Game theory final revision PDF

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GAME THEORY PROBLEM SOLVING 1. Mary Schultz is the incumbent Senator from Ohio and John Henderson is her opponent in the upcoming election.

John Henderson’s Offensive strategies

Mary Schultz's defensive strategies Strategy A Strategy B 3 7 6 10

Strategy 1 Strategy 2

Determine the optimal strategy and the optimal payoff for both players? Answer: John, strategy 2, Mary strategy A and the optimal payoff = 6 2. Given the following payoff/loss table for a mixed strategy game between two players: A 40 1 Player 1 90 2 80 3 60 4 Identify dominated strategy(s), if it (they) exist(s). Answer: B dominates A, B dominates D, 2 dominates 4 3.

Player 2 B 30 50 75 40

C 54 60 52 55

consider the following game b1

b2

a1

4

2

a2

3

5

NO saddle point, use mixed strategies Finding mixed strategies b1

b2

a1

4

2

p

0.5

a2

3

5

1-p

0.5

Q

1- Q

0.75

0.25 1

D 80 65 90 50

Player A Plays a1 with p probability Plays a2 with (1-p) probability Such that EV (A|b1) = EV (A|b2) EV (A|b1) = 4p + 3(1-p) EV (A|b2) = 2p + 5(1-p) Set equal and solve for p 4p + 3(1-p) = 2p + 5(1-p) p + 3 = 5 – 3p 4p = 2, then p = 0.5 and (1-p) = 0.5 The mixed strategy for A is (using strategy a1 50% of the time, and using strategy a2 50% of the time) Following the same reasoning, the mixed strategy for B is (0.75, 0.25) EV (B|a1) = 4Q + 2(1-Q) EV (B|a2) = 3Q + 5(1-Q) 4Q + 2(1-Q) = 3Q + 5(1-Q), then Q = 0.75 and (1-Q) = 0.25 The mixed strategy for B is (using strategy b1 75% of the time, and using strategy b2 25% of the time) Value of the game EV for (A): IF b1 = 0.5 (4) + 0.5 (3) = 3.5 IF b2 = 0.5 (2) + 0.5 (5) = 3.5 EV for (B): IF a1 = 0.75 (4) + 0.25 (2) = 3.5 IF b2 = 0.75 (3) + 0.25 (5) = 3.5 2

FILL IN THE BLANK 1.

_________________ addresses decision situations with two or more decision makers in competition. Answer: game theory

2. In a ______________________ game one player’s gains represent another player’s exact losses. Answer: zero-sum) 3. A ____________________ is a plan of action followed by a player. Answer: strategy 4. In a __________________ game each player adopts a single strategy as an optimal strategy. Answer: pure strategy 5. A __________________ game occurs when each player selects an optimal strategy that does not result in an equilibrium point when the minimax criterion is used. Answer: mixed strategy 6. In a pure strategy game the optimal strategy for each player results in the same payoff, called an ___________________ point. Answer: equilibrium or saddle 7. In a game the optimal strategy for each player results in the same payoff, called an equilibrium or saddle point. Answer: pure strategy 8. In a(n) ______________ game, players switch decisions in response to the decision of the other player and eventually return to the initial decisions, resulting in a closed loop. Answer: mixed strategy 9. A strategy is ______________________ and can be eliminated if all its payoffs are worse than the corresponding payoffs for another strategy. Answer: dominated 10. In game theory, each player seeks to minimize maximum possible losses with the ______________ criterion. Answer: minimax

MULTIPLE CHOICES 11. In a zero sum game one player’s gains represent another’s exact __________. a. gains b. losses c. gains and losses d. gains or losses Answer: B 12. The __________ is the offensive player’s gain and the defensive player’s loss in a zero-sum game. a. game situation b. value of the game c. strategy d. best strategy e. pure strategy Answer: B 13. A ____________ is a plan of action followed by a player. a. game situation b. strategy c. value of the game d. best strategy e. pure strategy

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Answer: B

14. In a ______________, it is assumed that the payoff table is known to all players. a. strategy b. game situation c. value of the game d. best strategy Answer: B 15. The _____________ criterion results in the maximum of the minimum payoffs. a. minimax b. maximax c. maximin d. minimin Answer: C 16. Regret is the difference between the payoff from the a. best decision and all other decision payoffs. b. worst decision and all other decision payoffs. c. best decision and the worst decision payoffs. d. none of the above. Answer: A 17. The ______________ minimizes the maximum regret. a. maximax regret criterion b. minimax regret criterion c. minimin regret criterion d. maximin regret criterion

Answer: B

18. Determining the worst payoff for each alternative and choosing the alternative with the best of the worst payoffs is called a ___________ criterion. a. maximin b. minimin c. maximax d. minimax Answer: A 19. The maximin approach to decision making refers to a. minimizing the maximum return. b. maximizing the minimum return. c. maximizing the maximum return. d. minimizing the minimum return. Answer: B 20. With the ___________ criterion each player seeks to minimize maximum possible losses. a. minimax b. maximax c. minimin d. maximin Answer: A 21. The _____________ is the expected value of the regret for each decision. a. expected value b. expected opportunity loss c. expected value of perfect information d. none of the above Answer: B 22. A tabular presentation that shows the outcome for each decision alternative under the various possible states of nature is called a ___________. a. decision tree b. payoff table c. feasible region d. payback matrix Answer: B 23. The expected value and expected opportunity loss criteria result in __________. a. the same decision b. different decisions c. similar decisions d. all of the above Answer: A 24. A(n) _____________ game occurs when each player selects an optimal strategy that does not result in an equilibrium point when the minimax criterion is used. a. optimal strategy b. pure strategy c. mixed strategy d. best strategy Answer: C

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