Existence of optimal strategies for dynamic zero-sum 2-person games As https://mathworld.wolfram.com/MinimaxTheorem.html says.

The fundamental theorem of game theory which states that every finite, zero-sum, two-person game has optimal mixed strategies. It was proved by John von Neumann in 1928.
Formally, let X and Y be mixed strategies for players A and B. Let A be the payoff matrix. Then
$$\max_X\min_YX^{T}AY=\min_Y\max_XX^{T}AY=v \  \ \ \ \ (1)$$
where v is called the value of the game and X and Y are called the solutions. It also turns out that if there is more than one optimal mixed strategy, there are infinitely many.

It looks like that the equation (1) is for static games rather than dynamic games.
For dynamic games, I mean the order in which the decisions are made is important, e.g., extensive games, differential games. So the strategy is a map from decision histories or system states to actions.
If we think about dynamic games, the optimal strategy we seek for is not a vector but a function which may be of infinite dimension, leading to the theorem's failing.
My questions are:

*

*Is it right that the theorem fails on dynamic form?

*Are there some theorems or research results about the existence of optimal strategies for zero-sum 2-person dynamic games?

 A: You should define carefully what you mean by dynamic games. If you are thinking of extensive form games, then the Sion minimax Theorem, together with the equivalence of general randomized strategies and behavioral strategies, will yield a positive answer.
https://en.wikipedia.org/wiki/Sion%27s_minimax_theorem
What is the difference between mixed strategy and behavioral strategy games?
https://www.yuval-peres-books.com/game-theory-alive/   Section 6.2.1 page 120
https://en.wikipedia.org/wiki/Strategy_(game_theory)#Behavior_strategy
See also: Solving zero-sum extensive-form games with arbitrary
payoff uncertainty models
https://arxiv.org/pdf/1905.03850.pdf
A: To question 1, the short answer is it depends.
For example, as for certain dynamic games with finite stage, finite action space and finite state space extensive,

*

*The strategy's (function) space is finite. So we have strategy vector $X$ and $Y$.

*Given strategies of each, finite stage gaurantees a payoff. So we have payoff matrix A.

The minimax theorem holds, so does the existence of the optimal strategies.
One example is Tic-tac-toe game.
