Title: Subgame-Perfect Epsilon-Equilibrium in Perfect Information Games
We consider multi-player perfect information games. These games are played on a tree of infinite depth, where each node is controlled by one of the players. Play of the game starts at the root. At every node that play visits, the player who controls this node has to choose one of the outgoing arcs. This induces an infinite sequence of nodes in the tree, and depending on this sequence, each player receives a payoff.
A strategy profile is called a Nash epsilon-equilibrium if no player can gain more than epsilon by a unilateral deviation. Such games are known to admit a Nash epsilon-equilibrium (Mertens and Neyman 1986) for all positive epsilon, provided that the payoff functions of the players are bounded and Borel measurable.
In this talk, we examine the existence of a subgame-perfect epsilon-equilibrium in these games. A subgame-perfect epsilon-equilibrium is a strategy profile that induces a Nash epsilon-equilibrum in every subgame (that is, not only from the root, but also from every other node in the tree). We provide an overview of recent existence and non-existence results. Particular attention is paid to the result (Flesch and Predtetchinski 2015) that a subgame-perfect epsilon-equilibrium exists, for every epsilon>0, if the payoff functions are bounded and exhibit common preferences at the limit. Moreover, if the payoff functions also have finite range, then there exists a pure subgame-perfect 0-equilibrium. These results extend and unify the existence theorems for bounded and semicontinuous payoffs in Flesch et al  and Purves and Sudderth .