Axiomatic Equilibrium Selection for Generic Two-Player Games

We impose three conditions on refinements of the Nash equilibria of finite games with perfect recall that select closed connected subsets, called solutions.
A. Each equilibrium in a solution uses undominated strategies;
B. Each solution contains a quasi-perfect equilibrium;
C. The solutions of a game map to the solutions of an embedded game, where a game is embedded if each players feasible strategies and payoffs are preserved by a multilinear map. We prove for games with two players and generic payoffs that these conditions characterize each solution as an essential component of equilibria in undominated strategies, and thus a stable set as defined by Mertens (1989).