Subgame-perfect $$\epsilon $$ ε -equilibria in perfect information games with sigma-discrete discontinuities
研究了在完美信息博弈中，当收益函数仅在可数多个离散点集上不连续时，子博弈完美ε-均衡仍然存在，并给出了一个部分逆命题。
Multi-player perfect information games are known to admit a subgame-perfect $$\epsilon $$ -equilibrium, for every $$\epsilon >0$$ , under the condition that every player’s payoff function is bounded and continuous on the whole set of plays. In this paper, we address the question on which subsets of plays the condition of payoff continuity can be dropped without losing existence. Our main result is that if payoff continuity only fails on a sigma-discrete set (a countable union of discrete sets) of plays, then a subgame-perfect $$\epsilon $$ -equilibrium, for every $$\epsilon >0$$ , still exists. For a partial converse, given any subset of plays that is not sigma-discrete, we construct a game in which the payoff functions are continuous outside this set but the game admits no subgame-perfect $$\epsilon $$ -equilibrium for small $$\epsilon >0$$ .