PPersuasion Produces the (Diamond) Paradox
Mark Whitmeyer ∗ December 8, 2020
Abstract
This paper extends the sequential search model of Wolinsky (1986) by allowing firms tochoose how much match value information to disclose to visiting consumers. This restoresthe Diamond paradox (Diamond (1971)): there exist no symmetric equilibria in which con-sumers engage in active search, so consumers obtain zero surplus and firms obtain monopolyprofits. Modifying the scenario to one in which prices are advertised, we discover that the no-active-search result persists, although the resulting symmetric equilibria are ones in whichfirms price at marginal cost.
Keywords:
Diamond Paradox, Sequential Search, Information Design, Bayesian Persuasion
JEL Classifications:
C72; D82; D83 ∗ Hausdorff Center for Mathematics & Institute for Microeconomics, University of BonnEmail: [email protected] am grateful to an anonymous referee for a different paper for the comment that sparked this idea. I also thank ÖzlemBedre-Defolie, Vasudha Jain, Stephan Lauermann, Benny Moldovanu, Joseph Whitmeyer, and Thomas Wiseman forhelpful comments and feedback. This work was generously funded by the DFG (German Research Foundation) underGermany’s Excellence Strategy-GZ 2047/1, Projekt-ID 390685813. a r X i v : . [ ec on . T H ] D ec ’m gonna stop, uh, shoppin’ around.Elvis Presley (G.I. Blues) The Diamond paradox (Diamond (1971)) is a stark result that highlights the importance of searchfrictions in models of price competition. Infamously, Diamond establishes that even in a marketwith a large number of firms, an arbitrarily small (yet positive) search cost ensures that the firmsbehave like monopolists and that consumers do not search. The intuition behind this result iswell-known: for any price strictly below the monopoly price, demand is locally perfectly inelasticand so a firm can always improve its lot by raising its price slightly.As a number of subsequent papers illustrate, small modifications to the model can overturnthe result. Two such works stand out in particular: Stahl (1989) assumes that a fraction of themarket’s consumers have zero search cost and therefore freely buy from whomever sets the lowestprice. As a result, demand for each firm slopes down again and they behave like monopolistsno longer. Wolinsky (1986) takes a different approach. In his model, although each consumerfaces a positive search cost, the firms’ products are differentiated and consumers have imperfectinformation about the products. Each consumer’s match value at any firm is an i.i.d. randomvariable, about which the firms and consumer are ex-ante uninformed. Upon visiting a firm, aconsumer discovers both the firm’s price as well as the realized match value. This uncertaintybegets equilibria with active search.In this paper, we revisit the framework of Wolinsky (1986) but alter it by assuming that eachfirm may choose how much information to provide to a consumer during her visit, in addition tosetting a price. Each firm commits to a signal that maps a consumer’s match value to a (condi-tional) distribution over signal realizations. In the spirit of the original Diamond paper, neitherthe signal nor the price chosen by a firm is observable until a consumer incurs the search cost.That is, after paying the search cost to visit a firm, a consumer then observes that firm’s signal,price, and draws a signal realization that yields her posterior belief about the firm’s match value.This modification has drastic consequences. We find that even in a model with product2ifferentiation and imperfect information, information provision restores the Diamond paradox .Namely, the main result of this paper, Theorem 2.5, states that there are no symmetric equilibriawith active consumer search. Each consumer visits at most one firm and purchases from it. Inall symmetric equilibria, firms leave consumers with zero rents.The intuition to this result is similar to that in the original paper by Diamond. There, in anypurported equilibrium with active search, firms can always exploit visiting consumers by raisingtheir prices slightly, which does not their purchase decisions due to the positive search cost.Here, firms deviate by providing slightly less information and pooling beliefs above consumers’stopping thresholds. Subsequently, given that any equilibrium must involve no search, the classicDiamond paradox incentive kicks in and firms must obtain monopoly profits.This incentive to pool beliefs is extremely strong, and continues to drive the results evenwhen prices are posted and so can be observed before consumers embark on their searches. Thesecond finding of this paper, Theorem 2.11, is that even when prices are posted–and thereforeshape consumers’ search behavior directly–there exist no symmetric equilibria with active search.Again, any purported equilibrium in which there is active search would allow firms to deviateprofitably by providing less information. In the unique symmetric equilibrium, because prices areposted, the usual Bertrand forces apply such that firms price at marginal cost and obtain no profits.Consumer surplus is merely the expected match value of the firm and no useful information isprovided. Note that none of the results of this paper require that there be a large number ofsellers, and all hold even in a duopoly setting.This paper belongs to the growing collection of papers that explore information design andpersuasion in consumer search settings. This literature includes Board and Lu (2018), who findan analog to the Diamond paradox. Their model is completely different, however: the uncertainstate of the world is common and so the competing sellers each provide information about thecommon state to prospective consumers. If sellers can perfectly observe a consumer’s currentbelief about the state upon her visit, or coordinate their persuasion strategies, then there is anequilibrium in which they provide the monopoly level of information and set the monopoly price.This equilibrium is not generally unique and requires additional assumptions that pertain to the There is a trivial equilibrium in which consumers conjecture extremely high prices and do not visit any firms.As is convention, throughout this paper, we ignore this equilibrium.
The foundation for this paper is the workhorse sequential search model of Wolinsky (1986). Thereare 𝑛 symmetric firms and a unit mass of consumers with unit demand. The match value of aconsumer at firm 𝑖 is an i.i.d. random variable, 𝑋 𝑖 , distributed according to (Borel) cdf 𝐺 on [0, 1] . Let 𝜇 denote the expectation of each 𝑋 𝑖 . Given a realized match value of 𝑥 𝑖 and price 𝑝 𝑖 , a4onsumer’s utility from purchasing from firm 𝑖 is 𝑢 (𝑥 𝑖 , 𝑝 𝑖 ) = 𝑥 𝑖 − 𝑝 𝑖 In contrast to the original model of Wolinsky (1986), a consumer does not directly observe hermatch value at firm 𝑖 , but instead observes a signal realization that is correlated with it. Each firmhas a compact metric space of signal realizations, 𝑆 , and commits to a signal, Borel measurablefunction 𝜋 𝑖 ∶ [0, 1] → Δ(𝑆) . As is well known, each signal realization begets–via Bayes’ law–aposterior distribution over values, and thus a signal begets a (Bayes-plausible) distribution overposteriors. Alternatively, a signal begets a distribution over posterior means, and the followingremark is now standard in the literature: Remark 2.1.
Each firm’s choice of signal, 𝜋 𝑖 , is equivalent to a choice of distribution 𝐹 𝑖 ∈ (cid:77) (𝐺) ,where (cid:77) (𝐺) is the set of all mean-preserving contractions of 𝐺 .Thus, each firm chooses a distribution over values, 𝐹 𝑖 , and sets a price 𝑝 𝑖 . Importantly, aconsumer only observes these choices upon her visit to a firm. Following Wolinsky (1986), searchis sequential and with recall. At a cost of 𝑐 > 0 , a consumer may visit a firm and observe its priceand realized draw from distribution 𝐹 𝑖 . As does Diamond (1971), we assume that a consumerincurs no search cost for her first visit. For simplicity, we impose that the marginal cost for eachfirm is .We look for symmetric equilibria in which each firm sets the same price 𝑝 and chooses thesame distribution over values 𝐹 . For simplicity, a consumer has an outside option of . It is astandard result that a consumer follows a reservation price policy. Indeed, a consumer’s searchproblem is a special case of the one explored in Weitzman (1979). Define the reservation value, 𝑧 , induced by the conjectured price ̃𝑝 and distribution ̃𝐹 by 𝑐 = ∫ (𝑥 − ̃𝑝 − 𝑧) 𝑑 ̃𝐹 (𝑥) ( ⋆ )In a symmetric equilibrium, a consumer’s optimal search protocol is to visit firms in randomorder and stop and purchase from firm 𝑖 if and only if the realized value at that firm, 𝑥 𝑖 , satisfies Wolinsky (1986) focuses on symmetric pure strategy pricing equilibria, which emphasis is echoed in this paper.Nevertheless, the results of this paper hold for all symmetric equilibria, in both pure and mixed strategies. 𝑖 − 𝑝 ≥ 𝑧 , where 𝑝 is the actual price set by firm 𝑖 . If 𝑥 𝑖 < 𝑧 + 𝑝 for all 𝑖 then a consumer selectsthe firm whose realized value 𝑥 𝑖 is highest. Observe that as the number of firms grows large thisrecall option becomes inconsequential. Naturally, at equilibrium, the conjectured price, ̃𝑝 , mustequal the actual price set by each firm, 𝑝 ; and the conjectured distribution, ̃𝐹 , must equal theactual distribution chosen by each firm, 𝐹 .Note that we are using Weitzman (1979)’s formulation of a consumer’s stopping problem,which is slightly nonstandard but equivalent to that of Wolinsky (1986). This allows for an easytransition into the next subsection, wherein we allow for advertised (posted) prices.Now, let us establish the main result. The first step is to derive the following lemma. Lemma 2.2.
There are no symmetric equilibria in which consumers visit more than one firm.Proof.
Suppose for the sake of contradiction that there is such an equilibrium. For expositionalease, let us begin by assuming that the firms’ choices of 𝐹 and 𝑝 are deterministic. Because 𝑐 > 0 and by the definition of the reservation value, the conjectured distribution, ̃𝐹 , must be suchthat both values strictly below and strictly above 𝑧 + ̃𝑝 occur with strictly positive probability.Accordingly, let [ ̄𝑎, ̄𝑎] and [ ̄𝑏, ̄𝑏] be intervals such that ∫ ̄𝑏̄𝑏 𝑑 ̃𝐹 (𝑥) > 0 and ∫ ̄𝑎̄𝑎 𝑑 ̃𝐹 (𝑥) > 0 , where ̄𝑎 ≤ ̄𝑎 < 𝑧 + ̃𝑝 and 𝑧 + ̃𝑝 < ̄𝑏 ≤ ̄𝑏 .Given price ̃𝑝 , a firm’s payoff from any value, 𝑥 , that is weakly greater than 𝑧+ ̃𝑝 is ̃𝑝 . Moreover,its payoff from any value, 𝑥 , that is strictly lower than 𝑧 + ̃𝑝 is strictly less than ̃𝑝 . Let 𝛼 < 1 denoteits average payoff (under ̃𝐹 ) for values in the interval [ ̄𝑎, ̄𝑎] .It is easy to see then that a firm can deviate profitably by choosing distribution ̂𝐹 , where ̂𝐹 isconstructed from ̃𝐹 by taking the measure on [ ̄𝑏, ̄𝑏] and some fraction 𝜖 > 0 of the measure on [ ̄𝑎, ̄𝑎] and collapsing them to their barycenter, and is set equal to ̃𝐹 everywhere else. ̂𝐹 will have a point mass on some ̂𝑥 , which will occur with probability ∫ ̄𝑏̄𝑏 𝑑 ̃𝐹 (𝑥) + 𝜖 ∫ ̄𝑎̄𝑎 𝑑 ̃𝐹 (𝑥) .For 𝜖 sufficiently small, ̂𝑥 > 𝑧 + ̃𝑝 , and by construction ̂𝐹 ∈ (cid:77) ( ̃𝐹 ) . Thus, since ̃𝐹 ∈ (cid:77) (𝐺) , ̂𝐹 ∈ (cid:77) (𝐺) . Finally, the net change in the firm’s payoff is 𝜖 ∫ ̄𝑎̄𝑎 𝑑 ̃𝐹 (𝑥) (1 − 𝛼 ) > 0 , which concludesthe proof.It is is simple to modify the proof to accommodate mixing by firms. In such a (purported)symmetric equilibrium, each firm chooses the joint distribution 𝐹 (𝑥, 𝑝) , over values and prices, In the parlance of Elton and Hill (1992), ̂𝐹 is a fusion of distribution ̃𝐹 . The notions of fusion and mean-preserving-contraction are (in this paper) equivalent. ̄𝑥, ̄𝑥] × [ ̄𝑝, ̄𝑝] , where each conditional distribution over values, 𝐹 𝑋|𝑃 (𝑥|𝑃 = 𝑝) ∈ (cid:77) (𝐺) . No ran-domness is resolved, however, until consumers visit firms and so we may define a new randomvariable
𝑌 ∶= 𝑋 − 𝑃 with distribution 𝐻 on some interval. Accordingly, the reservation value induced by a consumer’s conjectured ̃𝐻 is 𝑐 = ∫ (𝑦 − 𝑧) 𝑑 ̃𝐻 (𝑦) Note that the induced reservation values, 𝑧 , and distributions, 𝐻 , chosen by firms are determin-istic, and identical (because this a symmetric equilibrium).The remainder is (more-or-less) identical to the pure strategy case with some minor subtleties.Because consumers are searching actively, ̃𝐻 must be such that a consumer strictly prefers to stopand strictly prefers to continue with strictly positive probability. There are two cases: first, it ispossible that for some 𝑝 ′ in the support of a firm’s mixed strategy, the associated ̃𝐹 𝑋|𝑃 (𝑥|𝑝 ′ ) hasthe same property as ̃𝐻 –namely, both (strict) stopping and (strict) continuation values occur withstrictly positive probability. As in the pure strategy case, a firm can deviate profitably by poolingthese values carefully.On the other hand, it may be possible that there is no such 𝑝 ′ . However, then, there must existsome 𝑝 ′′ and ̃𝐹 𝑋|𝑃 (𝑥|𝑝 ′′ ) in the support of a firm’s mixed strategy such that, given 𝑝 ′′ , stoppingvalues occur with probability one and strict stopping values realize with strictly positive proba-bility. Consequently, a firm can deviate by providing no information (choosing the distribution 𝛿 𝑋 (𝜇) ) –which, at price 𝑝 ′′ , must leave the consumer with strict incentive to stop–and insteadcharging some price 𝑝 ′′ + 𝜖 , for a sufficiently small (yet strictly positive) 𝜖 . ■ In the standard Diamond paradox, firms have an incentive to raise prices slightly in order totake advantage of the quasi-monopoly power inferred upon them by the search cost. A similareffect occurs with regard to their information provision policies: they have an overwhelmingincentive to provide slightly less information to consumers in order to increase the probabilitythat they do not continue their searches.
Example 2.3.
To gain intuition, let us briefly investigate the following example. Suppose thereare infinitely many sellers and that the conjectured distribution over match values is the uniform Clearly, because prices are non-negative, the upper bound of the support must be (weakly) less than . 𝛿 𝑋 (𝜇) denotes the degenerate distribution ℙ (𝑋 = 𝜇) = 1 . ̃𝐹 (𝑥) = 𝑥 (in red) and a profitable deviation, ̂𝐹 (𝑥) (in blue), from Example 2.3. The grayarrows indicate the fusing of measures (or the contraction of the distribution).distribution on [0, 1] , and the search cost is 𝑐 = 1/32 . Given the conjectured distribution the con-jectured price must be √2𝑐 = 1/4 –indeed if this distribution over match values were exogenousand firms competed on price alone, this would be the unique symmetric equilibrium price (out-side of the trivial, no-visit equilibrium). Moreover, the associated reservation value 𝑧 = 1/2 andso a firm’s profit is .Now let a firm deviate in the manner proposed by the proof of Lemma 2.2. In particular,suppose it deviates by charging price and choosing a distribution that has a linear portionwith slope on the interval [0, 1/2] and a point mass of size on value . This brings thedeviating firm an improved profit of . In fact (following from Gentzkow and Kamenica (2016)),given a price , this particular deviation is the optimal distribution for a firm. The conjectureddistribution and the proposed profitable deviation are depicted in Figure 1.Next, we find that, given that no consumer visits more than one firm, there are no equilibriain which consumers obtain any rents. Lemma 2.4.
There are no symmetric equilibria in which consumers obtain any surplus.Proof.
From the previous lemma, we know that there are no symmetric equilibria in which con-8umers visit more than one firm. First, suppose that there is an equilibrium in which the reser-vation value chosen by each firm, 𝑧 , is greater than . Evidently, since a consumer visits just onefirm, the reservation value must equal 𝜇 − 𝑐 − ̃𝑝 , which is greater than by assumption.A firm’s profit, conditional on a consumer’s visit, is ̃𝑝 ≤ 𝜇 − 𝑐 . Clearly, then, a firm can deviateby charging price 𝑝 = ̃𝑝 + 𝑐 − 𝜖 . For 𝜖 > 0 sufficiently small, this is a profitable deviation. Thus,there are no equilibria in which firms induce non-negative reservation values.Second, suppose that there is an equilibrium in which the reservation value chosen by eachfirm, 𝑧 , is strictly less than . Now, each firm is facing a monopolist’s problem where a consumerhas an outside option of . Accordingly, a firm can extract all of a consumer’s surplus by providingno information (choosing the degenerate distribution 𝛿 𝑋 (𝜇) ) and charging 𝑝 = 𝜇 . ■ Combining the two lemmata, we have our main result
Theorem 2.5.
In any symmetric equilibrium in which firms are visited, each firm makes a salewith certainty, conditional on being visited, and obtains the monopoly profit of 𝜇 . There is no activesearch and consumer surplus equals . In general, there may exist a multiplicity of equilibria. For instance, if the prior distribution isthe Bernoulli distribution with mean 𝜇 , there are equilibria in which firms provide full informa-tion and charge price and those in which firms provide no information (choose the degeneratedistribution 𝛿 𝑋 (𝜇) ) and charge price 𝜇 . However, as stipulated by Theorem 2.5, all are equivalentin that consumer and producer surplus and consumer behavior are the same. Perhaps surprisingly, the no active search result continues to hold, even if prices are advertised and so search is directed. Specifically, we assume that a consumer observes the price set by eachfirm before starting her search, but still must visit a firm to observe its (expected) match value.We find that even when prices are posted, there are no symmetric equilibria with active search.
Lemma 2.6.
In any symmetric equilibrium in which firms make strictly positive profits, there is noactive search. That is, consumers visit no more than one firm. roof. In any symmetric equilibrium, each firm chooses a joint distribution,
𝐹 (𝑥, 𝑝) , over valuesand prices, [ ̄𝑥, ̄𝑥] × [ ̄𝑝, ̄𝑝] , where each conditional distribution over values, 𝐹 𝑋|𝑃 (𝑥|𝑃 = 𝑝) ∈ (cid:77) (𝐺) .Given a firm’s choice of (on-path) price 𝑝 ∗ ∈ [ ̄𝑝, ̄𝑝] , its (on-path) choice of distribution over values 𝐹 𝑋|𝑃 (𝑥|𝑃 = 𝑝 ∗ ) must yield a maximal payoff for the firm, given the strategies of the other firms.For each price, the corresponding conditional distribution over values corresponds to a reser-vation value, 𝑧 , which is pinned down by Equation ⋆ . Define (cid:90) to be the set of all reservationvalues that are induced on-path. Joint distribution 𝐹 (𝑥, 𝑝) induces a distribution over reservationvalues
Φ(𝑧) . Note that, in contrast to the previous section’s case, in which both prices and dis-tributions are hidden, the observability of firms’ price choices means that the reservation valueschosen by firms may be random.Since firms are making strictly positive profits, any on-path price, 𝑝 ∗ , must itself be strictlypositive. Denote by 𝑧 ∗ ∈ (cid:90) the corresponding reservation value. Evidently, a firm’s payoff,conditional on being visited, from any value 𝑥 ≥ max {0, 𝑧 ∗ } + 𝑝 ∗ , is 𝑝 ∗ .First, suppose that there exists an on-path price 𝑝 ∗ and conjectured distribution ̃𝐹 𝑋|𝑃 (𝑥|𝑝 ∗ ) thatinduces a 𝑧 ∗ that is weakly greater than . As in the proof for Lemma 2.2, suppose for the sakeof contradiction that conditional on her arrival at the firm a consumer strictly prefers to stopand strictly prefers to continue her search (or select her outside option) with strictly positiveprobability. However, all values in the stopping set yield a payoff of 𝑝 ∗ and all beliefs in thecontinuation set yield a payoff that is strictly below 𝑝 ∗ . Consequently, as in the proof for Lemma2.2, after posting price 𝑝 ∗ , a firm can (secretly) deviate by providing slightly less information andfusing a subset of the beliefs in the continuation and stopping sets.Second, suppose that there exists an on-path 𝑧 ∗ that is strictly less than . In such an equilib-rium, if a firm is visited, it is the first and only firm that is visited by the consumer, since the firstvisit is the only one that does not impose on the consumer a search cost. ■ The reason why there may exist an on-path 𝑧 ∗ that is strictly less than is because consumersdo not incur search costs at the first firm they visit. If they did incur such costs at the first firm,then all on-path 𝑧 ∗ would have to be weakly positive, since otherwise consumers would strictlyprefer their outside option to visiting firms with such 𝑧 values.Before proceeding on, let us glance at the following example, in which we use the concavifi-cation approach made famous by Kamenica and Gentzkow (2011) to illustrate a firm’s profitable10eviation in a purported equilibrium with active search. Example 2.7.
Let 𝑛 = 2 and the match value of each firm be a Bernoulli random variable withmean . Let the search cost, 𝑐 = 1/16 . If firms could not choose how much information toprovide–and were forced to provide full information–and could only compete by posting prices,then it is straightforward to see that there is an equilibrium in which each firm chooses a con-tinuous distribution over prices that induces the distribution over reservation values,
Φ (𝑧) , givenby
Φ (𝑧) = 77 − 8𝑧 − 1, on [0, 716 ] ( )The associated range of prices is [7/16, 7/8] . Now, let us maintain the assumption that pricesare posted but restore the ability of firms to choose the distribution over match values. We willestablish that given the conjectured distribution over reservation values stated in Expression ,there are on-path prices after which firms strictly prefer to deviate and choose a distribution otherthan the Bernoulli distribution.Indeed, let us characterize the optimal distribution over values following a choice of price 𝑝 ∗ = 7/16 by a firm. Given a consumer’s conjectured Φ , because the firm has chosen the loweston-path price, it will be visited first. This makes it easy to write the firm’s payoff as a function ofits posterior value, 𝑉 (𝑥) . It is
𝑉 (𝑥) = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩0, 0 ≤ 𝑥 < ( ) ( ) , ≤ 𝑥 ≤ , ≤ 𝑥 ≤ 1 From Kamenica and Gentzkow (2011), we know that the optimal distribution over values can begiven by concavifying 𝑉 . 𝑉 and its concavification, ̂𝑉 , are depicted in Figure 2, in which thedistribution over posteriors corresponding to full information (the Bernoulli distribution overvalues) is also portrayed. Evidently, the latter is strictly suboptimal.The next lemma restricts what can happen at equilibrium following firms’ price choices. Lemma 2.8.
In any equilibrium in which firms make strictly positive profits we must have thefollowing:
𝑉 (𝑥) (in black); its concavification, ̂𝑉 (𝑥) (in blue); and the splitting corresponding tothe Bernoulli distribution (in red); from Example 2.7.
1. If ∗ > 𝜇 is chosen on-path, distribution 𝐹 𝑋|𝑃 (𝑥|𝑝 ∗ ) has a mass point of size 𝑞 ∗ > 0 on 𝑝 ∗ and has no support strictly above 𝑝 ∗ ;2. If 𝜇 ≥ 𝑝 ∗ > 𝜇 − 𝑐 is chosen on-path, distribution 𝐹 𝑋|𝑃 (𝑥|𝑝 ∗ ) must have support entirely above 𝑝 ∗ . A firm is selected with certainty, conditional on being visited.3. If 𝜇 − 𝑐 ≥ 𝑝 ∗ > 0 is chosen on-path, distribution 𝐹 𝑋|𝑃 (𝑥|𝑝 ∗ ) must have support entirely above 𝑝 ∗ . A firm is selected with certainty, conditional on being visited.Proof. If ∗ > 𝜇 is chosen on-path, then if the associated reservation value, 𝑧 ∗ ≥ 0 , theconjectured distribution must be such that ∫ ̄𝑥̄𝑥 𝑑 ̃𝐹 𝑋|𝑃 (𝑥|𝑝 ∗ ) > 0 for some ̄𝑥 ≥ ̄𝑥 > 𝑧 ∗ + 𝑝 ∗ . Moreover,because 𝑝 ∗ > 𝜇 , the conjectured distribution must be such that ∫ ̄𝑤̄𝑤 𝑑 ̃𝐹 𝑋|𝑃 (𝑥|𝑝 ∗ ) > 0 for some ̄𝑤 ≤ ̄𝑤 < 𝑝 ∗ . A firm’s payoff from any belief 𝑥 ≥ 𝑧 ∗ + 𝑝 ∗ is 𝑝 ∗ and from any belief 𝑥 < 𝑝 ∗ is , soit can deviate profitably by fusing the measure on [ ̄𝑥, ̄𝑥] with a fraction, 𝜖 > 0 , of the measure on [ ̄𝑤, ̄𝑤] (collapsing them to their barycenter). We conclude that 𝑧 ∗ < 0 .Define 𝐹 −𝑋|𝑃 (𝑎|𝑝 ∗ ) ∶= sup 𝑤<𝑎 𝐹 𝑋|𝑃 (𝑤|𝑝 ∗ ) 𝐹 −𝑋|𝑃 (𝑝 ∗ |𝑝 ∗ ) < 1 , or else a firm would make profit from choosing 𝑝 ∗ . Furthermore,any values 𝑥 < 𝑝 ∗ yield a firm a profit of and since 𝑝 ∗ > 𝜇 , 𝐹 −𝑋|𝑃 (𝑝 ∗ |𝑝 ∗ ) > 0 . Consequently, a firmmust have 𝐹 𝑋|𝑃 (𝑝 ∗ |𝑝 ∗ ) = 1 since otherwise it could (as in the previous paragraph) fuse a positivemeasure of values strictly below 𝑝 ∗ with a positive measure of values strictly above 𝑝 ∗ and obtaina higher payoff.Define 𝑞 ∗ = 𝑞 (𝑝 ∗ ) ∶= 𝐹 𝑋|𝑃 (𝑝 ∗ |𝑝 ∗ ) − 𝐹 −𝑋|𝑃 (𝑝 ∗ |𝑝 ∗ ) i.e., 𝑞 ∗ is the size of the mass point that 𝐹 𝑋|𝑃 (𝑥|𝑝 ∗ ) places on 𝑝 ∗ . Accordingly, 𝑧 ∗ = −𝑐/𝑞 ∗ < 0 .If 𝜇 ≥ 𝑝 ∗ > 𝜇 − 𝑐 is chosen on-path, then if the associated reservation value, 𝑧 ∗ ≥ 0 , a firmmust be selected with certainty, conditional on being visited. This is because 𝑧 ∗ ≥ 0 > 𝜇 − 𝑐 − 𝑝 ∗ ,which is the minimal reservation value that can be induced. Thus, if a firm were not selectedwith certainty, conditional on being visited, it could always fuse values strictly below and strictlyabove 𝑧 ∗ + 𝑝 ∗ . Accordingly, all values must be weakly greater than 𝑝 ∗ , i.e., 𝐹 −𝑋|𝑃 (𝑝 ∗ |𝑝 ∗ ) = 0 . Ifthe associated reservation value, 𝑧 ∗ is strictly negative, the result is trivial since a firm couldalways just provide no information (choose the degenerate distribution 𝛿 𝑋 (𝜇) ) and be selectedwith certainty, conditional on being visited.Finally, if 𝜇−𝑐 ≥ 𝑝 ∗ > 0 , then the minimum reservation value that can be induced is 𝜇−𝑐−𝑝 ∗ ≥ 0 .Either 𝐹 −𝑋|𝑃 (𝑧 ∗ + 𝑝 ∗ |𝑝 ∗ ) = 0 , in which case 𝑧 ∗ = 𝜇 − 𝑐 − 𝑝 ∗ and so a firm is clearly selected for sure,conditional on being visited; or 𝐹 −𝑋|𝑃 (𝑧 ∗ + 𝑝 ∗ |𝑝 ∗ ) > 0 and so 𝑧 ∗ > 𝜇 − 𝑐 − 𝑝 ∗ and 𝐹 𝑋|𝑃 (𝑧 ∗ + 𝑝 ∗ |𝑝 ∗ ) < 1 .In that case either a firm is selected for sure or it can deviate profitably by fusing portions of themeasure strictly above and below 𝑧 ∗ + 𝑝 ∗ . ■ Next, because there is no active search, we find that firms cannot make positive profits in anysymmetric equilibrium.
Lemma 2.9.
There are no symmetric equilibria in which firms make nonzero profits. Equivalently,there are no symmetric equilibria in which firms charge any price other than .Proof. Let ̂𝑝 be the maximal price that is chosen on-path, with associated reservation value ̂𝑧 .First, suppose that ̂𝑝 ≤ 𝜇 . From Lemmata 2.6 and 2.8, following any on-path 𝑝 , a firm mustbe chosen for sure, conditional on being visited. Clearly, ̂𝑝 must be chosen with strictly posi-13ive probability on-path, since there is no active search. Otherwise, choosing that price wouldguarantee that that firm is never visited, yielding a profit of .Consequently, all values, 𝑥 , must be such that 𝑥 − ̂𝑝 ≥ max {0, ̂𝑧} . Thus, ̂𝑧 = 𝜇 − 𝑐 − ̂𝑝 . But thena firm can deviate profitably to some price ̂𝑝 − 𝜂 , where 𝜂 > 0 , and the degenerate distribution 𝛿 𝑋 (𝜇) . Because 𝜇 − 𝑐 − ̂𝑝 + 𝜂 is the minimum reservation value such a deviation could induce, thedeviating firm obtains a discrete jump up in its payoff.Second, suppose that ̂𝑝 > 𝜇 . The probability that firms are choosing prices that are weaklygreater than 𝜇 must be strictly positive (or else choosing ̂𝑝 would result in profit for a firm).From a consumer’s viewpoint, all on-path prices 𝑝 ∗ ≥ 𝜇 are equivalent. Indeed, for all such 𝑝 ∗ ≥ 𝜇 , 𝐹 𝑋|𝑃 (𝑝 ∗ |𝑝 ∗ ) = 1 .A firm’s payoff from choosing any 𝑝 ∗ ≥ 𝜇 , conditional on being visited, is 𝑞 (𝑝 ∗ ) 𝑝 ∗ , where,recall, 𝑞 (𝑝 ∗ ) is the size of the mass point on 𝑝 ∗ . Moreover, by definition, 𝑞 (𝑝 ∗ ) 𝑝 ∗ ≤ 𝜇 . But then,a firm can deviate profitably by choosing some price 𝑝 = 𝜇 − 𝜖 , for a small but strictly positive 𝜖 , and the degenerate distribution 𝛿 𝑋 (𝜇) . No matter a consumer’s belief about the firm’s distri-bution after observing this price, a deviating firm must still be visited (and thus purchased fromeventually) before any firm that is choosing 𝑝 ∗ ≥ 𝜇 . Accordingly, for a sufficient small 𝜖 > 0 , afirm will receive a discrete jump in its payoff and so there exists a profitable deviation. ■ Lemma 2.10.
There exist no symmetric equilibria with active search in which firms make zeroprofits.Proof.
Obviously, in any symmetric equilibrium in which firms make zero profits, then if a firmchooses any price 𝑝 > 0 it must be visited with probability . Thus, we impose that firms postprices 𝑝 = 0 . If 𝜇 − 𝑐 > 0 , it suffices to to show that there are no equilibria in which firms choosedistributions over values that induce reservation values strictly greater than 𝜇 − 𝑐 .Suppose for the sake of contradiction that such an equilibrium exists. Then, a firm’s distribu-tion over values must be such that the probability of a realization that is weakly below 𝜇 − 𝑐 − 𝛾 (with 𝛾 > 0 ) must occur with probability 𝛼 > 0 . However, then a firm can deviate profitably tosome price 𝑝 that satisfies and the degenerate distribution 𝛿 𝑋 (𝜇) . Evenshould a consumer assign it the most pessimistic belief about its distribution, yielding a reserva-14ion value of 𝜇 −𝑐 −𝑝 , the firm would still be visited (and selected) with strictly positive probability,yielding a positive profit.What if 𝜇 −𝑐 ≤ 0 ? From Au and Whitmeyer (2020a), the unique maximal reservation value thatcan be induced by a distribution with mean 𝜇 (and supported on [0, 1] ) is the Bernoulli distributionwith mean 𝜇 . Note that this is a MPC of 𝐺 if and only if 𝐺 is itself that Bernoulli distribution. Themaximal reservation value is , which is weakly bigger than if and only if 𝜇 ≥ 𝑐 . Moreover,given (an off-path) price 𝑝 > 0 , the maximal reservation value is , which is strictlynegative for all 𝜇 ≤ 𝑐 . Hence, if 𝜇 ≤ 𝑐 then no matter what distribution the consumer conjecturesafter observing a price deviation, the deviator will not be visited. Furthermore, if 𝜇 ≤ 𝑐 , there isno active search, on-path, since any value a consumer obtains at the first firm is at least weaklygreater than the reservation values of the other firms. ■ At last, we arrive at the second main result.
Theorem 2.11.
Any symmetric equilibrium must be one in which each firm posts price 𝑝 = 0 andchooses a distribution that induces the minimum reservation value, 𝜇 − 𝑐 .Proof.
First, we establish that such purported equilibria are indeed equilibria. Trivially, a firmcannot deviate profitably by choosing a different distribution over values (and keeping price 𝑝 =0 ). Next, we stipulate that a consumer assumes that any firm who deviates to a price 𝑝 ≠ 0 chooses a completely uninformative distribution. Accordingly, that firm will never be visited andtherefore such a deviation is not profitable.Second, uniqueness follows from Lemmata 2.6, 2.9, and 2.10. ■ In any symmetric equilibrium, each firm’s profit is and each consumer’s welfare is just 𝜇 (they do not incur a search cost, 𝑐 , since that applies only to searches beyond the first firm).Although consumers obtain some surplus, there is no benefit from increased competition, i.e., amarket with two firms is just as good for a consumer as one with infinitely many. Wolinsky (1986) provides a compelling resolution for the Diamond paradox. Product differen-15iation and imperfect information allow for more realistic and empirically-relevant equilibria inwhich consumers search and in which competitive forces have real effects on the pricing deci-sions of firms. Indeed, a key aspect of Wolinsky’s model is that consumers obtain informationabout their match values upon visiting firms.Crucially, that information must come from somewhere and in particular, firms have a say inhow much information a consumer’s visit will glean about their respective products. With thatin mind, the extreme results that we encounter in this paper are troubling, since the model itselfseems (in some sense) more realistic than Wolinsky (1986), yet restores the pathological (andostensibly unrealistic) result of Diamond (1971).Accordingly, this paper suggests a weakness in using imperfect information and horizontaldifferentiation alone to generate competitive pricing and active search in search models. Namely,if firms may choose how much information to provide about their products and that informationis unadvertised then the Diamond paradox reemerges, despite the market’s heterogeneity. Mod-ifications to the model–like, e.g., assuming that a fraction of consumers can search without costá la Stahl (1989)–are needed to restore active search.It is important to keep in mind that, just as the Diamond result requires that consumers learnprices only after paying a search cost, the hidden nature of the firms’ information choices isessential to the results that we encounter here. If information itself can be advertised–i.e. con-sumers can observe each firm’s chosen distribution over values without paying a visit cost–thereare equilibria in which consumers search and obtain positive surplus. This situation with postedinformation is the subject of Au and Whitmeyer (2020b). Furthermore, it seems quite plausible that although firms can advertise information policies partially , they have no way of fully specifying or committing to more information than a certain(minimal amount). Consequently, the strong incentives for firms to deviate and under-provideinformation we find here suggest that firms will not provide more information than they can (ex- See also Au and Whitmeyer (2020a), in which we explore pure informational competition between sellers inWeitzman (1979)’s classic model of directed sequential search. There, we compare the cases in which informationis hidden versus advertised and find that advertised information results either in full information or informationdispersion (randomization over information structures), depending on the prior distribution over match values. Incontrast, hidden information ensures that no (useful) information is provided in any (symmetric) equilibrium. or real-estate agents whiskingprogressive tenants through their apartment tours. References
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