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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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) Pure-strategy Nash equilibrium may NOT exist even if the game is …nite. (b) There MAY exist a subgame perfect Nash equilibrium which is not a Nash
equilibrium.
(c) Nash bargaining **solution** ALWAYS requires two players to divide the surplus equally.

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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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(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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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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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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(c) If a player randomizes pure strategies X and Y in a (mixed strategy) Nash equilibrium, she MUST be indi¤erent between choosing X and Y .
**2**. Monopoly (10 points)
Suppose a monopoly …rm operates in two di¤erent markets, A and B. Inverse demand for each market is given as follows.

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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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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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β(x i ) = c + θx i . (1)
Now suppose that player **2** follows the above equilibrium strategy, and we shall check whether player 1 has an incentive to choose the same linear strategy (1). Player 1’**s** optimization problem, given she received a valuation x 1 , is

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Rm In essence, the PBE is a **solution** concept that incorporates sequential rationality and consistency of beliefs.
The crucial new feature of PBE due to Kreps and Wilson (1982): Equilibrium no longer consists of just a strategy for each player but now also includes a belief for each player at each

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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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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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Suppose a seller auctions one object to two risk neutral buyers, = 1; **2**. The buyers submit bids simultaneously, and the buyer with higher bid receives the object. The loser pays nothing while the winner pays a weighted average of the two bids b+b **2** 0
where b is the winner’**s** bid, b 0 is the loser’**s** bid. Assume that the valuation of the

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

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◮ with probability p, a consumer with wealth x will receive a
times of her current wealth x
◮ with probability 1 − p she will receive b times of x.
Thm Assume that the assumptions of Pratt’**s** Theorem holds. Then, for any proportional risk, the decision maker 1 is more risk

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（一般に）ナッシュ均衡は複数存在する場合がある
プレイヤー全員にとってあるナッシュ均衡よりも別**の**ナッシュ
均衡**の**方が望ましい場合もある
良い均衡（Mac, Mac）ではなく悪い均衡（ Win , Win ）が選 ばれてしまう危険性がある

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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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