Subgame Perfect Equilibria in Infinite Stage Games: an alternative existence proof

Any stage-game with infinite choice sets can be approximated by finite games obtained as increasingly finer discretizations of the infinite game. The subgame perfect equilibrium outcomes of the finite games converge to a limit distribution. We prove that (i) if the limit distribution is feasible in the limit game, then it is also a subgame perfect equilibrium outcome of the limit game; and (ii) if the limit distribution prescribes sufficiently diffused behavior for first-stage players, then it is a subgame perfect equilibrium outcome of the limit game. These results are potentially useful in determining the existence of subgame perfect equilibria in applications. ∗I am grateful to Kim Border for helpful discussions and for bringing to my attention the Arsenin-Kunugui Selection Theorem. I also benefited from conversations with Andreas Blume, Hector Chade, Arthur Robson, Edward Schlee, and Jeroen Swinkels. Financial support from the National Science Foundation under Grant SBR-9810840 is gratefully acknowledged.