Chapter 12

1. Chapter 12 Strategy and Game Theory © 2004 Thomson Learning/South-Western

2. Basic Concepts • Any situation in which individuals must make strategic choices and in which the final outcome will depend on what each person chooses to do can be viewed as a game. • Game theory models seek to portray complex strategic situations in a highly simplified setting.

3. Basic Concepts • All games have three basic elements: • Players • Strategies • Payoffs • Players can make binding agreements in cooperative games, but can not in noncooperative games, which are studied in this chapter.

4. Players • A player is a decision maker and can be anything from individuals to entire nations. • Players have the ability to choose among a set of possible actions. • Games are often characterized by the fixed number of players. • Generally, the specific identity of a play is not important to the game.

5. Strategies • A strategy is a course of action available to a player. • Strategies may be simple or complex. • In noncooperative games each player is uncertain about what the other will do since players can not reach agreements among themselves.

6. Payoffs • Payoffs are the final returns to the players at the conclusion of the game. • Payoffs are usually measure in utility although sometimes measure monetarily. • In general, players are able to rank the payoffs from most preferred to least preferred. • Players seek the highest payoff available.

7. Equilibrium Concepts • In the theory of markets an equilibrium occurred when all parties to the market had no incentive to change his or her behavior. • When strategies are chosen, an equilibrium would also provide no incentives for the players to alter their behavior further. • The most frequently used equilibrium concept is a Nash equilibrium.

8. Nash Equilibrium • A Nash equilibrium is a pair of strategies (a*,b*) in a two-player game such that a* is an optimal strategy for A against b* and b* is an optimal strategy for B against A*. • Players can not benefit from knowing the equilibrium strategy of their opponents. • Not every game has a Nash equilibrium, and some games may have several.

9. An Illustrative Advertising Game • Two firms (A and B) must decide how much to spend on advertising • Each firm may adopt either a higher (H) budget or a low (L) budget. • The game is shown in extensive (tree) form in Figure 12.1

10. An Illustrative Advertising Game • A makes the first move by choosing either H or L at the first decision “node.” • Next, B chooses either H or L, but the large oval surrounding B’s two decision nodes indicates that B does not know what choice A made.

11. FIGURE 12.1: The Advertising Game in Extensive Form 7,5 L B H 5,4 L A L 6,4 B H H 6,3

12. An Illustrative Advertising Game • The numbers at the end of each branch, measured in thousand or millions of dollars, are the payoffs. • For example, if A chooses H and B chooses L, profits will be 6 for firm A and 4 for firm B. • The game in normal (tabular) form is shown in Table 12.1 where A’s strategies are the rows and B’s strategies are the columns.

13. Table 12.1: The Advertising Game in Normal Form

14. Dominant Strategies and Nash Equilibria • A dominant strategy is optimal regardless of the strategy adopted by an opponent. • As shown in Table 12.1 or Figure 12.1, the dominant strategy for B is L since this yields a larger payoff regardless of A’s choice. • If A chooses H, B’s choice of L yields 5, one better than if the choice of H was made. • If A chooses L, B’s choice of L yields 4 which is also one better than the choice of H.

15. Dominant Strategies and Nash Equilibria • A will recognize that B has a dominant strategy and choose the strategy which will yield the highest payoff, given B’s choice of L. • A will also choose L since the payoff of 7 is one better than the payoff from choosing H. • The strategy choice will be (A: L, B: L) with payoffs of 7 to A and 5 to B.

16. Dominant Strategies and Nash Equilibria • Since A knows B will play L, A’s best play is also L. • If B knows A will play L, B’s best play is also L. • Thus, the (A: L, B: L) strategy is a Nash equilibrium: it meets the symmetry required of the Nash criterion. • No other strategy is a Nash equilibrium.

17. Two Simple Games • Table 12.2 (a) illustrates the children’s finger game, “Rock, Scissors, Paper.” • The zero payoffs along the diagonal show that if players adopt the same strategy, no payments are made. • In other cases, the payoffs indicate a $1 payment from the loser to winner under the usual hierarchy (Rock breaks Scissors, Scissors cut Paper, Paper covers Rock).

18. TABLE 12.2 (a): Rock, Scissors, Paper--No Nash Equilibria

19. Two Simple Games • This game has no equilibrium. • Any strategy pair is unstable since it offers at least one of the players an incentive to adopt another strategy. • For example, (A: Scissors, B: Scissors) provides and incentive for either A or B to choose Rock. • Also, (A: Paper, B: Rock) encourages B to choose Scissors.

20. Two Simple Games • Table 12.2 (b) shows a game where a husband (A) and wife (B) have different preferences for a vacation (A prefers mountains, B prefers the seaside) • However, both players prefer a vacation together (where both players receive positive utility) than one spent apart (where neither players receives positive utility).

21. TABLE 12.2 (b): Battle of the Sexes--Two Nash Equilibria

22. Two Simple Games • At the strategy (A: Mountain, B: Mountain), neither player can gain by knowing the other’s strategy. • The same is true with the strategy (A: Seaside, B: Seaside). • Thus, this game has two Nash equilibria.

23. APPLICATION 12.1: Nash Equilibrium on the Beach • Applications of the Nash equilibrium concept have been used to analyze where firms choose to operate. • The concept can be used to analyze where firm’s locate geographically. • The concept can also be used to analyze where firm’s locate in the spectrum of specific types of products.

24. APPLICATION 12.1: Nash Equilibrium on the Beach • Hotelling’s Beach • Hotelling looked at the pricing of ice cream sellers along a linear beach. • If people are evenly spread over the length of the beach, he showed that each seller had an advantage selling to nearby consumers who incur lower (walking) costs. • The Nash equilibrium concept can be used to show the optimal location for each seller.

25. APPLICATION 12.1: Nash Equilibrium on the Beach • Milk Marketing in Japan • In southern Japan, four local marketing boards regulate the sale of milk. • It appears that each must take into account what the other boards are doing, since milk can be shipped between regions. • A Nash equilibrium similar to the Cournot model found prices about 30 percent above competitive levels.

26. APPLICATION 12.1: Nash Equilibrium on the Beach • Television Scheduling • Firms can also choose where to locate along the spectrum that represents consumers’ preferences for characteristics of a product. • Firms must take into account what other firms are doing, so game theory applies. • In television, viewers’ preferences are defined along two dimensions--program content and broadcast timing.

27. APPLICATION 12.1: Nash Equilibrium on the Beach • In general, the Nash equilibrium tended to focus on central locations • There is much duplication of both program types and schedule timing • This has left “room” for specialized cable channels to attract viewers with special preferences for content or viewing times. • Sometimes the equilibria tend to be stable (soap operas and sitcoms) and sometimes unstable (local news programming).

28. The Prisoner’s Dilemma • The Prisoner’s Dilemma is a game in which the optimal outcome for the players is unstable. • The name comes from the following situation. • Two people are arrested for a crime. • The district attorney has little evidence but is anxious to extract a confession.

29. The Prisoner’s Dilemma • The DA separates the suspects and tells each, “If you confess and your companion doesn’t, I can promise you a six-month sentence, whereas your companion will get ten years. If you both confess, you will each get a three year sentence.” • Each suspect knows that if neither confess, they will be tried for a lesser crime and will receive two-year sentences.

30. The Prisoner’s Dilemma • The normal form of the game is shown in Table 12.3. • The confess strategy dominates for both players so it is a Nash equilibria. • However, an agreement not to confess would reduce their prison terms by one year each. • This agreement would appear to be the rational solution.

31. TABLE 12.3: The Prisoner’s Dilemma

32. The Prisoner’s Dilemma • The “rational” solution is not stable, however, since each player has an incentive to cheat. • Hence the dilemma: • Outcomes that appear to be optimal are not stable and cheating will usually prevail.

33. Prisoner’s Dilemma Applications • Table 12.4 contains an illustration in the advertising context. • The Nash equilibria (A: H, B: H) is unstable since greater profits could be earned if they mutually agreed to low advertising. • Similar situations include airlines giving “bonus mileage” or farmers unwilling to restrict output. • The inability of cartels to enforce agreements can result in competitive like outcomes.

34. Table 12.4: An Advertising Game with a Desirable Outcome That is Unstable

35. Cooperation and Repetition • In the version of the advertising game shown in Table 12.5, the adoption of strategy H by firm A has disastrous consequences for B (-50 if L is chosen, -25 if H is chosen). • Without communication, the Nash equilibrium is (A: H, B: H) which results in profits of +15 for A and +10 for B.

36. TABLE 12.5: A Threat Game in Advertising

37. Cooperation and Repetition • However, A might threaten to use strategy H unless B plays L to increase profits by 5. • If a game is replayed many times, cooperative behavior my be fostered. • Some market are thought to be characterized by “tacit collusion” although firms never meet. • Repetition of the threat game might provide A with the opportunity to punish B for failing to choose L.

38. Many-Period Games • Figure 12.2 repeats the advertising game except that B knows which advertising spending level A has chosen. • The oral around B’s nodes has been eliminated. • B’s strategic choices now must be phrased in a way that takes the added information into account.

39. FIGURE 12.2: The Advertising Game in Sequential Form 7,5 L B H 5,4 L A L 6,4 H B H 6,3

40. Many-Period Games • The four strategies for B are shown in Table 12.6. • For example, the strategy (H, L) indicates that B chooses L if A first chooses H. • The explicit considerations of contingent strategy choices enables the exploration of equilibrium notions in dynamic games.

41. TABLE 12.6: Contingent Strategies in the Advertising Game

42. Credible Threat • The three Nash equilibria in the game shown in Table 12.6 are: • (1) A: L, B: (L, L); • (2) A: L, B: (L, H); and • (3) A: H, B: (H,L). • Pairs (2) and (3) are implausible, however, because they incorporate a noncredible threat that firm B would never carry out.

43. Credible Threat • Consider, for example, A: L, B: (L, H) where B promises to play H if A plays H. • This threat is not credible (empty threats) since, if A has chosen H, B would receive profits of 3 if it chooses H but profits of 4 if it chooses L. • By eliminating strategies that involve noncredible threats, A can conclude that, as before, B would always play L.

44. Credible Threat • The equilibrium A: L, B: (L, L) is the only one that does not involve noncredible threats. • A perfect equilibrium is a Nash equilibrium in which the strategy choices of each player avoid noncredible threats. • That is, no strategy in such an equilibrium requires a player to carry out an action that would not be in its interest at the time.

45. Models of Pricing Behavior: The Bertrand Equilibrium • Assume two firms (A and B) each producing a homogeneous good at constant marginal cost, c. • The demand is such that all sales go to the firm with the lowest price, and sales are evenly split if PA = PB. • All prices where profits are nonnegative, (P  c) are in each firm’s pricing strategy.

46. The Bertrand Equilibrium • The only Nash equilibrium is PA = PB = c. • Even with only two firms, the Nash equilibrium is the competitive equilibrium where price equals marginal cost. • To see why, suppose A chooses PA > c. • B can choose PB < PA and capture the market. • But, A would have an incentive to set PA < PB. • This would only stop when PA = PB = c.

47. Two-Stage Price Games and Cournot Equilibrium • If firms do not have equal costs or they do not produce goods that are perfect substitutes, the competitive equilibrium is not obtained. • Assume that each firm first choose a certain capacity output level for which marginal costs are constant up to that level and infinite thereafter.

48. Two-Stage Price Games and Cournot Equilibrium • A two-stage game where the firms choose capacity first and then price is formally identical to the Cournot analysis. • The quantities chosen in the Cournot equilibrium represent a Nash equilibrium, and the only price that can prevail is that for which total quantity demanded equals the combined capacities of the two firms.

49. Two-Stage Price Games and Cournot Equilibrium • Suppose Cournot capacities are given by • A situation in which is not a Nash equilibrium since total quantity demanded exceeds capacity. • Firm A could increase profits by slightly raising price and still selling its total output.

50. Two-Stage Price Games and Cournot Equilibrium • is not a Nash equilibrium because at least one firm is selling less than its capacity. • The only Nash equilibrium is which is indistinguishable from the Cournot result. • This price will be less than the monopoly price, but will exceed marginal cost.