In the previous blog I developed a new way of divining a bookmaker's probability assessments of the two teams by assuming that he believes his maximum calibration error - the (negative) difference between his probability assessment for a team and its true probability of victory - is the same for each team in percentage point terms, and that he levies overround on each team's price so as to ensure that it will still deliver an expected profit even if his probability assessment is maximally in error. This was in contrast to the typical assumption that he levies overround on each team equally.

I called this new approach "Risk-Equalising", a tangible example of which might help. Assume that the bookmaker assesses the Home team's probability of victory as being 25% and that he wants an aggregate overround across the two teams to be 5%. Given these numbers he will set the Home team price at about $3.64 and the Away team price at about $1.29. These prices will provide cover for 2.5% of calibration error (ie one half of the aggregate overround) on either team.

To prove this is the case, consider the expected return on the Home team if its true victory probability were the bookmaker's assessment (25%) plus the maximum calibration against which he is covered by the overround (2.5%). This return is given by:

Expected Return for Home Team: 1 - (25% + 2.5%) x $3.64, which is 0, or breakeven.

Similarly, for the Away team, we have:

Expected Return for Away Team: 1 - (75% + 2.5%) x $1.29, which is also 0, or breakeven.

Not only did we find that this approach to determining Implicit Probabilities made theoretical and practical sense, but we also found that the Implicit Probabilities so derived were, very mildly, better calibrated. 

To make this assertion of superior calibration, in the previous blog I used the Brier Score. Since then I've also assessed the quality of the predictions produced by the two approaches using the Logarithmic Probability Score, or LPS (a score we've used on MAFL for some time) and determined that the LPS using the Risk-Equalising approach is superior to the LPS using the Overround-Equalising approach for every season from 2007 to 2012 except season 2007, and for 21 of the 28 rounds that have comprised one or more of the past six seasons, including Finals rounds.

Pretty clearly then, the Risk-Equalising approach is superior to the Overround-Equalising approach on a number of objective and reasonable measures.

A FEW USEFUL RELATIONSHIPS

A little more tinkering with the algebra for this approach turns up a few identities that are worth discussing.

Identity #1

Price = 1 / (Implicit Probability + Maximum Calibration Error)

The "fair price" for any wager - the price at which the wager represents a zero expected return for bookmaker and punter - is the reciprocal of its true probability. The Risk-Equalising approach achieves its aims by adding a fraction to the outcome's probability equal to the maximum calibration error, as assessed by the bookmaker, and then taking the reciprocal of that sum.

One implication of this equation is that a bookmaker who's nervous about his ability to assess the chances of a particular team and who therefore believes his maximum calibration error to be higher, should lower the price that he offers for that team even though he hasn't altered his opinion about its Implicit Probability. We can think of this as his discounting for additional uncertainty, which seems like a very logical thing to do.

Identity #2

Maximum Calibration Error = Total Overround / 2

(note that here, as in the previous blog, I'm defining Total Overround as 1/Home Team Price + 1/Away Team Price - 1)

The implication of this identity is that the bookmaker can set either the Total Overround for the contest or the Maximum Calibration Error that his prices will shield him against; he can't set both independently. To the extent then that Total Overround is constrained or even capped by competitive forces, the bookmaker needs to be sufficiently well-calibrated that his maximum calibration error is below one half of the permissible Total Overround. If he's unable to achieve this level he'll face having wagers in the market with negative expected return.

Identity #3

Overround1 / Overround2  = Price2 / Price1

where the subscripts are used to denote each of the teams, and

Price1 = Overround2 / (Overround1 + Overround2)

Price2 = Overround1 / (Overround1 + Overround2)

So, we can motivate the Risk-Equalising approach by noting that it implies that the ratio of the overrounds on the two teams is equal to the reciprocal of their revealed prices, or that it implies that the revealed price of a team is equal to the proportion of the overround embedded in its opponent's price as a fraction of the sum of the overrounds on both teams.

(I probably should note here that the sum of the overrounds on both teams is not equal to the total overround on the market. Instead, it's equal to Price1(Overround1 - Overround2) + Overround2 and Price2(Overround2 - Overround1) + Overround1)

Identity #4

Under the Overround-Equalising approach we determine the bookmaker's Implicit Probability using this equation:

Implicit ProbabilityOE = 1 / (Price x (Overround + 1))

For the Risk-Equalising approach we use instead:

Implicit ProbabilityRE = (1 / Price) - (Overround / 2)

The difference between these is given by:

Implicit ProbabilityOE - Implicit ProbabilityRE = Overround x (Overound x Price + Price - 2) / (2 x Price x (Overround - 1))

This expression will be: 

• Positive when Price > 2 / (Overround + 1), ie for Underdogs
• Negative when Price < 2 / (Overround + 1), ie for Favourites
• Zero when Price = 2 / (Overround + 1), ie for Equal Favourites 

So, for a given price, the Overround-Equalising approach will come up with a larger Implied Probability than the Risk-Equalising approach for underdogs, a smaller Implied Probability for favourites, and the same Implied Probability for equal favourites.

TABLE OF IMPLIED OVERROUND

As one way of understanding the Risk-Equalising approach, I thought a table showing the overround embedded in each team's prices for a given level of Total Overround might be helpful.

The way to read this table is as follows:

• Lookup (or assume) the price of the team in question (ie choose the row)
• Calculate (or assume) the total overround in the market (ie calculate the sum of the reciprocal of the two team's prices and subtract 1).
• Look at the columns corresponding to that total overround. The percentage in the leftmost column corresponding to the team price and total overround is the overround embedded in that team's price; the percentage in the rightmost matching column is the overround embedded in the opponent team's price.

It's startling to see how large some of the overrounds are for rank longshots. Even a team priced at $7 in a market with just 5% total overround is encumbered with a 21% overround. That overround is protecting the bookmaker against the possibility that his assessment of the team as about an 11.8% chance is in error by 2.5% making the team, in reality, a 14.3% chance. At $7, this makes them only a breakeven proposition.