(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) We will get (B; Z) in the following iterated elimination process: Step 1: We can erase X since X is strictly dominated by Z.
Step **2**: Given step 1, we can erase A since A is strictly dominated by B.
Step 3: Given steps 1 and **2**, we can erase Y since Y is strictly dominated by Z. (c) Any combinations of x and y that satisfy x + y = 100 are Nash equilibria. Clearly, there are 101 such equilibria, i.e., (0; 100)(1; 99):::(100; 0).

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

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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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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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Signaling Game (**2**)
Figure: Signaling Game from Gibbons (1997)
Def Sender’**s** strategies are called (i) pooling when all types choose the same action, (ii) separating when each type chooses different actions, (iii) semi-separating when several actions are chosen but some action is chosen by more than one type.

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