Subgame-perfection in free transition games.

We prove the existence of a subgame-perfect epsilon-equilibrium, for every epsilon > 0, in a class of multi-player games with perfect information, which we call free transition games. The novelty is that a non-trivial class of perfect information games is solved for subgame-perfection, with multiple non-terminating actions, in which the payoff structure is generally not (upper or lower) semi-continuous. Due to the lack of semi-continuity, there is no general rule of comparison between the payoffs that a player can obtain by deviating a large but finite number of times or, respectively, infinitely many times. We introduce new techniques to overcome this difficulty.Our construction relies on an iterative scheme which is independent of E and terminates in polynomial time with the following output: for all possible histories h, a pure action a(1)(h) or in some cases two pure actions a(h)(2) and b(h)(2) for the active player at h. The subgame-perfect epsilon-equilibrium then prescribes for every history h that the active player plays a(h)(1) with probability 1 or respectively plays a(h)(2) with probability 1 - delta(epsilon) and b(h)(2) with probability delta(epsilon) Here, delta(epsilon) is arbitrary as long as it is positive and small compared to epsilon, so the strategies can be made "almost" pure. (C) 2013 Elsevier B.V. All rights reserved.