In the book _Freakonomics_ by Steven D. Levitt and Stephen J. Dubner, the authors showed that sumo wrestlers who threw matches could be detected from their stats. With this idea in mind, I developed a way of testing—at least a negative test—whether some kdice players are pga’s of others. Unlike the individual observation method that I described last month, this method relies on the players’ percentage profiles provided by Ryan.

Let us consider a particular player—let us call him R—whom we want to find pre-game associates of. A pre-game associate is one who tends to play games with R often, and can be a pre-game ally, pre-game enemy, pre-game secret admirer, or pre-game old time friend.

Let us say that R has played eight games, and his number of finishes, from first to seventh, is 2-3-2-0-0-0-1. Now if R has six pre-game associates who played those same eight games, their collective finishes would be 6-5-6-8-8-8-7, the 8-modulus of R’s finishes. Dividing by the number of players and games, the percentage profiles of the six pga’s would average out to 12%-10%-12%-16%-16%-16%-14%. So, to spot the 6 pga’s we would look for 6 players whose profiles most closely match 10%-12%-16%-16%-16%-14%.

If it were only so easy.

The profile developed above is under the assumption that the 6 pga’s played only 8 games, all of which were with player R. Obviously, anyone who has played more than 8 games will have played some of them without player R, and for those other games we may assume that they play their normal game. So the profile of any player will be the weighted average of the number of games played with R and the number of games played without R.

To measure the likelihood of a pga of a particular player, I have developed a new test called the Test Against Selected Modulus, or TASM. A player’s TASM against a particular other player is the chi-square goodness of fit test against the weighted average of the player’s ordinary game and the other player’s modulus.

So how do we determine a player’s ordinary game? That is the tricky part. To put it another way, impossible. The second best thing is use the game of ordinary players. For the TASM, the best indicator of the game of ordinary players in the top 25, who are the subject of study, would be the average profile of the top-25er’s. There will be some bias in the average profile, since suspected pga’s profiles contribute to it, but I can think of no better number to use.

For the 24 players in the study, player R excluded, the average percentage profile was 17%-15%-15%-15%-11%-12%-10%. For each player, the TASM is the chi-square fit of the weighted average of player R’s modulus, 10%-12%-16%-16%-16%-14%, and the ordinary game, 17%-15%-15%-15%-11%-12%-10%.

The following table shows the top 25 players in the study, their percentage profiles, and their TASM’s. The TASM is a two-tailed test. A very large TASM indicates that the particular player is not a pga of player R. A very small TASM (but I don’t know how small it would have to be) suggests that a player might be a pga of player R, but it cannot be used as a positive test since it is possible for a player never to have played player R and have a small TASM. So the TASM should be used more as an elimination test than an identification test. The players are arranged in order of increasing TASM.

Remember, the closer you are to the bottom of the list, the less likely you are a pga of Player R.

petomni -27 2152 (96th) 62 17% 16% 17% 12% 12% 9% 12% 228
MCiGGzy -22 2113 (154th) 120 12% 15% 17% 13% 13% 15% 13% 344
R.A.T.M. 68 2133 (123rd) 79 10% 15% 17% 17% 16% 11% 11% 415
uukrul 0 2174 (63rd) 82 14% 12% 17% 12% 12% 19% 12% 450
monkeymagic -41 2182 (54th) 76 14% 21% 11% 18% 11% 11% 10% 472
Tom-ster 0 2210 (26th) 54 18% 22% 14% 18% 11% 9% 5% 773
borsato 0 2200 (34th) 35 20% 14% 14% 8% 17% 20% 5% 786
Onimushaport -35 2152 (96th) 73 15% 10% 13% 13% 10% 15% 20% 875
[ju] 0 2243 (8th) 44 18% 18% 11% 18% 20% 6% 6% 923
SodaPop 64 2188 (47th) 100 15% 13% 19% 9% 17% 11% 16% 938
super strut 0 2192 (42nd) 38 15% 10% 15% 28% 10% 13% 5% 956
dakerzzz -20 2165 (79th) 47 19% 8% 12% 10% 14% 12% 21% 970
Phoenix37 -16 2163 (80th) 98 10% 22% 11% 11% 15% 16% 13% 1087
Vermont 0 2271 (5th) 14 21% 14% 14% 28% 0% 14% 7% 1199
sajin 0 2206 (30th) 21 14% 9% 28% 14% 19% 9% 4% 1216
RaccoonTail 0 2186 (49th) 44 22% 11% 25% 6% 9% 13% 11% 1218
Orlafede 22 2190 (44th) 39 25% 12% 5% 7% 12% 17% 17% 1275
Guit 0 2113 (154th) 47 14% 12% 12% 29% 4% 12% 12% 1292
leekstep -56 2237 (10th) 72 16% 26% 18% 6% 5% 12% 13% 1810
dasfury 0 2190 (44th) 14 14% 28% 0% 7% 21% 21% 7% 1871
Sinth 31 2253 (6th) 37 27% 24% 5% 18% 10% 5% 8% 1885
bcmatteagles -35 2234 (12th) 9 11% 22% 33% 11% 0% 22% 0% 2476
riser2 0 2289 (2nd) 20 35% 10% 15% 25% 10% 0% 5% 2538
Peter North 44 2275 (4th) 21 38% 19% 9% 9% 9% 0% 14% 2917
skrumgaer 0 1656th (3584th) 37 10% 8% 21% 18% 35% 2% 2% 3895
riser 0 2344 (1st) 8 25% 37% 25% 0% 0% 0% 12% 4224