Sulemental Material: Buyer-Otimal Learning and Monooly Pricing Anne-Katrin Roesler and Balázs Szentes February 3, 207 The goal of this note is to characterize buyer-otimal outcomes with minimal learning in environments where the seller s valuation is ositive and the buyer s valuation can be negative. Model There is a seller who has an object to sell to a single buyer. The seller s value for the object is c. The buyer s valuation, v, is distributed according to the CDF F suorted on, ]. Let µ denote the exected valuation, that is, vdf v = µ. The buyer observes a signal s about v. The 0 joint distribution of v and s is common knowledge. The seller then gives a take-it-or-leave-it rice offer to the buyer,. Finally, the buyer trades if and only if her exected valuation conditional on her signal weakly exceeds. If trade occurs, the ayoff of the seller is and the ayoff of the buyer is v ; otherwise, both the buyer s ayoff is zero and the seller s ayoff is c. 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. 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. The reason is that the buyer only needs to know E v s in order to decide whether to trade at a given rice, so it does not matter whether she observes s or E v s. structures. In what follows, we restrict attention to unbiased signal 2 Buyer-otimal Signal Structures First, we argue that the ayoffs of both the buyer and the seller are determined by the unconditional distribution of the signal. To this end, let D G, denote the demand at rice Roesler: Deartment of Economics, University of Michigan, 6 Taan Ave, Ann Arbor, MI-4809, USA, akroe@umich.edu. Szentes: Deartment of Economics, London School of Economics, Houghton Street, London WC2A 2AE, UK, b.szentes@lse.ac.uk. It is not hard but cumbersome to generalize our results for F s which are suorted on R.
if the signal s distribution is G, that is, D G, is the robability of trade at. Note that D G, = G + G, where G, s denotes the the robability of s according to the CDF G. 2 The seller s otimal rice,, solves max s s cd G, s and the buyer s ayoff is s dg s. Therefore, the roblem of designing a buyer-otimal signal structure can be reduced to identifying the unconditional 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 = 0. 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., 995. 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. 995 this set can be defined as follows { x x G F = G G : F v dv G s ds for all x [0, ], 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 s c D G, s. } sdg s = µ. In what follows, we call a air G, an outcome if G G F and arg max s s c D 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 c, ] and B [q, ] let the CDF G B q be defined as follows: 0 if s [0, q, G B q s = q c s c if s [q, B, if s [B, ]. Observe that the suort of G B q is [q, B] and it secifies an atom of size q c / B c at B. An imortant attribute of each CDF in this class is that the seller is indifferent between charging any rice on its suort. Notice that G q q is a degenerate distribution which secifies an atom of size one at q. Proosition Suose that G, is a buyer-otimal outcome and G involves minimum learning among all buyer-otimal signal structures. Let λ denote sdg s / [G G, ] if [, G > G, and zero otherwise. Then there exists a B [, ] such that i G s = 0 on s [0, λ and G s = G on [λ,. ii G B s = [G s G ] / [ G ] if s. 2 Formally, G, s = G s su x<s G x. If G has no atom at s then G, s = 0. 2 2
Part i states that there is a single signal, λ, which the buyer might observe below. In other words, with some robability, the buyer learns that her valuation is below the rice but receives no additional information regarding her valuation. If the buyer s value is very likely to be above the seller s cost c, then the buyer might never observes such a signal and under the buyer-otimal signal trade occurs with robability one. Otherwise, λ is the buyer s exected value conditional on s <. Part ii says that, conditional on observing a signal above, the signal is distributed according to G B for some B, that is, the seller is indifferent between any rice on [, B]. Proof. Let G, be a buyer-otimal outcome. We first argue that there exists a unique B such that G B generates the same exected value as G s. Since arg max s s c D G, s, s c D G, s c D G, for all s [0, ]. Using the definitions of D G,, this inequality can be rewritten as c D G, + G, s G s. s c Since G, s [0, ], the revious inequality imlies or equivalently, c D G, G s, s c D G, c s c G s D G,. Subtracting [G G, ] /D G, from both sides yields c s c G s G + G,. 3 D G, Note that the right-hand side of this inequality is the signal distribution conditional on s, so the revious inequality is just G s G s s, 4 that is, G first-order stochastically dominates G s. This imlies that sdg s sdg s s D G, = sdg s, where the first inequality follows from first-order stochastic dominance, and the last equality from G secifying an atom of size one at s =. Since sdgb s is continuous and strictly increasing in B, the Intermediate Value Theorem imlies that there exists a unique B [, ] such that sdg B s = sdg s s. 5 For each buyer-otimal outcome G, we can define a signal distribution G as follows 0 if s [0, λ, G G, if s [λ,, G s = D G, c s c + G G, if s [, B if s [B, ], 3
where λ = sdg s / [G G, ] and B is defined so that 5 is satisfied. Note that [, G, = 0 and G = G G,. First, we argue that the exected value of the signal conditional on s < is the same according to both distributions, G and G. Note that [, sd G s G = λ [G G, ] G where the second equality follows from G = G G,. Second, we show that = λ, 6 sd G s = µ. 7 To see this, note that sd G s = G [, sd G s G = [G G, ] λ + [ G + G, ] [ + G ] [,B] sd G s G sdg s [,] G + G, = sdg s = µ, where the first equality follows from 5 and 6 and the last equality follows from G G F. Third, we rove that if s [, B]. Note that G s = D G, where the inequality follows from 3. G s G s c + G G, G s, 8 s c Next, we show that G is a mean-reserving sread of G. If s λ then s G x dx s G x dx = 0. 9 If s [λ, then s G x dx G G, = λ λ G x dx s G G, G G, dx s = λ G G, s G x dx = G s G x dx G G,, where the inequality follows from G s G G, for s <, the second equality follows from G x = G G, on [λ, and the last equality follows from 6. The revious dislayed inequality chain imlies that if s [λ, then s G x dx s G x dx. 0 4
If s [, B], then s G x dx = [, [, G x dx + G x dx + [,s] [,s] G x dx G x dx, [, G x dx + G x dx [,s] where the first inequality follows from 0 and the second inequality follows from 8. s If s B then G x dx = µ s s G x dx µ s = µ G x dx = G x dx, 2 s where the first inequality follows from G x, the second equality from G x = if x B and the third equality from 7. Inequalities 9-2 imly that G is a mean-reserving sread of G. So far, we have roved that G is a mean-reserving sread of G, so G involves less learning than G. In fact, G is strictly less informative than G unless G = G. Note that the seller is indifferent between any rice on [, B]. If the seller charges, his rofit is c G = c D G,, that is, the same as in outcome G,. The buyer s ayoff is B s d G s = B sd G s G = sdg s D G,, [,] where the second equality follows from 5 and G = G G,. The revious inequality chain imlies that the buyer s ayoff is also the same as in the outcome G,. Also note that the seller must strictly refer setting rice to rice λ because otherwise the buyer would be better off in the outcome G, λ than in G,, contradicting the assumtion that G, is buyer-otimal. To summarize, if G, is a buyer-otimal outcome and induces minimal learning then G = G which imlies arts i and ii of the statement of the roosition. Next, we characterize the minimally informative buyer-otimal signal structure in an examle and show that trade is neither efficient nor does it occur for sure if v is likely to be smaller than c. Examle. Suose that 0 if v < 0 F v = /2 if v [0, if v, that is, the buyer s value is either zero or one with equal robabilities. Let c [0, /2]. For, c > 0, efficiency requires for trade to occur if and only if v =. So, the only signal structure that is comatible with efficient trade is the fully informative one, with s = v. But then, the seller charges = and the buyer s ayoff is zero. The seller s rofit is /2 c > 0. If the signal is ure noise, i.e. s µ, the seller charges the rice = µ = /2. The buyer s ayoff is zero and the seller s rofit is /2 c. 5
In general, for c > 0, the seller never charges a rice smaller than c. Hence, any information structure that results in sure trade must satisfy s c, s that occur with ositive robability. Next, we characterize the buyer-otimal signal structure, G c, in this examle. Proosition imlies that s = 0 with robability α [0, and, with the remaining robability α, the distribution of the signal is generated by G B. In what follows, we in down α, and B for some values of c. To this end, given c, let µ c, B denote the exected value generated by G B. Note that G c G F imlies α µ c, B = /2, and hence Therefore, the buyer s ayoff is α = 2µ c, B. α [µ c, B ] = 2 2µ c, B. 3 Note that since µ c, B is increasing in B, the buyer-otimal B is one. Therefore, solving for the buyer-otimal signal structure is reduced to the following roblem max [c,] µ c, 4 subject to µ c, 2. Next, we derive an exlicit formula for µ c, : [ c µ c, = s c 2 sds + c s c = c c s c 2 ds + [ = c log s c c ] s c + c [ c = c log c c c + c c + ] = c log c Therefore, the maximization roblem in 4 can be rewritten as c log max [c,] subject to c log c c c + c 2. ] c s c 2 ds + c The following table summarizes the solution to this roblem c {0, /8, /4, /2}: c +. c To summarize, for c = 0 or low costs, under the buyer-otimal signal structure the seller is indifferent between all rices [ c, ] and trade occurs with robability one. 3 For higher costs, for the buyer-otimal signal structure, the buyer learns that her valuation is zero with robability α > 0; with the remaining robability, the signal s is generated by the CDF G. 3 For the distribution in the examle the uer bound on costs for which this holds is around 0.33. 6
c α 0 0.9 0 8 0.25 0 4 0.40 0.22 2 0.64 0.39 We could have reached similar conclusions, i.e. trade is inefficient and does not occur for sure even if c = 0 but the buyer s value can be negative. The analysis of the next examle is analogous to the revious one, so we omit it. Examle 2. Suose that c = 0 0 if v < F v = /2 if v [, if v, that is, the buyer s value is either minus or one with equal robabilities. In the examle above, the allocation resulting from the buyer-otimal information structure is inefficient because some buyers whose valuation is below the cost of the seller end u buying the object. A natural question to ask is: Can the buyer-otimal information structure induce some buyers to refrain from trading even if their valuations are above the roduction cost? We rove that such an inefficiency can never arise. To this end, we first rove that each buyer with a valuation above the rice buys the object. Lemma Suose that G, is a buyer-otimal outcome and G G, > 0. F s < =. Then This lemma states that, if trade does not haen surely, then the robability that the buyer s true valuation is below the rice conditional on receiving a signal smaller than the rice is one. In other words, if a buyer does not trade at rice then her true valuation is below the rice. Proof. It is without loss of generality to assume that the buyer receives a single signal strictly below. Let λ denote the value of this signal, that is, λ = E G s s = λ. Furthermore, let F λ v denote F v s < = F v s = λ for all v. Note that vdf λ v = λ. Finally, let α denote the robability of receiving λ, that is, α = G λ. We rove the statement of the roosition by way of contradiction. That is, we assume that F λ = F s < < and construct an outcome in which the ayoff of the buyer is higher than in G,. In articular, we modify the signal distribution G conditional on s = λ and show that the buyer s ayoff strictly increases. To this end, fix a B, ] such that F λ B <. This is ossible if F λ <. Let the signal distribution G ε be defined such that the signal is generated 7
by G B with robability ε > 0. With the remaining robability ε, the signal is λ ε such that λ = ε sdg B s + ε λ ε. 5 Note that G ε converges to one on [λ, ] as ε goes to zero. Since F λ B <, ε can be chosen small enough so that F λ < G ε on [λ ε, ]. In addition, observe that G ε = 0 F λ on [0, λ ε. This imlies that F λ is a mean-reserving sread of G ε. Consider now the following modification of G denoted by G. The signal is generated by G. s > with robability α. With the remaining robability α, the signal is generated by G ε. In other words, instead of observing the signal λ, the buyer observes a more informative signal distributed according to G ε with robability α. Since F λ is a mean-reserving sread of G ε and G ε is a mean-reserving sread of G. s = by 5, the signal structure G can be interreted in the following sequential manner. First, the buyer receives a signal according to G. Conditional receiving the signal λ, the buyer receives an additional signal regarding her valuation according to G ε. By construction, F is a mean-reserving sread of G, so G G F. In what follows, G, is an outcome, that is, is a rofit-maximizing rice and and U G, > U G,. To show that G, is an outcome, first note that the seller would never set a rice below. To see this, note that the only signal below is λ ε, so λ ε generates higher rofit than any other rice on [0,. However, if the seller finds it otimal to set rice λ ε then he would also find it otimal to set rice λ when the buyer s signal is distributed according to G. Therefore, we can restrict attention to rices on [, ]. Note that max D G, v v c = max [αd v v Gε, v + α D G. s >, v] v c α max D v Gε, v v c + α max D G. s >, v v c. v Note that is a solution to both roblems on the right-hand side and hence, is also a solution to the roblem on the left-hand side. Therefore, G, is indeed an outcome. Finally, observe that U G, = α sdg s s > < α sdg s s > + αε sdg B s = U G,, that is, the buyer s ayoff is larger in the outcome G, than in G,. This contradicts the hyothesis that G, is buyer-otimal. We are ready to rove that a buyer with valuation above the seller s cost always trades in a buyer-otimal outcome. Proosition 2 Suose that G, is a buyer-otimal outcome and G G, > 0. Then F c s < =. This roosition states that if some buyer does not trade in the buyer-otimal outcome then her valuation must be smaller than the roduction cost. In other words, any efficiency loss in a 8
buyer-otimal outcome is due to too much trade, that is, a buyer might urchase the good even if her valuation below the cost. Proof. By Lemma, F s < =. It remains to show that F s < = F c s <, that is, each buyer whose valuation is in the interval [c, ] ends u buying the object. Note that it is without loss of generality to assume that those buyers who receive s < learn their valuation. The reason is that such a modification of the information structure can only lower the rice, which leads to more efficient urchasing decision of the buyer. We rove the statement of the roosition by way of contradiction. We assume that there is a ositive mass of buyers whose valuation is in [c, ] and who receive a signal smaller than, that is, F s < > F c s <. Since every buyer with s < observes her valuation, this inequality is equivalent to G > G c. Let F denote the distribution obtained from G by conditioning on [c, ] that is, F v = G v G c. G c Note that if the buyer s value distribution is given by F and the buyer learns her valuation, the seller finds it otimal to set rice, that is, F, is an outcome. Furthermore, since G > G c, trade is not efficient in this outcome. Let G, denote the buyer-otimal outcome if the buyer s valuation is distributed by F. Also observe that if the buyer s value-distribution is given by F, then v > c with robability one and hence, efficiency requires trade for sure. The analysis of our aer alies to this case. By Theorem of the aer, the outcome G, induces efficient trade and a strictly larger ayoff than the inefficient outcome F,, that is, U G, > U F,. 6 Let us now define a signal z as follows. If the buyer observed s < c then let z = s. If s > c then let the value of z determined by G. Let G denote the resulting CDF. Note that since F is a mean-reserving sread of G and G is a mean-reserving sread of F = G. s > c, F is also a mean-reserving sread of G. Therefore, G is a feasible signal distribution. Also note that G, is an outcome because G. z > c = G and G, is an outcome. Finally, note that U G, = G c U F, < G c U G, = U G,, where the inequality follows from 6. This inequality chain imlies that G, is not a buyerotimal outcome which is a contradiction. 3 Payoff Characterization This section is devoted to the analysis of the combinations of those consumer and roducer surlus which can arise as an equilibrium outcome for some signal s. First, we show that our analysis can 9
be used to characterize those equilibrium ayoff rofiles which are efficient. Second, we characterize the set of all ayoff-rofiles that can be imlemented by some signal structure Examle. Efficient Outcome Characterization. Efficiency requires a buyer with value v to trade if and only if v c. We call an outcome G, efficient if it induces efficient trade, that is, a buyer with valuation v receives a signal s if and only if v c. If the buyer s valuation is always weakly larger than c, F c = F, c, then the analysis of our aer directly alies with the only modification that the set of distributions to be considered is defined by 2 instead of the equal-revenue distributions considered in the aer. In articular, the set of ayoff-rofiles that can be imlemented by some signal structure can still be described by Figure, excet that µ must be relaced by µ c and must be relaced by c. Observe that among all these ayoff combinations, only the line segment connecting the oints 0, µ c and µ, c corresond to efficient outcomes. For the general case, in which the buyer s valuation may be smaller than c, let F c denote the distribution of the buyer s valuation conditional on the valuation being larger than c, that is, F c = F. v c. Let µ c denote the exectation generated by F c, that is, µ c = vdf c v. Furthermore, let c denote the equilibrium rice in the buyer-otimal outcome characterized by Theorem of the aer if the buyer s value distribution is F c. Proosition 3 There exists an efficient outcome such that the buyer s ayoff is u and the seller s rofit is π if and only if there exists an γ [0, ] such that u = F c γ µ c c and π = F c [γ µ c c + γ c c]. This roosition states that the set of efficient equilibrium rofiles for value distribution F can be obtained by characterizing the set of efficient ayoff rofiles for value distribution F c and by multilying each of these rofiles by F c. The two extreme oints in this set are suorted by the following information structures. Consider first the outcome 0, F c µ c c. This rofile arises if the buyer only learns whether her valuation is above or below c. The seller sets rice µ c and catures all the surlus. Consider now the oint F c µ c c, F c c c. This ayoff rofile arises if the buyer receives the buyer-otimal signal for F c conditional on v c and receives a different signal otherwise. The seller otimally sets rice c. Proof. First note that if the buyer s value distribution was given by F c then the combinations of those consumer and roducer surlus which can arise as an equilibrium outcome can be characterized as follows. There exists an outcome where the buyer s ayoff is u and the seller s rofit is π if and only if there exists an γ [0, ] such that u = γ µ c c and π = [γ µ c c + γ c c]. 7 This directly follows from the aragrah exlaining Figure in the aer. 0
Observe that if G, is an efficient outcome that the buyer must learn in equilibrium whether or not her valuation is above the seller s cost. Also note that the seller s rice only deends on G. s c if the signal is distributed according to G. Therefore, an outcome G, is efficient if and only if i a buyer with valuation v observes a signal s c if and only if v c and ii G. s c, corresonds to an efficient outcome if the buyer s value-distribution is F c. Finally, observe that U G, = F c U G. s c, 8 and D G, = F c D G. s c,, 9 in other words, the ayoffs of both the buyer and the seller in the outcome G, is F c as large as in the outcome G. s c, given that the buyer s value-distribution is F c. Noting that G. s c can be any CDF which yields an efficient outcome if the buyer s valuation is distributed according to F c, equations 7 9 yield the statement of the roosition. Payoff Characterization in Examle. Returning to our revious examle, we now characterize those ayoff-rofiles of buyer-surlus and seller-rofit that can arise as an equilibrium outcome for some signal structure in the setting of Examle. Recall that under ure noise, the ayoff rofile is 0, /2 c and under the fully informative signal it is 0, /2 /2c. Notice that the minimal rice that can be achieved with any information structure is the minimal rice that can be achieved with an information structure that induces trade with robability one. This rice c is given as the solution to µ c c, = 2. 20 The resulting ayoff of the buyer is u c = /2 c and the ayoff of the seller is π c = c c. The following roosition characterizes the set of all imlementable ayoff rofiles. Proosition 4 The set of imlementable ayoff rofiles is {x, π : x [0, u] and [ c, ]}, 2 with u = 2 2 c log c c, and π = + c 2 c log c c. + This result is illustrated in Figure, which shows the regions of the ayoff-rofiles that are achievable for some signal structure of the buyer. The decreasing art of each curve corresonds to {u, π } c for different values of c. The roosition states that the set of imlementable ayoff rofiles coincide with the convex hull of these curves. Proof. First, we rove that any oint in the set 2 can be imlemented by a signal structure. Note that for any [ c, ], µ c, /2. Therefore, the signal structure where s is distributed
seller's rofit 0.5 0.4 0.3 0.2 0. 0.05 0.0 0.5 0.20 0.25 0.30 buyer's ayoff Figure : Illustration of outcome regions deending on costs c: c = 0 region within the black triangle; c = /8 region within the blue curve; c = /4 region within the orange curve ; c = /2 region within the red curve. according to G with robability / [2µ c, ] and s = 0 otherwise is feasible. Let G denote this distribution. Given this signal structure, the seller finds it otimal to set rice, so G, is an outcome. Notice that U G, = u and Π G = π. It remains to show that x, π is imlementable for each x [0, u ]. If the buyer s signal is distributed according to G then the seller is indifferent between any rice on [, ]. Deending on which of these rices the seller charges, the buyer s ayoff can be anything on [0, u]. It remains to show that any oint which is imlementable is in 2. To this end, suose that G, is an outcome. We show that U G,, Π G is in the set 2. By the roof of Proosition, there is a signal structure where the buyer receives signal 0 with robability α and receives a signal distributed G B with robability α and the ayoffs of both the seller and the buyer are the same as in the outcome G,, that is, Π G = c 2µ c, B and U G, = 2 2µ c, B. Recall that, given rice the seller s minimum rofit is π = c 2µ c, = c 2 c log c c, + which is increasing in. Thus, the seller s minimum rofit is π c and the maximum rofit is π. Therefore, by continuity, there exists a [ c, ] such that Π G = π. In order to show that U G,, Π G = U G,, π is in the set 2, we must rove that U G, u. 2
First, observe that π = c 2µ c, c 2µ c, B = c 2µ c, = π, where the inequality follows from µ c, B being increasing in B. Since π is increasing in, we conclude from this inequality chain that, therefore, µ c, µ c, B, and hence 2µ c,b 2µ c,. Using 3, this imlies that U G, u. 3