The Chainstore game A national chain of electronics stores has franchises in shopping centers in ten different cities. In each shopping center, the chainstore's franchise is the only electronics store. Ten local competitors, one in each city, are each contemplating opening a rival electronics store in the local shopping center, in the following sequence.
The first competitor decides whether or not to open a rival electronics store in his city.
The second competitor checks whether or not the first competitor has opened an electronics store, and takes into account the national chainstore's response to this development, before deciding whether or not he will open a rival electronics store in his city.
The third competitor checks whether or not the first and second competitors have opened electronics stores, and takes into account the national chainstore's response to these developments, before deciding whether or not he will open a rival electronics store in his city, and so on.
If a competitor decides not to open a rival electronics store, the competitor's payoff is 0, and the national chain store's payoff is 5. If a competitor does decide to open a rival electronics store, his payoff depends on the response of the national chainstore. If the national chainstore responds by undercutting prices in that city, the competitor and the chainstore lose 1 each. If the national chainstore does not respond by undercutting prices in that city, the competitor and the national chainstore each receive a payoff of 3.
(a) Describe this situation as an extensive-form game.
(b) Find all the subgame perfect equilibria.
(c) Find a Nash equilibrium that is not a subgame perfect equilibrium, and explain why it fails to be a subgame perfect equilibrium.