Can the Henery Hypothesis Explain the Favourite-Longshot Bias?
The Favourite-Longshot Bias is the well-established tendency in most betting markets for bettors to over-bet ‘longshots’ (events with long odds, i.e. low probability events) and to relatively under-bet ‘favourites’ (events with short odds, i.e. high probability events).
Assume, for example, that Mr. Miller and Mr. Stiller both start with £1,000.
Now Mr. Miller places a level £10 stake on 100 horses quoted at 2 to 1
Mr. Stiller places a level £10 stake on 100 horses quoted at 20 to 1.
Who is likely to end up with more money at the end?
My Ladbrokes Flat Season Pocket Companion for 1990 provides a nicely laid out piece of evidence here for British flat horse racing between 1985 and 1989. The table conveniently presented in the Companion shows that not one out of 35 favourites sent off at 1/8 or shorter (as short as 1/25) lost between 1985 and 1989. This means a return of between 4% and 12.5% in a couple of minutes, which is an astronomical rate of interest. The point being made is that broadly speaking the shorter the odds, the better the return. The group of ‘white hot’ favourites (odds between 1/5 and 1/25) won 88 out of 96 races for a 6.5% profit. The following table looks at other odds groupings.
Odds Wins Runs Profit %
1/5-1/2 249 344 +£1.80 +0.52
4/7-5/4 881 1780 -£82.60 -4.64
6/4 -3/1 2187 7774 -£629 -8.09
7/2-6/1 3464 21681 -£2237 -10.32
8/1-20/1 2566 53741 -£19823 -36.89
25/1-100/1 441 43426 -£29424 -67.76
An interesting argument advanced by the Strathclyde-based statistician Dr. Robert Henery in 1985 is that the favourite-longshot bias is a consequence of bettors discounting a fixed fraction of their losses, i.e. they underweight their losses compared to their gains.
This argument also explains an observed link between the sum of bookmakers’ prices and the number of runners in a race. The prices being summed here are simply the odds. If, for example, odds of 3/1 (against) are offered about each of the five horses in a race, the implied probability of winning for each horse is ¼ and the sum of prices is 5/4.
In this context, an ‘over-round’ is defined as the excess of the sum of prices over 1, in this case ¼.
The rationale behind Henery’s hypothesis is that bettors will tend to explain away and therefore discount losses as atypical, or unrelated to the judgment of the bettor.
This is consistent with contemporaneous work on the psychology of gambling, such as Gilovich in 1983 and Gilovich and Douglas in 1986.
These studies demonstrate how gamblers tend to discount their losses, often as ‘near wins’ or the outcome of ‘fluke’ events, while bolstering their wins.
Let’s look more closely at how the Henery odds transformation works.
If the true probability of a horse losing a race is q, then the true odds against winning are q/(1-q).
For example, if the true probability of a horse losing a race (q) is ¾, the chance that it will win the race is ¼, i.e. 1- ¾. The odds against it winning are: q/(1-q) = 3/4/(1-3/4) = 3/4/(1/4) = 3/1.
Henery now applies a transformation whereby the bettor will assess the chance of losing not as q, but as Q which is equal to fq, where f is the fixed fraction of losses undiscounted by the bettor.
If, for example, f = ¾, and the true chance of a horse losing is ½ (q=1/2), then the bettor will rate subjectively the chance of the horse losing as Q = fq.
So Q = ½. ¾ = 3/8, i.e. a subjective chance of winning of 5/8.
So the perceived (subjective) odds of winning associated with true (objective odds) of losing of 50% (Evens, i.e. q=1/2) is 3/5 (60%), i.e. odds-on.
This is derived as follows:
Q/(1-Q) = fq/(1-fq) = 3/8/(1-3/8) = 3/8/(5/8) = 3/5
If the true probability of a horse losing a race is 80%, so that the true odds against winning are 4/1 (q = 0.8), then the bettor will assess the chance of losing not as q, but as Q which is equal to fq, where f is the fixed fraction of losses undiscounted by the bettor.
If, for example, f = ¾, and the true chance of a horse losing is 4/5 (q=0.8), then the bettor will rate subjectively the chance of the horse losing as Q = fq.
So Q = 3/4. 4/5 = 12/20, i.e. a subjective chance of winning of 8/20 (2/5).
So the perceived (subjective) odds of winning associated with true (objective odds) of losing of 80% (4 to 1, i.e. q=0.8) is 6/4 (40%).
This is derived as follows:
Q/(1-Q) = fq/(1-fq) = 12/20 / (1-12/20) = 12/8 = 6/4
To take this to the limit, if the true probability of a horse losing a race is 100%, so that the true odds against winning are ∞ to 1 against (q = 1), then the bettor will again assess the chance of losing not as q, but as Q which is equal to fq, where f is the fixed fraction of losses undiscounted by the bettor.
If, for example, f = ¾, and the true chance of a horse losing is 100% (q=1), then the bettor will rate subjectively the chance of the horse losing as Q = fq.
So Q = 3/4. 1 = 3/4, i.e. a subjective chance of winning of 1/4.
So the perceived (subjective) odds of winning associated with true (objective odds) of losing of 100% (∞ to 1, i.e. q=1) is 3/1 (25%).
This is derived as follows:
Q/(1-Q) = fq/(1-fq) = 3/4 / (1/4) = 3/1
Similarly, if the true probability of a horse losing a race is 0%, so that the true odds against winning are 0 to 1 against (q = 0), then the bettor will assess the chance of losing not as q, but as Q which is equal to fq, where f is the fixed fraction of losses undiscounted by the bettor.
If, for example, f = ¾, and the true chance of a horse losing is 0% (q=0), then the bettor will rate subjectively the chance of the horse losing as Q = fq.
So Q = 3/4. 0 = 0, i.e. a subjective chance of winning of 1.
So the perceived (subjective) odds associated of winning with true (objective odds) of losing of 0% (0 to 1, i.e. q=0) is also 0/1.
This is derived as follows:
Q/(1-Q) = fq/(1-fq) = 0 / 1 = 0/1
This can all be summarised in a table.
Objective odds (against) Subjective odds (against) | |
Evens 3/5 | |
4/1 6/4 | |
Infinity to 1 3/1 | |
0/1 0/1 |
We can now use these stylised examples to establish the bias.
In particular, the implication of the Henery odds transformation is that, for a given f of ¾, 3/5 is perceived as fair odds for a horse with a 1 in 2 chance of winning.
In fact, £100 wagered at 3/5 yields £160 (3/5 x £100, plus stake returned) half of the time (true odds = evens), i.e. an expected return of £80.
£100 wagered at 6/4 yields £250 (6/4 x £100, plus the stake back) one fifth of the time (true odds = 4/1), i.e. an expected return of £50.
£100 wagered at 3/1 yields £0 (3/1 x £100, plus the stake back) none of the time (true odds = Infinity to 1), i.e. an expected return of £0.
It can be shown that the higher the odds the lower is the expected rate of return on the stake, although the relationship between the subjective and objective probabilities remains at a fixed fraction throughout.
Now on to the over-round.
The same simple assumption about bettors’ behaviour can explain the observed relationship between the over-round (sum of win probabilities minus 1) and the number of runners in a race, n.
If each horse is priced according to its true win probability, then over-round = 0. So in a six horse race, where each has a 1 in 6 chance, each would be priced at 5 to 1, so none of the lose probability is shaded by the bookmaker. Here the sum of probabilities = (6 x 1/6) – 1 = 0.
If only a fixed fraction of losses, f, is counted by bettors, the subjective probability of losing on any horse is f(qi), where qi is the objective probability of losing for horse i, and the odds will reflect this bias, i.e. they will be shorter than the true probabilities would imply. The subjective win probabilities in this case are now 1-f(qi), and the sum of these minus 1 gives the over-round.
Where there is no discounting of the odds, the over-round (OR) = 0, i.e. n times correct odds minus 1. Assume now that f = ¾, i.e. ¾ of losses are counted by the bettor.
If there is discounting, then the odds will reflect this, and the more runners the bigger will be the over-round.
So in a race with 5 runners, q is 4/5, but fq = 3/4 x 4/5 = 12/20, so subjective win probability = 1-fq = 8/20, not 1/5. So OR = (5 x 8/20) – 1 = 1.
With 6 runners, fq = ¾ x 5/6 = 15/24, so subjective win probability = 1 – fq = 9/24. OR = (6x 9/24) – 1 = (54/24) -1 = 1_{1/4. }
With 7 runners, fq = ¾ x 6/7 = 18/28, so subjective win probability = 1-fq = 10/28. OR = (7 x 10/28) – 1 = 42/28 = 1_{1/2}
If there is no discounting, then the subjective win probability equals the actual win probability, so an example in a 5-horse is that each has a win probability of 1/5. Here, OR = (5×1/5) – 1 = 0. In a 6-horse race, with no discounting, subjective probability = 1/6. OR = (6 x 1/6) – 1 = 0.
Hence, the over-round is linearly related to the number of runners, assuming that bettors discount a fixed fraction of losses (the ‘Henery Hypothesis’).
If the Henery Hypothesis is correct as a way of explaining the favourite-longshot bias, the bias can be explained as the natural outcome of bettors’ pre-existing perceptions and preferences.
This is quite consistent with a market efficiently processing the information available to it. Moreover, there is little evidence that the market offers opportunities for market players to earn abnormal returns or positive profits. Thus although possibilities clearly exist for earning above-average returns on the basis of weak form information, there is no convincing evidence that this contradicts a wider conceptualisation of this type of information efficiency.
Are there other explanations for the favourite-longshot bias, and the observed link between over-round and runners, which do not rely on the Henery Hypothesis?
One explanation is based on consumer preference for risk. A seminal article by Richard Emeric Quandt in the Quarterly Journal of Economics in 1986 explains the existence of the bias as a natural and necessary consequence of equilibrium in a market characterised by risk-loving bettors with homogeneous beliefs. As such, this idea that bettors are risk-loving runs contrary to conventional explanations of financial behaviour which tend to assume risk-aversion. It is possible however, that bettors should be classified differently to participants in other types of financial market, not least because of consumption benefits from racetrack and other types of betting which may not be replicated elsewhere.
Joe Golec and Maurry Tamarkin (1998, Journal of Political Economy) seek to arbitrate between the hypothesis of risk-loving bettors and a hypothesis that bettors are in fact skewness-lovers, arguing in favour of the latter explanation for the existence of a favourite-longshot bias in betting markets.
William Hurley and Lawrence McDonough (1995, American Economic Review) propose a quite different theoretical model of the favourite-longshot bias, which requires neither a hypothesis of risk-loving nor skewness-loving behaviour. Instead, the bias can arise in a risk-neutral environment, populated by at least some uninformed bettors and unsophisticated bettors, as a consequences of positive transactions and/or information costs. Michael Smith, David Paton and Leighton Vaughan Williams (2006, Economica) compare the size of the bias in person-to-person betting exchanges (characterised by lower margins/transactions costs) and bookmaker markets (higher margins/costs). They find the bias to be lower in the former, a finding which is at least consistent with this explanation.
So far, it should be noted that these are all demand-side explanations.
A major challenge to demand-side explanations of the bias was proposed by Hyun Song Shin (1991, Economic Journal), based on the idea that odds-setters respond to the adverse selection problem posed by insiders (bettors with superior information to bookmakers) by artificially squeezing odds at the longer end of the market. The consequence of this price-setting behaviour is for the betting odds to relatively understate the winning chances of favourites and to overstate the winning chances of longshots. This is the traditional favourite-longshot bias. Another implication of this modelling of odds-setting is that the over-round (the sum of implied probabilities in the odds) will tend to be greater as the number of runners increases, because more runners implies higher odds.
While Shin’s modelling can explain a favourite-longshot bias in betting markets characterised by odds-setters, and also a link between the number of runners and the bookmakers’ over-round, it can be shown (Vaughan Williams and Paton, 1997, Economic Journal) that identical results may result from demand-side explanations. To help arbitrate between these competing hypotheses, Vaughan Williams and Paton employ a large data set to distinguish between two types of race, on the basis of their relative potential for insider trading. It is shown that the correlation between the number of runners and the sum of prices is restricted to those races in which there are clear possibilities for the use of inside information. This lends empirical support to Shin’s supply-side explanation of the phenomenon. Even so, the favourite-longshot bias continues to exist in pari-mutuel markets, in which there are no odds-setters, but instead a pool of all bets which is paid out (minus fixed operator deductions) to winning bets.
To the extent that the favourite-longshot bias cannot be fully explained by the adverse selection problem facing odds-setters (certainly the case in pari-mutuel betting markets), most explanations can be classified as either preference-based or perception-based. Risk love or skewness love are examples of preference-based explanations.
Discounting of losses or other explanations based on a miscalibration of probabilities can be categorized as perception-based explanations. Marco Ottaviani and Peter Sorensen (2009, American Economic Journal), for example, show that information asymmetries between bettors may lead to misperceptions of the true probabilities of horses winning.
Behavioural theories suggest that cognitive errors and misperceptions of probabilities play a role in market mispricing. These theories incorporate laboratory studies by cognitive psychologists which show that people are systematically poor at discerning between small and tiny probabilities, and hence price both similarly. Further, people express a strong preference for certainty over extremely likely outcomes, leading highly probable gambles to be under-priced. These results form an important foundation of Prospect Theory (Daniel Kahneman and Amos Tversky, 1979).
A number of papers seek to arbitrate between preference and perceptions based explanations of the favourite-longshot bias. An example is Erik Snowberg and Justin Wolfers (2010, Journal of Political Economy), who use a novel data set comparing behaviour in simple win pools and more complex compound bets (e.g. exactas, involving identification of first and second place) to seek to discriminate between these explanations. Their results, they argue, are more consistent with misperceptions rather than risk-love. The bias persists in equilibrium because misperceptions are not large enough to generate profit opportunities for unbiased bettors. That said, the cost of the bias is still large, and de-biasing an individual bettor could reduce their costs of betting substantially.
A more recent paper seeks to extend this analysis into the world of online poker. In ‘Towards an understanding of the origins of the favourite-longshot bias: Evidence from online poker markets, a real-money natural laboratory’, first published online in Economica in 2016, Leighton Vaughan Williams and others find a favourite-longshot bias in online poker play, especially in lower stakes games. “We find that misperception rather than risk-love offers the best explanation for the behaviour that we identify.”
In conclusion, the favourite-longshot bias is a well-established market anomaly in sports betting markets, which can be traced in the published academic literature as far back as Richard Griffith (1949, American Journal of Psychology). Explanations can broadly be divided into demand-based and supply-based, preference-based and perceptions-based. A significant amount of modern research has been focused on seeking to arbitrate between these competing explanations of the bias by formulating predictions as to how data derived from these markets would behave if one or other explanation was correct. A compromise position, which may or may not be correct, is that all of these explanations have some merit, the relative merit of each depending on the market context.