Author: Uri Blass
Date: 14:36:26 06/06/02
Go up one level in this thread
On June 05, 2002 at 00:05:34, Dann Corbit wrote: >On June 04, 2002 at 23:30:56, Stephen A. Boak wrote: > >>hi Dan, >> >>1. Since Elo's system defines (by design choice!) each specific rating >>difference in terms of a specific expected scoring percentage, regardless of >>where the two ratings fall on the scale, I suspect (but am not sure, not having >>worked out the math yet on paper) that the simple plotting of ratings in a >>histogram would not be a normal bell-shaped curve. > >The original question was whether or not computer strength was normally >distributed. This is a different random variable and is also normally >distributed. We can also plot win percentages multiplied by the opponent's >strength just as well: > >SELECT int((win_percentage * opponent_strength)/5000), count((win_percentage * >opponent_strength)/5000) >FROM SSDF >GROUP BY int((win_percentage * opponent_strength)/5000); > >Expr1000 Expr1001 >8 3 >9 3 >10 3 >11 6 >12 3 >13 12 >14 6 >15 9 >16 10 >17 15 >18 7 >19 12 >20 6 >21 18 >22 18 >23 20 >24 14 >25 11 >26 13 >27 13 >28 6 >29 10 >30 2 >31 4 >32 1 >33 7 >34 1 > >Or (squished a bit more): > >SELECT int(([win_percentage]*[opponent_strength])/15000), >count(([win_percentage]*[opponent_strength])/15000) >FROM SSDF >GROUP BY int(([win_percentage]*[opponent_strength])/15000); > >Expr1000 Expr1001 >2 3 >3 12 >4 21 >5 34 >6 25 >7 56 >8 38 >9 29 >10 7 >11 8 > >>Wouldn't some transformation be required to convert such ratings into >>'normalized' figures which *theoretically* might look more like a bell shaped >>curve? > >There is a surprising range of curve shapes that still fit the gaussian model >pretty well. > >>2. Over time, as new & improved program versions & ratings rise, due to advances >>in chess programming algorithms & techniques (and hardware improvements, >>perhaps), wouldn't the overall plotting of ratings on a histogram (roughly from >>older, weaker programs to newer, stronger programs) more closely follow the >>growth curve for average rating of each new crop of released program/hardware, >>rather than the normal bell curve. > >Since they are different hardware setups or different program versions, they are >treated as different organisms. The method you suggest should only be used to >model a single program, and then only changing one variable at a time (unless >you intend to generate a surface) > >>3. Perhaps any program crop released within a relatively short span of time (say >>a year or so) would have ratings plottable (with transformation, as noted above) >>that closely approximate the normal (bell) curve. > >I think probably the leptokurtotic shape is a function of reality. In other >words, if a program is dominatingly better, nobody would buy the others. If a >program is dominatingly weak, then nobody will buy it. So they are forced to be >fairly close in ability. There is a broad mass with nearly equal ability and a >few outliers with exceptional strength or weakness. > >In other words, if someone wrote a 3000 ELO program, it would be the only one >that people bought and got tested and we would see a spike. No If the price of the 3000 elo program will be 3000$ then I suspect that a lot of people will prefer to buy a program that is more than 200 elo weaker for 50$. Uri
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