Question

. Agent A and B would like to go on a date. They have two options: a quick dinner at BurgerKing or a romantic dinner at a Restaurant. Agent A first chooses where to go, and knowing where A went, Agent B also decide where to go. Agent A prefers BurgerKing, and Agent B prefers Restaurant. A player gets 3 out his/her preferred date, 1 out of his/her unpreferred date, and 0 if they end up at different places. All these are common knowledge.

(a) Find a subgame-perfect Nash equilibrium. Find also a non-subgame-perfect Nash equilibrium with a different outcome.

(b) Modify the game a little bit: Agent B does not automatically know where Agent A went, but she can learn without any cost. (That is, now, without knowing where Agent A went, Agent B chooses between Learn and Not-Learn; if she chooses Learn, then she knows where Agent A went and then decides where to go; otherwise she chooses where to go without learning where Agent A went.) Find a subgame-perfect equilibrium of this new game.

Answer 1

(a) SPE: Beatrice goes wherever A goes, and A goes to Burgerking . The outcome is both go to Burgerking . Non-subgame-perfect Nash Equilibrium: B goes to Restaurant at any history, so A goes to Restaurant . The outcome is each goes to Rastaurant . This is not subgame-perfect because it is not a Nash equilibrium in the subgame after A goes to Burgerking.

(b)  A plays Restaurant , and B plays Don’t and goes to Restaurant ; if B played Learn, then it would have played Burgerking if A played Burgerking and Restaurant if A played Restaturant. As in the non-subgame-perfect equilibrium, they both go to Restaurant at the end. This is a subgame-perfect equilibrium in the new game however. The only proper subgames are the two decision nodes where B moves after learning where A went, and B plays best response at these nodes, yielding a Nash equilibrium in these little subgames. As in the original game, the strategy profile is a Nash equilibrium of the whole game. Therefore, it is a subgame- perfect Nash equilibrium.