Buyer-Otimal Learning and Monooly Pricing Anne-Katrin Roesler and Balázs Szentes January 2, 217 Abstract This aer analyzes a bilateral trade model where the buyer s valuation for the object is uncertain and she observes only a signal about her valuation. The seller gives a take-it-orleave-it offer to the buyer. Our goal is to characterize those signal structures which maximize the buyer s exected ayoff. We identify a buyer-otimal signal structure which generates (i) effi cient trade and (ii) a unit-elastic demand. Furthermore, we show that every other buyer-otimal signal structure yields the same outcome as the one we identify, in articular, the same rice. 1 Introduction Raid develoments in information technology have given consumers access to new information sources that in many cases allow them to acquire roduct information rior to trading. For an examle, consider Yel, a mobile hone a that aggregates customer reviews and makes them available to new customers. Ceteris aribus, a single consumer is better off with such an a because she becomes more effi cient at choosing the store selling her most referred roducts. On the other hand, after the aearance of the a, stores will face customers with a different (and erhas higher) distribution of willingness-to-ay. Therefore, stores are likely to resond by increasing their rices. As a consequence, a new information source - even when freely available - might worsen the buyer s outcome if being better informed results in higher rices. The question then becomes: How does an information source affect rices and consumer welfare? Our aer contributes to this roblem by identifying information structures which are best for the buyers in a monoolistic market. Building on this result, we then characterize all the ossible combinations of consumer and roducer surlus which can arise from some learning. This aer suersedes the solo aer of the first author which was circulated with the title Is Ignorance Bliss? Rational Inattention and Otimal Pricing. We have benefited from discussions with Dirk Bergemann, Ben Brooks, Daniele Condorelli, Eddie Dekel, Yingni Guo, Johannes Hörner, Emir Kamenica, Daniel Krähmer, Benny Moldovanu, Tymofiy Mylovanov. Deartment of Economics, University of Michigan, Ann Arbor, MI 4819. E-mail: akroe@umich.edu. Deartment of Economics, London School of Economics, London, UK. E-mail: b.szentes@lse.ac.uk. 1
We consider a stylized model where the seller has full bargaining ower and the buyer receives a costless signal about her true valuation. The seller sets a rice knowing the joint distribution of the buyer s valuation and the signal but not their realizations. We assume that the buyer s valuation is always ositive and the seller s outside otion is zero, such that effi ciency requires trade with robability one. The aer s main theorem identifies the least informative buyerotimal signal distribution. This distribution generates a unit-elastic demand and the seller is indifferent between charging any rice on its suort. In equilibrium, the seller sets the lowest rice,, and, since this is the lowest ossible value estimate of the buyer, trade always occurs. We show that all buyer-otimal signal structures yield the same outcome, namely that the rice is and trade always takes lace. An imortant observation is that the buyer s signal does not enhance the effi ciency of her urchasing decision along the equilibrium ath. The buyer always buys the good at the equilibrium rice, irresective of her value estimate and tyically even if the rice exceeds her true valuation. The buyer s otimal learning is therefore driven exclusively by the goal of generating a demand function which induces the lowest ossible rice subject to effi cient trade. Finally, for any given value distribution of the buyer, we characterize the combinations of consumer and roducer surluses which can arise as an equilibrium outcome if the buyer has access to some information source. 1 We show that any consumer-roducer surlus air can be imlemented if and only if (i) consumer surlus is non-negative, (ii) roducer surlus is weakly larger than, and (iii) total surlus is weakly smaller than the first-best surlus. 2 Our observation that, in a contracting environment, an imerfectly informed agent can be better off than a erfectly informed one is also noticed by Kessler (1998). The author considers a rincial-agent model where, rior to contracting, an agent observes a ayoff relevant state with a certain robability. Of course, the rincial s contract deends on this robability. The author shows that the agent s ayoff is not maximized when this robability is one, that is, an agent might benefit from having less than erfect information. We not only confirm this result, but also characterize buyer-otimal learning without imosing any constraints on ossible signal structures. Several aers examine buyers incentives to acquire costly information about their valuations before articiating in auctions. The buyers learning strategies deend on the selling mechanism announced by the seller. Persico (2) shows that if the buyers signals are affi liated then they acquire more information in a first-rice auction than in a second-rice one. Comte and Jehiel (27) show that dynamic auctions tend to generate higher revenue than simultaneous ones. Shi (211) also analyses models where it is costly for the buyers to learn about their valuations and identifies the revenue-maximizing auction in rivate-value environments. In all of these setus, the 1 We are grateful to Ben Brooks for drawing our attention to this roblem. 2 In contrast, Bergemann et al. (215) characterize the ossible outcomes in a model where the buyer is fully informed and the seller observes a signal about the buyer s valuation. 2
seller is able to commit to a selling mechanism before the buyers decide how much information to acquire. In contrast, we characterize buyer-otimal learning in environments where the monoolist best-resonds to the buyer s signal structure. 3 Condorelli and Szentes (215) also consider a bilateral trade model. In contrast to our setu, the distribution of the buyer s valuation is not given exogenously. Instead, the buyer chooses her value-distribution suorted on a comact interval and erfectly observes its realization. The seller observes the buyer s distribution but not her valuation and sets a rice. The authors show that, as in our model, the equilibrium distribution generates a unit-elastic demand and trade always occurs. 2 Model There is a seller who has an object to sell to a single buyer. The buyer s valuation, v, is distributed according to the continuous CDF F suorted on [, 1]. Let µ denote the exected valuation, that is, vdf (v) = µ. The buyer observes a signal s about v. The joint distribution of v and s is common knowledge. The seller then gives a take-it-or-leave-it rice offer to the buyer,. 4 Finally, the buyer trades if and only if her exected valuation conditional on her signal weakly exceeds. 5 If trade occurs, the ayoff of the seller is and the ayoff of the buyer is v ; otherwise, both have a ayoff of zero. Both the seller and the buyer are von Neumann-Morgenstern exected ayoff maximizers. In what follows, we fix the CDF F and analyze those signal structures which maximize the buyer s exected ayoff. Since the buyer s trading decision only deends on E (v s), we may assume without loss of generality that each signal s rovides the buyer with an unbiased estimate about her valuation, that is, E (v s) = s. In what follows, we restrict attention to such signals and refer to them as unbiased signals. 3 Results First, we argue that the ayoffs of both the buyer and the seller are determined by the marginal distribution of the signal. To this end, let D (G, ) denote the demand at rice if the signal s distribution is G, that is, D (G, ) is the robability of trade at. Note that D (G, ) = 1 G ()+ (G, ) where (G, s) denotes the the robability of s according to the CDF G. 6 The seller s otimal rice,, solves max s sd (G, s) and the buyer s ayoff is (s ) dg (s). Therefore, the 3 Another strand of the literature analyzes the seller s incentives to reveal information about the buyers valuations rior to articiating in an auction, see for examle, Ganuza (24), Ganuza and Penalva (21) and Bergemann and Pesendorfer (27). 4 It is well-known that any rofit-maximizing mechanism can be imlemented by a take-it-or-leave-it offer. 5 Assuming that the buyer is non-strategic in the final stage of the game and buys the object whenever her value estimate weakly exceeds the rice has no effect on our results but makes the analysis simler. 6 Formally, (G, s) = G (s) su x<s G (x). If G has no atom at s then (G, s) =. 3
roblem of designing a buyer-otimal signal structure can be reduced to identifying the marginal signal distribution which maximizes the buyer s exected ayoff subject to monooly ricing. Of course, not every CDF corresonds to a signal distribution. In what follows, we characterize the set of distributions that do. For each unbiased signal structure, v can be exressed as s + ε for a random variable ε with E (ε s) =. This means that G is the distribution of some unbiased signal about v if and only if F is a mean-reserving sread of G (see Definition 6.D.2 of Mas-Colell et al., 1995). Let G F denote the set of CDFs of which F is a mean-reserving sread. By Proosition 6.D.2 of Mas-Colell et al. (1995) this set can be defined as follows { G F = G G : F (v) dv G (s) ds for all x [, 1], The roblem of designing a buyer-otimal signal structure can be stated as follows max G G F s.t. arg max s (s ) dg (s) sd (G, s). In what follows, we call a air (G, ) an outcome if G G F } sdg (s) = µ. (1) and arg max s sd (G, s). In other words, the air (G, ) is an outcome if there exists an unbiased signal about v which is distributed according to the CDF G and it induces the seller to set rice. Next, we define a set of distributions and rove that a buyer-otimal signal distribution lies in this set. For each q (, 1] and B [q, 1] let the CDF G B q be defined as follows: if s [, q), G B q (s) = 1 q s if s [q, B), 1 if s [B, 1]. Observe that the suort of G B q is [q, B] and it secifies an atom of size q/b at B. An imortant attribute of each CDF in this class is that the seller is indifferent between charging any rice on its suort. To see this, note that G B q generates a unit-elastic demand on its suort, D ( G B q, ) = q/, so that the seller s rofit is q irresective of the rice on [q, B]. Of course, the seller is strictly worse off by setting a rice outside of the suort. Notice that G q q is a degenerate distribution which secifies an atom of size one at q. The next lemma states that for each outcome, (G, ), there is another outcome ( G B q, q ) which makes the buyer weakly better off while generating the same rofit to the seller as G. Lemma 1 Suose that (G, ) is an outcome and let π denote D (G, ). Then there exists a unique B [π, 1] such that (i) G is a mean-reserving sread of G B π, (ii) ( G B π, π ) is an outcome and (iii) π (s π) dgb π (s) (s ) dg (s) and the inequality is strict if D (G, ) < 1. 4
Part (i) states that the signal distributed according to G B π is weakly less informative than the one distributed according to G. Parts (ii) and (iii) imly that this less informative signal generates the same ayoff to the seller and a higher ayoff to the buyer. Proof. We first argue that there exists a unique B [π, 1] such that G B π generates an exected value of µ. Since arg max s sd (G, s), sd (G, s) D (G, ) for all s [, 1]. Using the definitions of D (G, ) and π, this inequality can be rewritten as 1 π s + (G, s) G (s). Since (G, s) [, 1], the revious inequality imlies that is, G 1 π first-order stochastically dominates G. This imlies that sdg 1 π (s) G 1 π (s) G (s), (2) sdg (s) = µ D (G, ) = π = sdg π π (s), where the first inequality follows from first-order stochastic dominance, the first equality from G G F and the last equality from G π π secifying an atom of size one at s = π. Since sdgb π (s) is continuous and strictly increasing in B, the Intermediate Value Theorem imlies that there exists a unique B [π, 1] such that sdg B π (s) = µ. (3) We are ready to rove art (i). If x B, then (2) and G B π (s) = G 1 π (s) on [, B) imly that If x B then G (s) ds G B π (s) ds. (4) G (s) ds = 1 µ G (s) ds 1 µ (1 x) = 1 µ x x G B π (s) ds = G B π (s) ds, (5) where the inequality follows from G (s) 1 and the second equality from G B π (s) = 1 if s B. Equations (3)-(5) imly that G is a mean-reserving sread of G B π. To show art (ii), note that since F is a mean-reserving sread of G and G is a meanreserving sread of G B π, F is also a mean-reserving sread of G B π. Therefore, G B π G F. We have already argued that [π, B] = arg max s sd ( G B π, s ), so ( G B π, π ) is indeed an outcome. To rove art (iii), observe that (s π) dg B π (s) = µ π π (s ) dg (s), where the equality follows from (3) and the observation that the lower bound of the suort of G B π is π. The inequality follows from the fact that the buyer s ayoff cannot exceed the first-best 5
total surlus (µ) minus the seller s rofit (π). This inequality is strict whenever the total surlus in the outcome (G, ) is strictly less than µ, that is, D (G, ) < 1. Alying Lemma 1 to a buyer-otimal outcome (G, ) yields the result that there is also a buyer-otimal signal distribution in the set { } G B π. If the distribution of the buyer s signal is π,b G B π, it is otimal for the seller to set rice π which, in turn, rovides the buyer with a ayoff of µ π. A buyer-otimal signal distribution is therefore defined by the lowest π for which G B π G F. Let us define as this lowest value, that is, = min { π : B [π, 1] s.t. G B π G F }, (6) and let B denote the unique value for which G B G F. We are ready to state our main result. Theorem 1 The outcome ( G B, ) maximizes the buyer s ayoff across all outcomes. outcome (G, ) also maximizes the buyer s ayoff then (i) =, (ii) D (G, ) = 1 and (iii) G is a mean-reserving sread of G B. If the Parts (i) and (ii) of this theorem imly that trade occurs at rice with certainty in any buyer-otimal outcome. A consequence of this is that the buyer s ayoff is µ whenever the signal is buyer-otimal. In general, the buyer s otimal CDF is not determined uniquely because there might be many CDFs which second-order stochastically dominate F and induce the seller to set rice. However, according to art (iii), the CDF G B second-order stochastically dominates any other buyer-otimal CDF. This means that G B rovides the buyer with less information than any other buyer-otimal CDF. If F = G B, the otimal information structure is unique and it secifies erfect learning, that is, s = v. Proof. Let (G, ) be a buyer-otimal outcome and let π denote D (G, ). By Lemma 1, there exists a B [π, 1] such that ( G B π, π ) is an outcome. Then, (s ) dg (s) π (s π) dg B π (s) = µ π µ = (s ) dg B (s), where the first inequality follows from art (iii) of Lemma 1 and the second inequality follows from (6). Since (G, ) is buyer-otimal, the outcome ( G B, ) is also buyer-otimal and both inequalities are equalities. Part (iii) of Lemma 1 imlies that the first inequality is strict unless D (G, ) = 1, which roves art (ii). The second inequality is strict unless π =, which roves art (i). Finally, given that π =, art (i) of Lemma 1 imlies art (iii). Next, we solve for an otimal information structure in the case where the buyer s valuation is uniformly distributed, and comare the outcome to that realised in the full-information environment. 6
Examle. Suose that the buyer s valuation is distributed uniformly on [, 1], that is, F (v) = v and µ = 1/2. The exectation of G B sdg B (s) = B is s ds + = log s B + = log B log +. Therefore, in order to guarantee that the exected value generated by G B that log B = 1 2 + log is one half, it must be. (7) The CDF F is a mean-reserving sread of G B if and only if, for all x [, B] ( 1 ) ds = x ds = x log x + log s s sds = x2 2. (8) The smallest which satisfies these inequalities is aroximately.237. That is,.2 and, by (7), B.87. In other words, by designing the signal structure otimally, the buyer is able to trade at a rice just above.2. The buyer s ayoff is µ.3 and the seller s ayoff is.2. It is insightful to comare the outcome resulting from a buyer-otimal signal structure to that realized in the full information case, i.e. s = v. If the buyer observes her valuation rior to trade then she is willing to trade at rice with robability 1. Hence, the seller s otimal rice solves max (1 ), so the equilibrium rice is.5. In turn, the seller s ayoff is.25 since the buyer trades with robability one half at rice.5. The seller is therefore better off if the buyer receives a erfectly informative signal than if she receives a buyer-otimal one. The buyer s ayoff is v.5dv =.125 which is less than half of her ayoff under the otimal signal structure..5 The deadweight loss due to the buyer s erfectly informative signal is also.125. In the revious examle, the seller s rofit induced by the buyer-otimal signal structure is smaller than his rofit in the full-information environment. We now generalize this observation and show that a buyer-otimal signal structure is in fact a signal structure that minimizes the seller s rofit. Corollary 1 For each outcome (G, ), D (G, ). This corollary states that the seller s rofit from an outcome is at least. Moreover, by Theorem 1, the seller s rofit induced by any buyer-otimal signal structure is. The seller s minimum rofit is therefore. Proof. Let π denote D (G, ). By Lemma 1, there exists a B [π, 1] such that ( G B π, π ) is an outcome and, in articular, G B π G F. Hence, by (6), π. This corollary can be used to characterize the combinations of those consumer and roducer surlus which can arise as an equilibrium outcome for some signal s. We argue that the set of these ayoff rofiles is the convex hull of the following three oints: (, ), (, µ) and (µ, ) 7
rofit µ π * µ * buyer s ayoff Figure 1: Outcome Triangle which is reresented by the shaded triangle in Figure 1. It is not hard to show that for each π [, µ], there exists a B [π, 1], such that G B π G F. If the buyer s demand is generated by G B π, the seller s rofit is π and he is indifferent between any rice on [π, B]. Deending on which of these rices the seller sets, the buyer s surlus can be anything on [, µ π], see the horizontal dashed line-segment on Figure 1. 7 Hence, any oint in the triangle can be imlemented. A ayoff rofile outside of the triangle cannot be an equilibrium outcome because the seller s rofit must be at least (see Corollary 1) and the total surlus cannot exceed µ. 4 Conclusion The goal of this aer was to analyse a buyer s otimal learning when facing a monoolist. We characterized the buyer-otimal signal distribution which involves minimal learning. The demand induced by this signal is unit-elastic and makes the seller indifferent between setting any rice on its suort. We also roved that all buyer-otimal signal structures generate the same rice and effi cient trade. It is not hard to extend our characterization of the least informative buyer-otimal signal structure to environments where the buyer s valuation can be negative and the seller has a ositive valuation for the object. 8 There, the otimal signal structure may rovide the buyer with a single value estimate which is smaller than the equilibrium rice. Conditional on not getting this signal, 7 The characterization of Bergemann et al. (215) is also based on constructing information structures which make the seller indifferent on large sets of rices. 8 For further details, see htt://docs.akroesler.com/extensions.df 8
the otimal signal structure generates a demand which makes the seller indifferent between any rice on its suort, just like in our model. The buyer sometimes urchases the object even if her valuation is smaller than that of the seller, so trade is tyically ineffi cient. References [1] Dirk Bergemann, Benjamin Brooks, and Stehen Morris, 215, The Limits of Price Discrimination, American Economic Review, 15 : 921 957. [2] Dirk Bergemann and Martin Pesendorfer, 27, Information Structures in Otimal Auctions, Journal of Economic Theory, 137 (1): 58 69. [3] Olivier Comte and Philie Jehiel, 27, Auctions and Information Acquisition: Sealed-Bid or Dynamic Formats? RAND Journal of Economics, 38 (2): 355 372. [4] Daniele Condorelli and Balazs Szentes, 216, Buyer-otimal Demand and Monooly Pricing, Mimeo. [5] Juan Jose Ganuza, 24, Ignorance Promotes Cometition: An Auction Model with Endogenous Private Valuations, RAND Journal of Economics, 35 (3): 583 598. [6] Juan Jose Ganuza and Jose S. Penalva, 21, Signal Orderings Based on Disersion and the Suly of Private Information in Auctions, Econometrica, 78 (3): 17 13. [7] Anke Kessler, 1998, The Value of Ignorance, RAND Journal of Economics, 29 : 339 354. [8] Andreu Mas-Colell, Michael D. Whinston, and Jerry Green, 1995, Microeconomic Theory, Oxford University Press. [9] Nicola Persico, 2, Information Acquisition in Auctions, Econometrica, 68 : 135 148. [1] Xianwen Shi, 212, Otimal Auctions with Information Acquisition, Games and Economic Behavior, 158 : 666 686. 9