Disclosure of Product Information After Price Competition

1 Introduction

In many markets, consumers need product information that is hardly obtainable unless sellers disclose it, and sellers compete to attract consumers by providing this otherwise private information in verifiable manners: for instance, free samples, informative advertisements, third-party test results, and technical reports by independent laboratories. Laws against fraud may prevent deceptive advertisements. This article analyzes the disclosure of verifiable information in a competitive environment and investigates under what conditions the consumers are sufficiently informed about products. We focus on the case where the sellers inform the consumers after price competition. In the car industry, for instance, the manufacturers set the prices but leave it to the dealers to persuade potential clients with verifiable information such as test drives.

Much literature has dealt with the disclosure of verifiable information by a single seller. Under vertical differentiation, where all consumers agree on the ranking of the valuations for products as if they are differentiated only in terms of quality, the well-known unraveling argument establishes that full disclosure should be the unique outcome (see Grossman 1981; Grossman and Hart 1980; Milgrom 1981; Milgrom and Roberts 1986 for a few seminal works).^[1] Recently, several authors have investigated the disclosure of product information by allowing for horizontal differentiation, where the rankings of the valuations for products are not necessarily identical across the consumers as if the products differ only in colors. Sun (2011) considers the model of a multidimensional product space. In one dimension, products differ in quality. In the other dimension, they are horizontally differentiated. She shows that if product quality is known to the consumer, the seller with higher quality is less likely to disclose information about the horizontal attribute. In a spatial model, Celik (2014) shows that full disclosure obtains if and only if the consumer’s preference for her ideal variety is sufficiently strong. In a general framework allowing for both vertical and horizontal differentiation, Koessler and Renault (2012) provide pairwise monotonicity as a necessary and sufficient condition for full disclosure to be the unique outcome regardless of the prior. In a model similar to Koessler and Renault (2012) and Woo (2023) considers price-dependent orders over products, called the sales-dominance relations, and characterizes pairwise monotonicity as the completeness of the sales-dominance relations at all prices. He shows that if the sales-dominance relation is complete at every price, the seller of a sales-dominant product should not pool with other sales-dominated products. That is, the generalized unraveling argument runs based on the order over products defined in terms price-dependent sales rather than price-independent quality.

In this paper, two sellers know each other’s product characteristics and the consumer privately knows her preference for differentiated products. The sellers simultaneously set prices for their individual products. After observing the chosen prices, each seller sends a verifiable message about his and his opponent’s products, that is, the advertisements are comparative. Finally, the consumer makes a purchasing decision. We extend the results of Koessler and Renault (2012) and Woo (2023) in a duopoly model and explore under what conditions product information sufficiently unravels for the consumer to make only the purchasing decision that should be made as if she were to know perfectly about the sellers’ products.

Of course, we are not the first to study information disclosure in a competitive environment. Board (2009) considers a duopoly model where products are vertically differentiated and advertising is non-comparative (i.e. the sellers are not allowed to inform about their opponent’s product). In Hotz and Xiao (2013), products are characterized by quality and horizontal location, consumers know the locations of the sellers, and the duopolists provide information about the quality attributes of their products. Both works show that price competition between two firms can undermine the full unraveling result. Roughly, in these works, full disclosure fails because it triggers intense price competition, which would result in lower prices and profits for the sellers. Cheong and Kim (2004) consider a model of oligopoly where products are vertically differentiated and information revelation is costly. They show that no matter how small the disclosure cost is, no seller will reveal information if the number of firms is sufficiently large. We want to highlight that all aforementioned works adopt the disclosure-then-price setting. That is, the sellers set prices only after they provide product information. Janssen and Teteryatnikova (2016) consider a spatial model where a particular location of each seller represents his type, and the consumer purchases from the seller that she expects to be closest to her ideal position. They consider the four combinations of disclosure-then-price or price-then-disclosure settings and comparative or non-comparative advertisements. They show that while full disclosure is an equilibrium outcome in all cases, it is the unique outcome only under the price-then-disclosure timing and comparative advertisements. While they focus on comparing the equilibrium outcomes across different circumstances, we explore the conditions under which product information sufficiently unravels under the price-then-disclosure setting.

Several features distinguish our model from previous works. Typically, the literature assumes that sellers compete in prices after providing product information (see also Levin, Peck, and Ye 2009). In contrast, we consider the setting in which sellers make disclosure decision after they set prices. That is, we endogenize prices but prices are known before they provide product information. This timeline would be suitable when adjusting prices is less flexible than disclosing information. For instance, the posted prices are more or less stable in the retail stores. Moreover, this approach would enable us to focus on some reasons, if any, other than intense price competition that may impede information disclosure in a competitive environment. Secondly, we allow both vertical and horizontal preferences for the consumer, which is essential to argue that, in a competitive environment, whether products are differentiated vertically or horizontally does not really matter to obtain information unraveling at every outcome. Thirdly, we allow types to be correlated. In Janssen and Teteryatnikova (2016), in particular, types are assumed independent. After prices are selected, the sellers compete in a zero-sum game by providing information about each other’s products. Namely, an increase in the market share for one seller implies a decrease in the market share for the other. Therefore, if a pooling outcome exists, the market share at this pooling outcome should be the same as the market shares that would result from the full disclosure of the suppressed product information, which is proved to be non-generic in Janssen and Teteryatnikova (2016). When types are correlated, this intuition for unraveling does not apply simply because information disclosure may not be a zero-sum game in market shares. Lastly, we adopt, as a solution concept, prudent rationalizability that is a version of extensive-form rationalizability featuring cautious behaviors and forward induction, and an extensive-form version of iterated admissibility, one of the oldest solution concept in game theory. Heifetz, Meier, and Schipper (2021) show that the unraveling result is obtained by replacing sequential equilibrium and “skepticism” of the uninformed consumer with prudent rationalizability. Schipper and Woo (2019) apply prudent rationalizability to electoral campaigning games in which voters may be unaware of some political issues and uncertain about political positions of the candidates, and the candidates use the sophisticated campaign strategy, microtargeting. Most importantly, Li and Schipper (2020) experiment with persuasion games in which a single seller provides verifiable information to the consumer. They observe that participants play actions that are consistent with prudent rationalizable strategies.

Having described the framework, we now discuss the results and explain the intuition. A product state refers to a pair of product types, one type for each seller. We find that in a competitive environment, how the consumer evaluates one seller relative to the other over the product states matters for information unraveling to be the unique outcome. In other words, whether products are differentiated vertically or horizontally does not really matter to obtain the unraveling result in a competitive environment. For simplicity, suppose that products are vertically differentiated. By fully disclosing the realized product type, a monopolist informs that his product is of the highest quality among all other possible products the consumer may believe. In other words, the consumer’s valuations for products matter when the monopolist makes a decision regarding information disclosure. In a competitive setup, by fully disclosing the realized product state, seller a informs that he is the most competitive against seller b among all other possible product states the consumer may believe. Hence, the differences in the consumer’s valuations for the sellers’ products matter for their decisions about information disclosure. Given a product state, the relative competitiveness of seller a to b refers this difference. For instance, if each seller i ∈ {a, b} could have either a high-quality product H _i or low-quality product L _i , (H _a , L _b ) is the best product state to seller a since he is the most competitive against b in this product state. Likewise, (L _a , H _b ) is the worst product state for seller a. However, the ranking of the relative competitiveness of seller a for product states (H _a , H _b ) and (L _a , L _b ) is ambiguous. Intuitively, the “superiority” between these product states may not be well-defined as if products may not be well-ranked under horizontal differentiation.

Given a price pair, sales from seller a in a product state (say, X) refer to a set of consumer types who prefer to purchase from a in X. We say that (given the price pair) product state X sales-dominates product state Y for seller a if sales from a in X are larger than sales from a in Y. Sales from seller b in a product state and the sales-dominance relation for seller b are analogously defined.^[2] We show that the consumer is sufficiently informed at every outcome if and only if at every price pair, every two product states are well-ordered in terms of sales. This is reminiscent of the standard persuasion games under vertical differentiation. The single seller with the product of some quality sends a verifiable message that rules out the possibility of the consumer believing products of lower qualities, and full disclosure uniquely obtains. Under the complete sales-dominance relations, unraveling proceeds analogously except that two informed sellers are involved and that product states are well-ordered in terms of price-dependent sales. In a competitive environment, the extended unraveling argument runs as follows. Consider an arbitrary price pair. Each seller in the most sales-dominant product state discloses the realized product state. Each seller in the second-most sales-dominant product state would inform that some product state at least as “good” as the second-most sales-dominant state is realized. Inductive reasoning continues to apply until all relevant product information sufficiently unravels.

To get an intuition of why the complete sales-dominance relations are necessary for information unraveling to be the unique outcome, consider product states X and Y that are incomparable in terms of sales at some price pair.^[3] By the definition of the sales-dominance relations, at this price pair, some consumer types must prefer seller a in product state X and b in Y, while the optimal choices of some other consumer types should be seller b in product state X and a in Y, which are opposite to the choices of the first type. In each product state, information suppression can be rationalizable to every seller. For instance, to seller a in product state X, it affects the probability of the consumer purchasing his product. Firstly, it increases the probability since the second consumer type may purchase from a under uncertainty. Secondly, it decreases the probability since the first consumer type may purchase from b under uncertainty. If seller a in X optimistically believes that the gain outweighs the loss, it is rational for him not to reveal the realized product state X.

In the US, Federal Trade Commission has allowed to name competing brands in advertisements since 1970. In 1997, the EU legalized comparative advertising subject to the restriction that it should not be misleading. However, in some Asian markets, comparative advertising has not been profoundly common. For example, in Malaysia and China, comparative advertisements are legal but without directly naming their competitors, and the use of comparative advertisements was sanctioned in India by the Advertising Standards Council of India in the late 1990s. Comparative advertising has been allowed officially in Korea only since 2001 but has not been widely used. We consider an alternative model of non-comparative advertisements. Not surprisingly, the necessary and sufficient condition required to obtain information unraveling at every outcome is stronger under non-comparative advertisements. Consider arbitrary doubleton sets of product types, one set for each seller. As an concrete example, suppose that each seller i ∈ {a, b} could have either a high-quality product H _i or low-quality product L _i . Note that the products of each seller are vertically differentiated (see Proposition 2 for the detailed discussion). Let’s assume, for the moment, that the preference of the consumer is known to the sellers. Then, the consumer is sufficiently informed even without any information disclosure, for instance, if the price of a is sufficiently higher than the price of b so that she prefers b regardless of the product state. Otherwise, some seller i ∈ {a, b} improves his payoff through full disclosure of H _i . For instance, seller a should fully reveal product type H _a when the consumer prefers to purchase from a in product state (H _a , H _b ) since she prefers a even in (H _a , L _b ). Likewise, seller b should fully reveal H _b if she prefers b in (H _a , H _b ). Now, we assume that the preference of the consumer is not known to the sellers, as is in our setup. Under the condition for sufficient unraveling, the consumer types can be partitioned in the following way. Between any two groups, the values for the relative competitiveness of seller a to b are “separated”. Namely, all these values of the consumer types in one group are higher than those of all consumer types in the other. Moreover, the relative competitiveness of seller a takes the same value in the product state (H _a , H _b ) to all consumer types within a group. Then, either of the followings holds at every price pair, as does in the case with the known consumer type. The consumer is sufficiently informed even without any information disclosure or seller a of type H _a or b of H _b voluntarily disclose their product types.

Section 2 formalizes the baseline model, the one with comparative advertisements. In Section 3, we define the price-dependent sales-dominance relations over product states and show that product information sufficiently unravels at every prudent rationalizable outcome if and only if the sales-dominance relation is complete at every price pair. In Section 4, we consider the model of non-comparative advertisements and provide a necessary and sufficient condition required for every outcome to achieve information unraveling in this alternative circumstance. In Section 5, we provide an example in which competition enhances information disclosure when advertisements are comparative and the consumer has limited reasoning capability. In Section 6, we conclude and discuss some advantages of the solution concept, prudent rationalizability.

2 Baseline Model

Two male sellers a and b (called players a and b, respectively) compete for a single female consumer (called player c) by providing verifiable information about their products. The sellers have complete information about each other’s product attributes. The consumer has complete and private information about her preference for differentiated products. Let G _i denote a finite set of product types for seller i ∈ {a, b}. A product state refers to an ordered pair (g _a , g _b ) ∈ G _a  × G _b of product types. A set of product states is denoted in bold by G = G _a  × G _b and a typical product state by g ∈ G . Let T be a finite set of the consumer’s preference types.^[4]

The game proceeds through four stages (see Figure 1). In the information stage, nature chooses the type of each player k ∈ {a, b, c}. Each seller observes the realized product state but remains uncertain about the consumer’s preference type. The consumer learns only about her preference type. In the price stage, the sellers simultaneously set binding prices for their products. After observing each other’s prices, the sellers simultaneously inform the consumer of the product state through costless messages. The information provided in the message stage must be truthful, while it may not be totally detailed. In the purchasing stage, the consumer makes a purchasing decision given the offered prices and the received messages. After the purchasing stage, the payoffs of all players realize.

Figure 1:

Timeline of the game.

Let P denote a set of all prices a seller can charge the consumer. We assume that set P is finite with an arbitrary grid (e.g. like cents in the U.S.). This assumption is made for reality to retain a finite game. A set of price pairs (p _a , p _b ) ∈ P × P is denoted in bold by P and a typical price pair by p ∈ P . For every price pair p = (p _a , p _b ), Δp denotes the difference of p _a from p _b , that is, Δp = p _a − p _b .

For simplicity, we assume that the marginal cost of each seller is zero regardless of his product type. The payoff to seller i ∈ {a, b} is given by his profit. It is p _i if the consumer purchases his product at a price p _i ∈ P, while it is zero otherwise. If a consumer type t ∈ T purchases the product of a type g ∈ G _a ∪ G _b at a price p ∈ P, her payoff is v + u(g, t) − p, where v > 0 is a type-independent willingness-to-pay of the consumer and u : G a ∪ G b × T → R + . We assume that v is sufficiently large so that the consumer must purchase from either seller.^[5] For every product state g = (g _a , g _b ) ∈ G and consumer type t ∈ T, the relative competitiveness of seller a to b is denoted by Δ u g , t = u ( g a , t ) − u ( g b , t ) . Attention is restricted only to a consumer with generic preferences. Formally, her preference is generic if Δu( g , t) ≠ Δp for every g ∈ G , t ∈ T, and p ∈ P .^[6] Under this assumption, the consumer strictly prefers one seller over the other at every price pair if the realized product state fully unravels.

For every k ∈ {a, b, c}, H _k denotes a set of player k’s information sets. At every information set h _i ∈ H _i of seller i ∈ {a, b}, he has learned the realized product state. Let g _a (h _i ) and g _b (h _i ) respectively denote the product type of seller a and b realized on the path to h _i and g (h _i ) the realized product state g a h i , g b h i . At every information set h _i ∈ H _i of seller i ∈ {a, b} in the message stage, he has observed the prices chosen in the price stage. By p _a (h _i ) and p _b (h _i ), we respectively denote the price chosen by seller a and b on the path to h _i and by p (h _i ) the price pair p a ( h i ) , p b ( h i ) . Every information set h _c ∈ H _c of the consumer is identified with a tuple of her preference type, prices, and messages observed in the previous stages. Let t(h _c ) denote the preference type learned on the path to h _c . For every i ∈ {a, b}, p _i (h _c ) and m _i (h _c ) respectively denote the offered price and the message provided by seller i on the path to h _c . By p (h _c ), we denote price pair p a ( h c ) , p b ( h c ) .

A pure strategy s _i of seller i ∈ {a, b} prescribes what price to charge at every information set in the price stage and what message to send about the product state at every information set in the message stage.^[7] A seller at an information set in the price stage chooses a price in P. A message s _i (h _i ) chosen at an information set h _i in the message stage satisfies g (h _i ) ∈ s _i (h _i ) and s _i (h _i ) ⊆ G . The latter implies that advertisements are comparative and the provided information may be vague. The former requires that each seller provide only truthful information. Note that the real product state must be among the states mentioned in the message. A pure strategy s _c of the consumer specifies from which seller to purchase at every information set in the purchasing stage, that is, s _c : H _c → {a, b}. For every k ∈ {a, b, c}, S _k denotes the set of player k’s pure strategies.

The above mentioned are common knowledge among the players. As a solution concept, we use prudent rationalizability, which is an extensive-form version of iterated admissibility. To define prudent rationalizability, we introduce belief systems. At every information set of the consumer, she forms a belief about product states and strategies of the sellers. A belief system b _c of the consumer is a tuple

such that for every h _c ∈ H _c , belief b _c (h _c ) assigns probability 1 to the set of profiles ( g , s _a , s _c ) ∈ G  × S _a  × S _b that reach h _c . From now on, we reserve the notation i and j for one seller and his opponent seller, respectively. At every information set of a seller, he forms a belief about the consumer’s preference types and strategies of the other players. For every i ∈ {a, b}, a belief system b _i of seller i is a tuple

such that for every information set h _i ∈ H _i , belief b _i (h _i ) assigns probability 1 to the set of profiles (t, s _j , s _c ) ∈ T × S _j  × S _c that reach h _i . Moreover, Bayesian updating is applied whenever possible. Consider information sets h i p and h i m of seller i in the price and message stage such that g h i p = g h i m . Then, h i m follows h i p . For every profiles (t, s _j , s _c ) and t ′ , s j ′ , s c ′ that reach h i m , b i h i p ( t ′ , s j ′ , s c ′ ) b i h i p ( t , s j , s c ) = b i h i m ( t ′ , s j ′ , s c ′ ) b i h i m ( t , s j , s c ) whenever b i h i p ( t , s j , s c ) > 0 . For every k ∈ {a, b, c}, B _k denotes a set of player k’s belief systems.

For every k ∈ {a, b, c}, we say that a strategy s _k ∈ S _k of player k is rational at an information set h _k ∈ H _k with a belief system if there does not exist an action a _k such that only replacing action s _k (h _k ) with a _k yields player k a strictly higher expected payoff. We say that s _k is rationalizable at h _k if there exists a belief system with which s _k is rational at h _k .

Prudent rationalizability is an iterated elimination process. Roughly, in each round of elimination, player k ∈ {a, b, c} at an information set forms a full-support belief by taking into account the move of nature and the strategies of the other players that have survived all previous rounds. Moreover, a strategy of player k survives this round of elimination if it is rational at every information set of player k with some belief system formed in the aforementioned manner. Indeed, prudent rationalizability features cautious behaviors of the players (see Heifetz, Meier, and Schipper (2021) for more details on prudent rationalizability). In each round, the cautiousness of the players enters through the full-support beliefs over the unknown types and the surviving strategies of the other players. Namely, a player does not completely exclude any of the other players’ unknown types and not-yet-eliminated strategies. This feature would play an essential role for the results.

A prudent rationalizable outcome refers to a profile ( s a , s b , s c ) ∈ S a ∞ × S b ∞ × S c ∞ of prudent rationalizable strategies. The existence of prudent rationalizable outcomes follows from a result in Heifetz, Meier, and Schipper (2021). We say that product information sufficiently unravels at a prudent rationalizable outcome (s _a , s _b , s _c ) if for every product state g ∈ G and information set h _c ∈ H _c of the consumer reached by ( g , s _a , s _b ), s _c (h _c ) = a if and only if Δ u g , t ( h c ) > Δ p i ( h c ) . Note that at a sufficiently unraveling prudent rationalizable outcome, the consumer makes exactly the purchasing decision that she would make under perfect information of the product state. Note further that in contrast to the consumer at a full-disclosure outcome, she may not learn the realized product state precisely at a sufficiently unraveling outcome. However, she is more or less well-informed of the product state in that any further revelation of product information will not change her purchasing decision, that is, her choice must be ex post optimal regardless of the product state she believes as possible.

3 Sales-Dominance Relations and Sufficient Unraveling

For every price pair p ∈ P and non-empty set G ′ ⊆ G of product states, we define T _i ( p , G ′) as the set of consumer types who prefer to purchase from seller i ∈ {a, b} in some product state g ∈ G ′, i.e.

Suppose that the consumer is offered a price pair p and learns that all and only product states in G ′ are possible.^[8] If G ′ is singleton (say, G ′ = { g }), the consumer precisely learns the realized product state g and all and only consumer types in T _i ( p , { g }) purchase from seller i. We call T _i ( p , { g }) sales from seller i at price pair p in product state g .^[9] Otherwise, if G ′ is non-singleton, then the consumer remains uncertain about the product state and purchasing from seller i is rationalizable to all and only consumer types in T _i ( p , G ′).^[10] Firstly, every consumer type not in T _i ( p , G ′) must purchase from the opponent seller j with whatever full-support belief she forms over G ′. Secondly, purchasing from i is rational to a consumer type t ∈ T _i ( p , G ′) with a full-support belief on G ′ assigning a sufficiently high probability to some product state (g _i , g _j ) ∈ G ′ such that u(g _i , t) − p _i > u(g _j , t) − p _j . We interpret T _i ( p , G ′) as the highest possible sales from seller i when the consumer is offered a price pair p and believes that the true product state is in G ′.^[11]

For every price pair p ∈ P , the strict sales-dominance relation and the sales-equivalence relation are defined as usual. I.e. g strictly sales-dominates g ′ at p for seller i ∈ {a, b} (denoted by g ≻ p i g ′ ) if T _i ( p , { g }) ⊋ T _i ( p , { g ′}). Whenever we refer to sales-dominance without qualifying as strict or weak, the default interpretation will be weak sales-dominance. Product states g and g ′ are sales-equivalent for seller i ∈ {a, b} at p (denoted by g ∼ p i g ′ ) if T _i ( p , { g }) = T _i ( p , { g ′}).

We say that the sales-dominance relation ≿ p i for seller i ∈ {a, b} is complete at price pair p ∈ P if g ≿ p i g ′ or g ′ ≿ p i g for every product states g and g ′ in G . Under the assumption of generic preference, if the sales-dominance relation is complete at p for one seller, so should be for the other because they are dual to each other in that g ≿ p i g ′ if and only if g ′ ≿ p j g . We write that the sales-dominance relation is complete at a price pair if it is the case for each seller. We say that the sales-dominance relations are complete if it is complete at every price pair. Note that the completeness of the weak sales-dominance relations is the condition on the primitives of the model, namely the preferences of the consumer.

The sales-dominance relation is trivially reflexive and transitive at every price pair since it is defined using set inclusion. If the sales-dominance relation is complete at price pair p ∈ P , G is a completely-preordered set by the sales-dominance relation ≿ p i .^[12] It is well-known that a completely-preordered finite set has the greatest element. For example, the most preferred alternative exists over a finite choice set if the preference relation is complete, reflexive, and transitive. Formally, the sales-dominance relation is complete at price pair p if and only if a sales-dominant product state for seller i ∈ {a, b} exists at p over every non-empty set G ′ ⊆ G of product states.

Consider an arbitrary price pair p at which g ̄ is sales-dominant for seller i ∈ {a, b} over message m _i ⊆ G . Lemma 1 implies that sufficient-disclosure of g ̄ is at least as profitable as m _i at information set h _i in the message stage such that g ( h i ) = g ̄ and p (h _i ) = p . That is, the for-sure sales T i ( p , { g ̄ } ) achieved by sending a message m ̄ i such that g ̄ ∼ p i g for every g ∈ m ̄ i are never lower than the highest-possible sales achieved by sending m _i . Lemma 1 further implies that in any second-round prudent rationalizable outcome, if g ̄ ≻ p i g for some g ∈ m _i , seller i should not send message m _i at the above mentioned information set h _i .^[13] Suppose to the contrary that seller i sends such message m _i at h _i . Firstly, seller i believes with a positive probability that seller j sends message m _j with g ∈ m _j at information set h _j in the message stage such that g ( h j ) = g ̄ and p (h _j ) = p , which would cause the consumer to believe both g ̄ and g as possible. Secondly, it is rational for every consumer type t ∈ T i ( p , { g ̄ } ) ∩ T j ( p , { g } ) to purchase from seller j at p after receiving messages (m _i , m _j ) with a belief assigning a sufficient high probability to g . Thus, the resulting sales from seller i can be strictly lower than under sufficient-unraveling of g ̄ .

In Theorem 1, we claim that product information unravels sufficiently at every prudent rationalizable outcome if the sales-dominance relation is complete at every price pair. For a simple illustration of the proof, attention is restricted to a price pair p ∈ P at which no two product states are sales-equivalent. Then, we can order the product states in G according to the strict sales-dominance relation ≻ p i for seller i ∈ {a, b} at p and call the product state in the nth place according to ≻ p i the nth most sales-dominant product state for i at p . For every n ≥ 1, g i n denotes the nth most sales-dominant product state for seller i at p and h i n denotes his information set in the message stage such that g h i n = g i n and p h i n = p . Roughly, we show that every first-round prudent rationalizable strategy of the consumer prescribes to purchase from the most preferred seller at p in g i 1 if the received messages fully unravel g i 1 and that every second-round prudent rationalizable strategy of seller i prescribes to disclose g i 1 at h i 1 . For r > 1, every (2r − 1)th-round prudent rationalizable strategy of the consumer prescribes to purchase from the most preferred seller at p in g i r if the provided messages inform that the worst product state according to ≿ p i is g i r (for instance, if messages (m _a , m _b ) are received such that g i r ∈ m a ∩ m b and g ≿ p i g i r for every g ∈ m _a ∩ m _b ). Moreover, every 2rth-round of prudent rationalizable strategy of seller i prescribes to send at h i r a message including only product states g that are at least as good as g i r , i.e. g ≿ p i g i r . After a finite number of rounds, all “undisclosed” product states are sales-equivalent at p and the elimination process stops.

One may misleading believe that the number of rounds of prudent rationalizability required for sufficient unraveling is determined by the number of product states in G . Instead, it depends on the way how G is partitioned according to the sales-dominance relation. Pick an arbitrary price pair p ∈ P that can be chosen at a prudent rationalizable outcome. If G is partitioned into one equivalence class according to ≿ p a , all product states are sales-equivalent at p and the consumer is sufficiently informed without any information disclosure. Now, assume that G is partitioned into n ̄ equivalence classes according to ≿ p a , where n ̄ = 2 n or 2n + 1 for some n ≥ 1. Then, given p , product information unravels sufficiently after 2n rounds. In Example 2, the two products states are different in sales only at p with Δp ∈ (−4, −2) ∪ (1, 3) and some strategies of the sellers are eliminated until the second round.

What restriction would the complete sales-dominance relations impose on the consumer’s ranking of Δu in the pairwise comparison of product states? Consider t_wo product states g ¹ and g ² in G . Completeness of the sales-dominance relation at p requires that T _a ( p , { g ¹}) \ T _a ( p , { g ²}) or T _a ( p , { g ²}) \ T _a ( p , { g ¹}) are empty. Now, we consider consumer types t ₁ and t ₂ such that Δ u g 2 , t 1 < Δ u g 1 , t 1 and Δ u g 1 , t 2 ≤ Δ u g 2 , t 2 . For any price pair p with Δ p ∈ Δ u g 2 , t 1 , Δ u g 1 , t 1 , t ₁ ∈ T _a ( p , { g ¹}) \ T _a ( p , { g ²}), implying T _a ( p , { g ²}) \ T _a ( p , { g ¹}) = ∅. Therefore, Δu should be monotone with respect for consumer types in that either Δ u g 2 , t 1 < u g 1 , t 1 ≤ u g 1 , t 2 ≤ Δ u g 2 , t 2 or Δ u g 1 , t 2 ≤ u g 2 , t 2 ≤ u g 2 , t 1 < Δ u g 1 , t 1 .

For every non-empty sets G ′ ⊆ G and T′ ⊆ T, a saddle point is defined as a pair g * , t * ∈ G ′ × T ′ such that Δu( g *, t*) = max _{G ′} min _T′Δu( g , t) = min _T′ max _{G ′}Δu( g , t). According to Gurvich and Libkin (1990), a saddle point exists for every non-empty sets G ′ and T′ if and only if every doubleton subsets G ″ ⊆ G and T″ ⊆ T has a saddle point. Therefore, if no saddle point exists for some non-empty sets G ′ and T′, some doubleton sets G ″ = { g ¹, g ²} and T″ = {t ₁, t ₂} exist for which either case in Table 3 holds. We say that the set P of prices is sufficiently fine if for every different pairs ( g , t) and ( g ′, t′) in G  × T, there exist a price pair p ∈ P such that Δu( g , t) < Δp < Δu( g ′, t′) or Δu( g ′, t′) < Δp < Δu( g , t). We believe that this assumption would be appropriate in the model with a single consumer as prices would be more “densely” populated on the real line than the values of relative competitiveness.

The assumption that P is sufficiently fine is crucial for Proposition 1. Even without this assumption, statements (ii) and (iii) are equivalent and they imply statement (i). However, they are not necessarily implied by (i). Assume that P = {0, 3, 6} and that Δ u g 1 , t 1 = 2 , Δ u g 2 , t 2 = 1 , and Δ u g 1 , t 2 = Δ u g 2 , t 1 ∈ ( 0,1 ) for G = { g ¹, g ²} and T = {t ₁, t ₂}. While no saddle point exists for G and T, the sales-dominance relation is complete at every price pair.

According to Proposition 1, the completeness of the sales-dominance relations depends on how the consumer evaluates one seller relative to the other over the type space. Namely, sufficient disclosure of product information is not necessarily implied by vertical differentiation nor by horizontal differentiation. The valuation of t ₂ for product type g a 2 is denoted by x, i.e. x = u g a 2 , t 2 . Products are vertically differentiated if value x of t ₂ for g a 2 lies in the range (5, 9). If x takes some low value (for example, x = 3 or x = 6), the sales-dominance relation is complete at every price pair since Δu( g , t ₁) ≥Δu( g , t ₂) for every g . Otherwise, if x takes some high value (for instance, x = 8 or x = 11), the sales-dominance relation is incomplete at price pair p with Δp ∈ (5, 6) since g ¹ and g ⁴ are not well-ranked in terms of sales.

4 Non-Comparative Advertisements

Now, we consider the model with non-comparative advertisements. Advertisements could be non-comparative either because sellers have private information about their own products, respectively, or because comparative advertisement is forbidden by law. For direct comparison to the results in the previous section, we focus on the latter case and keep the assumption that all sellers have complete information about each other’s product attributes.

The alternative game proceeds analogously to the baseline model. In the information stage, nature chooses the type of each player k ∈ {a, b, c}. Each seller observes the realized product state. The consumer privately learns her preference for differentiated products. In the price stage, the sellers simultaneously set binding prices for their individual products. In the message stage, each seller informs the consumer through a costless message about his product but never about his opponent’s. In the purchasing stage, the consumer purchases from one and only one seller given the offered prices and the received messages. As for the preference of the consumer, we additionally assume that u(g, t) ≠ u(g′, t) for every consumer type t ∈ T and different product types {g, g′}⊆ G _i of seller i ∈ {a, b}, implying that between any two products of seller i, the consumer strictly prefers one over the other.

A pure strategy s _i of seller i ∈ {a, b} prescribes what price to charge at every information set in the price stage and what message to send about his realized product type at every information set in the message stage. A message s _i (h _i ) chosen at information set h _i in the message stage satisfies g _i (h _i ) ∈ s _i (h _i ) ⊆ G _i . Namely, the provided message may be vague but must be truthful in that the realized product type must be among the types mentioned in the message. The players’ belief systems and prudent rationalizability are defined according to these modified strategies.

It is not surprising that a stronger condition is required for sufficient disclosure to be unique outcome under non-comparative advertisements as each seller is allowed to inform only about one “side” of product states. For instance, even when a seller is in the sales-dominant product state, he cannot reveal the product type of the opponent seller and the consumer may believe in some other product states that are not sales-dominant. In Example 3, that is the underlying reason why suppressing information is rationalizable to seller a at h a 1 and to seller b at h b 4 .

By definition, if product type g ̄ i of some seller i is absolute sales-dominant at p over G a ′ × G b ′ , all product states in { g ̄ i } × G j ′ are sales-equivalent at p . If each seller has an absolute sales-dominant product type at p over G a ′ × G b ′ , it can be easily shown that all product states in G a ′ × G b ′ are sales-equivalent at p .

In Theorem 3, we provide a necessary and sufficient condition for sufficient unraveling to be the unique prudent rationalizable outcome in the model with non-comparative advertisements. The proof for Theorem 3 proceeds analogous to Theorems 1 and 2. For a simple illustration of the proof for sufficiency, consider an arbitrary price pair. We essentially show that in every odd round of prudent rationalizability, the consumer updates her belief and makes a purchasing decision accordingly and that in every subsequent even round, at least one seller sufficiently discloses his product type in the message stage. The elimination process continues until all “undisclosed” product states are sales-equivalent.

With slight modification of the proof for Theorem 3, we can show that the condition described in Theorem 3 is necessary and sufficient for sufficient unraveling even when the sellers have private information about their own products.^[15] Suppose that the consumer faces a price pair p and learns that all and only product states in G ′ = G a ′ × G b ′ are possible. By definition, if g ̄ a ∈ G a ′ is absolute sales-dominant for seller a at p over G ′, then product state ( g ̄ a , g b ) is sales-dominant for a for any g b ∈ G b ′ . Since it is rational for seller a to disclose his type g ̄ a regardless of the actual type g b ∈ G b ′ , his rational choice would not vary even if he remains uncertain about b’s product type.

Consider arbitrary non-singleton sets G a ′ ⊆ G a and G b ′ ⊆ G b and a consumer type t ̄ ∈ T . Let G ′ = G a ′ × G b ′ . To illustrate an intuition for the equivalence result in Proposition 2, we partition T into three groups: T + = t ∈ T : min g ∈ G ′ Δ u ( g , t ) ≥ max g ∈ G ′ Δ u ( g , t ̄ ) , T − = t ∈ T : max g ∈ G ′ Δ u ( g , t ) ≤ min g ∈ G ′ Δ u ( g , t ̄ ) , and T ̄ = T \ ( T + ∪ T − ) . By construction, t ̄ ∈ T ̄ .

Firstly, condition (b) of (ii) requires that an identical product type (say g ̄ i ) in G i ′ of seller i ∈ {a, b} be the most preferred to all consumer types in T ̄ . Moreover, in product state ( g ̄ a , g ̄ b ) , the relative competitiveness Δu takes the same value (say, Δ ̄ ) to all consumer types in T ̄ (i.e. Δ u ( g ̄ a , g ̄ b ) , t = Δ ̄ for every t ∈ T ̄ ). Statement (ii) further implies that the ranges of the values for the relative competitiveness are “separated” between any two groups. Namely, max G ′ × T − Δ u ( g , t ) ≤ min G ′ × T ̄ Δ u ( g , t ) and max G ′ × T ̄ Δ u ( g , t ) ≤ min G ′ × T + Δ u ( g , t ) . Otherwise, if there exist consumer types t ′ ≠ t ̄ in T ̄ and t ⁺ ∈ T ⁺ such that min g ∈ G ′ Δ u ( g , t ′ ) < max g ∈ G ′ Δ u ( g , t ̄ ) ≤ min g ∈ G ′ Δ u ( g , t + ) < max g ∈ G ′ Δ u ( g , t ′ ) , Δ u ( g ̄ a , g ̄ b ) , t = Δ ̄ for every t ∈ { t ̄ , t + , t ′ } due to condition (b), a contradiction.