ABSTRACT: In a 1987 article, Thomas Campbell argues that predation can be a credible threat in spatial markets. We contest this view.  [*1015]  INTRODUCTION

Markets institutionalize conflict. Competition goads peaceable businesses into price wars, and antitrust law deliberately aggravates matters by punishing truces. But antitrust law also worries if conflict becomes "predatory." Some have thought that antitrust has set itself upon a hopelessly paradoxical quest, seeking simultaneously to promote and yet to suppress conflict. n1 In the last few decades antitrust commentators thus have heaped more attention on the question of predatory business conduct than on any other.
 
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n1 See, e.g., R. Bork, The Antitrust Paradox: A Policy at War with Itself (1978).
 
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In the late 1950s, a reaction began in the academy to the view that predatory abuses were so widespread as not to require precise definition. Writing in the very first issue of the Journal of Law and Economics -- the house organ of the Chicago School of antitrust criticism -- Professor John McGee argued that a key antitrust precedent had mistaken desirable competition for predation. n2 Others soon chimed in, bagging one debunked case-law example of supposed predation after another by routinely finding only beneficial rivalry, whining losers, and perverse antitrust punishment. n3
 
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n2 McGee, Predatory Price Cutting: The Standard Oil (N.J.) Case, 1 J.L. & Econ. 137 (1958); see also McGee, Predatory Pricing Revisited, 23 J.L. & Econ. 289 (1980).

n3 See F. Scherer, Industrial Market Structure and Economic Performance 337 & nn. 11-12 (2d ed. 1980) (collecting studies). Scherer criticizes various aspects of these studies but calls them "persuasive" in support of the conclusion that predatory pricing is "rare, ineffective, or the symptom of a competitive struggle desired as little by the alleged predators as by its victims." Id. at 337. However, he also cites two published examples involving steamships -- one in the 19th century and one in the 20th -- for which inference of predation "appears indisputable." Id.; see also Memphis Steam Laundry-Cleaners v. Lindsey, 192 Miss. 224, 5 So. 2d 227 (1941).
 
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By the mid 1970s, this skepticism about predation had reached the East Coast. Harvard Professors Philip Areeda and Donald Turner published a famous article urging that courts constrict the definition of illegal predation to avoid deterring the very competition the antitrust laws  [*1016]  sought to promote. n4 These eminent scholars argued for a definition of predatory pricing that would make recovery more difficult for plaintiffs claiming predation. n5 By the early 1980s, the predation skeptics took the logical last step: complete rejection of "predatory pricing" as something relevant to antitrust law. Professor (now Judge) Frank Easterbrook argued that antitrust injured itself by worrying at all about predatory pricing. n6 Predation was difficult to distinguish from competition, he argued, and the blizzard of differing academic positions on predation arose "for the same reason that 600 years ago there were a thousand positions on what dragons looked like." n7
 
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n4 Areeda & Turner, Predatory Pricing and Related Practices Under Section 2 of the Sherman Act, 88 Harv. L. Rev. 697 (1975).

n5 Id. at 699-700. The Areeda & Turner average variable cost proposal loosed an avalanche of critical commentary. See, e.g., Liebeler, Whither Predatory Pricing? From Areeda and Turner to Matsushita, 61 Notre Dame L. Rev. 1052, 1097-98 (1986) (listing 25 articles). A court recently described their proposal as "like the Venus de Milo: it is much admired and often discussed, but rarely embraced." McGahee v. Northern Propane Gas Co., 858 F.2d 1487, 1495 (11th Cir. 1988) (footnotes omitted).

n6 Easterbrook, Predatory Strategies and Counterstrategies, 48 U. Chi. L. Rev. 263 (1981).

n7 Id. at 263-64.
 
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This increasing academic skepticism moved the law. A generation ago the Supreme Court disgusted contemporary Chicagoan critics with its willingness to say that low prices were predatory. n8 But by 1986, Matsushita Electric Industrial Co. v. Zenith Radio n9 suggested that a Supreme Court majority had become quite receptive to the skeptics' view. A Court majority reported "a consensus among commentators that predatory pricing schemes are rarely tried, and even more rarely successful," n10 and it cautioned that loose judicial indulgence of predation claims would "chill the very conduct the antitrust laws are designed to protect." n11 The following Term the Court reiterated these points and hinted that it would view future claims of predatory pricing with great disfavor. n12
 
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n8 See Utah Pie Co. v. Continental Baking Co., 386 U.S. 685, 702-04 (1967); Bowman, Restraint of Trade by the Supreme Court: The Utah Pie Case, 77 Yale L.J. 70 (1967).

n9 475 U.S. 574 (1986).

n10 Id. at 589.

n11 Id. at 594. Matsushita stopped well short of simple and wholesale espousal of Easterbrook's skepticism. The case dealt only with alleged predatory pricing by a group -- undoubtedly a less likely event than predation by a single dominant firm facing no collective-action problem -- and the Court stated flatly that it did not aim to resolve the debate over the proper legal definition of a predatory price. Id. at 584 n.8.

n12 Cargill, Inc. v. Monfort of Colo., Inc., 479 U.S. 104, 119-21 n.15, 121-22 n.17 (1986). The Court nonetheless commented that "[w]hile firms may engage in the practice [of predatory pricing] only infrequently, there is ample evidence suggesting that the practice does occur." Id. at 121 & n.16 (citing two economics articles). This claim of "ample evidence" is a bit ironic. One of the sources that the Court cited stresses the empirical scarcity rather than the definite existence of predation -- as is evident even from the article's title. See Koller, The Myth of Predatory Pricing: An Empirical Study, Antitrust L. & Econ. Rev., Summer 1971, at 105, 112 (of approximately 123 allegations of predation since the passage of the Sherman Act, only five produced sufficient evidence both to evaluate and to substantiate the claims); cf. Elzinga, Predatory Pricing: The Case of the Gunpowder Trust, 13 J.L. & Econ. 223 (1970) (largely contradicting one of Koller's five examples of predation). The Court's second citation supports the notion that litigants can use suit under the antitrust laws as a predatory tactic. Miller, Comments on Baumol and Ordover, 28 J.L. & Econ. 267, 267 (1985). But the author--chair of the Federal Trade Commission during the Reagan Administration -- makes clear his own dim view of the frequency of predatory pricing in another article published in the same volume. See Miller & Pautler, Predation: The Changing View in Economics and the Law, 28 J.L. & Econ. 495, 496 (1985) ("Of course, any decently trained expert -- lawyer or economist -- will counsel that even in the absence of laws against predation such a strategy would seldom be successful."); cf. Liebeler, supra note 5, at 1052 (examining predatory pricing decisions after 1975 and concluding that "[n]ot one of the cases is a real predatory pricing case").
 
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 [*1017]  Professor -- now Congressman -- Thomas J. Campbell recently disputed these views. In an article of potential importance, Campbell argues that the general skepticism about predation is significantly mistaken. n13 Campbell agrees with the skeptics that predation is unlikely in markets with standardized products. n14 But he argues that predation can succeed in markets in which products are just a little different from each other n15 -- which is to say in most real markets. Building upon the spatial oligopoly literature that followed Hotelling's famous work, n16 Campbell proposes a model in which a predator firm can successfully inflict disproportionate costs on rival firms by making their products similar to those of their prey, in terms of either quality or sales location. n17 On the basis of this analysis, Campbell recommends that judges expand antitrust liability to suppress this evil, n18 thus implying that predation is worrisomely common.
 
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n13 Campbell, Predation and Competition in Antitrust: The Case of Nonfungible Goods, 87 Colum. L. Rev. 1625 (1987).

n14 Id. at 1626-30.

n15 Id. at 1646-48.

n16 Hotelling, Stability in Competition, 39 Econ. J. 41 (1929); see also sources cited infra note 36.

n17 Campbell, supra note 13, at 1646.

n18 Id. at 1654-55.
 
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This Commentary shows that judges will err if they take Campbell's advice. His results rely upon a delicate, quirky, and incomplete model that yields drastically different and more conventional results under more realistic assumptions. Predation indeed may be common, but Campbell's model gives us no reason to think so.

We proceed as follows. Part I situates the debate within the economic literature on predation, recounting Hotelling's model and the use that Campbell makes of it. Parts II and III present two central criticisms of Campbell's model: it neglects the role of price, and it is too vaguely specified to be reliable. Part IV sketches the substantial problems that judges would confront in applying Campbell's model even if he were right.

 [*1018]  I. CAMPBELL'S ARGUMENT
 
A. How Predation Might Work

Predatory practices would warrant antitrust concern if they could produce inefficient or unfair monopoly pricing. n19 They might work in two different ways. First, a predatory firm might begin what amounts to an expensive money-losing contest with the prey -- for example, by setting price below cost for a lengthy period. The problem for the predator is to show that its threat to continue to lose money is credible: that the predator will maintain low prices until the prey exits the market. The predator for some reason might have more or cheaper capital available and hence be able to incur losses longer than the prey. n20 Or the predator might commit to incur those losses by some irreversible action. n21 Or the predator might wish to retain a reputation for ferocity. n22 Second, predation might work if the predator could find some method that inflicts costs on rivals at no cost to itself. An example is if at trivial cost the predator can convince a government to ban, tax, or regulate the prey's product. n23
 
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n19 There is controversy over the proper goal of antitrust law. For purposes of our argument here, we need not choose between the goals of economic efficiency or of distributive justice for consumers. See Wiley, "After Chicago": An Exaggerated Demise?, 1986 Duke L.J. 1003 (discussing the operational congruence between these two different antitrust goals).

n20 See Telser, Cutthroat Competition and the Long Purse, 9 J.L. & Econ. 259, 260-63 (1966).

n21 See Spence, Entry, Capacity, Investment, and Oligopolistic Pricing, 8 Bell J. Econ. 534 (1977).

n22 See Kreps, Milgrom, Roberts & Wilson, Rational Cooperation in the Finitely Repeated Prisoners' Dilemma, 27 J. Econ. Theory 245 (1982).

n23 See, e.g., Allied Tube & Conduit Corp. v. Indian Head, Inc., 108 S. Ct. 1931, 1935 (1988) (employees of steel conduit manufacturers pack a meeting of the National Fire Protection Association and vote against approving plastic conduit); see also Salop & Scheffman, Raising Rivals' Costs, Am. Econ. Rev. 267, 267-71 (1982) (analyzing strategies that are "virtually costless to the predator").
 
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Campbell agrees with critics who claim that the first costly kind of predation strategy confronts credibility problems strong enough to render its legal prohibition nearly redundant. n24 He therefore focuses  [*1019]  on the second kind of predation and claims to discover a new and troublingly effective costless predatory tactic. n25 It is simple: a predator firm changes either its product characteristics or its sales location to imitate its victim's product or location. n26 Campbell says that by moving in on a rival, a predator can keep its own revenues about or exactly the same and can return to its former style or location after the fight is over. n27 But by moving closer, it will reduce its rival's revenues, causing the rival to exit or go bankrupt. n28 If correct, Campbell's result is remarkable. But his analysis is correct only if the move-in-for-the-kill strategy truly offers a no or low cost weapon under real and common conditions. To show why it does not, we first explain the Hotelling model that Campbell takes as the foundation for his own work.
 
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n24 Campbell, supra note 13, at 1628. The central credibility problem is that it is costly for the incumbent firm to carry out the threat of predation. If the incumbent could bind itself to carry out the threat (using something like the "doomsday device" in Dr. Strangelove), the threat would dissuade rivals and the incumbent would not actually have to carry it out. But if the incumbent cannot so bind itself, then after the rival ignores the threat by entering, the incumbent has no incentive to follow through with an action that is then not only costly, but useless.

The relevant cost is the absolute sum borne by the threatening party, not the relative cost, which might be higher or lower than that inflicted on the threatened party. This point is not always well understood. See id. at 1653 ("Once it is granted that an incumbent has the power to impose asymmetric costs on an entrant, the deterrence of that entry in many cases can be expected.") (emphasis added). It is not enough for the entrant to suffer greater costs; what matters is that the incumbent can carry out an ignored threat at negligible cost. For elaboration on the credibility of threats, see E. Rasmusen, Games and Information: An Introduction to Game Theory 83-106 (1989) (citing further references); T. Schelling, The Strategy of Conflict 35-52 (1960).

n25 Campbell, supra note 13, at 1625.

n26 Id. at 1646-47.

n27 Id. at 1647-48.

n28 Id. at 1654.
 
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B. The Hotelling Approach

The question "what is the market" has long been central in antitrust. In most markets, products that compete with each other differ somewhat in their characteristics -- juice from real lemons versus juice from plastic lemons, for instance, or cornflakes versus oatmeal. Even when products are physically identical, they may be sold at different locations -- steel bars from Pittsburgh versus steel bars from Tokyo. This product differentiation complicates legal and economic analysis. When the difficulty is recognized, the solution has often been to redefine the market to include close substitutes, and then to analyze the market as if the substitutes were perfect rather than merely close.

As long ago as 1929, however, Hotelling introduced a way explicitly to model product differentiation. n29 The Hotelling approach treats any product characteristic as equivalent to geographical location. This approach is literally correct for the point of sale. For example, suppose Figure 1 (which resembles Campbell's Figure 1) n30 depicts a long beach with sunbathers located evenly along it and two hot dog vendors, A and B, respectively located at points 1/4 and 3/4.
 
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n29 Hotelling, supra note 16, at 45. Although Hotelling's article is justly famous, one section does contain a major error concerning the existence of equilibrium that is explained in d'Aspremont, Gabszewicz & Thisse, On Hotelling's "Stability in Competition," 47 Econometrica 1145 (1979). The error is that for a significant range of possible parameter values there is no certain equilibrium outcome in the model with fixed locations and variable prices; the firms rather choose prices randomly with probabilities that depend on the parameter values. Id. at 1147-48.

n30 See Campbell, supra note 13, at 1639. Following Hotelling, we put endpoints on the line -- an apparently trivial but actually significant matter to which we return. See infra text accompanying notes 52-56.
 
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If the prices were the same, sunbathers near Vendor A would buy from it. But because Vendor B sells a close substitute, Vendor A cannot raise its price too high or it will begin to lose customers to Vendor B. Vendor A has some -- but not much -- market power. n31
 
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n31 One can expand this simple one-dimensional model (in which each firm has two neighbors, one on each side) into more dimensions. As the model adds dimensions, each firm gains an increasing number of neighbors. See Campbell, supra note 13, at 1646 & n.81. The increasing number of neighbors in multidimension models does not imply, however, that firms in these models have less market power. Local proximity, not the global number of firms, determines the extent of each firm's market power in spatial models. Proximity in turn depends on equilibrium conditions. See infra text accompanying notes 57-60.
 
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The Hotelling approach can apply to product characteristics as well as to sales location. To use Hotelling's example, suppose that Figure 1 now represents a continuum of sweetness levels for cider. The ends of the continuum then could represent unbearably sweet and unbearably tart cider. Different consumers prefer different levels of tartness, and we can imagine them to be distributed evenly along the continuum. Suppose further that the cider companies produce different products: Company A produces a sweeter cider and Company B a tarter cider. n32
 
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n32 This transition from modeling location to modeling characteristics requires that "distance" be measured, not in units of literal distance like feet, but in the dollar value of consumer disutility of "traveling" to accept products that are not exactly what the consumer would most prefer. For instance, a consumer willing to pay $ 10/jug for very sweet cider might be willing to pay only $ 9 for a jug with a spoonful less sugar per jug. Other consumers, however, also might prefer very sweet cider but might be willing to pay only $ 8 or $ 7 for the less sweet jug. This issue -- that consumer preference rankings may be ordinal rather than cardinal -- does not affect the criticisms we present.
 
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Models like this can illuminate two questions: where the sellers choose to locate on the line of product characteristics, and how much they charge their customers. First, where will the sellers choose to locate? As a way of isolating the location decision, assume that both firms have identical prices. This rather drastic assumption frees competitors from concern about trading off a higher price against a lower market share, thus leaving the location decision as their sole choice. We can predict that in maximizing market share, both sellers will choose the same location: the midpoint 1/2. To see why, consider what happens if Vendors A and B choose two different points; say, 1/2 and 3/4. Vendor A will attract all of the customers from 0 to 1/2, plus half of the customers from 1/2 to 3/4, for a total of five-eighths of the market. This situation leaves Vendor B with only three-eighths of the market. Moreover,  [*1021]  if Vendor A were to move to just left of 3/4, it could capture almost three-fourths of the market unless Vendor B responded. If both vendors are next to each other at 1/2 and split the market, however, neither one can benefit by moving. The one that did not move would retain half of the entire market plus half of the distance to wherever the other seller had moved. Hence, if price is not a variable, both firms locate at 1/2.

Second, what prices will the sellers choose? If both sellers locate at the same place, they pose the standard question of what price duopolists will charge. n33 If the two sellers compete vigorously in price, their competition eventually will reduce price to marginal cost. If they do not, the model needs more structure to predict a result. The special feature of the location model is that if the two sellers have different locations, vigorous and self-interested competition does not reduce price to marginal cost. Assume that Vendors A and B for some reason locate at points 1/4 and 1 in Figure 1, and that their unit production costs do not increase with the quantity sold. Had these firms been unable to vary their prices, we could have predicted that Vendor A would have had five- eighths of the market -- all the customers to its left (one quarter of the market) plus half of the customers between it and Vendor B (half of three quarters). But that prediction no longer holds, for now the firms can change price and it is price that determines the market shares. If Vendor A sets prices only slightly higher than Vendor B, for instance, then Vendor A will retain all the customers in the segment from 0 to 1/4, plus those from 1/4 to something less than a distance of three-eighths to its right (something less, because Vendor A charges a higher price).
 
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n33 Duopoly issues (such as whether the firms compete in price or in quantities, whether the competition is static or dynamic, and whether competition will or will not drive prices down to costs) go back as far as the criticism by Bertrand of Augustin Cournot's model. See Bertrand, Book Review, 48 Journal des Savants 499, 503 (1883) (reviewing L. Walras, Theorie mathematique de la richesse sociale (1883) and A. Cournot, Recherches sur les principes mathematiques de la theorie des richesses (1838)). For a recent treatment, see J. Tirole, The Theory of Industrial Organization (1988).
 
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So far we have specified nothing about the intensity of customer preference for close location. If we did so, we could calculate how many customers Vendor A loses when it raises its price by some amount. Then we also could calculate an equilibrium pair of prices, and we would find that Vendor A indeed would choose a higher price than Vendor B. n34 The reason is that if Vendor A chooses a lower price it would trade off losses in its large, relatively safe market from 0 to 1/4 against gains in market share in the segment from 1/4 to 1, while Vendor B has no safe market to trade off against the contested market.
 
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n34 See E. Rasmusen, supra note 24, at 273; Hotelling, supra note 16.
 
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Both Vendor A and Vendor B will make supranormal profits, because  [*1022]  each is alone at a location and can raise its price above marginal cost without losing all its customers. This result is partly due to our assumption that there are only two sellers (otherwise, we would expect profits to attract entry), and it contrasts with the outcome when both sellers are located at 1/2. When both sellers are at the same location, each captures half the market -- but neither has supranormal profits, because the two compete prices down to marginal cost. n35 Thus, even Vendor B is better off when the locations differ. Two sellers close to each other compete prices down to a lower level than if they are further apart.
 
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n35 This conclusion is subject to the usual duopoly concerns. See supra note 33.
 
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It is apparent that any number of logical extensions can introduce bits of realism to the simple model that Hotelling offered. A good many commentators since 1929 have sought to do so. n36 This literature discloses that location models are tricky and delicate; one must specify assumptions carefully and consider whether they truly are appropriate to the situation at issue. Furthermore, key assumptions may not be plain. An example of one that is key but not obvious is the basic assumption that the characteristic at issue can be modeled as geographic location. This assumption may fit cider, but it is not appropriate for a characteristic like car color, because blue may be a close second to red in one consumer's ranking but a distant second to black for another consumer. We cannot aggregate consumer preferences about car color into a single continuum. The same limitation excludes product characteristics such as advertised image (scotch or soap that is "lighthearted" versus "sexy" versus "tasteful" versus "intellectual") and product names ("Bud" versus "Miller" versus "Foster's"). n37 Location models  [*1023]  require care. They are useful but abusable.
 
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n36 See, e.g., M. Greenhut & H. Ohta, Theory of Spatial Pricing and Market Areas 64, 152-55 (1975); E. Rasmusen, supra note 24, at 269-73, 282; Eaton & Lipsey, The Principle of Minimum Differentiation Reconsidered: Some New Developments in the Theory of Spatial Competition, 42 Rev. Econ. Stud. 27 (1975).

Location models have relevance to a number of different areas in antitrust, and were actually used, though unsuccessfully, in the FTC breakfast cereals case to try to show that sellers could have positive profits even in equilibrium. See Schmalensee, Entry Deterrence in the Ready-to-Eat Breakfast Cereal Industry, 9 Bell J. Econ. 305, 308-310 (1978). Schmalensee suggested that incumbent firms could crowd the product space with their own products, thus deterring entry. Id. at 314. Judd points out that the entrant could foresee that the incumbents would pull out some of their products to avoid price wars with the entrant's similar product. See Judd, Credible Spatial Preemption, 16 Rand J. Econ. 153, 154 (1985). The basis of Judd's criticism is that competition between products close to each other in product space drives down price--the point we emphasize. Id. at 153-54.

n37 By suggesting that this approach applies to product names generally, Campbell underestimates the full extent of these limitations:

In some industries the brand name of the product is a characteristic which consumers prize, independent of any other characteristic. . . . [A]ny industry is thus capable of fitting into heterogenous product analysis--so long as some product characteristic (even if only the name) is valued by consumers, and cannot be identically replicated by other firms.

Campbell, supra note 13, at 1640-41 (emphasis added).

Campbell correctly notes that "the characteristics must not be such that consumers agree on what is better and worse." Id. at 1639. The Hotelling approach could apply to trademarks or advertised images if (1) all consumers agree that marks or images represent different points along the continuum of a single characteristic, and (2) the characteristic is "not such that consumers agree on what is better and worse." These highly restrictive conditions do not utterly block the theoretical possibility that the Hotelling approach could apply to some trademarks or advertised images. But we can think of no real examples that survive these restrictions.
 
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C. Campbell's Analysis

We have explained how firms locate in a fixed-price model. Campbell asks what happens if an entrant tries to disrupt that equilibrium. Campbell argues that when a new firm moves in between two incumbents, one or both of the incumbents can reduce the entrant's market share--without losing any sales revenue themselves--by moving next to the entrant. n38 Seemingly, the tactic allows an existing firm costlessly to drive new entrants from the market. A market for differentiated products has a special property: at any one moment, only two of all the firms in the market compete for any one consumer. Consumers will choose between only the two products immediately to the right and left of their ideal preference locations (when all products carry the same price tag). Campbell argues that predation becomes credible once competition becomes local in this way.
 
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n38 Following Campbell, we generally limit our analysis to firms that make but a single product. In some instances, the possibility that a firm could make a range of products in the same market would complicate and further undermine Campbell's analysis. See infra text accompanying note 61.

Campbell expands his spatial treatment to two dimensions. See Campbell, supra note 13, at 1643-46. We stick to the simple case of a single dimension, which suffices to illustrate most of our criticisms. Cf. supra note note 31 (increasing dimensions increases number of neighbors each firm has but does not itself affect market power of firms or the results of the model). Regarding the two-dimensional case, we simply observe that expanding the Hotelling model beyond one dimension takes it into terra incognita. See infra note 63.
 
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As in Hotelling's first model, Campbell simplifies by implicitly assuming that all products sell at the same price. n39 Consumers consequently make their purchase choices entirely on the basis of the characteristic--here,location--that differentiates the products. Below, we explain why this fixed-price assumption is not harmless, n40 but first  [*1024]  we borrow Figure 2 from Campbell to explain his reasoning. n41
 
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n39 Campbell is not explicit about this assumption. At several points he possibly implies that firms in his model can charge different prices. See Campbell, supra note 13, at 1640 n.63, 1643. This interpretation of his article, however, is inconsistent with his key description of a predator that by moving closer to a rival, "[e]xcept for the cost of moving, . . . is just as well off." Id. at 1646. If prices were free to fall, the predator's profits would shrink.

n40 See infra notes 44-50 and accompanying text.

n41 See Campbell, supra note 13, at 1639 (Figure 1).
 
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In this diagram (the realism of which we later will dispute), n42 consider Vendor B. If Vendor B is located halfway between Vendors A and C, half of the consumers between A and C buy from B. In fact, B does equally well wherever it locates between A and C, not just at the midpoint between them. If B moves thirty feet further towards C, for example, B captures all the customers in those thirty feet, half of whom used to go to C, but B loses fifteen feet worth of customers on the other side to A, who becomes the closest vendor to them. As long as A and C do not move, B is indifferent about where it locates between the two. n43
 
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n42 See infra notes 52-56 and accompanying text.

n43 Suppose that the three vendors are at distances x[a], x[b], and x[c] from the furthest consumer to the left. Vendor B gets half the consumers in the gap (A,B) and half in (B,C), or (1/2)(x[b], - x[a]) + (1/2)(x[c], - x[b]). This expression equals (1/2)(x[c] - x[a]), which is independent of x[b].
 
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Now suppose that entrant E appears in the location shown in Figure 3. By the same argument as in the preceding paragraph, B is indifferent among all the locations between A and E. Campbell argues that B will move next to E--a move that is costless for B to make. By doing so, B causes E's market share to shrink to an unprofitably small segment. After E exits, B returns to its original position. Foreseeing B's strategy, E would decide not to enter in the first place. This threat of predation, Campbell argues, deters entry--a result of great significance for antitrust policy.

II. PROBLEM ONE: NO PRICES

The main problem with Campbell's model is that it is not really an  [*1025]  economic model. Campbell implicitly assumes away the role of price, but prices are a crucial aspect of market economies. n44 In remedying Campbell's neglect of price, we do not claim that price is the only (or even the most important) margin of competition in a given market. We assert only that prices will always play at least some part in the competitive process--whenever firms do not believe themselves free to raise prices to infinity. Campbell's omission of price thus confines the possible relevance of his model to markets in which the government controls prices. Allowing firms to vary their prices creates a fully determined model and leads to the accommodation of entry instead of successful predation, and to more intuitive and sensible results.
 
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n44 Cf. P. Ordeshook, Game Theory and Political Theory: An Introduction 166-75 (1986) (contrasting economic location models with political location models of electoral contests in which it is sensible to omit price).
 
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To introduce prices, we must specify something about the willingness of consumers to pay for a product. Suppose simply that consumers will pay a slight price premium for a product that more closely suits their tastes; in the hot dog example, they will choose a vendor ten feet away instead of a vendor two hundred feet away, even if the nearby price is somewhat higher. This bit of realism has a drastic effect on Campbell's analysis. Far from being indifferent to where it locates between Vendors A and C in Figure 2, Vendor B now has a strong preference. Assuming that A and C charge the same price, B wishes to locate halfway between them. That location offers B the maximum protection from competition. The consumers located exactly halfway between A and C are willing to pay the highest price to B of any in that interval. Even if B must charge a single price to everyone, B will choose that halfway location because it puts B in the middle of the consumers least willing to go to other vendors. n45
 
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n45 Cf. d'Aspremont, Gabszewicz & Thisse, supra note 29, at 1148-49 (two firms will tend to move as far as possible from each other when moving costs are quadratic).
 
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If Vendor E enters as depicted in Figure 3, B will no longer be indifferent among all locations between A and E. Rather, B will want to move away from the entrant--to halfway between A and E, so as again to locate in the center of the consumers furthest from competing vendors. Consider what would happen if B moved next to E, as in the Campbell argument. The closer B moves to E, the less relevant the locational model becomes to their competitive relationship. If B moves right next to E, the two sell a virtually identical product. Consumers would distinguish between them almost entirely on the basis of price. If their locations were exactly the same, their competition would reduce price to marginal cost--or, at any rate, the duopoly issues that arise have nothing to do with location or product differentiation. n46 If B moves slightly to the left of E, then B can charge a price slightly higher than E, but only slightly higher. If B raises price too much, B loses not  [*1026]  only all the consumers between its new location and E, but also all the customers to the left. If, for example, B charges two dollars and E charges one dollar, and B is ten feet to the left of E, even consumers to the left of B might consider it worth going to E for their hot dogs.
 
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n46 See supra note 33.
 
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The implication of price competition is that moving adjacent to an entrant is costly. If a predator moves next to an entrant, it must lower its price to keep customers. Because this predation is costly, the standard issue arises of whether such a threat is credible. n47 This issue is not particular to a location model, and Campbell does not claim to contribute to this more general debate about predation. Anything that enables the incumbent to make credible its threat to move next to the entrant also enables the incumbent to make credible its threat to lower price below cost. n48
 
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n47 See supra note 24 and accompanying text.

n48 We have assumed that each firm charges one price rather than charging different consumers different prices depending on the consumers' distance from the firm and from competing firms. If we allowed such price discrimination, our price-based argument against Campbell remains much the same. In moving towards its prey, the predator moves in a costly direction: toward the region of consumers to which it must charge low prices because of competition from the prey, and away from the region of consumers to which it can charge high prices because of their distance from the rival seller on the other side.
 
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Introducing price into Campbell's model also creates other problems for his conclusion that predators can costlessly engage in predation. Campbell assumes that a predator moving left can trust its neighbor on the right to stay put. But that neighbor will not be so neighborly in a model with price. Instead it will move toward the predator, to the midpoint between its neighbor to the right and the predator. This move adds to the costs of predation by stealing some customers from the predator.

Campbell recognizes this problem and tries to anticipate it: "The incumbent neighbors might recognize the benefit conferred on them by the moving incumbent. They may be hesitant to pick up its former customers, lest they chill action that they realize is in their own interest." n49 This "hesitant" response, however, is but one ad hoc choice among a host of possibilities. The prisoner's dilemma makes clear that two firms may have great difficulty coordinating to accomplish what is in their mutual long-run interest. n50 Here the predator moves first, then a neighbor must decide what to do. If the neighbor moves in on the predator's old turf, it benefits from added customers. The neighbor would benefit if the predator forces the prey out of the market, but benefits even more if it gets some of the predator's customers to boot--even if only temporarily. In short, the unreliability of neighbors' reactions is a cost  [*1027]  of a predatory strategy--another cost that Campbell unjustifiably dismisses.
 
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n49 Campbell, supra note 13, at 1653.

n50 See, e.g., Wiley, Reciprocal Altruism as a Felony: Antitrust and the Prisoner's Dilemma, 86 Mich. L. Rev. 1906, 1916-20 (1988).
 
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The model with price competition accords with everyday intuition. If a new vendor appears on the beach, what will the neighbors do? Common sense suggests that the neighbors would move away and slightly lower their prices, as in the discussion just above. Or if somehow they could engage in credible predation, they would move next to the entrant and slash prices to drive it away. Campbell instead suggests a peculiar compromise: they would move right next door yet also maintain their price. This happens only because the model is unrealistically simplistic: it does not allow prices to equilibrate supply and demand via competition.

III. PROBLEM TWO: POOR SPECIFICATION

Apart from its basic neglect of price, Campbell's version of Hotelling's model suffers from a second set of problems. Campbell claims his model initially is in stable equilibrium. n51 This claim is important to any modelling effort. If no stable equilibrium exists, then the model can make no predictions; the situation is too impermanent for theoretical analysis. Without a stable equilibrium one cannot say anything about the world before entry, much less afterwards. But Campbell's initial equilibrium has five problems.
 
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n51 Campbell, supra note 13, at 1640 n.63.
 
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First, it is hard to justify Campbell's starting point, even in the simple case of one dimension. Rather than being a mainstream approach accepted for its robust and reliable nature, the Hotelling model is a sensitive gizmo with some pronounced oddities. For technical reasons, it turns out to be critical whether the line n52 is infinite (or else a circle) or finite. On a line of finite length, the number of firms also is crucial. If the number is two, Hotelling argued that in the equilibrium of his fixed-price model the firms will not space themselves at equal intervals, as Campbell supposes. Rather, Hotelling claimed that both must locate halfway along the line, right next to each other -- in an adjacent or "paired" pattern that Campbell's analysis would make suspect. n53 With three firms along a finite strip, there is no equilibrium configuration at all; whatever configuration you pick, one of the vendors can benefit by moving. n54 With four firms, equilibrium returns--but again in the paired pattern that Campbell would declare to be questionable. n55
 
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n52 See supra note 38.

n53 Hotelling, supra note 16, at 51-52; see supra text following note 32.

n54 An exception exists if the firms use mixed (that is, probabilistic) location strategies. Shaked proves that an equilibrium does exist if three firms avoid the two end quarters and locate with equal probability at points in the middle half. See Shaked, Existence and Computation of Mixed Strategy Nash Equilibrium for 3-Firm Location Problem, 31 J. Indus. Econ. 93, 94 (1982).

n55 See Eaton & Lipsey, supra note 36, at 30. The four firms divide into two groups, with each group around the first and third quartiles. A similar pairing pattern occurs with five firms, but with more than five firms "the equilibrium configuration ceases to be unique." Id. These results are not robust, but rather are sensitive to a number of critical and restrictive assumptions. Id. at 28, 32-39.

Behavior in two dimensions is only poorly understood and may never settle into any equilibrium configuration, but does exhibit some dynamic tendency towards pairing. Id. at 39-46.
 
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 [*1028]  Campbell's model may seem to skirt these problems and achieve equilibrium by using lines of infinite length, but this depiction is unrealistic in most relevant applications. The Santa Monica beach does not go on forever, and products rarely can have infinitely more or less of a quality. To use Campbell's example of sugar in breakfast cereal, cereal can have neither less sugar than none nor more than Captain Crunch. n56
 
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n56 This example also shows why a circle is a poor model for this application.

Economists using the assumption of an infinite or circular line in theoretical exploration acknowledge this problem but justify their decision as analytically more convenient. E.g., Salop, Monopolistic Competition with Outside Goods, 10 Bell J. Econ. 141, 142, 155 (1979) ("neither assumption [about an infinite line or a circle] is realistic"). Such justifications may suffice in research journals that try bit by bit to understand the world but are inappropriate when advising policymakers.
 
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Second, Campbell begins by assuming that more than one firm populates each market. Yet the "moving in for the kill" strategy of predation logically works -- if it works at all -- against incumbent rivals as well as against new entrants. Before E entered, for instance, B in Figure 2 could have moved next to C and forced C from the market. Campbell's argument thus proves too much. If the tactic works as Campbell says, then he must explain why every market is not monopolized by a predator that has eliminated all competition.

Third, Campbell mistakenly states that his model is insensitive to different assumptions about whether the production technology requires sunk entry costs. n57 This factor is important, however, and indeed is a considerable problem for Campbell's analysis. If the costs of producing a new product are low, then Campbell's assumed pre-entry industry structure is implausible; if the cost is high, his argument about post-entry behavior does not apply. Entry can either be costless or costly. If entry is virtually costless and scale economies are negligible, then the only equilibrium contains exactly one firm per customer. Each firm earns a normal profit serving the exact taste of its customer, offering neither inducement nor room for new firms to enter. If there were fewer firms than customers, a new firm could safely enter and serve the neglected customer; after an attempt by the incumbent to move next door, the entrant could either happily serve its single customer or costlessly move to a more distant location. n58 Alternately, entry might  [*1029]  be costly. If establishing a new product is costly for the entrant, onc would expect that it is likewise costly (absolutely, not relatively) n59 for the incumbent. But if establishing a new product is costly for the incumbent, then Campbell's suggested method of predation is costly, and the incumbent's threat to use it is not credible. n60
 
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n57 "No other constraints are imposed on this model. Specifically, large entry or reentry costs, . . . or substantial sunk costs by incumbent firms, though commonly imposed by others on predation models, are not assumed here." Campbell, supra note 13, at 1642-43.

n58 Campbell assumes that the prey's neighbor on the far side from the predator will stay put. This assumption is highly questionable. If the predator's threat to move indeed is credible to the prey, the prey logically could react by moving away in the direction of this "innocent bystander." If the bystander recoils, its neighbor further along also might do the same. The consequence would be that all firms realign along the line, preserving the spacing that existed just before the predator's move and defeating the predator's effort. The predator's strategy would accomplish nothing.

n59 See supra note 24.

n60 See supra text accompanying note 24.
 
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Fourth, Campbell's assumptions fail to explain why the prey is a victim rather than an agressor itself. We have just recounted why it is not clear that anybody dies as a consequence of a firm moving in for the kill. If the tactic indeed works, however, one then asks further whether an entrant might purposely locate two products on each side of an incumbent to invade the market, or extort a payment not to do so. n61 Campbell's model thus does not have enough structure even to tell the predator from the victim.
 
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n61 See Rasmusen, Entry for Buyout, 36 J. Indus. Econ. 281 (1988). Campbell confounds matters by providing a defense for a firm that creates a "new" product rather than changing an "old" one. See infra notes 71-75 and accompanying text.
 
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Fifth, Campbell does not explain why a successful predator would return to its original location. As shown above, n62 a predator in Campbell's model is indifferent to its location between two fixed neighbors, and the cost of moving, while low, is probably positive. n63 Hence  [*1030]  under Campbell's model, the suspect pattern is a simple move toward a competitor -- not a move and then a return to the original position.
 
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n62 See supra note 43 and accompanying text.

n63 See Campbell, supra note 13, at 1642. Campbell does suggest a rationale for the predator's return in his analysis of behavior in two dimensions. He asserts that firms in two dimensions maximize their profits by locating at the center of a hexagonal area, thus implying that profit-maximizers will return to that center after they finish their predatory business at the boundry. Cf. id. at 1647 (when the mover leaves the center of the hexagon, "there is some net customer loss to the mover"). Critical to this claim is Campbell's assumption about the pattern of firms' initial location. See, e.g., id. at 1674-75 (offering geometric analysis that assumes a uniform hexagonal distribution of firm neighborhoods).

However, this key assumption contradicts the little we know about firm behavior in a finite two-dimensional world. Analytical investigation of equilibrium under these circumstances is "an almost impossible task." Eaton & Lipsey, supra note 36, at 41. This intractability has forced a reliance upon a simulation approach, which has solved only a handful of particular cases. These results show that the initial location of firms that Campbell assumes is not in equilibrium. See id. at 42-46. In fact, this investigation strongly suggests (although fails definitively to prove) that no equilibrium exists at all when more than two firms exist. Id. at 44. If that suggestion is correct, then any use of a two-dimensional approach--not just Campbell's particular and invalid application--would be useless for policy analysis of the real world. Campbell neither remedies nor even acknowledges these complexities. We therefore stick to the simple and tractable case of one dimension, and think Campbell should have done so as well.
 
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In sum, Campbell's model neither begins nor ends in equilibrium. It therefore cannot serve as a guide to antitrust policy, because dynamic behavior that looks predatory in an incomplete model might in reality be simply the normal ebb and flow of competitive disequilibrium dynamics. Because it fails to predict any particular outcome reliably, Campbell's model cannot furnish a compass for antitrust policy.

IV. PROBLEM THREE: COSTLY APPLICATION

Judges should not change law based on Campbell's analysis. Using Campbell's model to expand antitrust liability for predation would penalize beneficial conduct. Any number of product changes, promotional activities, or advertisements might qualify as "predatory" under Campbell's analysis if they affect consumer perceptions of product characteristics. But reasons other than predation might account for a competitor's decision to make its product more like a rival's. The rival's product, for instance, might be more intelligently located. In terms of the technical model, a firm that changes its product might be interpreting the success of the rival it approaches as a signal of greater density of consumer demand in that neighborhood.

Imitation is the lifeblood of competition. Campbell's proposal makes it legally suspect and necessary to defend in court. A cereal maker thus could be in court to defend its product's character as a consequence of adding sugar or removing it--or changing its product in any other way that might make the product more competitive. In spectacular fashion, Campbell's analysis recapitulates the standard antipredation problem of mistaking and punishing conduct that antitrust law instead should encourage. n64
 
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n64 See, e.g., Easterbrook, supra note 6, at 336 (1981) ("Any attempt to administer a rule against predation entails a significant risk of condemning the outcome of hard competition.").
 
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Campbell recognizes the impressive breadth of his logic and seems a bit appalled by it. n65 He tries to limit its sweep by adopting Baumol's suggested rule that singles out for liability "quasi-permanent" predation: predatory conduct that attacks a rival and then returns to the predator's original position. n66 Campbell's version would limit liability to firms that change to mimic an entrant and then return to their original position. This return requirement, however, is inconsistent with  [*1031]  Campbell's basic logic because the predator would have no incentive to return. n67
 
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n65 Campbell, supra note 13, at 1656 ("[T]his would punish a great deal of efficient behavior.").

n66 See Baumol, Quasi-Permanence of Price Reductions: A Policy for Prevention of Predatory Pricing, 89 Yale L.J. 1 (1979). Areeda remarks of the original Baumol article: "The ability of firms to live with the [Baumol] exception and the ability of courts to administer it are . . . doubtful and the subject of much dispute." P. Areeda, Antitrust Analysis 197 (3d ed. 1981).

n67 See supra note 63 and accompanying text.
 
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Moreover, even accepting Campbell's initial reasoning, the return requirement creates a loophole for possible predators and at the same time introduces costly inflexibility into non-predatory market dynamics. It creates a loophole because (supposing the Campbell tactic to work) a predator could avoid liability simply by avoiding a return to its precise original position. It introduces inflexibility because it raises the costs of perfectly legitimate product experimentation. n68 If Coca-Cola decides that it was wrong after all to mimic Pepsi's apparently more successful taste, it would face a treble damage suit for returning to "Classic Coke" using Campbell's test. Campbell mentions this famous marketing reversal in passing, diffidently suggesting that perhaps it was legitimate competition rather than predation. His basis for this suggestion is not clear, apart from his mention that Pepsi had not recently entered the market. n69 But whether the prey is a new entrant or an incumbent ought to be irrelevant to Campbell's logic. n70 In a caveat that discloses uneasiness with the scope of his logic, Campbell also qualifies his general test by creating a defense for a firm that introduces a "new" product rather than changes an "existing" one. n71 Campbell might excuse Coke on this other ground. Our point remains that Campbell's test would discourage desirable product experimentation by increasing its expected cost.
 
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n68 Campbell candidly concedes this point. Campbell, supra note 13, at 1656-57 & n.107. He responds by proposing that the defendant be permitted to raise a defense of "a marketing mistake." Campbell proposes no clear criteria for distinguishing "good faith mistakes" from disguised predation, cf. id. at 1657 (referring vaguely to a need for "planning documents and consumer surveys") and we see no reliable and administrable rule. Asking whether the defendant reduced its profits by changing its product would not work, because a predator that surprised itself by increasing profits through predatory changes would have little reason to return to its original position -- a point that Campbell also concedes. See id. The very fact that another firm makes a different product signals its belief that consumers like those differences and justifies the defendant's marketing experiment; the very fact that the firm later expires signals that the number of consumers is not impressive and justifies the defendant's return.

n69 Id. at 1658 n.111.

n70 See supra text following note 56.

n71 See Campbell, supra note 13, at 1666-70.
 
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The Coke episode also illustrates the problems of this defense, which makes treble damage liability turn on whether a product is "new" rather than "old but different." Coke originally announced that it was replacing Old Coke with New Coke. n72 Campbell's model logically should not protect Coke during this period. n73 A few months later, however, Coke reintroduced its old formula as "Classic Coke." n74 Perhaps  [*1032]  Campbell then would confer "new product" immunity upon Coke. If so, well-counseled defendants would have little trouble manipulating Campbell's artificial classification. If not, then Campbell is bravely cheerful to suggest that "cases calling for close judgment" might arise only "occasionally." n75
 
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n72 See M. Mitchell & D. Benjamin, Quality-Assuring Price Premia: Classic Evidence from the Real Thing 1 (Feb. 24, 1989) (unpublished manuscript on file at Columbia Law Review).

n73 See supra notes 62-63 and accompanying text.

n74 M. Mitchell & D. Benjamin, supra note 72, at 2.

n75 Campbell, supra note 13, at 1670 n.158.
 
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Finally, the Coke incident impugns Campbell's basic claim that "moving in for the kill" is a likely way to polish off rivals painlessly. Campbell's predatory tactic failed miserably. n76 And it was hardly costless: the best estimates are that the decision to change "The Real Thing" cost Coke's shareholders more than half a billion dollars. n77 In short, Campbell's model yields a test that either is stunning in its breadth and cost, or limited in a way that is illogical, ineffective, and still costly. In both events, this model fails to master the facts of the most prominent and recent real instance of a company changing its product to resemble its rival's.
 
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n76 See M. Mitchell & D. Benjamin, supra note 72, at 3.

n77 Id. at 24.
 
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CONCLUSION

Campbell invites courts to expand treble damage liability for predation under the Sherman Act. His justification for this invitation is a model that is fatally incomplete. This invitation is one courts should decline.