When the TAB Sportsbet bookie is framing a market for an upcoming game, clearly one set of data that he uses is the recent results of the participating teams. This raises two questions:
- What proportion of the variability in bookie prices can be explained solely by recent results?
- What relative weighting does the bookie ascribe to each previous game?
To investigate these questions I fed the following data into Eureqa for every home-and-away game starting with Round 1 of 2000 and finishing with Round 22 of 2009, which is 176x10 = 1,760 games:
- Bookie prices for each team (entered as implied probabilities)
- The average result for each of the participating teams across the most recent 1, 2, 3, 4 - and so on up to the most recent 22 - home-and-away games. That's 22 pieces of data for each team.
After a time, the best equation that Eureqa turned up was the following:
Predicted Bookie Probability = 0.5 + 0.000194583*Ave_Res_Last_1 + 0.00157292*Ave_Res_Last_6 + 0.00441631*Ave_Res_Last_22
(As it turned out I needn't necessarily have used Eureqa to find this simple linear equation, but using Eureqa at allowed for the possibility of finding a better fitting, non-linear solution.)
This equation explains only about one-third of the variability in bookie prices, which tells us that the vast majority of the variability in bookie prices comes from other than their simple consideration of the teams' previous results - presumably from things such as home ground status, player rosters, and an assessment of the teams' relevant abilities that is not reflected in the simplicity of recent results (for example, knowledge of the teams against which those most recent results were achieved).
The model presented above forms predictions by combining a team's most recent result with the average formed from its most recent 6 results, and with the average formed from its most recent 22 results. If you unravel all those averages you can convert this equation into a simpler (but longer) one that involves the results of individual games - that is, you can make it of the form 0.5 + a*Result from Previous Round + b*Result from the Round before Last + ... z*Result 22 rounds ago.
Doing this allows you to estimate the relative importance that the bookie ascribes to each result - the (normalised) a's, b's and z's in the equation above. In the TAB Sportsbet bookie's case, he weights the single most recent result most highly, assigning it a relative weight of about 11%. The next five most recent games - those from 2 to 6 rounds ago - each carry about a 7.5% weight, and then all games from between 7 and 22 rounds ago carry a weight of about 3.25%.
It's interesting to me firstly that the most recent game carries so much additional weight with the bookie, that the 6 most recent games carry about 50% of the weight, and that games as far as 22 rounds ago carry any weight at all. In fact, the weighting on games even older than 22 rounds ago might also be non-zero. In forming the average results series I arbitrarily imposed a maximum of 22 rounds, expecting that the weightings would taper off sharply beyond 10-12 ago rounds at most.
With that result in mind, and remembering that the Super Smart Model has shown that MARS Ratings have something to offer for predictive purposes in addition to what we can extract from bookie prices, I thought it would be interesting to subject MARS Ratings to the same sort of analysis. In other words, to find out how a team's most recent results are reflected in its MARS Rating.
To do this I proceeded in exactly the same way as I did in creating the earlier model except that I swapped MARS Ratings for Bookie Probabilities.
This time Eureqa came up with the following:
Predicted MARS Rating = 1 + 0.0000612456*Ave_Res_Last_3 + 0.000247029*Ave_Res_Last_10 + 0.000677933*Ave_Res_Last_22
This equation does a fine job of explaining variability in MARS Ratings. It accounts for over 90% of it.
A similar process for unwinding the averages in this equation allows us to determine that MARS Ratings weight games from 1 to 3 rounds ago equally at a little over 6% each. Games from 4 to 10 rounds ago are weighted only a little less, at just under 6%, and games from 11 to 22 rounds ago are weighted at about 3.3%, which is very similar to the weightings they receive from the bookie.
The chart below more readily facilitates a comparison of the weighting pattern of the TAB Sportsbet bookie compared with MARS Ratings.
The contribution of MARS Ratings to forming good prediction is then, perhaps, that it responds less rapidly to short term fluctuations in form, and the contribution of Bookie prices is that they incorporate a broader range of considerations than just recent results.