[Ruphin]

There is no hidden information and turns are sequentially. This means both players have full access to all information, and that implies there must be a single "best" move. If the board is at a certain state and one player is to play next, the strategy of the opponent is irrelevant for determining the optimal move.

Proof:

Suppose that it is possible that different strategies exist that outperform each other. Lets consider the case that there are three "optimal" strategies that counter each other, call them Rock Paper and Scissors. Since these are the three "optimal" strategies, both players know the exact move produced by these strategies. The first important part to note is that there is no cost to changing strategy during the game. If an opponent has one fixed strategy, say Rock, the "optimal" strategy is to change your strategy to Paper. Since you have full information on the state of the board, you do not have to guess the strategy of the opponent, because there is no hidden information. Now you could consider the opponent strategy to be hidden information, but this doesn't hold for the following reason: If knowing the opponent strategy would alter your choice of move (by for example changing to Rock that specifically counters his hidden Scissors), the opponent can freely change to Paper after you made your move, because he can see that you changed to Rock before making his next move. If it was impossible for the opponent to tell that you changed your strategy, that implies that each strategy produces the same board state, and that implies the strategies are identical.

[blharr]

Aren't the same conditions true for chess, though there is no proof that a best strategy exists?

What if it's not just rock/paper/scissors, and there are more than 3 strategies?

[Ruphin]

There is no fundamental difference between Chess and Tic-Tac-Toe other than the computational complexity. It is not feasible to compute all of the game states in Chess, and so we cannot determine (at this time) a single optimal strategy the same way we can for Tic-Tac-Toe. But that does not imply that it doesn't exist. From a pure game theoretical perspective, they are the same category of game.

If there are more than 3 strategies, the same argument stands. IF there are multiple "optimal" strategies, AND these strategies create different "optimal" moves on the same board state, the opponent can just play the optimal counter-strategy after you make your move, because your move is enough information to give away your strategy. If your opponent cannot determine your strategy after you make your move, that implies the different strategies have the same "optimal" move.

[sobellian]

At least one of the players has a strategy that cannot be countered though. That is, either white can force a win, black can force a win, or both players can force a draw. There is no situation where every strategy can be countered.