A function f : D (⊂ R n ) → R is called
**1** continuous at a point x 0 if, for all ε > 0, there exists δ > 0
such that d(x, x 0 ) < δ implies that d(f (x), f (x 0 )) < ε.
2 continuous if it is continuous at every point in its domain. **3** uniformly continuous if, for all ε > 0, there exists δ > 0 such

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vNM Utility Function (**1**)
Note the function U is a utility function representing the preferences on L(S) while v is a utility function defined over S, which is the building block for the construction of U (p). We refer to v as a vNM (Von Neumann-Morgenstern) utility function.

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where x is a vector of choice variables, and a := (a **1** , ..., a m ) is a vector of
parameters ( パラメータ ) that may enter the objective function and constraint.
Suppose that for each vector a, the solution is unique and denoted by x(a).
◮ A maximum-value function, denoted by M (a), is defined as follows:

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“Soon after Nash ’s work, game-theoretic models began to be used in economic theory and political science,. and psychologists began studying how human subjects behave in experimental [r]

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with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n +**1**
+ .
(a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. 5. Question 5 (4 points)

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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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with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n+**1**
+ .
(a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. 5. Question 5 (4 points)

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

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with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n+**1**
+ .
(a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. 5. Question 5 (4 points)

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Combination of dominant strategies is Nash equilibrium. There are many games where no dominant strategy exists[r]

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5. Production Economy (25 points)
Consider an economy with two firms and two consumers. Firm **1** is entirely owned by consumer **1**; it produces good A from input X via the production function a = 2x. Firm 2 is entirely owned by consumer 2; it produces good B from input X via the production function b = **3**x. Each consumer owns 10 units of X. Consumers’ preferences are given by the following utility functions:

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囚人**の**ジレンマ：注意点
このゲームでは個々**の**プレーヤーが最適戦略を持つ
【最適戦略（支配戦略）】 他**の**プレーヤーたちがどのような行 動を選択しても、自分がある特定**の**行動Aを選ぶことによって 利得が最大化されるとき、行動Aを「支配戦略」と呼ぶ。

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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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Klemperer (2002), “How (not) to Run Auctions: The European 3G Telecom Auctions,” European Economic Review. Milgrom (2004) Putting Auction Theory to Work Cambridge U Press[r]

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

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. Consumers buy at most one unit and have utility function u (**s**|θ) = θ**s** if they consume one unit of quality **s** and 0 if they do not consume. The monopolist decides on the quality and price that it is going to produce. Con- sumers observe qualities and prices and decide which quality to buy if at all.

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

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Price Discrimination (**1**)
A monopoly firm may further raise profit by charging different prices across consumers. This exercise is called price
discrimination. The traditional classification of the forms of price discrimination listed as follows is due to Pigou (1920):

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