Plackett-Luce and Harville Models explained for betting

In one sentence

Harville and Plackett-Luce models build the probability of any finishing order by repeatedly picking the next finisher in proportion to the remaining runners' strengths.

How it works

You have win probabilities for every horse. What is the chance a particular horse finishes second? Harville's 1973 idea: once the winner is removed, the rest of the field "races again" for second place, with probabilities rescaled so they add up to 100%. Then again for third, and so on.

Plackett-Luce is the general version used in statistics. Each runner has a strength, and at each step the next finisher is chosen in proportion to strength among those left. Harville is Plackett-Luce with the strengths set equal to win probabilities.

This lets you price place markets, forecasts (first and second in order) and tricasts from one set of win prices, and fit rankings from finishing orders in races, golf or motorsport.

The maths

P(i \text{ 1st}, j \text{ 2nd}) = p_i \times \frac{p_j}{1 - p_i}

P(\text{order } 1,2,\dots,n) = \prod_{k=1}^{n} \frac{w_k}{\sum_{m \ge k} w_m}

pi and pj are the win probabilities of runners i and j.
wk is the strength of the runner who finished in position k.
The denominator adds up the strengths of every runner not yet placed.

In words: the chance of an exact order is the chance of each finisher beating those still left, multiplied together.

Worked betting example

Racing, illustrative figures. Five runners with win probabilities A 35%, B 25%, C 20%, D 12%, E 8%. The place market pays the first two.

Forecast A then B: 0.35 × 0.25 ÷ 0.65 = 13.46%, fair odds 7.43. B then A: 0.25 × 0.35 ÷ 0.75 = 11.67%.
If A wins, C's chance of second is 0.20 ÷ 0.65 = 30.77%.
Summing over every possible winner, P(finishing in the first two): A 63.23%, B 50.29%, C 41.90%, D 26.51%, E 18.07%. These add up to 200%, as they should with two places.
Fair place odds for C = 1 ÷ 0.4190 ≈ 2.39.
The place market offers 2.60 on C. Backing £10: 0.4190 × £16 − 0.5810 × £10 ≈ +£0.89, or about +£0.76 after 2% commission.

Before trusting that, read the pitfalls: Harville is known to mis-order the minor places.

Where it's good

Pricing place, forecast, tricast and "without the favourite" markets from win prices.
Spotting inconsistencies between Betfair win and place markets.
Fitting horse, golfer or driver ratings from full finishing orders, not just winners.
Simulating full race results inside larger models.

Limitations and pitfalls

Harville tends to overstate how often favourites fill the minor places and understate outsiders. Many practitioners apply a discount, for example raising probabilities to a power below 1 for second and third, as popularised in Benter's work.
Win prices carry the favourite-longshot bias, and that bias flows into every derived place price.
It ignores race shape: a front-runner who fades and a hold-up horse who finishes fast do not behave like independent re-draws.
Place terms vary by field size, so check how many places the market pays.
Dead heats and non-runners change settlement; recompute probabilities after withdrawals.
A Plackett-Luce fit to finishing orders deep in the field is influenced by horses that were eased, which adds noise.

How to build it

Python: write the Harville sum directly with numpy, or use choix for Plackett-Luce fitting; R: PlackettLuce package.
Data: win probabilities from Betfair prices (margin removed) or your own model; full finishing orders for fitting.
Tip: backtest the place-price discount on historical Betfair place markets before using it live.

Conditional logit: the standard win-probability model that feeds it.
Bradley-Terry: the pairwise version of the same strength idea.
Favourite-longshot bias: a distortion that passes into place prices.
Dutching: spreading stakes across several forecast combinations.