Normalising PRís in Matches
Posted By: Rick Janowski
Date: Sunday, 30 June 2013, at 3:27 p.m.
Almost invariably it would appear, human players (i.e., non-bots) find cube-decisions tougher than checker play decisions. Investigation of the 50+ players on Ianchoís PR database show the following average PR values for checker play (PR_ch), cube play (PR_cu) and overall play (PR_xg):
PR_ch = 3.74 PR_cu = 7.29 PR_xg = 4.28
From these we can deduce that on average the PR for cube play is virtually twice (= 1.95 times) the value for checker play and the average cube play PR is 1.7 times average overall PR. I think these values may be fairly typical of world class and expert level players. From my experience, I think advanced and intermediate level players may generally have higher cube play error ratios, perhaps of the order of 2.5 to 3.0 times average checker play errors. However, I think extreme beginner level players may tend to have similar ratios as for world-class/expert players. One would expect that as the PR approaches zero (as it does with the best bots playing at their highest settings), this ratio will approach some limiting value. I thought first of all this would unity, but it now occurs to be that part of the amplification may simply be related to amplification caused by the raise in stakes, or sudden loss of a point Ė perhaps the limiting value is somewhere between 1 and 1.5.
From the three average values of PR It may also be deduced that the proportion of cube decisions made is about 15% from the following equation:
Cube decision proportion, CDP = (PR_cu Ė PR_xg) / (PR_cu Ė PR_ch) = 0.152.
This value of 15% is probably fairly typical for 7-point matches (the typical length of matches in Ianchoís database. For shorter matches it seems natural that the ratio will reduce as it approaches the limiting value of zero for DMP. For 3-point matches, I think the value might be between 10 and 12.5% as an increasing proportion of subsequent match scores reached are Crawford or Post-Crawford with generally trivial cube decisions. For longer matches than 7-points, the proportion will tend to increase as it approaches some limiting value, which sensibly would appear to be that for money games. I donít know what this is but I would guess perhaps 17.5-20%.
I do not think that skill level will have a massive effect on this ratio, although players who tend to double late will increase their ratios through unnecessary repeat decisions.
Because of the higher value of cube play PR in comparison to checker play PR, overall PR levels may be sensitive to the proportion of cube decisions made. An extreme case is DMP where no decisions are cube related. Assuming that DMP PRís are broadly equivalent to checker play PRís then one would expect the overall PR for matches to be a higher and indeed this is true in my own case and that of other players consulted. The 1.5 factor employed by XG I believe is wholly checker play related, seeking to make the scale of error equivalent to the scale in money play where cubeful equities are enhanced by cube availability. Moreover, there is additional amplification caused by gammons being realised. To obtain a PR value from DMP which be fairly typical for money play, I estimate that you should increase the PR value by between about 15 to 20%, with perhaps 17.5% being typical, to compensate for the higher cube PR and associated proportion of cube related decisions one would expect. Using this approach a DMP PR of 4.0 would equate to a money play PR of about 4.7, which I believe is realistic. PRís at other match scores will tend to be much closer to money values. In the example given above, using a similar approach, a 3-point match (or 4 point match) PR of 4.5 might transpose to the 4.7 value for money play. At other scores, I would expect the PRís to be between 4.6 and 4.7.
Another cause for concern is statistical variation in the proportion of cube decisions made in a single match or series, which will tend to yield unrealistic PR values in many cases. In Ianchoís PR database where there is a spread of values generally ranging from 10 to 20%, there are also four values significantly outside this range, with consequentially unrealistic PRs. Introducing a normalised approach of assuming each player having the same proportion of cube decisions (taking the average of 15%) provides some control on luck by proving a fair and even playing field, but also enhances the reliability by controlling a significant source of variation. I realise that late doubler will benefit a little by this adjustment process because their increase in decisions made caused by repeated missed cubes, will now disappear. However, this may not be undesirable, because it provides some small counterbalance to the somewhat over onerous approach of adding all the missed cube errors together, occasionally to the almost ridiculous extent that the player loses more equity than the whole game is worth.
Consequently, I believe the PR values outputted by XG would be improved in both fairness and reliability if they were normalised to be broadly equivalent to money game PRs using a suitable constant ratio of cube play errors to checker play errors. If anyone has PR data for money games or longer matches (say 11 points and above) with a fairly large sample size this would be very helpful for this purpose.
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