This program simulates the dice game left right center. Player 0 is the one who goes first up to the (n-1)th player. Players passing to the left goes to player i-1 except that player 0 passes to the (n-1)th player. Similarly, players passing to the right goes to player i+1 except player (i-1) goes to player 0.

Results 10M simulations:

8 players

player 0 win rate biased by -0.32916
player 1 win rate biased by -0.50054
player 2 win rate biased by -0.37765
player 3 win rate biased by -0.12424
player 4 win rate biased by 0.18864
player 5 win rate biased by 0.43943
player 6 win rate biased by 0.49795
player 7 win rate biased by 0.20557

9 players

player 0 win rate biased by -0.27304
player 1 win rate biased by -0.45511
player 2 win rate biased by -0.41751
player 3 win rate biased by -0.22497
player 4 win rate biased by 0.04331
player 5 win rate biased by 0.28722
player 6 win rate biased by 0.45322
player 7 win rate biased by 0.44902
player 8 win rate biased by 0.13787

10 players

player 0 win rate biased by -0.25588
player 1 win rate biased by -0.40888
player 2 win rate biased by -0.41768
player 3 win rate biased by -0.28712
player 4 win rate biased by -0.10212
player 5 win rate biased by 0.15494
player 6 win rate biased by 0.35212
player 7 win rate biased by 0.44370
player 8 win rate biased by 0.40755
player 9 win rate biased by 0.11337

11 players

player 0 win rate biased by -0.24240
player 1 win rate biased by -0.39904
player 2 win rate biased by -0.41065
player 3 win rate biased by -0.29988
player 4 win rate biased by -0.15216
player 5 win rate biased by 0.02343
player 6 win rate biased by 0.19159
player 7 win rate biased by 0.37264
player 8 win rate biased by 0.45194
player 9 win rate biased by 0.37623
player 10 win rate biased by 0.08829

12 players

player 0 win rate biased by -0.22356
player 1 win rate biased by -0.35986
player 2 win rate biased by -0.39305
player 3 win rate biased by -0.33734
player 4 win rate biased by -0.20592
player 5 win rate biased by -0.04231
player 6 win rate biased by 0.08839
player 7 win rate biased by 0.24749
player 8 win rate biased by 0.38176
player 9 win rate biased by 0.43635
player 10 win rate biased by 0.32904
player 11 win rate biased by 0.07905

13 players

player 0 win rate biased by -0.20531
player 1 win rate biased by -0.33983
player 2 win rate biased by -0.38259
player 3 win rate biased by -0.33013
player 4 win rate biased by -0.24159
player 5 win rate biased by -0.11175
player 6 win rate biased by 0.01930
player 7 win rate biased by 0.14441
player 8 win rate biased by 0.28193
player 9 win rate biased by 0.38350
player 10 win rate biased by 0.41646
player 11 win rate biased by 0.30641
player 12 win rate biased by 0.05916

14 players

player 0 win rate biased by -0.19733
player 1 win rate biased by -0.32906
player 2 win rate biased by -0.36071
player 3 win rate biased by -0.32419
player 4 win rate biased by -0.24272
player 5 win rate biased by -0.15278
player 6 win rate biased by -0.05542
player 7 win rate biased by 0.07286
player 8 win rate biased by 0.18924
player 9 win rate biased by 0.30050
player 10 win rate biased by 0.38155
player 11 win rate biased by 0.40408
player 12 win rate biased by 0.26727
player 13 win rate biased by 0.04667

15 players

player 0 win rate biased by -0.18254
player 1 win rate biased by -0.32462
player 2 win rate biased by -0.35268
player 3 win rate biased by -0.33645
player 4 win rate biased by -0.26377
player 5 win rate biased by -0.19125
player 6 win rate biased by -0.06974
player 7 win rate biased by 0.01645
player 8 win rate biased by 0.12153
player 9 win rate biased by 0.21161
player 10 win rate biased by 0.30326
player 11 win rate biased by 0.37635
player 12 win rate biased by 0.38672
player 13 win rate biased by 0.26152
player 14 win rate biased by 0.04356

16 players

player 0 win rate biased by -0.16897
player 1 win rate biased by -0.30013
player 2 win rate biased by -0.35063
player 3 win rate biased by -0.33360
player 4 win rate biased by -0.28075
player 5 win rate biased by -0.19879
player 6 win rate biased by -0.12409
player 7 win rate biased by -0.04682
player 8 win rate biased by 0.05494
player 9 win rate biased by 0.15734
player 10 win rate biased by 0.24197
player 11 win rate biased by 0.31631
player 12 win rate biased by 0.38060
player 13 win rate biased by 0.35291
player 14 win rate biased by 0.24882
player 15 win rate biased by 0.05089

17 players

player 0 win rate biased by -0.15958
player 1 win rate biased by -0.28816
player 2 win rate biased by -0.33123
player 3 win rate biased by -0.31931
player 4 win rate biased by -0.28446
player 5 win rate biased by -0.20252
player 6 win rate biased by -0.13856
player 7 win rate biased by -0.07975
player 8 win rate biased by 0.00532
player 9 win rate biased by 0.08628
player 10 win rate biased by 0.16661
player 11 win rate biased by 0.24262
player 12 win rate biased by 0.31867
player 13 win rate biased by 0.36634
player 14 win rate biased by 0.34036
player 15 win rate biased by 0.23328
player 16 win rate biased by 0.04414

18 players

player 0 win rate biased by -0.16251
player 1 win rate biased by -0.28960
player 2 win rate biased by -0.32390
player 3 win rate biased by -0.31285
player 4 win rate biased by -0.27717
player 5 win rate biased by -0.21816
player 6 win rate biased by -0.15642
player 7 win rate biased by -0.09721
player 8 win rate biased by -0.03534
player 9 win rate biased by 0.05022
player 10 win rate biased by 0.11944
player 11 win rate biased by 0.18879
player 12 win rate biased by 0.26144
player 13 win rate biased by 0.31683
player 14 win rate biased by 0.35890
player 15 win rate biased by 0.33425
player 16 win rate biased by 0.22474
player 17 win rate biased by 0.01847

19 players

player 0 win rate biased by -0.15406
player 1 win rate biased by -0.26654
player 2 win rate biased by -0.31279
player 3 win rate biased by -0.30630
player 4 win rate biased by -0.26733
player 5 win rate biased by -0.22656
player 6 win rate biased by -0.17300
player 7 win rate biased by -0.12053
player 8 win rate biased by -0.05583
player 9 win rate biased by 0.01234
player 10 win rate biased by 0.05504
player 11 win rate biased by 0.14355
player 12 win rate biased by 0.19991
player 13 win rate biased by 0.25564
player 14 win rate biased by 0.32406
player 15 win rate biased by 0.35499
player 16 win rate biased by 0.30442
player 17 win rate biased by 0.19314
player 18 win rate biased by 0.03981

20 players

player 0 win rate biased by -0.13614
player 1 win rate biased by -0.25365
player 2 win rate biased by -0.31003
player 3 win rate biased by -0.30393
player 4 win rate biased by -0.27492
player 5 win rate biased by -0.22951
player 6 win rate biased by -0.19274
player 7 win rate biased by -0.13258
player 8 win rate biased by -0.07409
player 9 win rate biased by -0.03061
player 10 win rate biased by 0.04771
player 11 win rate biased by 0.09637
player 12 win rate biased by 0.14240
player 13 win rate biased by 0.21438
player 14 win rate biased by 0.27413
player 15 win rate biased by 0.31282
player 16 win rate biased by 0.33620
player 17 win rate biased by 0.30359
player 18 win rate biased by 0.18279
player 19 win rate biased by 0.02781