The Favourite-Longshot Bias is the well-established tendency in most betting markets for bettors to bet too much on ‘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). This is strangely counterintuitive as it seems to offer a sure-fire way to make above-average returns in the betting booth. Assume, for example, that Mr. Smith and Mr. Jones both start with £1,000. Now Mr. Smith places a level £10 stake on 100 horses quoted at 2 to 1. Meanwhile, Mr. Jones 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? Surely the answer should be the same for both. Otherwise, either Mr. Smith or Mr. Jones would seem to be doing something very wrong. So let’s take a look.

The 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, but the same sort of pattern applies for any set of years we care to choose, or (with a few rare exceptions) pretty much any sport, anywhere.

In fact, 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 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, and this causes them to bias their perceptions of what they have won or lost in favour of longshots. 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. These studies demonstrate how gamblers tend to discount their losses, often as ‘near wins’ or the outcome of ‘fluke’ events, while bolstering their wins.

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. There is little evidence that the market offers opportunities for players to earn consistent profits, but they certainly do much better (lose a lot less) by a blind level-stakes strategy of backing favourites instead of longshots. Intuitively, we would think that people would wise up and switch their money away from the longshots to the favourites. In that case, favourites would become less good value, as their odds would shorten, and longshots would become better value as their odds would lengthen. But is doesn’t happen, despite a host of published papers pointing this out, as well as the Ladbrokes Pocket Companion. People continue to love their longshots, and are happy to pay a price for this love.

Are there other explanations for the persistence of the favourite-longshot bias? One explanation is based on consumer preference for risk. The idea here is that bettors are risk-loving and so prefer the risky series of long runs of losses followed by the odd big win to the less risky strategy of betting on favourites that will win more often albeit pay out less for each win. Such an assumption of risk-love by bettors, however, runs contrary to conventional explanations of financial behaviour which tend to assume people like to avoid risk. It’s also been argued that bettors are actually not risk-lovers but skewness-lovers, which would also explain a preference for backing longshots over favourites.

Another explanation that has been proposed for the existence of the bias is based on the existence of unskilled bettors in the context of high margins and other costs of betting which deter more skilled agents. These unskilled bettors find it more difficult to arbitrate between the true win probabilities of different horses, and so over-bet those offered at longer odds. One test of this hypothesis is to compare the size of the bias in person-to-person betting exchanges (characterised by lower margins) and bookmaker markets (higher margins). The bias was indeed lower in the former, a finding which is at least consistent with this theory.

So far, it should be noted that these are all demand-side explanations, i.e. based on the behaviour of bettors. Another explanation of at least some of the bias is the idea that odds-setters defend themselves against bettors who potentially have superior information to bookmakers by artificially squeezing odds at the longer end of the market. Even so, the favourite-longshot bias continues to exist in so-called ‘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 this odds-squeezing explanation, we can classify the remaining explanations 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 poor assessment of the true probabilities can be categorized as perception-based explanations.

The favourite-longshot bias has even been found in online poker, especially in lower-stake games. In that context, the evidence suggests that it was misperception of probabilities rather than risk-love that offered the best explanation for the bias.

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 1949. 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.

Appendix

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 = 11/4.

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 = 11/2

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’).

Exercise

Calculate the subjective odds (against) in this table assuming that f, the fixed fraction of losses undiscounted by the bettor, is a half.

Objective odds (against)       Subjective odds (against)

Evens

4/1

Infinity to 1

0/1