◮ If the two pennies match, then player 2 wins player 1’**s** penny; if the
pennies do not match, then 1 wins 2’**s** penny.
Although the existence of Nash equilibrium is not guaranteed, the natural extension of strategies, mixed strategies ( 混合戦略 ), will almost always assure the existence of equilibrium.

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Consider a consumer problem. Suppose that a choice function x(p; !) satis…es Walras’**s** law and WA. Then, show that x(p; !) is homogeneous of degree zero. **6**. Lagrange’**s** Method
You have two …nal exams upcoming, Mathematics (M) and Japanese (J), and have to decide how to allocate your time to study each subject. After eating, sleeping, exercising, and maintaining some human contact, you will have T hours each day in which to study for your exams. You have …gured out that your grade point average (G) from your two courses takes the form

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Q = K 1 =4
L 1 =8 Then, answer the following questions.
(a) In the short run, the …rm is committed to hire a …xed amount of capital K(+1), and can vary its output Q only by employing an appropriate amount of labor L . Derive the …rm’**s** short-run total, average, and marginal cost functions. (b) In the long run, the …rm can vary both capital and labor. Derive the …rm’**s**

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This is an advanced course in microeconomics, succeeding to Advanced Microeconomics I (ECO600E) in which we study individual economic decisions and their aggregate [r]

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2 units of the firm 1’**s** good and A − p 2 + p 1
2 units of the firm 2’**s** good. Assume that the firms have identical (and constant) marginal costs c(< A), and the payoff for each firm is equal to the firm’**s** profit, denoted by π 1 and π 2 .

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Introduction to Market Design and its Applications to School Choice.. Yosuke YASUDA.[r]

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Two neighboring homeowners, 1 and 2, simultaneously choose how many hours to spend maintaining a beautiful lawn (denoted by l 1 and l 2 ). Since the appearance of one’**s** property depends in part on the beauty of the surrounding neighborhood, homeowner’**s** benefit is increasing in the hours that neighbor spends on his own lawn. Suppose that 1’**s** payoff is expressed by

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Similarly, player 2 must be indi¤erent amongst choosing X and Y , which implies 4q + **6**(1 q) = 7(1 q)
, 5q = 1 , q = 1=5.
Thus, the mixed-strategy equilibirum is that player 1 takes A with probability 1=5 (and B with probability 4=5) and player 2 takes X with probability 3=4 (and Y with probability 1=4).

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安田予想で未受賞**の**候補者たち
Robert Barro (1944-, マクロ、成長理論) → イチオシ！
Elhanan Helpman (1946-, 国際貿易、成長) → 誰ともらう**の**か？
Paul Milgrom (1948-, 組織**の**経済学、オークション) → 今年は厳しい… Ariel Rubinstein (1951-, ゲーム理論) → 今年は厳しそう…

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(nw1) means student **s** prefers an empty slot at school c to her own assignment, and (nw2) 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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Both the Bertrand and Cournot models are particular cases of a more general model of oligopoly competition where firms choose prices and quantities (or capacities.). Ber[r]

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(a) If an agent is risk averse, her risk premium is ALWAYS positive.
(b) When every player has a (strictly) dominant strategy, the strategy profile that consists of each player’**s** dominant strategy MUST be a Nash equilibrium. (c) If there are two Nash equilibria in pure-strategy, they can ALWAYS be Pareto

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If the stage game has a unique NE, then for any T , the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the histor[r]

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3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game.. Thus, i[r]

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Prisoners’ Dilemma: Analysis
( Silent , Silent ) looks mutually beneficial outcomes, though
Playing Confess is optimal regardless of other player’**s** choice! Acting optimally ( Confess , Confess ) rends up realizing!!

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Prisoners’ Dilemma: Analysis (3)
(Silent, Silent) looks mutually beneficial outcomes, though
Playing Confess is optimal regardless of other player’**s** choice!
Acting optimally ( Confess , Confess ) rends up realizing!!

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payoff) while M gives 1 irrespective of player 1’**s** strategy.
Therefore, M is eliminated by mixing L and R .
After eliminating M , we can further eliminate D (step 2) and L
(step 3), eventually picks up ( U , R ) as a unique outcome.

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elimination of strictly dominated strategies can never be selected (with positive probability) in a mixed-strategy Nash equilibrium.[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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A tree starts with the initial node and ends at.. terminal nodes where payoffs are specified..[r]

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