---long before this classic match race between Man O

1
Many Years Ago

• ---long before this classic match race between
  Man OWar and Sir Bartona small group of
  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 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 schemewhich had
  since grown into a multi-billion dollar
  industrymight be a useful way to test their
  theories about how markets work and about how
  people deal with uncertainty.

2
Transactions Costs, Preferences and the
Favorite/Longshot Bias

• Michael L. Davis
• Department of Finance
• Cox School of Business
• Southern Methodist University
• Dallas, Texas
• (mldavis_at_mail.cox.smu.edu)

3
Questions That Studying Horse Race Betting Might
Help Answer

• Market microstructure
• What makes a market more efficient?
• How do transactions costs influence markets?
• How does the distribution of information
  influence markets?

• Psychology of risk-taking
• Are choices consistent with expected utility?
• Or are the choices seen in horse race betting
  best explained by preferences that are non-linear
  in probabilities?

4
A Brief Caricature of Part of the Literature

• The early literature on horse racing asked
  whether the market for these state-contingent
  claims was efficient. (Usually taken to mean
  that the odds were consistent with the liklihood
  of winning.)
• The answer is NO!!
• Horse racing is characterized by the
  favorite/longshot bias.
• Longshots (horses that go off at high odds) have
  significantly lower expected returns than
  favorites.

5
How To Explain the Bias Part 1(Gamblers Like
to Gamble)

• The bias is consistent with a market where the
  marginal bettor is rational (in the sense that
  preferences are consistent with expected utility)
  but just likes risk.
• Ali (1977)
• Quandt (1986)
• Bettors are rational but enjoy playing the game
• Ziemba

6
How To Explain the Bias Part 2(The bias is
evidence that people dont care about expected
utility.)

• The bias is consistent with a market where the
  marginal bettor has preferences that are
  described by one of the many models that are
  non-linear in probability (rank-dependent
  utility, cumulative prospect theory ..

7
How To Explain the Bias Part 3(Bettors are
Rational But the Market is Crazy)

• Key Insight Parimutual betting is both a
  contest against nature (pick the winner) but
  also against the other bettors (pick the horse
  offering the highest expected payoff).
• And so even if bettors were fully informed and
  risk neutral, the bias might arise as a
  consequence of this game
• Potters and Witt
• Ottaviani and Sorensen

8
Problem These Explanations of the Bias are all
Terrific

• That is, they are logical and seem to fit the
  data.
• And so how do we use racetrack data to really
  distinguish between models?
• Perhaps we should compare goodness of fit
  (Julliene and Salanie).
• Or maybe compare different types of bets
  contrasting, say, compound gambles with single
  gambles (Snowberg and Wolfers).

9
A Missing Link, Transactions Costs

• Parimutual pools are heavily taxed (somewhere
  between 14 and 25 at U.S. tracks).
• The tax rate varies between tracks, and types of
  bets.
• Even more intriguing, because of carryovers and
  guarantees the tax rate varies randomly from
  day-to-day.

10
Can This Variance in Tax Rates Help Us
Distinguish Models?

• First Step Include tax rates in the usual
  models.
• If it turns out that different models imply
  different reactions to changes in the tax rate,
  then maybe weve got a tool that could falsify
  one or more explanations.
• (Obvious) Second Step If the models suggest
  differences, get the data and do some tests.

11
Basics (Notation and Assumptions)

• Two-horse race between the favorite (f) and the
  longshot (l).
• pgt 0.5 is the objective probability that the
  favorite will win.
• Oh profit from 1 winning bet on horse h (the
  odds).
• In parimutal betting
• Oh (1-t)/Wh -1
• (t tax rate, Wh of pool bet on h)

12
Note to Eliminate Needless Confusion

• I have been told that the British state the odds
  differently than we do in the U.S.
• Throughout this presentation, I will follow the
  U.S. 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 U.S. parlance.
• The horse would (I think) be said to have odds
  of 1 to 2 on in Britain.
• I have also been told that the British have a
  different spelling of the word favorite.

13
Odds Are Constrained By the Tax on the Pool

• Since Wl1-Wf
• Ol (1-t)(Of1)/(Oft)-1 h(Of,t)
• Feasible Odds satisfy this constraint
• Increasing the tax rate shifts the feasible odds
  down

14
These odds are feasible given the tax rate
Odds on this curve are feasible when the there
are no taxes
Feasible odds when t15
15
A Wager Indifference Curve is one where the
bettor is indifferent between a bet on either
horse.

• Suppose the bettor is a risk-lover who cares
  about expected utility
• A winning bet on h gives utility U(Oh)
• Normalize the utility to zero if the bet is lost
  utility
• U(-1)0.
• In equilibrium, the bettor must be indifferent to
  a wager on either horse
• pU(Of) (1-p)U(Ol)

16
Wager indifference for a bettor whose utility is
U(x) (e2 -e-2x)/2 (A risk-lover with CARA, and
U(-1) 0)P.60
If odds are here, the bettor prefers the longshot
If odds are here, the bettor prefers the favorite
17
Wager Neutral Equilibrium The intersection marks
the only feasible odds where the marginal bettor
is just indifferent between the two horses
equilibrium
18
What happens when the tax rate changes?
19
Raising the tax rate will

• Lower the equilibrium odds on both horse
• A sensible conclusion since a higher tax rate
  means there is less in the pool to pay out to the
  winners
• Raise the odds on the longshot relative to the
  longshot
• That is increase the favorite/longshot bias

20
Concern The wager neutral equilibrium might
actually result in the favorite offering a
positive expected value bet.

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

21
Model 2 Preferences that are not linear in
probabilities

• Issue There are lots of models to pick from,
  which should be tested?
• Obvious answer (especially for a summer-time
  conference in a great city) the simplest one.

22
Risk-Neutral/Subjective Probability

• p(p) is the bettors subjective belief of the
  probability that the favorite will win.
• Suppose equilibrium is the point where the
  subjective expected gain from a bet on either
  horse is the same (same definition as Snowberg
  and Wolfers).
• p(p)(Of1) p(1-p)(Ol1)

23
Same thing, expressed in terms of proportion of
pool.

• Remember, if wh is the proportion of the total
  pool bet on horse h, then
• Oh1(1-t)/wh
• Thus, the equilibrium condition can be written as
  
• p(p)(1-t)/wf p(1-p)(1-t)/(1-wf), or simply
• (1-wf)/wf p(p)/ p(1-p)
• If this model is the right one, then the tax rate
  shouldnt matterthe proportion of the pool (and
  hence the odds) should depend only on subjective
  probabilities.

24
So far I have outlined median bettor models.
That is, the observed odds are assumed to be
consistent with the preferences of some typical
horse-player.

• If this subjective probability/risk-neutral
  model correctly describes how the market works,
  then the relative proportions bet on either horse
  should not vary with the tax rate.
• This is different than the risk-loving, expected
  utility model, which implied that as the tax rate
  changed, the bias in favor of the longshot should
  increase.

25
So far I have outlined median bettor
models--that is, the observed odds are assumed to
be consistent with the preferences of some
typical horse-player. Here is a summary

• Expected Utility Risk Love
• Increasing the tax rate on the betting pool
  should increase the bias in favor of the longshot

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

26
Model 3 (Some) Rational Bettors in an Irrational
Market

• Assume two types of bettors
• Informed (know pas well as some other stuff).
• Ithe proportion of the potential total pool who
  are informed
• uninformed (bets may not be consistent with p)
• Uf of uninformed betting on favorite

27
The Right Number of Informed Bettors Will Correct
the Market

• Assume
• p.75 (that is, odds of 1 to 3 would be a zero EV
  bet)
• Uf.50 (half of the uninformed bet the favorite)
• If only the uninformed bet, there will be a bias
• The favorite goes off at odds of 1 to 1 (that is,
  pays 2.00 on 1 bet).
• But the favorite wins 75 of the time
  (EV.75x21.5)

28
The Right Number of Informed Bettors Will Correct
the Market

• But now suppose that the informed make up half of
  the pool (I.50).
• If the informed all bet the favorite, then
• The favorite goes off at odds of 1 to 3
  1/(.5x.5.5)-1
• Thus, EV1.

29
But this argument depends on their being the
correct number of informed betters.

• If there are less than the correct number of
  informed bettors (equivalently, if the informed
  bettors face some sort of budget constraint),
  then the bias may still exist.
• To continue with the previous example if the
  informed make up only 20 of the total pool, then
  the favorite goes off at odds of 3 to 2 and still
  has a positive EVthe bias is reduced, but not
  eliminated.
• Even more striking, too many informed bettors can
  be a bad thing, in that if there are too many
  informed bettors, the odds might be even more
  distorted.

30
Why too many informed bettors might be bad

• Before making a bet, the informed know p, I and
  Uf
• They assume that all the other informed bettors
  know this too and so they assume that all the
  other informed will be doing whatever they do.
• If there were a lot of informed bettors, all of
  whom tried to take advantage of what appears to
  be a positive EV opportunity, they would actually
  lose money. To avoid this, they only bet on
  those races where the uninformed have gotten it
  so wrong that the bet is still a good deal even
  when all the other informed bettors recognize and
  act on the same thing.
• This means that sometimes a horse will go off at
  odds implying a positive expected value, but the
  informed wont bet.
• In game-speak, this is really just a kind of
  Cournot-Nash equilibrium, where there is no
  collusion between the players.

31
Formalize the story What the informed do
depends on how the uninformed bet

• Bet on the favorite if
• UltUf(P(1-t)-I )/(1-I)
• Bet on the longshot if
• UgtUl tp(1-t)/(1-I)
• Dont bet if
• UfU Ul

32
Results of Simulation Describing Relationship
Between Odds and EV of Bet (The exercise assumed
several different types of races, where the
uninformed under-bet the favorite.)
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
33
Things can seem even stranger if the informed bet
some races but not others. This is the previous
simulation exercise, except with a different
proportion of informed. When there are a greater
proportion of informed bettors, the relationship
between odds and EV appears almost random. This
is because the informed are only betting races
with a strong favorite (high p)
40 of bettors are informed.
20 of bettors are informed. They all bet and so
the bias is reduced.
34
There are certainly many objections to this
model. But lets ask how, if this does describe
whats going on, changes in the tax on the prize
pool would change the relationship between the
odds and the expected returns.

• A tax discourages the informed bettors from
  playing the more marginal races (that is, those
  races offering an EV only slightly greater than
  one.)
• If the tax rate were very low, the informed
  players might be betting on all the races, thus
  doing their bit to reduce the bias created by the
  uninformed.
• But as the tax goes up, the informed could drop
  out of more and more races, leaving an outside
  observer with the impression that odds and
  returns are unrelated.

35
This simulation compares two racing seasons. The
proportion of informed bettors is 20 in both
cases. In the first scenario, the tax rate is
zero, the informed bettors play every race and
so help reduce the bias. In the other, the tax
rate is 15, the informed bettors play only some
of the races and the odds/EV relationship appears
random. (Of course, the tax also shifts the
entire odds curve down.)
If t0, all informed bettors bet all races and so
help reduce the bias
If t15, informed bet only some races, meaning
that there appears to be no relationship.
36
Lets recap the predictions as to what would
happen if the tax rate goes up.

• If the odds are consistent with a representative
  bettor who likes risk and expected utility, then
• If taxes goes up, the favorite/longshot bias
  increases.
• If the odds are consistent with a representative
  bettor who is risk neutral but with biased
  assessment of probability, then
• Changes in tax rates will not change
  favorite/longshot bias.
• If the odds are consistent with the Cournot-Nash
  equilibrium of a game played among risk neutral,
  fully-informed players, then
• The bias may be present when tax rates are low,
  but as tax rates go up, the relationship between
  returns and odds may appear random.

37
Three types of data might be helpful in empirical
tests

• Comparisons across racetracks (Different
  racetracks have different tax rates.)
• Issue tracks are different in other regards as
  well
• Comparisons across types of wagers at the same
  track. (Different types of wagers have different
  tax rates.)
• Issue higher tax rates are imposed on very
  complex wagers that are difficult to handicap and
  may attract bettors with different attitudes
  towards risk.
• Comparisons across the same type of wager at the
  same racetrack, where the tax rate varies
  randomly. (Some racetracks are allowed to
  encourage certain types of wagers by injecting
  money into the wagering poolcarryovers and
  guarantees. )
• 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 possible outcomes.

38

• At present Ive only done some very preliminary
  comparisons between two tracks.
• But let me show you the analysis that I have so
  far. And even more important!!
• Shamelessly beg for suggestions and advice on
  where to go next
• mldavis_at_mail.cox.smu.edu

39
A Tale of Two Racetracks

• Belmont Downs
• Location New York
• Racing Dates/year 97
• Approx. Average Daily Total Wagers 1.5 million
• Tax rate on win bet 14-15

• Suffolk Downs
• Location Boston
• Racing Dates/year 117
• Approx. Average Daily Total Wagers 1.15
  million.
• Tax rate on win bet 19-20

40
In 2005 there were 35 days where Belmont and
Suffolk both held races. My data includes
information about every horse in entered in every
race on these days. Information includes.

• Race information Type of race (allowance, graded
  stakes, etc.), condition of race (maidens, 3
  year-olds, etc.) and purse.
• Horse information Jockey/Trainer, weight
  allowance, medication and equipment.
• Wager information types of wagers and odds.
• Results

41
Summary Statistics
  Combined Combined Belmont Belmont Suffolk 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 Breeders Cup races. The statistics in
this row exclude those races.
42
First Impression (or Maybe Just Wishful Thinking)

• Except for the differences in purses, and tax
  rates, Belmont and Suffolk are very similar.
• Remember, the purse goes to the owner of the
  winning horse, it has nothing to do with the
  return to the winning bettor. This means that
  Belmont horses should be faster than Suffolk
  horses (although it is not uncommon for horses to
  ship between tracks), But is there reason to
  think that horses at one track will be easier to
  handicap than horses at another track?
• The bettors are likely to be very similar as
  well.
• The vast majority of horse wagering in the U.S.
  does not come from those actually at the race.
  Most bets are made from off-track betting and
  account-wagering (i.e., internet). These are
  (usually) legal and the pools are combined.
• Casual empiricism (i.e., hanging out at the
  track) suggests that bettors will play several
  different tracks on the same day.

43
Comparing Odds
Belmont Belmont Belmont Suffolk Suffolk
Mean Std Dev Std Dev Mean Std Dev
14.59 14.59 16.38 13.87 16.80
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.
44
Comparing Distribution of Odds Between Tracks the
Proportion of Longshots and Favorites Appears to
be About the Same.
Quantile Suffolk Odds Belmont Odds
100 99.5 95
99 82 73.5
95 52 52.2
90 35.6 38.2
75 16.9 18.9
50 7.4 8.3
25 3.2 3.7
10 1.8 1.9
5 1.2 1.3
1 0.6 0.6
0 0.1 0.4
45
Do Favorites at the Two Tracks Win at the Same
Rate?
  Odds Rank Odds Rank Odds Rank Odds Rank Odds Rank Odds Rank Odds Rank Odds Rank Odds Rank Odds Rank
Track 1 2 3 4 5 6 7 8 9 10
Belmont 36.3 21.9 14.8 12.2 6.3 5.1 1.3 1.3 0.8 0
Suffolk 40.9 19.1 16.3 8.1 5.9 4.7 2.5 0.9 1.3 0.3
Combined 39 20.3 15.6 9.9 6.1 4.8 2 1.1 1.1 0.2
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 dont 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.
46
Logistic Regression Pr (win) f(odds)
  Belmont Suffolk Combined
Intercept -0.7466 -0.769 -0.7621
Intercept (0.0001) (0.0001) (0.0001)
odds -0.1468 -0.1581 -0.1522
odds (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.
47

• But this is circling around the really central
  question
• Is the relationship between odds and returns
  different at the two tracks?

48
Do Returns Vary With Odds Rank?
    Odds Rank Odds Rank Odds Rank Odds Rank Odds Rank Odds Rank Odds Rank Odds Rank
Track   1 2 3 4 5 6 7 8
Belmont odds 1.66 3.52 5.72 8.37 14 20.7 28.1 35.2
Belmont return -0.16 -0.12 -0.18 -0.01 -0.28 -0.21 -0.71 -0.32
Suffolk odds 1.52 2.98 4.69 7.69 11.7 19.2 26.2 38
Suffolk return -0.14 -0.31 -0.21 -0.37 -0.46 -0.15 -0.61 -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.
49
Censored RegressionReturn f(odds)
  Belmont Suffolk Combined
Intercept -0.053 -0.236 -0.15621
Intercept (0.5146) (0.0019) (0.0049)
odds -0.013 -0.005 -0.00835
odds (0.0005) (0.1751) (0.0011)
Betting on horses with higher odds lowers the
expected return at Belmont but not at Suffolk.
50
Summary (1) Are These Numbers Consistent With
The Odds Being Determined by Decision Makers Who
Like Risk and are Consistent With Expected
Utility?

• If this were the right model, we should expect
  the bias towards favorites to be greater at
  Suffolk than at Belmont.
• This was not found in the data. In fact,
  whatever differences there are seem to suggest
  that the bias is greater at the low tax track
  (Belmont) than the high tax track.
• But it is way to soon to reject the model
• The difference in tax rates between the two
  tracks may be too small to stand out from all the
  other differences.
• The way Im looking for bias may be too
  indirect to capture whats really going on.

51
Summary (2) Are These Numbers Consistent With
The Odds Being Determined by Decision Makers Who
Have Biased Subjective Probabilities but Care
About Subjective EV?

• If this is the correct model, we should expect
  there to be no difference in the bias between the
  two tracks.
• It is hard to make a strong case that there is a
  big difference between the two tracks.
• But the same cautions apply

52
But it is way to soon to reach any firm
conclusion about either median-bettor model

• The difference in tax rates between the two
  tracks may be too small to stand out from all the
  other differences.
• The way Im looking for bias may be too
  indirect to capture whats really going on.

53
Summary (3) Are These Numbers Consistent With
The Odds Being the Outcome of a Competition
Between Informed Bettors to Exploit Mistakes In
Odds Set by Uninformed Bettors?

• That model implied that as tax rates go up, the
  informed bettors will be discouraged from betting
  on some races.
• That means that at tracks with higher tax rates,
  we should see more erratic pricing and less
  clear-cut evidence of a bias.
• Im not yet ready to conclude that Ive found a
  favorite-longshot bias at Belmont and not at
  Suffolk, but some of the numbers are suggestive
  of that pattern.

54
What I Think Needs to be Done About the Theory

• Try to develop more general results for the
  expected utility models.
• Some of the conclusions were drawn from
  simulating specific forms of the utility
  function.
• Extend the non-expected utility models to include
  more general functions for valuing gains and
  losses.

55
What I Think Needs to be Done About the Empirical
Analysis

• Develop more precise ways of measuring bias.
• Consider using other conditioning variables to
  make sure that differences between tracks that
  are being attributed to different tax rates
  arent really reflecting something else.
• BEST IDEA Get the data on carryovers and
  guarantees so that we can actually make
  comparisons of the same types of bets at the same
  track.