Backward Induction

For game trees there is another, often used, solution method. The main idea is to start at the end of the tree, and then solve it backwards.

The last thing that happens in the game tree is that J chooses “price war” or “accept.” If she chooses the first option, she gets 50 and if she chooses the second, she gets 75. Obviously, it cannot be optimal to choose the first. Consequently, given that E has chosen “enter,” J will choose “accept.” We can then reduce the game tree by omitting the alternative that J will not choose, and just keep the payoffs of the alternative that she does choose. The game tree will then look like in Figure 13.4.

Given that J will chose “accept,” the choice for E is simpler. If she chooses “not enter” she will get 50 and if she chooses “enter” she will get 75. Consequently, she chooses the latter. The solution, using backward induction, is then

- E: "enter"; J: "accept.”

Comparing this solution to the one in Section 13.4.1 (that had two different solutions, one of which is the same as the one here) this one seems more reasonable. Earlier, we found a Nash equilibrium in which E chose “not enter” and that included a (never realized) threat from J to start a price war. Using backward induction, the threat reveals itself as being empty, and the only solution is that E establishes herself and J chooses to accept. The solution one obtains by using backward induction is called subgame perfect equilibrium.

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