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In this game, how many information sets are there for Cheezy?

ECON 402. Fall 2015 HOMEWORK 1 Due in class on Thursday, September 10th 1. Suppose a manager and a worker interact as follows. The manager decides whether to hire or not hire the worker. If the manager does not hire the worker, then the game ends. When hired, the worker chooses to exert either high effort or low effort. On observing the worker’s effort, the manager chooses to retain or fire the worker. In this game, does “not hire” describe a strategy for the manager? Explain. 2. Consider the following strategic setting. There are three players, numbered 1, 2, and 3. Player 1 has two cards, labeled King and Ace. At the beginning of the game, player 1 deals one of the cards to player 2 and the other card to player 3; that is, player 1 either gives the Ace to player 3 and the King to player 2 (call this the action A) or the King to player 3 and the Ace to player 2 (action K). Player 2 observes the card dealt to him; player 3 does not get to see the card dealt to her. Player 2 then must decide between switching cards with player 3 (S) or not (N). Player 3 observes whether player 2 made the switch, but does not see her card. Finally, player 3 responds to the question “Is your card the Ace?” by saying either “yes” (Y) or “no” (N). If player 3 correctly states whether her card is the Ace, then she obtains a payoff of 1 and the other players get 0; otherwise, players 1 and 2 both get a payoff of 1 and player 3 obtains 0. Represent this game in the extensive form. 3. Draw the extensive-form diagram for the following strategic setting. There are three people: Amy, Bart, and Chris. Amy and Bart each have two cards, one of which has “K” (for King) written on it and the other has “Q” (for Queen) written on it; that is, Amy and Bart both have a King and a Queen. At the beginning of the game, Amy must place one of her cards (either K or Q) into an envelope and then give the envelope to Bart. Bart sees the card that Amy placed into the envelope, and then he places one of his cards (either K or Q) into the envelope as well. The envelope is then given to Chris, who has not observed the moves of Amy and Bart. Chris opens the envelope. 1 Chris sees the two cards inside, but she does not know which card was placed there by Amy and which card was deposited by Bart. After observing the contents of the envelope, Chris selects “yes” (Y) or “no” (N). If Chris selects Y and Amy had put a King in the envelope, then Amy and Bart each get a payoff of 0 and Chris gets 1. If Chris selects N and Amy had put a Queen in the envelope, then, again, Amy and Bart each get a payoff of 0 and Chris gets 1. In all other outcomes, Amy and Bart each get a payoff of 1 and Chris gets 0. 4. Consider the following strategic setting involving a cat named Baker, a mouse named Cheezy, and a dog named Spike. Baker’s objective is to catch Cheezy while avoiding Spike; Cheezy wants to tease Baker but avoid get- ting caught; Spike wants to rest and is unhappy when he is disturbed. In the morning, Baker and Cheezy simultaneously decide what activity to engage in. Baker can either nap (N) or hunt (H), where hunting involves moving Spike’s bone. Cheezy can either hide (h) or play (p). If nap and hide are chosen, then the game ends. The game also will end immediately if hunt and play are chosen, in which case Baker captures Cheezy. On the other hand, if nap and play are chosen, then Cheezy observes that Baker is napping and must decide whether to move Spike’s bone (m) or not (n). If he chooses to not move the bone, then the game ends. Finally, in the event that Spike’s bone was moved (either by Baker choosing to hunt or by Cheezy moving it later), then Spike learns that his bone was moved but does not observe who moved it; in this contingency, Spike must choose whether to punish Baker (B) or punish Cheezy (J). After Spike moves, the game ends. In this game, how many information sets are there for Cheezy? How many strategy profiles are there in this game?

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