In 1975 the first author proved that every finite tight two-person game form $g$ is Nash-solvable, that is, for every payoffs $u$ and $w$ of two players the obtained game $(g;u,w)$, in normal form, has a Nash equilibrium (NE) in pure strategies. This result was extended in several directions; here we strengthen it further. We construct two special NE realized by a lexicographically safe (lexsafe) strategy of one player and a best response of the other. We obtain a polynomial algorithm computing these lexsafe NE. This is trivial when game form $g$ is given explicitly. Yet, in applications $g$ is frequently realized by an oracle $\cO$ such that size of $g$ is exponential in size $|\cO|$ of $\cO$. We assume that game form $g = g(\cO)$ generated by $\cO$ is tight and that an arbitrary {\em win-lose game} $(g;u,w)$ (in which payoffs $u$ and $w$ are zero-sum and take only values $\pm 1$) can be solved, in time polynomial in $|\cO|$. These assumptions allow us to construct an algorithm computing two (one for each player) lexsafe NE in time polynomial in $|\cO|$. We consider four types of oracles known in the literature and show that all four satisfy the above assumptions.

翻译：1975年，第一作者证明了每个有限紧致的双人博弈形式 $g$ 具有纳什可解性，即对于两位玩家的任意收益函数 $u$ 和 $w$，所得到的正则形式博弈 $(g;u,w)$ 存在纯策略纳什均衡（NE）。该结果已在多个方向上得到推广；本文进一步强化了它。我们构造了两个特殊的纳什均衡，分别由一个玩家的词典安全（lexsafe）策略和另一个玩家的最优反应实现。我们得到了一个计算这些词典安全纳什均衡的多项式算法。当博弈形式 $g$ 显式给出时，这一计算是平凡的。然而，在实际应用中，$g$ 常通过预言机 $\cO$ 实现，使得 $g$ 的规模在 $\cO$ 的规模 $|\cO|$ 上呈指数级。我们假设由 $\cO$ 生成的博弈形式 $g = g(\cO)$ 是紧致的，并且任意{胜负博弈} $(g;u,w)$（其中收益 $u$ 和 $w$ 为零和且仅取值为 $\pm 1$）可在 $|\cO|$ 的多项式时间内求解。这些假设使我们能够构造一个算法，在 $|\cO|$ 的多项式时间内计算出两个（每位玩家一个）词典安全纳什均衡。我们考虑了文献中已知的四类预言机，并证明它们均满足上述假设。