Backward Induction - Explained
What is Backward Induction?
Written by Jason Gordon
Updated at June 27th, 2021
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Table of Contents
What is Backward Induction?How does Backward Induction work?Example of Backward InductionAcademic Research on Backward InductionWhat is Backward Induction?
Backward induction is a reasoning process that is rooted in game theory. It is a repetitive reasoning process that involves reasoning backward in time. An individual or a player reasons from the end of a problem to determine sequential optimal actions in games. This process of reasoning allows a player to think of the end of a problem and then apply the sequence of happenings to decide what to do at a particular stage. Backward induction has been used as far back as 1944 when John von Neumann and Oskar Morgenstern published a book Theory of Games and Economic Behavior. This process of reasoning was also used by Selten in 1965 in his versions of game.
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How does Backward Induction work?
Backward induction is used by a player to decide on what move to make at a particular stage by a process of reasoning backward in time. The player considers the end of a problem and makes current decisions based on this. According to rational behavior highlighted in this theory sometimes reflect in real life but the theory does not precisely predict humans. In this theory, a player that makes the last move in a game uses an optimal strategy determined by this theory. Thereafter, the action of the player before the last one is determined by the last players action. The gaming process moves backward until the best optimal action for each subgame is determined.
Example of Backward Induction
This illustration is important for a better understanding of how a backward induction works. Player X plays first in the game and he needs to decide whether to take the stash worth $4 or pass it. If Player X takes it, the $4 will be shared between him and Player Y ($2 each), if he passes it, Player X will also need to decide whether to take or pass the stash. If Player Y passes, he gets an extra amount added to the existing $2 while Player X gets no amount. However, if both players cooperate and keeps passing, they receive an equal payoff at the end of the game, but the reverse is the case if they do not cooperate.
Academic Research on Backward Induction
- Backward inductionand common knowledge of rationality, Aumann, R. J. (1995). Backward induction and common knowledge of rationality.Games and Economic Behavior,8(1), 6-19.
- Thebackward inductionparadox, Pettit, P., & Sugden, R. (1989). The backward induction paradox.The Journal of Philosophy,86(4), 169-182.
- Detecting failures ofbackward induction: Monitoring information search in sequential bargaining, Johnson, E. J., Camerer, C., Sen, S., & Rymon, T. (2002). Detecting failures of backward induction: Monitoring information search in sequential bargaining.Journal of Economic Theory,104(1), 16-47. We did experiments in a three-round bargaining game where the (perfect) equilibrium offer was $1.25 and an equal split was $2.50. The average offer was $2.11. Patterns of information search (measured with a computerized information display) show limited lookahead rather than backward induction. Equilibrium theories which adjust for social utilities (reflecting inequality-aversion or reciprocity) cannot explain the results because they predict subjects will make equilibrium offers to robot players, but offers to robots are only a little lower. When trained subjects (who quickly learned to do backward induction) bargained with untrained subjects, offers ended up halfway between equilibrium and $2.11.Journal of Economic LiteratureClassification Numbers: C7, C9.
- Abackward inductionexperiment, Binmore, K., McCarthy, J., Ponti, G., Samuelson, L., & Shaked, A. (2002). A backward induction experiment.Journal of Economic theory,104(1), 48-88. This paper reports experiments with one-stage and two-stage alternating-offers bargaining games. Payoff-interdependent preferences have been suggested as an explanation for experimental results that are commonly inconsistent with players' maximizing their monetary payoffs and performing backward induction calculations. We examine whether, given payoff-interdependent preferences, players respect backward induction. To do this, we break backward induction into its components, subgame consistency and truncation consistency. We examine each by comparing the outcomes of two-stage bargaining games with one-stage games with varying rejection payoffs. We find and characterize systematic violations of both subgame and truncation consistency.Journal of Economic LiteratureClassification Numbers: C70, C78.
- Belief revision in games: forward andbackward induction, Stalnaker, R. (1998). Belief revision in games: forward and backward induction.Mathematical Social Sciences,36(1), 31-56. The rationality of choices in a game depend not only on what players believe, but also on their policies for revising their beliefs in response to surprising information. A general descriptive framework for representing belief revision policies in game situations is sketched, and the consequences of some assumptions about such policies are explored. Assumptions about epistemic independence and a rationalization principle are considered. It is argued that while such assumptions may be appropriate in some contexts, no substantive constraints on belief revision policies can be justified on the basis of the assumption of common knowledge of rationality.