Building a consensus fair price
Pool several bookmakers into one fair-price benchmark: remove each book's margin, average probabilities rather than odds, and avoid counting the same opinion twice.
Learning objectives
- De-vig each bookmaker separately and combine the results into a consensus probability for every outcome.
- Explain why averaging probabilities differs from averaging odds, and when a median is safer than a mean.
- Identify mirrored or stale quotes and remove or down-weight them before building the consensus.
- Compute the expected value of an outlier price against the consensus and state why that number is not proof of an edge.
Why one bookmaker is not enough
A single bookmaker's de-vigged price is one forecast with its own errors: the trader's opinion, the book's client mix, how quickly it reacts to news and how it spreads its margin. Pooling several books is the same idea as averaging several weather forecasts — independent errors partly cancel, so the pooled number is usually a steadier benchmark than any one input.
De-vig first, then average probabilities
Each book carries a different margin, so raw odds are not comparable. First remove each book's margin (see Removing the margin), giving fair probabilities p₍b,i₎ for book b and outcome i. Then combine those probabilities.
p̄ᵢ = Σ_b w_b × p₍b,i₎ / Σ_b w_bWhy not average the odds? Odds are 1/p, a curved function, so the average of odds is pulled towards the longer prices and the implied probabilities no longer sum to 100%. In the worked example below, averaging four books' fair odds gives implied probabilities that sum to 99.948% instead of 100%. The gap is small for a tight 1X2 market, but it grows with long shots and with disagreement between books.
Mean or median? A median ignores one wild quote, which helps when a book has a data error. Its drawback is that outcome-by-outcome medians need not sum to 100%: in the example the four-book medians sum to 100.054% and must be renormalised. With few books the two usually agree closely; with a suspected error, prefer the median or remove the bad quote explicitly.
Weights. Many analysts give more weight to books that take large stakes, run low margins or are known to move first, on the theory that their prices absorb more information. That is reasonable, but weights are model parameters: choose them in advance and test them out of sample (lesson 12), or the weighting itself becomes a source of overfitting.
Mirrors, shared feeds and stale prices
Several brands may take their odds from one supplier or simply copy a market leader. Five quotes can therefore be three opinions. Counting a mirrored pair twice doubles that opinion's weight without adding information. Clues: identical prices on every outcome and every line, and moves that happen at the same instant.
Staleness is the other trap. A quote that has not moved since before the team news is not a current opinion. Keep the capture time of every quote, use only quotes taken within a short window of each other, and drop markets that were suspended or re-opened in between.
| Consensus | Home | Draw | Away | Comment |
|---|---|---|---|---|
| Mean of all five | 46.37% | 28.09% | 25.54% | Mirrored D and E counted twice |
| Mean of A–D | 46.22% | 28.20% | 25.58% | One vote per independent opinion |
| Median of A–D, renormalised | 46.50% | 28.09% | 25.41% | Robust to one bad quote |
| Weighted (A = 2) | 46.19% | 28.25% | 25.56% | Weights must be fixed in advance |
The spread between books is information too. Here the single-book home probabilities run from 44.47% to 47.36%, a range of almost three points. A tight spread suggests the market has settled; a wide one suggests news, thin trading or a stale book, and it is a reason to be more cautious about any edge measured against the centre.
The consensus-outlier idea and its limits
De-vig each book separately before you average. Try Book C from the worked example.
Lab 2 · Remove the margin
Synthetic · deterministic| Method | Home | Draw | Away |
|---|---|---|---|
| Raw implied (with margin) | 46.51% | 30.30% | 27.78% |
| Proportional | 44.47% (2.249) | 28.97% (3.452) | 26.56% (3.765) |
| Additive | 44.98% (2.223) | 28.77% (3.476) | 26.25% (3.810) |
| Power k = 1.0438 | 44.98% (2.223) | 28.76% (3.477) | 26.26% (3.808) |
| Shin z = 0.0230 | 44.85% (2.230) | 28.82% (3.469) | 26.33% (3.798) |
- Proportional: Divides every implied probability by the book sum. Simple, but assumes the margin is spread in proportion to probability.
- Additive: Removes an equal amount from every outcome. Easy to explain; it can push long shots below zero.
- Power: Raises every implied probability to a common power. Takes relatively more margin from long shots.
- Shin: Derived from Shin's insider-trading model. Takes more margin from long shots; performed well in Štrumbelj (2014).
Reading the result. The methods agree on the favourite more than on long shots. None of them recovers the “true” probability: they are different assumptions about how the bookmaker spread its margin. Differences of a point or two can decide whether a price looks like value, so state which method you used.
Educational calculator. Uses only the numbers you enter or seeded synthetic data — no live prices, no staking, no real money.
Worked example
Worked example · A five-book consensus for Harbour City v Northvale
Hypothetical 1X2 quotes (home / draw / away): Book A 2.10 / 3.40 / 3.80; Book B 2.05 / 3.50 / 3.90; Book C 2.15 / 3.30 / 3.60; Book D 2.00 / 3.40 / 3.70; Book E 2.00 / 3.40 / 3.70. A sixth book, F, shows Harbour City at 2.30. All probabilities use proportional de-vigging.
- Book sums
Σ 1/o for each bookA 103.35%, B 102.99%, C 104.59%, D 106.44%, E 106.44% (overrounds 3.35%, 2.99%, 4.59%, 6.44%, 6.44%) - De-vig each book
pᵢ = (1/oᵢ) / book sumA 46.08 / 28.46 / 25.46%; B 47.36 / 27.74 / 24.90%; C 44.47 / 28.97 / 26.56%; D = E 46.98 / 27.63 / 25.39% - Spot the mirror
D and E match on every outcome and every priceTreat D and E as one opinion (likely a shared feed) - Naive mean of all five
Average the five de-vigged rows46.37 / 28.09 / 25.54% → fair odds 2.156 / 3.560 / 3.915 (D's view counted twice) - Mean of four independent books
Average A, B, C and D46.22 / 28.20 / 25.58% → fair odds 2.164 / 3.546 / 3.910 - Median of four, renormalised
Medians 46.53 / 28.10 / 25.43% sum to 100.054%; divide by the sum46.50 / 28.09 / 25.41% → fair odds 2.150 / 3.561 / 3.935 - Weighted mean
Weights A = 2 (assumed sharper), B = 1, C = 1, D and E share 1 (0.5 each)46.19 / 28.25 / 25.56% → fair odds 2.165 / 3.539 / 3.913 - Averaging odds instead (for contrast)
Mean of A–D fair odds: 2.165 / 3.547 / 3.912Implied 46.19 / 28.19 / 25.56%, summing to 99.948% — not a coherent probability set - Book F's outlier
2.30 → implied 43.48%; EV = 0.46221 × 2.30 − 1 against the four-book mean+6.31%. Using each single book's home probability (44.47% to 47.36%) instead: +2.28% to +8.93%
Every reasonable consensus puts Harbour City between 46.19% and 46.50%, so the method choice matters less than the question it raises: why is Book F at 2.30 when every other book's fair price is 2.111–2.249? Before calling it value, check whether F's quote is stale, whether it is the same market and settlement rule, and how much it will accept. The range of book opinions is not the full uncertainty, so a positive EV across that range is a reason to investigate, not a verdict.
Practice · De-vig one book three ways
Open the Remove the Margin lab with Book C (2.15 / 3.30 / 3.60). Record the proportional, Shin and power probabilities. Then repeat for Books A, B and D, average the Shin probabilities of the four independent books, and recompute Book F's EV at 2.30.
Lab 2 · Remove the margin
Synthetic · deterministic| Method | Home | Draw | Away |
|---|---|---|---|
| Raw implied (with margin) | 46.51% | 30.30% | 27.78% |
| Proportional | 44.47% (2.249) | 28.97% (3.452) | 26.56% (3.765) |
| Additive | 44.98% (2.223) | 28.77% (3.476) | 26.25% (3.810) |
| Power k = 1.0438 | 44.98% (2.223) | 28.76% (3.477) | 26.26% (3.808) |
| Shin z = 0.0230 | 44.85% (2.230) | 28.82% (3.469) | 26.33% (3.798) |
- Proportional: Divides every implied probability by the book sum. Simple, but assumes the margin is spread in proportion to probability.
- Additive: Removes an equal amount from every outcome. Easy to explain; it can push long shots below zero.
- Power: Raises every implied probability to a common power. Takes relatively more margin from long shots.
- Shin: Derived from Shin's insider-trading model. Takes more margin from long shots; performed well in Štrumbelj (2014).
Reading the result. The methods agree on the favourite more than on long shots. None of them recovers the “true” probability: they are different assumptions about how the bookmaker spread its margin. Differences of a point or two can decide whether a price looks like value, so state which method you used.
Educational calculator. Uses only the numbers you enter or seeded synthetic data — no live prices, no staking, no real money.
Show what you should find
Book C's overround is 4.59%. Proportional gives 44.47 / 28.97 / 26.56%, Shin 44.85 / 28.82 / 26.33% and power 44.98 / 28.76 / 26.26%: power moves about half a point of probability to the favourite (44.47% → 44.98%), taken from the draw and the long shot. Averaging Shin probabilities across A–D gives 46.64 / 28.03 / 25.33% (fair odds 2.144 / 3.567 / 3.948), so F's 2.30 shows +7.26% EV instead of +6.31%. The de-vig method changes the size of the apparent edge by about a point; it does not answer whether F's quote is real.
Common mistakes
- ✗ Averaging raw odds across books.Raw odds carry different margins and odds are a curved function of probability. De-vig each book, then average probabilities.
- ✗ Counting every brand as an independent opinion.Brands sharing one feed are one opinion. Collapse identical quote sets or give the group a single weight.
- ✗ Mixing quotes captured at different times.Store capture timestamps and use only quotes from a common window; a pre-news price is not a current forecast.
- ✗ Treating the consensus as the true probability.It is a benchmark built from forecasts that can share errors. Use it as the price to beat, then ask why an outlier exists.
When it fails
Supporting research
- Štrumbelj (2014): On determining probability forecasts from betting oddsInternational Journal of Forecasting · Peer-reviewed empiricalBookmakers differ as sources of probability and the conversion method affects forecast accuracy, which motivates de-vigging each book before pooling.
- Kaunitz, Zhong & Kreiner (2017): Beating the bookies with their own numbers – and how the online sports betting market is riggedarXiv preprint · Preprint · not peer reviewedThe consensus-outlier strategy and its reported (preprint, unreviewed) results, including account restrictions during a short live trial.
- Sauer (1998): The Economics of Wagering MarketsJournal of Economic Literature · Literature reviewTreating market prices as efficient-to-a-first-approximation forecasts, so a consensus is the benchmark to beat.
Check your understanding
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