Previously I've developed the notion of taking a Bookmaker's prices in the head-to-head market and using them to infer his opinion about the true victory probabilities of the competing teams by adopting an Overround-Equalising or a Risk-Equalising approach. In this blog I'll be summarising and generalising these approaches.
Let me start by restating the general terminology for the case of the head-to-head market:
- The market price for Team 1 is defined as m1 = 1/(p1(1+o1)), where p1 is the Bookmaker's assessment of the true victory probability (ignoring draws) of Team 1, and o1 is defined as the overround embedded in the price for Team 1.
- The market price for Team 2 is defined analagously as m2 = 1/((1-p1)(1+o2)), where (1-p1) is the Bookmaker's assessment of the true victory probability (ignoring draws) of Team 2, and o2 is defined as the overround embedded in the price for Team 2.
- In the Overround-Equalising case we set o1 = o2 = k, a constant, so the bookmaker embeds overround equally in the prices of both teams and the total overround in the head-to-head market T, given by 1/m1 + 1/m2 - 1, is equal to k.
- In the Risk-Equalising case we set o1 = e/p1 and o2 = e/p2, where e is a constant equal to the maximum calibration error against which the two team's prices are designed to protect. Specifically, e ensures that, even if the true victory probability of Team 1 is p1 + e or of Team 2 is p2 + e, wagers on either Team 1 or Team 2 have exactly zero expected return. If the true probabilities are less than p1 + e for Team 1 and less than p2 + e for Team 2, then the expectation for the Bookmaker on these wagers is positive. Here, the total overround T is equal to 2e.
There's no particular reason, a preference for parsimony aside, to believe that any given Bookmaker might determine to protect against the same calibration error for both teams, in which case we could define instead:
- o1 = e1/p1
- o2 = e2/(1-p1)
These definitions imply a total overround in the market, T, of e1 + e2, and prices m1 = 1/(p1+e1) and m2 = 1/(1 - p1 + e2).
To make this tangible, imagine you're a bookmaker and that the following is true:
- You believe Team 1's chance of victory are 60%, but you want to protect against the possibility they're as high as 65%. So, e1 = 5%.
- You believe Team 2's chance of victory are (by complementarity, ignoring draws) 40%, but you want to protect against the possibility they're as high as 47% So, e2 = 7%.
Based on these assumptions you'll price:
- Team 1 at 1/(60% + 5%) = $1.54
- Team 2 at 1/(40% + 7%) = $2.13
The total overround in that market is 1/1.54 + 1/2.13 - 1 = 12%, the sum of the two maximum calibration errors against which the team prices protect.
Two-Result Case: Team-Specific Overround a General Function of Price
Another obvious generalisation of the head-to-head approach is to assume that the overround on each team is determined as some function of that team's true probability:
- oi = fi(pi) for i = 1,2
Then, if we can re-express fi(pi) = 1/pi * gi(pi) for both values of i, the calibration error for Team i against which such a formulation would protect is given by gi(pi) and the total overround in the market, T, is given by g1(p1) + g2(1-p1).
Candidate functional forms for fi are:
- e/pi, with e a constant for both i, in which case we've the Risk-Equalising Approach
- k, a constant for both i, in which case we've the Overround-Equalising Approach
- |pi - 0.5|, so that overround is higher on teams whose probability is nearer 0 or 1
- (pi - 0.5)2, so that overround is much higher on teams whose probability is nearer 0 or 1
- 1 - pi so that overround is higher on underdogs (ie we have a favourite-longshot bias)
- pi so that overround is higher on favourites (ie we have a reverse favourite-longshot bias)
We might explore the empirical support for these and other functional forms in a later blog.
(In a similar vein, in this earlier blog I investigated different functional forms pi = h(mi,T) and what they implied for oi as a function of mi and T.)
Multiple-Result Overround and Risk-Equalisation
What about the case of a market where more than two outcomes are possible - say, for example, the market for the Flag?
Well, there's nothing preventing us from applying the Overround Equalising and Risk-Equalising frameworks to such situations.
So, for example, consider a Flag market with four teams priced at $2.25, $3.25, $3.50 and $9.00, for which we'd find that the total overround is 1/2.25 + 1/3.25 + 1/3.5 + 1/9 - 1 = 0.149.
- Adopting an overround-equalising approach we determine the implied probability for the ith team as 1/(mi*(1+T)) and the respective implied probabilities would be 39%, 25%. 27% and 10% (ie 1/(2.25 * 1.149), 1/(3.25 * 1.149), and so on).
- Adopting a risk-equalising approach we determine the implied probability for the ith team as 1/mi - T/n (where n is the number of possible results) and the respective implied probabilities would be 41%, 25%, 27% and 10% (ie 1/2.25 - 0.149/4, 1/3.25 - 0.149/4 etc).
Stated more generally,
- the Overround-Equalising approach can be used to calculate implicit probabilities in markets with multiple possible outcomes (and hence prices) by calculating the total overround, T, as the sum of the reciprocals of the prices less 1, and then "deflating" each price by using equations of the form 1/(mi*T) for all possible versions of i.
- the Risk-Equalising approach can be adopted by calculating implicit probabilities via 1/mi - T/n, where n is the number of possible outcomes.
Total Overround as Aggregate Miscalibration Protection
Staying on the multiple-result case and thinking most generally of all we can consider the price for every result as having been established by the Bookmaker to protect against some worst-case result-specific calibration error, so that the price for the ith result is determined by mi = 1/(1+ei/pi)) with ei the worst-case calibration error against which the price provides "insurance" and with pi the price-setter's assessment of the true probability of the ith outcome.
In this conceptualisation the total overround in the market, T, can be thought of as the aggregate calibration errors against which, collectively, the set of market prices protect. Such a view provides motivation for the empirical observation that total overround increases with the number of possible outcomes in a contest. If the Bookmaker wants to protect against some maximum calibration error in his probability assessments of any single team's chances, and he wants to apply this to every team, then the greater the number of teams, the larger will be the total overround.
It might also explain why the total overround in the TAB head-to-head market for every game has tended to contract as game time approaches and as, presumably, potential sources of calibration error such as the withdrawal of an important player or a change in the weather dissipate or crystalise.