Backward induction

Backward induction is the process of determining a sequence of optimal choices by employing reasoning backward from the end of a problem or situation to its beginning, choice by choice. It involves examining the last point at which a decision is to be made and identifying the most optimal choice of action at that point. Using this information, one can then determine what to do at the second-to-last point of decision. This process continues backward until the best action for every possible point along the sequence (i.e. for every possible information set) is determined. Backward induction was first utilized in 1875 by Arthur Cayley, who discovered the method while attempting to solve the Secretary problem.

In the mathematical optimization method of dynamic programming, backward induction is a method for solving the Bellman equation. In the related fields of automated planning and scheduling and automated theorem proving, the method is called backward search or backward chaining. In chess, it is called retrograde analysis.

In game theory, a variant of backward induction is a method used to compute subgame perfect equilibria in sequential games. The difference is that optimization problems involve just one decision maker who chooses what to do at each point of time, whereas game theory problems involve the decisions of several players interacting. In this situation it may still be possible to apply a generalization of backward induction, since by anticipating what the last player will do in each situation, it may be possible to determine what the second-to-last player will do, and so on. This variant of backward induction has been used to solve formal games from the very beginning of game theory. John von Neumann and Oskar Morgenstern suggested solving zero-sum, two-person formal games by this method in their Theory of Games and Economic Behavior (1944), the book which established game theory as a field of study.