Methodology

One representation

Every price is converted to an implied probability first. Decimal odds are 1 / p, American odds are (d − 1) × 100 above evens and −100 / (d − 1) below, fractional odds come from the traditional UK ladder or the closest fraction with a denominator up to 100, and a Polymarket price is the probability in cents. Intermediate values are never rounded; only the displayed figures are.

Commission

A bet's outcome is modelled as a pair: net profit per unit if it wins, net loss per unit if it loses. Commission on net winnings reduces the profit side; turnover commission reduces both. Everything downstream (arbitrage, expected value, Kelly) works on that pair, which is why turnover commission is handled exactly rather than approximated with an "effective" price.

Arbitrage

With one plain back bet per outcome the classic formula applies: sum the implied probabilities of the effective odds, and if the sum is below one, split the stake in proportion to each outcome's implied probability. With lay bets or turnover commission the calculator solves the equal-profit equations directly by Gaussian elimination. A lay at odds L is treated as a back on every other outcome at L / (L − 1), staked with the liability.

Devig

Five methods are offered: multiplicative, additive, power, Shin and worst case. Their formulas are shown on the devig calculator page. The power and Shin parameters are found by bisection to a tolerance of one part in ten billion.

Testing

The math lives in a separate package with over two hundred tests, including every worked example in the project specification and property-based tests that check invariants such as "every outcome pays the same profit" across thousands of random markets. The calculators on this site are then tested end to end in a browser against the same examples.

Two known corrections to the specification

While testing, two figures in the original specification turned out to be rounded from coarser intermediates: the equal-profit stakes for a 3.00 / 3.20 / 3.50 market, and the multiplicative fair probabilities for 2.20 / 3.40 / 3.60. The calculators use the exact values.