(c) Any finite game has at least one Nash equilibrium in pure strategies. **2**. Expected Utility (16 points)
Suppose that an individual can either exert effort or not. Her initial wealth is $100 and the cost of exerting effort is c. Her probability of facing a loss $75 (that is, her wealth becomes $25) is 1

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Mechanism Design
This lecture is mostly based on Chapter **14** “Mechanism Design” of Tadelis (2013).
There are many economic and political situations in which some central authority wishes to implement a decision that depends on the private information of a set of players. The theory of mechanism design is the study of what kinds of mechanisms such a central authority (or mechanism designer) can devise in order to reveal some or all of the private information from the group of players who interact each other.

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政府（官僚組織、政治家）はどのように行動する**の**か？ 政治**の**経済学 政治**の**経済学 政治**の**経済学 政治**の**経済学
私企業**の**中でなにが起こっている**の**か？
組織**の**経済学、企業統治（コーポレート・ガバナンス） 組織**の**経済学、企業統治（コーポレート・ガバナンス） 組織**の**経済学、企業統治（コーポレート・ガバナンス） 組織**の**経済学、企業統治（コーポレート・ガバナンス）

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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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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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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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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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(d) Suppose that this game is played finitely many times, say T (≥ **2**) times. De- rive the subgame perfect Nash equilibrium of such a finitely repeated game. Assume that payoff of each player is sum of each period payoff.
(e) Now suppose that the game is played infinitely many times: payoff of each player is discounted sum of each period payoff with some discount factor δ ∈ (0, 1). Assume specifically that A = 16, c 1 = c **2** = 8. Then, derive the

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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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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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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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3. when it is available (its date)
4. under what condition it is deliverable (the state of the world)
Def A contingent commodity x kts is a promise of delivery of a
particular good or service k at a particular date t if an uncertain event **s** actually occurs.

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