The theory of value betting

A betting event is a random experiment. Every outcome has some probability of occurring, even an unlikely one, such as more than ten goals in a soccer match, and nobody knows what those probabilities actually are.

Fair odds

The bookmaker estimates the probability \(p\) of an outcome and offers odds \(o\) on it. A bet of one unit returns \(o\) if the outcome occurs and nothing otherwise, so its expected profit is

\[ \mathbb{E}[\Pi] = p \, o - 1. \]

The odds are fair when this is zero, that is when

\[ o = \frac{1}{p}. \]

At fair odds, neither side makes profit in the long run.

The bookmaker's margin

Bookmakers do not offer fair odds. They shorten them, so that the implied probability \(1/o\) of every outcome is a little higher than the probability they estimated. Across the \(n\) mutually exclusive outcomes of an event, the implied probabilities therefore sum to more than one:

\[ \sum_{i=1}^{n} \frac{1}{o_i} = 1 + m, \qquad m > 0. \]

The excess \(m\) is the overround, and it is the bookmaker's margin. Notice what has not changed. The bookmaker still has to estimate \(p\). The margin protects a good estimate, it does not replace one.

Value bets

The bettor can estimate the probabilities too. Write the bettor's estimate as \(\hat{p}\). A bet is a value bet when the bettor's estimate exceeds the probability implied by the offered odds:

\[ \hat{p} > \frac{1}{o} \quad \Longleftrightarrow \quad \hat{p} \, o - 1 > 0, \]

that is, when the bet has positive expected profit under the bettor's own estimate. Selecting value bets is the only betting strategy that makes sense over the long run.

The caveat matters: neither side observes the true \(p\). A value bet is a claim that \(\hat{p}\) is closer to the truth than \(1/o\) is, and the bettor can be wrong, the bookmaker can be wrong, or both.

Is it hopeless?

Bookmakers have more data, more computing power and teams of analysts. It is tempting to conclude that competing with them is pointless, but that does not follow. Bookmakers balance many concerns beyond accuracy: their exposure, their competitors, the weight of public money, which is why the odds offered on the same event vary noticeably from one bookmaker to another. That variation is the opening.

The goal is therefore not to build an arbitrarily accurate model of football. It is to identify value bets systematically and backtest them honestly. A realistic aim, and the one sports-betting is built to serve.