HW3-answer - Dr. Huanxing Yang Econ 601 Game Theory...

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Dr. Huanxing Yang Econ 601 Game Theory Homework Assignment 3 Due April 23, Wednesday 1. Chapter 5, Problem 5 a) The payoff matrix is shown in Figure SOL5.5.1. There is a unique Nash equilibrium in which both order salmon. In fact, salmon is the dominant strategy. Note that they're worse off than if they could both somehow agree to order pasta. Figure SOL5.5.1 - Dining game with two players Diner 2 7.00, 7.00 3.50, 8.50 -1.00, 7.00 Diner 1 8.50, 3.50 5.00, 5.00 0.50, 3.50 7.00, -1.00 3.50, 0.50 -1.00, -1.00 Pasta Salmon Filet Mignon Pasta Salmon Filet Mignon b) Note that a diner cannot influence what others order and thus must pay 25% of the price of the meals they order. All that a diner can influence is what she orders. The key property to note is that whatever she orders, she only pays 25% of the price with the remaining 75% being paid by the other three diners. Once recognizing that this is the actual cost to her, not the price on the menu, a diner should choose the meal that maximizes her surplus. Taking all this into account, Table SOL5.5.2 shows the cost faced by a diner. For example, a diner who orders the pasta dish only pays 25% of it which is $3.50. We observe that each diner orders the filet mignon because it really only costs them $7.50 and the surplus is maximized with that order. The unique Nash equilibrium is then that all four diners order the steak. Hence, each gets a meal they value at $29, but end up paying $30! Table SOL5.5.2 Dish cos Pr $21.00 $3.50 $17.50 $26.00 $5.25 $20.75 $29.00 $7.50 $21.50 Value Actual t Surplus Pasta imavera Salmon Filet Mignon 2. Chapter 5, Problem 10 First note that one Nash equilibrium is the strategy profile in which all citizens choose
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not to protest. The payoff to protesting for citizen i is , i vc given no one else protests. Since citizen 1 is the most inclined to protest (that is, she receives the highest benefit) and 1 0 −< - so that citizen 1 prefers not to protest on her own - than no other citizen does either. If no one else plans to march against the government, an individual citizen will not want to do so as it just means getting thrown into jail. One equilibrium then has no one participating in a protest. Is there an equilibrium in which a protest emerges? We know there is no equilibrium in which all citizens protest because the payoff to citizen n from protesting is negative even when everyone else protests; it equals 0 cc n nn v −= . He would prefer to stay home and receive a zero payoff. What about an equilibrium in which some citizens protest? To answer that question, let's derive an important property. If m citizens protest then it is optimal for citizen i to be one of those protesting citizens when 0. (SOL5.10.1 i c v m ) Now consider citizen j where . j i < Since ji vv > (that is, citizen j values protesting more than citizen i ) then it follows from (SOL5.10.1) that 0.
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This note was uploaded on 07/17/2008 for the course ECON 601 taught by Professor Yang during the Spring '08 term at Ohio State.

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HW3-answer - Dr. Huanxing Yang Econ 601 Game Theory...

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