Both the Bertrand and Cournot models can be seen as particular cases of a more general model of oligopolistic competition where firms choose prices and quantities. Bertrand is more reaso[r]

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Substituting into p+q = 3=4, we achieve q = 1=**2**. Since the game is symmetric, we can derive exactly the same result for Player 1’**s** mixed action as well. Therefore, we get the mixed-strategy Nash equilibrium: both players choose Rock, Paper and Scissors with probabilities 1=4; 1=**2**; 1=4 respectively.

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Two players, 1 and **2**, each own a house. Each player i values her own house at v i
and this is private information. The value of player i’**s** house to the other player j(**6**= i) is 3
**2** v i . The values v i are drawn independently from the interval [0, 1] with uniform distribution. Suppose players announce simultaneously whether they want to exchange (E) their house of not (N). If both players agree to an exchange, the exchange takes place. Otherwise no exchange occurs.

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

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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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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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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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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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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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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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(d) The perfect Bayesian equilibrium puts NO restriction on beliefs at information sets that are not reached in equilibrium.
(e) In the simple moral hazard problem we studied in class, the optimal wage (= **s**( )) is NOT necessarily increasing in outcome (= x).

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

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A strategy in dynamic games is a complete action plan which prescribes how the player will act in each possible.. contingencies in future..[r]

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