3

1%1% 0.60.6 0.60.6

0%0% 0.10.1 0.40.4

Do Favorites at the Two Do Favorites at the Two Tracks Win at the Same Tracks Win at the Same

Rate?Rate?   Odds RankOdds Rank

TrackTrack 11 22 33 44 55 66 77 88 99 1010

BelmontBelmont 36.336.3 21.921.9 14.814.8 12.212.2 6.36.3 5.15.1 1.31.3 1.31.3 0.80.8 00

SuffolkSuffolk 40.940.9 19.119.1 16.316.3 8.18.1 5.95.9 4.74.7 2.52.5 0.90.9 1.31.3 0.30.3

Combined Combined 3939 20.320.3 15.615.6 9.99.9 6.16.1 4.84.8 22 1.11.1 1.11.1 0.20.2

The horses in each race were ranked by their odds (rank of 1 indicates the betting favorite). This table shows the percent of winners by odds rank. At Suffolk the favorite wins significantly more often than at Belmont. Longshots (those not ranked in the top three) win 27% of the races at Belmont and 23.7% of the races at Suffolk. While I don’t want to claim too much for this result, it does at least suggest that the high tax rate at Suffolk is not discouraging informed bettors.

Logistic Regression Logistic Regression [Pr (win) =f(odds)][Pr (win) =f(odds)]

   BelmontBelmont SuffolkSuffolk CombinedCombined

InterceptIntercept-0.7466-0.7466 -0.769-0.769 -0.7621-0.7621

(0.0001)(0.0001) (0.0001)(0.0001) (0.0001)(0.0001)

oddsodds-0.1468-0.1468 -0.1581-0.1581 -0.1522-0.1522

(0.0001)(0.0001) (0.0001)(0.0001) (0.0001)(0.0001)

The results of the logistic regression seem to suggest that overall odds, have the same relationship to winning at either track. That is, the coefficient on odds at Suffolk is not significantly bigger than at Belmont.

But this is circling around the really But this is circling around the really central question:central question:

Is the relationship between odds and Is the relationship between odds and returnsreturns different at the two tracks? different at the two tracks?

Do Returns Vary With Odds Rank?Do Returns Vary With Odds Rank?

      Odds RankOdds Rank

TrackTrack    11 22 33 44 55 66 77 88

BelmontBelmontoddsodds 1.661.66 3.523.52 5.725.72 8.378.37 1414 20.720.7 28.128.1 35.235.2

returnreturn -0.16-0.16 -0.12-0.12 -0.18-0.18 -0.01-0.01 -0.28-0.28 -0.21-0.21 -0.71-0.71 -0.32-0.32

SuffolkSuffolkoddsodds 1.521.52 2.982.98 4.694.69 7.697.69 11.711.7 19.219.2 26.226.2 3838

returnreturn -0.14-0.14 -0.31-0.31 -0.21-0.21 -0.37-0.37 -0.46-0.46 -0.15-0.15 -0.61-0.61 -0.31-0.31

These are the average profits on a $1 bet as well as the average odds grouped by odds rank (e.g., at Belmont, the average favorite went off at odds of 1.66 and always betting on the favorite would have result in a 16% loss). If you see a clear pattern, let me know.

Censored RegressionCensored Regression[Return = f(odds)][Return = f(odds)]

   BelmontBelmont SuffolkSuffolk CombinedCombined

InterceptIntercept-0.053-0.053 -0.236-0.236 -0.15621-0.15621

(0.5146)(0.5146) (0.0019)(0.0019) (0.0049)(0.0049)

oddsodds-0.013-0.013 -0.005-0.005 -0.00835-0.00835

(0.0005)(0.0005) (0.1751)(0.1751) (0.0011)(0.0011)

Betting on horses with higher odds lowers the expected return at Belmont but not at Suffolk.

Summary (1): Are These Numbers Summary (1): Are These Numbers Consistent With The Odds Being Consistent With The Odds Being Determined by Decision Makers Who Determined by Decision Makers Who Like Risk and are Consistent With Like Risk and are Consistent With Expected Utility?Expected Utility?

If this were the right model, we should expect If this were the right model, we should expect the bias towards favorites to be greater at the bias towards favorites to be greater at Suffolk than at Belmont.Suffolk than at Belmont.

This was not found in the data. In fact, whatever This was not found in the data. In fact, whatever differences there are seem to suggest that the differences there are seem to suggest that the bias is greater at the low tax track (Belmont) bias is greater at the low tax track (Belmont) than the high tax track.than the high tax track.

But it is way to soon to reject the modelBut it is way to soon to reject the model The difference in tax rates between the two tracks may The difference in tax rates between the two tracks may

be too small to stand out from all the other differences.be too small to stand out from all the other differences. The way I’m looking for “bias” may be too indirect to The way I’m looking for “bias” may be too indirect to

capture what’s really going on.capture what’s really going on.

Summary (2): Are These Numbers Summary (2): Are These Numbers Consistent With The Odds Being Consistent With The Odds Being

Determined by Decision Makers Who Determined by Decision Makers Who Have Biased Subjective Probabilities Have Biased Subjective Probabilities

but Care About Subjective EV?but Care About Subjective EV?

If this is the correct model, we should If this is the correct model, we should expect there to be no difference in the expect there to be no difference in the bias between the two tracks. bias between the two tracks.

It is hard to make a strong case that It is hard to make a strong case that there is a big difference between the there is a big difference between the two tracks.two tracks.

But the same cautions applyBut the same cautions apply

But it is way to soon to But it is way to soon to reach any firm conclusion reach any firm conclusion

about either median-bettor about either median-bettor modelmodel

The difference in tax rates between the The difference in tax rates between the two tracks may be too small to stand two tracks may be too small to stand out from all the other differences.out from all the other differences.

The way I’m looking for “bias” may be The way I’m looking for “bias” may be too indirect to capture what’s really too indirect to capture what’s really going on.going on.

Summary (3): Are These Numbers Summary (3): Are These Numbers Consistent With The Odds Being the Consistent With The Odds Being the Outcome of a Competition Between Outcome of a Competition Between Informed Bettors to Exploit Mistakes In Informed Bettors to Exploit Mistakes In Odds Set by Uninformed Bettors?Odds Set by Uninformed Bettors? That model implied that as tax rates go up, That model implied that as tax rates go up,

the informed bettors will be discouraged the informed bettors will be discouraged from betting on some races.from betting on some races.

That means that at tracks with higher tax That means that at tracks with higher tax rates, we should see more erratic pricing and rates, we should see more erratic pricing and less clear-cut evidence of a bias.less clear-cut evidence of a bias.

I’m not yet ready to conclude that I’ve found I’m not yet ready to conclude that I’ve found a favorite-longshot bias at Belmont and not a favorite-longshot bias at Belmont and not at Suffolk, but some of the numbers are at Suffolk, but some of the numbers are suggestive of that pattern. suggestive of that pattern.

What I Think Needs to be What I Think Needs to be Done About the TheoryDone About the Theory

Try to develop more general results Try to develop more general results for the expected utility models.for the expected utility models. Some of the conclusions were drawn Some of the conclusions were drawn

from simulating specific forms of the from simulating specific forms of the utility function.utility function.

Extend the non-expected utility Extend the non-expected utility models to include more general models to include more general functions for valuing gains and functions for valuing gains and losses.losses.

What I Think Needs to be What I Think Needs to be Done About the Empirical Done About the Empirical

AnalysisAnalysis Develop more precise ways of measuring Develop more precise ways of measuring

bias.bias. Consider using other conditioning Consider using other conditioning

variables to make sure that differences variables to make sure that differences between tracks that are being attributed between tracks that are being attributed to different tax rates aren’t really to different tax rates aren’t really reflecting something else.reflecting something else.

BEST IDEA: Get the data on carryovers BEST IDEA: Get the data on carryovers and guarantees so that we can actually and guarantees so that we can actually make comparisons of the same types of make comparisons of the same types of bets at the same track. bets at the same track.