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Game Theory
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Game Theory
Game Theory
Part II. Short Problems (71 total points available). Show your work. If not, you run the risk of not getting partial credit for wrong answers and not getting full credit for right ones. 1. Finitely Repeated Game. Suppose two firms are simultaneously introducing a new product and must choose a particular quality level for their product. Quality A means the product is of adequate (average quality), Quality B is the best quality, and Quality C is the worst quality. If a firm chooses quality C and the other chooses either A or B, then the better quality product is tainted by the presence of the quality product and both firms end up making zero profits. However, if both choose C, both can at least make some profit. If the firms match the other qualities (A or B), profits are higher for both. Otherwise, profits are unequal with the slight edge going to the one who chooses the best quality B. The single-stage game is depicted below. Assume that the game is played twice and that the players do not discount the future. Column Player Quality B Quality A Quality C 5,5 3,6 0,0 Quality A Row Player Quality B 6,3 4, 4 0,0 Quality C 0,0 0,0 1,1 (1.A) Find all the Nash Equilibrium for the single-stage game. Give these in terms of strategies and payoffs. (2 points) (1.B) Suppose the stage game above is repeated twice. If both players choose B each time, is this subgame perfect? If so what are the payoffs if this can be supported? Explain your answers. (2 points) (1.C) Again suppose the game is repeated two times. If (A, A) is played twice, is this strategy subgame perfect? If so what are the payoffs if this can be supported? Explain your answers. (2 points) (1.D) If(C, C) is played twice, is this subgame perfect? If so, what are the payoffs if this can be supported? Explain your answers. (2 points)
Part II. Short Problems (71 total points available). Show your work. If not, you run the risk of not getting partial credit for wrong answers and not getting full credit for right ones. 1. Finitely Repeated Game. Suppose two firms are simultaneously introducing a new product and must choose a particular quality level for their product. Quality A means the product is of adequate (average quality), Quality B is the best quality, and Quality C is the worst quality. If a firm chooses quality C and the other chooses either A or B, then the better quality product is tainted by the presence of the quality product and both firms end up making zero profits. However, if both choose C, both can at least make some profit. If the firms match the other qualities (A or B), profits are higher for both. Otherwise, profits are unequal with the slight edge going to the one who chooses the best quality B. The single-stage game is depicted below. Assume that the game is played twice and that the players do not discount the future. Column Player Quality B Quality A Quality C 5,5 3,6 0,0 Quality A Row Player Quality B 6,3 4, 4 0,0 Quality C 0,0 0,0 1,1 (1.A) Find all the Nash Equilibrium for the single-stage game. Give these in terms of strategies and payoffs. (2 points) (1.B) Suppose the stage game above is repeated twice. If both players choose B each time, is this subgame perfect? If so what are the payoffs if this can be supported? Explain your answers. (2 points) (1.C) Again suppose the game is repeated two times. If (A, A) is played twice, is this strategy subgame perfect? If so what are the payoffs if this can be supported? Explain your answers. (2 points) (1.D) If(C, C) is played twice, is this subgame perfect? If so, what are the payoffs if this can be supported? Explain your answers. (2 points)
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