The question
"The favourite is 3.1 to win. The place market has it at 1.80. Is that a fair price?"
How to price forecasts and place bets from win odds is a classic racing question. Statometrics itself focuses on football, and this lesson is here because the idea underneath, "given the winner, re-share the chances among the rest", is pure conditional probability. It transfers to any market with a finishing order, such as outright football markets.
The idea in one sentence
Once the winner is decided, treat second place as a fresh race among the horses left, with the same relative chances, and keep going for third.
The picture
Take a six-runner race. The win market gives each horse a chance. Harville's idea is that if A wins, the other five then run for second as if A had never been there, sharing out 100% in proportion to their original chances.
That gives the chance of every finishing order. From those you can price:
- Forecasts: A first and B second, in that order.
- Reverse forecasts: A and B in the first two, either order.
- Tricasts: the first three in order.
- Place markets: finishing in the first two or three.
The calculator below takes a set of win prices, strips the margin, and fills in the whole table.
| Runner | Price | Implied | Fair win | Top 2 | Fair |
|---|---|---|---|---|---|
| A | 3.1 | 32.3% | 31.90% | 58.09% | 1.721 |
| B | 4.2 | 23.8% | 23.54% | 46.65% | 2.143 |
| C | 5.5 | 18.2% | 17.98% | 37.22% | 2.687 |
| D | 8.4 | 11.9% | 11.77% | 25.38% | 3.940 |
| E | 11 | 9.1% | 8.99% | 19.69% | 5.078 |
| F | 17 | 5.9% | 5.82% | 12.96% | 7.717 |
| Won | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| A | · | 34.6 | 26.4 | 17.3 | 13.2 | 8.5 |
| B | 41.7 | · | 23.5 | 15.4 | 11.8 | 7.6 |
| C | 38.9 | 28.7 | · | 14.4 | 11.0 | 7.1 |
| D | 36.2 | 26.7 | 20.4 | · | 10.2 | 6.6 |
| E | 35.0 | 25.9 | 19.8 | 12.9 | · | 6.4 |
| F | 33.9 | 25.0 | 19.1 | 12.5 | 9.5 | · |
Worked Betfair example
A six-runner race with two places. The Betfair win market shows A 3.1, B 4.2, C 5.5, D 8.4, E 11, F 17. The place market has A at 1.80. (Illustrative prices.)
- Implied chances. 1 ÷ each price: 32.26%, 23.81%, 18.18%, 11.90%, 9.09%, 5.88%. They add to 101.13%.
- Remove the margin. Divide each by 1.0113: A 31.90%, B 23.54%, C 17.98%, D 11.77%, E 8.99%, F 5.82% (Lesson 2.2).
- A forecast. A first, B second = 0.3190 × 0.2354 ÷ (1 − 0.3190) = 0.3190 × 0.3457 = 11.03%, fair 9.07. B first, A second = 9.82%, fair 10.18. The reverse forecast is both: 20.85%, fair 4.80.
- A tricast. A, B, C in order = 11.03% × 17.98% ÷ (1 − 0.3190 − 0.2354) = 4.45%, fair 22.47.
- A's place chance. Add every way A finishes first or second: 58.09%, fair 1.721. All six place chances add to exactly 200%, as they should.
- EV at 1.80, after 2% commission, per £1. 0.5809 × 0.80 × 0.98 − 0.4191 = +3.6p. Break-even is 1.736. Harville says bet.
- Now apply a discount. A common fix raises the other runners' chances to a power below 1 before working out second place, which squeezes the favourite. With a power of 0.8, A's place chance falls to 55.47%, break-even 1.819, and EV at 1.80 becomes −1.0p.
Verdict: plain Harville says 1.80 is value; a discounted version says it isn't. Harville's known habit of flattering favourites for places is exactly the gap here, so the "value" is most likely the formula's bias, not the market's mistake.
Why Harville flatters favourites
Harville assumes that when the favourite doesn't win, it is still just as strong relative to the others in the race for second. Real races don't work that way. A favourite that gets beaten has often had a bad day: a poor start, the wrong ground, a horse that didn't stay. Those same problems usually cost it second place too.
Outsiders run the other way. Their win chance is small, but on a day when the favourite fails and the race is scrappy, they pick up places more often than their win price suggests. That's why the discount works: it shaves a little off the strong and hands it to the weak.
The formula
Harville: the chance of an exact order
- pᵢ, pⱼ, pₖ are the win chances of runners i, j and k, with the margin removed.
In plain English: win chance for the winner, then each later place is that runner's share of the chances still left. The general family is on the Plackett-Luce and Harville page.
Place chance (top two)
In plain English: either i wins, or someone else wins and i wins the "race for second" among the rest.
The discounted version
- γ (gamma) is a power below 1, fitted to past results.
In plain English: raising chances to a power below 1 flattens them, so outsiders get a bit more of second place and favourites a bit less. That matches what really happens.
Try it
Enter the six prices from the example and two places. Check A's place chance reads 58.09% and the column sums to 200%. Then slide the discount power from 1.0 down to 0.8 and watch A's place price drift out past 1.80.
Common mistakes
- Forgetting to remove the margin first. Harville needs chances that add to 100%. Feed it raw implied chances and every forecast is off.
- Trusting plain Harville for places. It's known to over-rate favourites for second and third. Use a discount and test it on past races.
- Checking only one runner. Place chances must add to the number of places (200% for two). If they don't, something is wrong.
- Assuming the win market is right. Harville is only as good as its inputs. It spreads the win market's views across the places; it doesn't add new information.
- Thinking this is racing-only. Any "finishing order" market, such as a football league's top two, can be priced the same way from win chances.
Formulas that are known to be biased, used without correcting for it, are one route to why good models still lose money.