Over on the Simulations blog I've been investigating how the returns to Kelly-staking and Level-staking respond to different levels of variability and bias in the bookmaker's team probability assessments, and to different levels of overround in that bookmaker's market prices.

In this blog I'll investigate, using a purely mathematical approach, how a punter's expected return varies as the overround varies, depending on the size of the bias in the bookmaker's probability assessment and in the true probability of the team being wagered on.

First, let's derive a formula for the punter's expected return: 

• The true probability of the team wagered on is True Probability
• The bookmaker, whose bias is Bookie Bias, assesses the team's probability as True Probability + Bookie Bias
• He therefore, applying his Overround, sets a price of 1/(Overround x (True Probability + Bookie Bias))
• The expected return is therefore True Probabilityx (1/(Overround x (True Probability + Bookie Bias)) - 1) - (1 - True Probability) 

This simplifies to yield:

Expected Return = (True Probability - Overround x (True Probability + Bookie Bias)) / (Overround x (True Probability + Bookie Bias))

The question is, how does expected return vary as we vary overround, and the answer is in the following table for various values of the team's true probability and bookie bias (we consider only negative biases, since expected return will be negative if the bias is towards the team wagered on, shortening their price).

The columns headed Chg (for Change) are those to focus on. They show the rate of change of expected return as we vary overround by 1% point (ie they are the value of the partial derivative of the expected return equation with respect to overround multiplied by 0.01).

Generally, every 1% point change in overround costs the punter about 1% point in expected return. So, if the punter can find a wager (or a bookie) with an overround of say 104% rather than 106%, she can expect to produce returns about 2% higher. The only exception to this is when she faces a bookmaker with a large bias and she is wagering on near favourites or underdogs, in which case she will enjoy up to a 1.5% point increase in returns for every 1% point decline in overround.