DEFINITION of 'Backward Induction'

Backward induction, in game theory, is an iterative process of reasoning backward in time, from the end of a problem or situation, to solve finite extensive form and sequential games, and infer a sequence of optimal actions.

BREAKING DOWN '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. Effectively, one is determining the Naish equilibrium of each subgame of the original game.

However, the results inferred from backward induction often fail to predict actual human play. Experimental studies have shown that “rational” behavior (as predicted by game theory) is seldom exhibited in real life. Irrational players may actually end up obtaining higher payoffs than predicted by backward induction, as illustrated in the in the centipede game.

In the centipede 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 less 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.

Example of Backward Induction

As an example, assume Player A goes first and has to decide if he should “take” or “pass” the stash, which currently amounts to $2. If he takes, then A and B get $1 each, but if A passes, the decision to take or pass now has to be made by Player B. If B takes, she gets $3 (i.e., the previous stash of $2 + $1) and A gets $0. But if B passes, A 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 A and B 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 lowest possible claim ($1 in this case).

The Nash equilibrium of this game, where no player has an incentive to deviate from his 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 Backward Induction

Below is a simple sequential game between two players. The labels with Player 1 and Player 2 within them are the information sets for players one or two, respectively. 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 its decision after Player 1 (up or down).

Chart example of backward induction.

Figure 1

Backward induction, like all game theory, uses the assumptions of rationality and maximization, meaning that Player 2 will maximize his 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 bold the lines that maximize the player's payoff at the given information set.

Second chart example of backward induction.

Figure 2

After this reduction, Player 1 can maximize its 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.

Backward induction diagram.

Figure 3

For example, one could easily set up a game similar to the one above using companies as the players. This game could include 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 forecasting sales of this new product in different scenarios, we can set up a game to predict how events might unfold. Below is an example of how one might model such a game. (For related reading, see: Why Is Game Theory Useful in Business?)

Example game model of forecasting sales of a new product in different scenarios.

Figure 4

RELATED TERMS
  1. Gaming Industry ETF

    Gaming ETFs (exchange-traded funds) invest in companies that ...
  2. John Harsanyi

    John Harsanyi was an economist and co-winner of the 1994 Nobel ...
  3. Dollar Auction

    A dollar auction is a non-zero-sum sequential game where the ...
  4. Payoff Statement

    A payoff statement is a statement prepared by a lender providing ...
  5. Robert J. Aumann

    Robert J Aumann is a mathematician and economist famous for his ...
  6. Mechanism Design Theory

    Mechanism design theory is an economic theory that seeks to study ...
Related Articles
  1. Insights

    The Basics Of Game Theory

    Take an introductory look at game theory and the terms involved. Get familiar with backwards induction, a simple method for solving games.
  2. Tech

    You Love Video Games, But Do You Know How The Industry Works?

    Traditionally, the video game industry was limited to consoles, such as Microsoft’s (MSFT) Xbox and Sony’s (SNE) PlayStation, but it now includes PC games, mobile games and, in the near future, ...
  3. Investing

    Alibaba Plans to Enter China's $11B Games Market

    Alibaba is doing an about face and is entering the games market with a new business unit.
  4. Investing

    How The Cloud May Boost Video Game Stocks

    Leading video gaming stocks have had a spectacular 5-year run, but cloud computing may propel them to even greater heights.
  5. Trading

    The Nash Equilibrium

    Nash Equilibrium is a key concept of game theory, which helps explain how people and groups approach complex decisions. Named after renowned mathematician John Nash, the idea of Nash Equilibrium ...
  6. Insights

    The 6 Most Famous Failed Video Game Makers (KING, ZNGA)

    The video-game industry pulls in $100 billion in revenue annually. Failed companies offer a cautionary tale.
  7. Insights

    Investors May Love the Fast Action in Video Games

    Video game companies that were once whipsawed by the "ship it and forget it" game cycle, are now thriving as they sell their games digitally.
  8. Investing

    Why Video Game Stocks' Hot Streak May Get Hotter

    Surging in-game spending by hardcore players should propel video game stocks even higher.
  9. Investing

    Power Up Your Portfolio With Video Game Stocks

    Level up your winnings by investing in this fast-paced, highly skilled industry.
  10. Insights

    3 Ways Companies Profit From Virtual Goods

    A report released in late 2011 predicts that the market for virtual goods will reach $2.9 billion this year.
RELATED FAQS
  1. What is the difference between a dominant strategy solution and a Nash equilibrium ...

    Dive into game theory and the Nash equilibrium, and learn why the equilibrium assumptions about information are less important ... Read Answer >>
  2. What are the most famous instances of backward integration?

    Learn more about backward integration in the supply chain and see how two famous examples, Carnegie Steel and Apple, used ... Read Answer >>
  3. How can traders use contango to take advantage of the storage shortage for crude ...

    Learn how traders can profit in oil markets with a contango forward term structure by storing oil, and understand how other ... Read Answer >>
Trading Center