21 Strategy and Game Theory

Transcription

1 CHAPTER 21 Strategy and Game Theory BASIC CONCEPTS In Chapter 20 we examined some of the problems that arise in modeling markets in which there are only a few firms. Perhaps the most difficult of these problems concerned questions of strategy-that is, with few firms each firm must, to some extent, be concerned with what its rivals will do. As we saw, making profitmaximizing decisions requires that each firm make some conjectures about its competitors' behavior. Under perfect competition such strategic thinking was unnecessary because the prevailing market price was assumed to convey all the external information that was relevant to the firm. With relatively few firms the situation may be more complicated since price-taking behavior is less likely. One of the primary tools that economists use to study strategic choices is game theory. This subject was originally developed during the 1920s and grew rapidly during World War II in response to the need to develop formal ways of thinking about military strategy.! Today the theory has applicability to problems as diverse as the development of optimal strategies for five-card stud poker and the analysis of antimissile defenses. In this chapter we will provide a brief introduction to game theory with a primary focus on its use in explaining pricing and entry behavior in oligopolistic markets. A few other applications will also be mentioned. Game theory models seek to portray complex strategic situations in a highly simplified and stylized setting. Much like the previous models in this book, game theory models abstract from most of the personal and institutional details of a problem in order to arrive at a representation of the situation that is mathematically tractable. This ability to get to the "heart" of the problem is the greatest strength of this type of modeling. 1 Much of the pioneering work in game theory was done by the mathematician John von Neumann. The main reference is J. von Neumann and O. Morgenstern, The Theory of Games and Economic Behal'ior (Princeton, N.].: Princeton University Press, 1944). 619

2 620 PART VI Models of Imperfect Competition 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. All games have three basic elements: (1) players; (2) strategies; and (3) payoffs. Games may be cooperative, in which players can make binding agreements, or noncooperative, where such agreements are not possible. Here we will be concerned primarily with noncooperative games. The basic elements listed below are included in such games. Players Strategies Payoffs Each decision-maker in a game is called a "player." These players may be individuals (as in poker games), firms (as in oligopoly markets), or entire nations (as in military conflicts). All players are characterized as having the ability to choose from among a set of possible actions they might take. 2 Usually, the number of players is fixed throughout the "play" of a game, and games are often characterized by the number of players (that is, two-player, three-player, or n-player games). In this chapter we will primarily study two-player games and will denote these players (usually firms) by A and B. One of the important assumptions usually made in game theory (as in most of economics) is that the specific identity of the players is irrelevant. There are no "good guys" or "bad guys" in a game, and players are not assumed to have any special abilities or shortcomings. Each player is simply assumed to choose the course of action that yields the most favorable outcome. Each course of action open to a player in a game is called a "strategy." Depending on the game being examined, a strategy may be a very simple action (take another card in blackjack) or a very complex one (build a laser-based antimissile defense), but each strategy is assumed to be a well-defined, specific course of action. J Usually, the number of strategies available to each player will be finite; many aspects of game theory can be illustrated for situations in which each player has only two strategies available. 4 In noncooperative games, players cannot reach binding agreements with each other about what strategies they will play-each player is uncertain about what the other will do. The final returns to the players of a game at its conclusion are called "payoffs." Payoffs are usually measured in levels of utility obtained by the players although frequently monetary payoffs (say, profits for firms) are used instead. In general, 'Sometimes one of the players in a game is taken to be "nature." For this player, actions are not "chosen" but rather occur with certain possibilities. For example, the weather may affect the out comes of a game, but it is not "chosen" by natute. Rather, particular weather outcomes arc assumed to occur with various probabilities. Games against nature can be analyzed using the methods developed in Chapter 10. \ In games involving a sequence of actions (for example, most board games such as chess), a speci fication of strategies may involve several decision points (each move in chess). Assuming perfect knowledge of how the game is played, such complex patterns can often be expressed as choices among a large but finite set of pure strategies, each of which specifies a complete course of action until the game is completed. See our discussion of "extensive" and "normal" forms and R. D. Luee and H. Raiffa, Games and Decisions (New York: John Wiley & Sons, 1957), chap. 3. Players may also adopt "mixed" strategies by choosmg to play their pure strategies randomly (say, by flipping a coin). We will analyze this possibility only briefly in footnotes.

3 CHAPTER 21 Strategy and Game Theory 621 it is assumed that players can rank the payoffs of a game ordinally from most preferred to least preferred and will seek the highest ranked payoff attainable. Payoffs incorporate all aspects associated with outcomes of a game; these include both explicit monetary payoffs and implicit feelings by the players about the outcomes such as whether they are embarrassed or gain self-esteem. Players prefer payoffs that offer more utility to those that offer less. EQUILIBRIUM CONCEPTS IN GAME THEORY In our examination of the theory of markets, we developed the concept of equilibrium in which both suppliers and demanders were content with the market outcome. Given the equilibrium price and quantity, no market participant has an incentive to change his or her behavior. The question therefore arises whether there are similar equilibrium concepts in game theory models. Are there strategic choices that, once made, provide no incentives for the players to alter their behavior further? Do these equilibria then offer believable explanations of market outcomes? Although there are several ways to formalize equilibrium concepts in games, the most frequently used approach was originally proposed by Coumot (see Chapter 20) in the nineteenth century and generalized in the early 1950s by J. Nash. s Under Nash's procedure a pair of strategies, say, (a*, b*), is defined to be an equilibrium if a* represents player A's best strategy when B plays b" and b * represents B's best strategy when A plays a". Even if one of the players reveals the (equilibrium) strategy he or she will use, the other player cannot benefit from knowing this. For nonequilibrium strategies, as we shall see, this is not the case. If one player knows what the other's strategy will be, he or she can often benefit from that knowledge and, in the process, take actions that reduce the payoff received by the player who has revealed his or her strategy. Not every game has a Nash equilibrium set of strategies. And, in some cases, a game may have multiple equilibria, some of which are more plausible than others. Some Nash equilibria may not be especially desirable for the players in a game. And, in some cases, other equilibrium concepts may be more reasonable than those proposed by Nash. Hence, there is a rather complex relationship between game theory equilibria and more traditional market equilibrium concepts. Still, we have an initial working definition of equilibrium with which to start our study of game theory: DEFINITION Nash Equilibrium Strategies A pair of strategies (a", b *) represents an equilibrium solution to a two-player game if a* is an optimal strategy for A against b* and b* is an optimal strategy for B against a*.6 'John Nash, "Equilibrium Points in n-person Games," Proceedings of the Natiolfal Academy of Sciences 36 (1950): Although this definition is stated only for two-player games, the generalization to If-persons is straightforward, though notationally cumbersome.

4 622 PART VI Models of Imperfect Competition Figure 21.1 The Advertising Game in Extensive Form In this game A chooses a low (L) or a high (H) advertising budget, then 8 makes a similar choice. The oval surrounding 8's nodes indicates that they share the same (lack of) information-8 does not know what strategy A has chosen. Payoffs (with A's first) are listed at the right. 7,5 5,4 A 6.4 6,3 AN ILLUSTRATIVE ADVERTISING GAME The Game in Extensive Form As a way of illustrating the game-theoretic approach to strategic modeling, we will examine a simple example in which two firms (A and B) must decide how much to spend on advertising. Each firm may adopt either a "high" (H) budget or a "low" (L) budget, and we wish to examine possible equilibrium choices in this situation. It should be stressed at the outset that this game is not especially realistic-it is intended for pedagogic purposes only. Figure 21.1 illustrates the specific details of the advertising game. In this game "tree," the action proceeds from left to right, and each "node" represents a decision point for the firm indicated there. The first move in this game belongs to firm A: it must choose its level of advertising expenditures, H or L. Because firm B's decisions occur to the right of A's, the tree indicates that firm B makes its decision after firm A. At this stage, two versions of the game are possible depending on whether B is assumed to know what choice A has made. First we will look at the case where B does not have this information. The larger oval surrounding B's two decision nodes indicates that both nodes share the same (lack of) information. Firm B must choose H or L without knowing what A has done. Later we will examine the case where B does have this information.

5 I CHAPTER 21 Strategy and Game Theory 623 Table 21.1 The Advertising Game in Normal Form B's Strategies L H A's Strategies L H The numbers at the end of each tree branch indicate payoffs, here measured in (thousands or millions of) dollars of profits. Each pair of payoffs lists A's profits first. For example, the payoffs in Figure 21.1 show that if firm A chooses H and firm B chooses L, profits will be 6 for A and 4 for B. Other payoffs are interpreted similarly. The Game in Normal Form Dominant and Nash Equilibria Although the game tree in Figure 21.1 offers a useful visual presentation of the complete structure of a game, sometimes it is more convenient to describe games in tabular (or "normal") form. Table 21.1 provides such a presentation for the advertising game. In the table, firm A's strategies (H or L) are shown at the left, and B's strategies are shown across the top. Payoffs (again with firm A's coming first) corresponding to the various strategic choices are shown in the body of the table. The reader should check that Figure 21.1 and Table 21.1 convey the same information about this game. Table 21.1 makes clear that adoption of a low advertising budget is a dominant strategy for firm B. No matter what A does, the L strategy provides greater profits to firm B than does the H strategy. Of course, since the structure of the game is assumed to be known to both players, firm A will recognize that B has such a dominant strategy and will opt for the strategy that does the best against it-that is, firm A will also choose L. Considerations of strategy dominance, therefore, suggest that the A:L, B:L strategy choice will be made and that the resulting payoffs will be 7 (to A) and 5 (to B). The A:L, B:L strategy choice also obeys the Nash criterion for equilibrium. If A knows that B will play L, its best choice is L. Similarly, if B knows A will play L, its best choice is also L (indeed, since L is a dominant strategy for B, this is its best choice no matter what A does). The A:L, B:L choice, therefore, meets the symmetry required by the Nash criterion. To see why the other strategy pairs in Table 21.1 do not meet the Nash criterion, let us consider them one at a time. If the players announce A:H, B:L, this provides A with a chance to better its position-if firm A knows B will opt for L, it can make greater profits by choosing L. The choice A:H, B:L is therefore not a Nash equilibrium. Neither of the two outcomes in which B chooses H meets the Nash criterion either since, as we have already pointed out, no matter what A does, B can improve its profits by choosing L instead. Since L

6 624 PART VI Models of Imperfect Competition 1 strictly dominates H for firm B, no outcome in which B plays H can be a Nash equilibrium.. Nature of Nash Equilibria Although the advertising game illustrated in Figure 21.1 contains a single Nash equilibrium, that is not a general property of all two-person games. Example 21.1 illustrates a simple game ("Rock, Scissors, Paper") in which no Nash equilibrium exists and. another game ("Battle of the Sexes") that contains two Nash equilibria. Hence the Nash approach may not always identify a well-defined solution to a game situation. 7 The Nash approach also may not yield an especially desirable equilibrium nor one that might be expected to persist if a game were played repeatedly. Some of these issues are illustrated by the Prisoner's Dilemma game that we take up in the next section. Example 21.1 Sample Nash Equilibria Table 21.2 illustrates two familiar games that reflect differing possibilities for Nash equilibria. Part (a) of the table depicts 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 loser to winner under the usual hierarchy (Rock breaks Scissors, Scissors cut Paper, Paper covers Rock). As anyone who has played this game knows, there is no equilibrium. Any strategy pair is unstable because it offers at least one of the players an incentive to adopt another strategy. For example, (A: Scissors, B: Scissors) provides an incentive for either A or B to choose Rock. Similarly (A: Paper, B: Rock) obviously encourages B to choose Scissors. The irregular cycling behavior exhibited in the play of this game clearly indicates the absence of a Nash equilibrium. Battle of the Sexes In the "Battle of the Sexes" game, a husband (A) and wife (B) are planning a vacation. A prefers mountain locations, B prefers the seaside. Both players prefer a vacation spent together to one spent apart. The payoffs in part (b) of Table 21.2 reflect these preferences. Here both of the joint vacations represent Nash equilibria. With (A: Mountain, B: Mountain) neither player can gain by taking advantage of knowing the other's strategy. Similar comments apply to (A: Seaside, B: Seaside). Hence this is a game with two Nash equilibria. Query: Are any of the strategies in either of these games dominant? Why aren't separate vacations Nash equilibria in the Battle of the Sexes? 7Nash equilibria can be shown always to exist in certain types of games. For example, in zero-sum games (where the payoffs sum to zero or any other constant), a Nash equilibrium always exists in mixed strategies (strategies that consist of various pure strategies played with certain probabilities). See, for example, Luce and Raiffa, Games and Decisions, appendices 2-5.

7 1 CHAPTER 21 Strategy and Game Theory 625 I Table 21.2 Two Simple Games (al Rock, Scissors, Paper-No Nash Equilibria B's Strategies Rock Scissors Paper Rock o o 1 -] 1 A's Strategies Scissors 1 o Paper 1-1 o o (b) Battle of the Sexes-Two Nash Equilibria B's Strategies Mountain Seaside A's Strategies Mountain Seaside THE PRISONER'S DILEMMA Applications The Prisoner's Dilemma game was first discussed by A. W. Tucker in the The title stems from the following game situation. Two people are arrested for a crime. The district attorney has little evidence in the case and is anxious to extract a confession. She separates the suspects and tells each, "If you confess and your companion doesn't, I can promise you a reduced (six-month) sentence, whereas on the basis of your confession, your companion will get 10 years. If you both confess, you will each get a three-year sentence." Each suspect also knows that if neither of them confesses, the lack of evidence will cause them to be tried for a lesser crime for which they will receive two-year sentences. The normal form payoff matrix for this situation is illustrated in Table The "confess" strategy dominates for both A and B. Hence these strategies constitute a Nash equilibrium and the district attorney's ploy looks successful. However, an agreement by both not to confess would reduce their prison terms from three to two years. This "rational" solution is not stable, and each prisoner has an incentive to squeal on his or her colleague. This then is the dilemma-outcomes that appear to be optimal are not stable. Prisoner's Dilemma-type problems may arise in many real-world market situations. Table 21.4 contains an illustration of the dilemma in a different advertising game. Here the twin L strategies are most profitable, but this choice is unstable. This situation resembles the situation discussed in Chapter 20 where we described why some advertising might be regarded as "defensive" in the sense that a mutual agreement to reduce expenditures would be profitable to both parties. Such an agreement in the situation of Table 21.4 would be unstable. Either firm could increase its profits even further by cheating on the agreement. Similar situations arise in the tendency for airlines to give "bonus mileage" (there would be larger profits if all firms stopped offering free trips, but such a solution is unstable) and in the instability of farmers' agreements to restrict

8 626 PART VI Models of Imperiect Competition Table 21.3 The Prisoner's Dilemma Confess B Not Confess A Confess Not Confess A: 3 years B: 3 years A: 10 years B: 6 months A: 6 months B: 10 years A: 2 years B: 2 years Table 21.4 An Advertising Game with a Desirable Outcome That Is Unstable B's Strategies L H A's Strategies L 3,10 H 5,5 output (it is just too tempting for an individual farmer to try to sell more milk). As these examples show, the difficulty of enforcing agreements may be very detrimental to the profits of an industry. Example 21.2 The Stackelberg Equilibrium In Example 20.2 we developed a numerical illustration of a duopoly market in which the outcomes depended on the strategic assumptions made by the competitors. Under the Stackelberg version of that model, each firm had two possible strategies-to be a "leader" (produce q 60) or a "follower" (produce q 30). Results from employing these strategies can be viewed as a 2 x 2 game, the payoff matrix for which is shown in Table Adoption of the leader-leader strategy is disastrous for the two firms, resulting in zero profits for each. A follower-follower strategy (the Cournot solution) is much more profitable for both firms as in the Prisoner's Dilemma, but this is unstable because it provides each firm with an incentive to cheat and grab the leader position. The game is not strictly a Prisoner's Dilemma game either, however, since the leader-leader solution is not a Nash equilibrium-if firm A knows that firm B will be a leader, it might as well be a follower and vice versa. Each of the leader-follower strategy pairs is an equilibrium, but, as before, the formal problem offers no guidance on which pair will be chosen. Presumably, it will

9 CHAPTER 21 Strategy and Game Theory 627 Table 21.5 Payoff Matrix for the Stackelberg Model B's Strategies Leader Follower (qn :::: 60) (qn = 30) A's Strategies Leader ITI :O A: $1,800 (q. :::: 60) B:O B: $ 900 Follower A: $ 900 A: $1,600 (q. :::: 30) B: $1,800 B: $1,600 depend on outside factors such as the history of the industry or the personalities of the firms' managers. Query: How do you think the Stackelberg game would evolve if it were played many times (see the discussion that follows)? Cooperation and Repetition Communication and cooperation between participants can be an important part of a game. In the Prisoner's Dilemma, for example, the inability to reach a cooperative agreement not to confess leads to a second-best outcome. If the parties could cooperate, they might do better. Similarly, in the Stackelberg game, an agreement to operate as a cartel would raise profits above any of those listed in Table As an example of how communications alone can affect the outcome of a game, consider the payoff matrix shown in Table In this version of the advertising game, the adoption of strategy H by firm A has disastrous consequences for firm B, causing a loss of 50 when B plays Land 25 when His chosen. Without any communication A would choose L (this dominates H) and B would choose H (which dominates L). Firm A would therefore end up with + 15 and B with However, by recognizing the potency of strategy H, A may be able to improve its situation. It can threaten to play H unless B plays L. If this threat is indeed credible (a topic we take up later), A can increase its profits from 15 to 20. If games are to be played many times, cooperative behavior may be fostered. In the Prisoner's Dilemma game, for example, it seems doubtful that the district attorney's ploy would work if it were used repeatedly. In this case, prisoners might hear about the method and act accordingly in their interrogations. In other contexts, firms that are continually exasperated by their inability to obtain favorable market outcomes may come to perceive the kind of cooperative behavior that is necessary. In antitrust theory, for example, some markets are believed to be characterized by "tacit collusion" among the participants. Firms act as a

10 628 PART VI Models of Imperfect Competition Table 21.6 A Threat Game in Advertising A's Strategies B's Strategies L H L 20,5 15,10 H 10, -50 5, -25 cartel even though they never meet to plot a common strategy. We will explore the formal aspects of this problem later. Finally, repetition of the threat game (Table 21.6) offers player A the opportunity to take reprisals on B for failing to choose L. Imposing severe losses on B for "improper" behavior may be far more persuasive than simply making abstract threats. ATWO-PERIOD ADVERTISING GAME These observations suggest that repeated games, perhaps with some types of communication or cooperation, may involve complex scenarios that better reflect real-world markets than do the simple single-period models we have studied so far. In order to illustrate the formal aspects of such games, we will return to a reformulated version of the advertising game presented at the beginning of this chapter. We present the game first in extensive form in order to understand its temporal aspects. Figure 21.2 repeats that game, but now we assume that firm B knows which advertising spending level A has chosen. In graphical terms, the oval around B's nodes has been eliminated in Figure 21.2 to indicate this additional information. B's strategic choices now must be phrased in a way that takes the information it has into account. In Table 21.7 we indicate such an extended delineation of strategies. In all, there are four such strategies covering the possible informational contingencies. Each strategy is stated as a pair of actions indicating what B will do depending on its information. The strategy (L, L) indicates that B chooses L if A chooses L (its first strategy) and L also if A chooses H (its second strategy). Similarly (H, L) indicates that B chooses H if A chooses Land B chooses L if A chooses H. Although this table conveys little more than did the previous normal form for the advertising game (Table 21.1), explicit consideration of contingent strategy choices does enable us to explore equilibrium notions for dynamic games in a simplified setting. There are three Nash equilibria in this game: (1) A:L, B:L, L; (2) A:L, B:L, H; and (3) A:H, B:H, L. Each of these strategy pairs meets the criterion of being optimal for each player given the strategy of the other. Pairs (2) and (3) are implausible, however, because they incorporate a noncredible threat that firm B would not carry out if it were in a position to do so. Consider, for example, the pair A:L, B:L, H. Under this choice B promises to play H if A plays H. A glance at Figure 21.2 shows that this threat is not credible. If B were presented with the fact of A having chosen H, it will make profits of 3 if it chooses H, but 4 if it chooses L. The threat implicit in the L, H strategy is therefore not credible.

11 CHAPTER 21 Strategy and Game Theory 629 Figure 21.2 The Advertising Game in Sequential Form In this form of the advertising game, firm B knows firm A's advertising choice. Strategies for B must be phrased taking this information into account. (See Table 21.7.) 7,5 5.4 A 6,4 6,3 Table 21.7 Contingent Strategies in the Advertising Game B's Strategies L, L L,H H,L H,H A's Strategies L 7,5 7,5 5,4 5,4 H 6,4 6,3 6,4 6,3 Even though B's strategy L, H is one component of a Nash equilibrium, firm A should be able to infer the noncredibility of the threat implicit in it. By eliminating strategies that involve noncredible threats, A can conclude that B would never play L, H or H, L.8 Proceeding in this way, the advertising game is reduced to the payoff matrix originally shown in Table 21.1 and, as we discussed previously, in that case L, L (always playing L) is a dominant strategy for B. Firm A can recognize this and will opt for strategy L. The Nash equilib 'The process of eliminating strategies involving noncredible threars is termed "backward induction." This method of solving games by "folding back the tree" was developed by H. Kuhn. See "Extensive Games and the Problem of Information," in H. Kuhn and A. Tucker, eds., Contributions to the Theory of Games (Princeton, N.].: Princeton University Press, 1953), pp

12 630 PART VI Models of Imperfect Competition rium A:L, B: L, L has therefore been shown to be the only one of the three in Table 21.7 that does not involve noncredible threats. Such an equilibrium is termed a "perfect equilibrium," which we define more formally as follows: DEFINITION Perfect Equilibrium A Nash equilibrium in which the strategy choices of each player do not involve 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. 9 By using the concepts of strategic dominance, Nash equilibrium, and perfect equilibrium, we are now in a position to examine a few game-theoretic models of firm behavior. MODELS OF PRICING BEHAVIOR The Bertrand Equilibrium We begin by illustrating some of the insights that game theory can provide to the analysis of pricing. As in Chapter 20, most of the interesting results can be shown for the duopoly case. Later in the chapter we briefly discuss some complications involved in extending game theory models to games involving many players. Suppose there are two firms (A and B) each producing a homogeneous good at constant marginal cost, c. The demand for the good is such that all sales go to the firm with the lowest price and that sales are split evenly if P A = P B The available pricing strategies here consist of all prices greater than or equal to c~no firm would choose to operate at a loss by choosing a price less than c. In this case the only Nash equilibrium is PA P B c. That is, the Nash equilibrium is the competitive solution even though there are only two firms. To see why, suppose firm A chooses a price greater than c. The profit-maximizing response for firm 8 is to choose a price slightly less than PA and corner the entire market. But 8's price, if it exceeds c, still cannot be a Nash equilibrium since it provides A with further incentives for price cutting. Only by choosing P A P B c will the two firms in this market have achieved a Nash equilibrium. This pricing strategy is sometimes referred to as a "Bertrand equilibrium" after the French economist who discovered it.lo 9 An alternative definition of perfection focuses on the "subgames" implicit in any extensive game. A "subgame" is a game that begins at one decision node and includes all future actions stemming from decisions at this node. For a Nash equilibrium choice of strategies to be a subgame perfect equilibrium, the strategies specified must constitute a Nash equilibrium in each subgame encountered during the play. In Figure 21.2 the Nash equilibrium A:L, 8:1-, L is a perfect equilibrium because once the game reaches 8's decision node, the choice B:L is a Nash equilibrium. The Nash equilibrium A:L, B:L, 11 is not a perfect equilibrium because the choice B:l1 is not a Nash equilibrium for the subgame starting at 8's decision node after A plays H. See R. Selten, "Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games," International Journal of Game Theory (March 1975): In this article Selten proposes another defimtion of a "perfect" equilibrium as a Nash equilibrium that is robust to errors made by the players. Here we adopt the earlier notion of (subgame) perfection. IOJ. Bertrand, "Theorie Mathematique de la Richesse Socia Ie," Journal de Savants (1883):

13 1 CHAPTER 21 Strategy and Game Theory 631 Two-Stage Price Games The simplicity and definiteness of the Bertrand result depend crucially on the assumptions underlying the model. If firms do not have equal costs (see problem 21.3) or if the goods produced by the two firms are not perfect substitutes, the competitive result no longer holds. Other duopoly models that depart from the Bertrand result treat price competition as only the final stage of a two-stage game in which the first stage involves various types of entry or investment considerations for the firms. In Example 20.1 we examined Cournot's example of a natural spring duopoly in which each spring owner chose how much water to supply. In the present context we might assume that each firm in a duopoly must choose a certain capacity output level for which marginal costs are constant up to that level and infinite thereafter. It seems clear that a two-stage game in which 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 since each firm correctly perceives what the other's output will be (see Equations and 20.17). Once these capacity decisions are made, the only price that can prevail is that for which total quantity demanded is equal to the combined capacities of the two firms. To see why Bertrand-type price competition will result in such a solution, suppose capacities are given by qa and qb and that where D I is the inverse demand function for the good. A situation in which (21.1) (21.2) is not a Nash equilibrium. With this price, total quantity demanded exceeds qa + qb so anyone firm could increase its profits by raising price a bit and still selling qa. Similarly, (21.3) is not a Nash equilibrium since now total sales fall short of qa +. At least one firm (say, firm A) is selling less than its capacity. By cutting price slightly, firm A can increase its profits by taking all possible sales up to qa' Of course, B will respond to a loss of sales by dropping its price a bit too. Hence the only Nash equilibrium that can prevail is the Cournot result: II P. (21.4) In general, this price will fall short of the monopoly price but will exceed marginal cost (as was the case in Example 20.1).12 Results of this two-stage game are therefore indistinguishable from those arising from the Cournot model of the previous chapter. The contrast between the Bertrand and Cournot games is striking-the former predicts competitive outcomes in a duopoly situation whereas the latter "For completeness, it should also be noted that no situation in which P A,. P B can be an equilibrium since the low-price firm has an incentive to raise price and the high-price firm wishes to cut price. 12Equation also suggests another source of inefficiency in the Cournot solution-except in special cases, marginal costs will not be the same among firms. See problem 21.4.

14 632 PART VI Models of Imperfect Competition predicts monopoly-like inefficiencies. This suggests that actual behavior in duopoly markets may exhibit a wide variety of outcomes depending on the precise way in which competition occurs. The principal lesson of the two-stage Cournot game is that, even with Bertrand price competition, decisions made prior to this final stage of a game can have an important impact on market behavior. This lesson will be reflected again in some of the game theory models of entry we describe later in this chapter. Tacit Collusion Our analysis of the Prisoner's Dilemma concluded that, if the game were played several times, the participants might devise ways to adopt more cooperative strategic choices. The same query might be raised about the Bertrand gamewould repetition of this game offer some mechanism for the players to attain supracompetitive profits by pursuing a monopoly pricing policy? One possibility, discussed in Chapter 20, would be for the players to establish a cartel and explicitly set price or output targets. As we showed, such explicit agreements are subject to a number of difficulties in enforcement. Here we adopt a noncooperative approach to the collusion question by exploring models of "tacit" collusion. That is, we use game theory concepts to see whether there exist equilibrium strategies that, though not explicitly coordinated, would allow firms to achieve monopoly profits. Our initial result from the Bertrand model poses a significant stumbling block to achieving tacit collusion. Since the single-period Nash equilibrium in this model results in P A P B = c, we need to ask whether this situation would change if the game were repeated during many periods. With any finite number of repetitions, it seems clear that the Bertrand result remains unchanged. Any strategy in which firm A, say, chooses P A > c during the final period (T) offers firm B the possibility of earning profits by setting P A > P B > c. The threat of charging Pi\. > c in period T is therefore not credible. Since a similar argument applies to any period prior to T, we can conclude that the only perfect equilibrium is one in which firms charge the competitive price in every period. The strict assumptions of the Bertrand model make tacit collusion impossible over any finite period. If firms are viewed as having an infinite time horizon, however, matters change significantly. In this case there is no "final" period so collusive strategies may exist that are not undermined by the logic of the Bertrand result. One such possibility is for firms to adopt "trigger" strategies in which each firm (again, say, firm A) sets P A = PM (where PM is the monopoly price) in every period for which firm B adopts a similar price, but chooses P A = c if firm B has cheated in the previous period. To determine whether these trigger strategies constitute a perfect equilibrium, we must ask whether they constitute a Nash equilibrium in every period. Suppose the firms have colluded for a time and firm A thinks about cheating in this period. Knowing that firm B will choose P B = PM, it can set its price slightly below PM and, in this period, obtain the entire market for itself. It will thereby earn (almost) the entire monopoly profits (7TM) in this period. But, by doing this, firm A will lose its share of monopoly profits (7TMI2) forever after because its

15 CHAPTER 21 Strategy and Game Theory 633 treachery will trigger firm B's retaliatory strategy. Since the present value (see Chapter 24) of these lost profits is given by 7TM 2 r (where r is the per-period interest rate), cheating will be unprofitable if 7TM 1 7TM < T'~' (21.5) (21.6) This condition holds for values of r less than!. We can therefore conclude that the trigger strategies constitute a perfect equilibrium for sufficiently low interest rates. The collusion implicit in these strategies is noncooperative. The firms never actually have to meet in seedy hotel rooms to adopt strategies that yield monopoly profits. 13 Example 21.3 Tacit Collusion in Steel Bars Suppose only two firms produce steel bars suitable for jailhouse windows. Bars are produced at a constant average and marginal cost of $10, and the demand for bars is given by Q = 5,000 loop. (21.7) Under Bertrand competition, each firm will charge a price of $10 and a total of 4,000 bars will be sold. Since the monopoly price in this market is $30, each firm has a dear incentive to consider collusive strategies. With the monopoly price, total profits are $40,000 (each firm's share of total profits is $20,000) so anyone firm will consider a next-period price cut only if 1 $40,000 > $20, (21.8) r If we consider the pricing period in this model to be one year and a reasonable value of r to be 0.20, the present value of each firm's future profit share is $100,000 so there is dearly little incentive to cheat on price. Alternatively, each firm might be willing to incur costs (say, by monitoring the other's price or by developing a "reputation" for reliability) of up to $60,000 in present value to maintain the agreement. Tacit Collusion with More Firms Viability of a trigger price strategy may depend importantly on the number of firms. With eight producers of steel bars, the gain from cheating on a collusive agreement is still $40,000 (assuming the IIIt seems clear from Equation 21.6 that other supracompetitive prices might also yield perfect equilibria depending on the value of r. For a discussion, see J. Friedman, Oligopoly and the Theory of Games (Amsterdam: North-Holland Publishing Co., 1977).

16 634 PART VI Models of Imperfect Competition cheater can corner the entire market). The present value of a continuing agreement is only $25,000 (= $40, /.2) so the trigger price strategy is not viable for anyone firm. Even with three or four firms or less responsive demand conditions, the gain from cheating may exceed the costs required to make tacit collusion work. Hence the commonsense idea that tacit collusion is easier with fewer firms is supported by this model. Generalizations and Limitations ENTRY, EXIT, AND STRATEGY Ouery: How does the interest rate determine the maximum number of firms that can successfully collude in this problem? What is the maximum if r = 0.2? How about the case when r = 0.1? Explain your results intuitively. The contrast between the competitive results of the Bertrand model and the monopoly results of the (infinite time period) collusive model suggests that the viability of tacit collusion in game theory models is very sensitive to the particular assumptions made. Two assumptions in our simple model of tacit collusion are especially important: (1) that firm B can easily detect whether firm A has cheated; and (2) that firm B responds to cheating by adopting a harsh response that not only punishes firm A, but also condemns firm B to zero profits forever. In more general models of tacit collusion, these assumptions can be relaxed by, for example, allowing for the possibility that it may be difficult for firm B to recognize cheating by A. Some models examine alternative types of punishment B might inflict on A-for example, B could cut price in some other market in which A also sells. Other categories of models explore the consequences of introducing differentiated products into models of tacit collusion or of incorporating other reasons why the demand for a firm's product may not respond instantly to price changes by its rival. As might be imagined, results of such modeling efforts are quite varied. 14 In all such models, the notions of Nash and perfect equilibria continue to play an important role in identifying whether tacit collusion can arise from strategic choices that appear to be viable. Our treatment of entry and exit in competitive and noncompetitive markets in previous chapters left little room for strategic considerations. A potential entrant was viewed as being concerned only with the relationship between prevailing market price and its own (average or marginal) costs. We assumed that making that comparison involved no special problems. Similarly, we assumed firms will promptly leave a market they find to be unprofitable. Upon closer inspection, however, the entry and exit issue can become considerably more complex. The fundamental problem is that a firm wishing to enter or leave a market must make some conjecture about how its action will affect market price in subsequent periods. Making such conjectures obviously requires the firm to consider I'See J. Tirole, The Theory of Industrial Organization (Cambridge, Mass.: MIT Press, 1988), chap. 6.

17 CHAPTER 21 Strategy and Game Theory 635 what its rivals will do. What appears to be a relatively straightforward decision comparing price and cost may therefore involve a number of possible strategic ploys, especially when a firm's information about its rivals is imperfect. Sunk Costs and Commitment Many game-theoretic models of the entry process stress the importance of a firm's commitment to a specific market. If the nature of production requires firms to make specific capital investments in order to operate in a market and if these cannot easily be shifted to other uses, a firm that makes such an investment has committed itself to being a market participant. Expenditures on such investments are called sunk costs, defined more formally as follows: DEFINITION Sunk Costs Sunk costs are one-time investments that must be made in order to enter a market. Such investments allow the firm to produce in the market but have no residual value if the firm exits the market. Investments in sunk costs might include expenditures such as unique types of equipment (for example, a newsprint-making machine) or job-specific training for workers (developing the skills to use the newsprint machine). Sunk costs have many characteristics similar to what we have called "fixed costs" in that both these costs are incurred even if no output is produced. Rather than being incurred periodically as are many fixed costs (heating the factory), however, sunk costs are incurred only once in connection with the entry process. 15 When the firm makes such an investment, it has committed itself to the market, and that may have important consequences for its strategic behavior. Sunk Costs, First Mover Advantages, and Entry Deterrence Although at first glance it might seem that incurring sunk costs by making the commitment to serve a market puts a firm at a disadvantage, in most models that is not the case. Rather, one firm can often stake out a claim to a market by making a commitment to serve it and in the process limit the kinds of actions its rivals find profitable. Many game theory models, therefore, stress the advantage of moving first. As an example, consider again the Stackelberg leadership game in Table Suppose we treat the output decision as reflecting commitments of the firms to a particular level of productive capacity, which they will maintain in future periods. With simultaneous moves, either of the follower-leader pairs in the payoff matrix represents a possible Nash equilibrium. However, if one firm (say, firm A) has the opportunity to move first in this game, it will choose to be a 15 Mathematically, the notion of sunk costs can be integrated into the per-period total cost function as TC t = S + F t + cqt, where $ is the per-period amortization of sunk costs (for example, the interest paid for funds used to finance specific capital investments), F is per-period fixed costs, c is marginal cost, and q, is perperiod output. If q, = 0, TC, $ + F t, but if the production period is long enough, some or all of F, may also be avoidable. No portion of S is avoidable, however.

18 636 PART VI Models of Imperfect Competition leader (q A = 60) and thereby limit firm B's options. Adoption of a relatively large initial capacity by firm A gives it an advantage-there is simply not much "room" left in the market for firm B. Given firm A's advantage in moving first, B's most profitable decision is to be a follower. Other situations in which a first mover might have an advantage include investing in research and development or pursuing product differentiation strategies. In international trade theory, for example, it is sometimes claimed that protection or subsidization of a domestic industry may allow it to enter an Industry first, thereby gaining strategic advantage. Similarly, pursuit of "brand proliferation" strategies by existing toothpaste or breakfast cereal companies may make it more difficult for those who come later to develop a sufficiently different product to warrant a place in the market. The success of such firstmover strategies is by no means assured, however. Careful modeling of the strategic situation is required in order to identify whether moving first does offer any real advantages. In some cases, first-mover advantages may be large enough to deter all entry by rivals. Intuitively, it seems plausible that the first mover could make the strategic choice to have a very large capacity and thereby discourage all other firms from entering the market. The economic rationality of such a decision is not clear-cut, however. In the Stackelberg model introduced in Chapter 20, for example, the only sure way for one spring owner to deter all entry is to satisfy the total market demand at the firm's marginal and average cost-that is, one firm would have to offer q 120 at a price of zero to have a fully successful entry deterrence strategy. Obviously, such a choice results in zero profits for the incumbent firm and would not represent profit maximization. Instead, it would be better for that firm to accept some entry by following the Stackelberg leadership strategy. With economies of scale in production, the possibility for profitable entry deterrence is increased. If the firm that is to move first can adopt a large enough scale of operation, it may be able to limit the scale of the potential entrant. The potential entrant will therefore experience such high average costs that there would be no advantage to its entering the market. Example 21.4 illustrates this possibility in the case of Cournot's natural springs. Whether this example is of general validity depends, among other factors, on whether the market is contestable. If other firms with large scales of operations elsewhere can take advantage of prices in excess of marginal cost to practice hit-and-run entry of the type described in Chapter 20, the entry deterrence strategy will not succeed. Example 21.4 Entry Deterrence in Natural Springs If the natural spring owners in our previous examples experience economies of scale in production, entry deterrence becomes a profitable strategy for the first firm to choose its quantity. The simplest way to incorporate economies of scale into the Cournot model is to assume each spring owner must pay a fixed cost of operations. If that fixed cost is given by $784 (a carefully chosen number!),

19 CHAPTER 21 Strategy and Game Theory 637 it is clear that the Nash equilibrium leader-follower strategies remain profitable for both firms (see Table 21.5). When firm A moves first and adopts the leader's role, however, B's profits are rather small ( = 116), and this suggests that firm A could push B completely out of the market simply by being a bit more aggressive. Since B's reaction function (Equation 20.18) is unaffected by considerations of fixed costs, firm A knows that and that market price is given by Hence A knows that B's profits are qa (21.9) P (21.10) 'TrB = PqB 784, (21.11) which, when B is a follower (that is, when B moves second) depends only on qa. Substituting Equation 21.9 into yields 20 'FrB = C ; qar (21.12) Consequently, firm A can ensure nonpositive profits for firm B by choosing (21.13) With qa = 64, firm A becomes the only supplier of natural spring water. Since market price is $56 (= ) in this case, firm A's profits are (56, 64) ,800, (21.14) a significant improvement over the leader-follower outcome. The ability to move first coupled with the fixed costs assumed here therefore makes entry deterrence a successful strategy in this case. Limit Pricing Query: Why is the time pattern of play in this game crucial to the entry deterrence result? How does the result here contrast with our analysis of a contestable monopoly in Example 20.4? So far our discussion of strategic considerations in entry decisions has focused on issues of sunk costs and output commitments. Prices were assumed to be determined through auction or Bertrand processes only after such commitments were made. A somewhat different approach to the entry deterrence question concerns the possibility of an incumbent monopoly accomplishing this goal through its pricing policy alone. That is, are there situations where a monopoly might purposely choose a low ("limit") price policy with the goal of deterring entry into its market?

20 638 PART VI Models of Imperfect Competition In most simple cases, the limit pricing strategy does not seem to yield maximum profits nor to be sustainable over time. If an incumbent monopoly opts for a price of P L < PM (where PM is the profit-maximizing price), it is obviously hurting its current-period profits. But this limit price will deter entry in the f~ture only if P L falls short of the average cost of any potential entrant. If the monopoly and its potential entrant have the same costs (and if capacity choices do not play the role they did in the previous example), the only limit price that IS sustainable in the presence of potential entry is P L = AC, adoption of which would obviously defeat the purpose of being a monopoly since profits would be zero. Hence the basic monopoly model offers little room for limit price behavioreither there are barriers to entry that allow the monopoly to sustain PM, or there are no such barriers, in which case competitive pricing prevails. Believable models of limit pricing behavior must therefore depart from traditional assumptions. The most important set of such models are those involving incomplete information. If an incumbent monopoly knows more about a particular market situation than does a potential entrant, it may be able to take advantage of its superior knowledge to deter entry. As an example, consider the game tree illustrated in Figure Here firm A, the incumbent monopolist, may have either "high" or "low" production costs as a result of past decisions. Firm A does not actually choose its costs currently but, since these costs are not known to B~ we must allow for the two possibilities. Clearly, the profitability of B~s entry into the market depends on A's costs-with high costs B's entry is profitable (7TH 3) whereas if A has low costs, entry is unprofitable (7TB = -1). What is B to do? One possibility would be for B to use whatever information it does have to develop a subjective probability estimate of A's true cost structure. 16 That is, B must assign probability estimates to the states of nature "low cost" and "high cost." If B assumes there is a probability p that A has high cost and (1 p) that it has low cost, entry will yield positive expected profits (see Chapter 9) provided which holds for E(7TB) = p(3) + (1 - ph -1) > 0, (21.15) p>!. (21.16) The particularly intriguing aspects of this game concern whether A can influence B's probability assessment. Clearly, regardless of its true costs, firm A is better off if B adopts the no-entry strategy, and one way to assure that is for A to make B believe that p < t. As an extreme case, if A can convince B that it is certainly a low-cost producer (p = 0), B will clearly be deterred from entry even if the true cost situation is otherwise. For example, if A chooses a low price policy when it serves the market as a monopoly, this may signal (see Chapter 10) 16Games that require one player to devise probability estimates for the other's situation are termed "Bayesian games of incomplete information" after the statistician Thomas Bayes who pioneered in the development of the mathematics of subjective probabilities. For further details on such games, see Tirole, The Theory of Industrial Organization, pp

21 CHAPTER 21 Strategy and Game Theory 639 Figure 21.3 An Entry Game Firm A has either a "high" or a "low" cost structure that cannot be observed by B. If B assigns a subjective probability (p) to the possibility that A is high cost, it will enter providing p *' Firm A may try to influence B's probability estimate. 1,3 B 4,0 A 3,-1 6,0 to B that A's costs are low and thereby deter entry. Such strategy might be profitable for A even though it would require it to sacrifice some profits if its costs are actually high. This provides a possible rationale for low limit pricing as an entry deterrence strategy. Unfortunately, as we said in Chapter 10, examination of the possibilities for signaling equilibria in situations of asymmetric information raises many complexities. Since firm B knows that A may create false signals and firm A knows that B will be wary of its signals, a number of solutions to this game seem possible. The viability of limit pricing as a strategy for achieving entry deterrence depends crucially on the types of informational assumptions that are made. P Predatory Pricing Tools used to study limit pricing can also shed light on the possibility for "predatory" pricing. Ever since the formation of the Standard Oil monopoly in the late nineteenth century, part of the mythology of American business has been that John D. Rockefeller was able to drive his competitors out of business by charging ruinously low (predatory) prices. Although both the economic logic and the "For an examination of some of these issues, see P. Milgrom and J. Roberts, "Limit Pricing and Entry under Conditions of Incomplete Information: An Equilibrium Analysis," Econometrica (March 1982):

22 640 PART VI Models of Impertect Competition empirical facts behind this version of the Standard Oil story have generally been discounted,18 the possibility of encouraging exit through predation continues to provide interesting opportunities for theoretical modeling. The structure of many models of predatory behavior is similar to that used in limit pricing models-that is, the models stress asymmetric information. An incumbent firm wishes to encourage its rival to exit the market so it takes actions intended to affect the rival's view of the future profitability of market participation. The incumbent may, for example, adopt a low price policy in an attempt to signal to its rival that its costs are low-even if they are not. Or the incumbent may adopt extensive advertising or product differentiation activities with the intention of convincing its rival that it has economies of scale in undertaking such activities. Once the rival is convinced that the incumbent firm possesses such advantages, it may recalculate the expected profitability of its production decisions and decide to exit the market. Of course, as in the limit pricing models, such successful predatory strategies are not a foregone conclusion. Their viability depends crucially on the nature of the informational asymmetries in the market. n-player GAME THEORY Coalitions All of the game theory examples we have developed so far in this chapter involve only two players. Although this limitation is useful for illustrating some of the strategic issues that arise in the play of a game (or the operation of a duopoly market), it also tends to obscure some important questions. In this final section, therefore, we briefly examine more general n-player games. The most important element added to game theory when one moves beyond two players is the possibility for the formation of subsets of players who agree on coordinated strategies. Although the possibility for forming such coalitions exists in two-player games (the two firms in a duopoly could form a carte!), the number of possible coalitions expands rapidly as games with larger numbers of players are considered. In some games, simply listing the number of potential coalitions and the payoffs they might receive can be a major analytical task. As in the formation of cartels in oligopolistic markets, the likelihood of forming successful coalitions in n-player games is importantly influenced by organizational costs. These costs involve both information costs associated with determining coalition strategies and enforcement costs associated with ensuring that a coalition's chosen strategy is actually followed by its members. If there are incentives for members to cheat on established coalition strategies, then monitoring and enforcement costs may be high. In some cases, such costs may be so high as to make the establishment of coalitions prohibitively costly. For these games, then, all n-players operate independently. IR J. S. McGee and others have pointed out that predatory pricing was a far less profitable strategy for Rockefeller than simply buying up his rivals at market price (which seems to have been what occurred). See J. S. McGee, "Predatory Pricing: The Standard Oil (NJ) Case," Journal of Law and Economics (1958): ; and "Predatory Pricing Revisited," Journal of Law and Economics (October 1980): Recent literature has examined whether predatory pricing can affect a rival firm's market value.

23 CHAPTER 21 Strategy and Game Theory 641 Game Theory, General Equilibrium, and the Core In its most abstract theoretical development, n-player game theory has many similarities to the type of general economic equilibrium theory described in Chapters 16 and 17. An economy of n individuals can be viewed as an n-player game, and the coalitions formed may be thought of as firms, local governments, or any other type of economic organization. Of course, this way of modeling the economy is quite abstract, and the results are therefore probably not appropriate for any kind of detailed empirical study. Nevertheless, It-player game theory has been widely used as a conceptual tool for understanding the nature of some types of economic activity. Central to these abstract uses of It-player game theory is the concept of the core of a game. This represents an attempt to generalize the notion of Pareto optimality to situations where subsets of individuals may form coalitions to improve the welfare of the subset's members. We have already illustrated some theoretical results related to the core in Chapter 8 when we discussed the Edgeworth model of exchange. Here we briefly illustrate how that concept has been adapted to game theory. We begin with a definition: 19 DEFINITION Core of an n-player Game The core ofan n-player game consists of that set of coalitions and strategies for which no subset of players would find it advantageous to seek improvements through further coalition activity. This core concept, therefore, includes the Pareto definition of optimality as a special case (since every individual in a game can form his or her own oneperson coalition and no unambiguous improvement in welfare is possible). The concept is more general than the Pareto notion, however, since it also allows for the balancing of power among multiplayer coalitions. Many games do not have a core under this definition-as in some of the duopoly models of the previous chapter, the play of such games represents an endless jockeying of the players (and coalitions) for favorable outcomes. Bur, for games that have one (or possibly many) allocations in the core, a wide variety of results relevant to economics has been obtained. Perhaps the most interesting of these results concern the relationship between the core of a game and market-type institutions. Several authors have demonstrated that core equilibria in n-player games can often be given a price system interpretation. to Other questions that have been examined include the modeling of public goods and externalities in game-theoretic contexts and ways of introducing money and financial institutions into economic games. 2l Such applica "This definition must necessarily be rather informal. For a complete development, see M. Shublk, Game Theory in the Social Sciences (Cambridge, Mass.: MIT Press. 1982), chap. 6. '" For an example, see R. M. Anderson, "An Elementary Core Equivalen~e Theorem," Econometriw (November 1978): For some references, see Shubik, Game Theory in the Social Sciences. chap. 12.

24 642 PART VI Models of Imperfect Competition tions serve to illustrate, in a conceptual way, how particular laws and institutions may arise to solve problems that are common to practically all economies. SUMMARY In this chapter we have briefly examined the economic theory of games with particular reference to the use of that theory to explain strategic behavior in duopoly markets. Some of the conclusions of this examination include: Concepts such as players, strategies, and payoffs are common to all games. Many games also possess a number of types of equilibrium solutions. With a Nash equilibrium, each player's strategic choice is optimal given its rival's choice. In dynamic games only Nash equilibria that involve credible threats are viable. The Prisoner's Dilemma represents a particularly instructive two-person game. In this game the most preferred outcome is unstable, though in repeated games the players may adopt various enforcement strategies. Game-theoretic models of duopoly pricing start from the Bertrand result that the only Nash equilibrium in a simple game is competitive (marginal cost) pricing. Consideration of possible output commitments and first-mover strategies may result in noncompetitive results, however. Tacit collusion at the monopoly price is sustainable in infinite-period games under certain circumstances. Much of the game-theoretic modeling of entry and exit stresses the importance of the informational environment. In situations of asymmetric information, incumbent firms may be able to capitalize on superior information by adopting strategies that result in entry deterrence. APPLICA TION The Game of Chicken In this chapter we looked at a number of simple twoperson games. There are 78 distinct variants of such games, many of which provide useful economic insights. Another example is provided by the game of "Chicken" for which a typical payoff matrix is shown in Table In this game each player has two strategies: (1) to "cooperate" or "not to cooperate." Payoffs to the players are recorded in the table with 0 being the least favorable outcome and 3 being the most favorable for each player. This game has two Nash equilibria points-the off-diagonal strategy pairs. Both of these solutions are, as we shall see, vulnerable to threats, however. It is the possibility of these threats that gives this game its name and makes the game empirically interesting. Basic Scenario of the Game Use of the title "Chicken" to describe this game derives from a "game" played by hot-rod gangs (for the true flavor of the game, see the great James Dean movie, Rebel without a Cause). The game's players start at opposite ends of a deserted road with their left wheels on the center line of the road. They then race toward each other with the one veering off first being branded the "chicken." In this game then "cooperation" consists of veering off the center line whereas "noncooperation" refers to a determined strategy of staying on that line. If both drivers "chicken out," they both live, but gain no social prestige from their peers. Hence, the twin cooperative strategy is unstable with each driver having an incentive to cheat on it (that is, stick to the center line). If one driver succeeds in cheating in this way, he

25 CHAPTER 21 Strategy and Game Theory 643 Table 21.8 The Game of Chicken B's Strategies Do Not Cooperate Cooperate A's Strategies Cooperate Do Not Cooperate or she receives considerable prestige (a "3") whereas the chicken loses face (a "1 "). A twin strategy of noncooperation is a disaster, however, since the cars crash and both drivers die (a "0"). Unlike the Prisoner's Dilemma, this noncooperative solution is unstable-all the entries in the table are Pareto superior to the crash solution. But each driver has an incentive to threaten to take a noncooperative strategy in the hope that the opponent will be cajoled into taking a cooperative strategy. It is in each player's interest to appear to be committed to a noncooperative strategy even if he or she intends to "chicken out" at some point along the way. Applications to Oligopoly Most applications of the game of Chicken to explaining oligopoly behavior focus on the stability of situations in which one firm could gain a clear advantage by altering its behavior. That is, they focus on explaining why "cooperative" solutions appear to be stable despite the fact that they do not meet the Nash criteria for stability. Consider, for example, the decision of U.S. auto makers to refrain from building small cars in the 1950s and early 1960s. Such restraint appeared to many observers to be unstable-a company that built small cars could have made a significant profit by doing so. But the threat that its competitors would quickly enter the market also (thereby seriously eroding profits on both small and large cars) kept anyone company from proceeding. Of course, this cooperative solution ultimately failed because the rapid growth of imported small cars forced the U.S. companies to respond. The paralysis induced by the strategic situation of the placed U.S. makers at a considerable disadvantage in later years because they had not developed the production expertise that they needed. A related situation concerns the pricing of gasoline on major highways. Although there are relatively many sellers, all seem to charge about the same price, and they change prices together, usually on the same day. Presumably, anyone seller faces a fairly elastic demand for its gasoline and could gain significant sales by dropping its price a few cents a gallon. But its competitors would probably retaliate, and since total demand for gasoline along the highway is relatively inelastic, all firms would be worse off. Gas "wars" arise because of the periodic breakdown of the cooperative solution, and prices drop dramatically (your author once purchased gasoline for $.12 per gallon in the 19605). But these price breaks are usually quite short-lived (the $.12 price lasted only about four hours), and discipline quickly returns to the market. Chicken, Nuclear Strategy, and the "Star Wars" Program Some authors claim that the game of Chicken provides the best representation of the strategic nuclear deterrence practiced by the United States and the Soviet UnionY In this game-theoretic interpretation, both superpowers refrain from the use of nuclear weapons because of the threat that the other side will use them too (similar arguments were also made about chemical and biological weapon use in World War II). Although this cooperative solution is unstable in that one party could gain by being the first to use such weapons, the threat that the other will follow suit maintains this unstable solution. Some authors have criticized this interpretation because, in their view, the threat of "Mutually Assured Destruction" is fundamentally "irrational" only a Dr. Strangelove would threaten to destroy the world if attacked. This has led to considerable analysis USee, for example, S. J. Beams, Superpower Games: Applying Game Theory to the Superpower Conflict (New Haven, Conn.: Yale University Press, 1985).

26 644 PART VI Models of Imperfect Competition of the nature of threats in the Chicken game including the issues of which threats are credible and whether threats can be probabilistic in nature. No simple, acceptable model of the nuclear stalemate has yet been developed, however. Because we do not have a complete model of nuclear deterrence, it is difficult to predict how technical innovations will affect the strategic balance. This is especially important in appraising the Strategic Defense Initiative (SOl or "Star Wars") proposed in the early 1980s. Antimissile defenses have both desirable and undesirable effects on the stability of strategic deterrence. The principal positive effect is that such defenses reduce the probability that accidents or third parties could initiate a major superpower confrontation. They offer the chance to destroy a few missiles that are launched in error. But such defensive systems also introduce some uncertainty into the deterrence calculation because they make retaliation more difficult. From the perspective of the Chicken game, they reduce the credibility of threats and therebv may make the cooperative solution less stable. Henc~ even discounting the cost and technical difficulties tha; SDI poses, the overall desirability of the system cannot be appraised fully until better theoretical representations of the strategic balance become available. PROBLEMS ~ ~-~ ~ Players A and B are engaged in a coin-matching game. Each shows a coin as either heads or tails. If the coins match, B pays A $1. If they differ, A pays B $1. a. Write down the payoff matrix for this game, and show that it does not contain a Nash equilibrium. b. How might the players choose their strategies in this case? 21.2 Smith and Jones are playing a number-matching game. Each chooses either 1, 2, or 3. If the numbers match, Jones pays Smith $3. If they differ, Smith pays Jones $1. a. Describe the payoff matrix for this game, and show that it does not possess a Nash equilibrium strategy pair. b. Show that with mixed strategies this game does have a Nash equilibrium if each player plays each number with probability!. What is the value of this game? 21.3 Suppose firms A and B each operate under conditions of constant average and marginal COSt, bur that MeA 10, MC B = 8. The demand for the firms' output is given by Qn P. a. If the firms practice Bertrand competition, what will be the market price under a Nash equilibrium? b. What will the profits be for each firm? c. Will this equilibrium be Pareto efficient? 21.4 Suppose the two firms in a duopoly pursue Cournot competition as described in Equation Suppose each firm operates under conditions of increasing mar ginal cost but that firm A has a larger scale of operations than does firm B in the sense that MC A < MC B for any given output level. In a Nash equilibrium, will marginal cost necessarily be equalized across the two firms? Will total output be produced as cheaply as possible? 21.5 In the ice cream cone stand example of Chapter 20, as sume each stand has five possible locational strategies locating 0, 25, 50, 75, or 100 yards from the left end of the beach. Describe the payoff matrix for this game, and explain whether it has an equilibrium strategy pair The world's entire supply of kryptonite is controlled by 20 people with each having 10,000 grams of this potent mineral. The world demand for kryptonite is given by Q = 10,000-1,000P, where P is the price per gram. a. If all owners could conspire to rig the price of kryptonite, what price would they set, and how much of their supply would they sell? b. Why is the price computed in part (a) an unstable equilibrium? c. Does a price for kryptonite exist that would be a stable equilibrium in the sense that no firm

27 CHAPTER 21 Strategy and Game Theory 645 could gain by altering its output from that required to maintain this market price? 21.7 Smith and Jones are stranded on a desert island with fixed initial endowments of clams and bread. Instead of seeking mutually beneficial trades directly, they opt for a bidding strategy in which they use their initial clam endowments to bid for bread. That is, the bread is deposited in a safe place, and each person states how many clams he or she will offer for the bread. When the bids are revealed, the available bread is split in proportion to each bid. If, for example, Smith bids one clam and Jones bids two clams, Smith gets! of the bread and Jones gets!. a. Sketch an Edgeworth box diagram that shows the initial endowments of Smith and Jones. b. On your Edgeworth box, show how a final al- location is determined given a particular set of bids by Smith and Jones. c. Use your construction from part (b) to show Smith's optimal reply to a particular bid by Jones. Develop a similar construction for Jones. d. Is there an equllibrium pair of strategies for the situation described in part (c)? That is, is there a bid for Smith that is optimal given Jones's bid and vice versa? e. Is the equilibrium described in parr (d) necessarily on the contract curve for this exchange economy? (Note: This is an example of a "strategic market game." For a further discussion, see M. Shubik, Game Theory in the Social Sciences [Cambridge, Mass.: MIT Press, 1982], pp ) EXTENSIONS StrategiC Substitutes and Complements One way to conceptualize the relationships between the choices of firms in an imperfectly competitive market is to introduce the ideas of strategic substitutes and complements. By drawing analogies to similar definitions from consumer and producer theory, game theorists define firms' activities to be strategic substitutes if an increase in the level of an activity (say, output, price, or spending on product differentiation) by one firm is met by a decrease in that activity by its rival. On the other hand, activities are strategic complements if an increase in an activity by one firm is met by an increase in that activity by its rival. To make these ideas precise, suppose that profits for firm A ('lt A ) depend on the level of an activity it uses itself (SA) and on use of a similar activity by its rival. The firm's goal, therefore, is to maximize 'IT,4(SA' S8)' E21.1 Optimality Conditions and Reaction Functions. The first-order condition for A's choice of its own strategic activity is 0, (i) where the subscripts for 'IT represent partial derivatives with respect to its various arguments. For a maximum we also require that Obviously, the optimal choice of SA specified by Equation (i) will differ for different values of S8' We can record this relationship by A's reaction function (R A ) (ii) (iii) The strategic relationship between SA and SB is implied by this reaction function. If R~ > 0,.)4 and S8 are strategic complements. If R~ < 0, SA and SB are strategic substitutes. E21.2 Inferences from the Profit Function. It is usually more convenient to use the profit function directly to examine strategic relationships. Substituting Equation (iii) into the first-order condition (i) gives o. (iv) Partial differentiation with respect to SB yields Therefore so, in view of the second-order condition (ii), 'lt~ > 0 implies R~ > 0 and 'lt~ < 0 implies R~ < O. Strategic (v)

28 - 646 PART VI Models of Imperfect Competition relationships can therefore be inferred directly from the derivatives of the profit function. E2 t.3 The Cournot Model. In the Coumot model (Equation 20.10) profits are given as a function of the two firms' quantities as In this case and 'r. A " TC' TC(qA). (vi) (vii) (viii) Since P' 0, the sign of will depend on the concavity of the demand curve (pn). With a linear demand curve, P" so TT~2 is dearly negative. Quantities are strategic substitutes in the COUrllot model with linear demand. This will generally be true unless the demand curve is relatively convex (P" > 0). E21.4 Strategic Relationship between Prices. If we view the duopoly problem as one of setting prices, both q, and qr will be functions of prices charged by the two firms: q., DA (P,\> PB) DR(P 4, PRJ. (ix) Using this notation, Hence and TTA P"q, TC(qA) TC'DA TC' D~\ - TC'D~lD ;'. (x) (xi) Obviously, interpreting this mass of symbols is no easy task. In the special case of constant marginal COSt ('TC: = 0) and linear demand (Dj\ 0), the sign of IS given by the sign of D;-that is how increases in PH affect qa. In the usual case when the two goods are themselves substitutes, D2' > 0, so TT'~2 > 0. That is, prices are strategic complements. Firms in such a duopoly would either raise or lower prices together. REFERENCES Bulow, J., Geanakoplous, G., and Klemperer, P. "Multimarket Oligopoly: Strategic Substitutes and Comple ments." Journal of Political Economy (June 1985): Tirole, J The Theory of Industrial Organization. Cambridge, Mass.: MIT Press, Pp Brams, J Superpower Games. New Haven, Conn.: Yale University Press, Application of game theory to the study of nuclear strategy ami arms control. Brams has several other books 011 game theory (including one on game theory and the Bible) that also make interesting reading. SUGGESTED READINGS Friedman, J. W. Oligopoly and the Theory of Games. Raiffa, H. Decision Analysis: Introductory Lectures 011 Amsterdam: North-Holland Publishing Co., Choices under Uncertainty. Reading, Mass.: Addison ExtC1lstve theoretical analysis of game-theoretic inter Wesleypretatio/ls of oligopoly models. Excellent treatment of repeated games. Kreps, D. M. A Course in Microeconomic Theory. Princeton, N.J: Princeton University Press, Part III contains a detailed summary of much recent work in game theory. Useful discusswn of a number of equilibrium concepts although examples are sometimes difficult to follow. Luce, R. D., and RaIffa, H. Games and Decisions. New York: John Wiley & Sons, Classic text 011 game theory. Very readable and com plete-but only through mid 1950s. Extended, easy-co-follow analysis ofa sillgle problem ill decision theory. For more formal treatment. see Raiffa and Schlaifler, Applied Statistical Decision Theory. Schelling, T. C. Micromotives and Macrohehavior. New York: Norton, 1978.