Information sharing and tacit collusion in laboratory duopoly markets.

Cason, Timothy N. ; Mason, Charles F.

I. INTRODUCTION

Economists have long believed that enhanced information about demand conditions or rival actions can play an important role in facilitating collusion, for example because it simplifies detection of chiselers (Stigler [1964]). An extensive theoretical literature examines the incentives for non-cooperative firms to share information through a trade association about some uncertain parameter (see Sakai [1990; 1991] for a survey). Firms in these models may "cooperate" by sharing information in the trade association, even though most authors assume that the firms choose non-cooperative strategies in the product market. Under these conditions, Vives [1984] shows that with demand uncertainty, substitute goods and quantity competition, each firm's dominant strategy is to conceal information from its rivals. Therefore, under these conditions non-cooperative firms will not find it in their interests to form an information-sharing arrangement such as a trade association. Because non-cooperative firms should not wish to share information, Clarke [1983, 392] concludes that "information-pooling mechanisms like trade associations can be considered prima facie evidence that firms are illegally cooperating to restrict output." However, Kirby [1988] shows that non-cooperative firms will want to share information if cost functions are sufficiently quadratic. Moreover, Hviid [1989] demonstrates that non-cooperative firms will prefer to share information if they are sufficiently risk averse.

These models typically assume that firms interact once and identify the static, subgame perfect Nash Equilibrium (NE) of the two-stage information sharing game. This is sensible when the goal is to focus on a narrow set of non-cooperative equilibria, because the set of non-cooperative equilibria becomes very large in these models in the more complex repeated game. However, repeated interaction between competing firms seems more relevant for empirical applications, especially for providing policy insight for trade associations that permit long-term interactions among firms.

Unfortunately, empirical work in this area based on field data is challenging because the information different decision-makers possess, and the corresponding residual uncertainty they face, are typically difficult to identify. This paper uses a different empirical approach. We use a series of laboratory duopoly markets to examine the impact of demand uncertainty on sellers' output choices. Specifically, we examine the importance of information sharing in facilitating tacit collusion under conditions of demand uncertainty.(1) Our experiment employs repeated interaction between fixed pairs of subjects, trading the ability to directly test the static models for a more realistic, policy-relevant study of outcomes in the repeated game.(2) Our results indicate the extent to which results from static theory carry over to the repeated environment, and can help guide future theory for dynamic oligopoly models of information sharing.

The experiment includes four distinct treatment conditions. In the primary treatment, sellers face uncertain demand but can eliminate uncertainty through voluntary, mutual information sharing. Three control treatments identify the reasons for voluntary information sharing and determine if it leads to more collusive outcomes. The first two control treatments eliminate sellers' information sharing opportunities. In the first control treatment sellers always remain uncertain about demand when choosing output, while in the second control treatment sellers always have perfect demand information. We refer to these control treatments respectively as Forced No Sharing and Forced Sharing throughout the paper. The third control treatment sharply reduces collusion incentives by truncating demand while still allowing voluntary information transmission. We refer to this control treatment as the Truncated Demand treatment.

The results, briefly stated, are as follows. First, sellers in these repeated laboratory markets generally shared information. Second, sellers who voluntarily shared information successfully restricted output below the static NE level in periods of high and low demand. Results were similar, however, in the Forced Sharing treatment. Third, sellers selected significantly greater output than the static NE in periods of low demand when information was not shared. Fourth, our analysis of the sharing decision indicates that increased rival output in previous periods modestly increases the likelihood that sellers conceal information. Fifth, information sharing was common in the Truncated Demand treatment. The second finding raises the specter that information sharing facilitates some tacit collusion, while the fourth finding suggests that some sellers employed a strategy of information concealment to punish non-colluding rivals. Collusive incentives are minimal in the Truncated Demand treatment, however, so the fifth finding suggests that risk aversion may be the primary explanation for information sharing.

Given the likelihood that to some extent sellers shared information to eliminate risk, but that the act of sharing modestly reduced output in some demand states, the policy implications are mixed. If we take the conventional view that welfare is measured by net benefits, then the apparent collusion observed at times in the low demand state due to information sharing is welfare reducing. The welfare impact of information sharing is more ambiguous, however, if welfare is better reflected by the sum of consumer surplus and the sum of firms' expected utility of profits. The distortionary effects of output reductions are not economically large in our experiment, and output choices were similar in the primary treatment and the Forced No Sharing and Forced Sharing treatments. This suggests a more sanguine view of the desirability of trade associations.(3)

II. THE MODEL AND LABORATORY ENVIRONMENT

Procedures and Experimental Design

Subjects in our experiment played a repeated Cournot competition game against the same rival throughout the session. Profits were represented using payoff matrices that varied across periods as demand conditions varied. In each session the experiment was subjected to a random termination rule. Starting with period 30, the monitor rolled a ten-sided die. The session ended the first time the die came up as a 0 or 1. This induces a 80% continuation probability, which is equivalent to a discount rate of 0.8 beginning in period 30 (with a higher discount rate for earlier periods).

Figure 1 presents a flow chart of an information sharing period. In the information sharing sessions subjects faced two tasks each period. In task 1 they chose whether or not to share a "coin flip" signal with their rival. After the task 1 choice the subjects learned their rival's sharing decision. Both signals were revealed to both subjects only if both agreed to share information; otherwise, subjects learned only their own signal (and so were imperfectly informed about demand). Having agents choose whether or not to share information prior to observing the demand signal is standard in the information sharing literature. It also simplifies subjects' tasks because they do not need to make inferences regarding the content of signals their rival may or may not wish to share. Following the completion of task 1 subjects faced a symmetric duopoly environment with homogeneous goods and linear demand. In task 2 they made what was framed as an "integer choice" that corresponded to a quantity choice. Costs were quadratic and represented increasing returns to scale in order to enhance the expected payoff differences for the two information sharing choices, as discussed below.

This study employs a total of 90 separate subjects (45 duopoly pairs) in 10 separate sessions summarized in Table I. Eighteen pairs participated in the information-sharing sessions described above. In addition, nine subject pairs participated in the Forced No Sharing treatment, ten subject pairs participated in the Forced Sharing treatment, and eight subject pairs participated in the Truncated Demand treatment. Although the realization of demand draws varied across duopoly pairs, we employed the same sequence of demand draws across the four treatments to minimize non-behavioral variance across treatments. In the Forced No Sharing control subjects only receive information about their own coin flip before making an integer choice. In the Forced Sharing control subjects receive both coin flips before making an integer choice, so they always have complete demand information. In all other respects the treatments are identical.4 With one exception, each session used 8 or 10 subjects.(5) Including instructions, each session lasted approximately 2 hours; subject earnings ranged from $5 to $54 with an average of about $31.

All sessions were conducted at the USC Experimental Economics Laboratory on a network of PCs. The computer recorded the data and handled the payoff accounting, but subjects also filled out detailed record sheets for each period at their computer so that they could always refer to their choices and the [TABULAR DATA FOR TABLE I OMITTED] choices of their rival in each previous period and information state. The software implements a simple command-line interface, which sequentially prompted subjects to input their task 1 (information sharing) and task 2 (integer) choice each period. The payoff matrices for the different demand states were taped to the partitions that separated each computer.(6) The high, medium and low demand states were referred to by colors for the subjects (green, blue and red, respectively). The expected payoff tables for the uncertainty information conditions (i.e., HEADS received, no information sharing) were also taped to the partitions. These matrices were referred to as blue-green (for HEADS) and purple (for TAILS). Instructions are available on request.

Costs and Demand

Demand each period is linear with additive shocks,

(1) [Mathematical Expression Omitted]

where P is the common price received by both firms, [Mathematical Expression Omitted] is a random demand intercept, [Beta] is a slope parameter and [q.sub.1] and [q.sub.2] are the quantity choices of firms 1 and 2, respectively. In our experiment the intercept [Mathematical Expression Omitted] takes on high and low values ([a.sub.H] and [a.sub.L], respectively) with probability 0.25 each, and it takes its mean value [Mathematical Expression Omitted] with probability 0.5 ([a.sub.h] and [a.sub.L] are distributed symmetrically about the mean). We refer to [Mathematical Expression Omitted] as medium demand. Costs are identical for the two firms and are given by

(2) [Mathematical Expression Omitted].

We employ increasing returns to scale (i.e., d [less than] 0 and F [greater than] 0) in order to increase the expected payoff differences at the static Nash equilibrium for the different information sharing choices. From the subjects' perspective the demand and cost structure were combined into payoff tables. The payoff tables indicate the profit for each demand state for each pair of integers chosen by the subject pair.

Table II provides payoff matrices for the three demand states. All payoffs are given in cents. The parameters underlying these payoffs for the model above are shown in Table III, along with the stage game Nash equilibrium and optimal collusive choices and expected profits. The "quantity choices" of the model that range between 1 and 9 are transformed [TABULAR DATA FOR TABLE II OMITTED] to "integer choices" between 0 and 8 for presentation to the subjects. Other asymmetric static NE exist, such as any combination of two integers that sum to 8 in the medium state. However, all these equilibria have the same aggregate pair output. Since our econometric model uses the aggregate pair choice as the unit of observation, our empirical analysis is unaffected by the multiplicity of stage game equilibria.

Information Structure

Each subject receives a fair "coin flip" each period. If both coin flips are heads, the demand state is [a.sub.H]; if both coin flips are tails, the demand state is [a.sub.L]. If the two coin flips do not match, the demand state is [Mathematical Expression Omitted]. If a subject learns both coin flips, she knows the true demand state for the period. However, if a subject receives only her own coin flip, this provides a noisy signal of the demand state. A signal of heads (resp., tails) by itself indicates that the state is "not [a.sub.L]" ("not [a.sub.H]"), so that there is a 50% chance of a and a 50% chance of [a.sub.H] ([a.sub.L]). The computer software running the experiment updates these Bayesian posterior assessments of the demand state for subjects and reminds them of the appropriate [TABULAR DATA FOR TABLE III OMITTED] payoff table(s). Subjects receive both coin flips only if both agree to share information. If either subject declines to share information with his or her rival, both subjects receive only their own coin flip. Also note from Figure 1 that sellers make information-sharing decisions prior to the coin flip revelation. This is consistent with the theoretical literature assessing (static game) incentives for information sharing and, as noted above, allows us to avoid the complications in the signaling game when sharing choices are based on an observed coin flip.

Task 2 Quantity Choice: Stage Game Equilibria

A straightforward application of the standard Cournot model yields the symmetric state-contingent stage game NE quantity choices shown in equation (1) of Table III, for the case when subjects share information. Although these static equilibria are a small subset of a large set of equilibria in the repeated game, they can play an important role in many equilibria for the repeated game. For example, the static NE often serves as a credible, subgame perfect punishment in trigger strategy equilibria. We also employ these static equilibria in the empirical analysis as null hypotheses and to normalize choices in the different demand states.

Without information sharing the subjects' have two signals upon which to base their quantity choices - tails and heads. If subjects maximize expected profit given Bayesian expectations of rival quantity choices, the static Nash equilibrium quantity choices for the two signals are shown in equations (2) and (3) of Table III. The first term is analogous to the right side of equation (1) of Table III, with the expectation replacing the known value of a. The second term represents an additional adjustment reflecting the residual uncertainty. If [Mathematical Expression Omitted], the optimal choice is greater than (less than) that implied by only the expectation of a as long as [absolute value of [Beta]][greater than][absolute value of d] (as it is for the experiment parameters).(7)]

Task 1 Information Sharing Choice: Stage Game Incentives

Because sellers make information sharing decisions before the demand state is revealed, in the subgame perfect equilibrium this decision is based on profit expectations from the quantity competition in Task 2. Firms will not wish to share information under Cournot quantity competition with substitute goods. Information sharing correlates quantity choices, and with substitute goods strategy correlation reduces expected profits (Vives [1984]). However, if costs are sufficiently quadratic under decreasing returns to scale (i.e., large positive values of d), then non-cooperative firms will find information sharing to be profitable (Kirby [1988]). With decreasing returns to scale, errors in production become more costly (in expected profit terms) than the cost of quantity correlation. In other words, the reduction in uncertainty is worth the increased strategy correlation occurring from information sharing.

Our objective was to provide incentives for subjects to conceal information if they behaved in accordance with the stage game noncooperative equilibrium in order to determine if information sharing might evolve as a strategy in the repeated game. Previous work has shown that a 15% payoff difference was sufficient and a 10% difference insufficient motivation for human subjects to learn optimal actions (Arthur [1991]; Cason [1994]). Since these payoff differences could not be obtained in our experimental design using constant returns to scale we employed increasing returns to scale (d [less than] 0 and F [greater than] 0). The increasing returns to scale used in the experiment were sufficient to generate expected profit differences of approximately 15% at the static NE - i.e., expected profit of 76.5 when sharing information versus 90 when not sharing information, as shown in Table III.

Although information sharing reduces expected profit in this setting, it also reduces profit variance. Therefore, risk averse subjects may find information sharing to be optimal. For example, a subject with constant absolute risk aversion whose Arrow-Pratt measure exceeded 0.0071 would prefer to share information. Our Truncated Demand treatment is designed to differentiate between the incentive to reduce risk and the incentive to facilitate collusion. In this treatment, prices could not rise above P = 48 in the high and medium demand states. Payoff matrices for this treatment are given in the appendix. This truncation effectively eliminated collusion opportunities in the high and medium demand states because it caused the symmetric joint profit-maximizing choice to equal the symmetric static NE, which remained unchanged from the primary treatment. The payoff matrix in the low demand state is the same in this treatment as the payoff matrix in the primary treatment because the low demand state provides minimal collusion opportunities; indeed, adjusting outputs from the Cournot/Nash level to the symmetric joint profit maximizing level would only raise profits by 3 cents. The Nash equilibrium quantity choices change slightly if information is concealed and subjects are risk neutral (i.e., to 8 for Heads and 0 for Tails), but only a slight amount of risk aversion leads to optimal choices of 7 for Heads and 1 for Tails as before. Therefore, we expect static, non-cooperative integer choices in this treatment to be indistinguishable from choices in the primary treatment. Correspondingly, and this is the key point, information sharing cannot be motivated in the Truncated Demand treatment by a collusion-facilitating strategy. Information sharing is, however, consistent with risk aversion.

Repeated Game Equilibria

Each session of the experiment ended probabilistically, which is mathematically equivalent to the infinitely repeated game with discounting. It is well known that the set of noncooperative Nash equilibria is greatly expanded with sufficiently high discount rates, and includes many cooperative-looking outcomes (Friedman [1983]).(8) We refer to these "cooperative" outcomes as collusive equilibria in the pursuant discussion. Theoretical models often construct these collusive equilibria with trigger strategies that threaten to punish non-collusive behavior of rival sellers. One possible strategy includes information sharing and collusive, restricted output as two components of a collusive equilibrium strategy:(9)

For demand states [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], and for coin flip signals [sig.sub.i] = Heads or Tails, share information and produce [Mathematical Expression Omitted] as long as my rival j and I have shared information and produced [Mathematical Expression Omitted] in every previous period; produce [Mathematical Expression Omitted] if my rival or I conceal information in the current period; and conceal information and produce [Mathematical Expression Omitted] if my rival or I conceal information or fail to produce [Mathematical Expression Omitted] in any previous period.

The [Mathematical Expression Omitted] could be a number of output levels lower than [Mathematical Expression Omitted] shown in Table III that generate payoffs which Pareto dominate the expected payoffs in the stage game equilibrium. As a benchmark, we define the symmetric joint profit maximizing output choices. Equation (4) of Table III presents the optimum quantity choices if both subjects share information and collude to maximize joint profits. For the discount rate induced in our experiment, combinations of any output between those defined in equations (1) and (4) of Table III can be an equilibrium outcome.(10)

Note that in the candidate repeated game strategy above, the punishment phase is triggered by either non-collusive output expansion or information concealment. While one can find collusive strategies where information concealment does not necessarily trigger punishment, we focus on the repeated game strategies with information sharing for three reasons. First, the empirical results below indicate that subjects generally shared information, and that they restricted output and earned greater profit in periods with information sharing for certain demand states. Second, our analysis of the sharing decision at the end of section III suggests that a number of subjects employed information concealment immediately following higher output choices by their rival, which is consistent with the punishment phase of these repeated game strategies. Third, both theoretical arguments (Green and Porter [1984]) and independent experimental evidence (Mason and Phillips [1997]) suggests that sellers collude less effectively when competing under conditions of incomplete payoff information, compared to complete information.

The multiplicity of collusive repeated game equilibria in this setting provides a wide range of possibilities and therefore produces a difficult coordination problem for subjects. Previous research with more simple games (e.g., Cooper et al. [1990]; Van Huyck et al. [1990]) demonstrates that coordination problems are difficult to overcome even when the Pareto optimal outcome is a Nash equilibrium of the stage game. Most experimental studies of collusion in repeated games share this coordination difficulty, and we do not expect (or observe) symmetric Pareto optimal equilibrium outcomes in our experiment. A more accurate characterization of behavior is that subjects are "groping" toward choices that improve their payoffs, similar in spirit to the theoretical model of Shapiro [1980]. Most subjects do not converge in any conventional sense to a specific equilibrium, so analysis of their dynamic behavior provides insight into their learning and the equilibrium selection process (Alger [1987]). We should emphasize, however, that these coordination difficulties are present in the primary information sharing treatment as well as our various control treatments, so the experimental design isolates any role that voluntary information sharing might play in facilitating collusion.

III. RESULTS

We begin this section with a summary of the integer choice distributions before turning to the econometric analysis. We then use two distinct econometric approaches to evaluate the impact of information sharing on behavior.

Qualitative Overview

By an overwhelming margin, subjects shared information when possible, contrary to the stage game non-cooperative equilibrium. We analyze the subjects' information sharing decisions in detail below. Figure 2 presents a frequency distribution of choices over all subjects in the high and medium demand states for the full information periods in which both sellers in a pair choose to share information. (To conserve space we omit the low demand state choice distribution.) The modal outcomes in the high and medium conditions are {8, 8} and {4, 4}, respectively, but substantial variation exists across pairs and across periods.(11) The modes remain unchanged but the variance decreases somewhat for the later periods. Despite the modal total output of 16, the substantial number of choices with total output less than the static Nash prediction of 14 in the high demand state are sufficient to reduce the average choice to less than 12 (see Table IV below). The asymmetric choices {0, 8} and {8, 0} in both demand states under this voluntary information sharing treatment are due primarily to two of the 18 pairs, who relied almost exclusively on this sophisticated collusive strategy. One additional pair employed this strategy with less success in the second half of the session, and the 15 other [TABULAR DATA FOR TABLE IV OMITTED] pairs never implemented this strategy. These collusive asymmetric choices are notably absent in the Forced Sharing control treatment (not shown here), occurring in only 2 of the 258 periods of high or medium demand.

Next compare Figure 2 with the choice frequency distribution in the Truncated Demand control treatment shown in Figure 3. This figure provides strong support for the static, symmetric Nash equilibrium of {7, 7} in high demand and {4, 4} in medium demand. More importantly, the comparison of Figures 2 and 3 dramatically illustrates the significant reduction in choice variance when collusion opportunities are removed in the Truncated Demand treatment. One interpretation is that the variance in choices in the primary treatment reflects a complex learning process, due in part to sellers' attempts to exploit collusion opportunities. As discussed above, sellers that attempt to collude face a complicated coordination problem, and they rarely coordinate from the beginning of the session. The econometric analysis below attempts to account for this dynamic coordination.

Table IV presents a summary of the paired integer choices and payoffs in the different demand states and treatment conditions. The table presents mean aggregate pair choice and standard errors, along with the static Nash prediction. Table IV shows that profits were higher than the static Nash prediction for the high demand state in all but the Truncated Demand treatment but were lower than the static Nash prediction for the medium and low demand states. The table also indicates that on average subjects restricted aggregate output in the primary treatment when they shared information, relative to all three control treatments. However, when compared to the Forced No Sharing treatment, the average impact on paired choices was not economically large except in the low demand state.(12)

Analysis of Aggregate Output Choices

In this subsection we use market behavior as the relevant statistic and analyze aggregate pair integer choices. These models interpret the data set as a pooled cross-section/time-series sample, where the dependent variable is subject pairs' summed choice.

Models and Estimation Results. The hypotheses we investigate in this subsection are the static Nash equilibrium output predictions:

Hypothesis H1: Conditional on sharing information, aggregate pair output equals the static NE.

Hypothesis H2: Conditional on not sharing information, aggregate pair output equals the static NE.

Using the integer choice transformation presented to subjects, the aggregate predictions are (a) 14 in the high demand state, (b) 8 in the medium demand state, and (c) 2 in the low demand state, both under Hypothesis H1 (when information is shared) and under Hypothesis H2 (when information is concealed). The alternative hypotheses to H1 and H2 that correspond to tacit collusion are aggregate choices below these static Nash equilibrium predictions.

Since we are treating our data set as a pooled cross-section time-series sample, we require an equal number of observations from each pair. Correspondingly, we consider the first 30 observations on each pair. This gives us 1110 data points - 540 from the primary treatment, 270 data points from the Forced No Sharing treatment, and 300 data points from the Forced Sharing treatment. This econometric approach cannot include the Truncated Demand treatment because it employed different payoff matrices, so we defer further analysis of this control treatment until the sharing choice analysis below.

Our goal is to identify tendencies towards more cooperative behavior, and to explore the link with the sharing decision. In order to make observations comparable across the demand states, we transform pair k's aggregate period t choice, [Q.sub.k](t), as follows:

(3) [Mathematical Expression Omitted],

where [Mathematical Expression Omitted] is the static Nash equilibrium joint pair output for the demand state pair k confronts in period t, and [Mathematical Expression Omitted] is the fully collusive, symmetric joint pair output for the demand state pair k confronts in period t. These outputs (transformed to our integer range) are shown in Table III.(13) This construct is larger the less cooperative is pair k; it equals zero if pair k is fully cooperative in period t, and it equals one if the pair plays the static non-cooperative Nash equilibrium.

Evaluation of the primary treatment is complicated by the fact that there could plausibly have been two behavioral regimes, one when information was shared and one where information was concealed. To handle this contingency, we use a switching regression model (Maddala [1986]). In the state where subjects have elected to share information, the model is

(4) [Y.sub.k](t) = [[Alpha].sub.1][H.sub.k](t) + [[Alpha].sub.2][M.sub.k](t) + [[Alpha].sub.3][L.sub.k](t) + c[Y.sub.k](t-1) + [u.sub.k](t),

where [Y.sub.k](t) is pair k's period t choice, transformed as discussed above, [H.sub.k](t) (respectively, [M.sub.k](t) or [L.sub.k](t)) is an indicator variable taking the value of 1 if pair k draws the high (respectively, medium or low) demand state, and 0 otherwise, and [u.sub.k](t) is a residual capturing variations about the equilibrium. In the state where subjects have elected not to share information, the model is(14)

(5) [Y.sub.k](t) = [[Beta].sub.1][H.sub.k](t) + [[Beta].sub.2][M.sub.k](t) + [[Beta].sub.3][L.sub.k](t) + d[Y.sub.k](t-1) + [u.sub.k](t).

These two formulae can be combined into the single relation

(6) [Y.sub.k](t) = [[Alpha].sub.1][H.sub.k](t) + [[Alpha].sub.2][M.sub.k](t) + [[Alpha].sub.3][L.sub.k](t) + c[Y.sub.k](t-1)]S(t) + [[[Beta].sub.1][H.sub.k](t) + [[Beta].sub.2][M.sub.k](t) + [[Beta].sub.3][L.sub.k](t) + d[Y.sub.k](t-1)][1 - S(t)] + [u.sub.k](t),

where S(t) is an indicator variable taking the value 1 if both subjects elect to share information, and 0 otherwise. Recall that the sharing decision precedes the output choice each period.

As we suggested above, there are reasons to expect a dynamic relation here if play is consistent with the use of trigger strategies (Friedman [1983]). Similarly, any attempts at signaling a desire to collude hinge on an intertemporal connection (Shapiro [1980]). Finally, any learning implies a connection between current and preceding choices. Taken together, these argue for including lagged values of [Y.sub.k] in equation (6). We also allow for systematic differences in behavior between demand states, and for differences in behavior based on the sharing decision. In this model, Hypothesis HI translates into a test of [a.sub.i] = 1 (i = 1, 2, 3), and Hypothesis H2 translates into a test of [[Beta].sub.i] = 1 (i = 1, 2, 3).

We report estimates of the parameters in equation (6) using ordinary least squares. We also estimated a variety of more elaborate models, such as allowing for heteroscedastic errors, an autocorrelated error structure, and fixed pair effects. The qualitative results are robust to these alternative specifications, so to conserve space we report only the OLS estimates.(15)

The estimates are summarized in the second column of Table V, and are labeled as regression 1. The first column gives the parameters that are estimated, which we write as [[Alpha].sub.1p], [[Alpha].sub.2p], [[Alpha].sub.3p], [c.sub.p], [[Beta].sub.1p], [[Beta].sub.2p], [[Beta].sub.3p], and [d.sub.p]; regression 1 corresponds to p = 1 in the subscripting. The results support three main conclusions. First, the point estimates are significantly less than one in the high demand state, whether or not information is shared. Second, the point estimate is significantly smaller than one in the low demand state if information is shared ([[Alpha].sub.31]), but is significantly larger than one if information is concealed ([[Beta].sub.31]). Third, in the middle demand state, the coefficient is not distinguishable from one whether or not information is shared. To summarize,

Result 1: In the primary treatment with voluntary information sharing, Hypotheses H1 and H2 are not rejected only in the medium demand state.

The pattern of rejection of Hypotheses H1 and H2, however, suggests that the information sharing decision has a relatively minor impact on output choices, except in the low demand state. When information is concealed output is above (respectively, below) the static NE in the low (high) demand state, which suggests that compared to this equilibrium subjects adjust their output insufficiently to their noisy signal.

In the Forced Sharing and Forced No Sharing control sessions, the subjects made no voluntary sharing decision, so that equation (6) does not apply for these sessions. Equation (4) is the relevant regression equation for the Forced Sharing sessions, while equation (5) is the relevant regression equation for the Forced No Sharing sessions. The results for the two control treatments are presented in the third and fourth columns of Table V, labeled as regressions 2 and 3. In the Forced Sharing treatment, choices are significantly more cooperative than the static Nash equilibrium (rejecting Hypothesis H1) in the high and low demand states, although only at the 10% level in the high demand state. Hypothesis H1 is not rejected in the medium demand state. In the Forced No Sharing treatment, the data fail to reject Hypothesis H2 in all three demand states at the 5% significance level, although in low demand for this treatment the Hypothesis H2 prediction of [[Beta].sub.33] = 1 is rejected at the 10% level. To summarize,

Result 2: In the Forced Sharing control treatment, Hypothesis H1 is rejected in the low and high demand states, and in the Forced No Sharing control treatment, Hypothesis H2 is (marginally) rejected in the low demand state.

Thus, choices in the Forced No Sharing treatment and in those periods where subjects concealed information in the primary treatment tend to be too large in the low demand state and too small in the high demand state. One interpretation of this pattern is that behavior is consistent with a version of prospect theory, discussed in detail in Cason and Mason [1997]. Myagkov and Plott [1997] also use evidence at an aggregated market level to test an "extended" version of prospect theory, and obtain findings that are consistent with their extended version.

Sharing Decision and Treatment Effect Hypothesis Tests. Our next three hypotheses concern the comparison of behavior under the two information sharing decisions and in the alternative designs, for each demand state. First, to identify the impact on choices in the primary treatment due to information sharing we test

Hypothesis H3: In the primary treatment, there is no difference between aggregate [TABULAR DATA FOR TABLE V OMITTED] output when information is shared and when information is concealed.

If information sharing plays a role in facilitating tacit collusion, the alternative to Hypothesis H3 is that aggregate output is lower when both subjects choose to share information. Second, the voluntary act of sharing information may affect choices, so we test

Hypothesis H4: There is no difference between aggregate output in the Forced Sharing treatment and aggregate output in the primary treatment when information is shared.

If voluntary information sharing facilitates collusion, we would expect subjects to behave more cooperatively in the primary treatment when they opted to share information, than in the forced sharing control treatment (i.e., the a coefficients from the former treatment are smaller than the [Alpha] coefficients from the latter treatment). Third, explicitly concealing information may affect choices, so we evaluate

Hypothesis H5: There is no difference between aggregate output in the Forced No Sharing treatment and aggregate output in the primary treatment when information is concealed.

If subjects withhold information to punish defection, they would be less cooperative in the primary treatment when they chose to not reveal information, than in the Forced No Sharing control treatment (i.e., the [Beta] coefficients from the former treatment are larger than the [Beta] coefficients from the latter treatment). In each comparison, any differences could be ascribed to the ability to share or conceal information, as opposed to the true demand state or subjects' knowledge of that state. If information sharing reduces output, irrespective of the motivation, then one can infer that information sharing has an anti-competitive effect. This effect would have to be balanced against any welfare gains from sharing, for example from the reduction in risk. We therefore are interested in three comparisons for each of the three demand states.

Table VI presents the numerical difference between the unrestricted parameter estimates for each of the comparisons and the t-statistics for these differences.(16) The construct for choices we use is larger the less cooperative is behavior, so that a negative difference represents less competitive behavior in the optional sharing treatment. The first column shows a statistically significant difference between behavior with sharing and concealment only in the low demand state. Nevertheless, we reject the joint hypothesis that output when information is shared equals output when information is concealed across all demand states; the chi-squared statistic for this test is 16.70, while the 95% critical value is 7.81.

Result 3: Hypothesis H3 is rejected only in the low demand state.

Since the parameter difference is negative in the low demand state, we infer that behavior was less cooperative when information was concealed. In part this is because choices were somewhat more cooperative than the static NE when information was shared, and in part the difference is tied to the substantially more competitive choices that obtained when information was concealed. Of these two effects, the most economically important appears to be the overproduction when information was concealed.

The second column shows that behavior in the Forced Sharing treatment does not differ from behavior in the primary treatment when information was shared in any demand state. A Chi-squared test fails to reject the joint hypothesis that behavior with optional sharing is identical to behavior in the Forced Sharing treatment in all demand states (here the test statistic is 2.77).

Result 4: Hypothesis H4 is not rejected in any demand state.

The third column shows that behavior in the Forced No Sharing treatment differed from the primary treatment when information was concealed only in the low demand state. The differences in the high and medium demand states are not statistically significant individually, [TABULAR DATA FOR TABLE VI OMITTED] and a Chi-squared test fails to reject the joint hypothesis that behavior with optional no sharing is identical to behavior in the Forced No Sharing treatment in all demand states (here the test statistic is 6.10).

Result 5: Hypothesis H5 is rejected only in the low demand state.

There appear to be two differences between choices in the Forced No Sharing treatment and choices in periods of the primary treatment where subjects chose to conceal information. While subjects tend to hedge their choices towards the middle in both cases, this effect appears to be smaller in the Forced No Sharing treatment. There is also a slight tendency towards larger choices in medium demand in the primary treatment. The overall impression is that sellers are more competitive when information is concealed by choice rather than by institutional design, at least in the medium and low demand states. This suggests the possibility that voluntary information concealment may have provided an additional signal to subjects.

The differences between the voluntary and forced concealment estimates suggest that subjects have different expectations and choose different outputs following the explicit act of concealment. Perhaps complex supergame strategies included information withholding as a means of punishing defections, with effects manifest in low (and perhaps medium) demand. The high demand impact conflicts with this explanation, however. This conflict suggests a more thorough analysis of the dynamic relationship between sharing decisions and previous output choices is in order. We carry out such an investigation in the next subsection.

Analysis of the Sharing Decision

Subjects earn greater expected profit in the static NE of the model with risk neutrality when they do not share information. The model therefore predicts

Hypothesis H6: Subjects do not share information in the primary treatment.

The incentive to conceal information in the stage game is identical in the Truncated Demand treatment, so we also test

Hypothesis H7: Subjects share information at the same rate in the primary treatment and the Truncated Demand treatment.

Rejection of Hypothesis H6 may suggest tacit collusion. But if the data also fail to reject Hypothesis H7, this instead provides evidence of risk aversion. This interpretation arises because the Truncated Demand treatment eliminates the collusion incentive, leaving risk aversion as the most plausible explanation for information sharing in the Truncated Demand treatment.

The 36 subjects in the primary information sharing treatment made a total of 1204 information sharing decisions. Of these, only 89 (7.4%) were to not share information, leading to

Result 6: Hypothesis H6 is strongly rejected.

Information is exchanged only if both subjects chose to share information, so subjects actually share information in 522 of the 602 market periods (86.7%) in this primary treatment.

The Truncated Demand treatment indicates that risk aversion is the most likely explanation for this sharing behavior. The 16 subjects in this control treatment made a total of 526 information sharing decisions. Information sharing was again the most frequent choice, and only 68 of these choices (12.9%) were to conceal information. Although this treatment nearly doubles the rate of information concealment, closer examination of the data indicates that this difference is due to one pair in this control treatment that almost never shared information. Consequently, a two-sample Mann-Whitney test that treats each pair as an independent observation (n = 18, m = 8) does not reject the null hypothesis that the two groups shared information at the same rate (U = 57.5; 5% critical value U* = 36).

Result 7: Hypothesis H7 is not rejected.

This result suggests that subjects shared information primarily to avoid the increased payoff variance arising from the imperfect demand information.

Nevertheless, the results from the previous subsection indicate that sellers succeeded in raising profits in the primary treatment when voluntary information sharing was possible in conditions of low demand. Therefore, risk avoidance may be only part of the explanation for information sharing. The remainder of this subsection explores the role of information sharing as a possible component of the collusive supergame strategies.

As discussed in section II, the two-stage information sharing game studied here has a richer strategy space than the standard repeated Cournot oligopoly model. Many strategies supporting a range of collusive output levels are subgame perfect Nash equilibria of the repeated game, including strategies with punishment periods of finite length. In addition to output strategy punishments in response to deviant behavior, sellers can punish their rivals by withholding information. To the extent that agents are risk averse, as our subjects appear to be, information concealment is a natural component of the punishment strategy.

If subject behavior is consistent with any of these trigger strategies, information concealment for subject i is more likely if the previous quantity choice for the rival subject j is high. Table VII presents a probit model that analyzes this hypothesis. The dependent variable of this model is the dichotomous decision of whether or not subject i shares information, where "1" represents information sharing and "0" represents information concealment. The explanatory variables are the rival subject j's integer choice in the previous period(s). The integer choice is normalized in a fashion analogous to equation (3) with individual quantity predictions replacing the joint pair predictions. After this normalization, the static Nash equilibrium integer choice is 1 and symmetric joint profit maximizing integer choice is 0, independent of the demand state. Subject-specific dummy variables are included (but not reported on the table for brevity) to account for different baseline information sharing probabilities across subjects.(17) In several unreported models we included variables representing time in the sessions (with a variety of specifications) to determine if sharing rates were different early in the sessions. Time was never significant, which indicates that sharing rates do not differ simply due to subject learning.(18)

Regression (1) lags the rival's choices one period only, and Regression (2) includes two-period lags to allow for more complex dynamic responses due to, for example, "groping" by subjects rather than only immediate and permanent responses.

The results provide some support for the interpretation of information concealment as a retaliatory or punishment strategy. The negative signs on the rival integer choice variables indicate that if the opponent chooses less cooperative output levels, then subjects are less likely to share information in subsequent periods. The significant rival integer choice lagged two periods in Regression (2) suggests that subjects' "punishment" responses are rather complex and dynamic, and not just short-term reactions to rival decisions in the immediately preceding period. Regardless of the interpretation of these regressions, our results point to the importance of preceding choices in explaining current sharing decisions.

Result 8: There is a slight tendency for subjects to react to past increases in the rival's output by concealing information.

It is difficult to interpret the economic importance of these feedback effects using the probit model coefficient magnitudes shown in Table VII. Inserting the estimates into the probit likelihood function, it is straightforward to show that increasing subject j's integer choice from the cooperative (normalized) choice of 0 to the non-cooperative (normalized) choice of 1 in regression (1) increase, the likelihood that subject i conceals information from roughly 0.1 to 0.14. Although statistically significant, this change in concealment likelihood is certainly small economically.

TABLE VII

Probit Models of Information Sharing Decision

Dependent Variable: Probability that Information is Shared

 Regression:

 (1) (2)

Mean of the Dependent Variable 0.925 0.923

Rival's Normalized Integer -0.238(**) 0.226(**)
Choice (1 period previous) (0.056) (0.056)

Rival's Normalized Integer - -0.153(**)
Choice (2 periods previous) (0.057)

Log-likelihood -219.31 -210.11

Restricted (Slopes = 0) -312.14 -306.80

Log-likelihood

Observations 1168 1132

Notes: Standard errors in parentheses. Subject-Specific dummies
were included in each specification.

** Significantly different from zero at the 1% level (two-tailed
test).

* Significantly different from zero at the 5% level (two-tailed
test).

IV. SUMMARY AND CONCLUSIONS

The demand variance and uncertainty in this experiment provided a challenging environment for subjects. Even so, average behavior corresponded rather accurately to the static NE in many cases. Nevertheless, subjects were successful in restricting output and receiving payoffs above the static NE under certain conditions. Overall, output restriction was more common when subjects shared information so that they were perfectly informed about payoffs, although there was virtually no evidence of tacit collusion in the medium demand state. Sellers were most successful in restricting output in the low demand state following information sharing. While at odds with the conventional wisdom that low demand conditions hinder collusion, this result is consistent with Rotemberg and Saloner [1986], who argue that collusion should be more effective in low demand periods. We also found that sellers were able to restrict outputs in the high demand state, particularly when they had the option to share or conceal information. The differences in outputs between the primary treatment and the control treatments were not significant in the high demand state, however.

This experimental environment has a unique non-cooperative equilibrium in the stage game under risk neutrality: information concealment followed by Bayesian-Nash Cournot competition. Subjects shared information more than 90% of the time, however, which is optimal if subjects are sufficiently risk averse or if they do not expect to play the static NE. Subjects also shared information in the vast majority of opportunities in the Truncated Demand treatment. Since sellers' ability to collude and increase profits are severely limited in this control treatment, we believe risk aversion is subjects' primary motive for sharing information. Applying laboratory results to address issues from the naturally-occurring economy should always be done with caution. With that caveat in mind, our result that information sharing is prevalent probably applies more to choices made by risk averse decision-makers, such as under-diversified managers whose compensation is tied to profits.

Antitrust authorities have traditionally been concerned that information sharing institutions, such as trade associations, might foster collusion. When challenged, trade associations have often argued that their information sharing mitigates uncertainty, which is a valuable service to the association's members. While our analysis does suggest that information sharing decisions can play a modest role in facilitating collusion in the low demand state, the welfare effect is relatively minor. Indeed, a reduction in industry output from the Cournot/Nash equilibrium level (of 4) to the joint profit maximizing level (of 2) in the low demand state reduces consumer surplus from 48 to 12 in our parameterization. By comparison, consumer surplus at the Cournot/Nash equilibrium in the medium demand state is 300, and we observed output in this medium demand state that was never statistically distinguishable from the Cournot/Nash equilibrium. On the other hand, our results suggest that subjects are primarily motivated to share information to reduce risk, since information sharing was very common in the Truncated Demand treatment where collusive possibilities were minimal. To the extent that firms are risk averse, sharing information may have important benefits. At least in the context of our experiments, these benefits would appear to be more significant than any welfare losses associated with output restrictions in phases of low demand. At the minimum, our results are unsupportive of an active antitrust stance against trade association information sharing arrangements.

[TABULAR DATA FOR APPENDIX OMITTED]

ABBREVIATION

NE: Nash Equilibrium

Financial support was provided by the Zumberge Faculty Research and Innovation Fund at the University of Southern California. Tim Adam, Charlie Plott, two anonymous referees, the managing editor and participants at the Economic Science Association meetings provided helpful comments. Errors remain our responsibility.

1. In a related experiment, Holcomb and Nelson [1991] study repeated experimental duopolies with and without perfect monitoring of their rival's output choices. Sessions began with written pre-play communication before each of the first five periods, and perfect monitoring of rival output choices after each of the initial 20 periods. After these first 20 periods, subjects were placed in an uncertain monitoring condition, in which it was common knowledge that there was a 50% probability that the reported output of their rival was randomly drawn. In this condition neither subject knew if a deviation from the collusive agreement occurs because of reporting error or because of cheating. Although a small number of sellers began cheating immediately when uncertain monitoring was introduced, the majority did not abandon the collusive agreement until the evidence that their rival was cheating became overwhelming. Thus, uncertainty made collusion unstable, but it often did not lead to an immediate breakdown.

2. Because of the multiplicity of equilibria in repeated games, one can think of our study as an exercise in "behavioral" equilibrium selection (Plott [1989]). Cason [1994] provides a more direct test of the static information sharing models. In his study one seller of each duopoly pair faced uncertainty, and results were generally consistent with the static models.

3. The U.S. Federal Trade Commission has carefully reviewed and regulated the information exchange programs of trade associations for many years (see, for example, the Federal Trade Commission Advisory Opinion Digest). In some cases, dating back to the early part of the century, antitrust authorities have directly intervened to limit industry information sharing arrangements (e.g., Maple Flooring Mfrs. Assn. v. United States, 268 U.S. 563, 45 Sup. Ct. 578, 1925). A more contemporary example is the Justice Department's investigation of the airline industry for illegal price fixing through information shared in their computer reservation systems (see Cason [1995] for a discussion).

4. We assigned about twice as many pairs to the primary information sharing treatment than to the control treatments because with voluntary information sharing subjects make decisions in five possible information states, compared to the two or three possible information states in the control sessions. In the Forced No Sharing treatment subjects faced the information states "Heads/State Unknown" and "Tails/State Unknown." In the Forced Sharing treatment subjects faced the information states "High State Known," "Medium State Known" and "Low State Known." More pairs are therefore required in the sharing sessions to obtain similar statistical power across treatments.

5. Several recruited subjects failed to appear at the first Truncated Demand control treatment, which only had six subjects. We considered eight pairs sufficient for this treatment because choice variance was very low across pairs for these payoff matrices [ILLUSTRATION FOR FIGURE 3 OMITTED] and our primary interest is in the comparison between information sharing frequency in this control and the primary information sharing treatment.

6. The instructions and the computer software used neutral wording such as "the person you are paired with" and "your integer choice" rather than possibly loaded terms such as "your rival," "the other seller" or "your quantity."

7. Optimal quantity choices in this uncertain environment can differ from this equation with risk aversion. For example, Hviid [1989] shows that output decreases with increased risk aversion in a model with constant absolute risk averse utility and normally-distributed random variables.

8. Of course, the subjects know that the experiment will not last forever even if the probabilistic stopping rule never terminates the session. Samuelson [1987] shows how incomplete information about the termination period can also expand the set of non-cooperative Nash equilibria.

9. Since information concealment is the subgame perfect sharing strategy in the stage game, this punishment strategy is subgame perfect in the repeated game because it corresponds to repeated play of the stage game Nash equilibrium. Because our repeated game is non-stationary (the induced discount rate is 80% following period 30, but is larger for earlier periods), more sophisticated strategies than the trigger strategies given in the text may be part of a subgame perfect equilibrium. Even so, relatively simple trigger strategies are still subgame perfect, because they are incentive compatible in the phase following period 30. Phillips and Mason [1996] show that behavior in a non-stationary design similar to ours is qualitatively indistinguishable from behavior in a parallel but stationary treatment. We note also that infinite repetition of the static NE strategy may not be an optimal punishment (Abreu [1986]).

10. Cason and Mason [1997] show that the symmetric joint profit-maximizing integer choice is always incentive compatible in the low and medium demand states for the stage game NE punishment strategy. Moreover, the symmetric joint profit maximizing integer choice (4,4) is incentive compatible in the high state for t [less than] 28. In the end periods of the session (t [greater than] 27), the integer pair (5, 5) is incentive compatible. Thus, like the Rotemberg and Saloner [1986] study of collusion over the business cycle, in these end periods only the binding incentive constraints in good demand conditions permit slightly less intense collusion.

11. The critical reader may be concerned about the concentration of choices at {8, 8} in the high demand state because these choices are on the boundary of the available choices. We chose to limit the number of entries on the payoff matrix to 81 because the information sharing and uncertainty introduced more complexity than usually found in duopoly experiments. Furthermore, we deliberately avoided placing the static NE the edge of the payoff matrices. As shown in Table IV, the average combined quantity in the primary information sharing treatment for the high demand state is less than 12. If the average combined choice was equal to 14 but subjects made random decision errors (McKelvey and Palfrey [1992]), the upper bound of 16 introduced by the constraint on the payoff matrix size could reduce the observed average below 14 because it limits the size of positive errors. Nevertheless, we do not interpret the low average as arising from this artificial limit of 16 because this constraint applies to each treatment, and there is no reason to believe that decision errors differ systematically across treatments. Average paired choices in the high demand state range from about 13 to up to 15 in the control treatments (see Table IV), so decision error is an unlikely explanation for lower choices in the primary treatment.

12. Referring to Table IV, the average output reduction is 1.42 units (13.33-11.91), or 10% of the static Nash equilibrium level in the high demand state. In the medium state, the reduction is even smaller, both in absolute terms and relative to the static NE level. Finally, in the low demand state output is sharply reduced in the primary treatment when information was shared. Overall, the average choices in all treatments are such closer to the static NE level than the joint profit-maximizing level in each demand state.

13. Recall that there exist multiple asymmetric static Nash equilibria, but the aggregate joint pair static NE output is unique. Our normalization of pair choices uses the collusive strategy pair (4, 4) that is incentive compatible in the high state for t [less than] 28. The conclusions are robust to the alternative collusive strategy pair (5, 5) in the normalization. Asymmetric collusive strategies provide even more profit because of the increasing returns to scale. If one subject chooses 0 and the other subject chooses 8, 7 and 1 in the high, medium and low demand states, respectively, the expected payoff is over 130 cents. As mentioned above, two of the 18 pairs implemented this sophisticated strategy of alternating the 0 choice, despite this strategy's payoff variance of nearly 30000. Collusive integer strategies are also possible with information concealment (e.g., 0 and 4 choices in the tails and heads information states, respectively), but they involved considerably more risk and were not apparently implemented by any subjects.

14. With imperfect information, choices must be made on the basis of a noisy signal, not the exact state. We did not change the dummy variables to the two signals from the three states because at the pooled pair level (the unit of analysis) the aggregate output and profit depend on states, not signals. Moreover, which seller of the pair receives which signal is irrelevant if subjects' strategies are symmetric on average. Recall that the individual symmetric static NE integer choices conditional on information sharing are 7, 4 and 1 in the high, medium and low demand states; and conditional on information concealment are 7 and 1 for the heads and tails signals, respectively. Therefore, at the aggregate level the static NE output are respectively 14, 8 and 2 in the high, medium and low demand states irrespective of the sharing decision.

15. OLS yields best unbiased estimates if the disturbance term is normally distributed. Alternatively, if the disturbances are symmetrically distributed, OLS yields best linear unbiased estimates. Even if the disturbances are asymmetric, as one might argue in the context of our data, one can regard OLS estimates as an approximation of a maximum likelihood estimator. Under conventional assumptions regarding the distribution of the disturbance, any biases associated with this approximation will vanish as the number of observations gets large. For a discussion of these points, see Fomby et al. [1988]. Because we have over 200 observations in each of the regressions, this asymptotic argument seems appropriate in our application.

16. Computation of the variance of this difference is simplified by the fact that the estimates from the primary treatment are statistically independent from the estimates under either control, so that they have no covariance. For the within-treatment comparison in the leftmost column of Table VI, we use the standard method of the variance calculation (i.e., the sum of the variances plus twice the covariance between the two coefficients).

17. Likelihood ratio tests reject the null hypothesis of no baseline (i.e., intercept) differences across subjects. For example, in the specification with one-period lags (model 1), we reject the hypothesis of equal intercepts across subjects at the 1% level ([Chi].sup.2](35) = 149.1).

18. The results are similar under other alternative specifications. For example, we estimated the models also including a (normalized) profit term, which is of course correlated with the rival's integer choice. This profit variable is insignificant. We also included lagged rival sharing choices, but these also generally add an insignificant amount of explanatory power. The results are also not sensitive to the error specification. Estimating with the logit instead of the probit model (i.e., assuming logistic rather than normally distributed errors) does not qualitatively change the results.

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