15.1 Two-Person Total-Conflict Games: Pure Strategies

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6. If a player has two strategies in a game and one is dominated, must the other strategy be dominant? Why?

7. If a player has more than two strategies in a game and one is dominated, it is possible that no single strategy is dominant. Fill in the payoffs in the following two-person zero-sum game in which Player I can choose between Top, Middle, and Bottom and Player II can choose between Left and Right so that both Top and Middle dominate Bottom, but neither Top nor Middle is dominant.

15.2 Two-Person Total-Conflict Games: Mixed Strategies

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10. It is possible for a player’s pure strategy to be dominated by a mixed strategy. For the following game, Player I’s strategy of playing Top and Middle, each with probability 1/2, dominates the pure strategy of playing Bottom. Calculate the expected value for each of these strategies against Player II playing the pure strategies Left and Right. Use your answer to explain why Bottom is dominated by the mixed strategy.

11. In the following game of batter-versus- pitcher in baseball, the batter’s batting averages are given in the game matrix. The batter tries to maximize his batting average, while the pitcher tries to minimize the batter’s batting average; this is a zero-sum game. The pitcher decides between throwing a fastball or a knuckleball. When the batter is in the batter’s box, he guesses which pitch is coming (either a fastball or a knuckleball). If he guesses correctly, then his batting average goes up. For example, it is easier for him to hit a fastball when he correctly guesses that one is coming. However, if he guesses incorrectly, then his batting average goes down. If the pitcher and the batter follow their mixed-equilibrium strategies, what will the batter’s batting average be?

12. A businessperson has the choice of either not cheating on his income tax or cheating and making $1000 if not audited. If caught cheating, he will pay a fine of $2000 in addition to the $1000 he owes. He feels good if he does not cheat and is not audited (worth $100). If he does not cheat and is audited, he evaluates this outcome as -$100 (for the lost day). Viewing the game as a two-person zero-sum game between the businessperson and the tax agency, what are the optimal mixed strategies for each player and the value of the game?

13. In American football, if a team tries for a 2-point conversion, then the ball is placed on the 2-yard line and the team has one play to try to get into the end zone. The quarterback can decide to run the ball or pass it. Similarly, the other team can commit itself to defend more heavily against a run or a pass. This can be modeled as a zero-sum matrix game, in which the payoffs are the probabilities of getting in the end zone and earning the 2 points. Find the value of the game.

14. Refer to Exercise 13 to answer the following two questions.

15. You have the choice of either parking illegally on the street or parking in the lot and paying $16. Parking illegally is free if the police officer is not patrolling, but you receive a $40 parking ticket if she is. However, you are peeved when you pay to park in the lot on days when the officer does not patrol, and you are willing to assess this outcome as costing $32 ($16 for parking plus $16 for your time, inconvenience, and grief). It seems reasonable to assume that the police officer ranks her preferences in the order (1) giving you a ticket, (2) not patrolling with you parked in the lot, (3) patrolling with you in the lot, and (4) not patrolling with you parked illegally.

16. Describe how a pure strategy for a player in a matrix game can be considered as merely a special case of a mixed strategy.

17. Describe in detail one pure strategy for the player who moves first in the game of tic-tac-toe. This strategy must tell how to respond to all possible moves of the other player. (Hint: You may wish to make use of the symmetry in the grid in this game; that is, there are one “center” box, four “corner” boxes, and four “side” boxes.)

Is your strategy optimal in the sense that it will guarantee the first player a tie (and possibly a win) in the game?

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18. In the Matching Pennies example, consider the case where Player I favors heads over tails . For example, assume that Player I plays three-fourths of the time and only one-fourth of the time—a nonoptimal mixed strategy. What should Player II do if he knows this?

19. Assume in the matching game of Example 5 that Player II is using the nonoptimal mixed strategy ; that is, he is playing and with the same frequency. What should Player I do in this case if she knows this?

20. You plan to manufacture a new product for sale next year, and you can decide to make either a small quantity, in anticipation of a poor economy and few sales, or a large quantity, hoping for brisk sales. Your expected profits are indicated in the following table.

If you want to avoid risk and believe that the economy is playing an optimal mixed strategy against you in a two-person zero-sum game, then what is your optimal mixed strategy and the resulting expected value? Discuss some alternative ways to go about making your decision.

21. Consider the following poker game with two players, I and II. Each antes $1. Each player is dealt either a high card or a low card , with probability 1/2. Player I then folds or bets $1. If Player I bets, then Player II either folds, calls, or raises $1. Finally, if II raises, Player I either folds or calls.

Most choices by the players are rather obvious, at least to anyone who has played poker: If either player holds , that player always bets or raises if he or she gets the choice. The question remains of how often one should bluff—that is, continue to play (by calling or raising) while holding a low card in the hope that one’s opponent also holds a low card.

This poker game can be represented by the following matrix game, in which the payoffs are the expected winnings for Player I (depending on the random deal) and the dominated strategies have been eliminated.

22. Considering the scenario described in Exercise 21, should either of the players ever bluff?

23. If Person threatens Person but does not intend to carry out his or her threat, we say he or she is bluffing. When is such a bluff rational?

24. (a) Describe in detail one pure strategy for the player who moves second in the game of tic-tac-toe.

(b) Is your strategy in part (a) optimal in the sense that it will guarantee the second player a tie (and possibly a win) in the game?

25. On an overcast morning, deciding whether to carry your umbrella can be viewed as a game between yourself and nature as follows:

Let’s assume that you are willing to assign the following numerical payoffs to these outcomes, and that you are also willing to make decisions on the basis of expected values (i.e., average payoffs):

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What strategy should you select if you wish to minimize your maximum possible regret?

15.3 Partial-Conflict Games

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27. Battle of the sexes:

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31. In Exercise 26, players maximize their possible gains by choosing their first strategies, but they minimize their possible losses by choosing their second strategies. Which strategy would you choose, and why?

32. Assume that two countries in an arms race assign points to all their own weapons so that the total for each is 1000. Each side can then designate weapons of the other side, totaling 100 points, that must be eliminated in the next year, thereby effecting a 10% reduction. Would these countries have any reason to lie about how they value their own weapons? Is this procedure practical as an arms-reduction scheme?

15.4 Mechanism Design and Larger Games

33. In Example 9 (page 644), Anneliese, Binh, and Charlie valued a stamp at $300, $200, and $100, respectively. Although it is a weakly dominant strategy for each player to bid his or her true valuation of the stamp under a Vickrey auction, there are many other equilibrium bids besides ($300, $200, $100). Determine which, if any, of the following bids are equilibrium bids for the stamp: ($250, $50, $50), ($150, $100, $50), and ($400, $299, $1).

34. There are other auction formats besides the second- price, or Vickrey, auction considered in Example 9. In a first-price auction, the highest bidder gets the item and pays his or her bid for the item. Explain why it would not be an equilibrium for Anneliese, Binh, and Charlie to bid their true valuations of $300, $200, and $100, respectively, for the stamp if a first-price auction were used in Example 9. (In a first-price auction, every player submits a bid at the same time and the person who bids the highest gets the item for his or her bid.)

35. Players Odd and Even play Low Person Wins, the rules of which are as follows:

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What is the Nash equilibrium of this game, based on the successive elimination of dominated strategies? Is there another Nash equilibrium? What are the similarities and differences between this game and the Prisoners’ Dilemma?

36. Return to the sequential model of Chicken from Skills Check 32. Use a game tree to model the situation in which Driver 2 goes first. Does the analysis change from your analysis in Skills Check 32?

37. In a sequential duel, why will the first player to act in a truel shoot in the air (if this option is allowed by the rules)? Is this choice optimal if a second player should succeed in firing in the air at the same time?

38. In a sequential truel with no firing in the air allowed, suppose , who hates , goes first; , who hates , goes second; , who hates , goes third. (If a player fires, he will shoot only his antagonist—the player he hates.)

Answer these questions (1) if each player can take only one turn (if alive), and (2) if, after one round, the game continues (if there is more than one player alive) and each player can take more than one turn.

15.5 Using Game Theory

39. Find a two-person zero-sum game with a saddlepoint in which the successive elimination of dominated strategies does not lead to the saddlepoint. (Hint: Restrict yourself to games. Can you construct such a game that has a saddlepoint but for which no strategies are dominated?)

40. Consider a free-rider problem in which two bus riders can choose to pay their fares or not. As long as at least one player pays his or her fare, then the bus continues to operate—though the rider who pays will have to pay twice as much for a ride. Each player ranks the outcomes from best to worst as follows:

This game is modeled by the following matrix. Use it to analyze the optimal behavior. Does this game share payoffs with any other game considered in this chapter?

Chapter Review

41. Every zero-sum game may be written using ordinal payoffs as used for the partial-conflict games analyzed in this chapter. Return to Example 1 (page 623) and change the payoffs to ordinal payoffs—remember to include payoffs for both Mark and Lisa. Do Mark’s and Lisa’s decisions to choose the Suburban hospital and the 8 A.M. to 4 P.M. shift remain an equilibrium? Explain, using ordinal payoffs.

42. Would a similar approach to Exercise 41 work for the kicker-goalie game of Example 3 (page 629)? Explain.

43. Use a game tree to model the Prisoners’ Dilemma in Table 15.9. Does your analysis depend on which country chooses to act first? Explain.