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Cooperation in symmetric duopolies with demand inertia
International Journal of Industrial Organization 18 (2000) 23–38 www.elsevier.com / locate / econbase
Cooperation in symmetric duopolies with demand inertia Claudia Keser a,b , * a
¨ Statistik und Mathematische Wirtschaftstheorie, Universitat ¨ Karlsruhe, Karlsruhe, Institut f ur Germany b ´ , Quebec ´ , Canada H3 A 2 A5 CIRANO, Montreal
Abstract We present an experiment on a price-setting duopoly with demand inertia and symmetric costs. Subjects gain experience in the game by playing it twice. In the first play, we observe prices above the subgame perfect equilibrium solution. All markets show considerable price instability. In the second play, the price level is higher than in the first play. About half of the markets exhibit price instability, while the remaining markets are characterized by high stable prices. Comparing these results with those of a similar duopoly with asymmetric costs, we conclude that markets with symmetric costs show significantly more cooperative behavior. 2000 Elsevier Science B.V. All rights reserved. Keywords: Experimental economics; Duopoly; Dynamic games JEL classification: C90; D43; C73
1. Introduction This article reports the results of an experiment on a multiperiod duopoly game with symmetric costs where prices in each period are the only decision variables. In this game, introduced by Selten (1965), demand is assumed to exhibit inertia.1
* Corresponding author. Tel.: 11-514-985-4000 ext. 3014; fax: 11-514-985-4039. E-mail address: [email protected] (C. Keser) 1 Also the experiment by Hoggatt (1959) is based on an oligopoly game with demand inertia. Fischer (1988) simulates oligopolistic market processes under the assumption of demand inertia. Phlips and Richard (1989) generalized Selten’s oligopoly model with demand inertia by allowing firms to carry inventories. 0167-7187 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 99 )00032-6
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This takes into account the idea that consumers do not instantaneously react to price changes.2 The multiperiod duopoly game has a unique subgame perfect equilibrium solution. In the experiment, we observe that the actual individual price-setting behavior of subjects is different from what is prescribed by the subgame perfect equilibrium solution. This is true on both the individual and the aggregate level. This is not a very surprising result, since the subgame perfect equilibrium solution is too complicated to be computed by a typical subject in an experimental game-playing situation (Simon, 1957). Another result of the experiment concerns the impact of experience on subject behavior in the multiperiod duopoly situation. Given the dynamic structure of the game, we cannot expect to identify learning effects within a game. For this reason, each subject played the multiperiod duopoly game twice. This allows us to evaluate the impact of subjects’ experience from the first to the second play of the dynamic game. We observe that, when subjects play the game for the first time, average prices over all markets are above the subgame perfect equilibrium prices in each period. When subjects play the game for the second time, the observed price level is even higher than the price level in the first play. Thus, the price level approaches the symmetric Pareto efficient price, which constitutes a prominent cooperative solution to the game. Furthermore, in the second play, subjects exhibit more stable price-setting behavior than in the first play. Consequently, the profits gained in the second play are significantly higher than the profits in the first play, and significantly higher than the profits in the subgame perfect equilibrium solution. We conclude that, in the aggregate, experience in playing the specific multiperiod duopoly game leads actual behavior away from the subgame perfect equilibrium solution toward a more cooperative solution. This result is in accordance with several earlier experimental studies on oligopolies (Stoecker, 1980; Benson and Faminow, 1988; Selten et al., 1997) and, in particular, with those of Keser (1992, 1993). In Keser (1992, 1993) we examined a very similar experimental duopoly which differed from this experiment in that it had an asymmetric cost structure. We observed there as well a significant tendency toward cooperation when subjects
2
Demand inertia may be caused by various factors, such as product differentiation, lack of market transparency, or customer loyalty. The interpretation of the multiperiod duopoly with demand inertia in terms of product differentiation is restricted by a property which Selten (1965) calls quasi-homogeneity: in the case of a constant market demand, price differences should cause an increase of the cheaper firm’s market share. The assumption of missing market transparency is fundamental to the models with stochastic buyer behavior by Schmalensee (1978) and Friedman (1991). Customer loyalty is an important phenomenon in economic life. This is shown in many empirical consumption investigations (see, for example, Kroeber-Riel and Trommsdorff, 1973). Selten (1970) observed considerable stability in the trading relationships in a market experiment where sellers and buyers could freely negotiate and trade a homogeneous good.
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were experienced. However, we observed there a significantly lesser degree of cooperation than in the symmetric cost situation. This difference can be explained by the fact that in the symmetric duopoly it is more obvious than in the asymmetric duopoly where cooperation should take place. Also Mason et al. (1992) provide experimental evidence that symmetry facilitates cooperation in repeated Cournot oligopoly games. The results of our experiment clearly indicate that the behavior of subjects in the dynamic duopoly situation is not guided by the principles of full rationality on which the game–theoretic solution is based. Selten (1990) argues that limits on human rationality need not only be cognitive; they might also be motivational. In the dynamic duopoly situation, motivational limits might arise because subjects recognize the possibility of higher profits for both firms through cooperation. In particular, if each firm charges in each period the price which maximizes its one-period profit, then a kind of cooperative outcome emerges. It is the symmetric Pareto efficient solution of the game, which we also denote as the myopic monopoly. At the same time, however, a firm’s interest in its own long-run profit may prevent cooperation. A firm might be tempted to set low prices in early periods in order to make big sales which, due to demand inertia, creates an advantage for the rest of the game. Low prices in early periods risk launching a price war and, thus, preventing cooperation. Indeed, we observe continual price wars in all markets of the first play of the experiment. Also in the second play, about half of the markets show price wars. However, in the other half of the markets in the second play, both firms coordinate from the beginning on prices close to the symmetric Pareto efficient price. Thus, the experiment provides us with evidence on how subjects deal with the conflict between short- and long-run interests inherent in the dynamic structure of the game. Given the importance of individual long-run considerations, the frequency of markets with high stable prices is surprising. In these markets, each of the two firms principally behaves in each period as if it were a monopolist, fixing a price which maximizes individual one-period profit and ignoring its individual interest in creating long-run advantages. The experiment on a dynamic duopoly situation presents an interesting expansion on experimental research on oligopolies. The dynamic structure of the game allows us to study aspects of subjects’ decision making which do not play a role in less complex oligopoly situations. Behavior in a more complex multiperiod oligopoly situation was, for example, analyzed by Selten (1967a,b,c) who observed no cooperation in his experiments. The bulk of oligopoly experiments, however, examines repetitions of the same simple game situation. There is no dynamic relationship between the repetitions. The extent to which cooperation is observed in these experiments depends, among others, on the subjects’ knowledge about the game situation, their experience in playing the specific game, and on the opportunities for communication. Surveys of this literature are given by, for example, Friedman (1969), Davis and Holt (1993, pp. 196–199), and Holt (1995).
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2. The model The model is one of a multiperiod price-setting duopoly with symmetric costs. A dynamic relationship between the periods arises because sales not only depend on current prices but also on past sales. This phenomenon can be explained by demand inertia: customers though tending to buy from the low price seller also tend to stay with the same seller. This model belongs to the class of games analyzed by Selten (1965). In his seminal paper, Selten introduced the notion of subgame perfect equilibrium to solve an oligopoly game with demand inertia. In our study, the parameters of the duopoly are chosen such that the game has a unique subgame perfect equilibrium solution. The first part of this section presents the game in the same way as it was presented to the subjects of the experiments. The second part of the section describes the subgame perfect equilibrium solution and the so-called myopic monopoly solution. The latter is a cooperative solution which is not based on game–theoretic considerations but on the concept of prominence (Schelling, 1960). Both solutions are considered as benchmarks to compare the observed behavior with.
2.1. The game There are two firms in the market offering a homogeneous product over 25 periods. The only decision variable of each firm is its price in each period. Let x ti be the quantity sold by firm i in period t. This quantity is assumed to depend linearly on the firm’s own price in period t, p it , and on a variable D it which is called the firm’s demand potential in period t: x ti 5 D ti 2 p ti
i 5 1,2; t 5 1, . . . ,25
(1)
In the first period, the demand potential is the same for each firm: D 1i 5 200
i 5 1,2
(2)
In the following periods, a firm’s demand potential is determined by its former demand potential plus half of the price difference of the two firms in the market in the previous period: D ti 5 D ti 21 1 ( p tj 21 2 p ti 21 ) / 2
i 5 1,2; j 5 3 2 i; t 5 2, . . . ,25
(3)
We see that if a firm has offered its product in period t21 at a lower price than the other firm in the market, then its demand potential for period t increases. The demand potential for the other firm decreases by the same amount. Obviously, the average demand potential remains constant over time; what might change from
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one period to another is how the demand potential is divided between the two firms. The firms are assumed to produce exactly as much as they can sell. Production costs result from constant unit cost, c, which is the same for both firms: c 5 71
(4)
Therefore, the profit g ti of firm i in period t is given by: g it 5 ( p it 2 c) x it
i 5 1,2; t 5 1, . . . ,25
(5)
This short-run profit is credited after each period to an account which yields an interest rate of 1 percent per period. Thus, firm i’s total profit after 25 periods is given by:
O 1.01 25
G 25 i 5
252t
g it
i 5 1,2
(6)
t 51
In the following, we call this firm i’s long-run profit. The procedure of the game is the following. In each period t, the two firms in the market independently decide on their prices for that period. Both firms are completely informed about the rules of the game. They know the formulas and also all parameters of the model. Furthermore, they have complete information on all that has happened in their market so far. Thus, they know their current individual demand potentials when they set their prices. They also know that the game ends after 25 periods. Each firm is assumed to maximize its long-run profit. The firms have to obey two limitations when they decide on their prices. First, they are not allowed to choose prices above their current individual demand potentials. This would yield negative sales according to formula (1). Secondly, to prevent dumping, they are not allowed to set prices below their own unit cost of production. However, in the case that a firm’s demand potential would fall below its unit cost, the price is automatically set at its demand potential so that the firm does not sell anything. Thus, neither a firm’s demand potential nor its short-run profits can ever become negative.
2.2. Theoretical features Applying the game–theoretic concept of subgame perfect equilibrium, the game is solved by backward induction. The result is a uniquely determined system of linear decision rules. The solution presented below follows directly from a translation of the results obtained by Selten (1965) to the specific game situation considered. The price paths in the subgame perfect equilibrium solution are prescribed by linear functions of the individual demand potentials: p ti * 5 a t (D ti 2 200) 1 b t
i 5 1,2; t 5 1, . . . ,25
(7)
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The parameters a t and b t are time-dependent but the same for both firms. They are determined by the market demand potential, the unit production cost common to both firms in the market, and the interest rate. They can be calculated by the recursive system given in the Appendix A. In the subgame perfect equilibrium solution, both firms set the same prices. Therefore, the demand potential is always shared equally. The equilibrium prices remain, during the first 20 periods, at a level of about 100 until they increase considerably because toward the end of the game long-run considerations become of minor importance. The equilibrium price in the last period is the myopic monopoly price, which is described by Eq. (8) below. The resulting long-run equilibrium profit is 85,336 for each firm. The subgame perfect equilibrium solution is not Pareto efficient. Thus, there is room for cooperation where both firms can make higher profits. We consider the symmetric Pareto efficient solution as a reference point of cooperation to compare the observed behavior with. We also call it the myopic monopoly solution because both firms in each period charge prices as if they were monopolists, without considering that there might be consequences for following periods. In each period, each firm maximizes its short-run profit (Eq. (5)), which depends on the firm’s demand potential and own current price but is independent of the other firm’s price in the current period. The resulting myopic monopoly price of firm i in period t is given by: p t,M 5 (D ti 1 c) / 2 i
i 5 1,2; t 5 1, . . . ,25
(8)
If both firms in the market behave according to the myopic monopoly solution, they set prices of 135.50 in each period. Demand potential is shared equally and the resulting long-run profit for each firm is 117,499.
3. Organization of the experiments The subjects were 48 students at the University of Bonn.3 We organized four experimental sessions with 12 subjects each. A session lasted 3–4 h. The subjects sat in separate cubicles and made their decisions via computer terminals. They did not interact in any other way. Before the experiment started, they were instructed about the game and the use of the computer program for about three quarters of an
3 They consisted of 37 economics students, eight law students, and one student each from mathematics, physics, and computer science. A few of the subjects had previous knowledge of oligopoly or game theory. None of the subjects had participated in an oligopoly experiment before.
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hour.4 Then they played the 25-period game twice. Communication was prohibited, including during the short break between the first and the second play of the game. For the first play of the multiperiod duopoly game, the 12 subjects of a session were divided into six groups of two firms each to form six duopoly markets. None of the subjects knew which other subject of the group was the firm he was interacting with in his market. In the second play, each subject interacted with a subject different from the one in the first play. Specifically, and this without the subjects’ knowledge, the organization of each experimental session was such that four subjects always formed an independent player group. Each player of such a group interacted only with other players of the same group. Thus, we obtained three independent observations per session and twelve independent observations in total. In each of 25 periods, the two firms in each market independently had to choose their prices. At the beginning of the next period, each firm was informed about its own and the other firm’s sales, profits, and new demand potential. At any time, a subject could review his previous results. Since in each period each subject knew his current demand potential, he could compute his sales and profits resulting from any price he was allowed to set. A calculation aid reported these figures for hypothetical prices to be selected by the subject. The subjects were paid in cash at the end of each session according to their balances after period 25 of each play. Each subject knew that his or her payment for a play was determined in the following way. We calculated the surplus of his total profit over the average total profit realized by all subjects in the other markets of the session. Obviously, the surplus must be negative for some subjects if it is positive for others. We multiplied the surplus by an exchange rate of 0.00033 and added it to 16 deutsche marks.5
4. Experimental results In this section, we analyze the average behavior over all subjects and compare it to the behavior prescribed by the theoretical solutions. It is, however, important to note that the behavior in the individual markets varies considerably. Therefore, in
4
The subjects did not receive any written instructions. This method of payment is strategically equivalent to simply multiplying a subject’s total profit by a fixed exchange rate. A subject’s profit is compared solely to profits of subjects with whom he is not interacting. Thus, the subject cannot have any influence on these other profits through his own decision-making. He has to take the other profits as given. We chose this method of payment because it allows one to fix in advance the total amount of money to be paid to all subjects. Obviously, we have to take into account the possibility that in spite of strategic equivalence, this method of payment may have led to different behavior than if we simply had multiplied each subject’s profit by a fixed exchange rate. 5
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Section 5 below, we will present an analysis of the individual market behavior. In both sections we use nonparametric statistics following Siegel (1987). The data are posted at URL http: / / www.econ.ku.dk / cie / ijio / exptsup.htm.
4.1. Prices and profits In Fig. 1 we compare, for the first and the second play, the realized prices on average over all firms in each period to the price paths in the equilibrium and the myopic monopoly solution. We see that in both the first and the second play the actual average prices are above the equilibrium prices. Furthermore, average prices in the second play are above those of the first one although they are still considerably below the myopic monopoly prices. To test whether the actual prices significantly tend to be above the equilibrium prices, we count the number of markets (in the first play) and the number of independent player groups (in the second play) where 50 percent or more of the prices are below or equal to the equilibrium prices.6 Among the 24 markets of the first play, we count seven with 50 percent or more of the prices below or equal to the equilibrium prices. Thus, the one-tailed binomial test rejects the null hypothesis at the five percent significance level. Among the twelve independent player groups of the second play, we find one with 50 percent or more of the prices below the equilibrium prices. The one-tailed binomial test rejects the null hypothesis at the one percent significance level. Next, we compare the actual prices with adjusted equilibrium prices. In each
Fig. 1. Dynamics of the realized average prices over all firms in the first play and in the second play, compared to the prices in the subgame perfect equilibrium and the myopic monopoly solution.
6
The smallest independent units to be considered are the markets in the first play and the independent player groups in the second play.
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period, we consider the actual individual demand potential and calculate, based on Eq. (7), the equilibrium price according to the actual demand potential. Obviously, the actual individual demand potential of the two firms in a market might differ from 200 in a given period, so that the adjusted equilibrium price in that period is off the subgame perfect equilibrium price path.7 We find the same result as before. Seven of the 24 markets in the first play and one of the 12 independent player groups in the second play have 50 percent or more of the prices below or equal to the respective adjusted equilibrium prices. Thus, we may conclude that prices tend to be significantly higher than the adjusted subgame perfect equilibrium prices. Comparing the first and the second play, we observe a significant tendency toward higher prices in the second play. Among the 12 independent player groups there are 11 for which we observe that the average prices increase from the first to the second play. The one-tailed binomial test rejects the null hypothesis at the one percent significance level. Comparing the actual prices to the myopic monopoly price, we observe in none of the 24 markets of the first play and in none of the 12 independent player groups of the second play, that 50 percent or more of the prices are above or equal to 135.50. The one-tailed binomial test rejects the null hypothesis at the one percent significance level for both the first and the second play. The same result holds if we compare the actual prices with the adjusted monopoly prices. The latter are, in each period, computed by Eq. (8), based on the actual demand potential in that period of the firm under consideration. The realized average long-run profit over all firms is 87,333 in the first play, while it is 102,511 in the second play. Thus, in the first play, the average profit is only two percent higher than the subgame perfect equilibrium profit, while it is twenty percent higher in the second play. In the first play, we observe 16 markets whose average profits are above the equilibrium profit, while there are eight markets with average profits below the equilibrium profit. At the five percent significance level, the one-tailed binomial test does not reject the null hypothesis that profits tend to be equally likely below and above the subgame perfect equilibrium profit. In the second play, we observe only one independent player group with an average profit below the equilibrium profit, while 11 independent player groups made profits above the equilibrium profit. The one-tailed binomial test rejects the null hypothesis at the one percent significance level. Thus, profits in the second play tend to be significantly higher than the equilibrium profit. Furthermore, profits realized in the second play tend to be significantly higher than profits in the first play. Among the 12 independent player groups there are 11 whose profit increases in the second play. The one-tailed binomial test rejects the
7 In this adjusted calculation, we take into account that the subjects in a market might not have behaved according to the subgame perfect equilibrium solution in any previous period but that they obey the equilibrium pricing rule (as given by Eq. (7)) for the current period.
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null hypothesis at the one percent significance level. Note that in neither play does the average realized profit come close to the myopic monopoly profit.
4.2. Price instability The individual pricing behavior in the second play appears more stable than in the first play. To illustrate this, we need a measure for the instability of the prices set by a firm in a given play. We define the price instability measure Si for a firm i as:
O (p 2p 25
Si 5
t i
t21 2 i
)
i 5 1, . . . ,48
(9)
t52
This measure computes the sum of a firm’s squared price deviations from its own price in the previous period. We add up the values of this measure for all firms in each independent player group in the first and second play and test whether prices in the second play are less unstable. In 11 of 12 independent player groups we observe a lower price instability in the second play. The one-tailed binomial test rejects the null hypothesis at the one percent significance level.
5. The behavior on the individual markets The various markets display a wide variety of price dynamics. In order to uncover some characteristic features of market behavior, we roughly classify the observed markets into strongly cooperative, weakly cooperative, and aggressive markets. This classification scheme follows the one given in Keser (1992, 1993) for a similar multistage duopoly with asymmetric costs. Somewhat arbitrarily, we use the price level (i.e. the average price of the two firms in the market over the 25 periods) as the only classification device. Markets with a price level above 125 are classified as strongly cooperative, markets with a price level below 100 are classified as aggressive, and all markets with a price level between these two benchmarks are classified as weakly cooperative. In the first play, we classify only four markets as strongly cooperative while 13 are weakly cooperative and seven are aggressive ones. With the exception of one market where prices are almost constantly at a price level close to 135 until period 15, the behavior in the strongly cooperative markets does not differ very much from the behavior in the weakly cooperative and aggressive markets, except for the level of the prices set. We generally observe considerable price movement in all three market types. In many markets, we actually observe abrupt price jumps, both up and down, which may be quite far-reaching. Subjects seem to compete for demand potential. Prices go down and up again, probably because the conflicting aims of increasing one’s demand potential and of gaining a high short-run profit alternately determine the price decision of a subject. We observe that the larger the
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difference between the initial prices set by the two firms in a market, the stronger will be the price instability in the subsequent periods (Spearman rank correlation coefficient of 0.77 in the first play and 0.49 in the second play). Note that market prices and profits are strongly correlated (Spearman rank correlation coefficient of 0.95 in the first play and 0.87 in the second play). Thus, it is not surprising that the three market categories differ in their profits: firms in the strongly cooperative markets gain higher profits than the firms in the weakly cooperative markets, and the firms in the weakly cooperative markets gain higher profits than those in the aggressive markets (one-tailed Mann–Whitney U-test, one percent significance level). Of the 24 markets in the second play, we classify 12 markets as strongly cooperative, 11 as weakly cooperative, and only one as aggressive. In the aggressive market the prices of both firms are permanently very low. This market makes the lowest profit among all markets. Again, the strongly cooperative markets make significantly higher profits than the weakly cooperative markets (one-tailed Mann–Whitney U-test, one percent significance level). Strongly and weakly cooperative markets in the second play show additional differences other than those with respect to price level and market profit. In eight of the 12 strongly cooperative markets, both firms set stable prices above 130 during most of the 25 periods. In some of these markets, we observe some price movement, but solely at the beginning of the game, caused by a difference in the initial prices of the two firms, or at the end of the game, a so-called end effect (Stoecker, 1983; Selten and Stoecker, 1986). Only four strongly cooperative markets show, over the whole play, some moderate ups and downs of prices. In the weakly cooperative markets, continual price movement is typical. In many of these markets, we observe even extreme price jumps. We can show that the prices on the weakly cooperative markets significantly tend to be more unstable than those on the strongly cooperative markets (one-tailed Mann–Whitney U-test based on the sum of the price instability measures of the two firms in a market, one percent significance level).
6. Comparison with Keser (1992, 1993) In Keser (1992, 1993) we presented the experiment on a related duopoly with asymmetric costs. Similarly to our results on the symmetric duopoly, we observed a significant tendency toward more cooperative behavior when subjects were experienced in playing the dynamic game. The degree of cooperation in the second play, however, appears higher in the symmetric situation than in the asymmetric one.8
8
Note also that, in contrast to the symmetric situation, first play prices in the asymmetric situation were not significantly different from the subgame perfect equilibrium solution.
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To give statistical support to this observation, we consider the play of experienced subjects in both situations. Note that all parameters except for the costs are the same. In the asymmetric duopoly, costs are 57 for the low cost firm, and 71 for the high cost firm. In the symmetric duoploy, costs are 71 for both firms. Table 1 shows for both situations the equilibrium profit (and price) on average over the two firms in a market, the average profit (and price) realized in the second play of the respective experiment, and the myopic monopoly profit (and price) on average over the two firms in a market. Let us designate the proportion of profit gained relative to the myopic monopoly profit as efficiency. The efficiency of the subgame perfect equilibrium solution is 75 percent in the asymmetric game and 73 percent in the symmetric game. Thus, in the subgame perfect equilibrium solution, the asymmetric game is slightly more efficient than the symmetric game. However, the efficiency of the actually realized profits is higher in the symmetric game than in the asymmetric game. On average over all subjects, it is 87 percent in the symmetric game and 77 percent in the asymmetric game. The one-sided binomial test based on the actual efficiencies of the independent player groups rejects the null hypothesis at the five percent level. We conclude that the actual efficiency is significantly higher in the symmetric situation than in the asymmetric situation. Similarly, we consider the relative closeness of the prices realized over the whole game to the respective myopic monopoly price. We find that the average realized prices are significantly closer to the myopic monopoly prices in the symmetric games (89 percent of the myopic monopoly price) than in the asymmetric games (80 percent of the myopic monopoly price). The significance level is five percent of the two-sided binomial test. Considering the sum of the instability measures of all subjects within an independent player group, we find no significant difference between the symmetric and the asymmetric game. In somewhat arbitrary ways, we classified both the symmetric and the asymmetric markets into strongly cooperative, weakly cooperative, and aggressive markets. The frequencies of each market type in both cost situations are presented in Table 2. In the asymmetric game, we classified about one third of the markets as strongly cooperative, another third as weakly cooperative, and the remaining third as aggressive. In the symmetric game, we classified one half of the markets as Table 1 Profits and prices in equilibrium, myopic monopoly, and the second play of the symmetric and the asymmetric duopoly Cost situation
Average profit in equilibrium
Average profit realized
Average profit in myopic monopoly
Average price in equilibrium
Average price realized
Average price in myopic monopoly
Symmetric Asymmetric
85,336 99,113
102,548 101,317
117,499 131,878
103.11 97.85
121.14 106.24
135.50 132.00
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Table 2 Classification of markets in the second play of the symmetric and the asymmetric duopoly Cost situation Symmetric Asymmetric
Number of markets classified as Strongly cooperative
Weakly cooperative
Aggressive
12 9
11 10
1 11
strongly cooperative and the other half (with one exception) as weakly cooperative. Applying a x 2 -test, we find that symmetric and asymmetric markets differ significantly with respect to their classification into strongly and weakly cooperative and aggressive markets (five percent significance level). Although the classification is largely based on ‘visual inspection of the graphs of the price dynamics’ (Keser, 1993, p. 143) we consider this difference as additional evidence for the stronger tendency in symmetric markets to cooperate than in asymmetric markets.
7. Conclusion In the experiment we observe that the actual behavior of subjects in the dynamic duopoly situation is different from what is prescribed by the subgame perfect equilibrium solution. Experience in playing the dynamic duopoly game moves subjects’ behavior even further away from the equilibrium toward more cooperation. When subjects play for the second time, there is a significant tendency toward higher prices and profits, and toward a more stable price behavior than in the first play. These results are similar to those in Keser (1992, 1993) where the same type of duopoly experiment with asymmetric costs is presented. We can show, however, that cooperation occurred there to a significantly lesser extent than in our experiment with symmetric costs, where it is more obvious how to cooperate. Although the behavior of individual markets varies considerably, we are able to identify general characteristics. In the first play, the most striking feature, common to almost all markets, is that prices permanently move up and down, often by a considerable amount. The profit of the two firms in a market is determined by their average price level. In the second play, we still find abrupt price movements in about half of the markets. These markets show relatively low price levels and profits. The other half of the markets in the second play show more or less stable prices at a high level. In these markets both firms make high profits. The dynamic structure makes the game difficult for the players to grasp because of conflicting short- and long-run considerations. A firm has the possibility to increase its future demand potential by setting low prices in the first periods. This behavior might lead to advantages having an effect on the whole remaining game,
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but it might also prevent any future cooperation. The existence of discounting on the profits weakens, to some extent, the long-run advantage. At the same time, it makes the conflict between short- and long-run considerations even more complicated. In the experiment, we observe that demand potential plays an important role in the decision making. Many markets exhibit the impact of the conflict between short- and long-run considerations in a permanent fight for demand potential. Such a fight typically takes the form of frequent downward jumps of prices. Often, a downward jump is followed by a price increase in the subsequent periods in order to gain a high short-run profit in these periods. Only in the second play, where subjects are experienced, do we observe in markets a cooperative behavior with stable prices on a high level. In these markets, both firms quickly agree on cooperation by setting prices which maximize individual short-run profits. The frequency (about 50 percent) of these strongly cooperative markets in the second play is an important result which has implications for competition policy. Even in the relatively complex dynamic duopoly situation, collusive outcomes may occur without explicit collusion. This is more likely in symmetric markets where firms face the same cost and demand conditions than in asymmetric markets. As pointed out by Mason et al. (1992), horizontal mergers may increase the likelihood of collusive outcomes not only due to a higher market concentration but probably also due to a reduction in the market asymmetry.
Acknowledgements I wish to thank Roy Gardner, Reinhard Selten, the editors and the referees for their valuable comments and suggestions. Financial support by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at the University of Bonn is gratefully acknowledged.
Appendix A. Calculation of the parameters for the subgame perfect equilibrium prices Following Selten (1965), the parameters a t and b t can be calculated by a system of recursive formulas. We present this system in a simplified way, adapted to our specific symmetric duopoly situation. The recursive system uses three ‘auxiliary’ variables, B t , Y t and z t . Note that, to determine the subgame perfect equilibrium solution of his general model, Selten makes use of a function that describes the equilibrium value of the sum of discounted future profits. The variable B t is one of the coefficients in this function. As we do not want to specify this function here, we need not calculate its other coefficients. The auxiliary variables Y t and z t are
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introduced by Selten to simplify notation in his presentation of the subgame perfect equilibrium solution. Beginning with the starting conditions a 25 5 0.5 b 25 5 135.5 B 25 5 64.5 1 Y 25 5 ]] 4.04 we can compute for t524 down to 1 1 2 Y t 11 a t 5 ]]] 2 2 Y t 11 1 b t 5 135.5 2 ]] B t 11 4.04 B t 5 64.5 1 z t B t 11 1 z t 5 ]](1.5 2 a t ) 2.02 1 Y t 5 ]] 1 z t (1 2 a t ) Y t 11 4.04 References Benson, B.L., Faminow, M.D., 1988. The impact of experience on prices and profits in experimental duopoly markets. Journal of Economic Behavior and Organization 9, 345–365. Davis, D.D., Holt, C.A., 1993. Experimental Economics, Princeton University Press. Fischer, K., 1988. Oligopolistische Marktprozesse: Einsatz verschiedener Preis-Mengen-Strategien ¨ ¨ unter Berucksichtigung von Nachfragetragheit, Physica, Heidelberg. Friedman, J.W., 1969. On Experimental Research in Oligopoly. Review of Economic Studies 36, 399–415. Friedman, J.W., 1991. Oligopoly with Price Inertia and Bounded Rationality, University of North Carolina, Discussion Paper. Hoggatt, A.C., 1959. An Experimental Business Game. Behavioral Science 4, 192–203. Holt, C.A., 1995. Industrial organization: a survey of laboratory research. In: Kagel, J.H., Roth, A.E. (Eds.), The Handbook of Experimental Economics, Princeton University Press, pp. 349–443. Keser, C., 1992. Experimental duopoly markets with demand inertia – game-playing experiments and the strategy method. In: Lecture Notes in Economics and Mathematical Systems,, Vol. vol. 391, Springer, Berlin. Keser, C., 1993. Some results of experimental duopoly markets with demand inertia. Journal of Industrial Economics 41, 133–151. ¨ Kroeber-Riel, W., Trommsdorff, V., 1973. Markentreue beim Kauf von Konsumgutern—Ergebnisse
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einer empirischen Untersuchung. In: Kroeber-Riel, W. (Ed.), Konsumverhalten und Marketing, Westdeutscher, Opladen. Mason, C.F., Philipps, O.R., Nowell, C., 1992. Duopoly behavior in asymmetric markets: an experimental evaluation. Review of Economics and Statistics 74, 662–669. Phlips, L., Richard, J.F., 1989. A dynamic oligopoly model with demand inertia and inventories. Mathematical Social Sciences 18, 1–32. Schelling, T.C., 1960. The Strategy of Conflict, Harvard University Press. Schmalensee, R., 1978. A model of advertising and product quality. Journal of Political Economy 86, 485–503. ¨ Selten, R., 1965. Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit. ¨ die Gesamte Staatswissenschaft 121, 301–324, see also p. 667–689. Zeitschrift fur ¨ zur Selten, R., 1967a. Investitionsverhalten im Oligopolexperiment. In: Sauermann, H. (Ed.), Beitrage ¨ experimentellen Wirtschaftsforschung, J.C.B. Mohr, Tubingen, pp. 61–103. Selten, R., 1967b. Eine Oligopolexperiment mit Preisvariation und Investition. In: Sauermann, H. (Ed.), ¨ zur experimentellen Wirtschaftsforschung, J.C.B. Mohr, Tubingen, ¨ Beitrage pp. 103–135. ¨ Selten, R., 1967c. Die Strategiemethode zur Erforschung des eingeschrankt rationalen Verhaltens im ¨ Rahmen eines Oligopolexperiments. In: Sauermann, H. (Ed.), Beitrage zur experimentellen ¨ Wirtschaftsforschung, J.C.B. Mohr, Tubingen, pp. 136–168. ¨ Selten, R., 1970. Ein Marktexperiment. In: Sauermann, H. (Ed.), Beitrage zur experimentellen ¨ Wirtschaftsforschung, Vol. vol. II, J.C.B. Mohr, Tubingen, pp. 33–98. Selten, R., 1990. Bounded rationality. Journal of Institutional and Theoretical Economics 146, 649–658. Selten, R., Stoecker, R., 1986. End behavior in finite prisoner’s dilemma supergames. Journal of Economic Behavior and Organization 7, 47–70. Selten, R., Mitzkewitz, M., Uhlich, G.R., 1997. Duopoly strategies programmed by experienced players. Econometrica 65, 517–555. ¨ ¨ Siegel, S., 1987. Nichtparametrische statistische Methoden (3. unverand. Aufl.). Fachbuchhandlung fur Psychologie, Verlagsabteilung, Eschborn, Frankfurt am Main. Simon, H.A., 1957. Models of Man, Wiley, New York. Stoecker, R., 1980. Experimentelle Untersuchung des Entscheidungsverhaltens im Bertrand-Oligopol, Pfeffer, Bielefeld. ¨ Stoecker, R., 1983. Das erlernte Schlußverhalten – Eine experimentelle Untersuchung. Zeitschrift fur die gesamte Staatswissenschaft 139, 100–121.
Cooperation in symmetric duopolies with demand inertia Claudia Keser a,b , * a
¨ Statistik und Mathematische Wirtschaftstheorie, Universitat ¨ Karlsruhe, Karlsruhe, Institut f ur Germany b ´ , Quebec ´ , Canada H3 A 2 A5 CIRANO, Montreal
Abstract We present an experiment on a price-setting duopoly with demand inertia and symmetric costs. Subjects gain experience in the game by playing it twice. In the first play, we observe prices above the subgame perfect equilibrium solution. All markets show considerable price instability. In the second play, the price level is higher than in the first play. About half of the markets exhibit price instability, while the remaining markets are characterized by high stable prices. Comparing these results with those of a similar duopoly with asymmetric costs, we conclude that markets with symmetric costs show significantly more cooperative behavior. 2000 Elsevier Science B.V. All rights reserved. Keywords: Experimental economics; Duopoly; Dynamic games JEL classification: C90; D43; C73
1. Introduction This article reports the results of an experiment on a multiperiod duopoly game with symmetric costs where prices in each period are the only decision variables. In this game, introduced by Selten (1965), demand is assumed to exhibit inertia.1
* Corresponding author. Tel.: 11-514-985-4000 ext. 3014; fax: 11-514-985-4039. E-mail address: [email protected] (C. Keser) 1 Also the experiment by Hoggatt (1959) is based on an oligopoly game with demand inertia. Fischer (1988) simulates oligopolistic market processes under the assumption of demand inertia. Phlips and Richard (1989) generalized Selten’s oligopoly model with demand inertia by allowing firms to carry inventories. 0167-7187 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 99 )00032-6
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This takes into account the idea that consumers do not instantaneously react to price changes.2 The multiperiod duopoly game has a unique subgame perfect equilibrium solution. In the experiment, we observe that the actual individual price-setting behavior of subjects is different from what is prescribed by the subgame perfect equilibrium solution. This is true on both the individual and the aggregate level. This is not a very surprising result, since the subgame perfect equilibrium solution is too complicated to be computed by a typical subject in an experimental game-playing situation (Simon, 1957). Another result of the experiment concerns the impact of experience on subject behavior in the multiperiod duopoly situation. Given the dynamic structure of the game, we cannot expect to identify learning effects within a game. For this reason, each subject played the multiperiod duopoly game twice. This allows us to evaluate the impact of subjects’ experience from the first to the second play of the dynamic game. We observe that, when subjects play the game for the first time, average prices over all markets are above the subgame perfect equilibrium prices in each period. When subjects play the game for the second time, the observed price level is even higher than the price level in the first play. Thus, the price level approaches the symmetric Pareto efficient price, which constitutes a prominent cooperative solution to the game. Furthermore, in the second play, subjects exhibit more stable price-setting behavior than in the first play. Consequently, the profits gained in the second play are significantly higher than the profits in the first play, and significantly higher than the profits in the subgame perfect equilibrium solution. We conclude that, in the aggregate, experience in playing the specific multiperiod duopoly game leads actual behavior away from the subgame perfect equilibrium solution toward a more cooperative solution. This result is in accordance with several earlier experimental studies on oligopolies (Stoecker, 1980; Benson and Faminow, 1988; Selten et al., 1997) and, in particular, with those of Keser (1992, 1993). In Keser (1992, 1993) we examined a very similar experimental duopoly which differed from this experiment in that it had an asymmetric cost structure. We observed there as well a significant tendency toward cooperation when subjects
2
Demand inertia may be caused by various factors, such as product differentiation, lack of market transparency, or customer loyalty. The interpretation of the multiperiod duopoly with demand inertia in terms of product differentiation is restricted by a property which Selten (1965) calls quasi-homogeneity: in the case of a constant market demand, price differences should cause an increase of the cheaper firm’s market share. The assumption of missing market transparency is fundamental to the models with stochastic buyer behavior by Schmalensee (1978) and Friedman (1991). Customer loyalty is an important phenomenon in economic life. This is shown in many empirical consumption investigations (see, for example, Kroeber-Riel and Trommsdorff, 1973). Selten (1970) observed considerable stability in the trading relationships in a market experiment where sellers and buyers could freely negotiate and trade a homogeneous good.
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were experienced. However, we observed there a significantly lesser degree of cooperation than in the symmetric cost situation. This difference can be explained by the fact that in the symmetric duopoly it is more obvious than in the asymmetric duopoly where cooperation should take place. Also Mason et al. (1992) provide experimental evidence that symmetry facilitates cooperation in repeated Cournot oligopoly games. The results of our experiment clearly indicate that the behavior of subjects in the dynamic duopoly situation is not guided by the principles of full rationality on which the game–theoretic solution is based. Selten (1990) argues that limits on human rationality need not only be cognitive; they might also be motivational. In the dynamic duopoly situation, motivational limits might arise because subjects recognize the possibility of higher profits for both firms through cooperation. In particular, if each firm charges in each period the price which maximizes its one-period profit, then a kind of cooperative outcome emerges. It is the symmetric Pareto efficient solution of the game, which we also denote as the myopic monopoly. At the same time, however, a firm’s interest in its own long-run profit may prevent cooperation. A firm might be tempted to set low prices in early periods in order to make big sales which, due to demand inertia, creates an advantage for the rest of the game. Low prices in early periods risk launching a price war and, thus, preventing cooperation. Indeed, we observe continual price wars in all markets of the first play of the experiment. Also in the second play, about half of the markets show price wars. However, in the other half of the markets in the second play, both firms coordinate from the beginning on prices close to the symmetric Pareto efficient price. Thus, the experiment provides us with evidence on how subjects deal with the conflict between short- and long-run interests inherent in the dynamic structure of the game. Given the importance of individual long-run considerations, the frequency of markets with high stable prices is surprising. In these markets, each of the two firms principally behaves in each period as if it were a monopolist, fixing a price which maximizes individual one-period profit and ignoring its individual interest in creating long-run advantages. The experiment on a dynamic duopoly situation presents an interesting expansion on experimental research on oligopolies. The dynamic structure of the game allows us to study aspects of subjects’ decision making which do not play a role in less complex oligopoly situations. Behavior in a more complex multiperiod oligopoly situation was, for example, analyzed by Selten (1967a,b,c) who observed no cooperation in his experiments. The bulk of oligopoly experiments, however, examines repetitions of the same simple game situation. There is no dynamic relationship between the repetitions. The extent to which cooperation is observed in these experiments depends, among others, on the subjects’ knowledge about the game situation, their experience in playing the specific game, and on the opportunities for communication. Surveys of this literature are given by, for example, Friedman (1969), Davis and Holt (1993, pp. 196–199), and Holt (1995).
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2. The model The model is one of a multiperiod price-setting duopoly with symmetric costs. A dynamic relationship between the periods arises because sales not only depend on current prices but also on past sales. This phenomenon can be explained by demand inertia: customers though tending to buy from the low price seller also tend to stay with the same seller. This model belongs to the class of games analyzed by Selten (1965). In his seminal paper, Selten introduced the notion of subgame perfect equilibrium to solve an oligopoly game with demand inertia. In our study, the parameters of the duopoly are chosen such that the game has a unique subgame perfect equilibrium solution. The first part of this section presents the game in the same way as it was presented to the subjects of the experiments. The second part of the section describes the subgame perfect equilibrium solution and the so-called myopic monopoly solution. The latter is a cooperative solution which is not based on game–theoretic considerations but on the concept of prominence (Schelling, 1960). Both solutions are considered as benchmarks to compare the observed behavior with.
2.1. The game There are two firms in the market offering a homogeneous product over 25 periods. The only decision variable of each firm is its price in each period. Let x ti be the quantity sold by firm i in period t. This quantity is assumed to depend linearly on the firm’s own price in period t, p it , and on a variable D it which is called the firm’s demand potential in period t: x ti 5 D ti 2 p ti
i 5 1,2; t 5 1, . . . ,25
(1)
In the first period, the demand potential is the same for each firm: D 1i 5 200
i 5 1,2
(2)
In the following periods, a firm’s demand potential is determined by its former demand potential plus half of the price difference of the two firms in the market in the previous period: D ti 5 D ti 21 1 ( p tj 21 2 p ti 21 ) / 2
i 5 1,2; j 5 3 2 i; t 5 2, . . . ,25
(3)
We see that if a firm has offered its product in period t21 at a lower price than the other firm in the market, then its demand potential for period t increases. The demand potential for the other firm decreases by the same amount. Obviously, the average demand potential remains constant over time; what might change from
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one period to another is how the demand potential is divided between the two firms. The firms are assumed to produce exactly as much as they can sell. Production costs result from constant unit cost, c, which is the same for both firms: c 5 71
(4)
Therefore, the profit g ti of firm i in period t is given by: g it 5 ( p it 2 c) x it
i 5 1,2; t 5 1, . . . ,25
(5)
This short-run profit is credited after each period to an account which yields an interest rate of 1 percent per period. Thus, firm i’s total profit after 25 periods is given by:
O 1.01 25
G 25 i 5
252t
g it
i 5 1,2
(6)
t 51
In the following, we call this firm i’s long-run profit. The procedure of the game is the following. In each period t, the two firms in the market independently decide on their prices for that period. Both firms are completely informed about the rules of the game. They know the formulas and also all parameters of the model. Furthermore, they have complete information on all that has happened in their market so far. Thus, they know their current individual demand potentials when they set their prices. They also know that the game ends after 25 periods. Each firm is assumed to maximize its long-run profit. The firms have to obey two limitations when they decide on their prices. First, they are not allowed to choose prices above their current individual demand potentials. This would yield negative sales according to formula (1). Secondly, to prevent dumping, they are not allowed to set prices below their own unit cost of production. However, in the case that a firm’s demand potential would fall below its unit cost, the price is automatically set at its demand potential so that the firm does not sell anything. Thus, neither a firm’s demand potential nor its short-run profits can ever become negative.
2.2. Theoretical features Applying the game–theoretic concept of subgame perfect equilibrium, the game is solved by backward induction. The result is a uniquely determined system of linear decision rules. The solution presented below follows directly from a translation of the results obtained by Selten (1965) to the specific game situation considered. The price paths in the subgame perfect equilibrium solution are prescribed by linear functions of the individual demand potentials: p ti * 5 a t (D ti 2 200) 1 b t
i 5 1,2; t 5 1, . . . ,25
(7)
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The parameters a t and b t are time-dependent but the same for both firms. They are determined by the market demand potential, the unit production cost common to both firms in the market, and the interest rate. They can be calculated by the recursive system given in the Appendix A. In the subgame perfect equilibrium solution, both firms set the same prices. Therefore, the demand potential is always shared equally. The equilibrium prices remain, during the first 20 periods, at a level of about 100 until they increase considerably because toward the end of the game long-run considerations become of minor importance. The equilibrium price in the last period is the myopic monopoly price, which is described by Eq. (8) below. The resulting long-run equilibrium profit is 85,336 for each firm. The subgame perfect equilibrium solution is not Pareto efficient. Thus, there is room for cooperation where both firms can make higher profits. We consider the symmetric Pareto efficient solution as a reference point of cooperation to compare the observed behavior with. We also call it the myopic monopoly solution because both firms in each period charge prices as if they were monopolists, without considering that there might be consequences for following periods. In each period, each firm maximizes its short-run profit (Eq. (5)), which depends on the firm’s demand potential and own current price but is independent of the other firm’s price in the current period. The resulting myopic monopoly price of firm i in period t is given by: p t,M 5 (D ti 1 c) / 2 i
i 5 1,2; t 5 1, . . . ,25
(8)
If both firms in the market behave according to the myopic monopoly solution, they set prices of 135.50 in each period. Demand potential is shared equally and the resulting long-run profit for each firm is 117,499.
3. Organization of the experiments The subjects were 48 students at the University of Bonn.3 We organized four experimental sessions with 12 subjects each. A session lasted 3–4 h. The subjects sat in separate cubicles and made their decisions via computer terminals. They did not interact in any other way. Before the experiment started, they were instructed about the game and the use of the computer program for about three quarters of an
3 They consisted of 37 economics students, eight law students, and one student each from mathematics, physics, and computer science. A few of the subjects had previous knowledge of oligopoly or game theory. None of the subjects had participated in an oligopoly experiment before.
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hour.4 Then they played the 25-period game twice. Communication was prohibited, including during the short break between the first and the second play of the game. For the first play of the multiperiod duopoly game, the 12 subjects of a session were divided into six groups of two firms each to form six duopoly markets. None of the subjects knew which other subject of the group was the firm he was interacting with in his market. In the second play, each subject interacted with a subject different from the one in the first play. Specifically, and this without the subjects’ knowledge, the organization of each experimental session was such that four subjects always formed an independent player group. Each player of such a group interacted only with other players of the same group. Thus, we obtained three independent observations per session and twelve independent observations in total. In each of 25 periods, the two firms in each market independently had to choose their prices. At the beginning of the next period, each firm was informed about its own and the other firm’s sales, profits, and new demand potential. At any time, a subject could review his previous results. Since in each period each subject knew his current demand potential, he could compute his sales and profits resulting from any price he was allowed to set. A calculation aid reported these figures for hypothetical prices to be selected by the subject. The subjects were paid in cash at the end of each session according to their balances after period 25 of each play. Each subject knew that his or her payment for a play was determined in the following way. We calculated the surplus of his total profit over the average total profit realized by all subjects in the other markets of the session. Obviously, the surplus must be negative for some subjects if it is positive for others. We multiplied the surplus by an exchange rate of 0.00033 and added it to 16 deutsche marks.5
4. Experimental results In this section, we analyze the average behavior over all subjects and compare it to the behavior prescribed by the theoretical solutions. It is, however, important to note that the behavior in the individual markets varies considerably. Therefore, in
4
The subjects did not receive any written instructions. This method of payment is strategically equivalent to simply multiplying a subject’s total profit by a fixed exchange rate. A subject’s profit is compared solely to profits of subjects with whom he is not interacting. Thus, the subject cannot have any influence on these other profits through his own decision-making. He has to take the other profits as given. We chose this method of payment because it allows one to fix in advance the total amount of money to be paid to all subjects. Obviously, we have to take into account the possibility that in spite of strategic equivalence, this method of payment may have led to different behavior than if we simply had multiplied each subject’s profit by a fixed exchange rate. 5
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Section 5 below, we will present an analysis of the individual market behavior. In both sections we use nonparametric statistics following Siegel (1987). The data are posted at URL http: / / www.econ.ku.dk / cie / ijio / exptsup.htm.
4.1. Prices and profits In Fig. 1 we compare, for the first and the second play, the realized prices on average over all firms in each period to the price paths in the equilibrium and the myopic monopoly solution. We see that in both the first and the second play the actual average prices are above the equilibrium prices. Furthermore, average prices in the second play are above those of the first one although they are still considerably below the myopic monopoly prices. To test whether the actual prices significantly tend to be above the equilibrium prices, we count the number of markets (in the first play) and the number of independent player groups (in the second play) where 50 percent or more of the prices are below or equal to the equilibrium prices.6 Among the 24 markets of the first play, we count seven with 50 percent or more of the prices below or equal to the equilibrium prices. Thus, the one-tailed binomial test rejects the null hypothesis at the five percent significance level. Among the twelve independent player groups of the second play, we find one with 50 percent or more of the prices below the equilibrium prices. The one-tailed binomial test rejects the null hypothesis at the one percent significance level. Next, we compare the actual prices with adjusted equilibrium prices. In each
Fig. 1. Dynamics of the realized average prices over all firms in the first play and in the second play, compared to the prices in the subgame perfect equilibrium and the myopic monopoly solution.
6
The smallest independent units to be considered are the markets in the first play and the independent player groups in the second play.
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period, we consider the actual individual demand potential and calculate, based on Eq. (7), the equilibrium price according to the actual demand potential. Obviously, the actual individual demand potential of the two firms in a market might differ from 200 in a given period, so that the adjusted equilibrium price in that period is off the subgame perfect equilibrium price path.7 We find the same result as before. Seven of the 24 markets in the first play and one of the 12 independent player groups in the second play have 50 percent or more of the prices below or equal to the respective adjusted equilibrium prices. Thus, we may conclude that prices tend to be significantly higher than the adjusted subgame perfect equilibrium prices. Comparing the first and the second play, we observe a significant tendency toward higher prices in the second play. Among the 12 independent player groups there are 11 for which we observe that the average prices increase from the first to the second play. The one-tailed binomial test rejects the null hypothesis at the one percent significance level. Comparing the actual prices to the myopic monopoly price, we observe in none of the 24 markets of the first play and in none of the 12 independent player groups of the second play, that 50 percent or more of the prices are above or equal to 135.50. The one-tailed binomial test rejects the null hypothesis at the one percent significance level for both the first and the second play. The same result holds if we compare the actual prices with the adjusted monopoly prices. The latter are, in each period, computed by Eq. (8), based on the actual demand potential in that period of the firm under consideration. The realized average long-run profit over all firms is 87,333 in the first play, while it is 102,511 in the second play. Thus, in the first play, the average profit is only two percent higher than the subgame perfect equilibrium profit, while it is twenty percent higher in the second play. In the first play, we observe 16 markets whose average profits are above the equilibrium profit, while there are eight markets with average profits below the equilibrium profit. At the five percent significance level, the one-tailed binomial test does not reject the null hypothesis that profits tend to be equally likely below and above the subgame perfect equilibrium profit. In the second play, we observe only one independent player group with an average profit below the equilibrium profit, while 11 independent player groups made profits above the equilibrium profit. The one-tailed binomial test rejects the null hypothesis at the one percent significance level. Thus, profits in the second play tend to be significantly higher than the equilibrium profit. Furthermore, profits realized in the second play tend to be significantly higher than profits in the first play. Among the 12 independent player groups there are 11 whose profit increases in the second play. The one-tailed binomial test rejects the
7 In this adjusted calculation, we take into account that the subjects in a market might not have behaved according to the subgame perfect equilibrium solution in any previous period but that they obey the equilibrium pricing rule (as given by Eq. (7)) for the current period.
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null hypothesis at the one percent significance level. Note that in neither play does the average realized profit come close to the myopic monopoly profit.
4.2. Price instability The individual pricing behavior in the second play appears more stable than in the first play. To illustrate this, we need a measure for the instability of the prices set by a firm in a given play. We define the price instability measure Si for a firm i as:
O (p 2p 25
Si 5
t i
t21 2 i
)
i 5 1, . . . ,48
(9)
t52
This measure computes the sum of a firm’s squared price deviations from its own price in the previous period. We add up the values of this measure for all firms in each independent player group in the first and second play and test whether prices in the second play are less unstable. In 11 of 12 independent player groups we observe a lower price instability in the second play. The one-tailed binomial test rejects the null hypothesis at the one percent significance level.
5. The behavior on the individual markets The various markets display a wide variety of price dynamics. In order to uncover some characteristic features of market behavior, we roughly classify the observed markets into strongly cooperative, weakly cooperative, and aggressive markets. This classification scheme follows the one given in Keser (1992, 1993) for a similar multistage duopoly with asymmetric costs. Somewhat arbitrarily, we use the price level (i.e. the average price of the two firms in the market over the 25 periods) as the only classification device. Markets with a price level above 125 are classified as strongly cooperative, markets with a price level below 100 are classified as aggressive, and all markets with a price level between these two benchmarks are classified as weakly cooperative. In the first play, we classify only four markets as strongly cooperative while 13 are weakly cooperative and seven are aggressive ones. With the exception of one market where prices are almost constantly at a price level close to 135 until period 15, the behavior in the strongly cooperative markets does not differ very much from the behavior in the weakly cooperative and aggressive markets, except for the level of the prices set. We generally observe considerable price movement in all three market types. In many markets, we actually observe abrupt price jumps, both up and down, which may be quite far-reaching. Subjects seem to compete for demand potential. Prices go down and up again, probably because the conflicting aims of increasing one’s demand potential and of gaining a high short-run profit alternately determine the price decision of a subject. We observe that the larger the
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difference between the initial prices set by the two firms in a market, the stronger will be the price instability in the subsequent periods (Spearman rank correlation coefficient of 0.77 in the first play and 0.49 in the second play). Note that market prices and profits are strongly correlated (Spearman rank correlation coefficient of 0.95 in the first play and 0.87 in the second play). Thus, it is not surprising that the three market categories differ in their profits: firms in the strongly cooperative markets gain higher profits than the firms in the weakly cooperative markets, and the firms in the weakly cooperative markets gain higher profits than those in the aggressive markets (one-tailed Mann–Whitney U-test, one percent significance level). Of the 24 markets in the second play, we classify 12 markets as strongly cooperative, 11 as weakly cooperative, and only one as aggressive. In the aggressive market the prices of both firms are permanently very low. This market makes the lowest profit among all markets. Again, the strongly cooperative markets make significantly higher profits than the weakly cooperative markets (one-tailed Mann–Whitney U-test, one percent significance level). Strongly and weakly cooperative markets in the second play show additional differences other than those with respect to price level and market profit. In eight of the 12 strongly cooperative markets, both firms set stable prices above 130 during most of the 25 periods. In some of these markets, we observe some price movement, but solely at the beginning of the game, caused by a difference in the initial prices of the two firms, or at the end of the game, a so-called end effect (Stoecker, 1983; Selten and Stoecker, 1986). Only four strongly cooperative markets show, over the whole play, some moderate ups and downs of prices. In the weakly cooperative markets, continual price movement is typical. In many of these markets, we observe even extreme price jumps. We can show that the prices on the weakly cooperative markets significantly tend to be more unstable than those on the strongly cooperative markets (one-tailed Mann–Whitney U-test based on the sum of the price instability measures of the two firms in a market, one percent significance level).
6. Comparison with Keser (1992, 1993) In Keser (1992, 1993) we presented the experiment on a related duopoly with asymmetric costs. Similarly to our results on the symmetric duopoly, we observed a significant tendency toward more cooperative behavior when subjects were experienced in playing the dynamic game. The degree of cooperation in the second play, however, appears higher in the symmetric situation than in the asymmetric one.8
8
Note also that, in contrast to the symmetric situation, first play prices in the asymmetric situation were not significantly different from the subgame perfect equilibrium solution.
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To give statistical support to this observation, we consider the play of experienced subjects in both situations. Note that all parameters except for the costs are the same. In the asymmetric duopoly, costs are 57 for the low cost firm, and 71 for the high cost firm. In the symmetric duoploy, costs are 71 for both firms. Table 1 shows for both situations the equilibrium profit (and price) on average over the two firms in a market, the average profit (and price) realized in the second play of the respective experiment, and the myopic monopoly profit (and price) on average over the two firms in a market. Let us designate the proportion of profit gained relative to the myopic monopoly profit as efficiency. The efficiency of the subgame perfect equilibrium solution is 75 percent in the asymmetric game and 73 percent in the symmetric game. Thus, in the subgame perfect equilibrium solution, the asymmetric game is slightly more efficient than the symmetric game. However, the efficiency of the actually realized profits is higher in the symmetric game than in the asymmetric game. On average over all subjects, it is 87 percent in the symmetric game and 77 percent in the asymmetric game. The one-sided binomial test based on the actual efficiencies of the independent player groups rejects the null hypothesis at the five percent level. We conclude that the actual efficiency is significantly higher in the symmetric situation than in the asymmetric situation. Similarly, we consider the relative closeness of the prices realized over the whole game to the respective myopic monopoly price. We find that the average realized prices are significantly closer to the myopic monopoly prices in the symmetric games (89 percent of the myopic monopoly price) than in the asymmetric games (80 percent of the myopic monopoly price). The significance level is five percent of the two-sided binomial test. Considering the sum of the instability measures of all subjects within an independent player group, we find no significant difference between the symmetric and the asymmetric game. In somewhat arbitrary ways, we classified both the symmetric and the asymmetric markets into strongly cooperative, weakly cooperative, and aggressive markets. The frequencies of each market type in both cost situations are presented in Table 2. In the asymmetric game, we classified about one third of the markets as strongly cooperative, another third as weakly cooperative, and the remaining third as aggressive. In the symmetric game, we classified one half of the markets as Table 1 Profits and prices in equilibrium, myopic monopoly, and the second play of the symmetric and the asymmetric duopoly Cost situation
Average profit in equilibrium
Average profit realized
Average profit in myopic monopoly
Average price in equilibrium
Average price realized
Average price in myopic monopoly
Symmetric Asymmetric
85,336 99,113
102,548 101,317
117,499 131,878
103.11 97.85
121.14 106.24
135.50 132.00
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Table 2 Classification of markets in the second play of the symmetric and the asymmetric duopoly Cost situation Symmetric Asymmetric
Number of markets classified as Strongly cooperative
Weakly cooperative
Aggressive
12 9
11 10
1 11
strongly cooperative and the other half (with one exception) as weakly cooperative. Applying a x 2 -test, we find that symmetric and asymmetric markets differ significantly with respect to their classification into strongly and weakly cooperative and aggressive markets (five percent significance level). Although the classification is largely based on ‘visual inspection of the graphs of the price dynamics’ (Keser, 1993, p. 143) we consider this difference as additional evidence for the stronger tendency in symmetric markets to cooperate than in asymmetric markets.
7. Conclusion In the experiment we observe that the actual behavior of subjects in the dynamic duopoly situation is different from what is prescribed by the subgame perfect equilibrium solution. Experience in playing the dynamic duopoly game moves subjects’ behavior even further away from the equilibrium toward more cooperation. When subjects play for the second time, there is a significant tendency toward higher prices and profits, and toward a more stable price behavior than in the first play. These results are similar to those in Keser (1992, 1993) where the same type of duopoly experiment with asymmetric costs is presented. We can show, however, that cooperation occurred there to a significantly lesser extent than in our experiment with symmetric costs, where it is more obvious how to cooperate. Although the behavior of individual markets varies considerably, we are able to identify general characteristics. In the first play, the most striking feature, common to almost all markets, is that prices permanently move up and down, often by a considerable amount. The profit of the two firms in a market is determined by their average price level. In the second play, we still find abrupt price movements in about half of the markets. These markets show relatively low price levels and profits. The other half of the markets in the second play show more or less stable prices at a high level. In these markets both firms make high profits. The dynamic structure makes the game difficult for the players to grasp because of conflicting short- and long-run considerations. A firm has the possibility to increase its future demand potential by setting low prices in the first periods. This behavior might lead to advantages having an effect on the whole remaining game,
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but it might also prevent any future cooperation. The existence of discounting on the profits weakens, to some extent, the long-run advantage. At the same time, it makes the conflict between short- and long-run considerations even more complicated. In the experiment, we observe that demand potential plays an important role in the decision making. Many markets exhibit the impact of the conflict between short- and long-run considerations in a permanent fight for demand potential. Such a fight typically takes the form of frequent downward jumps of prices. Often, a downward jump is followed by a price increase in the subsequent periods in order to gain a high short-run profit in these periods. Only in the second play, where subjects are experienced, do we observe in markets a cooperative behavior with stable prices on a high level. In these markets, both firms quickly agree on cooperation by setting prices which maximize individual short-run profits. The frequency (about 50 percent) of these strongly cooperative markets in the second play is an important result which has implications for competition policy. Even in the relatively complex dynamic duopoly situation, collusive outcomes may occur without explicit collusion. This is more likely in symmetric markets where firms face the same cost and demand conditions than in asymmetric markets. As pointed out by Mason et al. (1992), horizontal mergers may increase the likelihood of collusive outcomes not only due to a higher market concentration but probably also due to a reduction in the market asymmetry.
Acknowledgements I wish to thank Roy Gardner, Reinhard Selten, the editors and the referees for their valuable comments and suggestions. Financial support by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at the University of Bonn is gratefully acknowledged.
Appendix A. Calculation of the parameters for the subgame perfect equilibrium prices Following Selten (1965), the parameters a t and b t can be calculated by a system of recursive formulas. We present this system in a simplified way, adapted to our specific symmetric duopoly situation. The recursive system uses three ‘auxiliary’ variables, B t , Y t and z t . Note that, to determine the subgame perfect equilibrium solution of his general model, Selten makes use of a function that describes the equilibrium value of the sum of discounted future profits. The variable B t is one of the coefficients in this function. As we do not want to specify this function here, we need not calculate its other coefficients. The auxiliary variables Y t and z t are
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introduced by Selten to simplify notation in his presentation of the subgame perfect equilibrium solution. Beginning with the starting conditions a 25 5 0.5 b 25 5 135.5 B 25 5 64.5 1 Y 25 5 ]] 4.04 we can compute for t524 down to 1 1 2 Y t 11 a t 5 ]]] 2 2 Y t 11 1 b t 5 135.5 2 ]] B t 11 4.04 B t 5 64.5 1 z t B t 11 1 z t 5 ]](1.5 2 a t ) 2.02 1 Y t 5 ]] 1 z t (1 2 a t ) Y t 11 4.04 References Benson, B.L., Faminow, M.D., 1988. The impact of experience on prices and profits in experimental duopoly markets. Journal of Economic Behavior and Organization 9, 345–365. Davis, D.D., Holt, C.A., 1993. Experimental Economics, Princeton University Press. Fischer, K., 1988. Oligopolistische Marktprozesse: Einsatz verschiedener Preis-Mengen-Strategien ¨ ¨ unter Berucksichtigung von Nachfragetragheit, Physica, Heidelberg. Friedman, J.W., 1969. On Experimental Research in Oligopoly. Review of Economic Studies 36, 399–415. Friedman, J.W., 1991. Oligopoly with Price Inertia and Bounded Rationality, University of North Carolina, Discussion Paper. Hoggatt, A.C., 1959. An Experimental Business Game. Behavioral Science 4, 192–203. Holt, C.A., 1995. Industrial organization: a survey of laboratory research. In: Kagel, J.H., Roth, A.E. (Eds.), The Handbook of Experimental Economics, Princeton University Press, pp. 349–443. Keser, C., 1992. Experimental duopoly markets with demand inertia – game-playing experiments and the strategy method. In: Lecture Notes in Economics and Mathematical Systems,, Vol. vol. 391, Springer, Berlin. Keser, C., 1993. Some results of experimental duopoly markets with demand inertia. Journal of Industrial Economics 41, 133–151. ¨ Kroeber-Riel, W., Trommsdorff, V., 1973. Markentreue beim Kauf von Konsumgutern—Ergebnisse
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