> If N players use the same strategy then by symmetry each of them must win on average 1 time in N. The point here is that there's an option of tying, if all p

> If N players use the same strategy then by symmetry each of them must win on average 1 time in N. The point here is that there's an option of tying, if all players choose the same number. So P_1_win + P_2_win + P_3_win + P_tie = 1, means P_i_win < 1/3. > Also, if all players use the same strategy then it doesn't matter what the strategy actually is. The only requirement is that it have some probabilistic element, otherwise all players would choose the same number every time. A player choose its next move using some probabilities P, ie: it's going to choose randomly number 1 the 45% of times, 2 the 25% of times, etc. So there's a random element in the game, which makes players following the same strategy to not choose the same numbers every time.