# Does White have an advantage in chess?

It seems likely, doesn’t it? After all, as White, you get to develop your pieces first, putting them in position to better attack Black, or at least defend against his/her attacks. And the statistics seems to bear this out (at least, according to Wikipedia, my source for all things true). Though it could turn out that Black has an advantage – it might be that any move by White fatally weakens his/her position, so that they are at a comparative disadvantage to Black.

Whatever the outcome might be, it turns out that if either side has an advantage, that advantage is necessarily complete. Put more formally: either (a) there exists a strategy profile for White that guarantees victory, or (b) a strategy profile for Black that guarantees victory, or (c) strategy profiles for both White and Black that guarantee a draw. It sounds somewhat trivial, but it’s not: for example, if (a) is true, this means that no matter what Black does, the outcome of the game is not in doubt – White will win.

Results such as these are common among many games without exogenous uncertainty, and many games have been solved, so we know which of the analogues to the above possibilities are true. For example, checkers was recently shown to have strategy profiles for both players which guarantee a draw. So that this holds true for chess as well should not come as a surprise.

To show that one of these three possibilities must hold, we can draw a game tree which contains all the possible move sequences in chess[1]. This is because chess ends in a finite number of moves: a draw is automatically declared if the same position is reached three times, or if fifty moves have gone by without a pawn move or a piece capture. Since there are only a finite number of possible pawn moves (given the size of the board) and piece captures (since there are only 32 pieces), the game is finite.

Next, we can use backward induction (as in my post on tic-tac-toe) from each possible ending of a game to determine the outcome from the beginning. At each node, the player involved (White or Black) deterministically selects the branch that leads to the best final outcome for him/her (using tie-breakers if necessary if several outcomes are equally good). We proceed in this manner all the way up to the initial node, corresponding to the starting position of the game. We can then go back down the tree, and since we have already determined the best response to any position, we can deterministically get to the best outcome for Black or White. This automatically yields a win for one of them, or a draw.

Unfortunately, while this works well in theory, in practice it is virtually impossible. Given the combinatorial explosion of positions in chess, the computing necessary to determine which possibility is correct is infeasible. I guess we’ll be stuck with just a good game of chess.

[1] That is, theoretically; the actual game tree is WAY too big to actually depict