An Extended Duopoly Game

John C. Eckalbar

Abstract: The author shows how principles and intermediate economic studentscan gain a feel for strategic price setting by playing a relatively large oligopolygame. The author constructs a playoff matrix and discusses various strategies andoutcomes. The game extends to a continuous price space and outlines variousapplications appropriate for intermediate micro students. Finally, to make it eas-ier for others to tinker with the assumptions of the game, the author can providethe Mathematica code used to generate the table and figures.

Key words: duopoly, game theory, oligopoly, strategic behaviorJEL 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 areforced to think strategically. When they contemplate an action, such as a pricechange, they must consider the reactions of their rivals, because the effect of theprice change will depend critically on the behavior of their rivals after thechange. Making this fact come alive in a principles or intermediate micro classcan be a challenge.

Most texts present one or two oligopoly games to give the student a feel forthe issues. These games generally consist of two-by-two payoff matrices, and myexperience with them is that they are too small and simple to get the studentsinvolved.1 In this article, I report on the use of a much larger and richer payoffmatrix for the study of strategic duopoly price-setting behavior. I have found ituseful in both principles and intermediate micro classes, and I expect that itwould also be valuable in making tangible some of the more abstract issues in agame-theory class.2

I discuss the uses of a relatively large payoff matrix suitable for a principlesclass. I also consider some features of the underlying game in a continuous pricespace. 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 Iexplain that we are considering a duopoly, that is, an oligopoly with two firms inthe industry. The two firms produce similar products, which the buyers regard assubstitutes. 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 (e-mail: jeckalbar@aol.com).

Winter 2002 41

42 JOURNAL OF ECONOMIC EDUCATION

TA

BL

E1

Pay

off

Mat

rix:

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Pro

fit

(Top

Num

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91

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6.94

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1.56

3.

41

16.8

0 10

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11.6

9 12

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2.89

12

.67

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2 8.

65

6.87

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4.58

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2.40

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1.64

3.

48

5.28

16.6

0 9.

82

10.7

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11.8

1 *1

1.85

11

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8.90

7.

38

5.55

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2.42

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96

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96

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91

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72

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08

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53

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68

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39

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84

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59

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03

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54

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41

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6.

81

8.46

10

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11.6

5

Firm

B’s

pric

e

Winter 2002 43

15.6

05.

36

6.09

6.

51

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1 6.

41

5.89

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06

3.91

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46

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31

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62

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46

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26

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3.

80

2.61

1.

10

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66

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02

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23

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11

.30

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39

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0 4.

09

3.47

2.

53

1.28

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0.59

2.

27

3.91

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52

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8.

64

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48

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90

2.22

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1.04

2.

67

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83

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85

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1.

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88

4.42

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93

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86

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05

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04

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64

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8.

12

9.32

10

.50

Not

e:In

eac

h se

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2 nu

mbe

rs,t

he to

p nu

mbe

r is

fir

m A

’s p

rofi

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bot

tom

num

ber

is f

irm

B’s

pro

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sing

A’s

pri

ces

alon

g th

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p of

the

tabl

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d B

’s p

rice

s al

ong

the

left

sid

e. T

he *

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otes

A’s

max

imum

pro

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ell i

n ea

ch r

ow. T

he n

umbe

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re B

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func

tion.

and B’s price (Pb) in bold along the left side. Each firm’s sales depend on theprice 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 Paand Pb determine our “address,” or fix our position in the table. Each position inthe 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 priceshown along the top of the table, and B charges the price written at the left. Forexample, 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 Itell them that I made them up, or more accurately, I made up some demand andcost equations that seemed reasonable and then used Mathematica to crunch outthe numbers.4 Because the students are generally impatient to get on with thegame, I simply draw their attention to a few general features of the table. Forinstance, 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. Startingfrom 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 movefurther to the right). This is exactly what you would expect from looking at thestandard 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 out-put level where MR = MC, and as the price varies from the optimum, profitsdrop off. You can see the same pattern for B’s profit by looking up and down anycolumn. The other feature to notice is that A’s profit improves as we move upthe columns, and B’s profit rises as we move to the right. For example, when Braises price, some of B’s customers choose to buy from A, and this raises A’sdemand and A’s profit.

Given just these preliminaries, I divide the class into two groups, the As andthe Bs. I generally have a class of about 30 students, with about 15 per team. Inlarger classes, I might run several games concurrently, with opposing teams of10 to 15 students each. As long as the team is about 15 or fewer students, thedialogue flows easily among the team members. Students in larger groups areoften less willing to communicate with each other. Having several pairs ofsmaller opposing teams has its pros and cons—smaller teams mean more indi-vidual participation, but more simultaneous games tend to create bedlam in theclassroom.

I start them off at an arbitrary initial cell; I might say that Pa = $12.80 and Pb =$16.60. In the beginning, we will be civil and take turns adjusting price, as if wewere two gas stations across the street from one another posting prices and thenwaiting to see what our rival does. I let the As go first. Invariably the As talk itup among themselves and agree to cut the price to $12.00 in order to raise theirprofits from $8.90 to $11.85. When the Bs get their turn, they typically cut pricefrom $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 $14.60.Once we get to the $11.60/$14.60 cell, things come to a stop. Someone invari-ably suggests that we have found “the equilibrium,” but I caution that we willfind many different sorts of equilibria depending on the strategies they employ.

44 JOURNAL OF ECONOMIC EDUCATION

At the moment, I will only say that we have found the “follower/follower” orNash equilibrium—A is at an optimum, given B’s current choice, and B is like-wise at an optimum, given the behavior of A. Students are warned not to think ofthis point as the equilibrium.

I give them another starting point and let the game continue. Perhaps they startat Pa = $11.20 and Pb = $15.40. 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/$14.60. 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 max-imum-profit cell in each row, that is, for any given Pb, there will be a Pa thatmaximizes A’s profit. The set of all these cells is A’s reaction function. Thesecells are marked with an asterisk (*) in Table 1. If B were to fix price, A wouldlike to slide left or right to get onto A’s reaction function. Similarly, B has amaximum-profit cell in each column, and the collection of these cells (shown bythe boxes in Table 1) is B’s reaction function. The $11.60/$14.60 cell is at theintersection of the reaction functions, and the orientation of the reaction func-tions, 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 “follow-ers.” For instance, suppose prices sit at Pa = $12.60 and Pb = $16.20, and it is A’smove. Early in the game, firm A is likely to cut price to $11.80, thinking it will beincreasing its profits from $7.69 to $9.79. The problem is that firm A is not antic-ipating the reaction of its rival. If firm A were thinking strategically, it would com-pare its initial profits with the profit it would earn following B’s most likely reac-tion. For instance, if B cuts price to $14.60 following A’s cut to $11.80, A’s profitwould be only $0.93, which is far worse than its original profit. Once this point ismade in the classroom, the students generally behave quite differently, and muchmore 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 classschedule so that the class ends at this point, and the students are instructed toreturn 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. Ithen reform the teams and let the game begin. I will let the firms adjust prices atwill, no longer requiring that they take turns. It often happens that one of thefirms will discover there might be an advantage to behaving malevolently. Sup-pose, for instance, the class is at the follower/follower solution. Someone on theB 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 Asquickly realize that if they stay at Pa = $11.60 with Pb = $14.40, the As will gobroke, whereas the Bs continue to make a profit. This generally prompts the Asto cut price to $11.40, even though that makes their loss worse, because it nowcauses B to lose money. In fact, if B stayed there, B would now lose more thanA. 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 winthe price war, but at least it prompts a discussion of the subject. With just a little

Winter 2002 45

prodding, the students will bring up other factors, such as liquid assets, lines ofcredit, and the advantage of having low cost at the start of any price war. The caseof People’s Express and United Airlines might be mentioned. We could also talkabout an arms race.

When the students have tired of trying to ruin each other, I reposition them inthe top right portion of the table once again, for instance, at Pa = $12.80 and Pb =$16.20. This time, they are much more circumspect about their moves, and thereis a lot of discussion along the lines of, “If we do that, what are they going todo?” They are now learning the meaning of strategy. They have seen the gamedegenerate 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 nowprepared to see oligopoly price stickiness in a new light. Perhaps a fear of pre-cipitating a price war would justify a policy of leaving well enough alone whenconditions are tolerable. Generally, a little time draws us slowly back toward thefollower/follower solution, or worse.

When things again degenerate, I read them the now-famous transcript of theconversation between Robert Crandall, the President of American Airlines, andHoward Putnam, the President of rival Braniff. I warn the students (with a wink)that the conversation between these two CEOs contains some high level busi-ness-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 bothlive here and there ain’t no room for Delta. But there’s, ah, no reason that I cansee, 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 Iwill 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 notwhat the courts found.5 American Airlines is now the second largest airline in theworld, and Braniff is bankrupt.

Now I have the As take the role of American Airlines, and the Bs play the partof Braniff. I ask them to imagine that Mr. Putnam agreed to meet Mr. Crandalland discuss a price-setting deal. I ask each team to make a concrete proposal tothe other side, something like, “We will charge _____, if you will charge _____.”

The students seem to enjoy this simulated clandestine activity. They beginlooking in the upper right corner of the matrix, and inevitably each side looks fora cell that offers better profits for both of them, but gives a slight advantage tothe team making the proposal. For instance, the As might propose Pa = $12.80and Pb = $16.40, whereas the Bs favor something more like Pa = $12.80 andPb = $16.00. They typically wrangle back and forth, sometimes threatening the

46 JOURNAL OF ECONOMIC EDUCATION

other side with a price war if the last offer is refused. The cell at Pa = $12.80 andPb = $16.20 usually gets a lot of attention, because profits are about evenly splitat that location.

This is a good point to discuss cartels and joint profit maximum. The two firmsmight adopt a strategy of searching for the cell that brings in the largest totalprofit, that is, the cell that maximizes the sum of A’s profit and B’s profit. Oncethey have gotten the maximum from their customers, they might then decide toredistribute the subtotals, but at least then they would be arguing about cutting upthe biggest possible pie. The joint profit maximizing cell is at Pa = $13.00 andPb = $16.40, where A earns $6.09 and B gets $6.72. The Bs seem happy, but theAs argue about how big a kickback they deserve for going to this cell. In fact, thejealousy of the As seriously undermines the establishment of the joint profit out-come. 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 thetwo firms to reach an agreement. Each firm wants to bias the deal in its ownfavor, and neither is eager to concede the advantage to the other. Second, thesedeals are illegal, so they cannot be enforced in the courts, and there is a real riskof 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 inter-ests of the two firms pull them in other directions.

Suppose, for example, the firms settle at the point Pa = $12.80 and Pb =$16.20. Each is doing quite well in comparison with the follower/follower solu-tion or in comparison with a price war, but each is also tempted to cheat on thedeal, and there is no enforceable contract to prevent it. Firm A sees its profit ris-ing 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 prof-its for each firm falling. This is a good spot to bring up the history of OPEC, totalk about how the cartel has followed a pattern of formulating an agreement andholding prices and profits up for a while, only to be followed by an erosion oftheir 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 (upperright corner of the matrix), but inevitably it rolls back down.

I ask the students what other options are available, and someone always sug-gests that one of the companies might buy the other. This prompts some inter-esting discussions. Suppose, for instance, we are at the follower/follower solu-tion with A making $1.09 and B making $1.35. Suppose B wants to buy A andoperate it as a separate, autonomous division. What is a fair price for firm A? Afinance major might suggest that firms are “worth” about 25 times earnings thesedays, so the stock of firm A might be worth 25 × $1.09 = $27.25. Someone onthe A team will reply, “Yes, but once you buy us and we no longer compete, youcan set both prices, and earnings from the A division will be over $6, plus B divi-sion earnings will be much better without competition from A. So maybe a stockprice of at least 25 × $6 = $150 is in order.” That certainly leaves a lot of roomfor negotiation. It also provides an opportunity to discuss antitrust policy in lightof recent mergers and proposed mergers.

One strategy that students are unlikely to devise on their own is Stackelberg-

Winter 2002 47

type price leadership. I suggest the following: What if B decided to fix price atsome level and then simply refused to adjust it any further. At the same time, firmA decided to quit fighting and simply charge the best price it could, given B’sprice. Here B is the leader, and A is the follower. What price would B set? Withjust a little reflection, the students realize that B must search A’s reaction func-tion for the cell that yields the largest profit for B. This occurs when Pb = $15.20.If B sets its price at $15.20, A will charge $11.80. Profits now are $4.40 for Aand $2.27 for B. If A leads and B follows, the players go to Pa = $12.00 and Pb = $14.80.

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 con-tact with each other and negotiate the prices. There will be no smoking gun ortaped phone call. Second, both firms do better under price leadership than theydo under the follower/follower option. In this particular game, the followermakes out relatively better than the leader, and that may make both firms reluc-tant 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 ofmodels of price leadership by a dominant firm and cartel profit maximization,using familiar demand and cost curves. My impression is that these topics goover much better once the students have had some first-hand experience withthese 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-profitcurves and then find the reaction functions from them. A mathematical derivationof the iso-profit curves is generally beyond the level of an intermediate class, butthe table suggests that, for instance, A’s curves take on the appearance shown inFigure 1. Profits rise as we move up in the figure. A’s reaction function is thelocus of zero-slope points on A’s iso-profit curves, that is, the locus of points atwhich A’s profits are at a maximum given a fixed price from B. B’s iso-profitcurves and reaction function, not shown here, are similar to A’s, but rotated 90degrees clockwise.

If we combine the two reaction functions, we can see what generally happensin the early stages of the game. The two reaction functions together with somesample paths toward the follower/follower equilibrium are shown in Figure 2.The shaded areas in the acute regions between the reaction functions constitute astrongly attractive set. Given the way the students generally begin playing thegame, if we are not in the set, we are attracted to it, and once we are drawn intoit, 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 likepoint 2 better than point 1, they cut their price. But when B responds, we go topoint 3. If the A team were thinking strategically, it would compare point 3 withpoint 1, rather than point 2 with point 1.

48 JOURNAL OF ECONOMIC EDUCATION

Suppose B is the price leader. We then want to know where B’s profits reacha maximum along A’s reaction function. We can visualize this by superimposingB’s iso-profit curves over A’s reaction function. If B leads, then B is looking forthe point of tangency between its iso-profit curves and A’s reaction function (Figure 3).

Winter 2002 49

FIGURE 1Firm A’s Iso-Profit Curves and Reaction Function, Rfa

FIGURE 2 Movement Toward the Reaction Functions and Convergence to

the Follower/Follower Equilibrium

3

2

If we combine the two sets of iso-profit curves, we get Figure 4. When studentsfirst 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 twofirms are presently at point X in the figure. With appropriate price increases, theycould move into the shaded region where both firms will earn higher profits. Thisis reminiscent of a “better set” in an Edgeworth box. The dashed line runningthrough points T, Y, and V is analogous to a contract curve; that is, iso-profitcurves are tangent to each other at points on this line, so it is not possible for firmsto adjust prices so that both firms end up making higher profits. The cartel joint-profit maximum will be on this line. The equations used to generate our data give

50 JOURNAL OF ECONOMIC EDUCATION

FIGURE 3 Finding the Price Leadership Solution for B from A’s Reaction

Function and B’s Iso-Profit Curves

FIGURE 4Combining Both Iso-Profit Curves

a cartel maximum profit at point Y, with Pa = $12.97 and Pb = $16.2873. Noticethat both firms are tempted to cheat on the cartel prices. Firm A is enticed leftwardtoward higher iso-profit curves, and B is attracted downward.

EXTENSIONS

I believe that a large oligopoly game makes it possible for students to experi-ence a rich array of oligopoly strategies and outcomes. This is a good thing. Witha little coaching on the use of Mathematica, it is relatively easy for the instruc-tor 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 mightbe tried. For instance, the payoff matrix might be subdivided so that each teamsees only its own profits. How does this reduction in information affect the out-come? Carrying that a step further, the professor might not reveal any of the pay-off matrix in advance but rather provide profit information to the players onlyafter they have made their choices, leaving it to them to grope toward an accept-able outcome. Although it would be hard to present the game graphically, moreadvanced classes could play the game with three or more firms by generalizingthe demand equations and generating profit outcomes numerically with comput-ers or programmable calculators.

NOTES

1. Of course, there are exceptions, particularly in the experimental economics literature. Dolbear etal. (1968) report on an experimental pricing exercise involving a large payoff matrix, althoughtheir 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 anylarge payoff matrices, so they lose the chance to let the intuitive appeal of the discrete game illu-minate the more abstract content of the continuous function. With a large game it is relativelyeasy to see how a function, such as a reaction function, resides in the payoff matrix. This is like-ly to be helpful for students new to game theory.

3. An unmarked version of the table is available at http://www.csuchico.edu/econ/Faculty/Matrix.htm.

4. The functions used to generate the table and figures are as follows: A’s demand, Qa, is given by30 – 3Pa + 50Pb/(Pb + 15). Notice that as Pb approaches infinity, the final term in the equationapproaches 50, that is, as B raises price and chases away his own customers, there is a limit tohow 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 170 + Qa + Qa2/10, and profit has the obvious definition. The parallelfunctions for B are: Qb = 26 + 40Pa/(Pa+10) – 2Pb for demand and 205.5 + Qb + Qb2/8, for cost.The figures and table were generated using Mathematica, although a simple spreadsheet wouldwork just as well for the table. The Mathematica code used is available on request from theauthor.

5. This is not the place, and I am not the person, to wade into the legal details of the case, but thebrief history goes like this: The telephone conversation between Crandall and Putnam took placeFebruary 1, 1982. After Putnam turned over the tape, the U. S. Government brought suit againstAmerican 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 raiseprices, 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 theFifth Circuit vacated the lower court’s dismissal (743 f.2d 1114; 1984 U.S. app.). Americanappealed this decision to the U. S. Supreme Court, which dismissed the government’s caseagainst 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 2002 51

REFERENCES

Dixit, A., and S. Skeath. 1999. 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. 1968.

Collusion in oligopoly: An experiment on the effect of numbers and information. Quarterly Jour-nal of Economics 82 (May): 240–59.

Dutta, Prajit K. 1999. Strategies and games. Cambridge, Mass.: MIT Press.Kagel, J. H., and A. E. Roth, eds. 1997. Handbook of experimental economics. Princeton, N. J.:

Princeton University Press.

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