Minimax is a decision rule used in blah and philosophy for minimizing the possible loss for a worst case.
In Game theory, the maximin value of a player is the largest value that the player can be sure to get without knowing the actions of other players. Its formal definition is: \[\underline{v_i} = \mathop{max}_{a_i} \mathop{min}_{a_{-i}} \text{Value}_i(a_i, a_{-i})\] where blah blah blah, similarly for the minimax value \(\overline{v_i}\) of a player.
直观上来说， in maximin, the player \(i\) tries to maximize their value before knowing what the others will do; in minimax, the player maximize their value knowing what the others did. 所以有 \(\text{Maximin} \leq \text{Minimax}\)
Nash equilibrium 是什么？ In game theory, the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy. 纳什均衡是一个解的概念,当其他人策略不变时，任何人都不能够改变策略使得自己的收益增加。
In zero-sum games, the minimax solution is the same as the nash equilibrium. 这句话到底是什么意思呢？ zero-sum games 中的 minimax solution 是什么？ The name minimax arises because each player minimizes the maximum payoff possible for the other, 因为零和博弈，所以他也是 minimizes the maximum loss possible for himself. 看起来很神奇的结论，这样就意味着对于 player 1 的 minimax value 和 player 2 的 minimax value 只相差一个符号，并且有一个混合策略使得各自的期望收益对应 minimax value. 很神奇，怎么证明呢？
Maximin is distinct from minimax, maximin is a term for non-zero-sum games to describle the strategy which maximize one’s own minimum payoff.
In blah theory, there is a minimax algorithm for the game solutions. This algorithm is a recursive algorithm for choosing the next move in a n-player game. 我个人的看法是给定当前状态，每个player假定对手策略不变，选取一个最优策略去最大化下一轮的期望收益，是这样吗？零和博弈才会出现 minimax solution \(\equiv\) nash equilibrium, 那么这个算法是不是也只适用于此？ 居然有R软件包GNE可以直接计算
感觉博弈论水很深，我怕了，撤退，我找其他应用了。