What Is Backward Induction?

Backward induction is an iterative process of reasoning backward from the end of a problem or situation to solve finite extensive-form and sequential games and infer a sequence of optimal actions. It is used in game theory to determine the most optimal sequence of actions based on rational behavior༺.

Understanding Backward Induction

Backward induction has been used to solve games since John von Neumann and Oskar Morgenstern established game theory as an academic subject when they published their book, Theory of Games and Economic Behavior in 1944.

At each stage of the game, backward induction determines the optimal strategy of the player who makes the last move in the game. Then, the optimal action of the next-to-last moving player is determined, taking the last player's action as given. This process continues backward until the best action for every point in time has been determined.

However, the results inferred from backward induction often fail to predict actual human play. Studies have attempted to explain why “rational” behavior (as predicted by game theory) is seldom exhibited in real life; psychologists Daniel Kahneman and Amos Tversky, for example, have demonstrated that people's biases, faulty reasoning, and aversion to loss influence their behavior, leading to irrational decision-making.

Example of Backward Induction

Interestingly, irrational players may actually end up obtaining higher payoffs than predicted by backward induction, as illustrated in the centipede game. In this game, two players alternately get a chance to take a larger share of an increasing pot of money, or to pass the pot to the other player. The payoffs are arranged so that if the pot is passed to one's opponent and the opponent takes the pot on the next round, one receives slightly les🐽s than if one had taken the pot on this round. The game concludes as soon as a player takes the stash, with that player getting the larger portion and the other player getting the smaller portion.

As an example of both backward induction and a higher payoff for irrational decision-making, assume Izaz and Jian are playing the centipede game. Izaz goes first and has to decide if they should “take” or “pass” the stash, which currently amounts to $2. If they take, then Izaz and Jian get $1 each, but if Izaz passes, the decision to take or pass now has to be made by Jian. If Jian takes, they get $3 (i.e., the previous stash of $2 + $1) and Izaz gets $0. But if Jian passes, Izaz now gets to decide whether to take or pass, and so on. If both players always choose to pass, they each receive a payoff of $100 at the end of the game.

The point of the game is if Izaz and Jian both cooperate and continue to pass until the end of the game, they get the maximum payout of $100 each. But if they distrust the other player and expect them to “take” at the first opportunity, Nash equilibrium predicts the players will take the lowes🅠t possible claim ($1 in this case).

The Nash equilibrium of this game, where no player has the incentive to deviate from their chosen strategy after considering an opponent's choice, suggests the first player would take the pot on the very first round of the game. However, in reality, relatively few players do so. As a result, they get a higher payoff than the payoff predicted by the equilibria analysis.

Solving Sequential Games Using Back𒀰war𝔍d Induction

Below is a simple se⭕quential game between two players. The labels for Player 1 and Player 2 show the information sets for each. The numbers in the parentheses at the bottom of the tree are the payoffs at each respective point. The game is also sequential, so Player 1 makes the first decision (left or right) and Player 2 makes their decision after Player 1 (up or down).

Backward induction, like all game theory, uses the assumptions of rationality and maximization, meaning that Player 2 will maximize their payoff in any given situation. At either information set we have two choices, four in all. By eliminating the choices that Player 2 will not choose, we can narrow down our tree. In this way, we will mark the lines in blue that maximize the player's payoff at the given information set.

After this reduction, Player 1 can maximize their payoffs now that Player 2's choices are made known. The result is an equilibrium found by backward induction of Player 1 choosing "right" and Player 2 choosing "up." Below is the solution to the game with the equilibrium path bolded.

For example, one could easily set up a game similar to the one above using companies as the players. This game could include 澳洲幸运5开奖号码历史查询:product release scenarios. If Company 1 wanted to release a product, what might Company 2 do in response? Will Company 2 release a similar competing product? By 澳洲幸运5开奖号码历史查询:forecasting sales of this new product in diff🍰erent scenarios, we can set up a game to predict how events mig♛ht unfold. Below is an example of how one might model such a game.

The Bottom Line

For sequential games, backward induction provides a way to reason out the optimal actions throughout a game by beginning with the end. Although it can be useful for certain types of games, it becomes less reliable as the game increases in length and complexity; it also assumes perfect rationality, when real-life games don't always play out that way.