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Unformatted text preview: Game Theory Solutions to Exercises: Cooperative Games JanJaap Oosterwijk Winter 20072008 4 Cooperative Games 4.5.1 For the following bimatrix games, draw the NTU and TU feasible sets. What are the Pareto optimal outcomes? (a) (0 , 4) (3 , 2) (4 , 0) (2 , 3) Solution: The NTU feasible set is given in the figure below (figure) and has as Pareto optimal outcomes all vectors on the following three line segments: one joining the points (0 , 4) and (2 , 3), one joining the points (2 , 3) and (3 , 2), and one joinng the points (3 , 2) and (4 , 0). The TU feasible set is given in the figure below (figure) and has as Pareto optimal outcomes all vectors on the line of slope 1 through the points (2 , 3) and (3 , 2), i.e. { (1 t )(2 , 3) + t (3 , 2) : t R } = { (2 + t, 3 t ) : t R } . (b) (3 , 1) (0 , 2) (1 , 2) (3 , 0) Solution: The NTU feasible set is given in the figure below (figure) and has as Pareto optimal outcomes all vectors on the line segment joining the points (1 , 2) and (3 , 1), i.e. { (1 t )(1 , 2) + t (3 , 1) : 0 t 1 } = { (1 + 2 t, 2 t ) : 0 t 1 } . The TU feasible set is given in the figure below (figure) and has as Pareto optimal outcomes all vectors on the line of slope 1 through the point (3 , 1), i.e. { (3 + t, 1 t ) : t R } . 4.5.2 Find the cooperative strategy, the TU solution, the side payment, the optimal threat strategies, and the disagreement point for the two matrices (1) and (2) of Sections 4.1 and 4.2. Solution: Consider the TU game with bimatrix (1) of Section 4.1 (4 , 3) (0 , 0) (2 , 2) (1 , 4) ....
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This note was uploaded on 08/24/2009 for the course MATH 262447221 taught by Professor Weisbart during the Spring '09 term at UCLA.
 Spring '09
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