0Relationship between backward induction and other solutions
If you have gone through the exercises for the previous three sections, you will have seen that the backwards-induction solutions are also Nash equilibria in all those games. It is always true, as stated in the following theorem.
Theorem 3.4.1 Every backwards-induction solution of a perfect-information game is a Nash equilibrium of the associated strategic form. In some games, the set of backwards-induction solutions coincides with the set of Nash equilibria (see, for example, Exercise 3.9). Still, the set of Nash equilibria is typically larger than (is a proper superset of) the set of backwards-induction solutions. Nash equilibria that are not backwards-induction solutions often involve incredible threats.
To see this, consider the following game. The industry is currently a monopoly, and the incumbent monopolist is making a profit of $5 million. A potential entrant is considering whether or not to enter this industry. - If she does not enter, she will make $1 million in an alternative investment. If she does enter, then the incumbent can either fight entry with a price war whose outcome is that both firms make zero profits or accommodate entry by sharing the market with the entrant, in which case both firms make a profit of $2 million. This situation is illustrated in Figure 3.10 with the associated strategic form. We assume that each player is selfish and greedy; that is, it cares only about its profit and prefers more money to less. The backwards-induction solution is (in, accommodate), and it is also a Nash equilibrium. However, there is another Nash equilibrium, namely (out, fight). The latter should be discarded as a “rational solution” because it involves an incredible threat on the incumbent part, namely that it will fight entry if the potential entrant enters. - It is true that if the potential entrant believes the incumbent’s threat, then she is better off staying out. However, she should ignore the incumbent’s threat because she should realize that – when faced with the fait accompli of entry – the incumbent would not want to carry out the threat.
