An Efficient Optimal-Equilibrium Algorithm for Two-player Game Trees
  Two-player complete-information game trees are perhaps the simplest possible setting for studying general-sum games and the computational problem of finding equilibria. These games admit a simple bottom-up algorithm for finding subgame perfect Nash equilibria efficiently. However, such an algorithm can fail to identify optimal equilibria, such as those that maximize social welfare. The reason is that, counterintuitively, probabilistic action choices are sometimes needed to achieve maximum payoffs. We provide a novel polynomial-time algorithm for this problem that explicitly reasons about stochastic decisions and demonstrate its use in an example card game. 
Michael L. Littman, Nishkam Ravi, Arjun Talwar, Martin Zinkevich
cs.GT