To win the game, you must first imagine the end
In short: How do you make the perfect choice today when the consequences won't be felt for years? Backward induction suggests the answer lies in reversing time. By identifying the final optimal move and working toward the present, you can map out a sequence of perfect decisions for every step along the way.
Backward induction is a mathematical process used to determine a sequence of optimal choices by reasoning from the final endpoint of a problem back to its beginning. It involves identifying the best action at the very last point of a series of decisions and then moving one step backward to find the preceding optimal action. This technique is also known as backward search or backward chaining in the realms of automated theorem proving and scheduling. In the specific context of chess, the process is referred to as retrograde analysis.
This method was first utilized in 1875 by Arthur Cayley while he was attempting to solve the secretary problem. Sequential games, featuring decisions made by various players, employ a version of this technique to calculate subgame perfect equilibria within game theory. The method functions by identifying the highest payoff vector for each subgame. As the process moves toward the start, the extensive form of the game becomes progressively shorter.
While standard optimization involves one decision-maker, game theory assumes players will act rationally based on what they predict others will do at the final stage. This methodology was notably suggested by John von Neumann and Oskar Morgenstern in their 1944 book, Theory of Games and Economic Behaviour, which established game theory as a formal field of study. This approach follows the principle of sequential rationality. It requires dividing a game into distinct subgames and weighing expected payoffs at each information set.
In practical terms, this is often applied to 'optimal stopping problems,' such as job searches or shopping, where one must decide when to stop waiting for a better offer. In these scenarios, the logic dictates that if you expect to be committed to a choice for a long duration, it is mathematically worth being more selective in the early stages. A specific example involves a ten-year job search with a zero interest rate. In this model, a person might reject a bad offer at year nine to wait for a better one, but would accept any offer by year ten.
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- Summary by Intellect · source: Backward induction
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