Divining the Bookie Mind: Singularly Difficult

It's fun this time of year to mine the posted TAB Sportsbet markets in an attempt to glean what their bookie is thinking about the relative chances of the teams in each of the four possible Grand Final pairings.

Three markets provide us with the relevant information: those for each of the two Preliminary Finals, and that for the Flag.

From these markets we can deduce the following about the TAB Sportsbet bookie's current beliefs (making my standard assumption that the overround on each competitor in a contest is the same, which should be fairly safe given the range of probabilities that we're facing with the possible exception of the Dogs in the Flag market):

• The probability of Collingwood defeating Geelong this week is 52%
• The probability of St Kilda defeating the Dogs this week is 75%
• The probability of Collingwood winning the Flag is about 34%
• The probability of Geelong winning the Flag is about 32%
• The probability of St Kilda winning the Flag is about 27%
• The probability of the Western Bulldogs winning the Flag is about 6%

(Strictly speaking, the last probability is redundant since it's implied by the three before it.)

What I'd like to know is what these explicit probabilities imply about the implicit probabilities that the TAB Sportsbet bookie holds for each of the four possible Grand Final matchups - that is for the probability that the Pies beat the Dogs if those two teams meet in the Grand Final; that the Pies beat the Saints if, instead, that pair meet; and so on for the two matchups involving the Cats and the Dogs, and the Cats and the Saints.

It turns out that the six probabilities listed above are insufficient to determine a unique solution for the four Grand Final probabilities I'm after - in mathematical terms, the relevant system that we need to solve is singular.

That system is (approximately) the following four equations, which we can construct on the basis of the six known probabilities and the mechanics of which team plays which other team this week and, depending on those results, in the Grand Final: 

• 52% x Pr(Pies beat Dogs) + 48% x Pr(Cats beat Dogs) = 76%
• 52% x Pr(Pies beat Saints) + 48% x Pr(Cats beat Saints) = 63.5%
• 75% x Pr(Pies beat Saints) + 25% x Pr(Pies beat Dogs) = 66%
• 75% x Pr(Cats beat Saints) + 25% x Pr(Cats beat Dogs) = 67.5%

(If you've a mathematical bent you'll readily spot the reason for the singularity in this system of equations: the coefficients in every equation sum to 1, as they must since they're complementary probabilities.)

Whilst there's not a single solution to those four equations - actually there's an infinite number of them, so you'll be relieved to know that I won't be listing them all here - the fact that probabilities must lie between 0 and 1 puts constraints on the set of feasible solutions and allows us to bound the four probabilities we're after.

So, I can assert that, as far as the TAB Sportsbet bookie is concerned:

• The probability that Collingwood would beat St Kilda if that were the Grand Final matchup - Pr(Pies beats Saints) in the above - is between about 55% and 70%
• The probability that Collingwood would beat the Dogs if that were the Grand Final matchup is higher than 54% and, of course, less than or equal to 100%.
• The probability that Geelong would beat St Kilda if that were the Grand Final matchup is between 57% and 73%
• The probability that Geelong would beat the Dogs if that were the Grand Final matchup is higher than 50.5% and less than or equal to 100%.

One straightforward implication of these assertions is that the TAB Sportsbet bookie currently believes the winner of the Pies v Cats game on Friday night will start as favourite for the Grand Final. That's an interesting conclusion when you recall that the Saints beat the Cats in week 1 of the Finals.

We can be far more definitive about the four probabilities if we're willing to set the value of any one of them, as this then uniquely defines the other three.

So, let's assume that the bookie thinks that the probability of Collingwood defeating the Dogs if those two make the Grand Final is 80%. Given that, we can say that the bookie must also believe that:

• The probability that Collingwood would beat St Kilda if that were the Grand Final matchup is about 61%.
• The probability that Geelong would beat St Kilda if that were the Grand Final matchup, is about 66%.
• The probability that Geelong would beat the Dogs if that were the Grand Final matchup is higher than 72%.

Together, that forms a plausible set of probabilities, I'd suggest, although the Geelong v St Kilda probability is higher than I'd have guessed. The only way to reduce that probability though is to also reduce the probability of the Pies beating the Dogs.

If you want to come up with your own rough numbers, choose your own probability for the Pies v Dogs matchup and then adjust the other three probabilities using the four equations above or using the following approximation:

For every 5% that you add to the Pies v Dogs probability:

• subtract 1.5% from the Pies v Saints probability
• add 2% to the Cats v Saints probability, and
• subtract 5.5% from the Cats v Dogs probability

If you decide to reduce rather than increase the probability for the Pies v Dogs game then move the other three probabilities in the direction opposite to that prescribed in the above. Also, remember that you can't drop the Pies v Dogs probability below 55% nor raise it above 100% (no matter how much better than the Dogs you think the Pies are, the laws of probability must still be obeyed.)

Alternatively, you can just use the table below if you're happy to deal only in 5% increments of the Pies v Dogs probability. Each row corresponds to a set of the four probabilities that is consistent with the TAB Sportsbet markets as they currently stand.

I've highlighted the four rows in the table that I think are the ones most likely to match the actual beliefs of the TAB Sportsbet bookie. That narrows each of the four probabilities into a 5-15% range.

At the foot of the table I've then converted these probability ranges into equivalent fair-value price ranges. You should take about 5% off these prices if you want to obtain likely market prices.