5. Consider a game with the following payoff table for player 1: Strategy Player 1 1 2 3 1 3 -4 1 Player 2 2 -3 -2 3 3 -2 −1 2 (a) Use the graphical procedure to determine the value of the game and the optimal mixed strategy for each player. (b) Without eliminating any dominated strategies, follow the approach described in Sec 15.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. Please give the linear programming problem for each player. (c) Use Excel Solver (the simplex method) to find the optimal mixed strategy for each player.

5. Consider a game with the following payoff table for player 1: Strategy Player 1 1 2 3 1 3 -4 1 Player 2 2 -3 -2 3 3 -2 −1 2 (a) Use the graphical procedure to determine the value of the game and the optimal mixed strategy for each player. (b) Without eliminating any dominated strategies, follow the approach described in Sec 15.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. Please give the linear programming problem for each player. (c) Use Excel Solver (the simplex method) to find the optimal mixed strategy for each player.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.9P

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6. Consider the two-player Rock-Paper-Scissors game The payoff table is as follows. Player 2 Paper -1 0 1 Player 1 Rock Paper Scissors Rock 0 1 -1 Scissors 1 -1 0 (a) Follow the approach described in Sec 15.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. Please give the linear programming problem for each player. (b) Use Excel Solver (the simplex method) to find the optimal mixed strategies for each player. Also find the value of the game. Is this a fair game? If not, is this game in favor of player 1 or 2?
2. Consider the odds and evens game and whose payoff table is shown as below. Player 2 (odd player) 2 (fingers) Strategy Player 1 (even player) 1 (finger) 2 (fingers) 1 (finger) 3 -4 −4 3 (a) Use the graphical procedure described in Sec. 15.4 from the viewpoint of player 1(the evens player) to, first find the maximin point, then determine the optimal mixed strategy for player 1 (the even player) according to the minimax criterion. (Do not need to find the optimal mixed strategy for player 2.) (b) Use the Oddments Method directly to find the optimal mixed strategies for both player 1 and player 2. (c) What is the corresponding value of the game?
9. For the payoff matrix below, consider a sequential version of the game in which Player 2 moves first and then Player 1 moves second. Which of the following is not a correct statement? Player 1 Top Center Bottom Left 1,2 5,1 6,0 Player 2 Middle 7,1 11,2 -4,3 Right 10,4 1,0 8,5 The subgame perfect payoffs coincide with those of a Nash equilibrium b. Depending on Player 2's move, Player 1 will sometimes optimally choose Bottom c. Player 2 will choose Right in the subgame perfect equilibrium d. Player 2 would be willing to pay a positive $ amount to move first rather than second e. Player 1 would be willing to pay a larger $ amount than Player 2 to move first
5) Mixed strategy Nash equilibrium Consider a mixed strategy Nash equilibrium of the following coordination game: Player 2 Player 1 A B a 5.5 6.-2 b -2,6 1,1 a) In the above game, explain in words what condition player 1's probability p of playing strategy A must satisfy to induce player 2 to mix strategies between a and b in equilibrium. b) Solve for the mixed strategy Nash equilibrium where player 1 chooses A with probability P and player 2 chooses a with probability p, for 0 < p, P < 1.
Table 17-6 Two home-improvement stores (Lopes and HomeMax) in a growing urban area are interested in expanding their market share. Both are interested in expanding the size of their store and parking lot to accommodate potential growth in their customer base. The following game depicts the strategic outcomes that result from the game. Increases in annual profits (in millions of dollars) of the two home-improvement stores are shown in the following figure. Lopes Do notincreasethe size of store and parking lot Increase the sizeof store and parking lot Lopes = 1.0 Lopes = 0.4 Increasethe size of store and parking lot HomeMax = 1.5 HomeMax = 3.4 HomeMax Lopes = 3.2 Lopes = 2.0 Do notincreasethe size of store and parking lot HomeMax = 0.6 HomeMax = 2.5 Refer to Table 17-6. When this game reaches a Nash equilibrium, annual profit will grow by a. $0.6 million for HomeMax and by $3.2 million for Lopes. b. $2.5 million for HomeMax and by $2.0 million for Lopes. c. $3.4 million for HomeMax and…
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7. Consider a game with the following payoff table for player 1: Strategy 1 Player 1 2 3 1 523 2 042 Player 2 3 3 ww 3 124 2 (a) Follow the approach described in Sec 15.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. Please give the linear programming problem for each player. (b) Use Excel Solver (the simplex method) to find the optimal mixed strategies for each player. Also find the value of the game. Is this a fair game? If not, is this game in favor of player 1 or 2?
D and E
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9. Consider the following parlor game to be played between two players. Each player begins with three chips: one red, one white, and one blue. Each chip can be used only once. To begin, each player selects one of her chips and places it on the table, concealed. Both players then uncover the chips and determine the payoff to the winning player. In particular, if both players play the same kind of chip, it is a draw; otherwise, the following table indicates the winner and how much she receives from the other player. Next, each player selects one of her two remaining chips and repeats the procedure, resulting in another payoff according to the following table. Finally, each player plays her one remaining chip, resulting in the third and final payoff. Winning Chip Red beats white White beats blue Blue beats red Matching colors Payoff ($) 90 70 50 0 Formulate this problem as a two-person, zero-sum game by constructing a payoff table for Player 1. Do not need to solve this problem.
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