Competitive Disclosure of Multiple Product Attributes*
James A. Dearden

David Goldbaum

Ernest K. Lai§

Qichao Shi¶

April 4, 2022

Abstract
We analyze a duopolistic model of quality disclosure in which product differentiation can be either horizontal or vertical under two vertically differentiated productattributes. Disclosure is fully revealing in the absence of disclosure cost. When disclosure is costly, firms partially disclose. We apply the familiar economic propertiesof absolute and comparative advantages in a novel way to characterize partial disclosure under the multiple product attributes. The type of product differentiation andintensity of competition interact with absolute and comparative attribute advantages
in determining the equilibrium non-disclosure sets of the firms. When consumers fallshort of forming rational expectation after non-disclosure, firms may partially disclose even without disclosure cost, and the same properties continue to dictate firms’
disclosure behavior in the boundedly rational paradigm.
Keywords: Quality Disclosure; Multiple Product Attributes; Product Differentiation;
Absolute and Comparative Advantages; Boundedly Rational Consumers
JEL classification: C72; D21; D82; L13; L15* We thank conference participants at the 3rd Annual Meeting of the Institute of Urban Developmentof Nanjing Audit University and the 2019 Industrial Organization and Growth Conference, and seminar
participants at Lehigh University and Southwestern University of Finance and Economics for valuable
discussion and feedback.
 Department of Economics, Lehigh University, Bethlehem, PA 18015, USA. Email: jad8@lehigh.edu.
Economics Discipline Group, University of Technology Sydney, Ultimo, NSW, Australia. Email:david.goldbaum@uts.edu.au.
§ Department of Economics, Lehigh University, Bethlehem, PA 18015, USA. Email: kwl409@lehigh.edu.
¶ School of Economics and China Center for Behavioral Economics and Finance, Southwestern Universityof Finance and Economics, Chengdu, Sichuan, China. Email: shiqc@swufe.edu.cn.

1Introduction

Firms engaging in competition are motivated to make known to consumers the superiorityof their products relative to their competitors’. When products have multiple attributes,
this incentive often manifests as a tendency to underscore the product attributes that
represent one’s strength and to downplay the attributes that are not as remarkable. In a
familiar term, absolute and comparative advantages of some sorts are at work to influence
the marketing efforts of firms in deciding what to promote to, and what to hide from,consumers. This paper analyzes how this and other factors contribute to determine quality
disclosure in a competitive setting with multiple product attributes.
To demonstrate the superiority of one’s product is to establish the inferiority of thecompetitors’; the comparative information that firms reveal to consumers is bound tocover not only the qualities of their own products but also those of the rivals. One popular
medium that makes available this kind of competitive-product information is the productcategory reviews provided by third parties. Examples abound of this kind of reviews, whichhave been shown to influence consumer choices.1 There are third-party reviews provided bytraditional publishers (Consumer Reports, PC Magazine, Car and Driver, Runner’s World,
U.S. News & World Report’s Best Colleges), online platforms founded by individuals (The
Points Guy, Tom’s Guide), and resellers (TireRack ).
A classic example of third-party reviews with attribute breakdown is the “Best Cars
and Top 10 Lists” provided by the infomediary Kelley Blue Book. Their lists contain links
to detailed reviews of each selected vehicle, covering its “pros and cons” or the “what welike and don’t like,” which are essentially reviews of attributes vehicle shoppers care about.
The marketing campaign of an automobile manufacturer that promotes its products making the top lists effectively reveals to potential buyers, who otherwise may not be aware of
the lists, the attribute qualities of its products as well as the competitors’. What drives afirm’s disclosure decision when the quality information being disclosed is bundled, covering
one’s qualities as well as rivals’ ? What are the implications of absolute and comparative advantages in product attributes for the informational value of quality disclosure to
consumers? How do product differentiation and disclosure cost play a role?
We tackle these questions with a duopoly model with four key features: two products
each with two vertically differentiated attributes, consumers with heterogeneous preferences,
first-stage verifiable attribute-quality disclosure, and second-stage price competition. The
two vertically differentiated product attributes bring in a novel disclosure environment inwhich the products themselves can be either horizontally or vertically differentiated, and1See, e.g., the empirical studies by Pope (2009), Simonsohn (2011), and Luca and Smith (2013).

1

the key concepts in our characterizations of disclosure behavior, absolute and comparativeadvantages, are defined in terms of quality differences of these attributes. Absolute attributeadvantages refer to the quality difference between the two products with respect to agiven attribute, and comparative attribute advantages are differences (between attributes)
in differences (between qualities).
Consider a sports sedan that is far superior to a family sedan in terms of driving
performance and also offers a slightly more comfortable ride. Both performance and comfort
are vertically differentiated attributes, and the sports sedan has absolute advantages in both
attributes and comparative advantage in performance. Though the family sedan is inferior
in both attributes, it has comparative advantage in comfort because it is relatively lessinferior in this attribute. In terms of product differentiation, with the sports sedan having
two absolute advantages the two vehicles are vertically differentiated. On the other hand,
if the performance suspension of the sports sedan were to result in too stiff a ride thatwould make it less comfortable than the family sedan, then each vehicle would have its
own absolute advantage—the two sedans would be horizontally differentiated.
We characterize the equilibrium disclosure outcomes—loosely, what consumers know inequilibrium about the attribute-quality differences or attribute advantages—which reflect
the informational value of the disclosure to consumers. In a benchmark case with costless
disclosure, full revelation, which arises under the well-known unraveling result of Grossman(1981) and Milgrom (1981) in the canonical model of one firm with one-dimensional quality,
also arises in our duopolistic model with two-dimensional quality. The main insights of ourmodeling exercise, however, lie in the situations where at least some attribute advantages
are not disclosed by either firm so that consumers are only partially informed and quality
disclosure falls short of bringing the highest informational value to consumers. Such apartially revealing equilibrium arises when disclosure is costly.
We further develop the sedan example to exemplify. The sports sedan is in a sense theleading product of the two. Being in the performance genre, it certainly has advantage—
absolute and comparative—in driving performance, and the uncertainty faced by consumers
in respect of this attribute is limited to how much more powerful it is than the family sedan.
On the other hand, while the sports sedan may or may not offer a more comfortable ride
than the family sedan, the difference one way or the other would not be as dramatic as thetheir difference in performance. Our equilibrium analysis shows that the manufacturers of
the two vehicles, with their different prior positions in the market, would have different
considerations in their disclosure decisions under costly disclosure.
Selling the a priori trailing product, the family-sedan maker’s disclosure decision hinges
on comparative attribute advantages, which are characterized jointly by the absolute ad-2vantages in performance and comfort. Comparative attribute advantages weaken when (a)the sport sedan becomes less powerful (but still more so than the family sedan), in whichcase the family sedan would be facing the sports sedan priced more competitively, and (b)the sports sedan offers more comfortable (or less uncomfortable) ride relative to the family
sedan, in which case the family sedan would be losing consumers looking for comfort to thesports sedan. The effects on both prices and market shares resulting from weaker comparative advantages are unfavorable to the family-sedan maker. Accordingly, when disclosure
is costly so that there are some attribute qualities that firms find it not worthwhile to disclose, in equilibrium the firm selling the trailing product would not disclose the attribute
qualities at which comparative advantages are sufficiently weak.
Selling the a priori leading product, the sports-sedan maker’s disclosure decision depends on absolute attribute advantages, which in turn determine the degree of productdifferentiation. Regardless of whether the two vehicles are horizontally or vertically differentiated, the degree of differentiation decreases when (a′ ) the sports sedan becomes lesspowerful, in which case it would be losing consumers looking for performance to the family sedan, and (b′ ) the difference in comfort between the two vehicles in either direction
narrows, in which case the sports sedan would have to be priced more competitively tomaintain market share. For these reasons, when disclosure is costly, in equilibrium thefirm selling the leading product would not disclose the attribute qualities when absolute
attribute advantages are sufficiently weak and thus the products are sufficiently similar.
Our analysis of partially revealing equilibria, which represents the key contribution ofthis paper, generates three main insights about disclosure behavior and outcomes. The
sedan example above illustrates the first insight. In our model with two vertically differentiated product attributes, absolute and comparative advantages defined in terms of these
attributes fully characterize the “anchors” of the firms’ non-disclosure sets. The leading
firm’s non-disclosure set anchors at where absolute attribute advantages are weakest, whilethat of the trailing firm anchors at where comparative attribute advantages are weakest.
The second insight concerns product differentiation and the relationship between theanchors. If consumers know ex ante that each firm has an absolute advantage in one
attribute (i.e., products are horizontally differentiated), then the anchors of the firms’ nondisclosure sets coincide. However, if consumers ex ante do not know whether one firm
has an absolute advantage in both attributes (i.e., products are vertically differentiated)or each firm has an absolute advantage in one attribute (i.e., products are horizontallydifferentiated), then the anchors differ. In terms of the sedan example, conditions (a) and(a′ ) are equivalent, while (b) and (b′ ) may or may not be the same depending on consumer
priors about the type of product differentiation.

3

The third insight concerns the disclosure outcomes. While the two firms have theirown non-disclosure sets, consumers are only uninformed of the attribute qualities that lie
in their intersection; outside of the intersection, consumers learn the attribute qualities
from at least one firm. We find that even when the anchors of the two non-disclosure setsdiffer and the disclosure cost is vanishing, there are attribute qualities that neither firm
discloses. In equilibrium, firms engage in an optimal tradeoff between, on one side of theequation, letting consumers know precisely the attribute qualities and, on the other side,saving the disclosure cost at the expense of being pooled with the non-disclosed qualities.
The partially-revealing disclosure outcomes also arise in the absence of disclosure cost if
consumers fail to extract the informational content of non-disclosure. The presence of
disclosure cost or boundedly rational consumers limits the informational value that quality
disclosure delivers to consumers in our environment with multiple product attributes.
To the best of our knowledge, we are the first to analyze competitive disclosure with twodimensional vertical quality. Previous studies on quality disclosure have analyzed models
with single-dimensional quality (e.g., Board, 2009; Guo and Zhao, 2009; Koessler and Renault, 2012; Celik, 2014; Janssen and Teteryatnikova, 2016; Guo, 2020), multidimensional
quality under a monopoly (e.g., Sun, 2011), and duopolies differentiated by one verticaland one horizonal qualities (e.g., Levin et al., 2009; Anderson and Renault, 2009; Ghoshand Galbreth, 2013; Gu and Xie, 2013; Hotz and Xiao, 2013). The apparently slight difference of our environment leads to unique features of the equilibria. The strategic forcesthat drive non-disclosure in our model are multifaceted and differ from the typical resultsin the literature, where firms choose not to disclose either when quality is lowest (Board,
2009; Guo and Zhao, 2009; Levin et al., 2009) or when prices are lowest under minimal
product differentiation (Board, 2009). Simply put, our difference in this regard is that nondisclosure is characterized by, for one firm, minimal product differentiation but not lowest
prices, and, for the other firm, lowest prices but not minimal product differentiation.
Related Literature. We proceed to a more systematic and thorough review of the recent
literature on quality disclosure.2 Studies in this literature can be categorized based on
three dichotomies with respect to (a) market structures: monopoly or duopoly; (b) productquality: single-dimensional or two-dimensional; and (c) the unraveling result: whether itholds or fails.3 Our study belongs to the combination of duopoly, two-dimensional quality,and possible failing of unraveling. Naturally, previous studies in the same category (Levinet al., 2009; Anderson and Renault, 2009; Ghosh and Galbreth, 2013; Gu and Xie, 2013;
Hotz and Xiao, 2013) are closest to ours.
2

Dranove and Jin (2010) provide an excellent survey of the theoretical and empirical studies tracing to
the inception of the literature.
3
This last dichotomy correlates with, and thus largely covers, another important variation: whether
disclosure is costly or not.

4

In a costly-disclosure environment with two-dimensional quality, one horizontal andone vertical, Levin et al. show that a cartel is more likely to disclose than when the firms
are in duopoly.4 Anderson and Renault and Gu and Xie also consider horizontal-verticaldimensions of quality, but the vertical quality in their models is known to consumers.
Similar to our disclosure technology, Anderson and Renault allow disclosure of comparative
information interpreted as “comparative advertising.” The low-quality firm in their model
discloses, but neither firm would disclose if comparative advertising was banned. Disclosurein Anderson and Renault is costless; under costly disclosure and without the possibilityof disclosing comparative information, Gu and Xie obtain the opposite result, where thehigh-quality firm discloses and the low-quality firm can possibly free-ride. Free-riding on
the other firm’s disclosure also can arise in our model.
Ghosh and Galbreth study costly disclosure in the presence of partially informed consumers who can search for information at a cost. They show that firms disclose less information as there are more consumers who are informed about the qualities of one firm
but not the other’s, or when search costs increase. Hotz and Xiao consider horizontal andvertical qualities that are correlated and unknown to consumers, and the unraveling result
fails in their costless-disclosure environment due to the assumed correlation. Despite being
in the same broad category, as stated above our model differs from these studies in termsof both the environment and the results.5
In environments that intersect with ours in terms of market structures but not quality
dimensions, Board (2009) and Guo and Zhao (2009) analyze how competition drives disclosure under a single vertical quality. Board demonstrates that disclosure may intensify
price competition to such an extent that it outweighs the benefit of being separated fromlower-quality products, upsetting the unraveling even in the absence of disclosure cost. This
stands in contrast to our benchmark case with costless disclosure, in which the additional
dimension of vertical quality effectively recovers the unraveling. Guo and Zhao examine theeffects of timing. They find that the leader in sequential disclosure reveals less informationthat its counterpart in simultaneous disclosure.
Janssen and Teteryatnikova (2016) analyze quality disclosure in the same category as
Board and Guo and Zhao, but the single-dimensional quality is horizontal. As in Anderson
and Renault, they examine the role of comparative disclosure. A distinguishing feature of
their model is that prices may be announced as part of the product information. Theyfind that unraveling can fail when disclosure is non-comparative or when price information
4

In a note, Jansen (2017) extends the model in Levin et al. by allowing the degree of differentiation of
the horizontal quality to vary and evaluates the resulting impacts on disclosure.
5
Kuksov and Lin (2010) study strategic information provision in a related environment of multidimensional information and two vertically differentiated firms, but the two dimensions of their information
concern product quality and consumers’ uncertainty about their own preferences.

5

is not integral to the disclosure. Sun (2011) analyzes an environment that intersects withours in terms of the number of quality dimensions, but the dimensions she considers are
one horizontal and one vertical. She also considers a different market structure and showsthat unraveling can fail in a monopolistic model.6
There are other related studies outside of the three dichotomies. Our disclosure technology features comparative information. In a market environment similar to ours with
two firms and two product attributes, Shaffer and Zettelmeyer (2009) examine the conditions under which firms prefer comparative to non-comparative advertising. Iyer and Singh(forthcoming) investigate a persuasion contest in which a firm can disclose own positive information or negative information about the rival. They characterize the conditions underwhich positive or negative communication equilibria prevail.7
Finally, our consideration of boundedly rational consumers relates our paper to a line ofwork that studies how the unraveling result fares when consumers depart from the rational
ideals (e.g., Milgrom and Roberts, 1986; Fishman and Hagerty, 2003; Hirshleifer and Teoh,
2003; Hirshleifer et al., 2004; Zhang and Li, 2021). Closest to us is Hirshleifer et al., whofind in a single-dimensional environment that the presence of receivers who make naı̈veinferences about non-disclosure dampens disclosure. Under a different behavioral variation,
Zhang and Li find that consumer loss aversion, by contrast, could result in more disclosure.
The rest of the paper is organized as follows. Section 2 describes the oligopolistic
disclosure model. Section 3 analyze the simultaneous price competition in the second stageof the game. Section 4 contains the main analysis of the paper, in which we analyze theequilibrium disclosure decisions and outcomes. Section 5 concludes.

2

The Model

Firms and Consumers. Two firms, firm A and firm B, sell product A and productB respectively. The two products belong to the same product class, and each producthas two vertically differentiated attributes. The quality of attribute i ∈ {1, 2} of product
K, qK
K ∈ {A, B}, qiK , is distributed on [q Ki ] according to distribution function Fi . Thei
6

The disclosure environment with no intersection with ours in terms of the number of quality dimension
and market structures is monopoly with one single-dimensional quality. Two studies in this category are
Koessler and Renault (2012) and Celik (2014), who find conditions under which the unraveling results may
or may not hold with horizontal quality and price commitment. In a recent study, Guo (2020) considers
single-dimensional disclosure in a bilateral-monopoly model of upstream and downstream firms.
7
We use non-seller sources of information—third-party product reviews—as motivation for our disclosuretechnology of comparative information. In relation to this, Chen and Xie (2005) examine how firms adapt
their advertising and pricing strategies in the presence of third-party reviews, and Chen and Xie (2008)
analyze how firms’ marketing communication strategy responds to consumer reviews.

6exogenously determined profile of attribute qualities are known by both firms.
The market for the product class is made up of a unit mass of heterogeneous consumers,
each interested in purchasing one unit of one of the two products. Consumers are uncertain
about the attribute qualities of each product and are commonly known to use the quality
distributions as their priors. A consumer who purchases product K receives utility:
uK = r + θq1K + (1 − θ)q2K − pK ,where θ is a parameter measuring the relative weight of consumer’s taste toward attribute 1
and attribute 2, pK is the price of product K, and r is the reservation value of the productclass assumed to be large enough so that all consumers purchase one of two products. The
heterogeneity of consumers is captured by the distribution of θ, which is uniform on [0, 1].
While each attribute is vertically differentiated, the heterogeneity of consumer preferences
implies that the two products can be either horizontally or vertically differentiated.
Consumers may learn about attribute qualities from the firms. In the first of the twostages of the game, firms A and B simultaneously decide whether to disclose the qualities. A
firm incurs c ≥ 0 when it discloses. This cost parameter is common to both firms. Disclosureis credible and verifiable. All consumers learn all attribute qualities of both firms if at least
one firm discloses. In the second stage of the game, the firms simultaneously compete in
prices. If neither firm discloses in the first stage, consumers make purchase decisions basedon expected attribute qualities q̂iK , i ∈ {1, 2} and K ∈ {A, B}.
Attribute Advantages. The unit of analysis of our model is the difference in attribute′′
qualities, which we term attribute advantage. We denote by qiK-K = qiK − qiK the excess inthe quality of attribute i of product K over that of product K ′ ≠ K. The prior distribution′′of qiK-K , FiK-K , is derived accordingly as the convolutions of the primitive distributions,
′
′
′
K
K-K ′
K′
] = [q K, q K-K− qK, i ∈ {1, 2}, iswith support [q K-Kii , q i − q i ]. We assume that Fi
i
i
K-K ′
continuously differentiable with everywhere positive density fi.
We define two types of attribute advantages, absolute and comparative. Firm K is′
said to have absolute advantage in attribute i if qiK-K > 0, i.e., product K has a higher
quality than product K ′ ≠ K in respect of attribute i. We refer to a larger difference′
qiK-K as a greater absolute attribute advantage or the advantage improves.8 While absoluteadvantages are differences in qualities, comparative advantages are differences in differences;′′firm K is said to have comparative advantage in attribute i if qiK-K −qiK-K> 0, i.e., compared′
Since qiK-K = −qiK -K , one can also refer to qiK-K > 0, which is equivalent to qiK -K < 0, as firm K ′ having′
an absolute disadvantage in attribute i and a larger, i.e., less negative, qiK -K < 0 as a smaller absolutedisadvantage. Both a greater absolute attribute advantage and a smaller absolute attribute disadvantage
would be referred to as an improvement in absolute advantage.
8

′

′

′

′

7

to firm K ′ , firm K has a greater absolute advantage in attribute i than in attribute i′ ≠ i.
As with the term used in the context of trade, comparative advantages in our environment are mutual and reciprocal. When firm K has comparative advantage in attribute i,firm K ′ must have comparative advantage in attribute i′ . We refer to a larger difference
in differences in qualities as a stronger comparative attribute advantage or the advantage′′′′= qiK′ -K −qiK -K , as a firm’s comparative attribute advantage
strengthens. Since qiK-K −qiK-K′
strengthens, so does that of the other firm.
For expositional consistency, our notational default is to express attribute advantages by
qiA-B , supplemented with occasional uses of qiB-A to provide intuition. We make the following] × [q A-B], whichassumptions about the space of attribute advantages, Q = [q A-B, q A-B, q A-B12
12
delineate the absolute and comparative attribute advantages of the two firms:].
] ⊆ [−q A-B, q A-B, q A-B
Assumption 1. The space of attribute advantages Q satisfies [q A-B
2
2
1
1
Assumption 1 encompasses three sets of restrictions. First, firm A always has absoluteadvantage in attribute 1. Second, comparative advantage in attribute 1 generically rests
with firm A, which implies that comparative advantage in attribute 2 generically resides
with firm B. Third, we make no blanket assumption about which firm has absolute advantage in attribute 2, but there is a size restriction: the size of absolute advantage inattribute 2 is never greater than that of firm A’s absolute advantage in attribute 1.9 These
three restrictions give a sense that firm A is the firm with the leading product.
Demand and Product Differentiation. As a general formulation, we describe purchasedecisions and consumer demand in terms of expected qualities, which would coincide withthe actual values if consumers are perfectly informed. Given prices p = (pA , pB ) and apair of expected attribute advantages q̂ A-B = (q̂1A-B , q̂2A-B ), demand for the two products isdelineated by a consumer who is indifferent between purchasing products A and B andthus has a taste threshold:10
pA − pB − q̂2A-B
θ̃ = A-B.
(1)q̂1 − q̂2A-B
This indifferent consumer divides the unit mass of consumers into two groups, those withθ > θ̃ who prefer to purchase Product A and those with θ < θ̃ who prefer to purchase] that q A-B
To furnish the details, note that it is implicit in [−q A-B, q A-B> 0 so that firm A always
1
1
1
A-B
B-A
A-B
B-A
has absolute advantage in attribute 1; q 1 − q 2 = q 2 − q 1 ≥ 0 implies that comparative advantages
in attributes 1 and 2 generically rest with firms A and B respectively; finally, given that q A-B≥ −q 1A-B
2] ⊆ [−q A-B] implies that q A-B
is equivalent to q A-B≥ q B-A, the set inclusion [q A-B, q A-B, q A-B≥ q 2A-B
2
2
1
2
1
1
1
and q A-B
≥ q B-A, i.e., the smallest possible absolute advantage in attribute 1 is no smaller in size than the2
1
greatest possible absolute advantage in attribute 2, which may be held by either firm (there is no restriction
on the sign of q A-B). Note also that Assumption 1 places no direct restriction on the upper bound q A-B.
2
1
10
Throughout the paper, when referring to a pair of variables, we omit the subscript in the single variable.
98

product B. Accordingly, with q̂1A-B > q̂2A-B under Assumption 1, consumer demand for thetwo products, D(p, q̂ A-B ) = (DA (p, q̂ A-B ), DB (p, q̂ A-B )), is characterized by:⎧⎪(1, 0)
if pA − pB ≤ q̂2A-B ,
⎪
⎪
⎪
⎪
⎪
D(p, q̂ A-B ) = ⎨(1 − θ̃, θ̃) if pA − pB ∈ (q̂2A-B , q̂1A-B ),
⎪
⎪
⎪
⎪
⎪
⎪
if pA − pB ≥ q̂1A-B .
⎩(0, 1)(2)

The demand function makes clear that under the consumer priors both vertical andhorizonal expected product differentiations can arise in our model.It follows from Assumption 1 that firm A has expected absolute advantage in attribute 1 (q̂1A-B > 0). If expected
absolute advantage in attribute 2 also rests with firm A (q̂2A-B > 0), then there is expectedvertical product differentiation: per the first case in (2), all consumers prefer to purchaseproduct A when the two products are priced the same.11 If instead expected absoluteadvantage in attribute 2 resides with firm B (q̂2A-B < 0), then there is expected horizontal
product differentiation: the second case in (2) applies, where holding prices the same someconsumers prefer product A and some prefer product B.12
We assume that firms incur no costs of production, and thus profits equal revenues.13
The profit of firm K ∈ {A, B} is πK (p, q̂ A-B ) = pK DK (p, q̂ A-B ). We analyze how the instrument at firms’ disposal to maximize profits—disclosure, which dictates consumer expectations and, in turn, prices and demands—is determined in equilibrium. We begin our
analysis in the next section with the pricing decisions in the second stage of the game.

3

Equilibrium Analysis: Pricing

3.1

Equilibrium Prices

The priors and firms’ disclosure decisions in the first stage determine the consumer posterior expectations about attribute advantages, which we take as parameters for analysing
11Standard definitions of product differentiation (Cremer and Thisse, 1991) compare two products sold at
the same price. If each product priced the same has a positive demand, then the products are horizontallydifferentiated; if consumers demand only one product, then the products are vertically differentiated.
12
When q̂2A-B = 0, neither firm has expected absolute advantage in attribute 2. The horizontal differentiation captured in the second case of the demand function reduces to the first case. Note also that withq̂1A-B > 0, vertical expected product differentiation with product B being the dominant product never arises
because in the third case of (2) holding prices the same would require q̂1A-B ≤ 0.
13
Alternatively, we can assume that production costs exist but are sunk, and thus they are not relevantto the decisions being modeled.

9equilibrium pricing decisions in the second stage.14 The following proposition characterizes
the equilibrium prices of the products as a function of q̂ A-B under Assumption 1 (all proofs
are relegated to Appendix A):
Proposition 1. Given q̂ A-B = (q̂1A-B , q̂2A-B ) with q̂1A-B > 0 and q̂2A-B ∈ [−q̂1A-B , q̂1A-B ], theunique equilibrium prices set by firms A and B, p∗ (q̂ A-B ) = (p∗A (q̂ A-B ), p∗B (q̂ A-B )), are⎧
2q̂1A-B −q̂2A-B q̂1A-B −2q̂2A-B
⎪
,
)
(
⎪
⎪
3
3
p∗ (q̂ A-B ) = ⎨
⎪
A-B
⎪
⎪
⎩(q̂2 , 0)q̂ A-Bif q̂2A-B ∈ [−q̂1A-B , 12 ),q̂ A-B(3)

if q̂2A-B ∈ [ 12 , q̂1A-B ].

Proposition 1 captures the interplay of the firms’ attribute advantages, market shares,
and pricing. Figure 1 illuminates the various scenarios under different values of q̂2A-B fora given q̂1A-B > 0. With firm A being endowed with an absolute advantage in one of theattributes, it always has a positive share of the market. In fact, selling the leading product,firm A never has a smaller market share than the trailing firm B. Within this confine, howfirm B shares the market with firm A varies with q̂2A-B .
The array of scenarios can be compartmentalized using a cutoff at q̂2A-B = 0, which
divides the types of competition into whether the two firms compete under horizontal
or vertical product differentiation. At q̂2A-B = 0, neither firm has absolute advantage inattribute 2, and the products effectively have only one attribute. Except to the boundary
consumer who does not care about attribute 1 at all, product A is a better product than
product B. The competition environment reduces to the standard duopoly model of verticalproduct differentiation with one-dimensional quality. Firm B with the inferior productengages in pure price competition to secure a positive demand. In equilibrium, product B
is priced sufficiently low for firm B to obtain a market share of one third.15
When q̂2A-B > 0, absolute advantages in both attributes rest with firm A, and there are
two quality dimensions with which the products are vertically differentiated. While firm B
can continue to rely on lowering price to make up for the inferiority of its product, which
is now unequivocally inferior in both attributes, as q̂2A-B increases so that the margin of
superiority of product A widens, the room for firm B to stay competitive in terms of priceq̂ A-Bnarrows. When product B is sufficiently inferior in attribute 2, that q̂2A-B ≥ 12 , firm A isin a position where it can set a price high enough to maximize profit but low enough toeliminate any room for firm B to lower price further; any positive price charged by firm B
would result in zero market share, and firm A acts as a constrained monopolist.
14

For brevity, throughout this section we omit “expected” when referring to attribute advantages andproduct differentiation. Throughout the paper, we denote equilibrium variables or objects with an asterisk.
15
The equilibrium market share of firm B equals the taste threshold in (1) evaluated at the equilibriumq̂ A-B −2q̂ A-B

prices, which amounts to θ̃∗ = 3(q̂1 A-B −q̂2A-B ) .
1

2

10

Comparative attribute advantages strengthen𝑝∗ 𝑝∗ 0, and the two
firms equally split themarket𝑝∗ 𝑝∗ 0, and firm 𝐵 has apositive but smaller market share𝑞0

Horizontal Product Differentiation:
Absolute advantage in attribute
1 (2) held by Firm 𝐴 𝐵𝑝∗ 𝑝∗ 0, and firm 𝐵 isout of the market

𝑞
2

𝑞

𝑞

Vertical Product Differentiation:
Absolute advantages in both
attributes held by Firm 𝐴

Figure 1: Attribute Advantages, Market Shares, and Pricing
When q̂2A-B < 0, firm B holds absolute advantage in attribute 2. With each firm having
absolute advantage in one attribute, product differentiation becomes horizontal. As thedifferentiation deepens, i.e., as q̂2A-B decreases from zero, each firm enjoys stronger market
power over their “clienteles” with less reliance on pure price competition. For q̂2A-B ∈(−q̂1A-B , 0), firm B’s absolute advantage in attribute 2 is of a smaller size than its rival’s
absolute advantage in attribute 1, and firm B still uses a lower price to compensate forthe discrepancy. At the boundary where q̂2A-B = −q̂1A-B so that the two absolute attributeadvantages are of equal size, however, the two firms charge the same price and equally split
the market, not competing for the same consumer by means of price as the two products
are maximally differentiated horizontally.
The cases of the intensity of competition can be organized through the lens of absoluteand comparative attribute advantages. The competition faced by firm A from the trailingfirm B is most fierce when the absolute advantage in attribute 2, whether it is held by firm
A or B, is vanishing, i.e., when q̂2A-B approaches 0 from above or below. This is also thepoint where the two products are least differentiated within the confine of this illustration
with a fixed q̂1A-B > 0. Firm B, on the other hand, faces the most fierce competition fromq̂ A-Bthe leading firm A when q̂2A-B approaches 12 . Since the differences in differences thatcharacterize comparative attribute advantages decrease in q̂2A-B , the competition faced byfirm B intensifies as its comparative advantage in attribute 2 weakens.
We conclude this subsection by highlighting two facts of comparative statics about
prices stated in terms of absolute and comparative attribute advantages:

11

Corollary 1. In the price equilibrium,(a) as a firm’s absolute advantage in an attribute improves, the firm’s price increases if
the firm also has comparative advantage in that attribute but decreases if it does nothave comparative advantage in that attribute; and(b) conditional on positive market share, the price of a firm increases as comparative
attribute advantages strengthen.
Since a firm’s comparative advantage in an attribute strengthens when its absoluteadvantage in that attribute improves or when its absolute advantage in the other attributedeclines, Corollaries 1(a) and 1(b) are two sides of the same coin. Note also that sincecomparative attribute advantages weaken as q̂2A-B increases, Corollary 1(b) makes clearthat in our environment lowest prices do not coincide with minimal product differentiation,
which occurs at q̂2A-B = 0.16

3.2

Profits in Price Equilibrium

Our main goal is to analyze disclosure decisions. To that end, we further examine thecomparative statics of how consumer expectations about attribute advantages, which are
influenced by disclosure and taken as parameters in the second stage, affect the firms’ profits
in the price equilibrium. We consider the meaningful situations where the two firms shareq̂ A-Bthe market with competition, restricting attention to the cases where q̂2A-B ∈ [−q̂1A-B , 12 ].
The reduced-form profit of firm K as a function of q̂ A-B = (q̂1A-B , q̂2A-B ) is:∗πK(q̂ A-B ) = p∗K (q̂ A-B )DK (p∗ (q̂ A-B ), q̂ A-B ),(4)which, using the relevant cases in (1)–(3), can be computed explicitly as πA∗ (q̂ A-B ) =
A-B
A-B 2
(2q̂1A-B −q̂2A-B )2∗
A-B ) = (q̂1 −2q̂2 ) for firm B.
for
firm
A
and
π
(q̂
A-B
A-B
A-B
B
9(q̂
−q̂
)
9(q̂
−q̂ A-B )
1

2

1

2

It is constructive for isolating the different effects in the comparative statics by proceeding with the form in (4). We are interested in the signs and magnitudes of the derivatives:∗∂πK∂DK ∂DK ∂p∗K ′∗= pK ( A-B +),∂pK ′ ∂ q̂iA-B∂ q̂iA-B∂ q̂iK
is the direct demand effect of q̂iA-B on firm K’s
K ∈ {A, B} and i ∈ {1, 2}. The term ∂∂Dq̂ A-Bi

16

For expositional ease, Figure 1 and the discussion above have assumed a fixed q̂1A-B . As we will see
below, the properties in terms of absolute and comparative attribute advantages remain valid when q̂1A-B
varies.

12

∂p∗

K
K′
profit, and ∂D∂pK ′ ∂ q̂iA-B is the indirect strategic effect through the change in the price of firm
K ′ ≠ K.17 Each firm faces two sets of effects, one for each attribute. For firm A, they are:

Attribute 1:
Attribute 2:q̂1A-B − 2q̂2A-B∂DA ∂p∗B
1∂DA=> 0 and
=
> 0.
A-B
A-B
A-B 2
A-B
A-B
∂pB ∂ q̂1
∂ q̂1
3(q̂1 − q̂2 )3(q̂1 − q̂2A-B )2q̂1A-B − q̂2A-B∂DA∂DA ∂p∗B
2
=
>
0
and
= − A-B
< 0.
A-B
A-B
A-B 2
A-B
∂pB ∂ q̂2
∂ q̂2
3(q̂1 − q̂2 )3(q̂1 − q̂2A-B )

(5)
(6)

B
B
For firm B, the demand effects are the negatives of those faced by firm A, i.e., ∂∂D= − ∂∂D;q̂iA-Bq̂iA-Bfor the strategic effects, the equilibrium price function (3) implies that they are related to∗∗∗∗∂DA ∂pB∂DB ∂pA
1 ∂DA ∂pB
B ∂pA
firm A’s strategic effects by ∂D∂pA ∂ q̂ A-B = 2( ∂pB ∂ q̂ A-B ) and ∂pA ∂ q̂ A-B = 2 ( ∂pB ∂ q̂ A-B ).
1

1

2

2

The demand effect of q̂1A-B on firm A’s profit is positive, which is further reinforced by
the positive strategic effect, resulting in a higher profit as q̂1A-B increases. Product A attracts
more consumers when it has a relatively higher quality in attribute 1. As q̂1A-B increases,
the price of product B also increases as comparative advantages strengthen (Corollary 1),
and this further contributes to the relative desirability of product A.
Firm B’s profit likewise increases in q̂1A-B , but it is under a different interaction of theeffects. The demand effect, which is now negative, is offset by the positive strategic effect.
While the prices of both products increase as comparative advantages strengthen, the priceof product A increases more than that of product B as can be seen in the equilibrium pricefunction (1). Even though product B becomes more inferior in respect of attribute 1, it
also becomes relatively more price competitive. On balance, the strategic effect dominates.
Firm B’s profit decreases in q̂2A-B . The demand effect is negative, which is reinforcedby the negative strategic effect. When q̂2A-B increases, either because firm B’s absoluteadvantage in attribute 2 declines under horizontal differentiation or because firm A’s improves under vertical differentiation, product B appeals to fewer consumers. Furthermore,as q̂2A-B increases, firm A charges a lower price as it does not have comparative advantage
in attribute 2 (Corollary 1), and this also makes product B relatively less attractive.
The effect of q̂2A-B on firm A’s profit is distinct from the other cases, where the relationship is non-monotonic. It is apparent from the derivatives in (6) that, at q̂2A-B = 0,
the negative strategic effect cancels out the positive demand effect. When q̂2A-B > 0, thedemand effect dominates, and vice versa when q̂2A-B < 0.
We summarize the four cases in the following proposition:

17

The strategic effect is a cross-price effect, and by the envelope theorem own-price effects do not havea place in the profit derivatives. The interaction of the demand and the strategic effects solely determines
how changes in an attribute advantage affect profits.

13

Horizontal Differentiation

 A-B

q1

Vertical Differentiation

2π*

4  A-B
q
9 1

0
0

 A-B
-q
1

A-B

q 1

2

 A-B
q
2

πA*

πB*

Figure 2: Attribute Advantages and Profitsq̂ A-B

Proposition 2. For q̂ A-B = (q̂1A-B , q̂2A-B ) with q̂1A-B > 0 and q̂2A-B ∈ [−q̂1A-B , 12 ], the reducedform profits of the firms in (4) have the following properties:(a) Firm A: πA∗ (q̂ A-B ) is (i) increasing in q̂1A-B , (ii) increasing in q̂2A-B when productdifferentiation is vertical (q̂2A-B > 0), and (iii) decreasing in q̂2A-B when product differentiation is horizontal (q̂2A-B < 0).
(b) Firm B: πB∗ (q̂ A-B ) is (i) increasing in q̂1A-B and (ii) decreasing in q̂2A-B , irrespectiveof the type of production differentiation.
The comparative statics further implies the following properties, which, as we will seein the next section, are key to understand firms’ disclosure incentives:
Corollary 2. The firms’ profits attain the minimum under the following different scenarios:(a) Firm A: πA∗ (q̂ A-B ) is at the lowest when firm A’s absolute advantage in attribute 1
is smallest and when absolute advantage in attribute 2, be it held by firm A or B, issmallest; this is also the point where products A and B are least differentiated.
(b) Firm B: πB∗ (q̂ A-B ) is at the lowest when comparative advantages are weakest.
Corollary 2 resonates with the narrative about intensity of competition in Section 3.1.
The point where a firm’s profit is lowest is the point where it faces the most fierce competition from the other firm. Figure 2 provides further visualization of the connection, in which14q̂ A-B

we depict the two firms’ profit levels for varying values of q̂2A-B ∈ [−q̂1A-B , 12 ], assuming afixed q̂1A-B as we have done for Figure 1. Firm A earns the lowest profit at q̂2A-B = 0 andthe highest profit at the two end points at which competition is eliminated, in one caseunder maximal horizontal product differentiation, and in the other under extreme verticaldifferentiation where the market effectively becomes monopolistic. Firm B’s profit, on theq̂ A-Bother hand, is monotonically decreasing and reaches the lowest level at q̂2A-B = 12 .

4

Equilibrium Analysis: Disclosure

In the first stage of the game, the two firms simultaneously make their disclosure decisions.
The interpretation is that the firms disclose a third-party review which reports the attribute
qualities of both products. Since profits depend on attribute advantages, we assume without
loss of generality that the reported variables are the quality differences instead of directly
the four quality variables. The third-party review is subject to a verifiability constraint
where it reports q A-B = (q1A-B , q2A-B ) if and only if they are the true attribute advantages.
Accordingly, each firm’s decision is whether to credibly disclose the realized q A-B or not. We
continue to focus on the cases with competition, analyzing disclosure under the followingtightened version of Assumption 1:
q A-B] ⊆ [−q A-B, q A-B, 12 ].
Assumption 1′ . The space of attribute advantages Q satisfies [q A-B
2
2
1
Note that Assumption 1′ places no restriction on the sign of the upper bound q 2A-B .
When q A-B> 0, from the perspective of an uninformed consumer, product differentiation is2
indefinite: either horizontal or vertical differentiation can arise. On the other hand, whenq A-B
≤ 0, product differentiation can only be horizontal.
2
Let QK ⊆ Q be the set of attribute advantages that firm K discloses. We define a
disclosure outcome in terms of Q = QA ∪ QB , the set of attribute advantages disclosed byat least one firm. A disclosure outcome is (a) revealing if Q ≠ ∅, (b) partially revealing if,
in addition to Q being non-empty, Q ⊂ Q, or equivalently in terms of the complements,
Q = QA ∩ QB ≠ ∅, and (c) fully revealing if Q = Q, or equivalently, Q = ∅. We analyze theperfect Bayesian equilibria of the game that give rise to these outcomes.18
Consumer beliefs on the equilibrium paths are uniquely determined by Bayes’ rule. Inan equilibrium with outcome Q∗ ≠ ∅, all consumers are perfectly informed of the set of
attribute advantages disclosed by at least one firm, i.e., for all q A-B ∈ Q∗ , q̂ A-B = q A-B .
18

Note that a singleton Q is not defined because then the singleton would be revealed. An approach to
avoid this is to define disclosure outcomes in terms of probability measure. We use the current definition
for its intuitive appeal.

15

If the equilibrium is partially revealing, the event of non-disclosure is on the equilibrium
path, and consumers form expectations conditional on the non-disclosed set of attribute∗advantages so that q̂ A-B = E[q A-B ∣q A-B ∈ Q ]. Given the verifiability constraint, there areout-of-equilibrium events in which consumer beliefs can still be uniquely pinned down. In
a fully revealing equilibrium, however, the absence of disclosure is an out-of-equilibrium
event where Bayes’ rule cannot be applied; appropriate out-of-equilibrium beliefs will haveto be assigned to support a given equilibrium.19
Other than the standard analysis described above, we also analyze equilibria under abehavioral variation where firms face boundedly rational consumers who adhere to theirpriors in an otherwise equilibrium event of non-disclosure.

4.1

Full Disclosure

We begin with a benchmark case where there is no disclosure cost and consumers are
fully rational. The well-known unraveling result of disclosure games holds in this baseline
environment of our duopolistic disclosure model, as the following proposition shows:
Proposition 3. Suppose that Q satisfies Assumption 1′ . If the disclosure cost c = 0, thenthe unique equilibrium outcome is fully revealing.
The fully revealing outcome can be supported by multiple equilibria with different
disclosure strategies. In the proof, we construct two types of equilibria: in one type both
firms disclose all attribute advantages, and in the other only one firm discloses all attributeadvantages, with the other firm disclosing some or none. Both types are fully revealing
equilibria in light of the outcome. The multiplicity originates from the fact that consumers
learn everything when one firm discloses all attribute advantages—any disclosure strategy
by the other firm is then a best response.
While the equilibria are not unique, the outcome is. The non-existence of equilibriumoutcome other than fully revealing is a consequence of the fact that the set of attributeadvantages that renders a firm a lower profit than does a given q̃ A-B forms a convex set.
Such a set does not exist if q̃ A-B is the expected value of the attribute advantages inthe set. In other words, for any Q ⊆ Q, there will be attribute advantages in Q wherea firm earns a higher profit by disclosing them than leaving consumers to believe thatq̂ A-B = E[q A-B ∣q A-B ∈ Q]. This result stands in contrast to the monopolistic model of Sun
19

A full specification of equilibria also requires consumer beliefs to be specified after any out-ofequilibrium prices. We follow, e.g., Board (2009) and adopt the simplest assumption that consumer beliefs
are independent of prices. See Janssen and Roy (2014) for an analysis of a duopoly model where in addition
to disclosure, prices may signal quality.

16

(2011) and the oligopolistic model of Hotz and Xiao (2013) with horizontal and verticalattributes. In their models, partially revealing equilibria exist under costless disclosure andthe quality variables that are not disclosed form a disjoint set.

4.2

Partial Disclosure

Proposition 3 establishes a benchmark of full disclosure under the assumptions of costless
disclosure and fully rational consumers. Relaxing these two assumptions each provides anavenue for partial disclosure to emerge as an equilibrium outcome. The analysis of partial
disclosure brings to the fore some of the distinguishing features of our multi-attribute model,
in particular how disclosure decisions interact with absolute and comparative attributeadvantages as well as product differentiation.
Costly Disclosure. Each firm incurs a common c > 0 when choosing to disclose.20 The
presence of disclosure cost renders some attribute advantages not worthwhile to be disclosed. We use the concept of neighborhoods to characterize non-disclosure. For any q̃ A-B ∈Q and  > 0, we define the -neighborhood of q̃ A-B , N [q̃ A-B ] = {q A-B ∈ Q ∶ d(q A-B , q̃ A-B ) < },
where d is the Euclidean distance. It is intuitive to expect, and indeed observed in practice,that when selecting what to disclose and not disclose, firms highlight the favorable information and downplay the unfavorable. The following lemma substantiates this intuition
by establishing the properties of the non-disclosure set of each firm:
Lemma 1. Suppose that Q satisfies Assumption 1′ . If the disclosure cost c > 0,then there exists ∗ > 0 such that for any equilibrium outcome Q∗ = Q∗A ∪ Q∗B ,∗∗)] ⊂ QB .
, min{0, q A-B})] ⊂ QA and N∗ [(q A-B, q A-BN∗ [(q A-B
2
2
1
1
The non-disclosure set of each firm anchors at the attribute advantages that, if learnedby consumers, would put the firm at the most disadvantageous position facing the mostfierce competition and earning the lowest profit. Corollary 2 furnishes what these attributeadvantages are for each firm. Firm B is most disadvantaged at the “vertex” (q A-B, q 2A-B ),1
where the trailing firm’s comparative advantage in attribute 2 is weakest and any positive
price charged for Product B would render a zero sale. Firm A, on the other hand, ismost disadvantaged when absolute attribute advantages are smallest, which is also thepoint where the two products are least differentiated. Since firm A always holds absoluteadvantage in attribute 1, in respect of this attribute this occurs at q A-B.
1
20

We have interpreted the disclosure decisions as regarding a third-party review that contains quality
information on both products. In practice, firms typically publicize third-party reviews on their promotion
platforms such as company websites. An example of the disclosure cost could be opportunity cost resulting
from the scarcities of promotion spaces or consumer attention.

17

q1A-B

Horizontal Differentiation

q1A-B

Vertical Differentiation

Horizontal Differentiation

Q*A*

QA ("Dominant")
Firm A's non-disclosure anchor(products least differentiated)

q1A-B-q1A-B

Q*B

q1A-B

q1A-B

Vertical Differentiation

Firm B's non-disclosure anchor(weakest comparative
attribute advantages)

Q

0

*

QB ("Dominant")

*

Q

*

QB ("Free-Riding")

q1A-B
2

q1A-B
-q1A-B

0

*

q1A-B
2

q2A-B

q2A-B

(a) Firm A(b) Firm B

Figure 3: Disclosure Decisions: Indefinite Product Differentiation
For attribute 2, there are two cases. When product differentiation is indefinite (q 2A-B >, 0), where neither firm has absolute0), the least product differentiation occurs at (q A-B
1advantage in attribute 2. When the two products are always horizontally differentiated≤ 0), they are least so at (q A-B, q 2A-B ), where the smallest absolute advantage in(q A-B2
1
attribute 2 is held by firm B. Note that this coincides with weakest comparative attributeadvantages; under definite horizontal differentiation, the two firms’ non-disclosure setsanchor at the same point, although with firm A being the leading firm and firm B the
trailing the considerations behind their decisions are different.
∗∗

The purpose of Lemma 1 is to establish the “anchoring property” of QA and QB . The
-neighborhood N∗ [ ⋅ ] in general does not contain all the attribute advantages that a firm
does not disclose. In particular, unlike the benchmark case in Section 4.1, when disclosure
is costly, in equilibrium the two firms will not disclose the same attribute advantages since∗a firm can “free ride” on the other firm’s disclosure. There may be elements of QK thatfall under this category. This is in contrast to the non-disclosure of N∗ [ ⋅ ], which is a“dominant” decision in that, regardless of what firm K ′ discloses, firm K does not want todisclose those attribute advantages in the -neighborhood.
q A-B] × [−q A-B
Figure 3 depicts an example with Q = [q A-B, q A-B, 12 ], where product differ1
11
entiation is indefinite. The disclosure decisions of firms A and B are separately illustrated
in panels (a) and (b). The illustration assumes that firm B is the “free rider” so that firm
A discloses all the attribute advantages that are commonly preferred by both firms to bedisclosed. For firm A, the white region represents the disclosed Q∗A , and the shaded region∗
represents the non-disclosed QA . For firm B, the white region similarly represents the dis∗closed Q∗B , and the shaded region that represents the non-disclosed QB is further divided
into the “free-riding” region (lighter shade) and the “dominant” region (darker shade). The

18

free-riding region could be transferred to firm A, which would give rise to an alternativeequilibrium with the same disclosure outcome. By contrast, the dominant region cannot
be so transferred; even if it was not disclosed by firm A, firm B would still not disclose it.
∗∗
As stated in Lemma 1, QA and QB in the figure contain the respective anchoring points at(q A-B).
, 0) and (q A-B, q A-B
2
1
1
Lemma 1 presents the disclosure decisions of firms A and B separately. Whether aparticular set of attribute advantages is disclosed to consumers depends jointly on the∗decisions. Comparing Figures 3(a) and 3(b) shows that a subset of the non-disclosed QB∗
by firm B is disclosed by firm A, and vice versa for QA . The attribute advantages thatare not disclosed to consumers at all are those where there is a “double dominance” of∗∗∗non-disclosure—those that lie in the interaction of QA and QB indicated by Q .
∗∗As a general matter, the sizes of QA and QB depend on the prior distributions F A-B =(F1A-B , F2A-B ) and the disclosure cost c. The following proposition shows that so long as the∗∗disclosure cost is positive, QA intersects QB , and fully revealing outcome is out of reach as
an equilibrium outcome; as long as disclosure is not too costly given the distributions, apartially revealing equilibrium is guaranteed to exist; and when such an equilibrium exists,
product differentiation shapes the form of the partially revealing outcome:
Proposition 4. Suppose that Q satisfies Assumption 1′ with distributions F A-B . Thereexists cF A-B such that for any disclosure cost c ∈ (0, cF A-B ), the outcome of any equilibrium
Q∗ is partially revealing and depends on product differentiation as follows:(a) If q 2A-B > 0 so that product differentiation is indefinite, then there exists ∗hv > 0 such∗).
that N∗hv [(q A-B, q2A-B )] ⊂ Q for some q2A-B ∈ (0, q A-B
21(b) If q A-B≤ 0 so that product differentiation is horizontal, then there exists ∗h > 0 such2∗)] ⊂ Q .
that N∗h [(q A-B, q A-B
2
1
The two firms have perfectly aligned interests in not disclosing sufficiently low q1A-B .
But for the case in Proposition 4(a) where either horizontal or vertical differentiation canarise, their disclosure preferences in respect of attribute 2 are not as aligned: firm A prefers
not to disclose q2A-B in the neighborhood of 0, whereas firm B prefers not to disclose aroundthe upper bound q A-B> 0. In equilibrium, the set of non-disclosed attribute advantages,
2∗∗∗
Q , the intersection of QA and QB , anchors at a point made up of q A-B
and a value between
1
A-B
0 and q 2 , as illustrated in Figure 3.
A distinctive difference of the case in Proposition 4(b), in which product differentiation
is definitely horizontal, is that the disclosure interests of the two firms are perfectly aligned
in both attributes: they both prefer not to disclose q A-B with sufficiently low q1A-B and] × [−q A-B
sufficiently high q2A-B . Figure 4 displays an example with Q = [q A-B, q A-B
, 0].
1
1
1
19

q1A-B

q1A-B

Horizontal Differentiation

Horizontal Differentiation
*

QB
*

*

QB ("Dominant")*

QB ("Free-Riding")

QA

*

q1A-B

q1A-B

QA ("Dominant")
Firm A's non-disclosure anchor(products least differentiated)

q1A-B-q1A-BQ

Firm B's non-disclosure anchor(weakest comparative
attribute advantages)

*

0

q1A-B
-q1A-B

Q

*

0

q2A-B

q2A-B

(a) Firm A(b) Firm B

Figure 4: Disclosure Decisions: Horizontal Product Differentiation
Boundedly Rational Consumers. We further consider disclosure with boundedly rational consumers, who do not update beliefs in the equilibrium event of non-disclosure.
Instead, they adhere to the priors and make purchase decisions based on the unconditional
expected attribute advantages E[q A-B ] ∈ int(Q).21
Consumers making purchase decisions under the priors when neither firm discloses renders non-disclosure more attractive to firms, in a way similar to how economization of
disclosure cost makes non-disclosure a potentially profitable move. Proposition 3 has established that when disclosure is costless, full revelation is the unique equilibrium outcome
with fully rational consumers; the following proposition shows that a partially revealing
equilibrium exists even with costless disclosure if consumers are boundedly rational:22
Proposition 5. Suppose that Q satisfies Assumption 1′ with distributions F A-B . If consumers adhere to the prior F A-B when neither firm discloses, then there exists a partiallyrevealing equilibrium for disclosure cost c = 0. The outcome of this equilibrium depends on
product differentiation as follows:(a) If q 2A-B > 0 so that product differentiation is indefinite, then there exists ˜∗hv > 0 such∗).
that N˜∗hv [(q A-B, q2A-B )] ⊂ Q for some q2A-B ∈ (0, q A-B
21(b) If q A-B≤ 0 so that product differentiation is horizontal, then there exists ˜∗h > 0 such
2
∗
A-B
)]
that N˜∗h [(q A-B
⊂
,
q
Q
.
2
1
21

We would also assume that consumers use the priors in the out-of-equilibrium event of non-disclosure,but this is only for complete specification and is not relevant to the analysis below.
22
Note that a fully revealing equilibrium in which both firms disclose all attribute advantages also exists
with boundedly rational consumers and c = 0; partially revealing equilibria are not the unique type ofequilibria. Furthermore, a positive disclosure cost operates with boundedly rational consumers in a similarmanner as characterized in Proposition 4 for fully rational consumers. The proposition below focuses on
costless disclosure to highlight that boundedly rational consumers could supersede positive disclosure costin yielding a partially revealing equilibrium.

20q1A-B

Horizontal Differentiation

q1A-B

Vertical Differentiation

Horizontal Differentiation

Q*A

Q*B

*

E[q

A- B

Q

]

*

*

QB

q1A-B

q1A-B

QA

Vertical Differentiation

Q

]

*

Firm B's non-disclosure anchor(weakest comparative
attribute advantages)

Firm A's non-disclosure anchor(products least differentiated)

q1A-B-q1A-B

E[q

A- B

0

q1A-B
2

q1A-B
-q1A-B

0

q1A-B
2

q2A-B

q2A-B

(a) Firm A(b) Firm B

Figure 5: Disclosure Decisions: Boundedly Rational Consumers andIndefinite Product Differentiationq A-B]×[−q A-B
, q A-B, 12 ], where productFigure 5 illustrates Proposition 5(a) with Q = [q A-B
1
1
1
∗

∗

differentiation is indefinite. The boundaries of the firms’ non-disclosure sets, QA and QB ,pass through the unconditional E(q A-B ) that the boundedly rational consumers rely on
after non-disclosure. Comparing and contrasting Figure 5 and Figure 3 of costly disclosure
exemplify the similarities and differences between the disclosure decisions in the two en∗
vironments. For similarities, firm B’s non-disclosure set QB must contain, in both cases,), the point where comparative attribute advantages are weakest. In a partially(q A-B, q A-B
2
1∗
revealing equilibrium, this set intersects with firm A’s non-disclosure set QA to give rise to∗
Q , the set of attribute advantages that are not disclosed by either firm.
The differences are more subtle. The partially revealing outcome illustrated in Figure
5 is made up of (a) disclosed region in which both firms disclose (region that is white
in both panels), (b) disclosed regions in which only one firm discloses (regions that are
shaded in only one panel), and (c) the non-disclosed region (region that is shaded in both∗panels, indicated by Q ). Given that disclosure is costless, there is no longer dominant nondisclosure; similar to the benchmark in Section 4.1, there are multiple equilibria supporting
the same disclosure outcome that stem from a firm’s indifference when the other firm
discloses. One could construct other equilibria with the non-disclosed region unchanged
but each of the other three regions being disclosed by one or both firms.
Despite these differences, the behavioral environment with costless disclosure complements the costly-disclosure environment in that partial revealing equilibria arise in bothcases; when product differentiation must be horizontal (q A-B≤ 0), the insights from the two2
environments further align in which the partial revealing equilibria feature non-disclosureof attribute advantages where comparative advantages are weakest. Consumers falling21

short of applying Bayes’ rule and carrying out the iterative reasoning assumed by equilibrium are perhaps more descriptive of the consumers in the real world.23 Our behavioral
analysis suggests that the major insights from the costly-disclosure analysis regarding theinterplay of partial disclosure, absolute and comparative attribute advantages, and productdifferentiation remain relevant when one departs from the fully rational paradigm.

5

Concluding Remarks

This paper analyzes a novel variation of otherwise well-studied quality disclosure. In a
duopolistic environment with disclosure of comparative quality information, we show thatmodeling two vertical product attributes instead of one horizontal and one vertical as isdone in previous studies introduces unexplored considerations for analyzing competitive
disclosure. These strategically rich considerations, though new in the context of disclosure,surround the familiar notion of absolute and comparative advantages.
In our model, one of the two firms sells a product that is ex-ante leading in terms ofabsolute attribute advantages (Assumption 1). When disclosure is costly, firms strategically select what to disclose and what to withhold. The firm selling the leading productwithholds disclosure when it faces sufficiently intense competition from the firm selling
the trailing product, which occurs when absolute attribute advantages are smallest andthe two products are least differentiated. The trailing firm survives in the market on thecomparative attribute advantage of its product. Naturally, this firm has incentives not todisclose when comparative advantages are sufficiently weak; at the point where comparativeadvantages are the weakest, its sales drop to zero under any positive price.
Least product differentiation coincides with weakest comparative attribute advantages
when under the priors consumers consider the two products to be horizontally differentiated. When instead the two products can be either vertically or horizontally differentiatedfrom the ex-ante perspective, least product differentiation differs from weakest comparative advantages. In this case, the two firms’ interests in what to withhold are not aligned,
giving the hope that consumers may still be fully informed by combining the information
disclosed by each firm. However, we find that even for an infinitesimal amount of disclosure cost, the non-disclosure sets of the two firms, though possibly anchoring at different
points, intersect. Consequently, in equilibrium consumers are uninformed of at least some
attribute qualities regardless of the type of product differentiation. The partially revealing
disclosure outcome may also arise under costless disclosure when consumers fall short of23

While relevant field data are hard to come by, Jin et al. (2021), e.g., obtain laboratory evidence thatreceivers in a disclosure game fail to form correct beliefs about non-disclosed states.

22

forming rational expectation upon non-disclosure.
An important debate in the theory and practice of quality disclosure concerns the roleof mandatory disclosure. The unraveling result provides a basis to argue that mandatory
disclosure has no place under the profit incentives of firms: voluntary disclosure completely
resolves the problem of asymmetric information in the marketplace. Our paper contributes
to the debate by providing and analyzing a market environment under which mandatory
disclosure may be favored. In a duopolistic setting with two vertically differentiated productattributes, in which revelation of comparative quality information is practiced, relying on
the marketing incentives of firms to promote the attributes of their products may not
be enough to render consumers fully informed. Our analysis suggests that in such anenvironment mandatory disclosure has a role to play when, as is often the case in practice,disclosure is costly or consumers are not as intelligent as economic models would postulate.
One direction that future research can extend our analysis involves consumer search.
The presence of consumers who access third-party product reviews on their own (e.g.,
Ghosh and Galbreth, 2013) may possibly expand the set of product attributes that firms
choose not to disclose to consumers directly by themselves. If so, then active search bysome consumers might create a negative spillover on other consumers who do not searchby paradoxically reducing the amount of product information available to them. Anotherrelated direction involves the disclosure of a subset of product attributes. If third-party
reviews evaluate products only on certain attributes but not others, then depending on
absolute and comparative attribute advantages firms may have an incentive to expand theset of attributes that is made known to consumers. Formal analysis to substantiate orrefine this intuition would further enhance our understanding of quality disclosure.

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25

Appendix A

Proofs

Proof of Proposition 1. Consider first the case where q̂2A-B < pA − pB < q̂1A-B so that
there is a positive demand for each product. Maximizing the profit of firm K ∈ {A, B},
πK (p, q̂ A-B ) = pK DK (p, q̂ A-B ), with respect to pK yields best-response functions pA (pB ) =p −q̂ A-B
pB +q̂1A-B
and pB (pA ) = A 22 . Given that the functions are linear, there exist unique mutual2q̂ A-B −2q̂ A-B
2q̂ A-B −q̂ A-B
best responses p∗A = 1 3 2 and p∗B = 1 3 2 . It can be verified that the equilibriumq̂ A-B
prices satisfy q̂2A-B < p∗A − p∗B < q̂1A-B if and only if q̂2A-B < 12 , which is the first case in (3).
Consider next the case where pA − pB ≤ q̂2A-B so that there is a positive demand only for
product A. In any equilibrium under this case, we must have that pA − pB = q̂2A-B . Suppose
instead that pA −pB < q̂2A-B . Then by raising its price to pA +ε, with ε > 0 sufficiently small sothat the demand for product A remains to be DA (p, q̂ A-B ) = 1, firm A can increase its profit.
We next argue that in any equilibrium with pA − pB = q̂2A-B , it must be the case that pB = 0.
Suppose instead that pB > 0. Then by charging p′B = pB −δ so that pA −p′B = q̂2A-B +δ > q̂2A -B ,firm B secures a positive demand and earns positive profit rather than the zero profit fromcharging pB . We further identify the condition under which firm A has no incentive todeviate to charge higher than pA = q̂2A-B . For any such deviation, where product A is nolonger the only product with positive demand, the highest profit firm A can obtain is byq̂ A-B

1−q̂2A-Bq̂ A-Bq̂ A-B2), while the profit fromsetting p′A = 12 with resulting π(p′A , q̂ A-B ) = ( 12 )(1 − q̂A-B−q̂2A-B
1
A-B
A-B
charging pA = q̂2 is π(pA , q̂ A-B ) = q̂2 . It can be verified that π(pA , q̂ A-B ) ≥ π(p′A , q̂ A-B ) ifq̂ A-Band only if 12 ≤ q̂2A-B , and this is the second case in (3) with p∗A = q̂2A-B and p∗B = 0.

Finally, we argue that there is no equilibrium under the case where pA − pB ≥ q̂1A-B witha positive demand only for product B. Given that q̂1A-B > 0, we have that pA > pB . But
then firm B always has a profitable deviation by charging a slightly higher price.

Proof of Corollary 1. The corollary follows immediately from the derivatives of theprices in (3) and the definition of absolute and comparative advantages.

Proof of Proposition 2. The explicit expressions of the firms’ reduced-form profits are(2q̂1A-B − q̂2A-B )2, and
9(q̂1A-B − q̂2A-B )(q̂ A-B − 2q̂ A-B )2
πB∗ (q̂ A-B ) = 1 A-B 2A-B .
9(q̂1 − q̂2 )πA∗ (q̂ A-B ) =

26

(A.1)
(A.2)

∗
∂πA
(2q̂1A-B −3q̂2A-B )(2q̂1A-B −q̂2A-B )=, ∂ q̂A-B
A-B −q̂ A-B )2
9(q̂
1
1
2
2
∗
∗
A-B
A-B
A-B
A-B
A-B
A-B
A-B
A-B
A-B
A-B
∂πB
∂πB
q̂
(q̂1 −2q̂2 )(2q̂
−3q̂1 )(q̂1 −2q̂2 )q̂2 (2q̂1 −q̂2 )= 19(q̂A-B= 2 9(q̂A-B, ∂ q̂A-B, and ∂ q̂A-B, under the pa9(q̂1A-B −q̂2A-B )2−q̂2A-B )2−q̂2A-B )2
1
1
2
1
A-B
q̂
rameter restrictions that q̂1A-B > 0 and q̂2A-B ∈ [−q̂1A-B , 12 ] gives us the properties.

∂π ∗A
Evaluating the signs of the following four derivatives, ∂ q̂A-B=

Proof of Corollary 2. The corollary follows immediately from Proposition 2 and thedefinition of absolute and comparative advantages.

Proof of Proposition 3. We begin by constructing two types of equilibria that support
the outcome Q∗ = Q. Suppose that in a putative equilibrium only firm A discloses all
q A-B ∈ Q, while firm B may disclose some or none of the attribute advantages. Any eventof non-disclosure must be resulting from a deviation by firm A, and we assign consumer
beliefs so that in any of these out-of-equilibrium events the expected attribute advantages, 0). Since πA∗ (q A-B ) ≥ πA∗ (q A-B, 0) for all q A-B , firm A has no strict incentive toare (q A-B
1
1
deviate. Given that all q A-B are disclosed by firm A, any disclosure strategy by firm B isa best response. It follows that this is an equilibrium. By a similar argument, only firm B
disclosing all q A-B constitutes an equilibrium under out-of-equilibrium expected attribute, q A-B
advantages (q A-B) in the event of non-disclosure. These give us one type of fully
2
1
revealing equilibria where Q∗K ′ ⊂ Q∗K = Q∗ = Q, K, K ′ ∈ {A, B} and K ≠ K ′ . Another typeof equilibrium has both firms disclose all q A-B so that Q∗A = Q∗B = Q∗ = Q. To see that thisis an equilibrium, note that under the verifiability constraint consumers believe that the
attribute advantages are q A-B when one firm deviates and does not discloses q A-B .
We proceed to show that there exists no equilibrium in which Q∗ ⊂ Q. For a given∗∗q̃ A-B ∈ Q, define LPK (q̃ A-B ) = {q A-B ∈ Q ∶ πK
(q A-B ) ≤ πK(q̃ A-B )} to be the set of attributeadvantages that would give firm K ∈ {A, B} a weakly lower profit than would q̃ A-B . We firstestablish that LPK (q̃ A-B ) is a convex set. From the explicit expressions of the reduced-form
profits in (A.1) and (A.2), we obtain the Hessian matrix of firm K’s profit:
A-B q̃ A-B⎡ 2(q̃2A-B )2⎤
1
2⎢ A-B A-B 3 − 2q̃A-B⎥
A-B )3 ⎥
⎢
−q̃
)
−q̃
9(q̃
9(q̃
1
2
⎥,
HπK∗ (q̃ A-B ) = ⎢ 12q̃A-B q̃2A-B
A-B
2
2(q̃
)
⎥
⎢− 1 2⎢ 9(q̃1A-B −q̃2A-B )3 9(q̃1A-B1−q̃2A-B )3 ⎥⎦⎣∗which is positive semidefinite for all q̃ A-B ∈ Q under Assumption 1′ . It follows that πK
(⋅)
∗
A-B
is a convex function. Note that LPK (q̃ ) is a lower contour set for πK (⋅). The fact that∗
πK(⋅) is convex implies that LPK (q̃ A-B ) is a convex set for all q̃ A-B ∈ Q.

We utilize this convex property to show that in any equilibrium, if there exists a non-

27

∗

∗

empty QK ′ , then all q A-B ∈ QK ′ must be disclosed by firm K ≠ K ′ . Suppose not so that
∗
∗
∗
∗
∗
QK ′ = Q ⊆ QK and let q̂ A-B (Q ) = E[q A-B ∣q A-B ∈ Q ]. For this to constitute an equilibrium,∗∗∗
it must be that Q ⊆ LPK (q̂ A-B (Q )). The fact that LPK (q̂ A-B (Q )) is a convex set then∗∗∗implies that q̂ A-B (Q ) ∈ LPK (q̂ A-B (Q )). A contradiction follows immediately if q̂ A-B (Q ) ∈
∗
∗
∗
∗
∗
int(LPK (q̂ A-B (Q ))) because then πK(q̂ A-B (Q )) > πK(q̂ A-B (Q )). Given that the density∗∗
fiA-B , i ∈ {1, 2}, is everywhere positive, if q̂ A-B (Q ) ∈ ∂(LPK (q̂ A-B (Q ))), then there must
∗
∗
∗
∗
∗
(q ′A-B ) > πK(q̂ A-B (Q )) and thus q ′A-B ∉ LPK (q̂ A-B (Q )),exist a q ′A-B ∈ Q such that πK
again a contradiction. It follows that Q∗ = Q is the unique equilibrium outcome.

Proof of Lemma 1. We begin with the case where Q satisfying Assumption 1′ is suchthat q A-B> 0. Consider any equilibrium disclosure outcome Q∗ ⊆ Q. For any q A-B ∈ Q,
2there are three possibilities for firm K’s profit, K ∈ {A, B}: (a) if q A-B is revealed by firm∗
K, then its profit is πK(q A-B ) − c, (b) if q A-B is revealed only by firm K ′ ≠ K, then its profit
∗
∗
∗
is πK
(q A-B ), and (c) if q A-B is not revealed at all, then its profit is either πK(q̂ A-B (Q ))∗∗∗(q́ A-B )where q̂ A-B (Q ) = E[q A-B ∣q A-B ∈ Q ] or, if this is an out-of-equilibrium event, πK
where q́ A-B is some expected attribute advantages based on consumer out-of-equilibrium(2q A-B −q A-B )2(q1A-B −2q2A-B )2
beliefs. The reduced-form profits, πA∗ (q A-B ) = 9(q1A-B −q2A-B ) and πB∗ (q A-B ) = 9(qA-B −q A-B ) ,
1

2

1

2

are minimized at (q A-B, 0) and (q 1A-B , q 2A-B ) respectively. Since πA∗ (q A-B, 0) − c < πA∗ (q A-B )11, 0) regardless of whether firm B reveals
for any q A-B ∈ Q, firm A does not reveal (q A-B
1
∗
∗
A-B
(q 1 , 0) or not. It follows that we must have that (q A-B,
0)∈Q
A . The continuity of πA (⋅)
1∗, 0)] ⊂ QA . By a similar argument, for
implies that there exists ∗A > 0 such that N∗A [(q A-B
1
∗
A-B
∗ ∗
∗
)]
firm B, there exists ∗B > 0 such that N∗B [(q A-B
,
q
⊂
Q
2
B . Let  = min{A , B }, and1
the lemma follows. For the case where q A-B≤ 0, πA∗ (q A-B ) and πB∗ (q A-B ) are minimized at
2∗
A-B
A-B , 0) ∈ Q is replaced by).
(q
the same point (q A-B
,
q
Thus,
for
firm
A,
the
condition
2
A
1
1
∗
A-B
A-B
(q1 , q 2 ) ∈ QA . The lemma then follows by a similar argument.

Proof of Proposition 4. We prove the proposition by verifying five claims. Claims 1
and 2 establish two properties of equilibrium outcomes.
∗

Claim 1. If Q∗ is an equilibrium outcome for c > 0, then Q ≠ ∅.
Proof of Claim 1. We prove by contradiction. Suppose that c > 0 and there exists an∗equilibrium in which Q = ∅ or equivalently Q∗ = Q. Lemma 1 and the fact that the two
firms will not disclose the same q A-B in equilibrium when c > 0 implies that the equilibrium
must not involve only one firm disclosing all q A-B ∈ Q or both firms disclosing all q A-B ∈ Q,
the types of fully revealing equilibria constructed in the proof of Proposition 3 for costless
28

disclosure. We further rule out the remaining possibilities for a fully revealing outcome
where Q∗A ⊂ Q, Q∗B ⊂ Q, Q∗A ∪ Q∗B = Q, and Q∗A ∩ Q∗B = ∅. For a putative equilibriumwith these properties, the absence of disclosure is an out-of-equilibrium event in which
consumer beliefs are neither pinned down by Bayes’ rule nor restricted by the verifiability
constraint. Denote by q́ A-B an expected attribute advantages based on consumer out-ofequilibrium beliefs. With c > 0, firms A and B strictly prefer not to disclose any q A-B ∈LPA (q́ A-B ) ∩ LPB (q́ A-B ), and computation using the profit functions πA∗ (⋅) and πB∗ (⋅) showsthat LPA (q́ A-B ) ∩ LPB (q́ A-B ) ≠ ∅ for any q́ A-B ∈ Q. Hence, for any putative equilibriumwith the above properties supported by any q́ A-B , there exists a profitable deviation.

We introduce some notations and definition for Claim 2. Denote the unconditional
expected attribute advantages under distributions F A-B by q̂ A-B (F A-B ) and define cF A-B =∗min{π A − πA∗ (q̂ A-B (F A-B )), π B − πB∗ (q̂ A-B (F A-B ))}, where π K = max
πK(q A-B ), K ∈ {A, B}.
A-B
q

∈Q

Claim 2. Given F A-B , if Q∗ is an equilibrium outcome for c ∈ (0, cF A-B ), then Q∗ ≠ ∅.
Proof of Claim 2. We prove by contradiction. Suppose that c < cF A-B and there exists∗
an equilibrium in which Q∗ = ∅ or equivalently Q = Q. We show that firm K ∈ {A, B} hasa profitable deviation. Note that the unconditional q̂ A-B (F A-B ) is the expected attribute∗advantages conditional on non-disclosure given that Q = Q. By construction, c < cF A-B ≤∗∗π K − πK(q̂ A-B (F A-B )), which implies that π K − c > πK(q̂ A-B (F A-B )). Thus, firm K strictly∗prefers to disclose argmax πK(q A-B ).
q A-B ∈Q

Claims 1 and 2 have established that for any c ∈ (0, cF A-B ), if an equilibrium exists,then it is partially revealing. Claims 3 and 4 are about existence. We further introduce
additional notations and definitions. Let S(q A-B ) be the set of disclosure signals available toa firm when the realized attribute advantages are q A-B ∈ Q. Under the accurate verifiability
constraint, S(q A-B ) = {{q A-B }, Q}; a firm decides whether to send s = {q A-B }, disclosing therealized attribute advantages, or to send s = Q, which amounts to not disclosing. Define
S = ⋃qA-B ∈Q S(q A-B ). A pure disclosure strategy of firm K ∈ {A, B} is a mapping δK ∶ Q → S
with the restriction that for all q A-B ∈ Q, δK (q A-B ) ∈ S(q A-B ).
We specify a conditional best response of firm K, CBRK , relative to an arbitrary profit
threshold πK ≥ 0. Given πK , CBRK prescribes firm K the following response to δK ′ ,29

K ′ ≠ K: for every q A-B ∈ Q,

CBRK [qA-B, δK ′ (q

A-B

⎧
⎪
⎪
⎪q A-B)∣πK ] = ⎨
⎪
⎪
⎪
⎩Q

∗
if δK ′ (q A-B ) = Q and πK(q A-B ) − c ≥ πK ,∗if δK ′ (q A-B ) = q A-B or πK(q A-B ) − c < πK .

∗Claim 3. In any equilibrium, δK
satisfies: for every q A-B ∈ Q,
∗

∗
∗
A-B
∗
δK
(q A-B ) = CBRK [q A-B , δK)∣πK(q̂ A-B (Q ))],
′ (q
∗

∗

where K, K ′ ∈ {A, B}, K ≠ K ′ , and q̂ A-B (Q ) = E[q A-B ∣q A-B ∈ Q ].
∗Proof of Claim 3. By Claim 1, we can focus on putative equilibria in which Q ≠ ∅.
Suppose that firm K ′ does not disclose q A-B . If firm K discloses q A-B , then its profit
∗
∗
∗
is πK
(q A-B ) − c; if firm K does not disclose q A-B , then its profit is πK(q̂ A-B (Q )). It∗
A-B ) = Q, then δ ∗ (q A-B ) = q A-B if and only iffollows that if in a putative equilibrium δK
′ (q
K
∗
∗
∗
A-B
A-B
πK (q ) − c ≥ πK (q̂ (Q )). Note that specifying either disclosure or non-disclosure by∗∗∗firm K for πK(q A-B ) − c = πK(q̂ A-B (Q )) does not affect the measure of disclosed attribute∗
A-B ) = q A-B implies that δ ∗ (q A-B ) = Q follows
advantages. Finally, the condition that δK′ (qKfrom the fact that in equilibrium the two firms will not disclose the same q A-B when c > 0.

Claim 4. An equilibrium exists.
Proof of Claim 4. The set of non-disclosed attribute advantages by firm K can be characterized as QK = {q A-B ∈ Q ∶ CBRK [q A-B , δK ′ (q A-B )∣πK ] = Q}. It can be seen from thischaracterization that a pair of profit thresholds (πA , πB ) delineates the set of attributeadvantages not disclosed by either firm, Q = QA ∩ QB . The conditional expectation func∗tion, q̂ A-B (⋅) = E[q A-B ∣ ⋅ ], and then firm K’s profit function, πK(⋅), further map Q to∗πK(q̂ A-B (Q)). Let gK ∶ [π A , π A ] × [π B , π B ] → [π K , π K ] be this composite mapping for firm∗∗K ∈ {A, B}, where π K = A-B
min πK(q A-B ) and π K = max
πK(q A-B ). We further define a
A-B
q

q

∈Q

∈Q

function g ∶ [π A , π A ]×[π B , π B ] → [π A , π A ]×[π B , π B ] by g(πA , πB ) = gA (πA , πB )×gB (πA , πB ).
Claim 3 has established that if an equilibrium exists, then each firm’s disclosure strategy
must constitute a conditional best response. It follows that a fixed point of g, (π̃A , π̃B ) =∗g(π̃A , π̃B ), characterizes an equilibrium that induces the non-disclosure set Q . If a fixed
point exists, then an equilibrium exists. In the domain of g, the upper and lower boundsof the sets, π K and π K , K ∈ {A, B}, represent firm K’s lowest and highest profit in the] × [q A-B]. Therefore, the domain of g is non-empty, compact,
space Q = [q A-B, q A-B, q A-B1212and convex. The continuity of the distribution functions F A-B and of the profit functions30

πA∗ (⋅) and πB∗ (⋅) imply that g is continuous. Thus, with all the conditions of the Brouwer’s
fixed point theorem satisfied, there exists a fixed point for g.

∗

Claim 5 concerns the characterizations of Q in parts (a) and (b) of the proposition.
Claim 5. For any partially revealing equilibrium outcome Q∗ , if q A-B> 0, then there exists
2∗); if q A-B, q2A-B )] ⊂ Q for some q2A-B ∈ (0, q A-B
∗hv > 0 such that N∗hv [(q A-B≤ 0, then there
2
2
1
∗
A-B
∗
A-B
exists h > 0 such that N∗h [(q 1 , q 2 )] ⊂ Q .
Proof of Claim 5. With c > 0, the existence of a partially revealing equilibrium implies∗∗∗
that LPA (q̂ A-B (Q )) ∩ LPB (q̂ A-B (Q )) ⊂ Q ≠ ∅. Also recall that LPA (q A-B ) ∩ LPB (q A-B ) ≠∅ for any q A-B ∈ Q. It follows that for the case where q A-B> 0, since πA∗ (q A-B ) and πB∗ (q A-B )2) respectively, we must have that (q A-B, 0) and (q A-B, q A-B, q2A-B ) ∈are minimized at (q A-B
2
1
1
1
∗
∗
). The continuity of the profit
LPA (q̂ A-B (Q )) ∩ LPB (q̂ A-B (Q )) for some q2A-B ∈ (0, q A-B
2∗∗∗functions and the set inclusion LPA (q̂ A-B (Q )) ∩ LPB (q̂ A-B (Q )) ⊂ Q then imply that∗
A-B )]). For theQfor some q2A-B ∈ (0, q A-B⊂
there exists ∗hv > 0 such that N∗hv [(q A-B
,
q
2
2
1
), we
case where q A-B, q A-B≤ 0, since πA∗ (q A-B ) and πB∗ (q A-B ) are both minimized at (q A-B
2
2
1
∗
∗
A-B
must have that (q A-B, q 2 ) ∈ LPA (q̂ A-B (Q )) ∩ LPB (q̂ A-B (Q )). It follows that there exists1∗)] ⊂ Q .
, q A-B∗h > 0 such that N∗h [(q A-B
2
1

The proofs of Claims 1–5 combined to prove Proposition 4.

Proof of Proposition 5. We construct a partially revealing equilibrium where Q∗ ≠ ∅∗and Q ≠ ∅. We continue to use the notation for Claim 2 for the proof of Proposition
4, denoting the unconditional expected attribute advantages for distributions F A-B byq̂ A-B (F A-B ) (instead of E[q A-B ] in the main text). Upon non-disclosure, the expected
attribute advantages of the boundedly rational consumers are q̂ A-B (F A-B ). We assign∗
Q = LPA (q̂ A-B (F A-B )) ∩ LPB (q̂ A-B (F A-B )). Recall that LPA (q A-B ) ∩ LPB (q A-B ) ≠ ∅ for> 0,
any q A-B ∈ Q, and it follows from the proof of Claim 5 for Proposition 4 that, if q A-B
2
A-B
then (q A-B, q2A-B ) ∈ LPA (q̂ A-B (F A-B )) ∩ LPB (q̂ A-B (F A-B )) for some q2A-B ∈ (0, q 2 ), and,1) ∈ LPA (q̂ A-B (F A-B )) ∩ LPB (q̂ A-B (F A-B )). Furthermore, theif q A-B≤ 0, then (q A-B, q A-B2
2
1
proof of Proposition 3 has established that LPK (q A-B ), K ∈ {A, B}, is a convex set for
any q A-B ∈ Q. These properties imply that int(LPA (q̂ A-B (F A-B )) ∩ LPB (q̂ A-B (F A-B ))) ≠ ∅.
Thus, firms A and B prefer not to disclose any q A-B ∈ LPA (q̂ A-B (F A-B ))∩LPB (q̂ A-B (F A-B ))and strictly so for any q A-B ∈ int(LPA (q̂ A-B (F A-B )) ∩ LPB (q̂ A-B (F A-B ))), as we desired.
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∗We next argue that for Q = LPA (q̂ A-B (F A-B )) ∩ LPB (q̂ A-B (F A-B )), Q∗ ≠ ∅ or equiv∗∗
alently Q ⊂ Q. Suppose instead that Q = Q. Since the density fiA-B , i ∈ {1, 2}, is∗everywhere positive, π K > πK(q̂ A-B (F A-B )), and thus firm K strictly prefers to disclose∗∗
argmax πK(q A-B ), yielding a contradiction. The proof for the characterizations of Q inq A-B ∈Q

parts (a) and (b) of the proposition is similar to the proof of Claim 5 for Proposition 4 andis therefore skipped.

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