An Extended Duopoly Game

Transcription

1 An Extended Duopoly Game John C. Eckalbar Abstract: The author shows how principles and intermediate economic students can gain a feel for strategic price setting by playing a relatively large oligopoly game. The author constructs a playoff matrix and discusses various strategies and outcomes. The game extends to a continuous price space and outlines various applications appropriate for intermediate micro students. Finally, to make it easier for others to tinker with the assumptions of the game, the author can provide the Mathematica code used to generate the table and figures. Key words: duopoly, game theory, oligopoly, strategic behavior JEL code: A2 Firms in an oligopoly cannot maximize their profits by following simple rules, such as marginal revenue (MR) equals marginal cost (MC). Instead, they are forced to think strategically. When they contemplate an action, such as a price change, they must consider the reactions of their rivals, because the effect of the price change will depend critically on the behavior of their rivals after the change. Making this fact come alive in a principles or intermediate micro class can be a challenge. Most texts present one or two oligopoly games to give the student a feel for the issues. These games generally consist of two-by-two payoff matrices, and my experience with them is that they are too small and simple to get the students involved. 1 In this article, I report on the use of a much larger and richer payoff matrix for the study of strategic duopoly price-setting behavior. I have found it useful in both principles and intermediate micro classes, and I expect that it would also be valuable in making tangible some of the more abstract issues in a game-theory class. 2 I discuss the uses of a relatively large payoff matrix suitable for a principles class. I also consider some features of the underlying game in a continuous price space. The generalization can be studied in an intermediate micro class. USING A LARGE PAYOFF MATRIX I start the exercise by giving the students an unmarked version of Table 1. 3 I explain that we are considering a duopoly, that is, an oligopoly with two firms in the industry. The two firms produce similar products, which the buyers regard as substitutes. The table shows firm A s price (Pa) in bold along the top of the table John C. Eckalbar is a professor of economics at California State University, Chico ( Winter

2 TABLE 1 Payoff Matrix: A s Profit (Top Number) and B s Profit (Bottom Number), in Dollars Firm B s price A s Price * * * * * * * JOURNAL OF ECONOMIC EDUCATION

3 * * * * * * * * * Note: In each set of 2 numbers, the top number is firm A s profit and bottom number is firm B s profit using A s prices along the top of the table and B s prices along the left side. The * denotes A s maximum profit cell in each row. The numbers outlined in a box are B s reaction function. Winter

4 and B s price (Pb) in bold along the left side. Each firm s sales depend on the price it charges as well as the price its rival charges. When both prices are set, demand quantities, revenues, costs, and profits are determined. The values for Pa and Pb determine our address, or fix our position in the table. Each position in the table shows a pair of numbers; the top number of the pair is A s profit or loss, and the number on the bottom is B s profit or loss, when A charges the price shown along the top of the table, and B charges the price written at the left. For example, when Pa is $12.40 and Pb is $15.60, A s profit is $5.06 and B s is $5.98. The students are often curious about where these numbers come from, and I tell them that I made them up, or more accurately, I made up some demand and cost equations that seemed reasonable and then used Mathematica to crunch out the numbers. 4 Because the students are generally impatient to get on with the game, I simply draw their attention to a few general features of the table. For instance, suppose Pb is fixed at, say, $16. This locks us onto a row in the table, and by changing Pa, firm A has the ability to slide us left and right. Starting from a low price of $11.20, as firm A increases Pa, A s profit rises to a peak of $8.75 at Pa = $11.80 and then falls off as Pa continues to rise (i.e., as we move further to the right). This is exactly what you would expect from looking at the standard graphical presentations of demand, cost, and profits. When Pb is fixed, A s demand curve is locked in place. Then A has a single best price at the output level where MR = MC, and as the price varies from the optimum, profits drop off. You can see the same pattern for B s profit by looking up and down any column. The other feature to notice is that A s profit improves as we move up the columns, and B s profit rises as we move to the right. For example, when B raises price, some of B s customers choose to buy from A, and this raises A s demand and A s profit. Given just these preliminaries, I divide the class into two groups, the As and the Bs. I generally have a class of about 30 students, with about 15 per team. In larger classes, I might run several games concurrently, with opposing teams of 10 to 15 students each. As long as the team is about 15 or fewer students, the dialogue flows easily among the team members. Students in larger groups are often less willing to communicate with each other. Having several pairs of smaller opposing teams has its pros and cons smaller teams mean more individual participation, but more simultaneous games tend to create bedlam in the classroom. I start them off at an arbitrary initial cell; I might say that Pa = $12.80 and Pb = $ In the beginning, we will be civil and take turns adjusting price, as if we were two gas stations across the street from one another posting prices and then waiting to see what our rival does. I let the As go first. Invariably the As talk it up among themselves and agree to cut the price to $12.00 in order to raise their profits from $8.90 to $ When the Bs get their turn, they typically cut price from $16.60 to $14.80 in order to convert their $4.17 loss to a profit of $4.42. The A team then cuts price to $11.60, and the Bs follow with a cut to $ Once we get to the $11.60/$14.60 cell, things come to a stop. Someone invariably suggests that we have found the equilibrium, but I caution that we will find many different sorts of equilibria depending on the strategies they employ. 44 JOURNAL OF ECONOMIC EDUCATION

5 At the moment, I will only say that we have found the follower/follower or Nash equilibrium A is at an optimum, given B s current choice, and B is likewise at an optimum, given the behavior of A. Students are warned not to think of this point as the equilibrium. I give them another starting point and let the game continue. Perhaps they start at Pa = $11.20 and Pb = $ B goes first, and before long, we are back at the $11.60/$14.60 cell. After a few more tries, their attention is fixed on $11.60/$ At this point, they seem genuinely curious about what is going on. The next step is to define each firm s reaction function. Firm A has a maximum-profit cell in each row, that is, for any given Pb, there will be a Pa that maximizes A s profit. The set of all these cells is A s reaction function. These cells are marked with an asterisk (*) in Table 1. If B were to fix price, A would like to slide left or right to get onto A s reaction function. Similarly, B has a maximum-profit cell in each column, and the collection of these cells (shown by the boxes in Table 1) is B s reaction function. The $11.60/$14.60 cell is at the intersection of the reaction functions, and the orientation of the reaction functions, together with the students behavior have led to the $11.60/$14.60 cell. More on this shortly. The students are not yet thinking strategically. They are behaving as followers. For instance, suppose prices sit at Pa = $12.60 and Pb = $16.20, and it is A s move. Early in the game, firm A is likely to cut price to $11.80, thinking it will be increasing its profits from $7.69 to $9.79. The problem is that firm A is not anticipating the reaction of its rival. If firm A were thinking strategically, it would compare its initial profits with the profit it would earn following B s most likely reaction. For instance, if B cuts price to $14.60 following A s cut to $11.80, A s profit would be only $0.93, which is far worse than its original profit. Once this point is made in the classroom, the students generally behave quite differently, and much more discussion takes place within each team prior to a price move. The discussion up to this point takes about 45 minutes. I generally set my class schedule so that the class ends at this point, and the students are instructed to return to the next class with three written strategy suggestions for their teams. At the start of the next class, I review the rules and collect the homework. I then reform the teams and let the game begin. I will let the firms adjust prices at will, no longer requiring that they take turns. It often happens that one of the firms will discover there might be an advantage to behaving malevolently. Suppose, for instance, the class is at the follower/follower solution. Someone on the B team might notice that if the Bs cut price to $14.40, the As will lose money, even if they stay on their reaction function, but the Bs will make money. The As quickly realize that if they stay at Pa = $11.60 with Pb = $14.40, the As will go broke, whereas the Bs continue to make a profit. This generally prompts the As to cut price to $11.40, even though that makes their loss worse, because it now causes B to lose money. In fact, if B stayed there, B would now lose more than A. This may prompt a further cut by B. We are now in the midst of a price war, with each team heading toward the bottom left margins of the table. The table alone does not have enough information to determine who will win the price war, but at least it prompts a discussion of the subject. With just a little Winter

6 prodding, the students will bring up other factors, such as liquid assets, lines of credit, and the advantage of having low cost at the start of any price war. The case of People s Express and United Airlines might be mentioned. We could also talk about an arms race. When the students have tired of trying to ruin each other, I reposition them in the top right portion of the table once again, for instance, at Pa = $12.80 and Pb = $ This time, they are much more circumspect about their moves, and there is a lot of discussion along the lines of, If we do that, what are they going to do? They are now learning the meaning of strategy. They have seen the game degenerate to a point where neither group is doing well, and they are reluctant to rock the boat when things are going relatively well. At this stage in the course, we have already covered the theory of the kinked demand curve, and we are now prepared to see oligopoly price stickiness in a new light. Perhaps a fear of precipitating a price war would justify a policy of leaving well enough alone when conditions are tolerable. Generally, a little time draws us slowly back toward the follower/follower solution, or worse. When things again degenerate, I read them the now-famous transcript of the conversation between Robert Crandall, the President of American Airlines, and Howard Putnam, the President of rival Braniff. I warn the students (with a wink) that the conversation between these two CEOs contains some high level business-speak that they, as mere students, might not catch on to. Mr. Crandall says, I think it s dumb as hell... to sit here and pound the s out of each other and neither one of us making a (deleted) dime.... We can both live here and there ain t no room for Delta. But there s, ah, no reason that I can see, all right, to put both companies out of business. Mr. Putnam replies, Do you have a suggestion for me? Crandall s response is: Yes. I have a suggestion for you. Raise your goddamn fares 20%. I ll raise mine the next morning.... You ll make more money and I will too. Mr. Putnam says, We can t talk about pricing. Mr. Crandall replies, Oh, Bulls, Howard. We can talk about any g damn thing we want to talk about (quoted in the Wall Street Journal, February 24, 1983, p. 2). To the legally naive, this may sound like an attempt to fix prices, but this is not what the courts found. 5 American Airlines is now the second largest airline in the world, and Braniff is bankrupt. Now I have the As take the role of American Airlines, and the Bs play the part of Braniff. I ask them to imagine that Mr. Putnam agreed to meet Mr. Crandall and discuss a price-setting deal. I ask each team to make a concrete proposal to the other side, something like, We will charge, if you will charge. The students seem to enjoy this simulated clandestine activity. They begin looking in the upper right corner of the matrix, and inevitably each side looks for a cell that offers better profits for both of them, but gives a slight advantage to the team making the proposal. For instance, the As might propose Pa = $12.80 and Pb = $16.40, whereas the Bs favor something more like Pa = $12.80 and Pb = $ They typically wrangle back and forth, sometimes threatening the 46 JOURNAL OF ECONOMIC EDUCATION

7 other side with a price war if the last offer is refused. The cell at Pa = $12.80 and Pb = $16.20 usually gets a lot of attention, because profits are about evenly split at that location. This is a good point to discuss cartels and joint profit maximum. The two firms might adopt a strategy of searching for the cell that brings in the largest total profit, that is, the cell that maximizes the sum of A s profit and B s profit. Once they have gotten the maximum from their customers, they might then decide to redistribute the subtotals, but at least then they would be arguing about cutting up the biggest possible pie. The joint profit maximizing cell is at Pa = $13.00 and Pb = $16.40, where A earns $6.09 and B gets $6.72. The Bs seem happy, but the As argue about how big a kickback they deserve for going to this cell. In fact, the jealousy of the As seriously undermines the establishment of the joint profit outcome. Note that this issue would not arise in a symmetric payoff matrix. There are several lessons to be drawn at this stage. First, it is difficult for the two firms to reach an agreement. Each firm wants to bias the deal in its own favor, and neither is eager to concede the advantage to the other. Second, these deals are illegal, so they cannot be enforced in the courts, and there is a real risk of being caught and prosecuted. Third, and most important, once a deal is struck, it is unstable; that is, it tends to unravel of its own accord as the dissimilar interests of the two firms pull them in other directions. Suppose, for example, the firms settle at the point Pa = $12.80 and Pb = $ Each is doing quite well in comparison with the follower/follower solution or in comparison with a price war, but each is also tempted to cheat on the deal, and there is no enforceable contract to prevent it. Firm A sees its profit rising as it slides to the left, cutting price, while B is tempted to move downward. If both of them give in to the temptation, we move back down and left, with profits for each firm falling. This is a good spot to bring up the history of OPEC, to talk about how the cartel has followed a pattern of formulating an agreement and holding prices and profits up for a while, only to be followed by an erosion of their position, then perhaps another deal, followed by another erosion, and so on. It s like the myth of Sisyphus; they can roll the ball to the top of the hill (upper right corner of the matrix), but inevitably it rolls back down. I ask the students what other options are available, and someone always suggests that one of the companies might buy the other. This prompts some interesting discussions. Suppose, for instance, we are at the follower/follower solution with A making $1.09 and B making $1.35. Suppose B wants to buy A and operate it as a separate, autonomous division. What is a fair price for firm A? A finance major might suggest that firms are worth about 25 times earnings these days, so the stock of firm A might be worth 25 $1.09 = $ Someone on the A team will reply, Yes, but once you buy us and we no longer compete, you can set both prices, and earnings from the A division will be over $6, plus B division earnings will be much better without competition from A. So maybe a stock price of at least 25 $6 = $150 is in order. That certainly leaves a lot of room for negotiation. It also provides an opportunity to discuss antitrust policy in light of recent mergers and proposed mergers. One strategy that students are unlikely to devise on their own is Stackelberg- Winter

8 type price leadership. I suggest the following: What if B decided to fix price at some level and then simply refused to adjust it any further. At the same time, firm A decided to quit fighting and simply charge the best price it could, given B s price. Here B is the leader, and A is the follower. What price would B set? With just a little reflection, the students realize that B must search A s reaction function for the cell that yields the largest profit for B. This occurs when Pb = $ If B sets its price at $15.20, A will charge $ Profits now are $4.40 for A and $2.27 for B. If A leads and B follows, the players go to Pa = $12.00 and Pb = $ Why would price leadership emerge as a strategy? What are its advantages? First, it would be harder to prosecute, because the firms do not have to make contact with each other and negotiate the prices. There will be no smoking gun or taped phone call. Second, both firms do better under price leadership than they do under the follower/follower option. In this particular game, the follower makes out relatively better than the leader, and that may make both firms reluctant to lead, but other numerical assumptions could lead to other results. The second day of game playing takes about 30 minutes. In a principles class, I put the payoff matrix aside at this point and delve into a graphical analysis of models of price leadership by a dominant firm and cartel profit maximization, using familiar demand and cost curves. My impression is that these topics go over much better once the students have had some first-hand experience with these situations. In a more advanced class, I translate the game into a continuous (Pa, Pb) space. DUOPOLY GAME IN CONTINUOUS PRICE SPACE The first step in the continuous-case discussion is to derive the iso-profit curves and then find the reaction functions from them. A mathematical derivation of the iso-profit curves is generally beyond the level of an intermediate class, but the table suggests that, for instance, A s curves take on the appearance shown in Figure 1. Profits rise as we move up in the figure. A s reaction function is the locus of zero-slope points on A s iso-profit curves, that is, the locus of points at which A s profits are at a maximum given a fixed price from B. B s iso-profit curves and reaction function, not shown here, are similar to A s, but rotated 90 degrees clockwise. If we combine the two reaction functions, we can see what generally happens in the early stages of the game. The two reaction functions together with some sample paths toward the follower/follower equilibrium are shown in Figure 2. The shaded areas in the acute regions between the reaction functions constitute a strongly attractive set. Given the way the students generally begin playing the game, if we are not in the set, we are attracted to it, and once we are drawn into it, we remain within it and converge to the intersection of the reaction functions. When we begin at point 1, the A team is attracted to point 2. Because they like point 2 better than point 1, they cut their price. But when B responds, we go to point 3. If the A team were thinking strategically, it would compare point 3 with point 1, rather than point 2 with point JOURNAL OF ECONOMIC EDUCATION

9 FIGURE 1 Firm A s Iso-Profit Curves and Reaction Function, Rfa FIGURE 2 Movement Toward the Reaction Functions and Convergence to the Follower/Follower Equilibrium 2 3 Suppose B is the price leader. We then want to know where B s profits reach a maximum along A s reaction function. We can visualize this by superimposing B s iso-profit curves over A s reaction function. If B leads, then B is looking for the point of tangency between its iso-profit curves and A s reaction function (Figure 3). Winter

10 FIGURE 3 Finding the Price Leadership Solution for B from A s Reaction Function and B s Iso-Profit Curves FIGURE 4 Combining Both Iso-Profit Curves If we combine the two sets of iso-profit curves, we get Figure 4. When students first see Figure 4, someone is likely to say that it looks like an Edgeworth box. That is an insight that is worth building on. Suppose, for instance, that the two firms are presently at point X in the figure. With appropriate price increases, they could move into the shaded region where both firms will earn higher profits. This is reminiscent of a better set in an Edgeworth box. The dashed line running through points T, Y, and V is analogous to a contract curve; that is, iso-profit curves are tangent to each other at points on this line, so it is not possible for firms to adjust prices so that both firms end up making higher profits. The cartel jointprofit maximum will be on this line. The equations used to generate our data give 50 JOURNAL OF ECONOMIC EDUCATION

11 a cartel maximum profit at point Y, with Pa = $12.97 and Pb = $ Notice that both firms are tempted to cheat on the cartel prices. Firm A is enticed leftward toward higher iso-profit curves, and B is attracted downward. EXTENSIONS I believe that a large oligopoly game makes it possible for students to experience a rich array of oligopoly strategies and outcomes. This is a good thing. With a little coaching on the use of Mathematica, it is relatively easy for the instructor to tinker with, tune, and visualize a wide variety of situations. In a class with more than a day or two to cover the topic, other exercises might be tried. For instance, the payoff matrix might be subdivided so that each team sees only its own profits. How does this reduction in information affect the outcome? Carrying that a step further, the professor might not reveal any of the payoff matrix in advance but rather provide profit information to the players only after they have made their choices, leaving it to them to grope toward an acceptable outcome. Although it would be hard to present the game graphically, more advanced classes could play the game with three or more firms by generalizing the demand equations and generating profit outcomes numerically with computers or programmable calculators. NOTES 1. Of course, there are exceptions, particularly in the experimental economics literature. Dolbear et al. (1968) report on an experimental pricing exercise involving a large payoff matrix, although their focus is more on the number of firms in the industry and the extent of available information. See also Kagel and Roth (1997). 2. Game theory texts such as Dixit and Skeath (1999) and Dutta (1999) generally do not contain any large payoff matrices, so they lose the chance to let the intuitive appeal of the discrete game illuminate the more abstract content of the continuous function. With a large game it is relatively easy to see how a function, such as a reaction function, resides in the payoff matrix. This is likely to be helpful for students new to game theory. 3. An unmarked version of the table is available at Matrix.htm. 4. The functions used to generate the table and figures are as follows: A s demand, Qa, is given by 30 3Pa + 50Pb/(Pb + 15). Notice that as Pb approaches infinity, the final term in the equation approaches 50, that is, as B raises price and chases away his own customers, there is a limit to how much effect is felt by A. Also, as Pb heads toward zero, A s demand falls toward 30 3Pa. A s total cost is given by Qa + Qa 2 /10, and profit has the obvious definition. The parallel functions for B are: Qb = Pa/(Pa+10) 2Pb for demand and Qb + Qb 2 /8, for cost. The figures and table were generated using Mathematica, although a simple spreadsheet would work just as well for the table. The Mathematica code used is available on request from the author. 5. This is not the place, and I am not the person, to wade into the legal details of the case, but the brief history goes like this: The telephone conversation between Crandall and Putnam took place February 1, After Putnam turned over the tape, the U. S. Government brought suit against American Airlines (570 F. Supp. 654; 1983 U.S. Dist.). The suit was dismissed September 12, 1983, on the grounds that, although Mr. Crandall might have solicited Mr. Putnam to raise prices, this did not constitute an attempt to raise prices under Section 2 of the Sherman Act. The United States appealed, and on October 15, 1984, the United States Court of Appeals for the Fifth Circuit vacated the lower court s dismissal (743 f.2d 1114; 1984 U.S. app.). American appealed this decision to the U. S. Supreme Court, which dismissed the government s case against American Airlines without comment on November 22, 1985 (474 U.S. 1001; 106 S. Ct. 420; 1985). The text of these decisions is available on the Web via Lexis. Winter

12 REFERENCES Dixit, A., and S. Skeath Games of strategy. New York: W. W. Norton. Dolbear, F. T., L. B. Lave, G. Bowman, A. Lieberman, E. Prescott, F. Rueter, and R. Sherman Collusion in oligopoly: An experiment on the effect of numbers and information. Quarterly Journal of Economics 82 (May): Dutta, Prajit K Strategies and games. Cambridge, Mass.: MIT Press. Kagel, J. H., and A. E. Roth, eds Handbook of experimental economics. Princeton, N. J.: Princeton University Press. 52 JOURNAL OF ECONOMIC EDUCATION