Two-way markets are the easy case. Every formula in the arbitrage and devigging guides works for any number of outcomes, but the numbers behave differently when there are many of them: overrounds are larger, longshots dominate the error, and a single mispriced outcome can create or destroy a position. This guide works through a three-way market and a large field, and lists what to check before trusting the arithmetic.
The n-way arbitrage test
Nothing changes in the test. Take the best price on each outcome across all books, sum the implied probabilities, and compare with 100%.
S = Σ (i = 1..n) 1 / best_odds_i
arbitrage if S < 1; ROI = 1/S − 1
stake_i = total × (1 / best_odds_i) / S
Soccer 1X2 across two books
Book A: 2.20 / 3.40 / 3.60. Book B: 2.30 / 3.30 / 3.50. Best per outcome: 2.30 (B) / 3.40 (A) / 3.60 (A). Implied: 43.48% + 29.41% + 27.78% = 100.67%. Not an arbitrage; the books disagree, but not by more than their margins. Had Book B offered 2.40 on the home side, the sum would be 98.86% and the position would return 1.15%: on 1,000, stake 421.5 on home at B, 297.4 on the draw at A, 281.1 on away at A. The
arbitrage calculator
shows the split.
Three-way arbitrages are more common than two-way ones because books disagree more about draws, and harder to execute because three legs must be placed before any price moves.
Where the margin lives in a large field
A tennis match at 1.45 / 2.90 has an overround of 3.4%. A golf tournament with 60 players can have an overround of 40%. The margin per outcome is not larger; there are just more outcomes, and each carries its share. That has two consequences.
First, an arbitrage across a field needs many books, because no single book's set of prices comes close to 100%. Taking the best of six books on each of sixty players might sum to 105%; the best of twelve might sum to 99%. The bookkeeping is the hard part.
Second, the margin is not spread evenly. Bookmakers shade the longshots hardest, so the implied probability of a 100/1 shot might be double its fair chance while the favourite's is only a few percent too high. Devigging a field with the multiplicative method, which assumes an even proportional cut, gives longshots far too much credit.
A worked field
Eight outcomes: seven named players at 8.00 / 10.00 / 12.00 / 15.00 / 18.00 / 22.00 / 25.00 and "the field" at 1.50. Implied probabilities sum to 118.3%, an overround of 18.3%.
| Outcome | Implied | Multiplicative | Additive | Power (k = 1.13) | Shin (z = 2.8%) | Worst case |
|---|---|---|---|---|---|---|
| 8.00 | 12.5% | 10.6% | 10.2% | 9.5% | 10.3% | 9.5% |
| 10.00 | 10.0% | 8.5% | 7.7% | 7.4% | 8.0% | 7.4% |
| 12.00 | 8.3% | 7.0% | 6.0% | 6.0% | 6.5% | 6.0% |
| 15.00 | 6.7% | 5.6% | 4.4% | 4.6% | 4.9% | 4.4% |
| 18.00 | 5.6% | 4.7% | 3.3% | 3.8% | 3.9% | 3.3% |
| 22.00 | 4.5% | 3.8% | 2.3% | 3.0% | 3.0% | 2.3% |
| 25.00 | 4.0% | 3.4% | 1.7% | 2.6% | 2.6% | 1.7% |
| Field 1.50 | 66.7% | 56.4% | 64.4% | 63.2% | 60.8% | 56.4% |
The 25.00 shot is worth 3.4% under multiplicative and 1.7% under additive: a factor of two. The field is 56% or 64% depending on the method. On a two-way market the methods differed by tenths of a point; here they differ by whole points on every outcome, and by a factor of two on the longshots. A bettor pricing a longshot from a multiplicative devig will systematically overpay. Run it in the devig calculator to see all five columns.
Which is right? Power and Shin are built to model the longshot shading and are the usual choice for fields; between them, power tends to move more probability toward the favourite. Additive is the most aggressive on longshots and can go negative on larger fields. Multiplicative is the one to avoid here. Worst case gives the number a bettor can defend under any of them.
Equal-profit stakes with many legs
With every leg a plain back bet, the stake formula above works for any n. With lay bets or turnover commission on some legs it does not, because a lay on one outcome pays on all the others; the calculator then solves the equal-profit equations directly. For n outcomes that is n equations (one per outcome, each saying the total profit equals a common value) plus one for the total stake, solved by Gaussian elimination. The result is the same kind of stake plan, and the calculator reports which method it used.
A lay in a multiway market is a leg that wins on a set of outcomes. Laying the draw in a 1X2 market pays if either team wins, so it can replace both the home and away back bets in a position, or combine with one of them. The solver handles any such leg; the constraint is that the number of legs must equal the number of outcomes, so that the system is square.
Rounding and thin margins
Field arbitrages are usually thin, 1–2%, and spread over many stakes. Rounding each of twelve stakes to a whole unit can move the profit on some outcomes below zero even when the unrounded plan was positive. The calculator's rounding mode recomputes every outcome after rounding and reports the worst case; check it before placing anything. Below about 0.5% the position is not worth the execution risk.
Settlement rules that break multiway positions
Multiway markets have more ways to settle than two-way ones, and a position that relied on all of them being consistent can come apart.
- Dead heats. Two players tie for first in a tournament; each book applies its dead-heat rule, usually halving the stake. A position sized for a single winner pays less than planned.
- Each-way terms. Outright markets are often bet each-way; the place terms differ by book and change the value of the price.
- Non-runners and Rule 4. A withdrawn horse triggers deductions from every remaining price at the books where the bet was placed before withdrawal. The deduction differs by the withdrawn runner's price and can turn an arbitrage into a loss.
- Regulation versus including overtime. Hockey and basketball offer both a two-way market including overtime and a three-way regulation-time market. Legs from different market types do not form an arbitrage.
- Push rules. A whole-number spread or total can land exactly and refund the stake at one book while the other leg stands.
Before sizing a multiway position, confirm that every book settles the same event under the same rules. The maths assumes exactly one outcome pays, exactly once.
Each-way and place terms
Outright markets are usually bet each-way: half the stake on the win, half on a place, with the place part paid at a fraction of the win odds (a quarter or a fifth) for a fixed number of places. That turns one price into two bets with different probabilities, and neither the arbitrage test nor a devig applies to the win price alone. To price an each-way bet, devig the win market for the win probability, estimate the place probability separately (roughly the sum of the top-n fair probabilities the runner competes for), and value the two halves at their own odds. Books with better place terms (more places, larger fraction) can offer a worse win price and still be the better each-way bet, which is a common source of value that a win-only comparison misses.
Bookkeeping across many books
A field arbitrage might need twelve prices from eight books. Two habits keep it manageable. First, work in decimal odds and implied probabilities in a single table, one row per outcome, one column per book, with the best price and its book marked; the sum of the best-price column is the arbitrage test. Second, record the settlement terms next to each book's name once, not once per bet: dead-heat rule, each-way terms, Rule 4 handling. The arbitrage calculator accepts up to twenty legs and labels for each, so the table can be entered directly; the outcome table then shows, for every winner, which legs pay and what the position nets.
Devigging a field with a "field" outcome
Bookmakers sometimes offer "any other" or "the field" as a single price covering every unnamed runner. For devigging that is convenient, since the market is complete and the sum can be compared with 100%; for arbitrage it is a trap, because the field outcome at one book covers a different set of runners than at another, and backing the field at both does not cover the same event. Treat a field price as its own outcome only within one book's market. Across books, an arbitrage needs every named runner priced explicitly at whichever book gives the best price, and the field, if used, must be defined identically. The devig methods handle a field outcome like any other; note that the field's fair probability moves most between methods, as the worked example shows, because it is where the shading on all the unnamed longshots accumulates.
Checklist for multiway markets
- Sum implied probabilities across the best prices; the number of books needed grows with the field.
- Devig with power or Shin, not multiplicative; use worst case for longshots.
- Size stakes in proportion to implied probability; round and recheck every outcome.
- Match settlement rules: dead heats, each-way, Rule 4, overtime, pushes.
- Place the legs most likely to move first, and expect at least one to move before you finish.