### Abstract

We prove the existence of a subgame-perfect e-equilibrium, for every e > 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 ah1${a}_{h}^{1}$ or in some cases two pure actions ah2${a}_{h}^{2}$ and bh2${b}_{h}^{2}$ for the active player at h. The subgame-perfect e-equilibrium then prescribes for every history h that the active player plays ah1${a}_{h}^{1}$ with probability 1 or respectively plays ah2${a}_{h}^{2}$ with probability 1 - d(e) and bh2${b}_{h}^{2}$ with probability d(e). Here, d(e) is arbitrary as long as it is positive and small compared to e, so the strategies can be made “almost” pure.

Original language | English |
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Pages (from-to) | 201-207 |

Journal | European Journal of Operational Research |

Volume | 228 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2013 |

## Cite this

Flesch, J., Kuipers, J., Schoenmakers, G. M., & Vrieze, K. (2013). Subgame-perfection in free transition games.

*European Journal of Operational Research*,*228*(1), 201-207. https://doi.org/10.1016/j.ejor.2013.01.034