Bookmaking, odds notation and the overround

Betting without a sample space

A wheel or a die supplies its own probabilities: count the outcomes and the arithmetic is settled. A contest between competitors supplies nothing of the kind. There is no list of equally likely outcomes to count, only an event that will happen once, and the person offering a price has to invent a number where the game itself provides none.

The earliest organised betting on contests worked around the problem by not solving it. Wagers were matched directly between two parties who disagreed, at terms they negotiated between themselves, with a third party holding the stakes. That arrangement needs no probabilities at all, but it scales badly: every wager requires a counterparty who wants the opposite side at the same terms and for the same amount.

The bookmaker is the institution that removes that constraint. By standing ready to take either side of any outcome at posted prices, one party replaces the search for a counterparty. The consequence is that the bookmaker accumulates a position across all outcomes simultaneously, and managing that position, rather than forecasting the result, becomes the actual occupation.

Making a book

The word book is literal. The bookmaker records what has been staked on each outcome and what would have to be paid if each one occurred. A book is balanced when the payout is the same whichever outcome wins, which means the operator's result no longer depends on the contest at all.

Consider a contest with two outcomes. If the operator prices both at decimal 2.00, each implying a probability of one half, then the two implied probabilities sum to exactly one. Balancing the book by taking one hundred units on each side returns two hundred units and pays out two hundred units. The operation has covered its liabilities and earned nothing, which is not a business.

Price both outcomes at decimal 1.90 instead and the position changes. Each price implies a probability of one divided by 1.90, which is 0.5263, and the two implied probabilities sum to 1.0526. Take one hundred units on each side and the operator holds two hundred units while owing one hundred and ninety whichever way the contest goes. The retained ten units on two hundred staked is 5.0 per cent.

The overround

That excess above one is called the overround, and it is the bookmaking equivalent of house edge. The method for finding it is the same in every market regardless of how many outcomes there are: convert each price to its implied probability, add them together, and subtract one.

The margin retained is the overround divided by the total, so a book summing to 1.0526 retains 0.0526 divided by 1.0526, which is 5.0 per cent of the money staked. This is why implied probabilities read off a list of prices cannot be treated as forecasts without adjustment: they have been inflated deliberately, and stripping the inflation out means dividing each one by the total.

An important practical point follows. The overround only guarantees a margin if the book is balanced, and the book is balanced by the distribution of stakes, not by the prices alone. When money arrives unevenly the operator carries genuine exposure to the result, and the usual response is to move prices to attract stakes onto the light side. Price movement in a betting market is therefore driven by the flow of money at least as much as by any change in view about the contest.

The pool as the alternative

Pool betting solves the same problem by a different route and is worth setting beside bookmaking. All stakes on an event go into a common fund, a fixed percentage is deducted, and the remainder is divided among the holders of winning tickets in proportion to their stakes. No prices are posted in advance and the operator forecasts nothing.

The structural difference is where the risk sits. A pool operator's return is fixed by the deduction and is unaffected by the result, since it only ever distributes what remains; the participants bear all the uncertainty between themselves, and the effective price of any outcome is only known once the pool has closed. A bookmaker, by contrast, fixes the price in advance and takes on the balancing risk in exchange. Both retain a margin, and in both cases the arithmetic in the entry on expected value applies unchanged to the participant.