Using this minimax theorem, answer the following questions.
(b) Show that Nash equilibria are interchangeable; if and are two Nash equilibria, then and are also Nash equilibria.
(c) Show that each player’**s** payo¤ is the same in every Nash equilibrium.

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(b) Revenue equivalence theorem claims that the equilibrium bidding strategy under the …rst-price auction is ALWAYS identical to the one under the second- price auction.
(c) EVERY perfect Bayesian equilibrium is a weak perfect Bayesian equilibrium. **2**. Dynamic Game (14 points)

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(d) What is the Nash equilibrium of this game? 4. Mixed Strategy (15 points)
Three …rms (1, **2** and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A …rm advertises exactly once per day. If more than one …rm advertises at the same time, their pro…ts become 0. If exactly one …rm advertises in the morning, its pro…t is 1; if exactly one …rm advertises in the evening, its pro…t is **2**. Firms must make their daily advertising decisions simultaneously.

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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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3. Auction (9 points)
Consider a “common-value auction” with two players, where the value of the object being auctioned is identical for both players. Call this value V and suppose that V = v 1 + v **2** , where v i is independently and uniformly distributed between 0 and 1,

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Prisoners’ Dilemma: Analysis
( Silent , Silent ) looks mutually beneficial outcomes, though
Playing Confess is optimal regardless of other player’**s** choice! Acting optimally ( Confess , Confess ) rends up realizing!!

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(c) Confirm that by choosing the tax t appropriately, the socially optimal level of pollution is produced.
(d) Add a second firm with a different production function. Now the consumers observe a pollution level b = b 1 + b **2** . Show that the social optimum can still

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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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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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Axiomatic Approach (**2**)
PAR (Pareto Efficiency) Suppose hU, di is a bargaining problem with v, v ′ ∈ U and v ′
i > v i for i = 1, **2**. Then f (U, d) 6= v. The axioms SYM and PAR restrict the behavior of the solution on single bargaining problems, while INV and IIA require the solution to exhibit some consistency across bargaining problems.

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1 It may stay out (O),
**2** prepare itself for combat and enter (“Ready” = R),
3 or enter without making preparations (“Unready” = U ).
Preparation is costly but reduces the loss from a fight. The incumbent may either fight (F ) or accommodate (A) entry. Depending on the payoffs, we consider two cases.

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A belief about other players’ types is a conditional probability distribution of other players’ types given the player’**s**
knowledge of her own type p i (t −i |t i ).
When nature reveals t i to player i, she can compute the belief p i (t −i |t i ) using Bayes’ rule:

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First-Price: General Model (1)
Consider a first-price auction with n bidders in which all the conditions in the previous theorem are satisfied.
Assume that bidders play a symmetric equilibrium, β(x). Given some bidding strategy b, a bidder’**s** expected payoff becomes

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Proof Sketch (**2**): Existence of Pivotal Voter Lemma 3 (Existence of Pivotal Voter)
There is a voter n ∗ = n(b) who is extremely pivotal in the sense that by changing his vote at some profile he can move b from the very bottom of the social ranking to the very top.

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Explain.
(b) Show that any risk averse decision maker whose preference satisfies indepen- dence axiom must prefer L **2** to L 3 .
3. Question 3 (4 points) Suppose a monopolist with constant marginal costs prac- tices third-degree price discrimination. Group A’**s** elasticity of demand is ǫ A and

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b + (1 )b 0
where b is the winner’**s** bid, b 0 is the loser’**s** bid, and is some constant
satisfying 0 1. (In case of ties, each company wins with equal probability.) Assume the valuation of the spectrum block for each company is independently and uniformly distributed between 0 and 1.

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Problem Set **2**: Due on May 14
Advanced Microeconomics I (Spring, 1st, 2013)
1. Question 1 (6 points)
(a) Suppose the utility function is continuous and strictly increasing. Then, show that the associated indirect utility function v(p, ω) is quasi-convex in (p, ω). (b) Show that the (minimum) expenditure function e(p, u) is concave in p.

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すべて**の**プレーヤーに支配戦略が無いゲームでも解け る場合がある
「支配される戦略**の**逐次消去」（後述）
（お互い**の**行動に関する）「正しい予想**の**共有＋合理性」 によってナッシュ均衡は実現する！

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How to Measure Welfare Change | 厚生**の**変化をどうはかるか？
When the economic environment or market outcome changes, a consumer may be made better off ( 改善 ) or worse off ( 悪化 ). Economists often want to measure how consumers are affected by these changes, and have developed several tools for the assessment of welfare ( 厚生 ).

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

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