Refinements of subgame-perfect ε-equilibrium in games with perfect information

We consider games with perfect information and deterministic transitions. A common solution concept is the concept of subgame-perfect ε-equilibrium, where ε ≥ 0, which is a strategy profile such that no player can improve his payoff in any subgame by more than ε. We propose and examine a number of refinements of this concept. A major emphasis lies on existence results. Roughly speaking, the most important refinements require the following respective properties: (1) no player makes a big mistake with positive probability, (2) the mistakes vanish as the horizon approaches infinity, i.e. ε depends on the subgame and goes to 0 as play proceeds, and (3) for pure strategy profiles, the induced play paths are continuity points of the payoff functions. The game: We consider games with perfect information and deterministic transitions, i.e. without chance moves. Such a game can be given by a triple G = (N, T, u = (ui)i∈N), where • N is a finite and nonempty set of players. • T is a directed tree with a root, in which each node is associated with a player, who controls this node. We assume that, at each node in the tree, the number of outgoing arcs is finite and is at least one. This implies that there are no terminal nodes and the tree has an infinite depth.1 Let P denote the set of all infinite paths (often called plays) starting at the root. We endow P with the topology induced by the cylinders.2 • ui : P→ R is a payoff function for player i. We assume that ui is bounded and Borel measurable.3 Play of the game starts at the root, and at any node z that play visits, the player who controls z has to choose one of the outgoing arcs at z, which brings play to a next node. This induces an infinite path p in the tree, and each player i ∈ N receives the corresponding payoff ui(p). 1This way we allow for fairly general games, since every tree with finite depth can be easily transformed into a strategically equivalent tree with infinite depth. Indeed, we can extend a finite tree by simply adding one infinite sequence of arcs to every terminal node. So, instead of termination, play will continue along a unique path in which the players have no further strategic choices. 2If z is a node in the tree, then the cylinder set corresponding to z is the set of all infinite paths from the root that go through z. 3Note that these payoffs are very general. Indeed, only a very special case would be a common situation when these payoffs arise by aggregating certain daily payoffs in the game, possibly by taking the total discounted sum.