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Revised 21 Ats Opening Roll Heuristic
Posted By: AdamStocks
Date: Saturday, 10 May 2014, at 8:07 a.m.
In Response To: Revised 21 Ats Opening Roll Heuristic (scotty)
The principle of the methodology is pretty straightforward. Construct a spreadsheet with a 15x15 grid of scores. Let's for example take splitting as a default option. Look at the XG equity differences between splitting and slotting (i.e. at 1away,1away, splitting is 'best' by minus 14.6 millipoints EMG (i.e. it is worse than slotting by 0.0146 EMG). So put 14.6 into the grid. Repeat for all scores in the first 15x15 grid, which will be for nonCrawford scores. Thus you will have 225 entries. Do another grid for Crawford scores (only the 1away scores will have entries). There are 29 of these including 1away,1,away, but 1away,1away is already included as a nonCrawford score, so really you have only 28 Crawfords, plus the 225 nonCrawfords making 253 altogether in the 2 grids.
The easiest way to do the next step is to make another 15x15 grid next to the nonCrawford grid and make a spreadsheet formula of the type If(equity entry in the first grid is less than zero then put a "$" here, Else put nothing here), and fill this grid with that formula, thereby creating a nice visual representation of which scores are splits and which scores are slots. Then you look for any pattern that might be there. With 21, the pattern is visually obvious  the slots are bunched into one area of the grid and the splits are the empty bits of the grid. The division between these two areas is more or less a straight line with some bumpy bits on it, but it's straight enough to be able to get an approximate straight line mathematical formula of the type y=ax+c that cuts the grid into 2 convenient parts. For 21, this is roughly y=7x+3.5. x is actually A (player on roll's awayscore as in Aaway,Baway), and y is really B. So where 7A+3.5 is greater than B, you will have the "$" entries, indicating that a slot play is preferred. To find the best constants (the 7 and the 3.5), I made a graph from the grid data and played around with various constants to draw a straight line that best fits the division between splits and slots that minimises the overall error. 7 and 3.5 were the best I could find without resorting to 2 decimal places. There are bound to be some entries that fall on the wrong side of the line, hence the 16 errors my rule produces. You will need to make yet another parallel grid that keeps track of the size of these errors so that you can know how accurate or otherwise your constants are. Eventually you won't be able to minimise the errors any more, so therefore those will be the constants to use.
As for the Crawford grid, the odd/even effect of what is the correct play at Crawford scores will 'pepper' one edge of your grid with alternate 'holes', hence the need for the first condition in my rule (concerning free drop scores) which makes these 'holes' an exception to the basic division line.
I don't have a dropbox account, but if you Email me I'll send you my .xls file for you to peruse. If you have any questions about it, just ask. Have fun.

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