For the uninitiated, LCR is a game in which each player starts with three "tokens" and rolls up to three dice (at most as many as tokens they have). Each die has three sides which indicate that nothing happens and one spot apiece for left, center, and right; left and right indicating that they pass one token in that direction and center indicating that a token goes to the "center," where it is removed from the game. The game ends when only one player has any tokens.
I was wondering what position would be best, so I created the following Python script to do this for me:
from random import * def LCRRound(players): global playerset playerset =  * players while not GameOver(): for player in range(len(playerset)): for i in range(min(3, playerset[player])): Move(player) if GameOver(): break return [playerset.index(p) for p in playerset if p!=0] def L(player): playerset[player]-=1 playerset[player-1]+=1 def C(player): playerset[player]-=1 def R(player): playerset[player]-=1 playerset[(player+1)%len(playerset)]+=1 def GameOver(): return playerset.count(0) == len(playerset)-1 def Move(player): tmp = randrange(6) if tmp==3: L(player) if tmp==4: C(player) if tmp==5: R(player) for x in range(2, 11): wins =  * x for y in range(100000 * x): wins[LCRRound(x)]+=1 print(wins)
Which tests for randomly generated games with 2 to 10 players, playing 100,000 games for each player in each set (so 200,000 games for the two player tests, 300,000 for three, and so on up to 1,000,000 games for ten players). This generated the following output (and similar output other times I ran it), with the numbers being the number of times that player one (players are in order of who rolls):
[76233, 123767] [91720, 98359, 109921] [95913, 97396, 101796, 104895] [96340, 97629, 99926, 103080, 103025] [96985, 96768, 98607, 101509, 103283, 102848] [97557, 96211, 97613, 100659, 102562, 103595, 101803] [97636, 95984, 96652, 99220, 101364, 103619, 104070, 101455] [98000, 95559, 96338, 97589, 99600, 102966, 104767, 104061, 101120] [97355, 95754, 95876, 97163, 99303, 101537, 103103, 104583, 103943, 101383]
For two and three players, the optimal position is last. However, after that, the player who wins the most becomes the second to last and then the third to last (and would presumably continue in this motion as more players are added). This is contrary to what I expected, since I thought the optimal position would be the player immediately before, after, or opposite the starting player, but it is none of these. In face, the second player seems to be the most disfavorable rather than the person who starts. Which leads me to my question:
What is the mathematical explanation for what position is the best in LCR?