Polynomial algorithms computing two lexicographically safe Nash equilibria in finite two-person games with tight game forms given by oracles
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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 such that size of g is exponential in size || of . We assume that game form g = g() generated by is tight and that an arbitrary win-lose game (g;u,w) (in which payoffs u and w are zero-sum and take only values ± 1) can be solved, in time polynomial in ||. These assumptions allow us to construct an algorithm computing two (one for each player) lexsafe NE in time polynomial in ||. We consider four types of oracles known in the literature and show that all four satisfy the above assumptions.
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