If my understanding of weak duality is correct, I can prove that any symmetric, alternating, turn-based, two-player game favors the second mover. (For example, in chess, white can have no forced mate at the beginning of the game--either chess is inherently drawish or black has a forced mate.) I'm asking a question here because I take this as evidence that my understanding of weak duality is incorrect (see also https://xkcd.com/675/).
A sketch of my "proof" follows. Weak duality was introduced to me in a machine learning class as the inequality $$\max_x \min_y f(x, y) \leq \min_y \max_x f(x, y),$$ though this may more properly be the Max-min inequality. This holds for any function $f \colon \mathbb{X} \times \mathbb{Y} \to \mathbb{R}$.
Take chess as our example. Let $x$ and $y$ represent the first moves white and black, respectively, make, and let $f$ be the board evaluation after the first move (assuming perfect play, and we'll assume that this evaluation function is perfect/searches the entire chess game tree). $x$ seeks to maximize it, so we will say that $x$ wins if $f(x, y) > 0$, while $x$ and $y$ tie if $f(x, y) = 0$; otherwise, $y$ wins. The LHS of the above equation implies that white moves first; that is, $x$ must consider what move $y$ will be given that $y$ will be the best move among those possible for any given $x$. Meanwhile, the RHS represents the scenario where black moves first. It was in this sense that my ML professor captioned this inequality as showing the "second-mover advantage".
With that settled, the inequality directly states that the best white can hope to achieve by going first is no better than what it could achieve by going second. "Going second" is the definition of black in chess, so we conclude that white cannot hope for a better outcome than black can; thus, either the game is a draw or black has a forced win.
Where is the error? The sketch above is pretty high level, so I assume it's buried in the weeds of abstraction, but I've not been able to find it, and neither (for what it's worth) has ChatGPT.
$+1$ for the apt xkcd reference :-)
The error is that you confused the order in which the players have to decide on their moves with the order in which the moves are evaluated. It’s always good to be able to decide on your moves as late as possible, but it’s also usually good to be able to actually make them as soon as possible.
For instance, one simple aspect of white’s advantage is that with symmetric play, white can never lose, because any mate will first be a mate for white, and then the symmetric mate for black never happens. Allowing white to decide on their moves after knowing what black will move doesn’t change the fact that white will mate before black with symmetric play.
There’s a minor fixable flaw as well. You only considered the first move, whereas even if switching the decision order implied switching the evaluation order, you’d have to switch it for the entire game and not just for the first pair of moves. For subsequent pairs of moves the domain is no longer a Cartesian product because black’s possible moves depend on white’s move. But you could get around that by allowing illegal moves and defining the evaluation function so that illegal moves lose.