On Forward Induction

We examine Hillas and Kohlberg’s conjecture that invariance to the addition of payoff-redundant strategies implies that a backward induction outcome survives deletion of strategies that are inferior replies to all equilibria with the same outcome. That is, invariance and backward induction imply forward induction. Although it suffices in simple games to interpret backward induction as a subgame-prefect or sequential equilibrium, to obtain general theorems we use a quasi-perfect equilibrium, viz. a sequential equilibrium in strategies that are admissible continuations from each information set. Using this version of backward induction, we prove the Hillas-Kohlberg conjecture for two-player extensive-form games with perfect recall. We also prove an analogous theorem for general games by interpreting backward induction as a proper equilibrium, since a proper equilibrium is equivalent to a quasi-perfect equilibrium of each extensive form with the same normal form, provided beliefs are justified by perturbations invariant to inessential transformations of the extensive form. For a two-player game we prove that if a set of equilibria includes a proper equilibrium of every game with the same reduced normal form then it satisfies forward induction, i.e. it includes a proper equilibrium of the game after deleting strategies that are inferior replies to all equilibria in the set. We invoke slightly stronger versions of invariance and properness to handle nonlinearities in an N-player game.