Nash equilibrium and buyer rationing rules: experimental evidence.

Kruse, Jamie Brown

I. INTRODUCTION

This paper presents the results of laboratory experiments comparing the effect of two buyer rationing rules on capacity-constrained duopoly markets (Bertrand-Edgeworth games). This setting offers an excellent opportunity to compare the stage-game Nash prediction with other equilibria of a repeated game. Under otherwise identical conditions, the rationing rule generates either a unique pure strategy equilibrium or a unique mixed strategy equilibrium of the stage game. Further, collusive outcomes of the repeated game are supported by a wider range of discount factors for one rationing rule than the other. This study presents evidence that the single-stage Nash equilibrium has predictive power for both rationing rules even in repeated settings. This result appears to be robust whether the Nash equilibrium is in pure strategies or exists only in mixed strategies.

In modeling capacity-constrained price games, the choice of rationing rule is possibly the most arbitrary decision a theorist makes. In the literature of single-shot Bertrand-Edgeworth games, we can find examples of nearly identical models which yield substantively different results due to differences in the rationing rule assumed. Examples are: Beckmann's |1965~ model and Levitan and Shubik |1972~; Allen and Hellwig |1986~ and Vives |1986~; and Kreps and Scheinkman |1983~ and Davidson and Deneckere |1986~. Laboratory experiments provide an interesting opportunity to examine the empirical consequences of alternative buyer rationing rules. Knowledge of the type of buyer rationing scheme involved with field data is usually unavailable to a researcher. In laboratory experiments, the researcher can control the type of buyer rationing rule to be used and compare the effects of alternative rules. This study provides a first step toward an empirical understanding of the effects of rationing rules.

One of the rationing rules that I study, the value queue,(1) is mathematically more tractable than an alternative, the random queue.(2) Hence, the authors are able to generalize the value queue models in other dimensions. However, the value queue assumption (reservation values that comprise demand are ordered from highest to lowest) should be viewed as a serious restriction on the applicability of these models.

Section II relates laboratory and field experiments to single-shot Bertrand-Edgeworth Nash equilibria. Section III describes the development and recent contributions to Bertrand-Edgeworth oligopoly theory, and demonstrates the effect of rationing rule assumptions on theoretical predictions. I describe the experimental design in section IV and identify the competitive, monopoly, joint-profit maximum and single-shot Nash predictions. Additionally, qualitative differences predicted by single-shot and repeated-game Nash equilibria are discussed. Experimental results follow in section V with concluding remarks in section VI.

II. LABORATORY AND FIELD EXPERIMENTS AND THE NASH EQUILIBRIUM

Existing laboratory evidence for capacity-constrained price games has been compared to the single-shot Nash equilibrium only. Ketchum, Smith and Williams |1984~ and Kruse, Rassenti, Reynolds and Smith |1993~ provide initial results as to the predictive strength of the single-stage Nash equilibrium in a laboratory environment. Ketchum, Smith and Williams compare the prices resulting from four-seller Plato computer-assisted posted-offer markets. The comparison is between one treatment group in which the competitive equilibrium is also the single-stage Nash equilibrium and another treatment group in which the competitive equilibrium is not a Nash equilibrium. They find significantly different market outcomes between the two groups. Kruse, Rassenti, Reynolds and Smith |1987~ test a four-seller constant marginal cost Bertrand-Edgeworth model. Using a Komogorov-Smirnoff test, they reject the hypothesis that the actual price distribution matches the mixed strategy price distribution predicted by the theory. However, the Nash prediction does provide the correct qualitative ranking of the price distributions for the capacities studied and predicts better than all alternative hypotheses. A field experiment by Domowitz, Hubbard, and Petersen |1987~ uses panel data to test the price path predictions of two supergame models (Green and Porter |1984~, Rotemberg and Saloner |1986~). However, they find that the price-cost margins resemble a one-shot Cournot-Nash outcome more than the collusive outcomes predicted by the repeated game models.

The experiments I report in the following pages represent a design similar in spirit to that suggested by Hugo Sonnenshein in Frontiers of Economics. Sonnenshein calls for theoretical and experimental research to facilitate the basic conceptualization of oligopolistic markets. He proposes that duopolists with U-shaped average costs be examined in the context of a repeated game with discounting.

III. LITERATURE OF BERTRAND-EDGEWORTH GAMES

Joseph Bertrand first proposed a duopoly model of firms whose decision variable is price in 1883 for comparison with Augustin Cournot's |1838~ quantity-choice model. Bertrand criticized Cournot's pioneering work on the basis that it lacked a plausible explanation of price formation. In Bertrand's model, producers simultaneously and independently choose prices. Demand is allocated to the lower-price seller who then produces the quantity demanded at the stated price. Bertrand showed that the only equilibrium must be at the price which equals (constant) marginal cost.

Francis Edgeworth's 1925 critique of Bertrand demonstrates that, except for the case of constant marginal cost, there are existence problems even in the homogeneous good case. Edgeworth modified the original Bertrand model to introduce the possibility of limited capacity. He characterized a set of demand and cost conditions in which one firm cannot serve the entire market. Edgeworth suggested that prices might be expected to fluctuate over a range whose upper and lower bounds depend on the two firms' respective capacities. Appealing to an ad hoc dynamic argument, Edgeworth envisioned an unstable market which alternates between a price war phase and a relenting phase. The price path manifested by this model is referred to as an "Edgeworth Cycle."

A majority of recent papers on Bertrand-Edgeworth oligopolies utilize one of two rationing rule assumptions. The rationing rule specifies the order in which buyers whose reservation values represent the demand curve are allowed to purchase one or more units. One rationing rule, hereafter the value queue, allocates demand by "marching down the demand curve." Under the value queue, highest reservation price purchases are from the lowest-price seller. Whereas under the random queue, components of demand are realized in random order.

The reason why equilibrium predictions will differ becomes clearer by studying the residual demand under the two rationing rules. For the duopoly case, if the price a seller has chosen turns out to be the lowest, the rationing rule has no effect on her payoff. Likewise when the two sellers match prices, the rationing rule is innocuous. However, if the price chosen by a seller is the higher of the two, the residual demand left after the low-price seller has exhausted his capacity is quite different under the two assumptions.

Consider linear demand curve, d(P), shown in Figure 1. Under the value queue, the residual demand curve (GD) is determined by a parallel downward shift of the original market demand curve by the low-price firm's capacity (|k.sub.j~). Therefore, the high-price firm's profit function depends on its own price choice and the other firm's capacity, which is parametric.

The random queue is generated by letting the fraction of consumers left to the higher priced firm be a random sample of the population. Referring to Figure 1, given |p.sub.j~, the residual demand possibilities resulting from a random assignment of buyers to the queue could range from GD to HF with an expectation of HD. The residual demand under the random queue yields greater potential profit to the high-price seller. Thus the incentive for price cutting deteriorates with the random queue as compared to the value queue.

Theorists are aware of the impact of different rationing assumptions on Bertrand-Edgeworth models. The following theoretical research provides numerous examples of models which differ only in the rationing assumption and, as a result, produce different equilibrium predictions.

Shubik |1959~ discusses the structure of a Bertrand-Edgeworth game and three candidate rationing rules which give rise to different contingent demand for the high-price firm. Martin Beckmann |1965~ chose one of Shubik's suggested rules, a random queue, and explicitly calculated pure and mixed strategy equilibria for an example with two symmetric capacity-constrained firms and normalized linear demand. For explanatory purposes, let D:|R.sup.+~ |right arrow~ |R.sup.+~ give the quantity demanded at all positive prices and let c be the constant unit cost of production up to firm i's capacity limit |k.sub.i~. Beckmann showed that when each firm's capacity is 1/4 |Alpha~, where |Alpha~ = D(c), the Bertrand-Edgeworth Nash equilibrium is in pure strategies at the monopoly price. When each firm has capacity equal to |Alpha~, the pure strategy Bertrand-Edgeworth Nash equilibrium is at the efficient price. For symmetric capacities between 1/4 |Alpha~ and |Alpha~, Beckmann solves for the mixed strategy equilibrium distribution. Levitan and Shubik |1972~ adopt the value queue in a model otherwise similar to Beckmann's. They find equilibria in mixed strategies for identical capacities between 1/3 |Alpha~ and |Alpha~ and pure strategy equilibria for capacities outside this range.

Kreps and Scheinkman |1983~ attempt to resolve the quantity-choice price-choice dilemma by arguing that capacity is a long-run decision and price, a short-run decision. They demonstrate that if firms choose capacity first and then prices in a second stage, a Cournot pure strategy equilibrium obtains (assuming a value queue). Davidson and Deneckere |1986~ show that the Kreps and Scheinkman result is contingent upon the rationing rule assumed. In fact, the Cournot equilibrium found by Kreps and Scheinkman does not exist for any perturbation of the value queue.

Brock and Scheinkman |1985~ present a repeated game of capacity-constrained, price-setting firms. They focus on the role of industry capacity in enforcing collusive behavior using a value queue assumption.

IV. RATIONING RULE EXPERIMENTS

The purpose of the experiments reported in this study is to determine the behavioral effect of different rationing rules on market outcomes. Eight experiments are conducted involving two sellers with identical cost structures trading in the Plato posted-offer institution with simulated demand.(3) Figure 2 illustrates the theoretical model and its laboratory interpretation. Each seller has a U-shaped average cost curve with minimum at Q*. Demand will just accommodate two sellers at the efficient price (|P.sub.e~) that corresponds with AC(Q*). Thus the efficient outcome involves both sellers producing Q* and receiving |P.sub.e~ for their output. In the laboratory model, each seller has a discrete units version of a "U-shaped" average cost curve, with minimum at four units. The market clears with each firm selling four units at a price up to $0.15 greater than minimum average cost |A|C.sub.i~(4)~. This yields a profit of $0.60 per period. In treatment I, fictitious buyers are randomly queued. Treatment II buyers are queued from highest to lowest values (value queue).

Each subject seller is given a table for calculating total and average costs from the marginal cost schedule given on the PLATO screen. The demand curve is not provided to the subject/sellers. Subjects are told that demand will be simulated and, in the event of matched prices, each subject will sell about half of the quantity demanded at that price. A copy of the instruction sheet is contained in appendix B. Entry cost for additional sellers is infinite, consequently the duopolists operate in a market with zero threat of further entry. Each experiment lasts twenty trading periods with duration not announced to control for end-period effects. Laboratory market results are tested against the theoretical predictions summarized in Table I. All prices are reported in deviation from minimum average cost. The competitive and monopoly predictions for this model form benchmarks that we may compare with Nash predictions and laboratory market results. The competitive outcome corresponds with total surplus maximizing exchange of eight units. Maximum efficiency is attained at prices up to $0.15 above A|C.sub.i~(4) with profit of $0.60 per period. Price outcomes within the range of $0.00 to $0.15 with eight units traded will be considered competitive results. The monopoly price of $1.15 and quantity of four units yields a profit of $4.60. If the duopolists attempt to maximize joint profits, they will choose a price of $0.90 and split the market to earn expected profit of $1.38 per period.

The form of the rationing rule has no effect on the nature of the uniform price predictions described above. However, the Nash prediction for a single-period game does depend on the rationing rule chosen. When we examine the price reaction functions for the laboratory value queue in Figure 3a, we can identify the intersection ($0.15) as a pure strategy Nash equilibrium. The random queue reaction functions in Figure 3b do not intersect. No pure strategy equilibrium exists for this case. When its opponent chooses $0.22, either firm's best reply is to choose $1.15. Numerical solution for the symmetric mixed strategy Nash equilibrium of the laboratory random queue case yields the price frequency distribution illustrated in Figure 4 with expected equilibrium profit of $1.05.(4)

The static Nash equilibrium predicts the following differences under the two rationing rules:

1. Random queue experiments should exhibit more price dispersion than the value queue counterparts.

2. Mean prices should be higher under the random queue treatment than the value queue.

The theoretical predictions in the context of a repeated game are not as well defined.(5) The standard folk theorem is not falsifiable since any set of price observations support the hypothesis that (almost) "anything is possible."

Though many equilibrium prices are sustainable under tacit collusion, I will concentrate on the joint profit maximizing price of $0.90. The profit under simple joint profit maximization is $1.38 per period. This is the profit that each firm may expect from cartelization of the industry at a uniform price under either rationing rule. The single-shot Nash equilibrium is the credible punishment for defection. The parameterization chosen coupled with the choice of rationing rule has the property that the single-shot Nash profit of $0.60 per period for the value queue is substantially different from the joint profit maximum of $1.38. Thus, a (tacit) cartel is enforceable via the credible threat of reducing a cheater's profit by over 50 percent each ensuing period. However, for the random queue, the single-period (mixed strategy) Nash expected profit ($1.04 per period) is only $0.34 less than the joint profit maximum of $1.38 per period. There is less incentive to maintain tacit collusion under the random queue. We would then expect by the argument TABULAR DATA OMITTED put forth by Brock and Scheinkman |1985~ that value queue duopolists are more likely to sustain tacit collusion than random queue duopolists. We can calculate the minimum discount parameter, |Delta~, that would sustain joint profit maximization as one trigger strategy equilibrium. Value queue collusion is sustainable for 0.82 |is less than or equal to~ |Delta~ |is less than or equal to~ 1, whereas |Delta~ must be at least 0.91 to sustain collusion under a random queue.

V. RESULTS AND DISCUSSION

The observations from eight PLATO posted-offer duopoly experiments are reported in this section. Buying order is the treatment variable, all other procedures and parameters are held constant.

Foremost is the compelling result that buying order has a significant effect on the market outcome. The pooled mean prices under a random queue and value queue are illustrated in Figure 5. Using a Wilcoxon nonparametric test for matched pairs, we can reject the hypothesis (1 percent significance level) that both sets of prices originated from the same distribution. A t-test for dependent samples yields similar results.

The single-shot and repeated-game models would lead us to different qualitative predictions on prices. Value queue prices are expected to be lower than random queue prices in a single-shot game whereas in the context of a repeated game, this is not the case. Value queue cartels would have greater enforcement strength and thus better ability to maintain (tacitly) collusive prices than random queue cartels. Failure to reject the hypothesis that value queue price observations are at least as high as random queue prices would support the repeated game prediction. Rejection of the hypothesis would support the prediction of the single-shot game. Thus the hypothesis tested is:

|H.sub.0~: Mean |P.sup.V~ |is greater than or equal to~ Mean |P.sup.R~

|H.sub.a~: Mean |P.sup.V~ |is less than~ Mean |P.sup.R~

A t-test for dependent samples yields a test statistic of 7.92 allowing rejection of |H.sub.0~ in favor of the alternate hypothesis that random queue mean prices are higher than value queue mean prices at a 1 percent level of significance.

The frequencies of observed prices from all periods of the experiments are shown in Figure 6. The mode for value queue experiments is $0.15 with the probability density function skewed to the right. The modal price observed in random queue experiments is $0.50, with more weight on the monopoly price. The average income per period in the value queue treatment is $0.636 which is not significantly different from the single-shot Nash prediction of $0.60. Random queue sellers earned an average of $1.03 per period which is not significantly different from the single-shot prediction of $1.04.

Tacit collusion is not observed in either treatment. We can reject the hypothesis that mean prices match the theoretical collusive prediction of $0.90 with t-statistics of -28.76 and -27.05 for the random and value queue respectively.

Recognizing that pooling prices tends to suppress interesting characteristics of individual experiments, the results of experiments P245 and P249 are presented. The random queue experiment in Figure 7 and value queue experiment in Figure 8 help us appreciate the commanding differences due to the treatment variable. The first four periods of P245 in Figure 7 are characterized by competitive undercutting behavior. In period 5 one seller abandons the price war and chooses a high price. Subsequent periods exhibit price dispersion with no discernable trend. This price pattern is strikingly different from value queue experiment P249 in Figure 8. In P249, the sellers lock on the single-shot pure strategy Nash equilibrium price of $0.15. Costly signals in periods 10, 13 and 15 go unanswered. The results of the two experiments discussed above are robust under replication. This can be verified by studying the results of all experiments contained in appendix C.

VI. CONCLUSIONS

The multiplicity of equilibria that arise from repeated games make falsifying evidence difficult to identify. The static Nash prediction is one of several equilibria of the repeated game. However, it is considered a less likely result since profits can be increased by tacit collusion. In his discussion of repeated games, Tirole wrote, "Although it is hoped that the testing of full-fledged dynamic models will develop, it must be acknowledged that such models are complex, and little attention has been paid to testable implications" |1989, 245~. The results reported here, in addition to the field experiments by Domowitz, Hubbard and Petersen |1987~ and laboratory experiments by Kruse, Rassenti, Reynolds and Smith |1993~ and others, indicate that the single-shot unique Nash equilibrium has predictive power, even in repeated settings.

The accumulation of evidence in this study shows that the impact of different rationing rule assumptions in Bertrand-Edgeworth models is not trivial. The value queue has often been adopted as a mathematical convenience in numerous Bertrand-Edgeworth models including a high proportion of the theoretical pieces cited here. The theoretical results from these models do not generalize to more believable queuing rules. These experiments show the magnitude by which market outcome can change with a change in the ordering of buyers. Theorists must move toward a more general characterization of demand allocation or at least recognize the limitations on the models that have been proposed.

The theory of Bertrand-Edgeworth competition has advanced at an extraordinary rate in the last few years. Behaviorally, the outcome of laboratory Bertrand-Edgeworth duopoly competition is significantly different with different rationing rules. The direction of the difference agrees with the predictions of the singleshot Nash Prediction. The experimental evidence that I report: 1) underscores the need for a generalization of buyer rationing assumptions in theoretical models and 2) indicates that the appropriate role of the static Nash equilibrium in a repeated context has not been fully appreciated.

TABULAR DATA OMITTED

APPENDIX B
Auxiliary Instructions
Seller Cost Table

# of Units Additional Total Per Unit Cost
Sold Cost Cost (total cost/# units)

1
2
3
4
5

IN THIS EXPERIMENT, THE DEMAND FOR YOUR PRODUCT WILL BE
SIMULATED. IN THE EVENT THAT YOU MATCH PRICES, YOU WILL EACH
SELL APPROXIMATELY 1/2 OF THE TOTAL QUANTITY SOLD IN THAT
PERIOD.

1. This is also referred to as parallel rationing or efficient rationing. The former name is due to a geometric interpretation of the rule's effect on residual demand. The latter name was used since it maximizes consumer surplus. However, in a total surplus sense, the name is inappropriate.

2. This is also referred to as the proportional rationing rule or Beckmann rationing. This rule was first proposed by Shubik |1959~. Beckmann |1965~ presents a closed-form solution using the random queue.

3. The Plato posted-offer program allows sellers and buyers (simulated or human) to make exchanges of a fictitious commodity on the Plato computer system. Sellers post a price to buyers that arrive to purchase according to their rank in a queue. See Ketchum, Smith, and Williams |1984~ for a detailed description of the PLATO posted-offer protocol.

4. The solution was achieved using a special application of the quadratic programming option on LINDO and checked using a spreadsheet program.

5. Our laboratory experiments have an uncertain endpoint. However, the continuation probability was not controlled, so could vary widely across subjects.

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