# Inferring 2015 Grand Final Prices

/This time of year it's always fun to consider what the current wagering markets imply about the likely Grand Final prices. We can, as I explained in this blog post from a few years ago, make inferences about these prices by combining the current head-to-head prices for the two Preliminary Finals with current Flag prices.

The first thing we need to do is convert these prices into implicit probabilities, which requires that we make assumptions about how the bookmaker has levied overround on the teams in each of the markets. For this blog I'll assume either that he's adopted an Overround Equalising approach in both Preliminary Finals and in the Flag market, or that he's instead adopted a Risk Equalising approach.

Both of these approaches are discussed in this blog post from 2013 but there I talk only about their application in head-to-head markets, so let me spend a moment here to explain how they can be generalised to the Flag markets.

Let me do that in the context of the actual current prices, which appear in the table at right.

We see there that West Coast are $2.50 for the Flag, Hawthorn $2.75, Fremantle $4.50, and the Kangaroos $11. The total overround in the Flag market is therefore (1/2.5 + 1/2.75 + 1/4.5 + 1/11 - 1), which is about 7.7%.

To apply the Overround Equalising approach, we assume that the market price we see (M) is equal to the inverse of the bookmaker's assessment of a team's true chances (P) multiplied by one plus the overround (O), or:

- M = 1 / [P x (1+O)]

In the Overround Equalising case we assume that O is the same for all teams, which gives us, for West Coast in the Flag market 2.50 = 1 / (P x 1.077). Solving for P provides our estimate of the bookmaker's implied West Coast Flag probability, which comes out at abut 37%. Similar calculations for the other teams yield probabilities of 34% for Hawthorn, 21% for Fremantle, and 8% for the Kangaroos.

If, instead, we want to apply the Risk Equalising approach, we recognise that the 7.7% total overround in the Flag market represents the sum of the maximum calibration errors that the bookmaker can afford to have in relation to the four teams, and then assume that he applies this "insurance" equally to the four teams. In other words, his prices are assumed to allow him to be in error with his probability estimates by, at most, 7.7%/4 or about 1.9% points on each team.

We can therefore calculate a team's implicit Flag probability by calculating the inverse of its market price (M) less one-quarter of the total overround in the Flag market (O). So, as an equation:

- M = 1/P - O/4.

For West Coast this equation yields an estimate of the implied Flag probability of 1/2.5 - 7.7%/4, which is about 38%. Similar calculations yield estimates for Hawthorn of 34%, Fremantle 20%, and the Kangaroos 7%.

The differences in probability estimates that the two approaches yield are small in absolute terms (between 0.3% and 1.2% points), but they are large in relative terms for the Kangaroos, which we'll see has a relatively large affect on our estimates of the likely prices for Grand Finals involving the Roos.

### OVERROUND EQUALISING

Now, as I laid out in that blog from 2012, calculating the implicit probabilities in the two Preliminary Final and Flag markets does not allow us to uniquely infer implicit probabilities for all possible Grand Final pairings. In mathematical terms, the system is not fully determined, and the only way to progress is to set the probability for one of the pairings, after which the three other probabilities become fully determined.

I've chosen to set the probability for the Kangaroos v Hawthorn Grand Final and to constrain this to the 25% to 40% range. The results of doing this appear in the table at left.

What we see there, for example under Option C, is that a 35% probability for the Roos toppling the Hawks in the GF is consistent with a 50% probability of them toppling the Dockers should they meet them instead, and is consistent also with the Eagles starting as narrow underdogs should they end up taking on either the Dockers or the Hawks.

For me, the most plausible of the options shown is Option D and the probabilities there would imply Grand Final prices similar to those shown in the table at right (assuming a 5% total overround in each market).

So, rounding a little, we might see a $2.40/$1.60 market if the GF was a Roos v Hawks affair, a $2.30/$1.65 market if it were a Roos v Dockers matchup, a $1.90/$1.90 market if the Eagles played the Dockers, and a $2.10/$1.75 market if the Eagles played the Hawks.

### RISK EQUALISING

For the Risk Equalising case we proceed as we did for the Overround Equalising case, exploring the same four options defined by varying the Roos' chances in a Roos v Hawthorn GF matchup.

What alters most in the results here compared to those for Overround Equalising is the implied probabilities for Roos v Dockers Grand Finals, which are all about 8 to 9% points lower. This is because, under a Risk Equalising approach, the Roos' implied Flag chances drop from the 8.4% they are under an Overround Equalising approach to 7.2%. So, for a given probability of beating the Hawks in a GF and for beating West Coast this weekend, the probability of beating the Dockers must fall to produce a lower overall Flag probability for the Roos.

Here I find the results for Option C most plausible, and they can be used to infer the Grand Final prices shown in the table at right (again assuming a total overround of 5% in each market). Option C has the Roos as 5% points less likely to beat the Hawks compared to the Overround Equalising Option D, which lifts their price in that contest from about $2.40 to about $2.65, but only about 1% point less likely to beat the Dockers, which keeps their price for that contest at about $2.30.

The prices for the two other possible GF matchups are also largely unchanged.

### SUMMARY AND CONCLUSION

This analysis suggests that a Kangaroos v Hawks Grand Final would be seen by the bookmakers as the most potentially lop-sided, with a price for the Roos somewhere in the $2.35 to $2.70 range and for the Hawks in the $1.50 to $1.60 range. A Kangaroos v Dockers GF would likely be a little more competitive, the Roos being offered at around $2.30 and the Dockers at around $1.65.

West Coast, should they meet the Hawks, would likely start as around $2.10 underdogs with the Hawks about $1.75 favourites. If, instead, the Eagles faced the Dockers, the contest would likely start with both teams roughly equal-favourites.

There is, of course, no guarantee that the TAB Bookmaker consistently follows an Overround Equalising or a Risk Equalising approach in setting his prices, so other sets of prices could, instead, match his actual current beliefs. It's difficult though, I've found, to generate plausible GF price quartets that are much different from those shown here and still consistent with Flag probabilities in the ballpark of those derived using the two approaches from this blog.