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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タカ ―ハト・ゲーム（チキンゲーム）
交渉事に強気（タカ戦略）でのぞむか弱気（ハト戦略）で**の**ぞ
むかをそれぞれ**の**プレーヤーが決定
ナッシュ均衡は非対称な行動**の**組：（ ハト 、 タカ ）と（ タカ 、 ハト ） 現実に、ハト戦略とタカ戦略**の**どちらも観察されることに対応

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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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◮ 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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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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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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(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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(a) Write the payoff functions π 1 and π **2** (as a function of p 1 and p **2** ).
(b) Derive the best response functions and solve the pure-strategy Nash equilib- rium of this game.
(c) Derive the prices (p 1 , p **2** ) that maximize joint-profit, i.e., π 1 + π **2** .

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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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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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For a singleton information set, i.e., x = h(x), the player’**s** belief puts probability one on the single decision node.
(**2**) Given their beliefs, the players’ strategies must be
sequentially rational. That is, at each information set, the action taken by the player must be optimal given

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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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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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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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Connection between UMP and EMP | UMP と EMP **の**関係
There is a strong link between the utility maximization problem (UMP, 効用最 大化問題 ) and the expenditure minimization problem (EMP, 支出最小化問題 ). Let us first consider the following practice question.

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ローチ ローチ
ローチ にある。単なる現状分析や、選択制を導 入あるいは廃止すべきか、という是非論にとど まらず、 制度をデザインするという視点 制度をデザインするという視点 制度をデザインするという視点 制度をデザインするという視点 から、望 ましい学校選択制**の**制度設計について、 ゲーム ゲーム ゲーム ゲーム

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