Multinomial and ordinal logistic regression explained for betting

In one sentence

These models extend logistic regression to outcomes with more than two categories, producing a full set of probabilities that add up to 100%.

How it works

Football's match odds market has three outcomes, and fitting two separate yes/no models gives probabilities that do not add to one. Multinomial and ordinal logistic regression solve this in one model.

Multinomial logistic regression treats the outcomes as unordered categories. It gives each outcome its own set of weights and turns the scores into probabilities with the softmax formula, which divides each score's exponential by the total.

Ordinal logistic regression uses the natural order away win, draw, home win. Imagine a hidden "home superiority" score: below one threshold it is an away win, between the thresholds a draw, above the upper threshold a home win. One set of weights shifts the score while the thresholds stay fixed, and fewer parameters means less overfitting, which suits football well.

The maths

\text{Multinomial:} \quad P(Y = k) = \frac{e^{\beta_k \cdot x}}{\sum_{m} e^{\beta_m \cdot x}}

\text{Ordinal:} \quad P(Y \le k) = \frac{1}{1 + e^{-(c_k - \eta)}}, \qquad \eta = \beta_1 x_1 + \dots + \beta_j x_j

Y: the outcome category (away, draw, home).
x: the inputs, such as rating gap.
β_k: weights for outcome k (multinomial); β: a single set of weights (ordinal).
η: the hidden home-superiority score.
c_k: the thresholds (cut-points) between categories, learned from data.

In words: the ordinal version slides a single score up or down and reads off how much probability falls into each band between the thresholds.

Worked betting example

An ordinal model: η = 0.9 × (rating gap, home minus away). Cut-points c₁ = −0.55 (away/draw boundary) and c₂ = 0.75 (draw/home boundary).

Tonight's rating gap is 0.6, so η = 0.54.
P(away) = 1 ÷ (1 + e^(0.54 + 0.55)) ≈ 25.2%.
P(away or draw) = 1 ÷ (1 + e^(0.54 − 0.75)) ≈ 55.2%, so P(draw) ≈ 30.1%.
P(home) = 1 − 55.2% ≈ 44.8%. The three add to 100%. Fair prices: home 2.23, draw 3.33, away 3.97.
Betfair back prices: home 2.00, draw 3.60, away 4.20 (implied total ≈ 101.6%).
Draw looks like value. £10 stake, 2% commission: a win pays £26 × 0.98 = £25.48. EV ≈ 0.301 × £25.48 − 0.699 × £10 ≈ +£0.68.
Away at 4.20 is also slightly positive: EV ≈ 0.252 × £31.36 − 0.748 × £10 ≈ +£0.42. Home at 2.00 has EV ≈ −£1.13.

Where it's good

Football match odds, where draws are frequent and need their own probability.
Ordered outcomes: winning margin bands, number-of-sets markets in tennis, rugby winning margins.
Converting a rating gap (Elo, pi-ratings) into coherent home/draw/away prices.
Horse racing position bands when finishing order is coarse.

Limitations and pitfalls

The ordinal version assumes each input shifts all thresholds equally (the proportional odds assumption); check it, as it often fails for draws.
Multinomial models have many more weights and overfit small datasets.
Draw probability depends on things the rating gap misses, such as low-scoring teams or end-of-season incentives.
Neither model knows about scorelines, so they cannot price correct score or totals.
Draw prices on Betfair are efficient in major leagues; a model edge on the draw is often an error in the model.
Treat three-way calibration carefully: use the ranked probability score or multi-class log loss, not accuracy.

How to build it

Python: statsmodels OrderedModel for ordinal, scikit-learn LogisticRegression (multinomial) or statsmodels MNLogit; R: MASS::polr and nnet::multinom.
Data: results with pre-match ratings and context features.
Validate walk-forward and compare with market implied probabilities after removing the margin.
Tip: add features that specifically move the draw, such as total-goals expectation, rather than relying on rating gap alone.

Logistic regression: the two-outcome base model.
Conditional logit: multinomial choice for fields of runners.
Multinomial distribution: the probability model behind several outcomes.
Ranked probability score: the right score for ordered outcomes.
Elo ratings: a common input to ordinal football models.