Finitely Repeated Games (**2**) Theorem **2**
If the stage game G has a unique Nash equilibrium, then, for any finite T , the repeated game G(T ) has a unique subgame perfect Nash equilibrium: the Nash equilibrium of G is played in every stage irrespective of the past history of the play.

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]

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A bargaining situation is described by a tuple hX, D, % 1 , % **2** i: X is a set of possible agreements: a set of possible consequences that the two players can jointly achieve.
D ∈ X is the disagreement outcome: the event that occurs if the players fail to agree.

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where u i (x, θ i ) is the money-equivalent value of alternative x ∈ X.
This assumes the case of private values in which player i’**s** payoff does not depend directly on other players’ types. If it does, then it is called common values case. The outcome (of the mechanism) is described by

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]

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(a) Find a Bayesian Nash equilibrium of the game in pure strategies in which each player i accepts an exchange if and only if the value v i does not exceed some
threshold θ i
(b) How would your answer to (a) change if the value of player i’**s** house to the other player j becomes 5

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]

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(c) There are two pure-strategy Nash equilibria: (A; X) and (B; Y ).
(d) Let p be a probability that player **2** chooses X and q be a probability that player 1 chooses A. Since player 1 must be indi¤erent amongst choosing A and B, we obtain
**2**p = p + 3(1 p) , 4p = 3 , p = 3=4.

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Proof of Pratt’**s** Theorem (1) Sketch of the Proof.
To establish (i) ⇔ (iii), it is enough to show that P is positively related to r. Let ε be a “small” random variable with expectation of zero, i.e., E(ε) = 0. The risk premium P (ε) (at initial wealth x) is defined by

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(nw1) means student **s** prefers an empty slot at school c to her own assignment, and (nw**2**) and (nw3) mean that legal constraints are not violated when **s** is assigned the empty slot without changing other students’ assignments.
The second property is about no-envy, which is also widely used in the context of school choice. But due to the structure of controlled school choice, as in Definition 1, even when a student prefers a school to her own and there is a student with lower priority in the school, the envy is not justified if the student’**s** move violates the legal constraints. Definition **2** formally states the condition for a student to have justified envy.

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るい ひとみ ひとみ ひとみ ひとみ あい あい あい あい
1 位 位 位 位 ともき ともき ともき ともき ともき ともき ともき ともき だいき だいき だいき だいき **2** 位 位 位 位 こうき こうき こうき こうき こうき こうき こうき こうき ともき ともき ともき ともき 3 位 位 位 位 だいき だいき だいき だいき だいき だいき だいき だいき こうき こうき こうき こうき

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Three firms (1, **2** and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits become 0. If exactly one firm advertises in the morning, its profit is 1; if exactly one firm advertises in the evening, its profit is **2**. Firms must make their daily advertising decisions simultaneously.

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(d) Zermelo’**s** theorem assures that the first mover has a winning strategy in ANY perfect information game with strictly opposite interests.
(e) The weak perfect Bayesian equilibrium puts NO restriction on beliefs at the information sets that are not reached in equilibrium.

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(a) Derive each partner’**s** payo¤ function.
(b) Derive each partner’**s** best reply function and graphically draw them in a …gure. (Taking m in the horizontal axis and n in the vertical axis.)
(c) Is this game strategic complementarity, strategic substitution, or neither of them? Explain why.

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5. Bayesian Game (20 points)
There are 10 envelopes and each of them contains a number 1 through 10. That is, one envelope contains 1, another envelope contains **2**, and so on; these numbers cannot be observable from outside. Suppose there are two individuals. Each of them randomly receives one envelope and observes the number inside of her/his own envelope. Then, they are given an option to exchange the envelope to the other person; exchange occurs if and only if both individuals wish to exchange. Finally, individuals receive prize ($) equal to the number, i.e., she receives $X if the number is X. Assume that both individuals are risk-neutral so that they maximize expected value of prizes.

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Hint: Your answers in (a) – (c) may change depending on the value of θ.
4. Duopoly (20 points)
Consider a duopoly game in which two firms, denoted by firm 1 and firm **2**, simul- taneously and independently select their own price, p 1 and p **2** . The firms’ products are differentiated. After the prices are set, consumers demand 24 − p i +

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4. Auctions (30 points)
Suppose that the government auctions one block of radio spectrum to two risk neu- tral mobile phone companies, i = 1, **2**. The companies submit bids simultaneously, and the company with higher bid receives a spectrum block. The loser pays nothing while the winner pays a weighted average of the two bids:

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A good is called normal (resp. inferior) if consumption of it increases (resp. declines) as income increases, holding prices constant.. Show the following claims.[r]

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two or three highly proitable RAM producers left. '
During the decade of the 1990**s**, however, both justifications for targeting RAMs technological externalities and excess returns-apparently failed to materialize. On one side, Japan'**s** lead in RAMs ultimately did not translate into an advantage in other types of semiconductor: For example, American irms retained a secure lead in microprocessors. On the other side, instead of continuing to shrink, the number of RAM producers began to rise again, with the main new entrants from South Korea and other newly industrial izing economies. By the end of the 1990**s**, RAM production was regarded as a "com modity" business: Many people could make RAMs, and there was nothing especially strategic about the sector.

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elimination of strictly dominated strategies can never be selected (with positive probability) in a mixed-strategy Nash equilibrium.[r]

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