Section Strictly Determined Games, Dominated Rows, Saddle Points

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1 Finite Math B Chapter 11 Practice Questions Game Theory Section Strictly Determined Games, Dominated Rows, Saddle Points MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Decide on the payoff in this game when the specified strategy is used. 1) (1, 1) 1) A) $5 from A to B B) No payoff C) $5 from B to A D) $2 from A to B 2) (1, 2) 2) A) $5 from B to A B) $4 from B to A C) No payoff D) $2 from A to B 3) (1, 3) 3) A) No payoff B) $5 from A to B C) $10 from B to A D) $5 from B to A 4) (2, 1) 4) A) $5 from B to A B) $7 from B to A C) $4 from B to A D) $2 from A to B 5) (2, 2) 5) A) $2 from A to B B) No payoff C) $4 from B to A D) $1 from A to B 6) (2, 3) 6) A) No payoff B) $10 from B to A C) $4 from B to A D) $7 from B to A 7) (3, 1) 7) A) $2 from A to B B) $2 from B to A C) $10 from B to A D) $5 from B to A 8) (3, 2) 8) A) $4 from B to A B) $2 from A to B C) $1 from A to B D) $10 from B to A 9) (3, 3) 9) A) $2 from B to A B) $2 from A to B C) No payoff D) $10 from B to A

2 Remove a dominated strategy if one exists in the game. 10) 10) C) D) no dominated rows or columns 11) 11) C) D) no dominated rows or columns 12) 12) C) D) no dominated rows or columns 13) 13) C) D)

3 14) 14) C) D) no dominated rows or columns If a saddle point exists, identify it along with the value of the game. 15) 15) A) (1, 1), value 4 B) Does not exist C) (2, 1), value 3 D) (1, 2), value 6 16) 16) A) (1, 2), value 7 B) Does not exist C) (1, 1), value 6 D) (2, 1), value 7 17) 17) A) (2, 4), value 4 B) Does not exist C) (1, 1), value -3 D) (1, 4), value -6 18) 18) A) (2, 2), value -4 B) (1, 2), value -5 C) (3, 2), value -2 D) Does not exist 19) 19) A) (2, 4), value -8 B) Does not exist C) (2, 3), value -9 D) (1, 1), value -7 Find the saddle point and value of the game. 20) Two countries are involved in a border war. Each country has 3 strategies with payoffs in square miles of land. Positive numbers represent gains by A. B 20) A A) (1, 3), value -7 B) (3, 2), value 7 C) (1, 3), value -10 D) (3, 3), value -11

4 21) Suppose a rugby team with the ball (team A) can choose from three plays while the opposing team (B) has four possible defenses. The numbers in the payoff matrix represent yards gained by A. B 21) A A) (2, 2), value 9 B) (1, 1), value 19 C) (3, 4), value 15 D) (2, 3), value 3 22) The U.S. Army is playing a war game with side A trying to capture side B's headquarters. Side A has three strategies and side B has four defenses. The payoff matrix shows side A's percentage chance of winning. B 22) A A) (2, 2), value 80 B) (1, 4), value -28 C) (3, 1), value 36 D) (1, 2), value 40 Set up the payoff matrix and decide on the best strategy. 23) Promoters preparing for an outdoor concert could also set up an alternative indoor site in case of rain. It costs $ 15,000 to set up outdoors and $ 13,000 to set up indoors. With clear weather the expected gate is $ 123,000. If it rains, the expected gate is $ 77,000. How should they set up? 23) C) D) 24) A state is considering conducting an anti-smoking campaign for adults, youth, or both. Its costs, if there is a cause-effect relationship between smoking and cancer, are $ 896,000 for adults, $ 618,000 for youth and for both. If there is no cause-effect relationship, then the cost is $ 745,000 for youth, $ 736,000 for adults, and $ 1,157,000 for both. Whom should the state target in the campaign? A) 24) B) C) D)

5 25) A student can bring either a calculator or a reference book to an exam. The exam might stress calculations or definitions. If the exam stresses calculations then the student could gain 40 points with a calculator or 22 points with a reference book. If the exam stresses definitions, then a calculator would give the student 49 points but a reference book would give 15 points. Should the student bring a reference book or a calculator to the exam? A) 25) B) C) D) Section 11.2 Mixed Strategies for 2x2 Games (Formula sheet!) M Find the optimum strategies for player A and player B in the game. 26) 26) A) A: 1: 5/12, 2: 7/12 B: 1: 7/12, 2: 5/12 C) A: 1: 7/12, 2: 5/12 B: 1: 5/12, 2: 7/12 B) A: 1: 5/6, 2: 1/6 B: 1: 1/6, 2: 5/6 D) A: 1: 5/12, 2: 7/12 B: 1: 1/6, 2: 5/6 27) 27) A) A: 1: 4/13, 2: 9/13 B: 1: 9/13, 2: 4/13 C) A: 1: 9/13, 2: 4/13 B: 1: 8/13, 2: 5/13 B) A: 1: 5/13, 2: 8/13 B: 1: 5/13, 2: 8/13 D) A: 1: 9/13, 2: 4/13 B: 1: 5/13, 2: 8/13 28) 28) A) A: 1: 2/3, 2: 1/3 B: 1: 8/15, 2: 7/15 C) A: 1: 7/15, 2: 8/15 B: 1: 1/3, 2: 2/3 B) A: 1: 11/15, 2: 4/15 B: 1: 2/3, 2: 1/3 D) A: 1: 8/15, 2: 7/15 B: 1: 2/3, 2: 1/3

6 Solve the problem. 29) Mary has the flu and her doctor has told her it could be one of two viruses. There are two medicines available with various degrees of success shown in the matrix below. Find the optimum strategy to choose the most effective medicine. 29) A) Medicine 1 with probability 1/5 and medicine 2 with probability 4/5 B) Medicine 1 with probability 3/5 and medicine 2 with probability 2/5 C) Medicine 1 with probability 2/5 and medicine 2 with probability 3/5 D) Medicine 1 with probability 4/5 and medicine 2 with probability 1/5 30) Mr. Barnes sells raincoats and sun hats at a concession stand at an amusement park. His profit matrix is based on rainy and sunny days. Find the best mixed-sales strategy for Mr. Barnes. 30) A) Buy for rain: 4/7 Buy for shine: 3/7 C) Buy for rain: 3/4 Buy for shine: 1/4 B) Buy for rain: 15/28 Buy for shine: 13/28 D) Buy for rain: 17/28 Buy for shine: 11/28 Section Game Theory & Linear Programming Non-Strictly Determined, Non-2x2 Games 31. a) Set up a linear programming problem that could be used to find the optimum strategies for the payoff matrix above. Set up the initial simplex tableau. b) After all row operations have been completed, you have the following simplex tableau. What are the optimum strategies for the row player? What are the optimum strategies for the column player? What is the value of the game

7 32. a) Set up a linear programming problem that could be used to find the optimum strategies for the payoff matrix above. Set up the initial simplex tableau. b) After all row operations have been completed, you have the following simplex tableau. What are the optimum strategies for the row player? What are the optimum strategies for the column player? What is the value of the game Use the simplex method to find the optimum strategy for players A and B and the value of the game for the payoff matrix. 33) ) A) A: 1: 2/3, 2: 2/3 value = - 13/12 C) A: 1: 1/2, 2: 1/2 value = 5/72 B) A: 1: 2/3, 2: 1/3 value = 5/72 D) A: 1: 5/13, 2: 8/13 B: 1: 4/13, 2: 9/13, 3: 0 value = -6/13 34) ) A) A: 1: 1/3, 2: 0, 3: 2/3 value = 5/72 C) A: 1: 2/3, 2: 0, 3: 1/3 value = -65/8 B) A: 1: 5/13, 2: 0, 3: 8/13 B: 1: 4/13, 2: 9/13, 3: 0 value = -45/13 D) A: 1: 1/2, 2: 0, 3: 1/2 value = 5/72 Provide an appropriate response. 35) In a mixed strategy game, should either player favor a certain strategy always? 35) A) No B) Yes 36) Is it necessary that the simplex tableau be used to solve the game for every 3 3 payoff matrix? 36) A) No B) Yes 37) If a constant is added to every element of a payoff matrix and then one proceeds to solve the game, should an adjustment be made to the expected value? A) No B) Yes, the constant should be subtracted from the one that is found. 37) 38) If a constant is added to every element of a payoff matrix, should the best strategies be adjusted? 38) A) Yes B) No

8 1) C 16) C 2) D 17) D 3) A 18) C 4) C 19) B 5) D 20) A 6) D 21) D 7) A 22) B 8) A 23) B 9) D 24) C 10) C 25) C 11) B 26) A 12) B 27) C 13) C 28) D 14) A 29) A 15) A 30) B 31. a) 31b) 32. a) b) 33) D 34) B 35) A 36) A 37) B 38) B