Many Years Ago …

1. Many Years Ago … ---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. 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 scheme—which had 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.

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@mail.cox.smu.edu)

3. 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? Questions That Studying Horse Race Betting Might Help Answer

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 don’t 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 we’ve 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). • p> 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. Since Wl=1-Wf Ol = (1-t)(Of+1)/(Of+t)-1 = h(Of,t) “Feasible Odds” satisfy this constraint Increasing the tax rate shifts the feasible odds down Odds Are Constrained By the Tax on the Pool

14. These odds are feasible given the tax rate Feasible odds when t=15% Odds on this curve are feasible when the there are no taxes

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 Indifference Feasible odds Wager Neutral Equilibrium The intersection marks the only feasible odds where the marginal bettor is just indifferent between the two horses equilibrium

18. Feasible odds (t=20%) Feasible odds (t=0) 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 • π(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). • π(p)(Of+1) = π(1-p)(Ol+1)

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 Oh+1=(1-t)/wh • Thus, the equilibrium condition can be written as • π(p)(1-t)/wf= π(1-p)(1-t)/(1-wf), or simply • (1-wf)/wf= π(p)/ π(1-p) • If this model is the right one, then the tax rate shouldn’t matter—the 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. 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. 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

26. Model 3: (Some) Rational Bettors in an Irrational Market • Assume two types of bettors • Informed (know p—as well as some other stuff). • I=the 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=.75x2=1.5)

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, EV=1.

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 EV—the 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 won’t 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 U<Uf=(P(1-t)-I )/(1-I) • Bet on the longshot if U>Ul =[t+p(1-t)]/(1-I) • Don’t bet if Uf≤U ≤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 let’s ask how, if this does describe what’s 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 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.

36. Let’s 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 pool—”carryovers” 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 I’ve 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@mail.cox.smu.edu

39. 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% A Tale of Two Racetracks

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 *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.

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 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.

45. Do Favorites at the Two Tracks Win at the Same Rate? 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.

46. Logistic Regression [Pr (win) =f(odds)] 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? 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 Regression[Return = f(odds)] 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 I’m looking for “bias” may be too indirect to capture what’s really going on.