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Unformatted text preview: Problem Set 2 (Lecture 2, March 9) ECOS3012 - Strategic Behavior, Semester 1, 2011 Note. I have given a number of problems here concerning the Nash Bargaining Solution and on taking a non-cooperative game situation and representing it as a strategic/normal form game. Some of these give you practice and others anticipate what I will teach in the next class. I do not obviously expect you to solve all the problems by next week, but I do expect you to have tried enough of them by the time I go over their solutions in next week’s (16 March) class. Calculating Nash Bargaining Solutions 1. ( Bargaining over a pie of unit size ) In class, we studied how to compute the Nash Bargaining Solution when splitting a pie of unit size and the utilities of a player from getting x units of the pie is u 1 ( x ) = x if it is Player 1 and u 2 ( x ) = √ x if it is Player 2. ( d 1 = d 2 = 0 was assumed.) Now, replace the utility functions with u 1 ( x ) = x α and u 2 ( x ) = x β where 0 < α,β < 1. (a) Compute the Nash Bargaining solution. How does the solution vary with respect to α and β ? Ans Let N ( x ) denote the Nash product (i.e. the product of the two players utilities (net of their disagreement payoffs) when they split the pie so that Player 1 gets a share x and Player 2 gets a share 1- x . That is, N ( x ) = ( u 1 ( x )- d 1 )( u 2 ( x )- d 2 ) = x α (1- x ) β In order to find the Nash Bargaining Solution, we need to maximize N ( x ) as x ranges from 0 to 1. Equivalently, we may maximize log ( N ( x )) = α log ( x ) + β log (1- x ), which upon differentiating gives us the first order condition at the maximum (denote by x * ) to be α x *- β 1- x * = 0 from which we get, x * = α α + β (1) So u * 1 = α α + β α and u * 2 = α α + β β is the Nash Bargaining solution. (b) If you are familiar (from other your past courses) with how α,β determine risk attitudes, does being relatively more risk averse benefit you or hurt you? Does this feel intuitive? (Do not worry if you cannot answer this. I will explain when I go over the solutions.) 1 Ans With a vNM utility of u ( x ) = x α , a higher α means the agent is less risk averse. Therefore when α > β , Player 2 is more risk averse than Player 1 and yet in this case, x * > 1 / 2. Therefore, one prediction of the Nash Bargaining Solution is that the more risk averse you are, the smaller will be your share of the pie. (This is something that you can check by running experiments or empirically and forms a test of the Nash Bargaining Theory)....
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This note was uploaded on 08/20/2011 for the course ECON 101 taught by Professor Etw during the Spring '11 term at UniversitÃ di Bologna.
- Spring '11