Cost Structures and Nash Play in Repeated Cournot Games * by Douglas D. Davis Department of Economics Virginia Commonwealth University Richmond, VA 23284-4000 Robert J. Reilly Department of Economics Virginia Commonwealth University Richmond, VA 23284-4000 and Bart J. Wilson Interdisciplinary Center for Economic Science George Mason University Fairfax, VA 22030 August 1, 2002 Abstract This paper reports an experiment designed to assess the effects of a rotation in the marginal cost curve on convergence in a repeated Cournot triopoly. Increasing the cost curve’s slope both reduces the serially-undominated set to the Nash prediction, and increases the peakedness of earnings. We observe higher rates of Nash equilibrium play in the design with the steeper marginal cost schedule, but only when participants are also rematched after each decision. Examination of response patterns suggests the treatment with a steeper marginal cost curve and with a re-matching of participants across periods induces the selection of Nash Consistent responses. JEL Classifications: D43, C72, C91 Keywords: experimental economics, Cournot competition, serially undominated equilibria. *

Financial support from the National Science Foundation and the Virginia Commonwealth University Faculty Excellence Fund are gratefully acknowledged. Experimental data and a sample copy of instructions are available at http://www.people.vcu.edu/ ~dddavis.

1. Introduction Cournot interactions are a central assumption in industrial organization, and economists have been concerned with the organizing power of Cournot-Nash equilibrium predictions at least as long as they have been doing experiments (see, e.g., Fouraker and Seigel, 1963). A general conclusion from the ensuing literature is that the behavioral drawing power of Cournot predictions is notably weaker than some other equilibrium concepts in market games, such as the Bertrand prediction in homogenous-product posted price games (Holt, 1995 reviews the relevant literature). In Cournot markets with three or more players, quantities tend to exceed the Cournot prediction. Further, outcomes tend to remain volatile, even after extensive experience.1 Recent results by Huck, Normann and Oechssler (1999) indicate that information conditions may explain a considerable portion of the tendency for high quantities. Although information about underlying cost and demand conditions tends to move outcomes away from Walrasian (competitive) predictions and toward the Cournot outcome, information about the actions of others tends to push outcomes toward the Walrasian prediction.2

Nevertheless, little progress has been made in

understanding the persistent variability of Cournot outcomes. The variability is typically nontrivial, and in applied contexts can rival the magnitude of predicted comparative statics effects (see, e.g., Davis, 2002, Phillips and Mason, 1992 and Wellford, 1990). More fundamentally, the persistent volatility of decisions raises questions about the adjustment processes participants use, and the conditions, if any, under which these responses can lead to stable outcomes. Longstanding theoretical results make the observed variability of results less than entirely surprising. Theocharis (1960) established that “linear” Cournot games (those characterized by constant marginal costs and linear demand) with three or more players do not converge dynamically to the Cournot outcome under a “Best Response” adjustment process where each seller optimizes each period on the assumption that rivals repeat the choices that they made in the previous period. Altering the underlying conditions, however, can make the game stable. In particular, Fisher (1961) shows that multiple-player Cournot games become stable to a Best-Response adjustment process if the marginal costs curves are sufficiently upward sloping.

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Davis (1999) and Rassenti, Reynolds, Smith and Szidarovszky (2000) observe explicitly the persistent variability of outcomes. Other recent papers do not comment on outcome variability directly, but variability is suggested from reported quantity standard deviations. For some recent examples, see e.g., Huck, Normann and Oechssler (1998, 1999). 2 The conclusion that information about others’ actions tends to generate more nearly competitive outcomes confirms a similar conjecture by Fouraker and Seigel (1963).

In fact, altering the slope of marginal cost curves may affect the stability of Cournot markets under less specialized circumstances, and particularly under more general dynamic response patterns. Cost curves impact on the stability of Cournot predictions in a number of ways. First, increasing the slope of the marginal cost curve reduces the size of the serially undominated set. Milgrom and Roberts (1991) show that a wide variety of dynamic responses that satisfy an adaptive learning criterion will converge to the serially undominated set.3 Second, steeper marginal cost curves increase the peakedness of earnings about the Cournot outcome. Finally, rotating the marginal cost curve while holding the Cournot prediction constant reduces the distance in quantity space between the competitive, Cournot and joint profit maximizing outcomes. This paper reports an experiment designed to analyze the sensitivity of a Cournot triopoly to a relatively subtle change in the cost schedule. Cost conditions may vary substantially across “Cournot” markets, and the first two of the above arguments suggest that even relatively subtle changes may prominently affect the stability of equilibrium predictions in those markets.

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secondary objective is to examine the effects of leaving participants matched in the same groups throughout a session. In market experiments with more than two players, participants are almost always kept in the same group throughout the course of a session, presumably because this condition most closely parallels naturally occurring markets. Nevertheless, in Cournot duopolies and other two person games, rematching players according to a predetermined schedule tends to eliminate a tendency toward collusion (see, e.g., Holt, 1985). Unlike Cournot duopolies, little evidence suggests that aggregate output is less than the Cournot prediction in repeated Cournot games with more than two players because sellers are engaging in strategic efforts to impose and enforce collusive intentions. However, some players may try to signal cooperative intentions. Further, both Davis (1999) and Rassenti, Reynolds, Smith and Szidarovszky (2000) report some evidence of strategic efforts by players to “bully” others into accepting asymmetric outcomes in multi-player Cournot games. Such strategic behavior and the concomitant responses of rivals may

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The intuition driving the proofs is straightforward. Consider a collection of “realized” strategies for a static game that includes some outcomes outside the serially undominated set. The strategy that deviates furthest from the serially undominated set will be strictly dominated, and will thus yield less than maximal payoffs regardless of the choices of others. Any modification of this static strategy set in the direction of others’ plays will reduce the range of outcomes for realized strategies toward those outcomes consistent with the serially undominated set. Further, unless realized strategies include only those in the serially undominated set, at least one of the new realized strategies would again be strictly dominated. Reasoning iteratively, play decays to the serially undominated set, as long as players don’t return to choices they have already excluded.

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potentially explain the persistent volatility observed in multi-player Cournot games. Strategic efforts to “bully” rivals may also contribute to the tendency for supra-Cournot quantities.4 The paper is organized as follows. Following a discussion of the experimental design in section 2, we present the experimental procedures in section 3. Section 4 contains the results, followed by a brief conclusion in section 5. 2. Experimental Design Consider the supply and demand arrays illustrated in the upper and lower panels of Figure 1. Each panel characterizes a triopoly facing a step-wise linear demand curve that decays in 3-cent increments from a vertical intercept of 60 cents. As is evident from the unit endowments indicated by the seller identifiers S1, S2 and S3 printed below the cost curve in each panel, the symmetric sellers are each endowed with seven units: three low cost units, an intermediate cost unit, and three high cost units. The designs differ only in the slope of the aggregate supply schedule. As illustrated by the cost entries printed on the right side of the figure, in moving from the Flat Design in the upper panel to the Steep Design in the lower panel, low cost units fall from 11 cents to 8 cents, and high cost units increase from 13 cents to 17 cents. In each case unit costs for intermediate units are constant at 12 cents. Notice further in each panel that each seller offers four units in the Cournot equilibrium, and that the twelve units offered in aggregate clear the market at price Pc = 24 cents. The existence and uniqueness of the Cournot equilibrium in each case is established by examining the payoff tables for seller S1, presented as Tables 1(a) and 1(b). Each table lists the profits for seller S1 (columns), given possible choices for seller S2 (rows) and seller S3 (the different panels). Thus, for example, the shaded entry 21 in the lower right corner of the first (upper left) panel of Table 1(a) represents the profit for seller S1 when seller S3 plays 1 (the first panel), seller S2 plays 7 (seventh row) and seller S1 also picks 7 (seventh column). For ease of reference, vertical lines printed to the right of each entry highlight seller S1’s profit maximizing responses to possible choices by the others. By symmetry, seller S2’s best responses to plays of S1 and S3 are highlighted as underlines, and S3’s best responses as shaded

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Theoretical results by Vega-Redondo (1997) suggest that efforts by one player to secure an advantageous outcome relative to others might generate larger than Cournot outcomes, if others respond by copying the actions of the quantityincreasing seller. Vega-Redondo shows that Cournot markets almost always evolve toward the Walrasian prediction if sellers adopt the strategy of copying the actions of the most successful player each period, and if decisions are made with some small possibility of an error.

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areas.5 Nash equilibria are the outcomes that are simultaneously shaded, underlined and have a bolded vertical line on the right. As is evident from examination of Tables 1(a) and 1(b) the outcome (4, 4, 4) is the unique Nash equilibrium for each design. Turning back to Figure 1, notice also that steepening the supply schedule alters slightly the competitive (Walrasian) and joint profit maximizing predictions. Relative to the Flat Design, the aggregate cost schedule in Steep Design raises the joint profit maximizing quantity from 8 to 9 units and lowers the Walrasian outcome from 15 to 14 units. Behaviorally, these differences are expected to be relatively minor. In particular, an aggregate outcome of 9 units represents a reasonable reference for optimal tacitly collusive behavior in both designs, in that 9 units is the symmetric joint profit-maximizing outcome in both designs.6 But even with relatively similar reference predictions, other features potentially affecting the stability of Cournot predictions are altered prominently. Notice first the costs of deviating from the Cournot prediction. Consider, for example, the costs to seller S1 of unilaterally deviating from the Cournot outcome in the Flat Design. As seen by moving across columns in row 4 of panel 4 of Table 1(a), a single unit deviation costs seller S1 3 or 4 cents (51-48 or 51-47) while two unit deviations cost 9 or 14 cents (51-38 or 51- 37). Turning to the same entries for Steep Design, in Table 1(b), observe that the cost of single unit deviations increase to 3 or 8 cents (60-57 or 60-52) and two-unit deviations cost 16 or 22 cents (60-44 or 60-38). Second, again comparing across Tables 1(a) and 1(b) observe that changing the cost schedule prominently affects the serially undominated set. Consider first the payoffs for the Flat Design, summarized in Table 1(a). As suggested by the shaded boxes in the lower right corner of panel 1, and in the upper left corner of panel 7, even the most extreme quantity choices of 1 and 7 are a best response to some combination of others’ choices. Thus, no outcomes may be eliminated via the iterated deletion of dominated strategies, and the serially undominated set includes all possible positive quantity choices. In contrast, panels 1 and 7 of the Steep Design, shown in Table 1(b), contain no shaded areas, suggesting that they are dominated. More directly, observe in each panel of Table 1(b) that 5

The horizontal lines represent optimal choices for seller S2 since seller S2 ’s payoff table is the same as seller S1’s but with the rows and columns inverted. Optimal choices for seller S3 can be derived by examining the instances where seller S1 finds a particular play optimal. Consider for example the shaded box in the bottom right corner of panel 1 of Table 1(a). As seen from the bolded vertical line aside column 1 in panel 7, seller S1 finds a play of 1 optimal only if the other sellers each play 7.

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payoffs in column 6 strictly dominate those in column 7, and payoffs in column 2 strictly dominate those in column 1. By the symmetry of players, both panels 1 and 7 and rows 1 and 7 are also strictly dominated. The serially undominated set thus is a subset of the boxes encircled by lightdotted lines in panels 2 to 6. Notice further, however, that within the dotted boxes, payoffs in column 5 strictly dominate payoffs in column 6 and payoffs in column 3 strictly dominate payoffs in column 2. Thus, reasoning similarly, columns, rows and panels 2 and 6 may be eliminated, and the serially undominated set lies within the bold dotted lines in panels 3, 4 and 5. Iteratively applying the same reasoning, column, row and panel 5, and then column, row and panel 3 may be eliminated, leaving outcome (4, 4, 4) as the unique serially undominated strategy. Considerable experimental evidence suggests that making the Nash equilibrium the only serially-undominated outcome does not guarantee convergence. Persistent deviations from a unique serially undominated Nash prediction have been observed in repeated versions of a variety of games including the centipede game (McKelvey and Palfrey 1992), the Traveler’s Dilemma (Capra, Goeree, Gomez and Holt, 1999), and a large variety of Public Goods Games (Ledyard, 1995, reviews much of the relevant literature).7

Nevertheless, the Nash prediction for these games is

typically a corner solution. Further, any convergence-enhancing effects of reducing the serially undominated set should be enhanced by a simultaneous increase in the peakedness of earnings about the Nash prediction that occurs when the marginal cost curve is made steeper. 3. Experimental Procedures The experiment consisted of 10 nine-person sessions. In each session participants made a sequence of 45 triopoly decisions. In four of the sessions individuals were given marginal cost schedules that induced the supply schedule shown in the Flat Design, while in the remaining six sessions unit cost schedules in the Steep Design were induced. In order to enhance the likelihood of observing near Cournot outcomes, participants were provided complete and common information

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Observe in Table 1(a) that moving from (3,3,3) to the asymmetric (2,3,3) outcome increases joint earnings by only 2 cents (198 to 200) cents, while making the distribution of earnings highly asymmetric ((50,75,75) rather than (66,66,66)). 7 Indeed Capra, Goeree, Gomez and Holt (1999) drive behavior away from a uniquely rationalizable Nash outcome by manipulating the costs of deviations.

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about the structure of the demand and cost (in the form of tables), but only aggregate quantity information.8 To examine the effects of strategic play across periods participants were anonymously reshuffled after each decision-period in six of the sessions.

In the remaining four sessions,

participants remained in the same group throughout the session. To the extent that strategic considerations motivate non-Cournot plays, we should expect both lower and more stable outcomes in the sessions where participants are re-matched after each decision period. Table 2 summarizes the experimental design.

As suggested by the row and column

headings, sessions are identified by a three part mnemonic: First, F and S indicate use of the Flat and Steep Designs, respectively. Second, C and R indicate, respectively, that participants remained in constant groups throughout each session, or that participants were rematched into new groups after each decision period. The identifier ends with a number that indicates the session number in sequence. Thus, for example, FC-1, listed in the upper left corner of Table 3 indicates the first of two sessions conducted under the Flat design parameters, and where participants remained in the same group throughout the sessions. Procedures: At the outset of each session the nine participants were randomly seated at visually isolated booths.9 They then took a printed set of common instructions from their folders and followed along as a monitor read the instructions aloud. After reading instructions, participants made a sequence of 45 decisions, with the total number of periods not announced in advance. Each trading period commenced with participants privately recording a quantity choice on their record sheets.10 A monitor walked behind the participants, copied these quantity choices, and then input them into an Excel spreadsheet on a computer in the laboratory room. The spreadsheet returned the aggregate quantity choices for each seller, which the monitor then privately transmitted to each participant. Finally, participants used the aggregate quantity information to determine the market 8

Providing complete market information, but no information about private actions corresponds to the BEST information, described by Huck, Normann and Oechssler (1999). Of the various information conditions examined by these authors, the BEST condition generated mean outcomes closest to the Nash-Cournot prediction. 9 In order to ensure that enough participants would appear for a session, we typically recruited 2 “alternates” for each session. Alternates were paid $10 for meeting their scheduled appointment, and were free to leave after the requisite 9 students appeared. Alternates were also invited to participate in future. For this reason they were not allowed stay and listen to the instructions, or otherwise watch the session. 10 We allowed participants select integer quantities {0, 1, 2, 3, 4, 5, 6 , 7}. A choice of zero is strictly dominated in both the flat and in the steep designs, but does not affect the analysis in Tables 1(a) and 1(b). In any case, no player ever selected a quantity of zero.

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price, along with period and cumulative earnings. (To verify individual calculations the monitor’s spreadsheet also maintained automatically an earnings record for each participant.) This process is repeated exactly as described until the session’s termination, at which time participants were privately paid, and left the room. Participants were undergraduate students enrolled in undergraduate economics courses at Middlebury College late in the fall semester of 1997 and early in the spring semester of 1998. No participant had any previous experience with an economics experiment, and no one participated in more than one session. Earnings for the 1¾ to 2¼ hour sessions ranged from $20.50 to $36.75 (inclusive of the $6 appearance fee). Per participant earnings averaged $30.60. 4. Results To compare results with outcomes in related similar experiments, we consider first performance relative to market equilibrium predictions. To generate market observations in the SR and FR treatments, we collect participant decisions in these sessions into constant “pseudo-markets” that consist of the same pairs used in the SC and FC treatments each period. These pseudo-markets, of course, do not reflect actual aggregate market quantity realizations, but they usefully provide a non-arbitrary way to evaluate the progress of constant groups as each session progresses. Figure 2 summarizes aggregate results for the 30 markets and “pseudo-markets”. In each panel of Figure 2 the dots illustrate the mean market outcome for an eleven period session-quarter.11 The bars extending above and below the dots demark the range of an approximate 95% confidence interval about the mean outcome. Solid dots highlight instances where mean market outcomes fall closer to Qc, the Nash-Cournot prediction than to the Walrasian (competitive) reference Qw or the joint profit maximizing prediction Qm. The market outcomes shown in Figure 2 reflect a broad consistency with market outcomes observed in Cournot experiments under similar information conditions (see e.g., Huck, Normann and Oechssler, 1999): Average market outcomes tend to fall closer to Qc than to Qw or Qm . Even in the first quarter, 23 of the 30 mean outcomes fall closer to Qc than Qw or Qm. In quarters 2, 3, and 4, either 26 or 27 of the 30 market means are closer to Qc than to Qw or Qm. Nevertheless, outcomes 11

The 30 aggregate market outcomes illustrated in each quarter reflect results of 3 markets (or pseudo markets) in each of the 10 sessions. To divide the session into equal-sized quarters, we omit the first period from the data-set. Thus, we analyze here the same 44 periods used to analyze response patterns later in this section. Quarter 1 consists of periods 2 to 12, quarter 2 periods 13 to 23, quarter 3 periods 24 to 34 and quarter 4 periods 35 to 45.

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remain persistently variable. For example in Quarter 3, just ten of the thirty 95% confidence bands uniquely capture Qc. Even in Quarter 4 confidence bands uniquely capture Qc in less than half of instances (in 14 of 30 outcomes). Finally, looking across treatments in the bottom panel suggests an SR treatment effect:

Confidence bands about mean outcomes in the SR pseudo-markets are

generally narrower than bands in markets for the other treatments. For example, seven of the eight narrowest confidence bands are about SR pseudo-market means. Market results, however, reflect imperfectly Nash equilibrium behavior, because group outcomes consistent with a market equilibrium may not indicate individual equilibrium choices. For example, the zero-variance market equilibrium outcome for the 4th quarter of a session in FC treatment shown at the left of the bottom panel of Figure 2, consists of three individuals persistently playing {3,4,5}. To more precisely evaluate the evolution of equilibrium play, we focus on individual decisions relative to the Nash equilibrium for the remainder of this section. The analysis consists of two parts. First, we consider individual decisions relative to the individual Nash Equilibrium outcome. Then we examine individual response patterns relative to a set of identifiably “Nash Consistent” patterns. a. Individual Cournot-Nash Equilibrium Play Rates: Table 3 summarizes the tendency of individuals to select the Nash equilibrium play of 4. Each entry in the table lists the percentage of possible instances per session quarter where individuals made the individual Nash equilibrium quantity choice. The entries listed in Table 3 illustrate two primary results. First, looking at the bolded treatment averages for the FC and SC markets printed in top two row blocks, observe that Nash equilibrium play rates are uniformly low in both treatments, with session-averages failing to exceed 48% in any quarter. Notice further, however, that increasing the slope of the cost curve does not appear to improve the drawing power of Nash predictions when participants are continuously paired. To the contrary, for the 2nd and 3rd quarters, the mean incidence of Nash Equilibrium plays in the FC treatment slightly exceeds that in the SC treatment, and Nash Equilibrium play rates are virtually identical in the 4th quarter. A second observation pertains to the FR and SR sessions, displayed in the bottom half of Table 3. Notice here that although the incidence of Nash equilibrium play is again uniformly low in the first quarter of the sessions, substantially higher rates of Nash play occur in the final periods of the SR treatment. In the 4th quarter the incidence of Nash equilibrium play is 32.6 percentage points higher in the SR treatment (77.0%) than in the FR treatment (44.4%).

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To support the above observations statistically, we use a linear mixed effects model to estimate the percentage of instances that individuals in each session selected the Nash equilibrium outcome (pij). As a control for potential interdependencies across individual decisions, we break the data into the four session quarters illustrated in Table 3, and use the percentage of Nash plays per quarter by an individual as the unit of observation. To control for interdependencies within sessions we define sessions as panels, and then assume a random effects error structure within panels. Specifically, for individuals j = 1, 2,…,9 in sessions i = 1, 2,…,10 we estimate pij = β FC FC i + β SC SC i + β FR FRi + β SR SRi + ν i + ε ij ,

(1)

where ν i ~ N (0, σ ν2 ) , ε ij ~ N (0, σ ε2 ) , and FC, SC, FR, and SR are indicator variables for the treatments. Table 4 summarizes results of generalized least squares estimates of equation (1) for each session quarter. Looking over the treatment estimates in the upper portion of the table, the uniformly significant estimates essentially recover the treatment averages in Table 3.

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coefficient comparisons listed at the bottom of the table summarize results of χ2 tests for the equality of coefficients across treatments. These tests support the following two findings. Finding 1: When participants remain paired in the same triopolies throughout the sessions, increasing the slope of the marginal cost curve does not improve convergence to the Nash equilibrium. Support: The third row from the bottom in Table 4 lists χ2 (1) tests for the equality of βFC and βSC parameters throughout session quarters. Given the similarity of βFC and βSC estimates shown in the upper portion Table 4, the insignificance of differences is unsurprising. Nevertheless, the extremely low χ2 (1) statistics, particularly in quarters 2, 3 and 4 are notable. No evidence suggests that these variables are different. Finding 2: When participants are rematched into new markets after each period, increasing the slope of the cost curve improves conformance with Nash predictions, toward the end of sessions. Support: The penultimate row of Table 4 lists χ2 tests for the equality of βFR and βSR estimates. Notice from the p-values that these variables differ significantly in quarters 2, and 4. As observed previously, the difference in Nash play rates across treatments is numerically the largest in quarter 4.

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b. Individual Response Patterns: Considering the evolution of individual response patterns as the sessions progress lends insight into why the SR treatment generates higher Nash play rates than the other treatments. It has become standard to focus on Best Response (B) and Fictitious play (F) response patterns when evaluating convergence tendencies in Cournot markets. Recently, Huck, Normann and Oechssler (1998, 1999) suggest that a third “Imitate-the-Average” strategy (M), may explain a considerable portion of decisions under the information conditions used here. Mechanically, a Best Response is the profit maximizing response to the play of rivals in the previous period.12 Fictitious play is the best response choice to the historical mean of rivals’ plays in preceding periods. Finally, play is consistent with an Imitate-the-Average response pattern when a player selects the average of the choices made by rivals in the previous period.13 A response is classified as Nash Consistent if the response conforms to at least one of these three patterns. As it turns out, none of these response patterns is a dominant characterization of behavior in our design.14 Our inability to distinguish between actions may in part be a consequence of the limited number of actions available to sellers in our design (which was not constructed with the intent of distinguishing between response patterns). In what follows we pool B, F and M responses into a single “Nash Consistent” classification, a pooling that is legitimate in that repeated Nash choices by other players would eventually lead to Nash equilibrium play by a player following one of the included strategies.15 12

In both the Flat Design and the Steep Design, players following a Best Response process are indifferent between a response of 3 or 4 when others play 9. In what follows we consider either choice consistent with a Best Response to a play of 9. 13 In the case that either Fictitious play or Imitate the Average response rules generate a fractional result, the nearest integer choice is used as the response. In the event that the fraction is ½ either integer about the result is regarded as being consistent with the rule. 14 For example, consider 4th quarter SR data, and define data points as the percentage of instances in the quarter that an individual’s responses were Nash Consistent. Using a Kolmogorov Smirnov test, we are unable to reject the null hypothesis that any pair-wise comparison of Fictitious, Best Response and Imitate the Average play differ significantly. Relevant test statistics are as follows. F v. M, K= .166; F v. B, K=.139; B v. M, K=.083. These test statistics are all far below the 90% c.v., of .287 (36, 36 d.f., direction not predicted). Pair-wise comparisons of empirical densities of F, B and M for the other quarters yield similar results. Note that this test (and those that follow) assumes the independence of average individual responses. Assuming the independence of response patterns is far less problematic that assuming independent Nash play rates. Although a single deviator can induce many players to make non-Nash plays, such deviations do not affect directly the consistency of individual responses. 15 It would be perhaps more interesting to identify a class of Nash Convergent response patterns, or patterns of play, that, when used in conjunction with others would lead to the Nash outcome. But the identification of such a class is problematic. Although the adaptive learning criterion of Milgrom and Roberts (1991) could be used to classify Best Response and Fictitious play as Nash Consistent in the Steep Design, adaptive learning does not necessarily generate convergent behavior in the Flat Design. Further, while common use of an “Imitate the Average” response pattern would not be Nash Consistent in either the Steep Design or the Flat Design, in at least some instances markets do converge to the Nash prediction if only a subset of participants use an Imitate the Average response pattern. The identification of

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The eight panels of Figure 3 illustrate the empirical densities of Nash Consistent plays by session quarters for the SR and for the non-SR sessions.16 In each panel, the vertical bars illustrate the percentage of players in a treatment that made Nash Consistent choices in the frequency interval shown on the horizontal axis. For example, the right-most vertical bar in the upper left corner of the figure indicates that in the 1st quarter of the SR treatment, roughly 10% of players made Nash Consistent choices 90% or more of the time. Looking down the two columns of Figure 3 notice the increased evolution of Nash Consistent play in the SR sessions.

Although the distribution of Nash Consistent Plays does not

differ importantly across treatments in the first session quarters, Nash Consistent play rates steadily pick up density as the sessions progress. By the fourth quarter, roughly 75% of responses by SR players are Nash Consistent at least 80% of the time. This compares with only about 30% of 4th quarter responses by non-SR players. The increased evolution of Nash Consistent Play in the SR treatment relative to the other treatments is easily established with the Kolmogorov-Smirnov test. Although the empirical distributions of Nash Consistent plays for the SR and non-SR treatments do not differ significantly for quarters one and two, the differences in the distributions become significant at a 97.5% confidence level in quarter three and at a 99.9% confidence level in quarter four.17 The increased tendency toward Nash Consistent behavior is perhaps unsurprising in light of the increased rates of Nash equilibrium play observed in Table 3. However, examining outcomes in terms of individual response patterns rather than aggregate outcomes allows us to emphasize two features of the convergence process. First, the bulk of players do not go into an experiment with a pre-established response pattern. Rather, at the outset participants most typically test out a variety of responses, which are narrowed down as the sessions progress. Second, the underlying structure

Nash Convergent response patterns becomes still more problematic when markets involve rematching across periods, since in this case players no longer respond to actions of the same group. 16 Pooling of the FR, FC and SC treatments is innocuous here, since the Nash Consistent play rates do not differ significantly across treatments. For example, a Kolmogorov-Smirnov test on 4th quarter data, yields the following test statisitics, FR v. FC, K=.388; FC v. SC, K-.167, FR v. SC, K=.222. These are all less than a 90% c.v., of .407 (16, 16 d.f., direction not predicted). Comparisons of data for the other quarters yields similar results. The homogeneity of response patterns across treatments is unsurprising in light of the similarity of Nash Play rates, shown in Table 4. Notice in the bottom row of Table 4 that rejection of the joint hypothesis βSC =βFC =βFR is not possible for any session quarter at conventional significance levels. 17 K-S test statistics for difference the SR vs. non-SR sessions are as follows: Quarter 1, K=.231, Quarter 2, K=.212, Quarter 3, K=.343 and Quarter 4, K=.424. Relevant critical values include 90%, .262; 97.5% .318; and 99.9% , .419 (36, 54 d.f., direction not predicted).

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of the sessions importantly influences the development of Nash Consistent patterns. This is our third finding. Finding 3: Independent of the underlying structure, players start sessions with heterogeneous response patterns. Over time, the SR treatment induces the selection of Nash Consistent Play. Observe finally that structural factors alone do not dictate the convergence to Nash equilibrium outcomes. In particular, even under the best of circumstances individual differences can importantly affect the convergence process.

Notice, for example, the variability of Nash

equilibrium play rates across the SR sessions, shown in the bottom of Table 3. In session SR-1, a single player persistently selected a quantity of “2” throughout the session (two units away from the Nash equilibrium quantity choice). As is evidenced by the 4th quarter Nash Equilibrium play rate of 56.6% for session SR-1, the deviant choices of this player undermined critically the development of Nash outcomes for the group.18 Similarly, the relatively low 69.73% Nash Equilibrium Play rate in the 4th quarter of session SR-3 may be largely driven by a single player who persistently chose “3” throughout most of the session. The capacity of single individuals to deviate and so undermine the development of Nash outcomes is consistent with the intuition driving the result by Milgrom and Roberts (1991) that adaptive learning will drive play to the set of serially undominated outcomes. Convergence under adaptive response patterns requires that everyone play adaptively. Even a single, relatively infrequently encountered deviator who persistently selects a strictly dominated strategy can disrupt the convergence process. 5. Conclusions These experimental results clearly indicate that relatively subtle structural details can affect the stability of Cournot markets. When participants are rematched after each period, we observe markedly improved rates of Nash play in treatments with a steeper marginal cost schedule, but only after participants have considerable experience with the game. However, we also find that the structural manipulations explored here do not improve stability in the perhaps more standard contexts where participants remain matched throughout a market session.

Changes in static

incentives caused by manipulating cost schedules appear to be dominated by “gaming” behavior 18

A single individual accounts for only 11.1% of (1/9) of plays. A 4th quarter Nash play rate o 56.6% suggests that this deviator may exerted considerable external effects.

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when participants remain paired repeatedly. Finally, even under the best of circumstances (e.g., when players are rotated across periods), deviant individual choices can undermine the development of Nash Consistent responses.

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References Bernheim, Douglas (1984) “Rationalizable Strategic Behavior,” Econometrica 52, 1007-1028. Capra, Monica, Jacob K. Goeree, Rosario Gomez and Charles A. Holt (1999) “Anomalous Behavior in a Traveler’s Dilemma?” American Economic Review 89, 678-690. Davis Douglas D. (1999) “Advance Production and Cournot Outcomes: An Experimental Investigation,” Journal of Economic Behavior and Organization 40, 59-79. Davis, Douglas D. (2002) “Strategic Interactions, Market Information and Mergers in Differentiated Product Markets: An Experimental Investigation,” International Journal of Industrial Organization, forthcoming. Fisher, F. (1961) “The Stability of the Cournot Solution: The Effects of Speeds of Adjustment and Increasing Marginal Costs,” Review of Economic Studies 74, 125-135. Fouraker, Lawrence E., and Sidney Siegel (1963) Bargaining Behavior (McGraw-Hill, New York). Holt, Charles A. (1985) “An Experimental Test of the Consistent Conjectures Hypothesis” American Economic Review 75, 314-25. Holt, Charles A. (1995) “Industrial Organization: A Survey of Laboratory Research,” in Alvin Roth and John Kagel, eds., Handbook of Experimental Economics (Princeton University Press, Princeton). Huck, Steffen, Hans-Theo Normann and Jorg Oechssler “Best Reply Behavior and Imitation in a Cournot Experiment,” (1998) working paper, Humboldt University, Berlin Germany. Huck, Steffen, Hans-Theo Normann and Jorg Oechssler (1999) “Learning in Cournot Oligopoly – An Experiment” Economic Journal 109, c80-c95. McKelvey, Richard D. and Thomas R. Palfrey (1992) “An Experimental Study of the Centipede Game,” Econometrica 60, 803-836. Milgrom, Paul and Roberts John (1991) “Adaptive and Sophisticated Learning in Normal Form Games” Games and Economics Behavior 3, 82-100. Ledyard, John O. “Public Goods: A Survey of Experimental Research” in Alvin Roth and John Kagel, eds., Handbook of Experimental Economics (Princeton University Press, Princeton). Phillips, Owen R., and Charles F. Mason (1992) “Mutual Forbearance in Experimental Conglomerate Markets,” RAND Journal of Economics 23, 395-414.

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Rassenti, Stephen, Stanley S. Reynolds, Vernon L. Smith and Ferenc Szidarovszky (2000) “Adaptation and Convergence of Behavior in Repeated Experimental Cournot Games,” Journal of Economic Behavior and Organization 41(2), 117-146. Theocharis, R. D. (1960) “On the Stability of the Cournot Solution to the Oligopoly Problem,” Review of Economic Studies 27, 133-134. Wellford, Charissa P. (1990) “Horizontal Mergers: Policy and Performance,” Ph.D. Dissertation, University of Arizona. Vega-Redondo, Fernando (1997) “The Evolution of Walrasian Behavior,” Econometrica 65, 37584.

15

Table 1(a). Payoffs for S1 (column), Flat Design Qty

S2

S3= 1

1 2

3

4

5

6

7

Qty

S2

S3= 1

5 2

3

4

5

6

7

1 2

40 37

74 68

102 93

123 111

137 122

145 127

147 126

1 2

28 25

50 44

66 57

75 63

77 62

73 55

63 42

3

34

62

84

99

107

109

105

3

22

38

48

51

47

37

21

4

31

56

75

87

5

28

50

66

75

92

91

84

4

19

32

39

39

32

19

0

77

73

63

5

16

26

30

27

17

1

-21

6

25

44

57

63

62

55

42

6

13

20

21

15

2

-17

-42

7

22

38

48

51

47

37

21

7

10

14

12

3

-13

-35

-53

Qty

S2

S3= 1

2 2

Qty

S2

S3= 1

6 2

3

4

5

6

7

1 2

37 34

68 62

93 84

111 99

122 107

127 109

126 105

3

4

5

6

7

1 2

25 22

44 38

57 48

63 51

62 47

55 37

42 21

3

31

56

75

87

92

91

84

3

19

32

39

39

32

19

0

4

28

50

66

75

77

5

25

44

57

63

62

73

63

4

16

26

30

27

17

1

-21

55

42

5

13

20

21

15

2

-17

-42

6

22

38

48

51

47

37

21

6

10

14

12

3

-13

-35

-63

7

19

32

39

39

32

19

0

7

7

8

3

-9

-28

-53

-84

Qty

S2

S3= 1

3 2

Qty 3

4

5

6

7

S2

S3= 1

7 2

3

4

5

6

7

1

34

62

84

99

107

109

105

1

22

38

48

51

47

37

21

2 3

31

56

75

87

92

91

84

2

19

32

39

39

32

19

0

28

50

66

75

77

73

63

3

16

26

30

27

17

1

-21

4

25

44

57

63

62

55

42

4

13

20

21

15

2

-17

-42

5

22

38

48

51

47

37

21

5

10

14

12

3

-13

-35

-63

6

19

32

39

39

32

19

0

6

7

8

3

-9

-28

-53

-84

7

16

26

30

27

17

1

-21

7

4

2

-6

-21

-43

-71

-103

Qty

S2

S3= 1

4 2

3

4

5

6

7

1 2 3

31 28 25

56 50 44

75 66 57

87 75 63

92 77 62

91 73 55

84 63 42

4

22

38

48

51

47

37

21

5

19

32

39

39

32

19

0

6 7

16 13

26 20

30 21

27 15

17 2

1 -17

-21 -42

16

Table 1(b). Payoffs for S1 (column), Steep Design Qty

S2

S3= 1

1 2

3

4

5

6

7

Qty

S2

S3= 1

5 2

3

4

5

6

7

1 2 3

43 40 37

80 74 68

111 102 93

132 120 108

142 127 112

146 128 110

144 123 102

1 2 3

31 28 25

56 50 44

75 66 57

84 72 60

82 67 52

74 56 38

60 39 18

4

34

62

84

96

97

92

81

4

22

38

48

48

37

20

-3

5

31

56

75

84

82

74

60

5

19

32

39

36

22

2

-24

6

28

50

66

7

25

44

57

72

67

56

39

6

16

26

30

24

7

-16

-45

60

52

38

18

7

13

20

21

12

-8

-34

-66

Qty

S2

S3= 1

2 2

Qty 3

4

5

6

7

S2

S3= 1

6 2

3

4

5

6

7

1 2

40 37

74 68

102 93

120 108

127 112

128 110

123 102

1 2

28 25

50 44

66 57

72 60

67 52

56 38

39 18

3

34

62

84

96

97

92

81

3

22

38

48

48

37

20

-3

4

31

56

75

84

82

74

60

4

19

32

39

36

22

2

-3

5

28

50

66

72

67

56

39

5

16

26

30

24

7

-16

-24

6

25

44

57

60

52

38

18

6

13

20

21

12

-53

-34

-54

7

22

38

48

48

37

20

-3

7

10

14

12

0

-53

-52

-66

Qty

3 2

Qty

S2

S3= 1

3

4

5

6

7

S2

S3= 1

7 2

3

4

5

6

7

1 2

37 34

68 62

93 84

108 96

112 97

110 92

102 81

1 2

25 22

44 38

57 48

60 48

52 37

38 20

18 -3

3

31

56

75

84

82

74

60

3

19

32

39

36

22

2

-24

4

28

50

66

72

67

56

39

4

16

26

30

24

7

-16

-45

5

25

44

57

60

52

38

18

5

13

20

21

12

-8

-34

-66

6

22 19

38 32

48 39

48 36

37 22

20 2

-3 -24

6 7

10 7

14 8

12 3

0 -12

-23 -38

-52 -70

-87 -108

Qty

S2

S3= 1

4 2

3

4

5

6

7

1 2

34 31

62 56

84 75

96 84

97 82

92 74

81 60

3

28

50

66

72

67

56

39

4

25

44

57

60

52

38

18

5

22

38

48

48

37

20

-3

6 7

19 16

32 26

39 30

36 24

22 7

2 -16

-24 -45

17

Table 2. Matrix of Treatments by Session Dynamic Group Structure Design

Constant (C) Matching

Rematched (R) each period

Flat (F) Design

FC–1 (3 markets) FC –2 (3 markets)

FR–1 (3 markets each pd.) FR –2 (3 markets each pd.)

Steep (S) Design

SC –1 (3 markets) SC –2 (3 markets)

SR –1 SR –2 SR –3 SR –4

18

(3 markets each pd.) (3 markets each pd.) (3 markets each pd.) (3 markets each pd.)

Table 3. Individual Nash Equilibrium Play Rates for Sessions, by Quarter Session (1) Session FC-1 FC-2 FC

(2) 1st Qtr.

(3) 2nd Qtr.

(4) 3rd Qtr.

(5) 4th Qtr.

41.4 25.3 33.3

54.5 38.4 46.5

48.5 39.4 43.9

60.6 35.4 48.0

SC-1 SC-2 SC

41.4 38.4 39.9

43.4 52.5 48.0

42.4 51.5 47.0

45.5 48.5 47.0

FR-1 FR-2 FR

15.2 41.4 28.3

22.2 47.5 34.8

34.3 60.6 47.5

30.3 58.6 44.4

SR-1 41.4 49.5 50.5 56.6 SR-2 44.4 75.8 82.8 92.9 SR-3 29.3 59.6 49.5 69.7 SR-4 35.4 41.4 49.5 88.9 SR 37.6 56.6 58.1 77.0 Key: Each entry lists the percentage of total possible instances in the session portion where a player made a Nash equilibrium selection (4).

19

Table 4. Estimation of Individual Nash Play Rates 1st Qtr. 2nd Qtr. 3rd Qtr. Coefficient (p-valuea) βFC

4th Qtr.

0.33 (0.001)

0.46 (0.001)

0.44 (0.001)

0.48 (0.002)

βSC

0.40 (0.000)

0.48 (0.001)

0.47 (0.000)

0.47 (0.002)

βFR

0.28 (0.002)

0.35 (0.004)

0.47 (0.001)

0.44 (0.003)

βSR

0.38 (0.000)

0.57 (0.000)

0.58 (0.000)

0.77 (0.000)

0.00 0.24 1.82

0.05 0.28 -14.32

0.04 0.31 -23.78

0.08 0.30 -22.60

Wald χ2(4) N

202.77* 90.00

210.39* 90.00

211.08* 90.00

230.88* 90.00

Coefficient Comparisons βSC =βFC χ2(1) (p-value)

0.69 (0.406)

0.02 (0.873)

0.07 (0.773)

0.01 (0.936)

βSR =βFR χ2(1) (p-value)

1.86 (0.172)

7.00 (0.008)

1.36 (0.243)

13.02 (0.000)

βSC =βFC =βFR χ2(3) (p-value)

2.17 (0.338)

2.30 (0.317)

0.13 (0.936)

0.12 (0.941)

σv σe Log likelihood

* p-values based on 10 (sessions) – 4 (treatments) = 6 degrees of freedom. (2-tailed tests, 3 significant digits)

20

Cents 60

Flat D esign

Pm =36

S Pc=24 13=ch 12=cm 11=cl

Pw =15 S1 S1 S1 S2 S2 S2 S3 S3 S3 S1 S2 S3 S1 S1 S1 S2 S2 S2 S3 S3

MR Q m =8

Cents 60

D

Q c=12

Q ty.

Q w =15

Steep Design

Pm =33

S Pc=24 Pw =18 S1 S1 S1 S2 S2 S2 S3 S3 S3 S1 S2 S3 S1 S1 S1 S2 S2 S2 S3 S3 S3

Q m =9

D

MR Q c=12 Q w =14

Figure 1. Supply and Demand Arrays

21

Q ty.

17=ch 12=cm 8=cl

Q ty. 21 18 Q w 15 Q c 12 Qm 9 6

FC

SC

3

FR

SR

FR

SR

FR

SR

FR

SR

Q uarter1

Q ty. 21 18 Q w 15 Q c 12 Qm 9 6

FC

SC

3

Q uarter 2

Q ty. 21 18 Q w 15 Q c 12 Qm 9 6

FC

SC

3

Q uarter 3

Q ty. 21 18 Q w 15 Q c 12 Qm 9 6 3

FC

SC Q uarter 4

Figure 2. Market and “Pseudo-Market” Mean Quantities. Key: In each panel dots illustrate mean outcomes for one market in a treatment. Solid dots indicate instances where the mean outcome is closer to the Cournot Nash prediction, Qc than to the joint profit maximizing outcome Qm or the Walrasian outcome, Qw. Bars illustrate 95% confidence bands about the means. Outcomes shown in the FR and SR sessions illustrate “pseudo-market” outcomes, or outcomes that would have been observed had participants remained in the same nominal markets throughout a session.

N on SR Sessions

SR Sessions 60%

1stQ uarter

50%

60%

1stQ uarter

Players

Players

50% 40% 30% 20%

40% 30% 20%

10% 10% 0%

[0 ] (0 ,.1 ] (.1 ,.2 ] (.2 ,.3 ] (.3 ,.4 ] (.4 ,.5 ] (.5 ,.6 ] (.6 ,.7 ] (.7 ,.8 ] (.8 ,.9 ] (.9 ,1 )]

[0 ] (0 ,.1 ] (.1 ,.2 ] (.2 ,.3 ] (.3 ,.4 ] (.4 ,.5 ] (.5 ,.6 ] (.6 ,.7 ] (.7 ,.8 ] (.8 ,.9 ] (.9 ,1 )]

0%

Percentage of N ash ConsistentPlays Percentage of N ash ConsistentPlays 60%

60%

2nd Q uarter

50%

Players

40% 30% 20%

30% 20%

10%

10%

0%

0%

[0 ] (0 ,.1 ] (.1 ,.2 ] (.2 ,.3 ] (.3 ,.4 ] (.4 ,.5 ] (.5 ,.6 ] (.6 ,.7 ] (.7 ,.8 ] (.8 ,.9 ] (.9 ,1 )]

Percentage of N ash ConsistentPlays 60%

Percentage of N ash ConsistentPlays 60%

3rd Q uarter

50%

40%

Players

Players

50%

30% 20% 10%

40% 30% 20%

[0 ] (0 ,.1 ] (.1 ,.2 ] (.2 ,.3 ] (.3 ,.4 ] (.4 ,.5 ] (.5 ,.6 ] (.6 ,.7 ] (.7 ,.8 ] (.8 ,.9 ] (.9 ,1 )]

[0 ] (0 ,.1 ] (.1 ,.2 ] (.2 ,.3 ] (.3 ,.4 ] (.4 ,.5 ] (.5 ,.6 ] (.6 ,.7 ] (.7 ,.8 ] (.8 ,.9 ] (.9 ,1 )]

0%

Percentage of N ash ConsistentPlays

Percentage of N ash ConsistentPlays 60%

60%

4th Q uarter Players

40%

50%

30% 20%

4th Q uarter

40% 30% 20% 10%

0%

0%

[0 ] (0 ,.1 ] (.1 ,.2 ] (.2 ,.3 ] (.3 ,.4 ] (.4 ,.5 ] (.5 ,.6 ] (.6 ,.7 ] (.7 ,.8 ] (.8 ,.9 ] (.9 ,1 )]

10%

[0 ] (0 ,.1 ] (.1 ,.2 ] (.2 ,.3 ] (.3 ,.4 ] (.4 ,.5 ] (.5 ,.6 ] (.6 ,.7 ] (.7 ,.8 ] (.8 ,.9 ] (.9 ,1 )]

Players

3rd Q uarter

10%

0%

50%

2nd Q uarter

40%

[0 ] (0 ,.1 ] (.1 ,.2 ] (.2 ,.3 ] (.3 ,.4 ] (.4 ,.5 ] (.5 ,.6 ] (.6 ,.7 ] (.7 ,.8 ] (.8 ,.9 ] (.9 ,1 )]

Players

50%

Percentage of N ash ConsistentPlays

Percentage of N ash ConsistentPlays

Figure 3. Em piricalD ensities of N ash ConsistentPlays by Session Q uarters.

2

(1) Session FC-1 FC-2 FC

Table A1. Nash Equilibrium Outcomes by, Quarter Session (2) (3) (4) 1st Qtr. 2nd Qtr. 3rd Qtr. 9.09 18.18 15.15 0.00 12.12 3.03 4.55 15.15 9.09

(5) 4th Qtr. 21.21 3.03 12.12

SC-1 SC-2 SC

6.06 9.09 7.58

9.09 18.18 13.64

3.03 12.12 7.58

12.12 9.09 10.61

FR-1 FR-2 FR

0.00 3.03 1.52

0.00 9.09 4.55

3.03 18.18 10.61

3.03 18.18 10.61

3.03 12.12 9.09 9.09 SR-1 3.03 42.42 54.55 80.81 SR-2 0.00 24.24 15.15 26.26 SR-3 6.06 3.03 10.10 68.69 SR-4 SR 3.03 20.45 22.22 46.21 Key: Each entry lists the percentage of total possible instances in the session portion where a player made a Nash equilibrium selection (4), and where the other players also each selected (4).

3

Table A2. Instances of Nash Equilibrium Payoffs by, Quarter Session (1) Session FC-1 FC-2 FC

(2) 1st Qtr.

(3) 2nd Qtr. 14.1 3.0 8.6

27.3 15.2 21.2

(4) 3rd Qtr. 20.2 12.1 16.2

(5) 4th Qtr. 25.3 16.2 20.7

SC-1 SC-2 SC

8.1 11.1 9.6

13.1 19.2 16.2

12.1 14.1 13.1

15.2 15.2 15.2

FR-1 FR-2 FR

3.0 4.0 3.5

1.0 12.1 6.6

6.1 23.2 14.6

9.1 19.2 14.1

7.1 18.2 12.1 15.2 SR-1 6.1 43.4 55.6 81.8 SR-2 3.0 30.3 21.2 29.3 SR-3 9.1 8.1 11.1 70.7 SR-4 SR 6.3 25.0 25.0 49.2 Key: Each entry lists the percentage of total possible instances in the session portion where a player made a Nash equilibrium selection (4), and where the other players combined made the joint Nash equilibrium selection (8).

4