Backward Induction

In some games, there are multiple Nash equilibria, but not all of them are realistic. In dynamic games, backward induction can be used to eliminate unrealistic Nash equilibria. Backward induction assumes that players are rational and will make the best decisions based on their future expectations. This eliminates noncredible threats, which are threats that a player would not carry out if they were ever called upon to do so.

For example, consider a dynamic game with an incumbent firm and a potential entrant to the industry. The incumbent has a monopoly and wants to maintain its market share. If the entrant enters, the incumbent can either fight or accommodate the entrant. If the incumbent accommodates, the entrant will enter and gain profit. If the incumbent fights, it will lower its prices, run the entrant out of business (incurring exit costs), and damage its own profits.

The best response for the incumbent if the entrant enters is to accommodate, and the best response for the entrant if the incumbent accommodates is to enter. This results in a Nash equilibrium. However, if the incumbent chooses to fight, the best response for the entrant is to not enter. If the entrant does not enter, it does not matter what the incumbent chooses to do. Hence, fight can be considered a best response for the incumbent if the entrant does not enter, resulting in another Nash equilibrium.

However, this second Nash equilibrium can be eliminated by backward induction because it relies on a noncredible threat from the incumbent. By the time the incumbent reaches the decision node where it can choose to fight, it would be irrational to do so because the entrant has already entered. Therefore, backward induction eliminates this unrealistic Nash equilibrium.