That is, given that the incumbent’**s** information set is reached, choosing A is clearly optimal irrespective of her belief over nodes. → Choosing F looks non-credible.
SPNE cannot eliminate this since there is no proper subgame. Weak perfect Bayesian equilibrium (explain in detail later) is enough to exclude (O, F ) in this case.

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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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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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(b) Let p be a probability that player **2** would choose Rock, and q be a probability that she chooses Paper. Note that her probability of choosing Scissors is written as 1 p q. Under mixed strategy Nash equilibrium, player 1 must be indi¤erent amongst choosing Rock, Paper and Scissors, which implies that these three actions must give him the same expected payo¤**s**. Let u R ; u P ; u S be his expected payo¤**s** by selecting

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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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or
u i ( i ; i ) u i (**s** i ; i ) for all **s** i **2** S i . (**2**)
7. Mixed strategies: Application
A crime is observed by a group of n people. Each person would like the police to be informed but prefers that someone else make the phone call. They choose either “call” or “not” independently and simultaneously. A person receives 0 payo¤ if no

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

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Let w = (w 1 , w **2** , w 3 , w 4 ) ≫ 0 be factor prices and y be an (target) output.
(a) Does the production function exhibit increasing, constant or decreasing returns to scale? Explain.
(b) Calculate the conditional input demand function for factors 1 and **2**. (c) Suppose w 3 >

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

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i = α i v i + β i for i = 1, **2**, where α i > 0 for i = 1, **2**. Then, for i = 1, **2**,
f i (U ′ , d ′ ) = α i f i (U, d) + β i .
SYM (Symmetry) If the bargaining problem hU, di is symmetric, i.e., d 1 = d **2** and (v 1 , v **2** ) ∈ U if and only if (v **2** , v 1 ) ∈ U , then

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(a) Derive firm 1’**s** payoff function and the best reply function.
(b) Solve the pure-strategy Nash equilibrium of this game. How much profit does each firm earn?
(c) Now suppose that firms decide prices sequentially: firm 1 sets its price p 1 first, and firm **2** chooses price only after observing firm 1’**s** price. Find the subgame perfect equilibrium of this game. How much profit does each firm earn?

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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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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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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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Edgeworth Box | エッジワース・ボックス
The most useful example of an exchange economy is one in which there are two people and two goods. This economy’**s** set of allocations can be illustrated in an Edgeworth box ( エッジワース・ボックス ) diagram.

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