Solution Concept - Backward Induction

Backward Induction

There are games that have multiple Nash equilibria, some of which are unrealistic. In the case of dynamic games, unrealistic Nash equilibria might be eliminated by applying backward induction, which assumes that future play will be rational. It therefore eliminates noncredible (or incredible) threats because such threats would be irrational to carry out if a player was ever called upon to do so.

For example, consider a dynamic game in which the players are an incumbent firm in an industry and a potential entrant to that industry. As it stands, the incumbent has a monopoly over the industry and does not want to lose some of its market share to the entrant. If the entrant chooses not to enter, the payoff to the incumbent is high (it maintains its monopoly) and the entrant neither loses nor gains (its payoff is zero). If the entrant enters, the incumbent can fight or accommodate the entrant. It will fight by lowering its price, running the entrant out of business (and incurring exit costs – a negative payoff) and damaging its own profits. If it accommodates the entrant it will lose some of its sales, but a high price will be maintained and it will receive greater profits than by lowering its price (but lower than monopoly profits).

If the entrant enters, the best response of the incumbent is to accommodate. If the incumbent accommodates, the best response of the entrant is to enter (and gain profit). Hence the strategy profile in which the incumbent accommodates if the entrant enters and the entrant enters if the incumbent accommodates is a Nash equilibrium. However, if the incumbent is going to play fight, the best response of the entrant is to not enter. If the entrant does not enter, it does not matter what the incumbent chooses to do (since there is no other firm to do it to - note that if the entrant does not enter, fight and accommodate yield the same payoffs to both players; the incumbent will not lower its prices if the entrant does not enter). Hence fight can be considered as a best response of the incumbent if the entrant does not enter. Hence the strategy profile in which the incumbent fights if the entrant does not enter and the entrant does not enter if the incumbent fights is a Nash equilibrium. Since the game is dynamic, any claim by the incumbent that it will fight is an incredible threat because by the time the decision node is reached where it can decide to fight (i.e. the entrant has entered), it would be irrational to do so. Therefore this Nash equilibrium can be eliminated by backward induction.

See also:

• Monetary policy theory
• Stackelberg competition