Implied Market Shares and Antitrust Markets as Fuzzy Sets

Abstract

Market definition is an important aspect of antitrust analysis, but the exercise requires designating each potentially relevant good as either completely in or out of the market. In a market with differentiated goods, this often necessitates making unsatisfactory designations. This article introduces the concept of the antitrust market as a fuzzy set, where each relevant good can have a degree of membership in the market, rather than only the traditional binary in or out designation. A product’s degree of membership in the market is assigned based on its closeness of substitutability to the focal products. We then show how implied market shares can be calculated from a defined fuzzy antitrust market. We illustrate an implementation of these concepts using data from the grocery retail market and demand estimations from prior literature. We also discuss how this concept serves as a middle way between supporters and detractors of market definition, as implied market shares are objects that not only reflect demand substitutability but also allow for the rhetorical benefits of having delineated a market to focus discussion and analysis of competitive issues.

Keywords

antitrust market definition differentiated products

I. Introduction

Market definition is central and even pivotal to merger investigations in the United States.¹ Case law and antitrust agency guidance has enshrined market definition as an important step in merger analysis.² As a consequence, economists are often, if not always, asked to define markets and calculate corresponding concentration measures in their expert testimony on merger challenges.

The market definition exercise traditionally requires designating each potentially relevant good as either completely in or out of the antitrust market. In a market with differentiated goods, this often necessitates making unsatisfactory designations. This article introduces the concept of the antitrust market as a fuzzy set, where each relevant good can have a degree of membership in the market. This is in contrast to a crisp set, where each element of the set can have only a binary in or out designation. We propose that a product’s degree of membership in an antitrust market could be assigned based on its closeness of substitutability to the focal products. In the case of a merger, the products of the merging firms are the focal products; other products that are close substitutes to those focal products would have a high degree of membership in the antitrust market, while more distant substitutes would have a lower degree of membership.

Characterizing the antitrust market as a fuzzy set allows for mapping market substitution patterns into objects we call implied market shares. Implied market shares resulting from an analysis based on demand substitutability can be an effective way to present market shares that also reflect economic relationships in the market. The degree of membership in the market would determine how much weight a product receives in determining the total market size. This total market size serves as the denominator in market share calculations. Once the size of the implied market is determined, implied market shares can be calculated analogously to traditional market shares.

Agency merger guidelines reflect that real markets have an inherent fuzziness to them. The 2010 Horizontal Merger Guidelines recognized that “Relevant markets need not have precise metes and bounds.” The 2023 Merger Guidelines maintain precisely the same language and quote court precedent that “fuzziness would seem inherent in any attempt to delineate the relevant . . . market.”³ The approach to market definition proposed here formalizes that fuzziness and makes transparent how it can be reflected through implied market shares.

Conceptualizing antitrust markets as fuzzy sets also serves as a middle way between supporters and detractors of market definition, as implied market shares are objects that both reflect economic substitutability and allow for the rhetorical benefits of having a delineated market, that is, to focus discussion and analysis of competitive issues. We argue that this formal generalization of the traditional market definition practice can help clarify the discussion and bring the legal analysis into better harmony with economic analysis. Implied market shares, which provide a richer and more perspicacious picture of the market, can be calculated based on a demand analysis and will better reflect the differential degrees of substitutability of relevant products.

Implied market shares offer a viable alternative to the hypothetical monopolist test (HMT), which is sometimes treated as practically synonymous with market definition. While critics of market definition like Louis Kaplow suggest dispensing with market definition entirely, an alternative would be to still define markets but without the HMT. This can be appealing if there are legal requirements to define a market, or if market definition does provide some benefit in designating the relevant domain of inquiry for a particular competitive issue. Moreover, Crane notes that “If market definition falls, so does the entire structure of analysis built on top of it—which is to say, a whole lot of antitrust law—unless a suitable replacement can be found” (33).⁴

This article is the answer to such a call for a replacement that is conceptually sound and practically feasible. The approach to market definition described here is entirely based on demand side (buyer) substitution. Although there has been some discussion about incorporating supply side factors into market definition, many experts advocate for a sole focus on demand substitution in the market definition stage of analysis.⁵ This parsimony simplifies the methodology for defining a market. The HMT, which is a leading approach to defining an antitrust market, is quite a bit more complicated as it requires not only estimates of demand substitution but also information on product margins and a specification of how firms competitively interact.⁶ The implied market shares approach thus has the feature of being less informationally demanding than the HMT while still reflecting some relevant underlying forces of the market in question. We propose the implied shares approach as a contender for defining a market that may, in some cases, be more attractive than the options currently available.

Finally, we illustrate an implementation of these concepts using data from the grocery retail market and demand estimations from prior literature. In that example, an analysis of a merger in grocery retail might include traditional supermarkets (e.g., Harris Teeter, Publix), club stores (Costco, Sam’s Club, BJ’s), steep discount stores (Aldi, Dollar General), and premium stores (Whole Foods, Wild Oats, Wegman’s). While traditional market definition would require that each grocery retailer be designated as completely in or out of the relevant market (a crisp set), by generalizing market definition so that the relevant market can include all of the grocery retailers, and by designating their degree of membership in the market based on their level of differentiation (a fuzzy set), the resulting implied market shares will reflect that the different store formats are substitutes to different degrees. Instead of a binary designation of being either in or out of the market, the framework allows for a representation of the market that is more faithful to the economic realities of the market at hand.

II. Definitions and Background

The recent 2023 Merger Guidelines offer the following definition: “a relevant antitrust market is an area of effective competition, comprising both product (or service) and geographic elements” (at 40). The exercise of market definition is undertaken to determine the set of included products for the purpose of calculating market shares, which can then be compared to presumptions provided in merger guidelines or case law. There is a lively debate over whether market definition is a useful exercise at all, with strong proponents on both ends of the spectrum. A tremendous amount of ink has been spilled explaining, defending, and attacking the exercise of defining a relevant market for purposes of antitrust enforcement.

The concern is not new. Richard S. Markovits expressed concern as early as 1978 about whether the notion that markets can be defined non-arbitrarily is warranted at all.⁷ But the discussion of market definition, its role in antitrust enforcement, and how to actually define markets changed in the 1980s with the rising emphasis on unilateral competitive effects and the use of the HMT to define markets. The 1992 Horizontal Merger Guidelines introduced unilateral effects into federal guidelines, marking the beginning of a shift in emphasis from coordinated effects to unilateral theories of anticompetitive harm.⁸ That shift ran its course over the next decades; Malcolm Coate and Shawn Ulrick document that 76 percent of government merger investigations between 2011 and 2014 involved unilateral effects theories.⁹

With a focus on unilateral effects, the suitability of market definition and tests like the HMT came into question. Even soon after the release of the 1992 Guidelines, James Keyte worried about the “lack of any clear standard for defining the relevant product,” especially in cases where differentiated products are involved.¹⁰ Adriaan ten Kate and Gunnar Niels pointed out the 1992 Guidelines contradiction that described market definition as a solely demand side approach, but also defined the HMT which depended on supply side factors.¹¹ According to Sean Sullivan, while the HMT may have been the “darling of 1980s antitrust” when coordinated effects were primarily of interest, two decades later it “had morphed from darling to demon” as unilateral effects took center stage.¹² Indeed, “The HMT does not typically define relevant markets that are helpful for evaluating differentiated-product unilateral effects.”¹³ Sullivan goes on to further claim that “For economists and practitioners looking for unilateral effects in mergers, time spent on the HMT was time wasted.”¹⁴ In the provocatively titled “Why (Ever) Define Markets?” Louis Kaplow proposes that the HMT should be discarded altogether.¹⁵

Worse than being unneeded, market definition in differentiated product markets can be misleading. Valletti and Zenger note that “it is well-known that market shares can be off the mark in trying to account for consumers’ heterogeneous switching patterns between differentiated products.”¹⁶ Johnny Shaw elaborates further: “the binary nature of this market definition approach does not reflect the reality of economic competition. Products are labeled either inside or outside the relevant market, and once they are found to be inside it, the level of competition each product provides to the defendant’s product is inferred from its market share. This fails to recognize that products can have different levels of substitutability.”¹⁷ This sort of binary thinking yields that in practice, litigants on opposing sides can offer very different yet reasonable definitions of the relevant market, “[creating] uncertainty for both plaintiffs and defendants.”¹⁸ This makes decision-making error prone, either by excluding relevant products or including irrelevant ones. Hovenkamp summarizes the difficulty that “one characteristic of unilateral effects merger cases in product differentiated markets is that the market boundaries themselves tend to be both uncertain and shifting.”¹⁹

Despite these critiques, market definition is typically invoked anyway in litigated cases, and is often dispositive. Recently, Shapiro and Shelanski noted that “the agencies have continued, without exception, to define relevant markets and measure market concentration when they challenge mergers in court.”²⁰ One reason may be that the exercise of market definition does have some conceptual utility. Coate and Simons conclude from their analysis of the handling of FTC cases that “As a bottom line, market definition, aided by the hypothetical monopolist analysis and critical loss, is likely to remain the starting point for a broad range of antitrust analyses”²¹ and they maintain that “market definition remains useful in a full competitive effects evaluation as an organizing structure for the economic analysis.”²²

III. Defining Markets Without Hypothetical Monopolists

The market definition exercise is typically an implementation of the HMT. When implementing the HMT, one first locates the smallest market of substitutable goods where a hypothetical monopolist could profitably raise prices by some designated amount. Then, one calculates the Herfindahl-Hirschman Index (HHI) as well as the change in HHI that would result from a merger. Markets with a very high HHI where a merger would increase it by more than 100 points are “presumed to substantially lessen competition or tend to create a monopoly.”²³

A prominent critic of market definition, Louis Kaplow maintains that the only point of conducting market definition is to assess firms’ market share. His core concern is that “there does not exist any coherent way to choose a relevant market without first formulating one’s best assessment of market power, whereas the entire rationale for the market definition process is to enable an inference about market power.”²⁴ Two important elements of his critique are that (1) although in markets with homogeneous goods, market shares are correlated with market power, that relationship does not hold in differentiated product markets; and (2) that using the HMT to define a relevant antitrust market is logically incoherent and unnecessary, since measures of market power are a required input to the test, but also the purported goal is to show market power through high market shares.

With respect to the first point, Kaplow illustrates his concerns formally by considering the usefulness of the Lerner index, which is equivalent to the inverse of a firm’s elasticity of demand. It can be used to show that in a market of homogeneous goods there is a correlation between market share and market power. With some assumptions, one can derive “an explicit indication of the functional relationship between a firm’s market share and its market power.”²⁵ However, for heterogeneous goods, “there is no such formula” that is available to use.²⁶ Thus, “there is no economic method of interpreting shares in [heterogeneous goods] markets.”²⁷ That is, if what one really cares about is inferring market power, in the broad class of situations where products are differentiated, market shares are not the right place to look and so a market defined through the HMT is not useful.²⁸

With respect to the second point, Kaplow asks the reader to consider two potential ways of defining the market: Narrow and Broad. Naturally, we would wish to pick the market that estimates market power to be as close as possible to the true answer. But in order to adjudicate between those two cases, we must already have some idea of what the best estimate of market power already is. He indicates that we might as well, in fact, choose the narrowly drawn market. This is why the HMT itself is circular: an estimate of market power is an input into the HMT, and the subsequent defined market is then used to calculate market shares, which are then the basis for the demonstration of market power. Kaplow concludes that “the diversion ratios and the [cost-price] margin” are the only things that matter, rather than the HMT’s requirement.²⁹

There have been many responses to Kaplow’s critique. Some criticism addresses specific details of Kaplow’s argument.³⁰ Greg Werden argues by countering that the input information for conducting the HMT is different from the informational requirements of assessing market power, and that market power is inextricably linked to market definition. So the input information is not the same as the output information we seek, and in fact as a conceptual matter market power is something that cannot be assessed without presupposing something about market definition.³¹

More general responses contend that market definition might serve purposes in antitrust analysis beyond only allowing for the calculation of market shares. For example, Werden takes issue with Kaplow’s insistence that market definition is only useful for calculating market power from market share, where market power is interpreted as pricing over marginal cost. Indeed, “Market shares can never be better than crude indicators of market power.”³² Rather, it is “to identify, in terms of products and places, the locus of active competitive forces central to the case.”³³ Because the market definition exercise as deployed via the HMT is a comparative statics exercise, it is crucial to uncover which causal forces should be treated as operative within some time (and spatial) scale and which ones can be held fixed or neglected.³⁴ Coate, Ulrick, and Yun summarize the aim of antitrust analyses is to “consider the responses of competitors within the market more closely than the responses from those rivals outside the market.”³⁵

And while there are cases in which market definition is not necessary (as an example, Werden cites Evanston Nw. Healthcare Corp 2007-2, a post-merger case decided on direct effects evidence), Werden does not agree that the existence of differentiated goods should motivate abandoning market definition altogether. For instance, the quantitative tools such as merger simulation used for assessing such mergers may not be sufficient on their own, such as when there is not enough data to estimate relevant demand elasticities or the model does not fit the industry appropriately.³⁶ In a review of sample cases over a period of twenty years, Coate and Fischer make a similar point that game-theoretical models and empirical estimation are useful in some situations, but that market definition is still useful in others.³⁷ Furthermore, market definition serves a key role in constructing narratives around a certain merger or conduct, which is useful for presenting a case: “Separating active from passive competitive forces by delineating the relevant market is a rhetorical and analytical device for bringing order to real-world chaos.”³⁸

Others, such as Conlon and Mortimer, note that in principle, one could forego the step of market definition if information such as precise diversion ratios is available.³⁹ In such settings, such as when one is interested in merger effects, market definition could be seen as redundant, calculating implied shares will not add additional information to those diversion ratios. Despite this theoretical possibility, in practice market definition is nearly always invoked, and for a variety of reasons.

In much of this debate, critiques of market definition are typically aimed at the HMT in particular, while defenses of market definition are of the concept more generally. And yet market definition is often applied through the HMT. One reason this continues to be the case is that while the HMT has some conceptual problems, it may also be the best approach available.

We propose implied market shares as a framework that stays true to the original motivation of traditional market definition but can also accommodate, in a fairly nuanced way, the kind of market demand information that is used in competitive effects analysis. In particular, this article proposes that generalizing the concept of a relevant market from a crisp set to a fuzzy set provides the additional flexibility that allows a defined market to be both conceptually useful and also connected to the relevant economic demand. Including a set of firms that are relevant for analysis without necessarily implying that they should all be included to the same degree in the denominator of a share calculation can be a way to reconcile some of the tensions that have been noted since the onset of tests like the HMT. Implied market shares can bridge this divide by offering an alternative method to the HMT that is still grounded in the economic forces of demand substitution.

We focus on market definition as a conceptually distinct activity from an effects analysis (whether unilateral or coordinated) that may follow or complement the market definition exercise.⁴⁰ Rather than completely discarding the endeavor of market definition, we can account for Kaplow’s concern that we ought primarily to be attending to information such as diversion ratios, while recognizing Werden’s point that there is an epistemic and rhetorical benefit to having a delineated market and some notion of shares in that market. That is, we still ought to keep market definition on the table even if diversion ratios are already available. After all, “caselaw and academic theory have not yet provided a robust and comprehensive approach to proving market power and that antitrust law will be largely starting from scratch, with a few scattered and incomplete tools, if it moves away from the market definition/market share paradigm” (Crane 2014, 35).

Insofar as one is interested in market definition as a distinct exercise from or precursor to effects analysis, the implied shares approach is an attractive framework to use when analyzing a market with product differentiation.

IV. Determining Implied Market Shares

There are three steps to determining implied market shares. First, one must delineate the set of products or services that will be included in the analysis. Second, some analysis must be done to determine the substitution patterns among those products. Finally, a membership function is specified to reflect that each product’s degree of membership in the relevant fuzzy set reflects its closeness of substitutability to the focal products. That degree of membership will then determine how much the product contributes to the total antitrust market size, with more distant substitute products contributing only a fraction of their total revenue to the relevant antitrust market.

To express this idea formally, consider a set of products that are relevant to a competitive issue and denote that set by $J$ . Some product in that set is designated as the focal product; in the case of a merger, the focal product will be a product owned by one of the merging companies.⁴¹ Each product $j \in J$ gets assigned a membership weight $a_{j} \in [0, 1),$ which captures how close of a substitute each product in $J$ is to the focal product. If a product is determined to have a membership weight $a_{j} = 1,$ that means it is a very close substitute to the focal products and it has full membership in the market. When $a_{j} = 0,$ the product is a distant enough substitute that it is determined to be completely out of the market. Intermediate values of the assignment weight allow for some products to be partly in the market.

In order to implement this approach, it is necessary to first estimate demand, which is a familiar task that is undertaken in nearly every antitrust matter. Demand estimates can come from econometric demand estimation, documentary evidence, and testimony that illuminate which products are close substitutes to each other and which are more distant substitutes. Documentary evidence and testimony in particular can be important sources of information about product characteristics and how market participants view the degree of substitutability between the focal products and other potentially relevant products.

With membership weights, implied market shares can then be calculated as long as a measure of each product’s revenue or output be known. Denote product $j$ ’s revenue by $r_{j}$ . Then, the total size of the implied antitrust market can be calculated as $M = \sum_{j \in J} a_{j} \cdot r_{j}$ . The implied share for each good $j$ can then be calculated as $s_{j}^{i m p l i e d} = \frac{a_{j} r_{j}}{M}$ . Implied versions of concentration measures such as HHI or a change in HHI due to a merger can then be calculated based on the implied shares and implied total market size. When antitrust markets are a crisp set, $a_{j}$ must be equal to 0 or 1 for each product. By allowing the generalization that the membership degree $a_{j}$ can also be in between 0 and 1, differentiated products can be assigned a degree of membership depending on how related they are to the focal product.

One key challenge for implementing this approach is to determine the appropriate membership weights for each product in a non-arbitrary way. The guiding principle we propose is to assign membership weights $a_{j}$ to goods such that the resulting market reflects diversion proportional to implied market shares from the focal products. Such a principle will yield membership weights that fully tie the implied market to economic demand for products in the market. Rather than assigning membership weights in an ad hoc manner, a given demand system will map into exactly one best assignment of membership weights.

The next section uses data from the grocery retail industry to show how implied market shares can be calculated and presented. In particular, we show how to translate econometric estimates of demand into membership weights and thus implied market shares in a market. We show how the resulting implied market shares are between potential narrow and broad markets by reflecting the degree of substitutability between the focal products and differentiated alternatives.

V. An Illustrative Example: Grocery Store Markets

This section illustrates how econometric demand estimates can be used to obtain implied market shares. We use the example of grocery retail and borrow econometric estimates of demand for that market from academic literature. Hosken and Tenn detail the difficulties courts have had in determining market definition in differentiated retail markets.⁴² This section illustrates how implied shares can be one way to resolve those difficulties.

We use data from Ellickson, Grieco, and Khvastunov 2020 (henceforth EGK), which uses data from grocery retail stores from calendar year 2006.⁴³ Although geography is a very important dimension of retail markets, we suppose for simplicity in this exercise that there is only a single national market and focus on how the issues relate to questions of the product market. Table 1 shows the name, segment, and number of stores of large national chains; these data come directly from a table in EGK. The revenue column of Table 1 shows the average weekly revenue, in millions, of the entire chain.⁴⁴

Table 1.

Calculation of Shares in a Narrow Market, a Broad Market, and Implied Shares That Reflect Demand by Satisfying Diversion Proportion to Implied Share in the Market.

Name	Segment	Stores	Revenue	Narrow	Broad	Implied
Ahold	chain	604	341.9	8.7	4.8	5.9
Albertsons	chain	510	182.5	4.7	2.5	3.2
Aldi	chain	615	47.4	1.2	0.7	0.8
Delhaize	chain	1,057	226.7	5.8	3.1	3.9
Great A&P Tea Co.	chain	161	54.9	1.4	0.8	1.0
HE Butt	chain	227	184.7	4.7	2.6	3.2
Kroger	chain	1,973	914.3	23.3	12.7	15.9
Lone Star Funds (Bi-Lo)	chain	238	53.6	1.4	0.7	0.9
Publix	chain	845	354.6	9.0	4.9	6.2
Safeway	chain	1,339	569.0	14.5	7.9	9.9
Save A Lot	chain	715	82.2	2.1	1.1	1.4
Smart & Final	chain	217	31.9	0.8	0.4	0.6
Stater Bros	chain	162	62.9	1.6	0.9	1.1
SuperValu	chain	1,194	550.1	14.0	7.6	9.6
Trader Joes	chain	236	71.3	1.8	1.0	1.2
Whole Foods	chain	159	81.4	2.1	1.1	1.4
Winn-Dixie	chain	451	113.1	2.9	1.6	2.0
Meijer	super	159	131.3		1.8	1.3
Target	super	160	84.2		1.2	0.8
Walmart	super	1,385	1,474.0		20.5	14.3
BJ’s	club	122	97.3		1.4	0.9
Costco	club	311	702.7		9.8	6.8
Sam’s Club	club	540	783.9		10.9	7.6

Note. Ahold includes the Giant and Stop & Shop banners, and Delhaize includes the Food Lion and Hannaford Bros banners. Revenue is average weekly revenue from 2006, given in millions of dollars.

An important element of these data is that grocery retailers are allocated to different market segments based on the format of the stores: large chains (e.g. Harris Teeter and Publix), superstores (Meijer, Target, Walmart), and club stores (Sam’s Club, BJ’s, Costco). While the formats are certainly differentiated, they also serve some common customer needs. This creates a conundrum for traditional market definition, which requires that every seller be defined as either in or out of the market. In the columns “Narrow” and “Broad” of Table 1, we calculate market shares both ways. The Narrow column calculates market shares supposing that all large national chains (those with greater than 150 locations) are in the market, while club stores and superstores are out of the market. The Broad column calculates shares based on including club stores and superstores in the relevant market; as expected, the market shares for each chain decrease under the broader market definition.

A. Determining Implied Market Shares

Allocating the club and superstore segments completely in or out of the market is likely too rigid. Instead, economic demand can be a guide for indicating the degree to which differentiated formats should be included in the relevant market, based on the degree to which they serve as substitutes to the products of interest. Specifically, we propose that implied market shares be calculated such that the focal products reflect diversion to other products that is proportional to implied market shares.⁴⁵ This principle allows for the direct translation of any economic demand system into implied market shares. We show the relevant calculations for the nested logit demand system, which is the demand system that is estimated in EGK for grocery stores.

Logit demand (or “standard logit”) is a demand framework where the preferences of the consumer are assumed to have a distribution such that substitution across products in the market is proportional to market shares. This demand system is often used in antitrust analysis.⁴⁶ The nested logit demand model generalizes the standard logit, by placing goods into mutually exclusive nests, and then allowing a separate “nesting parameter” to govern how closely goods in different nests are to one another.⁴⁷

The nested logit demand system requires the designation of products into nests. For grocery stores, the nests could correspond to the segments based on store formats: traditional supermarket chain, club store, and superstore. We will denote the nesting parameter by $σ \in [0, 1) .$ When the nesting parameter equals 1, substitution stays within nest only, and the market shares implied by the model would equal to the Narrow column shares in Table 1 for supermarkets. When the nesting parameter equals 0, the model is equivalent to a standard logit model, where substitution occurs across all stores regardless of nest, in proportion to the overall share; each store format is thus equally in the market, and this model would have implied shares equivalent to those shown in the Broad column of Table 1. And for any value of the nesting parameter in between 0 and 1, the implied shares will be something in between the two extremes of the Narrow and Broad shares. The nesting parameter gives some degree of weight to options in other nests based on substitution patterns, and implied market shares are essentially a corresponding weight between the narrow and broad markets.

Conlon and Mortimer provide the formula for a diversion ratio in a nested logit demand system.⁴⁸ For two goods $j$ and $k$ in the same nest, the diversion ratio from $j$ to $k,$ denoted by $D_{j k}^{*},$ is given by

D_{j k}^{*} = \frac{s_{k | g} \cdot [σ + (1 - σ) s_{g})}{1 - s_{j | g} \cdot [σ + (1 - σ) s_{g})}

where $s_{g}$ is the share of nest $g$ to the overall market and $s_{k | g}$ is good $k$ ’s share of nest $g .$ What this suggests is that in a nested logit demand system, assigning implied market shares that reflect diversion proportional to implied share means that the implied share for good $k$ in the focal nest should be

s_{k}^{i m p l i e d} = s_{k | g} \cdot [σ + (1 - σ) s_{g})

Intuitively, this formula scales the implied share of the good to be in between the narrow market share, $s_{k | g},$ and the broad market share, $s_{k} = s_{k | g} \cdot s_{g}$ .

For goods outside of the focal nest, we need the formula for a diversion for when goods $j$ and $k$ are in different nests, denoted $D_{j k}^{* *}$ . This is given by

D_{j k}^{* *} = \frac{s_{k} \cdot (1 - σ)}{1 - s_{j | g} \cdot [σ + (1 - σ) s_{g})}

This suggests that for goods outside of the focal nest, the implied share should be given by

s_{k}^{i m p l i e d} = s_{k} \cdot (1 - σ)

Intuitively, this means that the implied share is given by taking the overall market share of the good and scaling it down by the nesting parameter $σ .$

Ellickson, Grieco, and Khvastunov use data on grocery stores and find estimates of the nesting parameter ranging from 0.2 to 0.3, suggesting that club store and superstores are differentiated from supermarkets, but that they are still very important alternative options. Table 1 calculates implied shares using a value of the nesting parameter of 0.3. The implied shares for stores in the supermarket chain segment must necessarily fall in between the narrow and board market shares. They are closer to the broad than the narrow shares, reflecting that superstores and club stores are fairly strong substitute for traditional supermarkets and thus they have a large degree of membership in the implied market.

The same exercise can be undertaken for demand systems other than the nested logit. Basing the calculations on the principle that the implied shares should reflect diversions proportional to implied shares will yield an implied market that directly and fully reflects the underlying demand of the market.

B. Assignment of Membership Weights

Once the correct implied market shares are known for each good in the market, the corresponding membership weights can be calculated. The advantage of framing the implied market in terms of membership weights is that it provides a more straightforward and intuitive way to arrive at the implied market shares through discounting differentiated product revenues based on their degree of differentiation. That is, it makes clear to what degree each product is a member of the implied market.

There always exist membership weights that will yield the same implied market shares as results from matching implied shares to diversion proportional to share. For this grocery store example, we use numerical calculations to determine that the corresponding membership weights are $a_{j} = 0.56$ for each store in the superstore and club store segments, and $a_{j} = 1$ for the traditional grocery chains.⁴⁹ This means that traditional grocery chains are fully in the market, while superstores and club stores are discounted so that a little more than half of their revenues are included in the implied market. Such a membership weight assignment is thoroughly tied to underlying econometric demand estimates because the resulting implied market shares reflect the estimated substitution patterns.

Table 2 shows the resulting implied revenue, which is discounted by the membership weight, and the implied market shares. The implied market shares resulting from the membership weighting described above very closely match the implied shares based on the previous calculations. The primary appeal of presenting the implied market as shown in Table 2 is that it is connected to the fuzzy market definition and membership weighting, and it shows transparently how much differentiated products are discounted when arriving at the total market size. Following the method described in Section 3, the implied market size $M$ can be calculated by summing up the discounted revenues. Implied shares are then calculated using this market size $M$ as the denominator and discounted revenues as the numerator.

Table 2.

Calculation of Implied Shares Based on Discounted Revenue.

Name	Segment	Stores	Revenue	Discounted Rev	Implied Share
Ahold	chain	604	341.9	341.9	5.9
Albertsons	chain	510	182.5	182.5	3.2
Aldi	chain	615	47.4	47.4	0.8
Delhaize	chain	1,057	226.7	226.7	3.9
Great A&P Tea Co.	chain	161	54.9	54.9	1.0
HE Butt	chain	227	184.7	184.7	3.2
Kroger	chain	1,973	914.3	914.3	15.9
Lone Star Funds (Bi-Lo)	chain	238	53.6	53.6	0.9
Publix	chain	845	354.6	354.6	6.2
Safeway	chain	1,339	569.0	569.0	9.9
Save A Lot	chain	715	82.2	82.2	1.4
Smart & Final	chain	217	31.9	31.9	0.6
Stater Bros	chain	162	62.9	62.9	1.1
SuperValu	chain	1,194	550.1	550.1	9.6
Trader Joes	chain	236	71.3	71.3	1.2
Whole Foods	chain	159	81.4	81.4	1.4
Winn-Dixie	chain	451	113.1	113.1	2.0
Meijer	super	159	131.3	73.5	1.3
Target	super	160	84.2	47.1	0.8
Walmart	super	1,385	1,474.0	825.0	14.3
BJ’s	club	122	97.3	54.5	0.9
Costco	club	311	702.7	393.3	6.8
Sam’s Club	club	540	783.9	438.7	7.6

Note. Revenue for traditional grocery chains is fully included in the implied market, while superstore and club stores are discounted by multiplying their revenue by 0.56, the membership weight that reflects their degree of differentiation. Implied shares are calculated as the share in a market based on discounted revenues.

The implied market size $M$ must necessarily fall between the total market size of the narrow market and the market size of the broad market. Consequently, the implied market shares for firms in the focal product segment, in this case the “supermarket chain” stores, must necessarily be between the narrow and broad market shares. That is because the nesting structure essentially takes the broad market shares for supermarket chains, and scales them up to reflect that other supermarket chains are closer substitutes than the differentiated grocery retail formats. The quantified degree of demand substitution determines by how much to scale up the shares.

The implied shares of stores in the club and superstore segment are based on their discounted revenue, which scales down the values from the broad shares to reflect that differentiation. In the extreme case where $σ = 1$ , the implied shares of the club and superstores would all be scaled down to zero, and the implied shares of the supermarket chains would be equal to the “narrow” market shares.

These implied shares could then be compared to some standards such as merger guideline presumptions or case law, as often required in legal analysis, but in a way that is closely tied to the economic conditions of the market.

C. Interpreting Implied Market Shares

Implied market shares lie between the narrow and broad market shares, and this can be helpful when the narrow market appears too narrow and the broad market appears too broad. An example of a hypothetical merger of grocery chains can demonstrate how one might interpret implied markets.

Suppose we need to analyze a proposed merger of two supermarket chains in the data. To best illustrate the usefulness of this approach, suppose it is a merger of the two largest supermarket chains that needs to be analyzed. Table 1 shows that the two largest traditional grocery chains in the 2006 data are Kroger and Safeway. We can calculate the metrics typically of interest, such as HHI, change in HHI, and the share of the merged firm, based on the implied market.

In these data, the post-merger HHI in the narrow market is 1,980, for the broad market it is 1,201, and for the implied market the post-merger HHI is 1,191.⁵⁰

For merger analysis, the change in HHI due to a merger and the shares of the merging firms are often of interest. For metrics such as these, the implied values will be between the narrow and broad markets, and will fall along that spectrum depending on the closeness of substitution of the differentiated products. The narrow market definition yields a combined firm share of 37% and a change in HHI of 676. At the other end, the broad market has a combined firm share of 20% and a change in HHI of 201. The implied market has a combined share of 26% and a change in HHI of 314. Because the club stores and superstores are quite important alternatives to grocery chains, the implied market yields values that are more similar to the broad market than the narrow market. With more differentiated products, the implied values would be more similar to the narrow values.

This approach has enormous potential to resolve some of the persistent difficulties in defining markets in differentiated markets such as retail. Tenn and Hosken describe the issues as they applied to the merger cases FTC v. Staples and FTC v. Whole Foods. They describe that in the Staples case, one question was whether the market should be narrow and include only the large office superstores OfficeMax, Office Depot, and Staples, in which case there would high concentration, or whether other retail formats that also sold some office supplies such as Walmart, Costco, CVS, and Best Buy should be included, in which case the concentration would be low. They write that “Clearly, both proposed market definitions fail to make use of important information about how firms are differentiated, and may provide the decision maker (the antitrust agency or a court) with misleading information as to how a merger is likely to affect competition. For this reason, many economists argue that the government and courts should dispense with market definition altogether . . .”⁵¹

Our goal has been to illustrate through this grocery example how such tensions can be resolved without completely dispensing with market definition altogether. This method proposes that one solution to the “Costco problem” for traditional grocery markets is to include Costco in the market, but at a discount of 0.56 (though of course this exact value is only illustrative, as this example has completely ignored geographic factors which are critical in any actual retail market analysis).

We also note that this method could be applied without describing competitors as partially in the market if an “outside option” can be included in the implied market in place of the differentiated competitors. For example, the shares and revenues shown in Table 2 can also be presented with the exact same values, but with superstores and club stores all aggregated into an outside option with a share equal to the sum of the implied shares of those firms. If one is uncomfortable presenting a market with competitors having partial membership, then such a presentation of the market would allow for only presenting shares for firms that are fully included—in this case, the traditional grocery supermarkets. The key is that the size of the outside option reflects the amount of substitution to all the differentiated firms, which is accomplished by assigning the outside share, and therefore the total size of the market, in such a way that the focal products exhibit diversion proportional to implied share.

VI. Conclusion

We have proposed the use of implied market shares as an alternative method to the HMT of obtaining economically grounded market shares. With the additional flexibility offered by conceptualizing the relevant market as a fuzzy set rather than a crisp set, implied market shares can reflect rich substitution patterns among differentiated products, and thus a more meaningful assessment of some aspect of the market of interest. This resolves some of the longstanding difficulties faced when implementing market definition with a traditional binary in/out designation.

The implied market share approach is an appealing alternative to the HMT, which is subject to critique of being circular, unnecessary, and even misleading. Implied market shares also have lower information requirements than the HMT. While both approaches require knowledge of demand and substitution patterns, the HMT additionally requires both knowing firms’ economic profit margins and making an assumption about how they interact competitively. Both those objects are difficult to obtain. It has long been recognized that accounting margins and economic margins are not the same, and that additional work and assumptions are required to determine economic margins even when accounting information is available. Moreover, predictions about the profitability of a hypothetical monopolist to raise prices will depend on the model of competition assumed, and it is not always obvious which model of competition is best suited to a particular industry.⁵² Thus, it can be appealing to have a method to determine markets that is focused exclusively on demand and does not require additional information burden beyond demand-side factors.

The main benefit of this framework is transparency: it is clear at which junctures exactly which choices are being made in order to facilitate analysis. Market definition, as many commentators admit, is ambiguous. The fuzzy set framework here has two clear advantages. First, it is an explicit representation of that ambiguity and helps locate possibilities between narrow and broad market definitions that might be alternative plausible ways of defining the market of interest. Second, the formal apparatus presented in this paper takes seriously what all parties to this debate seem to agree on, despite being ostensibly at odds with one another, which is that “deciding where to look for effects on market outcomes requires some implicit judgments about the relatedness of goods that traditional market definition might class as in separate markets.”⁵³

The upshot of this formalism is primarily conceptual. While it offers no particular policy advice, nor does it imply any particular market as the relevant market for litigation purposes, it allows us to frame the relevant domain of inquiry in a transparent and general way. How this would be implemented in decision-making procedures has been left to future work. For example, since currently used concentration thresholds are based on the traditional approaches to market definition, additional research would be required to know what values would be appropriate to compare against implied market shares and implied HHI’s.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Authors’ Note

The views expressed in this article are those of the authors and do not necessarily reflect those of the Federal Trade Commission or any individual Commissioner.

ORCID iD

Matthew T. Panhans

1.

Jonathan B. Baker, Market Definition: An Analytical Overview, 74 Antitrust Law J, 129–173 (2007) at 129 (“Throughout the history of U.S. antitrust litigation, the outcome of more cases has surely turned on market definition than on any other substantive issue.”); Julie Carlson et al., Economics at the FTC: Deceptive Claims, Market Definition, and Patent Assertion Entities, 51.4 Rev. Ind. Organ., 487–513 (2017) at 496 ([Describing the hospital merger cases Hershey/Pinnacle and Advocate/Northshore,] “the FTC first lost, and then won, both cases based on how it defined the relevant set of competitors and competition. In other words, the cases turned on ‘market definition.’”)

2.

United States v. Phila. Nat’l Bank, 374 U.S. 321, 363 (1963) (“[A] merger which produces a firm controlling an undue percentage share of the relevant market, and results in a significant increase in the concentration of firms in that market, is so inherently likely to lessen competition substantially that it must be enjoined in the absence of evidence clearly showing that the merger is not likely to have such anticompetitive effects.”); U.S. Dep’t of Justice & Fed. Trade Comm’n, Horizontal Merger Guidelines § 4.1.2 (2010) [hereinafter 2010 Horizontal Merger Guidelines]; U.S. Dep’t of Justice & Fed. Trade Comm’n 2023 Merger Guidelines [hereinafter 2023 Merger Guidelines].

3.

Phila. Nat’l Bank, 374 U.S. at 360 n.37.

4.

Daniel A. Crane, Market Power Without Market Definition, 90 Notre Dame L. Rev. 31 (2014).

5.

Baker, supra note 1 at 132 (“U.S. courts have long emphasized that markets should be defined with respect to the economic force of demand (buyer) substitution.”); Baker, supra note 1 at 138 (“But the better approach, as described above, is to account for only one economic force, demand substitution, in the market definition stage of the competitive effects analysis, and to account for supply substitution in one or more other steps of the analysis (identifying market participants, analyzing competitive effects, and evaluating entry conditions).”)

6.

Upward pricing pressure analysis and merger simulations also require information on price-cost margins and taking a stand on the form of strategic interaction among the firms in the market (i.e. the model of competition).

7.

Richard S. Markovits, Predicting the Competitive Impact of Horizontal Mergers in a Monopolistically Competitive World: A Non-Market-Oriented Proposal and Critique of the Market Definition-Market Share-Market Concentration Approach, 56 Tex. Law Rev. 587 (1978); Richard S. Markovits, International Competition, Market Definition, and the Appropriate Way to Analyze the Legality of Horizontal Mergers under the Clayton Act: A Positive Analysis and Critique of Both the Traditional Market-Oriented Approach and the Justice Department’s Horizontal Merger Guidelines, 64 Chi. Kent. L. Rev. 745 (1988).

8.

Robert D. Willig, who was Deputy Assistant Attorney General in the U.S. Department of Justice Antitrust Division at the time, wrote in 1991: “ignoring the general language of the 1984 guidelines and focusing on collusion as the only competitive effect of concern leads to a confining and confusing framework for merger analysis . . . It may be useful as a matter of terminology to divide theories of possible anticompetitive effects of mergers into two categories: unilateral effects and coordinated effects. Unilateral effects are changes in the actions of the merging firms that would be profitable for them as a result of the merger if the nonparties did not alter their actions or if the nonparties reacted unilaterally themselves.” Robert D. Willig, Merger Analysis, Industrial Organization Theory, and Merger Guidelines, Brookings Papers on Economic Activity—Microeconomics, 281–332 (1991).

9.

Malcolm B. Coate & Shawn W. Ulrick, Does Merger Enforcement Depend on the Portion of the Merger Associated With the Competitive Concerns? 5.3 J. Antitrust Enforc. 407–33 (2017).

10.

J. A. Keyte, Market Definition and Differentiated Products: The Need for a Workable Standard, 63 Antitrust Law J. 697 (1995).

11.

Adriaan ten Kate & Gunnar Niels, The Relevant Market: A Concept Still in Search of a Definition, 5.2 J Compet. Law Econ. 297–333 (2009).

12.

Sean P. Sullivan, Modular Market Definition, 55 UC Davis L. Rev. 1091–1148, 1111 (2021).

13.

Sean P. Sullivan, Seven Myths of Market Definition, CPI Antitrust Chronicle, 2022, at 5.

14.

Sean P. Sullivan, supra note 12, at 1115.

15.

Louis Kaplow, Why (Ever) Define Markets? 124.2 Harv. Law Rev. 81 (2010).

16.

T. Valletti & H. Zenger, Mergers with Differentiated Products: Where Do We Stand? 58 Rev. Ind. Organ. 179–212 (2021), at 182.

17.

Johnny Shaw, Avoiding Market Definition Under Section 1 of the Sherman Act, 88.3 Fordham L. Rev. pp. 1133–71 (2019), at 1150.

18.

Id. at 1150.

19.

Herbert J. Hovenkamp, Analyzing Horizontal Mergers: Unilateral Effects in Product-Differentiated Markets, Faculty Scholarship at Penn Carey Law 1802 (2009) at 15; Hovenkamp also provides illustrative examples showing that a relevant market may look any number of ways, despite its being the case that we know what the “closest substitutes” are.

20.

Carl Shapiro & Howard Shelanski, Judicial Response to the 2010 Horizontal Merger Guidelines, 58 REV. IND. ORGAN. 51–79 (2021) at 60.

21.

Malcolm B. Coate & Joseph J. Simons, In Defense of Market Definition, 57.4 Antitrust Bullet. 667–717 (2012), at 689.

22.

Id. at 669.

23.

2023 Merger Guidelines at 6.

24.

Kaplow, supra note 15, at 440.

25.

Louis Kaplow, Market Definition and the Merger Guidelines, 39.1/2 Rev. Ind. Organ. 107–25 (2011), at 112.

26.

Louis Kaplow, Market Definition Alchemy, 57.4 Antitrust Bullet. 915–52 (2012), at 921.

27.

Kaplow, supra note 15, at 110.

28.

For another critical view, see Herbert J. Hovenkamp, Markets in Merger Analysis, 57.4 Antitrust Bullet. 887–914 (2012).

29.

Kaplow, supra note 26.

30.

Coate & Simon, supra note 21 (n 15); Malcolm B. Coate, Market Definition is Not Alchemy, SSRN Scholarly Paper (2014); Gregory J. Werden, The Relevant Market: Possible and Productive, Antitrust Law J. Online (2014) (addressing Kaplow’s use of the Lerner index). Werden explicitly counters Kaplow that “the formula is fully applicable to multi-good industries . . . Moreover, neither economists nor courts use the formula much or care much about the amount a defendant sets its price above short-run marginal cost. Market shares are used to answer different questions, which depend on the case” (p. 4).

31.

Id. at 3.

32.

Id. at 4.

33.

Id. at 2.

34.

Gregory J. Werden, Why (Ever) Define Markets? An Answer to Professor Kaplow, 78.3 Antitrust Law J. 729–46 (2013) at 746 (“Separating active and passive competitive forces is part of economic analysis because economic models distinguish the strategic action of competitors from background influences on them.”).

35.

Malcolm B. Coate et al., Tailoring Critical Loss to the Competitive Process, 65 Int’l Rev. L. & Econ. (2021) at 3.

36.

Werden, supra note 34 at 737 (n 27).

37.

Malcolm B. Coate & Jeffrey H. Fischer, Is Market Definition Still Needed After All These Years? 2.2 J. Antitrust Enfor. 422–50 (2014). Among three settings of homogeneous goods, static differentiated goods, and dynamic differentiated goods, Coate and Fischer conclude that “Standard game-theoretic models or empirical estimation is most useful in the second structure, especially when the results of the model can be validated with other evidence. Market definition is more useful in the first and third structure” (p. 423). And “that the choice between these methodologies depends on the institutional realities and the facts available to the analyst” (p. 430).

38.

Werden, supra note 30, at 2.

39.

See Christopher Conlon & Julie Holland Mortimer, The Role of Diversion Ratios and Surveys in Horizontal Mergers (2022), .

40.

This also accords as well with the structure of the newly released 2023 Merger Guidelines. Guideline 1 concerns market concentration and shares, stating that the reason they matter is because high concentration “creates significant risk that the merger may substantially lessen competition or tend to create a monopoly.” There is no connection to any specific type of effect, whether unilateral or coordinated. Rather, Guideline 2 (“Mergers Can Violate the Law When They Eliminate Substantial Competition Between Firms.”) covers unilateral effects and Guideline 3 (“Mergers Can Violate the Law When They Increase the Risk of Coordination”) covers coordinated effects. These are separate aspects of merger analysis today, and we are not proposing to change that.

41.

By requiring the designation of a focal product, this method makes explicit that relevant antitrust markets are defined with respect to some specific competitive issue or theory of harm, and mergers between different pairs of companies could therefore implicate different relevant markets. See David Glasner and Sean Patrick Sullivan, The Logic of Market Definition, 83.2 Antitrust Law J. (2020) Glasner and Sullivan refer to this issue as the ‘independent market fallacy’: “As purely analytical constructs, markets do not exist independent of a problem or inquiry but must be defined in terms of a problem or inquiry. Specifically, antitrust markets are defined in terms of specific theories of anticompetitive harm” (p. 298). See also Brief for the Federal Trade Commission as Amicus Curiae, Staley v. Gilead Scis., 19-cv-02573-EMC (N.D. Cal. Apr. 11, 2022): “As these cases demonstrate, the relevant product market may vary considerably, depending on the alleged anticompetitive effects at issue. And if there are multiple theories of harm in the same case, as here, the case may implicate multiple relevant markets. In such cases, the court’s task is to assess each alleged market on its merits.” Available at .

42.

Daniel Hosken & Steven Tenn, Chapter 11: Horizontal Merger Analysis in Retail Markets. In Handbook on the Economics of Retailing and Distribution (Cheltenham, UK: Edward Elgar Publishing 2016).

43.

Paul B. Ellickson et al., Measuring Competition in Spatial Retail, 51.1 RAND J. Econ. 189–232 (2020).

44.

The revenue reported in EGK is the average weekly revenue at each location, which we aggregate up to the chain level. We also aggregate the Ahold and Delhaize banners into a single observation for the parent company, and show only chains with greater than 150 locations. Otherwise, this table is identical to Table 3 in that paper.

45.

See also Valletti & Zenger, supra note 16; Valletti and Zenger have proposed calculating implied market shares from diversion ratios assuming logit demand. We generalize this logic to be applicable to any demand system and extend the application to the market definition exercise through an implied market from which implied shares can be calculated.

46.

See Gregory J. Werden & Luke M. Froeb, The Antitrust Logit Model for Predicting Unilateral Competitive Effects, 70.1 Antitrust Law J. 257–60 (2002).

47.

For the details on the nested logit model, see Chapter 4 of Kenneth E. Train, Discrete Choice Methods with Simulation. 2nd ed. (Cambridge: Cambridge University Press 2009).

48.

Christopher Conlon & Julie Holland Mortimer, Empirical Properties of Diversion Ratios, 52.4 RAND J. Econ. 693–726 (2021).

49.

Specifically, set aj = 1 for the traditional grocery chains, and find remaining aj by scaling down revenues until implied shares are matched, that is find a_j that solves $s_{j}^{i m p l i e d} = \frac{a_{j} r e v_{j}}{\sum a_{k} r e v_{k}}$ .

50.

The implied HHI is not necessarily in between the HHI of the narrow and broad markets. In this case, WalMart is the largest firm in the data set. Because the implied market discounts WalMart revenue, as it is differentiated from the supermarket chains, this causes the implied HHI to actually fall compared to the broad market.

51.

Hosken & Tenn, supra note 42 at 253.

52.

See Matthew T. Panhans & Charles Taragin, Consequences of Model Choice in Predicting Horizontal Merger Effects, 89 Int. J. Indust. Org. 102986 (2023).

53.

Mark A. Lemley & Mark P. McKenna, Is Pepsi Really a Substitute for Coke? Market Definition in Antitrust and IP, 100.6 Georgetown L. J. 2055–118 (2012) at 2100.