The impact of information sharing opportunities on market outcomes: an experimental study.

Cason, Timothy N.

I. Introduction

Information regarding uncertain demand conditions or industry costs, as well as the opportunity to share this information, can have an important impact on firm behavior. A series of recent game-theoretic models have characterized the incentives for firms to share information regarding uncertain parameters [3; 7; 8; 13; 14; 16; 22; 26; 20; 21 provide a comprehensive survey]. Most of the models assume that behavior in the output market is noncooperative, regardless of the firms' information sharing decisions. This paper reports a series of 15 laboratory sessions that test the noncooperative, backward induction approach of these models. The results suggest that pricing behavior may be influenced by the information sharing decision: Under most conditions the non-cooperative Nash model accurately describes average behavior, although in certain conditions information sharing appears to facilitate tacit collusion. The paper also estimates a simple behavioral model of learning that describes how experimental subjects learn the optimal information sharing decision as an alternative to the theories' backward induction assumption.

The information sharing models tested here have been interpreted as identifying the incentives for competitive firms to form a trade association that reduces their uncertainty. Thousands of trade and professional associations are active worldwide, and the majority collect information from their members that is aggregated and distributed through association statistical programs. The theoretical models demonstrate how three variables influence the information sharing incentives: 1) The type of competition (Bertrand or Cournot); 2) the nature of the goods (substitutes or complements); and, 3) the source of uncertainty (demand or cost). The first two variables determine the slope of the firms' reaction functions, and the uncertainty source along with the information sharing decision determines the degree of correlation among the firms' strategies. The incentive for noncooperative firms to share information shifts as these three variables change because reduced correlation has a negative or positive effect on profit depending upon the slope of the reaction functions.

The models usually have two stages. In the first stage, firms make a decision regarding the amount of private information they wish to truthfully reveal to other firms. This can be thought of, for example, as the establishment and funding decision of a trade association statistical program. This decision is made before any uncertainty is resolved. In the second stage, firms choose noncooperative output or price strategies based upon the information revealed in stage one. No additional firm interaction occurs after stage two. The equilibrium concept is Nash subgame perfection, so that the first-stage information sharing decision is optimal given the second-stage (subgame) equilibrium.

The models typically identify unique dominant strategy information sharing equilibria assuming Nash competition in the second stage. Vives [26] shows that with an uncertain common demand intercept and quantity competition, firms' dominant strategy is to reveal (conceal) demand information if the goods are complements (substitutes). Vives also demonstrates that the incentives reverse with price competition so that firms should reveal (conceal) demand information if the goods are substitutes (complements). Gal-Or [8] shows that if costs are uncertain but are conditionally independent across firms, firms' dominant strategy is to reveal (conceal) cost information under quantity (price) competition if the goods are substitutes. Cason [2] demonstrates that with price competition and perfectly correlated but uncertain costs, firms' dominant strategy is to reveal (conceal) private cost information if the goods are complements (substitutes). The experiment reported here tests these qualitatively distinct dominant strategy predictions by using the uncertainty

source and demand structure as treatment parameters.

Antitrust authorities have been concerned for many years with the behavior of trade and professional associations.(1) One focus of this concern has been the publishing of information (such as prices) that may make a cooperative agreement easier to enforce. Clarke [4], in contrast, argues that information sharing can facilitate collusion because it eliminates disagreements based on private information and it homogenizes firms' perceptions. He concludes that "information-pooling mechanisms like trade associations can be considered prima facie evidence that firms are illegally cooperating to restrict output [4, 392]." Kirby [13], however, shows that in a model very similar to Clarke's, noncooperative firms will want to share information if cost functions are sufficiently quadratic. Kirby concludes that "Trade associations, therefore, are not prima facie evidence of collusion [13, 145]." Whether information sharing facilitates tacit collusion is an empirical question. The results presented here provide limited empirical support for the hypothesis that a relationship exists between information sharing and cooperative pricing behavior. In the sessions with cost uncertainty and complement goods, collusive behavior is observed when information is shared. However, in the demand uncertainty sessions and in all sessions with substitute goods, the noncooperative Nash model describes average price behavior more accurately.

This static, two-stage model is the dominant approach employed in the extensive information sharing literature, although it does not capture fully the incentives for firms which interact repeatedly in the same industry. This limits the application of the available theoretical work for policy toward trade association activities. Nevertheless, the experiment implements a test of the existing static theory by randomly re-pairing subjects with new rivals each period. This is therefore a theory-testing experiment that evaluates this noncooperative, static approach on the theory's domain as much as possible. (See Smith [23, 940-942] for a methodological classification of experiments.) Future experiments can extend this test to include more realistic features of trade associations that are not present in existing theory (called boundary experiments), such as repeated interactions among the same sellers.

We wish to emphasize that this experiment is not intended to test subjects' ability to solve strategic problems like game theorists. Like many models in industrial organization, the problem described in the next section is clearly too complicated for undergraduates (or, for that matter, firm managers) to solve without training in game theory. Instead, we are interested in testing if human decision-makers can develop heuristic mechanisms that lead them to eventually behave according to the predictions economists make using game-theoretic tools. The results provide support for the subgame perfect Nash equilibrium in this environment, and identify simple trial-and-error learning rules that lead to optimal behavior. Therefore, this research offers empirical support for a class of industrial organization models and the information-sharing literature in particular.(2)

II. The Model

The model developed here and tested by the experiment is a simplified version of the demand uncertainty model from Vives [26] and the (perfectly correlated) cost uncertainty model from Cason [2]. (The model in Cason [2] is very similar to the model in Gal-Or [8]; the substantive difference is that Gal-Or's model assumes that the random costs are uncorrelated across firms, in contrast to the perfectly correlated costs in Cason's model.) Two major simplifications are required to operationalize the model in a form simple enough to be understood by the experimental subjects. First, the information structure is modified so that information quality is asymmetric; i.e., one firm has more accurate information. Second, the firm with high-quality information is given perfect information regarding the realization of the random variable.(3)

The model contained here is representative of many of the information sharing models cited in the introduction. Among the shared features are the following: Firms make information sharing decisions before the realization of the random variables, output market competition is noncooperative, and the subgame perfect equilibria require firms to understand the implications of their first stage information sharing decision on output market profit. This operational model captures the essential features of the more complex theoretical models and yields testable predictions about the effects of uncertainty and information sharing opportunities on market outcomes.

We develop the cost uncertainty and demand uncertainty versions of the model in parallel to highlight how firms have opposite information incentives in the two cases. In addition to making price predictions, the model implies that in a subgame perfect equilibrium, firms will share (conceal) cost information if the goods are complements (substitutes), and will conceal (share) demand information if the goods are complements (substitutes). Figure 1 summarizes this shifting information sharing result.

Demand and Costs

In this duopoly model, the two firms choose prices and each faces a linear demand curve given by

[q.sub.i] = a - [bp.sub.i] + [dp.sub.j], i = 1, 2 and i [is not equal to] j (1)

Assume that a, b [is greater than or equal to] 0 and that b [is greater than] [absolute value of] d. The latter assumption requires that "own-price" effects dominate "cross-price" effects. Clearly, the goods are substitutes if d [is greater than] 0 and are complements if d [is less than] 0. If d = 0, the goods are independent. Each firm faces the same constant unit costs c for each unit sold and production is made to order (i.e., there are no inventories).

Uncertainty

The source of uncertainty is critical for determining the incentive to share information and is a treatment parameter for the experiment.

Cost Uncertainty. Unit cost c is the same for both firms and is drawn from a known discrete uniform distribution [f.sub.c] before prices are chosen. Denote the finite mean and finite variance of the cost as [[Mu].sub.c] and [Mathematical Expression Omitted], respectively. The mean [[Mu].sub.c] and variance [Mathematical Expression Omitted] and all other parameters are common knowledge.

Demand Uncertainty. The common demand intercept a is drawn from a known discrete uniform distribution [f.sub.a] before prices are chosen. Denote the finite mean and finite variance of the intercept as [[Mu].sub.a] and [Mathematical Expression Omitted], respectively. The mean [[Mu].sub.a] variance [Mathematical Expression Omitted] and all other parameters are common knowledge.

Timing and Information

Firm 1 receives the exact value drawn from the distribution prior to setting its price. Before the value is drawn and firms choose prices, firm 1 can instruct the "trade association" (i.e., the experimenter) to provide perfect information regarding the drawn value to firm 2. Thus firm 1 has complete control over firm 2's information, and firm 1's information sharing decision is constrained to be "all or nothing." This timing is illustrated in Figure 2.

To derive the model in this section we do not use the uniform distribution assumption. The experiment uses the uniform distribution because it is easy to explain and visually operationalize for subjects. The polar case of perfect information is the easiest to implement and avoids "compound hypotheses" problems that arise if we need to assume that subjects update posterior probabilities using Bayes rule.(4) Asymmetric information quality is necessary with perfect information; otherwise, both subjects would have perfect information, removing uncertainty from the model. The "all or nothing" information sharing decision constraint is never binding in equilibrium because, as shown below, partial sharing is never optimal.

Firms maximize expected profit, and the model is solved recursively in order to identify the subgame perfect equilibrium. In stage two, firm i's strategy is a mapping from its information I (either [f.sub.c]/[f.sub.a] or the realized value of c/a, along with the knowledge of whether or not the other firm is informed) into a price [p.sub.i] [is an element of] R+, for i = 1, 2. An equilibrium in stage two is a set of prices [[p*.sub.1], [p*.sub.2]] [is an element of] R+ such that (for i = 1, 2) no other [p.sub.i] [is not equal to] [p*.sub.i] has higher expected profit given the information [I.sub.i] and [p*.sub.j] (j [is not equal to] i). In stage one, firm 1's information decision maximizes expected profit given the noncooperative equilibrium strategies derived for stage two.

Stage Two--The Pricing Decision

Given the information structure described above, two information conditions are possible at the stage two pricing decision. First, if firm 1 reveals its information to firm 2, both firms will have perfect information. Call this the "Reveal" equilibrium. Second, if firm 1 does not reveal its information, firm 1 will have perfect information and firm 2 will have no information other than the probability distribution. Call this the "Conceal" equilibrium. We denote the equilibrium parameters in these two cases with r and c superscripts, respectively.

It is straightforward to calculate, assuming Nash conjectures, the risk neutral equilibrium prices in these two cases.

[Mathematical Expression Omitted], i = 1, 2, (2)

Cost Uncertainty

[Mathematical Expression Omitted],

[Mathematical Expression Omitted].

Demand Uncertainty

[Mathematical Expression Omitted],

[Mathematical Expression Omitted].

Equation (2) is the familiar equilibrium without uncertainty, since both firms have perfect information. Firm 1's concealment price, represented in equations (3) and (3'), has an additional term that reflects its information advantage. The term in brackets is positive, zero or negative as the goods are substitutes, independent or complements. Therefore, if the cost or the demand intercept is lower than its expectation, firm 1 will set a price higher (lower) than the "reveal" price [Mathematical Expression Omitted] if the goods are substitutes (complements). Finally, when firm 2 has no information, its pricing rule (represented in equations (4) and (4')) is analogous to the perfect information case with the expectation terms [[Mu].sub.c] and [[Mu].sub.a] replacing the actual values c and a.

The equilibrium is illustrated in Figure 3 for a simple two-state example with cost uncertainty. This figure displays two reaction functions for each firm (corresponding to two possible cost values) and equilibrium prices for the case of substitutes (panel a) and complements (panel b). For both diagrams, the reveal equilibria are determined by the intersection of the certainty reaction functions and are marked with A for high costs and B for low costs. In the conceal equilibria, although firm 1 still chooses prices on its certainty reaction functions, firm 2 does not have any additional cost information. Therefore, firm 2 must choose a price (denoted [Mathematical Expression Omitted]) based on its expected value of the cost and its expected value of firm 1's price. Firm 1 takes this into account and chooses prices on its certainty reaction functions, depending on the cost value. These equilibria are labeled C for high cost and D for low cost in each diagram.

Note that if the goods are substitutes (panel a), firm 1's price is lower (higher) in the conceal equilibrium than the reveal equilibrium if the cost is high (low). If the goods are complements (panel b), this is reversed and firm 1's price is higher (lower) in the conceal equilibrium than the reveal equilibrium if the cost is high (low). This difference arises from the differing reaction function slopes. Finally, note that in the conceal equilibrium the covariance of the two firms prices is zero, and in the reveal equilibrium the covariance is positive [It is easy to verify that the covariance in this case is [b.sup.2][[Sigma].sub.c]/[(2b - d).sup.2]]. This combination of price correlation and reaction function slope is what determines the relative profitability to firm 1 of revealing or concealing information from firm 2, to which we now turn.

Stage One--The Information Sharing Decision

Firm 1 can choose the equilibrium, conceal or reveal, that provides the highest expected profit. The expectation is taken in the first-stage before the drawn value is revealed. Using a version of expected profit obtained from the first order conditions, E([[Pi].sub.i]) = bE[[([p*.sub.i] - c).sup.2]], it is straightforward to simplify the firms' expected profit as follows:

Cost Uncertainty

[Mathematical Expression Omitted], i = 1, 2, (5)

[Mathematical Expression Omitted],

[Mathematical Expression Omitted].

Demand Uncertainty

[Mathematical Expression Omitted],i = 1, 2, (5')

[Mathematical Expression Omitted],

[Mathematical Expression Omitted].

Firm 1 compares the expected profit in equations (5) and (6) (or, for demand uncertainty, equations (5') and (6')), and makes the revelation decision corresponding to the highest expected profit. Its optimal decision can be summarized by the following propositions.

PROPOSITION 1 (Cost Uncertainty). Suppose unit costs are random but perfectly correlated across firms. If the goods are substitutes, firm 1's optimal stage one policy is to conceal the cost information; if the goods are complements, its optimal policy is to reveal the cost information.

Proof. The difference in expected profit between concealing and revealing information, [Mathematical Expression Omitted] (which is equation (6) minus equation (5)) can be simplified to

[Mathematical Expression Omitted].

Since b [is greater than] [absolute value of] d, the term in brackets and the denominator [(2b - d).sup.2] are positive. Thus concealing cost information is more profitable than revealing cost information if and only if d [is greater than] 0, which is true if the goods are substitutes.

PROPOSITION 2 (Demand Uncertainty). Suppose the common demand intercept is random. If the goods are substitutes, firm 1's optimal stage one policy is to reveal the demand information; if the goods are complements, its optimal policy is to conceal the demand information.

Proof. The difference in expected profit between concealing and revealing information, [Mathematical Expression Omitted] (which is equation (6') minus equation (5')) can be simplified to

[Mathematical Expression Omitted].

Since b [is greater than] [absolute value of] d, the term in brackets is negative and the denominator [(2b - d).sup.2] is positive. Thus concealing demand information is more profitable than revealing demand information if and only if d [is less than] 0, which is true if the goods are complements.

Propositions 1 and 2 indicate that the model predicts qualitatively opposite information sharing decisions depending on whether the goods are substitutes or complements and whether cost or demand is uncertain. Referring again to the reaction functions of Figure 3 for the simple cost uncertainty example, if the goods are substitutes firm 1 prefers zero correlation of prices and chooses points C and D of panel a. If the goods are complements, firm 1 prefers prices to be positively correlated and chooses points A and B of panel b. The experiment determines if this prediction implied by standard game-theoretic techniques and the subgame perfection equilibrium refinement is supported empirically.(5)

Joint Profit Maximization

As discussed in the introduction, allowing communication in the first stage regarding the stochastic parameters of the market may facilitate cooperative pricing behavior in the second stage. Cooperation requires firm 1 to reveal its information to firm 2. The (full information) prices that maximize joint profits are given by

[Mathematical Expression Omitted], i = 1, 2 for both cost and demand uncertainty. (9)

The data analysis below uses this price level as a benchmark representing perfect price coordination.

Baseline Periods

The model above describes the experimental environment in the "treatment" periods. In other periods (the "baseline" periods), neither firm is given information other than the probability distribution of the random variable. The baseline periods test the predictive ability a symmetric Nash model under uncertainty. The prices that constitute the equilibrium, derived under the assumption of risk neutrality (i.e., expected profit maximization) are

[Mathematical Expression Omitted], i = 1, 2, for cost uncertainty, and (10)

[Mathematical Expression Omitted], i = 1, 2, for demand uncertainty. (10')

Alternatively, sellers that collude perfectly may choose prices that maximize joint profits:

[Mathematical Expression Omitted], i = 1, 2 for cost uncertainty, and (11)

[Mathematical Expression Omitted], i = 1, 2 for demand uncertainty. (11')

Experimental Hypotheses

We summarize the experimental hypotheses generated by this model as follows:

HYPOTHESIS 1 (Revelation decisions): When costs are uncertain and the goods are complements, or when demand is uncertain and the goods are substitutes, firm 1 will reveal its information to firm 2. In the other cases firm 1 will conceal its information from firm 2.

HYPOTHESIS 2 (Treatment period prices): If Firm 1 reveals information, prices will converge to the full information noncooperative Nash equilibrium given by equation (2). If Firm 1 conceals information, prices will converge to the asymmetric information noncooperative Nash equilibrium given by equations (3), (4), (3') and (4'). Note that these prices are dependent on the realization of the random variable.

Hypothesis 3 is an alternative to Hypothesis 2. It states that prices, conditional on cooperative revelation decisions, will be consistent with joint profit-maximizing behavior rather than non-cooperative behavior.

HYPOTHESIS 3 (Treatment period cooperative prices): If Firm 1 reveals information, prices will converge to the level that maximizes joint profits given by equation (9).

Finally, Hypothesis 4 concerns the price choices in the (no information) baseline periods.

HYPOTHESIS 4 (Baseline period prices): When neither firm receives information other than the distribution of the random variable, prices will converge to the (no information) noncooperative Nash equilibrium given by equations (10) and (10').

The analysis will also compare the baseline period prices with the joint profit-maximizing level given in equations (11) and (11').

III. Experimental Design

Our interest is in seller behavior for this experiment, so we employ a duopolistic environment with the demand curve revealed to the firms. Sellers post a single offer price in each period. Several studies have found noncooperative Nash models to be good predictors of behavior in similar markets [6; 15]. The goal here to make the pricing task simple, so that subjects are not confused by the information revelation decision added in the initial stage of each period. The economic incentives must overcome the subjective costs of making these decisions if we are to have a genuine test of the theory. Recent results summarized by Harrison and McCabe [10] indicate that two-stage games (and, under certain conditions, three-stage games) are not beyond the cognitive abilities of experimental subjects.

We chose the demand and cost parameters to sufficiently separate the various alternative price predictions (i.e., info-sharing Nash, info-concealing Nash, joint profit-maximizing), and to increase the profit from cooperating substantially over the (subgame perfect) Nash profit. This provides a test of the drawing power of the Nash equilibrium and investigates the possibility that (fairly restrictive) information sharing opportunities can facilitate tacit collusion. The parameters and equilibrium predictions are provided in Table I for the cost uncertainty sessions and in Table II for the demand uncertainty sessions. The term Joint Max (or JM) on all tables and figures refers to the theoretical predictions under joint profit-maximizing behavior. The theoretical increase in expected profit from making the optimal revelation decision ranges between 8 and 15 percent.

We implement uncertainty by drawing one of five possible balls each period from a bingo cage.(6) For the baseline periods, no seller observes the ball until after selecting his or her price offer. In the treatment periods, "firm 1" sellers observe the ball before selecting price offers; in addition, prior to observing the ball realization "firm 1" sellers determine if the "firm 2" seller in their market is allowed to observe the ball's value before making his or her price decision.

Table I. Parameters and Predicted Equilibrium Prices and Profits for Cost
Uncertainty Experiments

 Substitutes Complements
 (Series 1) (Series 2)

 J.M. J.M.
Information Cost Value N.E. N.E. Both N.E. N.E. Both
Condition (Sub/Comp) Firm 1 Firm 2 Firms Firm 1 Firm 2
Firms

 Panel A: Prices

No Information All 14,15 14,15 24 24 24
21 0/0 8 8 20 20 20
 15 Complete 4/6 11,12 11,12 22 22 22
 18 Information 8/12 14,15 14,15 24 24
24 21 12/18 17,18 17,18 26 26
 26 24 16/24 20,21 20,21 28 28
 28 27 0/0 10,11 14,15 NA 18
 24 NA Asymmetric 4/61 12,13 14,15 NA
21 24 NA Information 8/12 14,15 14,15 NA
 24 24 NA 12/18 16,17 14,15 NA
 27 24 NA 16/24 18,19 14,15 NA
 30 24 NA

 Panel B: Profits

No Information All 328 328 512 288 288
324 0/0 512 512 800 800 800
 900 Complete 4/6 415 415 648 512 512
 576 Information 8/12 328 328 512 288
288 324 12/18 251 251 392 128
 128 144 16/24 184 184 288 32
 32 36 Expected Value 338 338 528
352 352 396 0/0 865 392 NA
 648 864 NA Asymmetric 4/6 565 408 NA
 450 540 NA Information 8/12 328 328
NA 288 288 NA 12/18 155 152
 NA 162 108 NA 16/24 46 -120
 NA 72 0 NA Expected Value 392 232
 NA 324 360 NA

Note: All values given in experimental "francs." N.E. denotes non-cooperative
Nash equilibrium, J.M. denotes joint profit maximum, and NA denotes not
applicable.

Demand Curves: Substitutes [q.sub.i] = 80 - 8[p.sub.i] + 6[p.sub.j], i [is not
equal to] j
Complements [q.sub.i] = 120 - 2[p.sub.i] - 2[p.sub.j],i [is not equal to] j

Five Cost Values: Substitutes {0, 4, 8, 12, 16}
Complements {0, 6, 12, 18, 24}

To summarize, the steps of each treatment period are as follows:

Step 1: "Firm 1" sellers use a written form to indicate if the experimenter should reveal the value on the ball to the "firm 2" seller in their market.

Step 2: The experimenter draws the ball and reveals its value (via a written message) to all "firm 1" sellers and those "firm 2" sellers who have been authorized by the "firm 1" seller in their market to receive the value.(7)

Step 3: Based on their private information, sellers select prices with written messages. These messages are collected by the experimenter, who records the prices (privately) and returns to each seller the price selected by his or her rival. This allows each seller to calculate his or her profit.

Table II. Parameters and Predicted Equilibrium Prices and Profits for Demand
Uncertainty Experiments

 Substitutes Complements
 (Series 3 & 4) (Series 5)

 J.M. J.M.
Information Bonus N.E. N.E. Both N.E. N.E. Both
Condition Value Firm 1 Firm 2 Firms Firm 1 Firm 2 Firms

 Panel A: Prices

No Information All 12,13 12,13 19 15 15 12,13
 0 4,5 4,5 6 5 5 4
Complete 25 8,9 8,9 12,13 10 10 8
Information 50 12,13 12,13 19 15 15 12,13
 75 16,17 16,17 25 20 20 17
 100 21 21 31 25 25 21
 0 6 12,13 NA 2,3 15 NA
Asymmetric 25 9,10 12,13 NA 9 15 NA
Information 50 12,13 12,13 NA 15 15 NA
 75 15,16 12,13 NA 21 15 NA
 100 19 12,13 NA 27,28 15 NA

 Panel B: Profits

No Information All 625 625 703 450 450 469
 0 67 67 78 50 50 52
Complete 25 278 278 312 200 200 208
Information 50 625 625 703 450 450 469
 75 1111 1111 1250 800 800 833
 100 1736 1736 1953 1250 1250 1302
Expected Value 764 764 859 550 550 573
 0 156 0 NA 12 0 NA
Asymmetric 25 352 234 NA 153 169 NA
Information 50 625 625 NA 450 450 NA
 75 977 1016 NA 903 731 NA
 100 1406 1406 NA 1512 1012 NA
Expected Value 703 656 NA 606 472 NA

Note: All values given in experimental "francs." N.E. denotes non-cooperative
Nash equilibrium, J.M. denotes joint profit maximum, and NA denotes not
applicable. The "Bonus" value is added to the demand intercept (25), and costs
are zero.

Base Demand Curves: Substitutes [q.sub.i] = 25 - 4[p.sub.i] + 2[p.sub.j], i
[is not equal to] j
Complements [q.sub.i] = 25 - 2[p.sub.i] - 1[p.sub.j],i [is not equal to] j

Five Bonus Values: Substitutes {0, 25, 50, 75, 100}
Complements {0, 25, 50, 75, 100}

TABULAR DATA OMITTED

Eight subjects (four duopoly pairs) participated in each session and no subject knew the identity of the other seller in her market.(8) As demonstrated by Roth and Murnighan [18], knowing the identity of one's opponent can have an important impact on outcomes in bargaining experiments. Each seller was paired with a different seller each period (randomly determined prior to the experiment), so that the single period Nash model is appropriate, rather than the repeated supergame.(9) Two trial periods, one baseline and one treatment, were conducted (without payoffs) at the start of the session in order to familiarize the subjects with the experimental procedures. No reference was ever made to a "competitive" relationship among sellers. Instead, the opponent in the market was always referred to as "the other seller." Subjects were paid three dollars upon arrival to the session and were paid their trading profit at the its conclusion. Individual subject payments averaged slightly under 20 dollars for the two-hour sessions.

A total of 15 sessions with 117 different subjects were conducted, and each session was conducted for 15 periods. Of these 15 periods, five periods were the baseline type and ten periods were the treatment type. Each period within a type (either baseline or treatment) was identical in every respect. All subjects were inexperienced in the sense that they had never participated in this kind of experiment, and the subjects were recruited from undergraduate economics courses. The sessions are organized into five series, summarized in Table III. The Cost Uncertainty sessions (Series 1 and 2) were conducted at the University of Arizona and the Demand Uncertainty sessions (Series 3, 4 and 5) were conducted at the University of California at Berkeley. Note that the order of the baseline and treatment periods was systematically varied across sessions. As discussed in the next section, this ordering does not have a measurable impact on the market outcomes.

Experiment Series 3 and 4 were conducted to investigate the role of the firm 1 assignment method and payoff conversion rate information on seller behavior. This additional design treatment has the potential to complicate the interpretation of the results; however, we detected no differences in seller decisions between Series 3 and 4, indicating that neither the assignment method nor the payoff information affected behavior in this experimental design. Nevertheless, we describe this innocuous design treatment (indicated in the final column of Table III) as follows. In Series 3 (as well as Series 1 and 2), subjects were randomly assigned the role of "firm 1" sellers, who control the information, and all sellers were given private conversion rates of experimental francs into U.S. dollars. In Series 4 (and Series 5), subjects were randomly paired into groups of two at the beginning of the session to play a simple game called Nim. Winners in each pair were awarded the role of "firm 1" sellers for the duration of the session. In these latter sessions, conversion rates were common knowledge. The game played was identical to the game used in Hoffman and Spitzer [11]. In that study, the authors find that when one subject was arbitrarily assigned to a powerful bargaining position, he or she never received the large individually rational share of the payoff. In contrast, when the game of Nim was used to assign the powerful bargaining position and the instructions emphasized that this position was "earned," two-thirds of the advantaged players successfully bargained for the larger individually rational share. Our goal was to determine whether the private conversion rates (which would make it impossible for subjects to identify a "fair" allocation) or the use of the assignment game reduced the impact of "fairness" considerations in this experiment.(10) As shown in the next section, the comparison of Series 3 and 4 results indicates that seller decisions were invariant to this design treatment.(11)

IV. Experimental Results

We present the experimental results in three subsections. The first subsection presents the revelation decisions in the treatment periods and the second subsection examines the price choices in the treatment periods. The third subsection summarizes the baseline period price choices. We conducted a series of non-parametric tests to determine if the treatment/baseline order conditions affected price choices. With the exception of the lowest cost draws in the Series 2 sessions with both sellers informed, we are unable to identify any impact of the treatment/baseline order conditions (details and the raw data are available from the author). Therefore, the analysis below pool observations across the set of three sessions within each of the five experiment series.

Treatment Periods--The Revelation Decision

Hypothesis 1 states that information will be revealed in Series 2, 3 and 4. Figure 4 illustrates the percentage of sellers choosing to reveal information in each treatment period.

The top panel of this figure provides solid support for the model when costs are uncertain. The cost uncertainty revelation decisions show a clear tendency to conform to the Nash predictions as the periods progress and nearly all revelation decisions are predicted by the model in the final period. Only 12 percent of the decisions are non-optimal in the final period. In the (lower panel) demand uncertainty sessions, however, many sellers persist in making non-optimal revelation decisions even in the later periods. In the final period, 35 percent of the information revelation decisions are not consistent with the Nash model predictions.(12)

TABULAR DATA OMITTED

A simple model of subject learning is useful to provide insight into the asymmetry between the cost uncertainty and the demand uncertainty revelation frequencies. The model of section II assumes that sellers determine their optimal revelation decision immediately by backward induction; however, a more realistic alternative is that they use information from second-stage profit outcomes of previous periods when making the current period's first-stage revelation decision. TABULAR DATA OMITTED High profit provides positive feedback for a particular information revelation strategy, and low profit provides negative feedback that may lead to an adjustment in the information revelation strategy in the following period. This model is similar to psychologists' models of "trial and error" learning; for example, see Einhorn and Hogarth [5], in which positive feedback has greater reinforcement value than negative feedback.(13)

Table IV provides evidence that subjects use this kind of trial and error procedure to learn the optimal revelation strategy. The binomial logit model shown in this table estimates the probability of changing one's revelation decision as a function of the previous period's profit. The coefficient on lagged profit is always negative and highly significant, and ranges within a narrow interval [-0.81, -0.63] for the five series of experiments. This implies, as illustrated by the example in the lower haft of the table, that the probability of changing one's information revelation strategy is near zero when profit in the previous period is significantly above average. In other words, high profits provide positive feedback for a revelation strategy.

According to this model, more subjects choose the optimal revelation strategy only if the optimal strategy leads to significantly higher profit. Recall from Tables I and II that the optimal decision is rewarded on average with an 8 to 15 percent increase in profit if prices correspond exactly to the Nash predictions. However, since the uncertainty is resolved each period with a new draw from the bingo cage, the experimenter has no control over the actual draws. Table V presents the realized average profit levels in each series for the optimal and non-optimal revelation strategies. This table indicates that the positive feedback required in the trial and error model was not nearly as strong in the demand uncertainty sessions. This occurred because of a combination of poorly-timed bonus draws and non-Nash pricing. Given the strong predictive ability of the trial and error model, it is not surprising that the weaker profit signals led to more non-optimal revelation decisions in the demand uncertainty sessions.(14)

Treatment Periods--The Pricing Decision

Figures 5 through 7 display the cost or bonus draw-contingent Nash and Joint Max price predictions and the actual mean prices for the optimal revelation decisions (i.e., conceal for Series 1 and 5 and reveal for Series 2, 3 and 4). Five categories are listed across the bottom of each figure, corresponding to the five possible draws in each treatment. These graphs show the cost-contingent or bonus draw-contingent mean prices separated into the first 5 and the last 5 treatment periods of the sessions to indicate the general pattern of learning. Two neighboring parallel dashed lines are often indicated for the theoretical price predictions because the discreteness of the problem often makes the theoretical solutions non-unique. Note that the Joint Max price lies below the Nash price in the complement demand conditions.

In four of the five series (1, 3, 4 and 5), mean prices are well-predicted by the noncooperative Nash model, especially in the final 5 periods. Although the variance of the price choices is significant, the Nash model is an excellent predictor of the central tendency of these prices. In contrast, in Series 2, the empirical final price distributions lie mostly below the Nash price predictions and near the level that maximizes joint profits.

Table VI presents Wilcoxon signed rank tests of the hypothesis that the observed distribution of prices has a median equal to the Nash prediction (and Joint Max prediction when both sellers are informed). Because of non-independence, formal hypothesis tests cannot use multiple observations from a single seller. Instead, we use each sellers' final price choice in each cost or demand realization. The maximum number of observations for the "Both Sellers Informed" column is n = 24 because up to 8 X 3 = 24 separate sellers can provide data from each series. TABULAR DATA OMITTED The maximum number of observations for the "One Seller Informed" column is n = 12 because only one-half of the 24 available sellers can be asymmetrically informed. The observations for each test vary because not all sellers faced every possible cost/demand realization in each information condition. For example, 13 sellers made a symmetrically informed final price choice in the series 1 sessions when the cost draw was 0. Table VI indicates that these price choices were not significantly different from the Nash price (8), but they were significantly different from the joint profit-maximizing price (20).

In the substitutes/demand uncertainty sessions (Series 3 and 4), the hypothesis that the median price is given by the Nash level cannot be rejected for eight out of nine bonus draws.(15) The TABULAR DATA OMITTED Joint Max prediction can be rejected in eight out of nine cases. In the optimal concealment sessions (Series 1 and 5), the Nash prediction can be rejected in only two out of ten cases.(16) For Series 1, 3, 4 and 5, we therefore conclude that the Nash model is effective in describing the central tendency of the observed price distributions.(17) However, in the Series 2 sessions, the Nash model can be rejected in all five cost draws, while the price that maximizes joint profits cannot be rejected for any cost draw. In these complement goods/cost uncertainty sessions, prices are consistent with joint profit maximization.

Baseline Periods

Recall that in the baseline periods, neither subject is aware of the cost or bonus draw when making his or her price decision. The baseline period data generally provide support for the Nash price predictions of Hypothesis 4. When mean prices are away from the Nash predictions in the early periods, they tend towards the Nash price as the session progresses. Table VII presents the results of Wilcoxon signed rank tests of the hypothesis that the final period prices have a median equal to the Nash prices (equations (10) and (10[prime])) or joint profit maximizing prices (equations (11) and (11[prime])). The test rejects strongly rejects the Joint Max price in all series, and the test only marginally rejects the Nash prediction in one series (series 3).(18)

V. Summary and Extensions

This paper has studied the realized incentives for competing sellers to share information in order to reduce uncertainty and correlate their price choices. The environment analyzed here is analogous to trade associations that publish demand or cost information for their members. The results generally support the theoretical models' Nash subgame perfect equilibrium predictions, although they also suggest conditions that may lead to collusive pricing behavior--namely when the industry is characterized by cost uncertainty and complement products.

While this experiment has tested a specific simplified information sharing model, its conclusions are relevant for the broad range of information sharing models indicated in the references. All of the models postulate that firms account for the implications of their information sharing decisions on output market profit. If they are to be useful, these models must be able to explain the behavior of individuals making decisions in an environment consistent with their assumptions. The subjects are definitely not "solving the game" as we do in game theory; the problem is too complex given their training. However, the subjects are able to successfully develop heuristic mechanisms--such as adjusting information sharing decisions based on previous period outcomes, as suggested by the logit model--to enable them to learn optimal strategies as if they have solved the game through backward induction.

Our conclusions can be summarized as follows:

* Hypothesis 1 (Revelation Decisions) is well-supported in the cost uncertainty sessions, but only weakly supported in the demand uncertainty sessions. We estimate a "trial and error" learning model and show that profit signals are weaker in the demand uncertainty sessions to explain this result.

* Hypothesis 2 (Nash prices in treatment periods) is supported for four out of the five series.

* Hypothesis 3 (Joint Max prices in treatment periods) is supported for one series--the complement goods/cost uncertainty sessions.

* Hypothesis 4 (Nash prices for the no-information periods) is supported in four out of the five series; where it is rejected, prices are below the Nash prediction.

We conjecture that the collusive pricing is facilitated by the availability of information sharing opportunities. While the existing models (with the exception of Clarke [4]) assume that the price or output strategy is noncooperative regardless of the information sharing decision, it may be possible to construct a model in which cooperation depends on the information sharing decision. Information sharing is a form of non-market interaction that firms may be able to use to alter the competitive environment in which they operate. For example, collusion may be more likely when information is shared merely because information makes it easier for firms to identify collusive strategy choices. We leave the development of a richer model that accounts for this relationship to future research.

1. For example, the FTC has published guidelines for trade associations to help their directors identify potential antitrust liabilities; see The Federal Trade Commission Advisory Opinion Digest.

2. See Plott [17] for a survey of experimental results in industrial organization.

3. We do not specialize the model in this way to mimic real-world uncertainty; indeed, it is unlikely that one firm will have all of the relevant cost or demand information. The asymmetry is introduced for practical reasons. It gives the theory its "best shot" by eliminating the need for subjects to use Bayes Rule and rely on beliefs about the information sharing strategies of their rival.

4. Evidence that subjects do not properly use Bayes rule is provided by Grether [9]. Camerer [1] has demonstrated that these biases in probability assessments persist even in repeated market environments.

5. The asymmetric access to information in the model suggests that alternative asymmetric pricing strategies may be relevant. The most obvious candidate is Stackleberg price leadership for firm 1. For the parameters chosen in the experiment, the theoretical Stackleberg prices are distinguishable from the theoretical Nash prices only in the complement goods/cost uncertainty series. However, the Stackleberg equilibrium prices have no ability to predict observed price choices, so we do not consider Stackleberg price leadership further.

6. Often, researchers use a predetermined sequence of draws for the realization of random variables. The problem with operating the bingo cage throughout the experiment is that it can lead to a nonrepresentative sequence of draws; however, this procedure allows subjects to observe the operation of the randomizing device and provides an opportunity for subjects to verify the prior probabilities of each draw.

7. The "firm 2" sellers who are not authorized to receive information receive a message that says "NO INFO."

8. Because of subject no-shows, one session in Series 4 had seven participants and one session in Series 5 had six.

9. Van Huyck, Battalio, and Beil [25] have explicitly examined random versus fixed pairings as a treatment in their study of tacit coordination games and they find very little impact. We conjecture that using fixed pairings would increase the level of cooperation observed in our experiment; of course, cooperation may be a Nash equilibrium to this repeated game. Some researchers have implemented the one-shot game in the laboratory by using exactly one more subject than the (publicly announced) number of periods and pairing each subject with every other subject exactly once. In the current design subjects may encounter each other more than once; however, it is impossible to identify when the repeated interactions occur, so we feel comfortable interpreting our results as arising from the one-shot game.

10. For a discussion of the role of fairness in experimental bargaining, see Kahneman, Knetsch, and Thaler [12]. The present experiment instructions do not inform subjects that they earned the firm 1 position after the assignment game. Vernon Smith points out that "the Hoffman-Spitzer result . . . shows . . . that equal split bargaining results [typically found in bargaining experiments with a first-mover advantage] may be due, generically, to an important treatment thought to be benign, namely, the standard use of random devices to allocate subjects to initial conditions" [24, 883 n. 8]. Series 3 and 4 were intended to test this conjecture within the context of this information sharing experiment. The results indicate that random assignment of seller roles is irrelevant for this design.

11. A complete design to test the role of conversion rate information and firm 1 assignment method in the four primary treatment cells (two demand conditions by two uncertainty sources) would require 16 experimental series. Since neither the firm 1 assignment method nor the conversion rate information had a significant impact on behavior, we expect that filling all the suggested 16 treatment cells with additional sessions isolating the impact of the assignment method from the conversion rate information would not provide important additional insight.

12. We initially considered risk aversion to explain the differences between the cost uncertainty and demand uncertainty results. In all cases the "correct" revelation decision increases expected profit but also increases risk (i.e., increases payoff variance). However, the increase in risk from making the correct revelation decision is greater in the cost uncertainty sessions than in the demand uncertainty sessions, so risk aversion does not seem to explain the better model performance in the cost uncertainty sessions. Recall that the cost uncertainty series were conducted at the University of Arizona while the demand uncertainty series were conducted at Berkeley, which is another possible explanation of this difference. We believe the learning model presented below to be a more plausible explanation.

13. The learning model proposed here is distinct from the learning literature often referred to as the "two-armed bandit framework" [19]. In bandit models, agents are uncertain about parameters in the model (such as purchase probabilities) and strategies are chosen to generate information about these parameters. In the current experiment, this type of uncertainty does not exist; all parameters are known, including the exact distribution of the random variables. Instead, the behavioral model estimated below assumes that subjects are uncertain about what strategy is optimal.

14. One design improvement suggested by this result would be to use multiple bingo cages--one for each duopoly pair--in order to reduce the likelihood of "unrepresentative" draws and induce independence of draws across subjects. To help interpret the current results, we can isolate the impact of the non-Nash prices from the impact of non-representative draws by using actual mean rather than theoretical prices and calculating the difference in expected profit (giving each realization a weight of 0.2) from making the correct revelation decision. For the cost uncertainty sessions, the increase in expected profit from choosing the correct revelation decision is 26-30 percent using this calculation, while for the demand uncertainty sessions this calculation method indicates only a 10-17 percent increase in expected profit from choosing the correct revelation decision.

15. The Series 3, bonus = 100 distribution rejects the Nash prediction because prices are too low. Note that several cases in this table do not have enough observations to conduct a meaningful test.

16. In Series 1, prices are too low in cost draws of 4 and 12. For these concealment tests, no Joint Max price exists because joint profit maximization requires information revelation.

17. We also conducted a series of Mann-Whitney tests to determine if informed price choices were different between Series 3 and 4. Since we were unable to detect a significant difference for any bonus draw, we conclude that the firm 1 assignment method and conversion rate information that varied between these series did not affect informed price choices.

18. In series 3, prices are below the Nash prediction of equation (10[prime]). A Mann-Whitney test rejects the hypothesis that the final baseline period price choices in Series 3 and 4 have the same median (Statistic = 276 [is greater than] five percent critical value = 185). Sellers are symmetric in the baselines, however, so this difference between the Series 3 and Series 4 sessions cannot be due to the firm assignment method. We conjecture that the depressed baseline prices in Series 3 was due to an unusually low series of demand draws during the early learning periods.

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