Generalizing Gale's theorem on backward induction and domination of strategies

In 1953 David Gale noticed that for every n-person game in extensive form with perfect information modeled by an arborescence (a rooted tree) some special Nash equilibrium in pure strategies can be found by an algorithm of successive elimination of leaves, which is now called the backward induction. (The result can be easily extended from the trees to the acyclic directed graphs.) He also noticed the same procedure, performed for the normal form of this game, turns into successive elimination of dominated strategies of the players that results in a single strategy profile (x1, . . . , xn), which is called a domination equilibrium (DE) and appears to be a Nash-equilibrium (NE) too. In other words, the game in normal form obtained from a positional game with perfect information is dominance-solvable (DS) and also Nash-solvable (NS). Yet, an arbitrary game in normal form may be not DS. We strengthen Gale’s results as follows. Consider several successive eliminations of dominated strategies that begins with X = X1× . . .×Xn and ends in X ′ = X ′ 1× . . .×X ′ n. We will call X ′ a D-box of X. Our main (although obvious) lemma claims that for any i ∈ I = {1, . . . , n} and for any strategy xi ∈ Xi its “projection” to a D-box X ′ of X is dominated by a strategy xi ∈ X ′ i. It immediately follows that any DE is an NE and, hence, DS implies NS. It is enough to apply the lemma in the case when X ′ consists of a single strategy profile. Also this lemma implies that, in general, the domination procedure is well-defined in the following sense. A D-box X ′ of X is called terminal if it is domination-free, that is, it contains no pair of strategies such that one of them is dominated by the other. Any two terminal D-boxes X ′ and X ′′ of X are equal. More precisely, there exist n permutations π = (π1, . . . , πn), with πi : Xi → Xi for i ∈ I, that transform X ′ into X ′′, that is, π(X ′) = X ′′ and the payoffs are respected. We also recall some published results on dominance-solvable game forms.