The matching problem (and inventories) in private negotiation.

The experimental design captures the matching problem in private negotiation trading with advance production (see figure 1 for the organization of an experimental session). Each trading period begins with a production decision, followed by several rounds of bargaining for price. A baseline treatment has four buyers and four sellers to be randomly matched/paired at the beginning of each of five bargaining rounds. Random re-matching at the beginning of each bargaining round can result in the same buyer and seller being matched in a subsequent round. Buyers and sellers are randomly and anonymously matched in order to avoid the formation of agreements and reputation building among agents. Three other treatments make matching more difficult. Reducing the number of bargaining rounds from five to three in the experiments, as well as creating asymmetry in the number of buyers or sellers in the market, both increase the matching problem. Four treatments make up the experimental design (table 1).

Designated as 5M, the baseline treatment has four buyers and four sellers with five matches during each of 20 trading periods. Treatment (3M) reduces the number of matches from five to three, again using four buyers and four sellers. A third treatment (2B5M) reduces the number of buyers to two, who are randomly matched with two of the four sellers during five bargaining rounds. Two sellers, therefore, did not trade during each of the bargaining rounds. Hence their expected number of matches is 2.5, while buyers have five matches. The final treatment (2S5M) consists of two sellers randomly matched with four buyers for five bargaining rounds per trading period. In this treatment, two buyers did not trade during each of the bargaining rounds. The 2S5M treatment is designed to provide insight into the amount of bargaining power sellers might gain as they become more concentrated, e.g., through a bargaining association or cooperative. The expected number of matches is 2.5 for buyers and five for sellers.

Subjects were recruited, primarily from undergraduate business and economics classes. A list of participant names was kept to minimize the chances of subjects participating more than once in the experiments. (6) The participants randomly drew a slip of paper that designated them as either a buyer or seller when they entered the computer laboratory. Buyers and sellers were asked to sit in separate sections of the room and each participant was seated in a different row. This procedure minimized visual interaction of participants. The instructions for the experiment were then read and followed by a practice session, which included as many production/trading periods as were necessary for all participants to become familiar and comfortable with the procedures (typically two to three periods). After the production decision, there were one-minute bargaining rounds (three or five) during which a buyer and seller exchanged "units" from a computer station through private, bilateral negotiation. Buyers were supplied with redemption values for units they could purchase. Sellers were given production costs for units they could produce and then sell. Unit values and costs were different in the practice session than in the actual experiment. Participants were told to keep their values and costs private. An artificial currency called "tokens" was used, with an exchange value of one cent per token. The unit values and unit costs, which were the same for each of the four buyers and four sellers, respectively, are presented in table 2. Each of the four treatments was replicated three times, i.e., there were three separate sessions of twenty trading periods for each treatment. Participants were unaware that trading would be terminated at the end of period twenty and also were not informed about how long the session would last.

Buyers waited while sellers made their production decisions. Once all sellers completed a production decision, which was private information, the trading began. For each one-minute bargaining round, buyers and sellers sequentially traded as many units as they could to make money. The matched buyer/seller pairs made bids and offers, respectively, until bids and offers were equal, or until the buyer or seller accepted the existing bid or offer. Following each trading period an individual's earnings were posted on their computer screen. Buyers earned the sum of the difference between what they paid for a unit ([P.sub.i]) and the given redemption value for that unit, i.e.,

(1) BuyerEarnings

= [n.summation over (i=1)](Redemption[Value.sub.i] - [P.sub.i])

where j = number of units purchased. Sellers earned the sum of the difference between unit price ([P.sub.i]) and its unit cost, i.e.,

(2) Seller Earnings = [k.summation over (i=1)] ([P.sub.i] - Unit[Cost.sub.i])

where k = number of units produced. If sellers did not trade a unit that was produced, [P.sub.i] = 0 and the cost of the unit was lost. There was no inventory carryover. Earnings accumulated over the sequence of trading periods and were displayed on the individual computer screens at the end of each period. Participants could view only their own information. Average participant earnings across all treatments were about $29 for 1 1/2 to 2 hours of participation.

Each participant was given an initial endowment of $7.00 or 700 tokens at the beginning of each session. The initial endowment was necessary because sellers incurred costs associated with advance production prior to being given the opportunity to earn profit from sales. Another concern was that the initial token balance be large enough to preclude the possibility of bankruptcy early in the session for individual sellers. This initial balance was given to both buyers and sellers in order to maintain symmetry.

The cost schedule for sellers ranged from thirty tokens for the first unit produced to a hundred for the eighth unit produced, as seen in table 2, for treatments 1 and 2. Redemption values for buyers ranged from 130 tokens for the first unit purchased to sixty tokens for the eighth unit in these two treatments. In treatment 3 (with two buyers), each buyer was able to buy sixteen units and the unit values were 130 tokens for the first two units, 120 for the third and fourth units, etc. Similarly, in the fourth treatment (with two sellers), each of the two sellers can produce up to sixteen units each. The unit cost schedule had two units costing thirty tokens, two units at forty tokens, etc. These schedules roughly (due to their discrete nature) translate to the individual supply schedule p = 25 + 10q and the individual demand p = 135 - 10q.

Horizontally summing the unit values and unit costs for the four buyers and four sellers results in a predicted competitive equilibrium price of eighty tokens and quantity twenty of to twenty-four units. The Cournot solution (four sellers) is 86.11 tokens and 19.56 units traded. The predicted monopsony price is sixty tokens with sixteen units traded. These serve as base values in the analysis that follows.

Methods of Analysis and Results from Laboratory Markets

Data collected from the laboratory markets include quantities traded and trade prices. Descriptions of the characteristics of each of these market outcomes over the twenty trading periods and four primary treatments are provided by means of a graphical analysis and a convergence model (Noussair, Plott, and Reizman 1995). The former offers a description of the general tendencies and the latter allows for tests of statistical inferences regarding differences between convergence levels across treatments relative to baseline predictions and between treatments. The following general convergence model is estimated for quantities traded and prices, [Z.sub.i], from the alternative treatments using the alternative base category predictions:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Z.sub.it] = average sale price (or units traded) across the replications of the treatment and all trades for each of the trading periods in cross-section treatment i; [B.sub.0] = the predicted convergence level of the dependent variable for the base category (competitive, Cournot, or monopsony prediction); [B.sub.1] = predicted starting level of the data for the base category; t = trading periods 1, ..., 20; [D.sub.j] = dummy variable separating the j treatments and [u.sub.it] = error term. Six equations were estimated; there is a price and trade equation for each of the three base categories. The base price (tokens) and quantity trade (units) values for the competitive, Cournot, and monopsony equilibria are, respectively, 80 and 20, 86.11 and 19.56, and 60 and 16, as previously reported.

The dummy variables ([D.sub.j]) take on the value of one when the dependent variable is from the jth treatment (3M, 5M, 2B5M, and 2S5M) and are otherwise zero. For the base, the convergence level of the dependent variable is given by [B.sub.0], while [B.sub.1] is the estimated origin (starting level) of the time series. If t = 1, then the value of the dependent variable is equal to [B.sub.1] for the base treatment. As t gets large, the weight of [B.sub.1] is small, because 1/t approaches zero, while the weight of [B.sub.0], (t - 1)/t approaches 1. The base treatment holds [B.sub.0] and [B.sub.1] fixed, but these values are adjusted by [[alpha].sub.j] and [[GAMMA].sub.j], respectively, for other treatments. In this study, the estimated [[alpha].sub.j] (the asymptote coefficients) are the parameters of interest, because they measure how trade or price convergence levels for the treatments deviate from the base category prediction (table 3). (7)