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### 6001(10)solassign6

Course: ECON 6001, Spring 2010
School: Sydney
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Word Count: 598

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6001: ECON Solutions to Assignment 6 (1) For each of the following games, nd out a) the set of dominated strategies (strict and weak) for each player and b) the set of Nash equilibria. Player 1 Player 2 L R 4, 2 3, 2 2, 0 3, 3 T B For player 1, T weakly dominates B and for player 2, R weakly dominates L. Nash equilibria: (T, L), (T, R) and (B, R). Player 1 T B Player 2 L M R 2, 4 3, 3 2, 3 1, 3 4, 0 2, 5...

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6001: ECON Solutions to Assignment 6 (1) For each of the following games, nd out a) the set of dominated strategies (strict and weak) for each player and b) the set of Nash equilibria. Player 1 Player 2 L R 4, 2 3, 2 2, 0 3, 3 T B For player 1, T weakly dominates B and for player 2, R weakly dominates L. Nash equilibria: (T, L), (T, R) and (B, R). Player 1 T B Player 2 L M R 2, 4 3, 3 2, 3 1, 3 4, 0 2, 5 Player 1 has no dominated strategies. For player 2, strategy M is strictly dominated. Nash equilibria: (T, L) and (B, R). T Player 1 C B L 1, 1 1, 2 0, 0 Player M 0, 0 0, 0 1, 3 2 R 1, 1 0, 4 2, 2 For player 1, strategy C is weakly dominated. For player 2, strategy L is weakly dominated. Nash Equilibria: (T, L) and (B, M ). (2) Consider a two player game with player 1 having two strategies A and B while player player 2 has three strategies {a, b, c}. If player 1 plays A, then irrespective of player 2s strategy, player 1 receives 0 and 1 player 2 receives 4. If player 1 choose B , the payos are (4, 3) if player 2 chooses a or b and (3, 3) if player 2 chooses c. (i) Determine the set of dominated strategies (both weak and strict) for each of these players. For player 1, no strategies are dominated. For player 2, strategy c is weakly dominated. (ii) What is the set of pure strategy (Nash) equilibrium? (A, c), (B, a) and (B, b). (iii) Does this game have an equilibrium in mixed strategies? There is no mixed strategy equilibrium where both players mix. To see this, rst note that if player 1 is playing both of his strategies with positive probability, then player 2 can never use strategy c. This is because c is weakly dominated by a or b. Thus, in this equilibrium, player 2 must mix between a and b. But if player 2 never plays c, player 1 is strictly better o playing . B Thus, if there is a mixed strategy equilibrium, it must be of the form that player 2 mixes between strategies a and b (any mix will do) while player 1 plays B with probability 1. (3) Consider the game with two players where the strategy of each player is to pick an integer in the set A = {0, 1, 2 . . . 100}. Given a strategy choice (s1 , s2 ), the payos are as follows: the player who chooses the larger integer gets one dollar from the other. If both players choose the same integer, each gets zero. (a) What are the strictly (respectively weakly) dominated strategies for each player? Strategy 0 is strictly dominated by the strategy 100. and strategy 100 weakly dominates every other strategy. (b) What is the set of pure strategy equilibrium for this game? Is this equilibrium unique? The unique Nash equilibrium of this game is the choice of 100 for each player. 2 (4) Two players are contributing towards a joint project. Player i chooses gi 0 to contribute. If g1 + g2 C , the project is successfully completed and player i receives a payo of Vi gi . If however g1 + g2 < C , the project fails and player i s payo is gi . Assume that Vi < C < V1 + V2 (i) What are the set of strictly dominated strategies for each player? For agent i, since his valuation is Vi , the maximum he will contribute is Vi . Thus any gi > Vi is strictly dominated. (ii) Describe the set of pure strategy equilibrium for this game. Since Vi < C , there is always an equilibrium where both players contribute zero. However, since C < V1 + V2 , there also exists a whole host of other equilibria in which both players contribute and the joint project is completed. In any such equilibrium (g1 , g2 ), the following conditions must hold: i) g1 + g2 = C and ii) g1 V1 and iii) g2 V2 . 3
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