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-perfect 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 justifed 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.
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Paper provided by Stanford University, Graduate School of Business in its series Research Papers with number
1955.
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Kyle Bagwell & Garey Ramey, 1987.
"Advertising and Limit Pricing,"
Discussion Papers
729, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
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Lawrence M. Ausubel & Peter Cramton & Raymond J. Deneckere, 2002.
"Bargaining with Incomplete Information,"
Papers of Peter Cramton
02barg, University of Maryland, Department of Economics - Peter Cramton, revised 12 Mar 2001.
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Other versions:
Wilson, Robert B. & Govindan, Srihari, 2007.
"On Forward Induction,"
Research Papers
1955, Stanford University, Graduate School of Business.
[Downloadable!]
Other versions:
Cited by: (explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)
Wilson, Robert B. & Govindan, Srihari, 2007.
"On Forward Induction,"
Research Papers
1955, Stanford University, Graduate School of Business.
[Downloadable!]