Subgame-perfection in free transition games

We prove the existence of a subgame-perfect ε-equilibrium, for every ε > 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 subgameperfection, with multiple non-terminating actions, in which the payoff structure is generally not 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 ε and terminates in polynomial time with the following output: for all possible histories h, a pure action ah or in some cases two pure actions a 2 h and b 2 h for the active player at h. The subgame-perfect ε-equilibrium then prescribes for every history h that the active player plays ah with probability 1 or respectively plays a 2 h with probability 1 − δ(ε) and bh with probability δ(ε). Here, δ(ε) is arbitrary as long as it is positive and small compared to ε, so the strategies can be made “almost” pure.