> Also, if all players use the same strategy then it doesn't matter what the strategy actually is. But it's not very interesting to add "all players use the sa

> Also, if all players use the same strategy then it doesn't matter what the strategy actually is. But it's not very interesting to add "all players use the same strategy" as an assumption, because yes, obviously, if every player randomly picks from the set {33, 247, 17855433344} each has a 29.6% chance of winning. But that's a different game. In fact the author doesn't assume everyone will use the same strategy; rather, the author simply asks what optimal play is, when you don't know the others' strategies. The fact that all players choose the same strategy (if they're playing optimally) is a consequence of this. Optimal play has to work regardless of the strategy your opponents pick; assuming they'll play optimally too is just a shortcut to figuring out what optimal play is. For example, let's say you knew the other two players' strategy is to pick from this distribution. If you simply always pick 1, you win any time neither of the others picked 1, which happens with probability (1-0.456)^2 ~ 0.296 -- the same probability of winning as the optimal strategy! So why bother with the distribution? Precisely because you don't know whether your opponents are playing optimally or not. If another player gets the same bright idea as you, you counteract each other and the 3rd player wins 54% of the time (the rest are ties). An optimal strategy has to be optimal even if opponents are not using it; otherwise it's not really optimal (one may not exist). So all players using the same strategy is not an assumption, but a consequence, and therefore much more interesting.