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Fall 2007 G AME T HEORY IN THE S OCIAL S CIENCES Problem Set 5 (Due in Lecutre, Tuesdsay, Deecember 4) 1. As part of her research, an anthropologist visits a small agarian village. The land is not very good and the villagers have to work very hard from a very early age in order to scratch out a living. Indeed, life is so hard that by the time a person is forty, that person can do no useful work. Nevertheless, the younger people in the village continue to take care of these elders. Of course one could explain this observation by appealing to altruism, family ties, etc. But the anthropologist wonders if one could also explain this observation in a more self- interested way. To that end he constructs an over-lapping generation model of a repeated game. Each individual lives for two periods and is young (Y) in the first and old (O) in the second. Each individual produces two units of food when young and nothing when old. The demographics of the model are that there there are always two individuals alive, a young individual and an old one. When an old individual dies after two periods, a young one takes its place and so on forever. These demograhics are illustrated below: YO ... O Y ... ... ... During an individual’s life, four units of food are produced: two when that individual is young and two when that individual is old. Thus an individual can only have 9 consumption patterns which are ranked from best to worst as follows where the numbers under the pattern indicate its rank. {2,2}>{2,1}={1,2}> {1,1}>{2,0}={0,2} >{1,0}={0,1}>{0,0}. 5 > 4 = 4 > 3 > 2 = 2 > 1 = 1 > 0 The idea behind this rank is that an individual prefers more consumption to less, but as the boldface indicates that person also prefers “smooth” consumption. The anthropologist’s observation that the young support the old corresponds to the following equilibrium path in the overlapping-generations game: each individual gives one unit of food to an old person. (a) One strategy profile that would produce this path is simply for each young individual to give one unit to the old individual. Is this strategy profile a Nash equilibrium of the game? Explain why or why not? 1
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Fall 2007 (b) Suppose that each individual follows the strategy: “When young, share one unit with the old individual unless that individual did not share when young.” Is this a Nash equilibrium of the game? Be sure to explain why or why not. (c) Show that the strategies in (b) are not subgame perfect. Be sure to explain why they are not. (d) Find a subgame perfect equilibrium of the game. (Hint: In light of your answer in (c), what can you do to give the “punisher” an incentive to follow through on the punishment.) 2. In class we have analyzed the problem of brinkmanship when the states have complete information about each other’s payoffs and therefore can determine how much risk the other is willing to run. There are two game trees below.
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This note was uploaded on 12/25/2010 for the course PO 137 taught by Professor Power during the Fall '10 term at Berkeley.

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