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=0.2), 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 20% (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.
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? Any coherent theory of the favourite-longshot bias should be able to explain both observed regularities. That is a topic for another time.