# minmax optimal strategies

consider we have game between adversary and player. Player can make some actions, let' call $w$ and adversary can make actions $\ell$. Then player get loss function $L(w, \ell)$. Adversary trying maximize $L$ and the player trying minimize the $L$. What is defeniniton of minmax optimal strategies in this case?

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Minimax attempts to minimise the worst case.

So, the player will seek to choose a strategy where the worst case is the least harsh.

So, for each possible strategy w, choose the opponent's strategy to give the worst value of L. Now the strategy we'll choose is the one where the worst value of L is the highest.

Suppose strategy $v$ gives a minimax strategy. Then, for all strategies $w$: $$\min_{\forall{l}}(L(v,l)) \geq \min_{\forall{l}}(L(w,l))$$ In other words, the worst case for the strategy $v$, has a better (at least as good) outcome than the worst case for any other strategy.

This is a risk-averse strategy: the player is imagining that the adversary will choose the nastiest possible option and thus protecting themselves accordingly. A good way of thinking about this is - if I know the adversary will pick after seeing what I picked, what should I do? (although we usually imagine that the two players pick simultaneously)

The adversary can also play a minimax strategy, symmetrically. I'm not sure that this is relevant to your question.

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Do you have some kind of 'conditions' on w and l... like do they come from finite sets or how did this question come up. Yet generally you might say a minimax strategy for the player is a w such that Max L(l,v) =< L(l,w) for every l that the adversary can choose. A similar application can find minimax strategies for the adversary (IE maximizing L given certain conditions). But now it's important to specify what the conditions on l and w are.

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