I've found yet another MAFL-related use for the Eureqa tool, this time to determine the precise relationship between a team's head-to-head price and the start it's giving or receiving on line betting.
A simple plot of the history of a team's head-to-head price (or the probability that can be inferred from it) versus its start on line betting makes it obvious that there's a relationship between the two and that it's a non-linear one, but in the past I've been constrained by my own (lack of) ingenuity and persistence in generating sufficient possibilities to find its exact nature.
Eureqa, with minimal guidance from me, found this equation fairly rapidly:
Predicted Start = 22.3 * ln((1-Prob)/Prob)
(There's something deeply satisfying about stumbling across a log-odds ratio - which is what the ln((1-Prob)/Prob) term is - unexpectedly in the current context.)
As you can see from the following chart this equation does a great job of fitting the historical data. In fact, the R-squared between the Actual and Predicted Starts is over 98%.
One use for this equation is in determining the start that a team should have received in games where it is a narrow favourite. For the past few years TAB Sportsbet have offered a minimum 6.5 points start for underdogs however narrow their underdoggedness, choosing to adjust the price offered for each team on line betting away from the standard $1.90 to reflect the fact that 6.5 points start doesn't create a 50-50 proposition for both teams in this situation.
(I realise underdoggedness is not a word, but if favourites enjoy favouritism what do underdogs endure?)
For example, this week Melbourne are priced at $1.80 to Essendon's $1.95 on the head-to-head market. This means that they should have been giving 22.3*ln(0.48/0.52) or about 1.5 points start, not the 6.5 points that they're actually giving, albeit with a $2.05 line price.
It's also interesting to note the outliers in the chart above for large values of actual start. Those outliers at the top left of the chart are teams where the points start presumably truly reflects the team's superiority but where this dominance isn't reflected in their head-to-head price due to TAB Sportsbet's reticence to offer prices under $1.05.
Consider, for example, a team giving 80.5 points start. Using the equation above suggests that this team's victory probability is about 97.35%, which implies a head-to-head price - ignoring any adjustment for vig - of 1/0.9735 or about $1.03. On TAB Sportsbet this team will be priced at $1.05 and the vig will presumably be made up by shortening the underdog's price.