Simulating the Finalists for 2013 : Post Round 22

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What a strange and intriguing time it is to be following - and trying to predict - AFL football. As I write this, just three days from the start of the final home-and-away series of games, and just one week more distant from the Finals, there's still no complete resolution of the Essendon debacle and no definitive statement about whether they will or will not play in those Finals.
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Simulated Performance of Head-to-Head Algorithm vs TAB Bookmaker

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In the previous blog I reported that:

the TAB Bookmaker can be thought of as a Bookmaker with zero bias and a 5-5.5% sigma, and the Head-to-Head Probability Predictor can be thought of as a Punter with a +1-2% bias and a 10-12% sigma

Simulating these parameter ranges, with an LPSO-like Bookmaker and with his overround varying in the range 5% to 6.5%, reveals that the return to Kelly-staking for a Punter with Bias and Sigma in these ranges (who, unlike MAFL's Head-to-Head Fund, wagers on Away as well as Home teams) is positively related to Bookmaker Sigma ...

(Note that if Bookmaker Sigma is around 5% the expected return to Kelly-staking is negative.)

The Punter's ROI is negatively related to his Sigma ...

... and pretty much unrelated to his Bias.

These relationships are all similar to what we found in earlier blogs in which the parameter space investigated was much larger.

What's also interesting is that the variability of the returns to Kelly-staking is positively related to Bookmaker Sigma, broadly unrelated to Punter Sigma, and negatively related to Punter Bias.

(Note that these images can be clicked for larger versions.)

 

We also find that, for every set of parameters in the simulation, the expected return to Kelly-staking exceeds that for Level-staking, which is broadly consistent with what we found when exploring the larger parameter space, but the correlation between the average Log Probability Score and the ROI to Kelly-staking is, in absolute terms, always lower than the correlation between the average Brier Score and the ROI to Kelly-staking, which is contrary to what we found when exploring the larger scenario space.

Using RWeka to create simple rules for when to Kelly-stake and when to beat a considered retreat from wagering, the first few rules we're offered are:

  • If the Total Overround is less than 5.88% and the Bookmaker Sigma exceeds 5.24%, then the best strategy is to Kelly-stake, otherwise
  • If the Total Overround is less than 5.63% and the Punter Sigma is less than 11.00%, then the best strategy is, again, to Kelly-stake, otherwise
  • If the Total Overround is greater than 6.18% and the Bookmaker Sigma is less than 5.27%, then the best strategy is not to bet, otherwise
  • If the Total Overround is greater than 5.34%, the Punter Sigma is less than 10.73%, and the Bookmaker Sigma exceeds 5.14%, then the best strategy is to Kelly-stake, otherwise
  • If the Total Overround is less than 5.44%, then the best strategy is to Kelly-stake

For about 70% of the remaining scenarios the recommendation is not to bet.

Using Eureqa's Formulize to build a model of the ROI to Kelly-staking, we find that one of the best-fitting models, with an R-squared in excess of 85%, is:

  • Expected Kelly ROI = 2.143*Bookie Sigma - Bookie Total Overround - Bookie Sigma*Punter Bias - 0.463*Punter Sigma

This suggests that, as we've found previously, the ROI to Kelly-staking is heavily dependent on the Bookmaker's precision and, to a lesser extent, on the Punter's. As well, the ROI to Kelly-staking drops percent-for-percent with the Bookmaker's Total Overround.

And, finally, using Eureqa's Formulize to build a model of the standard deviation of the ROI to Kelly-staking, we find that one of the best-fitting models, but with an R-squared of only around 16%, is:

  • Expected SD Kelly ROI = 0.067 + 0.415*Bookie Sigma - 3.600*Punter Bias*Bookie Total Overround

So, not only does the expected return to Kelly-staking rise with the Bookmaker's Sigma, so too does the variability of that return. In addition, variability falls with Punter Bias and with the Bookmaker's Total Overround.

SUMMARY

This blog addresses Bookmaker vs Punter scenarios that we are, based on empirical data for the TAB Bookmaker and MAFL's Head-to-Head Fund algorithm, most likely to encounter in practice, and shows that there is a fairly narrow range of scenarios - where Bookmaker Sigma is sufficiently high, and Bookmaker overround and Punter Sigma are sufficiently low - for which the expected profit to Kelly-staking is positive.

It also suggests that, within the range of parameter values explored, the variability to Kelly-staking grows with Bookmaker imprecision and shrinks with the product of Punter Bias and Bookmaker Overround.

Bookmaker vs Punter Simulations Revisited : Risk-Equalising and LPSO-Like Bookmakers

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In 2011 I introduced the five-parameter model, which I used to simulate the contest between Bookmaker and Punter by making different assumptions about the relative precision and unbiasedness with which they estimated the Home team's victory probability. The fifth parameter in the model was the total overround embedded in team prices by the Bookmaker and, as was my custom at the time, I assumed that the Bookmaker levied this total overround equally on both teams. Now, I think differently.
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2012 - Final Simulations : Week 2

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We lost Geelong and the Roos in Week 1 of the Finals and MARS re-rated the six remaining teams, which leaves us (using the same model that we used for Week 1 of the Finals) with the following team-versus-team probability matrix.

Broadly, Hawthorn is expected to beat everyone else fairly handily, except Sydney, which they're expected to beat less convincingly but still beat nonetheless.

Using this probability matrix for 1 million simulations yields the following team-by-team Finals outcome probabilities:

Hawthorn are now estimated to win two-thirds of the time and to make the Grand Final almost 9 times in 10.

Sydney are rated about 4/1 chances for the Flag but are about even money to play in the GF.

The remaining teams are mostly there to mop up the residual probabilities, with none rated better than about 30/1 chances for the Flag and about 5/1 or longer chances to even make the Grand Final. Adelaide, despite finishing 2nd on the ladder in the home-and-away season, are now rated only about 32/1 Flag chances and 15/1 chances of even making the Grand Final.

Simulated Grand Final quinella probabilities appear in this next table. A Hawks v Swans matchup is rated by far the most likely pairing, with a probability approaching 60%.

The next most-likely pairings are Hawks v Pies and Hawks v Eagles, which carry probabilities in the 10-15% range, then Swans v Freo and Swans v Crows matchups, which carry probabilities of around 5%.

Only four other pairings are possible, none of which include the Hawks or the Swans, and none of which are assessed by the simulations as being greater than about 1% prospects.

Amongst these quinellas only the Hawthorn v Sydney pairing at $1.90 offers any value on the TAB AFL Futures market. This wager is assessed as having a 14% edge.

Other TAB AFL Futures market wagers with a positive expectation this week are the Hawks to win the Flag at $1.65 (10% edge), the Hawks to play in the GF at $1.25 (7% edge), and the Swans to play in the GF at $1.55 (8% edge).

Adding these bets to those we've identified in AFL Futures markets in previous weeks yields the following picture:

As you can see, I've closed out those wagers whose fate has already been determined, the net return from which, it turns out, is marginally positive. The ROI from level-staking the identified opportunities is just over 1%.

Having locked in, last week, what look like very promising wagers on the Hawks and Swans at attractive prices on various AFL Futures markets, the current expectation must be for that ROI to grow.

At this point, however, all our eggs are very firmly in two baskets each, appropriately enough, avian based. We'll not know anything more about the fate of these wagers for at least another fortnight.