midterm-practice questions solutions (1).pdf

# 1nothing changes in the last stage of the game except

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1.Nothing changes in the last stage of the game except Alice’s utility becomes 600. 2. In the second stage of the game, Alice accepts an offer if and only if: 1 · (1 - y 2 ) · 1000 600 which implies y 2 2 5 . Hence, in any SPE, Bob offers y 2 = 0 . 4, Alice will accept and the payoffs are u Alice = 1 · (1 - 0 . 4) · 1000 = 600 and u Bob = 0 . 6 · 0 . 4 · 1000 = 240. 3. In the first stage, Bob will accept an offer if and only if: (1 - y 1 ) · 1000 240 which implies y 1 0 . 76. Hence, in any SPE, Alice offers y 1 = 0 . 76, Bob accepts and the payoffs are u Alice = 0 . 76 · 1000 = 760 and u Bob = 0 . 24 · 1000 = 240. As we see, Alice has a higher payoff which is very intuitive. Since the discount factor for Bob is fixed, being more patient creates an advantage for Alice to offer more in the first round and note that her payoff is larger if game reaches to the end stage where Bob rejects the final offer. Problem 4. Consider the two player perfect information game given in the tree (arrows point to the next move if the corresponding action is taken). a. Find a (pure strategy ) subgame perfect equilibrium. Solution. Player 1 will pick L and R respectively in the right subgames and player 2 will pick b since 1 > 0 and thus, if player 1 picks r, the payoffs are (2 , 1). Player 2 will pick R and she is indifferent between L and R respectively in the left subgames and player 1 will pick m over l since 0 > - 1 and thus, if player 1 picks m, the payoffs are (0 , 1). Hence, the subgame perfect equilibrium is the path r,b,R. b. Find a (pure-strategy) NE that is not subgame perfect or claim that such a NE does not exist. Solution. Note that we can present the game in 12 by 8 matrix since Player 1 has 3 · 2 · 2 = 12 and Player 2 has 2 3 = 8 pure strategies. The strategy Player 1 plays rLL and Player 2 plays LLa is a NE with the payoffs (1,0). Note that given that Player 1 selects r and plays L no matter what Player 2 plays, Player 2 has no incentive do deviate (her payoff is 0 in all possibilities). Similarly given that player 2 plays L if Player 1 plays l or m, player 1 has no incentive to deviate to R. And given that player 2 plays a when Player 1 plays r, Player 1 has no incentive to deviate. c. Suppose player 1 cannot predict what player 2 will do and just wants to guarantee himself the highest possible payoff (i.e. no matter what player 2 does, player 1 will get at least that payoff). How should player 1 play (using only pure-strategies) and how much can he guarantee?
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• Spring '14
• HauLLee
• Game Theory, Alice, pure-strategies of each players

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