Over the past few blogs I've been investigating the relationship between the result of each quarter of an AFL game and the pre-game head-to-head prices set for that same game. In the most recent blog I came up with an equation that allows us to estimate the probability that a team will win a quarter (p) using as input only that team's pre-game Implicit Victory Probability (V), which we can derive from the pre-game head-to-head prices as the ratio of the team's opponent's price divided by the sum of the two teams' prices.

The equation relating p to V was the following quintic:

p = 0.031720 + 2.27862*V - 7.65809*V2 + 17.19818*V3 - 18.12259*V4 + 7.24110*V5

It was derived as the empirical inverse of the relationship V = p2(3-2p).

EMPIRICAL VALIDITY

It's always fair to ask: in practice, how good is this equation?

One way to answer this is to use it to predict the expected number of quarters won by the favourite in each of the 1,329 games across the period 2006 to 2012 and then to compare those predictions to the actual outcomes of those games.

Each row of this table summarises the predictions and outcomes for a number of games, grouped according to the Implicit Probability of the Home team. So, for example, the first row summarises the 24 games in which the Home team's Implicit Probability was under 10%.

The left hand block of data summarises the predictions for these games and the right hand block summarises the actual outcomes. In the first row we find that, amongst the 24 games where the Home team was rated less than a 10% chance, about half of them (11.8) were expected to end with the Home team having lost all 4 quarters. What actually happened was that only 6 games finished this way. (Note that for games where quarters were drawn I've included one half of them in the next lower and one half in the next higher count. So, for example, a game where the Home team won 3 quarters and drew the fourth would count as half a game in the 3 qtrs column and half a game in the 4 qtrs column.)

Rightmost in the table is a column that shows the average Actual number of quarters won less the average Expected number of quarters won. This shows that Home teams with Implicit Probabilities less than 30% have won, roughly, about 0.3 quarters more than our equation would suggest.

We could interpret this difference in a number of ways including that:

• the equation relating p to V breaks down for Home teams at these prices
• the empirical estimate of the inverse of the equation relating V to p is relatively less accurate in this range. We know this is likely to be true for probabilities under 10% (and over 90%) as the empirical equation was fitted only to the range 10% to 90%.
• these Home teams were mispriced in the head-to-head markets. There is some evidence for this explanation, as the average Implicit Probability for Home teams with such probabilities below 30% is about 20% yet they've won 22% of the time. A couple of percentage points of average mispricing is not, however, sufficient to explain the entirety of the 0.3 quarters per game difference.

More generally there's evidence for the mild mispricing of Home teams at any price based on the fact that Home teams win, on average, 0.1 quarters per game more than the equation suggests they should.

Those issues aside, the performance of the equation, based as it is on such simple assumptions, seems adequate. 

PRACTICAL USE

One possible use of the equation would be to set prices for markets on the number of quarters that, say, the favourite might win in an upcoming game solely on the basis of the prevailing prices in the head-to-head market.

Each row of this table pertains to a game with the pre-game prices as specified in the first two columns.

So, for example, for a game where the prices are $1.70/$2.15, which means that the favourite's Implicit Probability is 55.8%, the estimated probability of the favourite winning any quarter is 54.2%.

This probability can be used to derive the probabilities in the next 5 columns, assuming that the result of each quarter is independent and that the probability of the favourite winning a quarter is constant. The most likely result in this game is that the two teams win 2 quarters each, which is an outcome carrying a 37% probability.

The final block of data converts these probability estimates into prices, assuming a 6% vig. For the game just highlighted where the relevant probability was 37% this equates to a price of $2.55.

One of the interesting features of this table is that it highlights that the most likely outcome is for the teams to win 2 quarters each even if the favourite's head-to-head price suggests that it's about a 63% chance of winning the game. For prices shorter than this, a 3-1 result for the favourite is the most likely outcome unless the favourite has about a 90% overall victory probability. 

If you look at the actual results provided in the earlier table you can see that this is borne out in practice. Favourites rated 85-90% chances, for example, won 3 quarters of their games at a rate about 1.5 times the rate at which they won all 4 quarters. 

There's certainly the makings of a proposition bet in that ...