Theory and experiments on spatial competition.

Brown-Kruse, Jamie ; Cronshaw, Mark B. ; Schenk, David J. 等

I. INTRODUCTION

Why do convenience stores tend to choose dispersed locations close to consumers, whereas fast food restaurants tend to locate near each other? Why do automobile dealers locate close to one another, yet go to great effort to differentiate their products through advertising? It is clear that firms can compete on many dimensions including price, location and product characteristics. Previous work that has sought to explain competition on many dimensions has been less than successful (see Osborne and Pitchik |1987~).

In our investigation, we allow competition to occur only in the form of spatial product differentiation. In the spirit of Hotelling |1929~, each firm selects a location at some point on a line segment. Identical consumers with downward sloping demand curves are uniformly distributed over the market. Customers incur a transportation cost which varies linearly with their distance from the closer firm. The FOB (free on board) price charged by each firm is the same and exogenous. The price paid by a consumer is the sum of the fixed FOB price and the transportation cost. This gives a negative relationship between a customer's distance from the nearest supplier and the quantity demanded.

A firm has two considerations when selecting a location: market share and proximity to consumers. When the firms locate symmetrically, each at a distance of one-half from the center of the market, they split the market in half. Each firm obtains a 50 percent market share and the most remote consumer is a distance of only one-fourth of the total market space away from one of the firms. Because of this, by locating at the quartiles of the market the firms maximize combined sales. However, in this situation a firm can typically increase it sales by moving toward its rival, thereby increasing its market share as buyers switch firms.

The decision to locate away from the center of the market carries a degree of risk. Any position away from the center leaves "unprotected" market share. The other firm has an opportunity to gain market share (and profit) by locating close by, but a little closer to the center. By locating at the center of the market, a firm assures itself of at least one-half of the market regardless of the other firm's location. Thus market share is secure. In this case, the firms share the market equally, the most remote customer is a distance of one-half from each firm, and combined sales are relatively low compared to a quartile outcome. This corresponds to the notion of minimum differentiation that is attributable to Hotelling. However, if the transportation cost is sufficiently high, this is unlikely to be the outcome.

The theory of location choice has been considered by economists and political scientists. Smithies |1941~, Eaton |1972~ and Eaton and Lipsey |1975~ all consider a uniform distribution of consumers over a linear market. Using verbal arguments Smithies shows that for a duopoly facing linear demand, the type of equilibrium depends on the ratio of the transportation cost to the choke price(1) on the demand curve. Eaton provides the mathematical details. Eaton and Lipsey consider unit demand and extend the analysis to n-firms and a two-dimensional market. In a duopoly on a linear market they conclude that the firms will locate at the center.

Hinich and Ordeshook |1970~ consider a competition for votes between political candidates.(2) They show the candidates will choose the same location in equilibrium if each candidate's objective is to maximize plurality (the excess of its votes over its opponent). However, if each attempts to maximize votes, the candidates may choose different positions in equilibrium, depending on the sensitivity of voter preferences. Much of their analysis is analogous to the case of unit demand and linear transportation cost.

Gabszewicz and Thisse |1986~ maintain unit demand and conclude that due to the harsh competition that such an environment fosters, "the only location equilibrium consists of both firms clustered at the center of the market." (Gabszewicz and Thisse |1986, 37~ Telser |1988~ arrives at the opposite conclusion. He contends that in the n-firm case, firms would be equispaced within the market, corresponding to a quartile equilibrium in the duopoly case. Within our theoretical framework we show that these disparate views both represent possible equilibria.

Our work both extends the theory and presents experimental results on location choice. The experiments support the idea that a wide variety of outcomes wil occur. Our theoretical contributions show that, depending on the transportation cost, the equilibrium may or may not be at the center if consumers have a downward sloping demand curve. We also show that a central location is consistent with max min behavior.(3) Finally we solve for symmetric equilibria in an infinitely repeated location choice game with linear demand for any discount factor.(4)

In order that our theory and experiments fit together we must be relatively confident that the restriction to symmetric equilibria is appropriate. Our first tests of experimental results focus on the hypothesis of symmetry for all treatments. We also examine two communication environments. In one case we allowed no communication between subjects. In the other, we allowed anonymous nonbinding communication to occur before sellers made their location choice. The nonbinding communication is in the spirit of Farrell's "cheap talk" |1987~. Such announcements do not directly affect payoffs.(5) However, sellers could suggest that they would each be better off if they locate symmetrically away from the center.

The next section summarizes previous experiments that examine the effect of communication. This is followed by a theory of nonprice spatial competition. In section IV we present the experimental design. Section V contains our experimental results. Finally, we give concluding remarks in section VI.

II. PREVIOUS EXPERIMENTS WITH COMMUNICATION

Isaac and Plott |1981~, Isaac, Ramey and Williams |1984~, and Isaac and Walker |1985~ examine cartel formation and sustainability under double auction, posted offer and sealed bid auction institutions. Subjects had limited success in coordinating profit-increasing cartels in the repeated settings. Their ability to coordinate depends on the trading institution and market information available. All of these experiments allowed repeated free-form face-to-face communication. Subjects were monitored during communication sessions in which neither threats nor side payments were allowed. However, the effect of physical attributes of subjects cannot be controlled for under this type of communication. The possibility of future retribution for cheating on a collusive arrangement still exists since subjects can identify their cohort.

Cooper, DeJong, Forsythe and Ross |1990a; 1990b~ tested the effect of stylized forms of anonymous communication on the outcomes of a series of single-shot 2 x 2 and 3 x 3 games. Under one-way communication, the row player was allowed to send a non-binding message of what he planned to play. Under two-way communication both players simultaneously sent a prospective move to each other. Simultaneous choice of actions followed immediately with no opportunity for reply or negotiation. These experiments demonstrated that the ability to coordinate at desirable actions was enhanced by limited communication, but was far from completely successful.

Communication in our experiments captures features of both sets of experiments described above. Using the VAX Electronic Phone system, we could allow free-form communication with reply and negotiation and at the same time control for any distortions that might occur due to physical appearance and voice of the subject participants (since all messages were typed and subjects did not know the identity of their rival). Interestingly, it appears that profit-enhancing coordination in our experiments is strikingly prevalent with the demarcation between communication and noncommunication outcomes more distinct than in any of the experiments discussed above.

III. A THEORY OF SPATIAL COMPETITION

Model Structure

We consider a market consisting of identical consumers uniformly distributed along a line segment. Each consumer has an identical non-negative valued demand function q which is strictly decreasing when it is positive and |Mathematical Expression Omitted~ where |Mathematical Expression Omitted~ is a choke price. There are two firms which choose where to locate on the line segment. Each firm charges the same FOB price |Mathematical Expression Omitted~ and has a constant marginal cost. Price and marginal cost are exogenous. Since the FOB price and marginal cost for each firm is constant, each firm wishes to maximize the quantity which it sells.

Each consumer buys from the closer of the two firms. Consumers incur a transportation cost of t |is greater than~ 0 per unit of distance. Thus, a consumer who is a distance from the closer firm demands an amount q(|p.sub.0~ + |xi~).

It is convenient to represent the market by the interval |-1,1~. We will refer to a point on this interval as a location. We can exploit the symmetry of the model with an appropriate definition of each firm's choice variable. We let firm 1 choose a position (not location) x. A position x=0 corresponds to locating at the center of the interval. We let x increase to the right. Thus, a position of x=1 represents firm 1 locating at the right end of the market. We let firm 2 choose a position y. We let y increase to the left. Positions of y=1 or 0 correspond to locations at the left or center of the market, respectively. Figure 1 shows the position of firms and how demand varies across the market. Let Q(x,y) be the quantity sold by firm 1 when the firms' positions are x and y.

Single Period Game

Since Hotelling |1929~, economists have been accustomed to the principle of minimum differentiation. We can build a great deal of intuition about the situation by considering circumstances under which minimum differentiation does not hold, namely when the transportation cost is sufficiently high.

Suppose that both firms are located at the center. We assume the firms share the market equally with firm 1 serving all customers to the right of the center and firm 2 all those to the left. Suppose that firm 1 moves a small distance to the right as shown on Figure 2. Before the move, the length of the firm's market to the right was 1 and to the left was 0. After the move the length to the right decreases by |epsilon~ to 1 - |epsilon~, but increases to |epsilon~/2 on the left. (Firm 2 gets the customers in the remaining |epsilon~/2 market length.) Thus the total length served by the firm falls to 1 - |epsilon~/2. However, the total quantity sold by the firm may either increase or decrease since the average distance of a consumer from the firm also falls, causing a drop in transportation cost. The customers lost due to the decrease in the length at the right were a distance of about 1 from the firm (since is small). Hence the corresponding decrease in quantity sold is approximately q(|p.sub.0~ + t)|epsilon~. The customers on the left of the firm after the move perceive a price close to |p.sub.0~. Hence the total amount demand by them is q(|p.sub.0~|epsilon~/2).

Such a move will lead to a net increase in sales if and only if q(|p.sub.0~)|epsilon~/2 |is greater than~ q(|p.sub.0~ + t)|epsilon~. Assuming that q is strictly decreasing at |p.sub.0~ + t, this condition is equivalent to t |is greater than~ |t.sub.0~ where |t.sub.0~ = p(q(|p.sub.0~)/2) - |p.sub.0~ and |Mathematical Expression Omitted~ is the inverse demand function. For a linear demand curve, |Mathematical Expression Omitted~, this critical transportation cost is |Mathematical Expression Omitted~. That is, if the transportation cost is sufficiently high it is not an equilibrium for both firms to locate at the center.

In appendix A we sharpen this intuition by proving the following theorem

Theorem: For the single-period location choice game

a) If 0 |is less than~ t |is less than or equal to~ |t.sub.0~ there is a unique Nash equilibrium, x = y = 0

b) If |Mathematical Expression Omitted~ there is a unique Nash equilibrium, x = y |is greater than~ 0

c) If |Mathematical Expression Omitted~ there are multiple Nash equilibria in which the firms are local monopolists.(6)

Note that the equilibrium is unique and symmetric for any downward sloping demand curve, providing that the firms are not local monopolists. Note also that in the case of a linear demand curve the nature of the Nash equilibrium depends on the ratio of the transportation cost across the entire market to the residual price |Mathematical Expression Omitted~ as pointed out by Smithies |1941~. However, for non-linear demand curves, |t.sub.0~ will not, in general, be a multiple of|Mathematical Expression Omitted~, and thus the nature of the equilibrium is not determined by the ratio described above.

If the transportation cost is high enough, it is not an equilibrium for the firms to locate at the center. However, there is another behavioral assumption, rather than Nash equilibrium, which does lead the firms to locate at the center. We show in appendix A that a firm can best protect itself from low sales by locating at the center. Formally the security level (or max inf) payoff is Q(0,0).(7) This is very intuitive. If a firm locates at the center, it is guaranteed to sell to at least half of the market. If it locates anywhere else, its rival can locate close to the firm on the long side of the market leaving that firm with less than half of the market.

If the firms locate symmetrically about the center, they split the market at the center. In particular they do so if they each locate a distance of one-half from the center, referred to as the quartiles of the market. We show in appendix A that they maximize the sum of their sales by locating at the quartiles. We will refer to this as the collusive outcome. Note that the consumers also benefit from these positions since the most remote consumer is only a distance of one-quarter from a firm.

We now turn to an infinitely repeated version of this game.

Infinitely Repeated Game with Linear Demand

It is well known that repetition can greatly expand the set of equilibria. According to the Folk theorem (Fudenberg and Maskin |1986~) any individually rational and feasible payoff is an equilibrium if agents maximize the present discounted value of their payoffs when the discount factor is large (i.e., the interest rate is low.) What is not well known is how large the discount factor must be for this result to hold, or what is the set of equilibria for an arbitrary discount factor.

This section uses the theory presented in Cronshaw |1989~, and Cronshaw and Luenberger |1990~ to solve for the set of strongly symmetric subgame perfect equilibria (SSE) in an infinitely repeated game with linear demand. In an SSE each firm chooses the same action as the other on and off the equilibrium path. The restriction to symmetric equilibria is made for tractability.

However, subject behavior that we observed was symmetric (see section V). Thus, the set of SSEs is an interesting subset of the equilibrium that appears to be appropriate.

Furthermore, the payoff, which is the lowest in any equilibrium (symmetric or not), can be achieved by a symmetric equilibrium if the discount factor is high enough. Thus, in the spirit of Abreu |1986~ we are able to find a globally optimal punishment for a large discount factor.

We let the demand function be |Mathematical Expression Omitted~. It is convenient to define a parameter |Mathematical Expression Omitted~, the distance from a firm at which demand falls to zero.

We show in appendix B that the set of average discounted SSE payoffs is as shown in Figure 6 (in appendix B), providing that 5/4 |is less than or equal to~ |psi~ |is less than or equal to~ 2. The figure shows the set of SSE payoffs as a function of the discount factor. For |delta~ = |is greater than or equal to~ ||delta~.sub.2~ = |(2 |psi~ - 1).sup.2~/(12 |psi~ - 5) we get a symmetric Folk theorem-like result that any individually rational and feasible payoff is an SSE. For smaller discount factors the set of SSE payoffs shrinks. In the limit of a zero discount factor only the stage game equilibrium remains.

In our experimental design the game was played with an uncertain endpoint. The probability of continuing next stage was fixed at 7/8. The design resulted in a parameter value for |psi~ of 1.894 at which the discount factor threshold ||delta~.sub.2~ is 0.438. So in our experiments all individually rational and feasible symmetric payoffs are SSEs. It would be interesting in future work to consider other designs with a smaller set of SSEs.

Each such payoff can be achieved by one or more pairs of strategies for the two firms. These strategies define both an equilibrium path and behavior that would occur after any deviation from that path. In theory only the equilibrium path would be observed since each firm can anticipate that any deviation from the equilibrium path would be unprofitable.

Equilibrium paths can in principle be complicated. However, in the experiments we observed that subject firms usually settled down to some steady state or stationary behavior after a few periods. In appendix B we show that locating at the quartiles or stage game Nash position forever is an equilibrium path; but locating at the center forever is not an equilibrium path. If both firms locate at the center forever they get the worst SSE payoffs. Yet, within our design if an opponent locates at the center, the best response in the stage game is not at the center. A firm could earn a single-period gain by moving away from the center and would be no worse off after that period, since it is already being punished by staying at the center.

Of course, a firm wishing to protect against losses could select the center in every period to achieve at worst its security level payoff (see the section Single Period Game). But it is not an SSE for both firms to do so.

IV. EXPERIMENTAL DESIGN

All experiments were conducted at the Laboratory for Economics and Psychology (LEAP) at the University of Colorado using a subject pool drawn from undergraduate intermediate economics classes. Subject sellers were randomly paired within each session and were told that they would remain in those pairings throughout the experiment. The laboratory conditions were controlled so that no one was able to detect the identity of the other seller in his/her market. Subjects were told that their market consisted of a road that was 100 miles long and that they could locate at the mile post of their choice, i.e. in the set {0,1,2,...,100}. They were informed that simulated buyers, one located at each mile marker, would choose to purchase some quantity of the hypothetical good. The FOB price they could charge was fixed at |p.sub.0~ = $0.53 per unit, and the transportation cost to the consumer was set at $0.10 per mile per unit of the good. The result in a value of t equal to 5 (=0.10x50). Subjects were given this information and told that the amount a consumer will demand falls with price, and that the price "seen" by the consumer is dependent on the FOB price of the good and the cost of transporting purchases home.(8) In addition, they were given the costs of production they would incur each period, which included a fixed cost of $10.00 and a $0.50 constant marginal production cost.

Each consumer's demand function was q(p) = 10 - p, where p = |p.sub.0~ + 0.1xd, and d is the distance from the seller chosen. Thus, in terms of the variables we defined in section III, |Mathematical Expression Omitted~ and |psi~ = 1.894. The profit received by the ith firm is |P.sub.i~ = 0.03|Q.sub.i~ - 10, where |Q.sub.i~ is the sum of demands for individual consumers who buy from the ith firm. If a buyer was indifferent between the two sellers, given their location choices, each seller sold one-half of the indifferent buyer's demand. The set of SSE payoffs for the experimental design is shown in Figure 3.

To begin each period, subjects were asked to simultaneously write their location choice on a record sheet which the monitor would then collect. All location pairs were entered in a spreadsheet program on an IBM PC which calculated quantities sold, market share, and profits. This information was then recorded on the appropriate seller's record sheet. Thus, the sellers were informed of the quantity they sold, respective percentages of total sales, and profit for the period. At the start of the session, subjects were told that there would be three practice rounds that did not count toward their earnings.

The subject were informed of the procedure for ending the experiment, described as follows: After record sheets were returned, a ball was drawn (with replacement) from a bingo cage containing fourteen white balls and two red balls. The game terminated if a red ball was drawn, otherwise it continued. After giving a one-minute warning, we required sellers to enter a location decision on their record sheets. Thus, the probability of the game ending in the current period was 1/8. Subjects were shown the balls before they were placed in the bingo cage and draws were made in full view with the color of the ball clearly revealed and announced. Our laboratory market with a random endpoint can be viewed as an operationalization of an infinitely repeated game with a discount factor 7/8.

For the parameters of this experimental design, the set of SSE payoffs are the entire interval of individually rational and feasible symmetric payoffs, depicted in Figure 3. There are also asymmetric equilibria in the uncertain endpoint game, but we have not computed them.

This means that in terms of solutions, anything from the quartile locations to any pair of symmetric locations nearby but not including the center constitute a Nash equilibrium. At this point the Nash conjecture gives us no guidance as to which of these equilibria will arise or if there is any reason to expect any element of the set with greater frequency than the others. Table I summarizes the outcomes predicted for the various solution characteristics described above.

There were two experimental treatments. Within Treatment I, no communication was allowed between subjects. Treatment II subjects operated in a market that was identical to Treatment I, with a single important difference. Under Treatment II each subject was allowed to engage in anonymous nonbinding communication with the other seller in his/her respective market. This was accomplished using the VAX PHONE facility on terminals in the LEAP laboratory. The VAX PHONE facility concurrently displayed the messages of a pair of sellers in a particular market in two "windows" on each seller's terminal screen. Through this facility, a dialogue was possible with no "sequencing" rules, yet we could maintain anonymity. Communication was allowed to be continues and voluntary throughout the course of the session. Thus, any communication was nonbinding and market specific.

V. EXPERIMENTAL RESULTS

The results we report are from twenty-four duopoly markets which involved forty-eight subject-firms. The duration of a market was probabilistic and ranged from four to fifteen trading periods. Under both treatments, subjects made simultaneous location choices in a fixed price environment.

Before reporting the observed subject choices, we first establish whether our theoretical model which focuses on symmetric equilibria is appropriate. We use two versions of a Mann-Whitney nonparametric test |Chou 1969~ to conclude whether the sellers in a market were pursuing the same strategies. The first hypothesis we tested was whether there were statistical differences in the means of the "position" choices made by the two sellers in a given market. If we could not reject the hypothesis that the positions chosen by each subject had the same mean, we tested the null hypothesis that observations also had the same dispersion.(9) If we failed to reject both hypotheses, then we interpreted this as support of the conclusion that the strategies of both subjects in a market were the same. Table II shows the two test statistics for each duopoly market.

Our group 5 markets lasted only four trading periods. Since a statistical test with so few observations would have very little meaning, we do not report test statistics for these pairs.(10) For the remaining twenty-one duopoly markets, we reject the null hypothesis of equal population means (significance level = 0.05) in only three cases (two for Treatment I and one for Treatment II) which will be discussed below. Of the eighteen cases that survived the mean test, we reject the null hypothesis of like variances in one case. Of the twenty-one markets we tested, seventeen survived the two tests. Thus, we conclude that a comparison of our results with a symmetric Nash equilibrium prediction is appropriate.

Treatment I

All of our experiments were in a repeated setting with a probabilistic endpoint. As demonstrated in sections II and III, the single-shot equilibrium and the joint-profit-maximizing solution are both elements of the set of symmetric subgame perfect equilibria. Under this treatment both firms chose locations very near the center. The average of all locations chosen under this treatment is 50.668. Figure 4, which shows the frequency with which all possible locations are observed, illustrates the concentration of locations around the midpoint.

TABULAR DATA OMITTED

There are two possible interpretations for such behavior. The firms could be playing max inf strategies. However, they may also be playing the stage game Nash equilibrium positions of roughly two units either side of the center. Note that the stage game Nash equilibrium forever is Pareto dominated by other equilibria.

Under this treatment, the central positions predominate.(11) The central locations prevail even when there were several instances in which one seller unilaterally chose a quartile location for several periods hoping his/her compeer would do likewise. Precisely those pairs in which we are able to reject symmetry in Treatment I often had one seller acting in this way. Hereafter, we shall refer to such behavior as signalling. (The progression of location choices by pair are presented in appendix C.) In these cases, one seller chooses a noncentral location and the other seller either does not respond or chooses an advantageous location very close to the signaller and gains market share. Either response significantly damages the signaller's profit for the period. Sellers 3 and 6 in Group 2 serve as an example. Seller 3 signals the 75 location in periods 11, 12, and 13, with coincident replies of 45, 45, and 60 by the other seller.

In the fourteenth period the signaller chooses 70 and the other seller picks 69 with market shares of 37 percent and 63 percent and profits of -$2.59 and $2.64 respectively. With the costly failure of his attempt to signal a quartile equilibrium, the signaller chooses a "50" location in the fifteenth and final period.

Treatment II

Introducing the opportunity for nonbinding communication between sellers in the same market seems to provide a catalyst for coordination at the joint-profit-maximizing "quartile" equilibrium. The preponderate result under this treatment was for a pair of sellers to line up on the quartile equilibrium and sustain it.

Symmetric strategies are statistically supported in all but one case of those with a sufficient number of observations. The market in which we reject symmetry provides us with an interesting "outlier." Sellers 1 and 4 in group 4 used their communication capability to speculate that we, the experimenters, would not require a subject to actually reimburse us if accumulated profits were negative. Their coordination took the form of one seller locating at an endpoint allowing his/her counterpart to monopolize a large portion of the market. In essence, one subject chose to declare bankruptcy at the end of the experiment. After the subjects were paid, they sought to identify each other and split the lucrative firm's profits. They were able to coordinate on a solution with side payments. Since they were clearly following a different strategy than all other pairs, we decided to exclude the observations from this pair in our Treatment II pool for statistical tests.

The average location choice of our Treatment II pool is 50.870. Figure 5 presents a strikingly different result than Figure 4. The predominant location pair is 25 and 75, which maximizes the point profits of the two sellers.

Two versions of a Mann-Whitney non-parametric test were again used to assess the relative characteristic of the pool of observations from each of the two treatments. When observations by treatment are pooled, the means should not be very different if we have predominantly symmetric strategies in both treatments. If Treatment II sellers are able to sustain greater product differentiation than Treatment I, we would also expect a difference in the dispersion of the two samples. This is precisely the result that our Mann-Whitney Z statistics indicate. We fail to reject the hypothesis that the two samples have the same mean (test statistic of -0.079). However, we reject the hypothesis that the samples have the same dispersion with a test statistic of 12.279. The dominant result is that two very different equilibria occur depending on whether nonbinding communication is allowed.

VI. CONCLUDING REMARKS

For the case of linear demand, we have found the set of strongly symmetric subgame perfect equilibria. We show how this set varies with transportation cost, with reservation price in demand, and with discount factor. Hotelling's concept of "minimal differentiation," though not supported by his own model, is one of many outcomes that can be supported by ours. With the introduction of nonbinding communication, we also find experimental support for the conjectures and findings of the work of Telser |1988~, Gabszewicz and Thisse |1986~, and Eaton and Lipsey |1975~. That is, the set of equilibria contains both collusive and competitive outcomes if the discount factor is sufficiently large.

Without communication, subjects clustered near the center of the market. This occurs despite the fact that there are much more lucrative equilibria. We conjecture that this result is due to the failure of sellers to coordinate when they are unable to communicate. This conjecture is clearly supported by the second set of experiments where we allowed voluntary nonbinding and anonymous communication. Coordination at the joint-profit-maximizing quartile equilibrium was the overwhelming result.

This provides some anecdotal support for institutions like professional trade associations. While price-fixing agreements among individual competitors in such a setting is per se illegal, communication of product attributes is not. We have shown that under certain circumstances nonprice competition should be allowed and even encouraged. Communication facilitated a welfare improving outcome. It allowed firms to coordinate on an equilibrium in which they differentiate their locations, thus more nearly matching consumers' desires.

APPENDIX A

Stage Game

It is convenient to express the function Q(x,y) in terms of a function I: |R.sub.+~ |right arrow~ |R.sub.+~ defined by

|Mathematical Expression Omitted~

This function represents the total quantity sold to consumers who are within a distance of |eta~ one side of a firm. Then

Q(x,y) = I (1 - x) + I |1/2 (x + y)~ if x |is greater than~ -y

and

Q(-y,y) = 1/2 { I (1 - y) + I (1 + y) }.

Note that we assume that the firms share the entire market equally if they are at the same location.

LEMMA: The function I is concave and non-decreasing.

Proof: We will show that for any |eta~ |is greater than or equal to~ ||eta~.sub.0~ |is greater than or equal to~ 0 the function I satisfies |Mathematical Expression Omitted~. The proof for ||eta~.sub.0~ |is greater than or equal to~ |eta~ |is greater than or equal to~ 0 is similar. From the definition of I we see that

|Mathematical Expression Omitted~

where the inequality follows because q is non-increasing by assumption.

I is non-decreasing because q is non-negative valued.

The following theorem demonstrates that the stage game has a unique Nash equilibrium if the transportation cost is not too high. Let |t.sub.0~ = p|q(|p.sub.0~/2)~ - |p.sub.0~. Note that |t.sub.0~ |is greater than~ 0 since q and p are non-increasing and |Mathematical Expression Omitted~.

THEOREM 1: For the single-period location choice game:

a) If 0 |is less than~ t |is less than or equal to~ |t.sub.0~ there is a unique Nash equilibrium, x = y = 0.

b) If |Mathematical Expression Omitted~ there is a unique Nash equilibrium, x = y |is greater than~ 0.

c) If |Mathematical Expression Omitted~ there are multiple Nash equilibrium in which the firms are local monopolists.

Proof: Consider first cases (b) and (c) in which the firms have distinct locations in equilibrium. A necessary condition for such a situation to be an equilibrium is that neither firm can increase sales by changing location slightly, i.e.,

|Q.sub.1~(x,y) = |Q.sub.1~(x,y) = 0,

where |Q.sub.1~ denotes the partial derivative with respect to the first argument. This pair of equations is also sufficient given the concavity of the function I. Computing the derivatives and rearranging gives

(A.1) q||p.sub.0~ + t(x + y)/2~ = 2q||p.sub.0~ + t(1-x)~ = 2q||p.sub.0~ + t(1-y)~

There are two types of solution to these equations. Either each term is zero or each is positive.

Suppose that each term is zero. Then the price at each location exceeds the choke price, i.e.

|Mathematical Expression Omitted~.

In particular, from (A.2) demand at the midpoint between the firms is zero, so each firm is a local monopolist.

Simple algebra reveals that these three inequalities can only be satisfied if |Mathematical Expression Omitted~. Since the inequalities are linear, the set of values of (x,y) which satisfy them is convex.

Suppose instead that each term in equation (1) is positive. This can only occur if |Mathematical Expression Omitted~. Since q is strictly decreasing when it is positive valued, it must be the case that x = y. We need only show that the equation q(|p.sub.0~ + tx) = 2q(|p.sub.0~ + t(1 - x)) has a unique solution. When x = 1/2, the right-hand side strictly exceeds the left since |Mathematical Expression Omitted~. For values of x between 0 and 1/2 the left-hand side is strictly decreasing, while the right is strictly increasing. Since q is continuous, this equation has a unique solution x |is greater than~ 0 providing that q(|p.sub.0~) |is greater than~ 2 q(|p.sub.0~ + t), or equivalently if t |is greater than~ |t.sub.0~.

Finally consider case (a), in which 0 |is less than~ t |is less than or equal to~ |t.sub.0~. Since I is concave it is an equilibrium for both firms to locate at the center (i.e., x = y = 0) providing that |Q.sub.1~(0,0) |is less than or equal to~ 0. Evaluating the derivative gives

1/2 q(|p.sub.0~) - q(|p.sub.0~ + t) |is less than or equal to~ 0,

or equivalently t |is less than or equal to~ |t.sub.0~.

We now show that a firm achieves its security level by locating at the center.

THEOREM 2: The security level |Mathematical Expression Omitted~ is Q(0,0).

Proof: One firm can most adversely affect the sales of another by locating a small distance away on the long side of the market. That is:

|Mathematical Expression Omitted~

From the lemma this infimum is maximized by choosing x = 0.

The following theorem shows that the highest aggregate sales are achieved when the firms locate at the quartiles of the market.

THEOREM 3: The joint-sales-maximizing positions are x = y = 1/2.

Proof: We must show that x = y = 1/2 solves

|Mathematical Expression Omitted~

Since Q is concave and the feasible region is convex the first-order condition is sufficient for a maximum. The interior first-order conditions are

q||p.sub.0~ + t(1 - x)~ = q||p.sub.0~ + t(x + y)/2~

q||p.sub.0~ + t(1 - y)~ = q||p.sub.0~ + t(x + y)/2~.

These are satisfied by x = y = 1/2.

A lower bound on average sales in an infinitely repeated game is |Mathematical Expression Omitted~. The next theorem shows that a firm can punish its rival by choosing y = 0.

THEOREM 4: The worst punishment a firm can inflict on its rival is Q(0,0), achieved by locating at the center.

Proof: The proof is somewhat delicate since Q is not continuous. The supremum of Q(x,y) with respect to x is

|Mathematical Expression Omitted~

Since I is non-decreasing this supremum is minimized by choosing y = 0.

APPENDIX B

Infinitely Repeated Game With Linear Demand

This appendix solves for the set of strongly symmetric subgame perfect equilibria (SSE) in an infinitely repeated version of the game in which each firm discounts its payoff with a discount factor of |delta~|is an element of~(0,1).(11) The reader can refer to Cronshaw |1989~ and Cronshaw and Luenberger |1990~ for details of the methods used here.

Let |Mathematical Expression Omitted~. Then the demand at a distance |xi~ from the firm is |Mathematical Expression Omitted~, where |Mathematical Expression Omitted~. We will restrict attention to the case |Mathematical Expression Omitted~ or equivalent 1/2 |is less than~ |psi~ |is less than~ 2. For this demand curve the function I is given by

(B.1) I(|eta~) = t { |psi~|eta~ - (1/2) ||eta~.sup.2~ for 0 |is less than or equal to~ |eta~ |is less than or equal to~ |psi~.

The payoff(12) to each firm if they both locate a distance x from the center is

(B.2) |pi~(x) = (1/t) {I(1-x) + I (x)} = |psi~ - (1/4) - ||x - (1/2)~.sup.2~.

Note that |pi~ is unimodal with a maximum at x = 1/2.

The single-period best response payoff to a position of y |is greater than or equal to~ 0 by a firm's opponent is

|Mathematical Expression Omitted~

It is straight forward to show that the best response position is x(y) = (1/5){4 -2|psi~ - y} and that the corresponding payoff is

|Mathematical Expression Omitted~.

Note that |pi~* is increasing between 0 and 1/2 given the parameter restrictions on |psi~.

The temptation to cheat is the difference between |pi~* (x) and |pi~(x), namely

|delta~(x) = |pi~*(x) - |pi~(x) = (9/10) |(x - |x.sup.N~).sup.2~

where

|x.sup.N~ = (1/3) (2 - |psi~)

is the Nash equilibrium position in the stage game.(13)

We can readily obtain bounds on the set of SSE payoffs. As shown in appendix A, the largest possible payoff is that obtained by each firm choosing a position of 1/2. Thus the average discounted payoff in any path through the game (equilibrium or not) is bounded above by |pi~(1/2)=(1/t) Q(1/2,1/2).

In any period of the game a firm can always play a single-period best response to its opponent's position in that period. Therefore in any equilibrium of the repeated game the average discounted payoff of each firm must be at least |Mathematical Expression Omitted~. Since |pi~* is increasing, this means that the set of SSE payoffs is bounded below by |pi~*(0).

Thus the set of SSE payoffs is contained in the interval ||pi~*(0), |pi~(1/2)~. This interval is often referred to as the set of individually rational and feasible symmetric payoffs.

Since |delta~ is convex, the set of SSE payoffs is the interval bounded by the largest fixed pair (see Cronshaw and Luenberger |1990~ Theorem 5). We find the largest fixed pair by first finding the functions f and g defined below and then solving for the maximal level of deterrence ||beta~.sub.|delta~~ (see Cronshaw and Luenberger |1990~ Lemmas 3 and 5).

The function f:|R.sub.+~ |right arrow~ R gives the largest symmetric stage game payoff when the firms are restricted to choosing a position x for which the temptation to cheat is not too large. The function g:|R.sub.+~ |right arrow~ R gives the min-max payoff with the same restriction. That is, for |beta~ |is greater than or equal to~ 0,

|Mathematical Expression Omitted~.

Since |delta~ is convex, the constraint |delta~(x) |is less than or equal to~ |beta~ defines an interval of incentive-compatible actions ||x.sub.-~(|beta~),|x.sub.+~)|beta~)~ = {x / |delta~(x) |is less than or equal to~ |beta~}

where

|x.sub.-~(|beta~) = |x.sup.N~ - |square root of~10|beta~/9 and |x.sub.+~(|beta~) = |x.sup.N~ + |square root of~10|beta~/9.

Let ||beta~.sub.1~ be the value of |beta~ at which |x.sub.-~(|beta~) = 0. Let ||beta~.sub.2~ be the value of |beta~ at which |x.sub.+~(|beta~) = 1/2.

Since |pi~* is increasing on the interval |0,1/2~, the minimum in the definition of g is achieved at max{|x.sub.-~(|beta~),0}. Thus, g(|beta~) = |pi~*(max{|x.sub.-~(|beta~),0}).

Now consider the solution for f. From equation (B.2) the function |pi~ is concave with a maximum at x = 1/2, the collusive position. Given the parameter restriction, the stage game Nash position |x.sup.N~ is between 0 and 1/2. Thus, for 0 |is less than or equal to~ |beta~ |is less than~ ||beta~.sub.2~, we have |x.sup.N~ |is less than or equal to~ |x.sub.+~(|beta~) |is less than~ 1/2, so that the maximum in the definition of f is attained when x = |x.sub.+~(|beta~). For |beta~ |is greater than or equal to~ ||beta~.sub.2~, |x.sub.+~(|beta~) |is greater than or equal to~ 1/2, so that the maximum is attained when x = 1/2. That is, f(|beta~) = |pi~(min{|x.sub.+~(|beta~), 1/2}).

For a given discount factor |delta~, the set of SSE payoffs is the interval |g(||beta~.sub.|delta~~, f(||beta~.sub.|delta~)~ where ||beta~.sub.|delta~~ is the largest scalar that solves

(B.4) ||beta~.sub.|delta~~ = ||delta~/(1-|delta~)~ |f(||beta~.sub.|delta~~) - g(||beta~.sub.|delta~~)~,

(Cronshaw and Luenberger, Theorem 5). Furthermore, ||beta~.sub.|delta~~ is increasing in the discount factor (Cronshaw and Luenberger, Lemma 7). There are three classes of solutions to equation (B.4), depending on the discount factor.(14) Let

||delta~.sub.1~ = 9(2-|psi~)/20 (2 |psi~ - 1) and ||delta~.sub.2~ = |(2|psi~-1).sup.2~ / (12 |psi~ - 5).

For 0 |is less than~ |delta~ |is less than~ ||delta~.sub.1~, ||beta~.sub.|delta~~ is small enough that |x.sub.-~(||beta~.sub.|delta~~) |is greater than~ 0 and |x.sub.+~(||beta~.sub.|delta~~) |is less than~ 1/2. In this case, the set of SSE payoffs is strictly contained in the open interval (|pi~*(0),|pi~(1/2)). For ||delta~.sub.2~ |is less than or equal to~ |delta~ |is less than~ 1,||beta~.sub.|delta~~ is large enough that |x.sub.-~(||beta~.sub.|delta~~) |is less than or equal to~ 0 and |x.sub.+~(||beta~.sub.|delta~~) |is greater than or equal to~ 1/2. In this case the set of SSE payoffs is the entire interval ||pi~*(0),|pi~(1/2)~. For intermediate values of |delta~ between ||delta~.sub.1~ and ||delta~.sub.2~, |x.sub.-~(||beta~.sub.|delta~~) |is less than or equal to~ 0 but |x.sub.+~ (||beta~.sub.|delta~~) |is less than~ 1/2, so that the set of payoffs is the interval ||pi~*(0),|pi~||x.sub.+~(||beta~.sub.|delta~~)~~ where |pi~||x.sub.+~(||beta~.sub.|delta~~)~ |is less than~ |pi~(1/2).

Table III shows the solution for several values of the parameter |psi~, including the value used in the experiments. Figure 6 shows the set of SSE payoffs as a function of the discount factor for the parameters of the experimental design.

Stationary SSE:

An SSE is called stationary if the firms choose the same location in every period on the equilibrium path. It is well known that playing the stage game equilibrium in every period is a stationary SSE, and that the best SSE is also stationary. We will now show that the worst SSE is not stationary.(15)

Let |Mathematical Expression Omitted~ be the worst SSE payoff. The average payoff from a stationary position x is |pi~(x). A position x is a stationary SSE if and only if

|Mathematical Expression Omitted~

This follows from Abreu's |1986~ notion of supporting an equilibrium by the worst possible punishment. From Cronshaw and Luenberger there is a position |Mathematical Expression Omitted~ such that

|Mathematical Expression Omitted~

Suppose that |Mathematical Expression Omitted~ is a stationary SSE. Then since |pi~*(x) |is greater than or equal to~ |pi~(x) for any x

|Mathematical Expression Omitted~

That is, |Mathematical Expression Omitted~ is the Nash equilibrium in the stage game, and |Mathematical Expression Omitted~. However, this is a contradiction since for any positive discount factor, |Mathematical Expression Omitted~.

JAMIE BROWN-KRUSE, MARK B. CRONSHAW, and DAVID J. SCHENK Department of Economics, University of Colorado, Boulder, CO 80309. We would like to acknowledge the financial support of the Laboratory for Economics and Psychology (LEAP) of the University of Colorado, Boulder, and the helpful suggestions and contributions of Michael McKee and William Schulze and anonymous referees. Of course, any errors that remain are the sole responsibility of the authors.

1. The choke price is the vertical intercept of the demand curve, i.e., the price at which demand is zero.

2. We are grateful to a referee for bringing this reference to our attention.

3. Actually max inf due to discontinuities.

4. The symmetry restriction is for tractability. We do not know of any solution technique for asymmetric equilibria for an arbitrary discount factor.

5. Farrell showed that such communication produces a degree of coordination that would otherwise be much more difficult to obtain.

6. A local monopolist is a firm whose market area does not abut that of its rival. One suspects that entry by a third firm would be profitable in this case. Eaton and Lipsey consider the case with more than two firms.

7. We use the word "payoff" to mean sales quantity, since, as noted above, firms want to maximize the sales quantity.

8. The subject sellers are not provided with the exact characterization of the individual consumer's demand curve.

9. The Mann-Whitney test for differences in means is a fairly standard nonparametric test. See Chou |1969~ for a description. u = |n.sub.1~|n.sub.2~ + |n.sub.1~(|n.sub.1~ + 1)/2-|R.sub.1~ where |n.sub.1~, |n.sub.2~ are the sizes of the two samples.

By simply changing the rule for assigning ranks we can also test the null hypothesis of identical populations against the alternative that they have unequal dispersions. As before, the values are arranged in increasing (or decreasing) order, but are ranked from both ends to the middle. The smallest value receives a 1, the largest and second largest assigned a 2 and a 3, respectively, the second and third smallest get a 4 and a 5, respectively, and so on. Using the sums from this alternative ranking we can construct a test statistic to test for unequal dispersion.

10. We did however, include group 5 in our pool of observations for statistical tests between treatments.

11. Of the locations 48 to 52 that were chosen, 50 was chosen 49 percent of the time. The other locations in this set were chosen less frequently (52--9 percent, 51--22 percent, 49--17 percent and 48--3 percent).

11. In a strongly symmetric subgame perfect equilibrium firms locate symmetrically about the center both on and off the equilibrium path.

12. It is convenient to work in terms of the actual sales divided by t. We will refer to the actual sales divided by t as the payoff.

13. Note that the parameter restriction on |psi~ ensures that |x.sup.N~ is between the center and the quartile.

14. The following solution is valid providing that 5/4 |is less than or equal to~ |psi~ |is less than~ 2, which ensures that ||delta~.sub.1~ |is less than or equal to~ ||delta~.sub.2~. If 1/2 |is less than~ |psi~ |is less than~ 5/4 the solution is similar, but for an intermediate range of discount factors collusion is an SSE while the min-max payoff is not.

15. Abreu has shown that the worst SSE is not stationary in quantity setting oligopolies with identical firms and constant average cost.

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