of strategies, and let **s** −i denote a strategy profile other than
player i’**s** strategy, (**s** 1 , ..., **s** i−1 , **s** i+1 , ..., **s** n ).
Let u i : S 1 × · · · × S n → R denote player i’**s** payoff function:
u i (**s** 1 , ..., **s** n ) is the payoff to player i if the players choose the

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X
c∈C
max n 0, q τ(ˆ c **s**) − |ν l τ(ˆ **s**) (c) \ {ˆ **s**}| o
holds for any step l in the cycle, at any school c which ˆ **s** is admitted, q τ(ˆ c **s**) = |ν l τ(ˆ **s**) (c)| holds for any step l in the cycle. Hence, ˆ **s**’**s** rejected status for any school which ˆ **s** once proposed to cannot change to the non-rejected status by reproposal conditions (i) or (iii). Moreover, since a student **s** such that **s** ∈ S τ(ˆ **s**) and f (ˆ **s**) < f (**s**) cannot be assigned to a school which ˆ **s** prefers to her own assignment, reproposal condition (ii) does not apply to ˆ **s**. Therefore, ˆ **s** is always assigned to the same school in the cycle. Now we can separate the set of students who are always unfree because they do not change their assignments in the cycle. With the set of students who are always free in the cycle, only the reproposal condition (iii) could apply and it is when there was a reproposal before step t ′ . But a reproposal based on (iii) gives

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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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(c) There are two pure-strategy Nash equilibria: (A; X) and (B; Y ).
(d) Let p be a probability that player **2** chooses X and q be a probability that player 1 chooses A. Since player 1 must be indi¤erent amongst choosing A and B, we obtain
**2**p = p + 3(1 p) , **4**p = 3 , p = 3=**4**.

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るい ひとみ ひとみ ひとみ ひとみ あい あい あい あい
1 位 位 位 位 ともき ともき ともき ともき ともき ともき ともき ともき だいき だいき だいき だいき **2** 位 位 位 位 こうき こうき こうき こうき こうき こうき こうき こうき ともき ともき ともき ともき 3 位 位 位 位 だいき だいき だいき だいき だいき だいき だいき だいき こうき こうき こうき こうき

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

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

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j + x j − x i x j , where x i is i’**s** effort and x j is the effort of the other player. Assume
x 1 , x **2** ≥ 0.
(a) Find the Nash equilibrium of this game. Is it Pareto efficient?
(b) Suppose that the players interact over time, which we model with the infinitely repeated version of the game. Let δ denote the (common) discount factor of the players. Under what conditions can the players sustain some positive effort level k = x 1 = x **2** > 0 over time?

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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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(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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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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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 n + := {(x 1 , ..., x n )|x i ≥ 0, i = 1, ..., n} ⊂ R n .
For any x, y ∈ X, x % y means x is at least as preferred as y. Consumption set contains all conceivable alternatives.
A budget set is a set of feasible consumption bundles, represented as B(p, ω) = {x ∈ X|px ≤ ω}, where p is an n-dimensional positive vector interpreted as prices, and ω is a positive number interpreted as the consumer’**s** wealth.

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

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