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

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Outcome of JRMP May Violate Weak Stability
The same example as before: There are two hospitals h 1 , h **2** in one
region with regional cap 10.
Each hospital has a capacity of 10 and a target capacity of 5. There are 10 doctors, d 1 , . . . , d 10 such that

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is strictly increasing in its first two arguments and strictly decreasing in b. Thus by increasing pollution, the firm can produce more output (or use less input). The consumer has a concave utility function U (y 1 , y **2** , b ) that is also increasing in its first
two arguments and decreasing in b.

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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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3. Nash Equilibrium (16 points)
Monica and Nancy have formed a business partnership. Each partner must make her e¤ort decision without knowing what e¤ort decision the other player has made. Let m be the amount of e¤ort chosen by Monica and n be the amount of e¤ort chosen by Nancy. The joint pro…ts are given by 4m + 4n + mn, and two partners split these pro…ts equally. However, they must each separately incur the costs of their own e¤ort, which is a quadratic function of the amount of e¤ort, i.e., m **2** 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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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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However, it is difficult to assess how reasonable some axioms are without having in mind a specific bargaining procedure. In particular, IIA and PAR are hard to defend in the abstract. Unless we can find a sensible strategic model that has an equilibrium corresponding to the Nash solution, the appeal of Nash’**s** axioms is in doubt.

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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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R i (or % i ) An individual i’**s** preference relation on X (an binary relation satisfying completeness and transitivity. → Let P i (or ≻i) and I i (or ∼i) be the associated relations of strict individual preference and indifference, respectively.

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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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1 Nature draws a type t i for the Sender from a set of feasible
types T = {t1 , ..., t I} according to a probability distribution
p(ti), where p(ti) > 0 for every i and p(t 1 ) + · · · + p(tn) = 1.
**2** Sender observes ti and then chooses a message mj from a set

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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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object for each buyer is independently and uniformly distributed between 0 and 1. (a) Suppose that buyer **2** takes a linear strategy, b **2** = v **2** . Then, derive the
probability such that buyer 1 wins as a function of b 1 .
(b) Solve a Bayesian Nash equilibrium.

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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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Open Set and Closed Set (**2**)
Boundary and interior
◮ A point x is called a boundary point of a set S in R n
if every ε-ball centered at x contains points in S as well as points not in S. The set of all boundary points of a set S is called boundary, and is denoted ∂S .

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合理的な豚：分析
子豚には最適戦略（支配戦略）が存在する！ 大豚**の**行動によらず「 待つ 」**の**が常に最適
子豚が合理的ならば絶対にスイッチを押さない 子豚**の**「 スイッチを押す 」は可能性から消去される

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