In a closely related paper, Eső and Szentes (2007) study discriminatory mechanisms in a
“valuation-rank” framework and obtain two main qualitative results. First, they show that full information disclosure is generally optimal. Second, they show that the seller cannot benefit from conditioning the price on the disclosure realization.
In this section, I illustrate that both of these results do not hold in the attribute setting.
Consider the example in which there are two attributes J = 2, x1 ∼ U[0,1], x2 ∼ U[0,2], independently distributed. Let there be two equally likely orthogonal types: θ1 values only the first attribute and θ2 values only the second attribute.
An optimal menu can be calculated by Theorem 3. The menu can be implemented by a single disclosure that reveals whether the first attribute is above or below 1/2 followed by a posted price of 3/4. This optimal mechanism obtains revenue 9/16.
The revenue is strictly higher compared to that under full disclosure. In that case, if the seller does not use upfront payments, then the optimal object price can be calculated to be 2/3, translating into revenue of 1/3. The seller can improve it by using upfront payments.
The optimal mechanism can be easily calculated to be r1 = r2 = 1/2, p1 = p2 = 0. That is, anticipating full disclosure, the seller prefers to not discriminate between the types at all and sell the object in advance through an upfront fee. The corresponding maximal revenue under full disclosure is 1/2 falling short off optimal 9/16.
At the same time, the seller can strictly improve the revenue if she could condition the price on the disclosure realization. In fact, she can improve the revenue even under full disclosure. Consider the following mechanism. The seller provides full disclosure, observes the attributes, and chooses the price optimal for the realized valuation distribution. Under this scheme, the seller obtains the revenue:
Π =
Naturally, observing the disclosure realization reduces the buyer informational advantage and helps the seller to better screen the types.
These observations highlight the distinction between the attribute setting and the “valuation-rank” setting of Eső and Szentes (2007). In their framework, the seller informs the buyer about an “orthogonal shock” ξ(θ), defined as the type’s valuation percentile. By construc-tion, these percentiles are uniformly distributed:
ξ(θ)∼U[0,1] ∀θ ∈Θ. (42)
The implicit assumption of the valuation-rank framework is that these shocks in fact equal each other, ξ(θ) ≡ ξ. This assumption is suitable for environments in which, conditional on the object’s state, the buyer’s type does not affect his valuation percentile. As Eső and Szentes (2007) discuss, these environments include the cases of additive valuations and Gaussian learning.
However, despite having the same distribution, the shocks ξ(θ) are generally different
random variables. This distinction is particularly prominent in the case of a single-minded buyer with orthogonal types. Different shocks correspond to different attributes and thus are independently distributed. In the example above, these shocks are ξ(θ1) = x1 and ξ(θ2) =x2/2.
6 Conclusion
I studied a monopolist who sells a multi-attribute object to a privately informed buyer and showed that the seller can benefit from disclosure of attribute information. The benefit comes through two channels. First, disclosure can be used as a screening device, leveraging the fact that different buyer types prefer learning about different aspects of the object.
Second, disclosure can lift the buyer’s expectations and persuade him to buy the object at a higher price. Both channels are important. However, I show that if each type values a single attribute and attributes are independent, then screening is not beneficial and information should be disclosed partially and free of charge. That is, in such settings, the choice of information content is more important than the choice of its pricing.
In this paper, I deliberately focused on the simplest model of pricing and information control. In practice, additional details may be important and should be accounted for. The seller may be restricted in what kinds of information she may provide. The buyer may feature heterogeneity in his ability to process data. The market may involve imperfect competition.
Each of these extensions can be approached within the multi-attribute disclosure framework that I have outlined.
7 Appendix
Proof of Proposition1. Consider an arbitrary menuM = (r(i), E(i), p(i))i∈I. For any typeθ, the menu induces the allocation distributionµ(θ) :X →∆ (A),A ={buy,not buy}, the expected upfront payment ˆr(θ), and the expected object payment, conditional on a trade, ˆp(θ). Consider a direct responsive menu M′ = (r′(θ), E′(θ), p′(θ)) with r′(θ) = ˆ
r(θ), p′(θ) = ˆp(θ), and E′(θ) = (A, µ(θ)). If all types report truthfully and follow the recommendations, then the menuM′ results in the same allocation distribution and the same expected payments as the menu M. At the same time, any deviation under M′ is available to the buyer under M. Hence, reporting truthfully and following the recommendations is incentive-compatible for all types under M′.
Proof of Proposition 2. Towards a contradiction, assume that a responsive menu M = (r(θ),X (θ),Q(θ), p(θ))θ∈Θ is optimal yet every type does not buy the object with some strictly positive probability. Construct a new menu M′ as follows. Pick a type θ with the highest expected payment T =rθ+pθQθ. As Θ is finite, this type exists. Change this type’s item to no disclosure followed by an object price as follows:
r′θ,X′θ,Q′θ, p′θ=0,E[x],1, T +θ·E[x]− Xθ.
Keep all other items the same. In this menu, type θ chooses the new item and always buys the object. This strategy gives him exactly the same payoff as the original menu:
θ· X′θ−p′θ=θ· X θ−pθQθ−rθ.
As X ⊆ RJ++, the uninformative experiment achieves a maximal attribute surplus, E[x] =
R
x∈XxdG > X(θ). As Θ ⊆ RJ++, the new expected payment from type θ is strictly higher than in the original menu, p′θ> T.
The new menu M′ is not necessarily direct. The no disclosure item may be attractive to some types other than θ. However, the only profitable strategy under no disclosure is always buying. Such deviation would only increase the seller’s profit as T was chosen to be the highest expected payment. Hence, the menuM′ brings strictly greater revenue than the menu M. Contradiction. The result follows.
Proof of Proposition 3. As the expected valuations of all types exist, the set F is a continuous image of a compact set. Hence, F is compact.
The set F is convex because for any trade functions q1, q2 their convex combination is a
feasible trade function that achieves a convex combination of their attribute surpluses and trade probabilities. Indeed, take any two points (X1,Q1),(X1,Q2) ∈ F and γ ∈ [0,1]. By construction, there exist trade functions q1, q2 : X → [0,1] that generate these two points.
Then, the function q3 , γq1 + (1−γ)q2 is an admissible trade function that generates the
The same argument can be applied to the trade probability. Hence, (X3,Q3) is a convex combination of (X1,Q1) and (X2,Q2) and belongs to the feasibility set F.
The boundary of F is spanned by linear disclosures. Indeed, as F is a finite-dimensional closed set, the Supporting Hyperplane theorem (Rockafellar(1970), Theorem 11.6, Corollary 11.6.1, p. 100) can be applied.13 In particular, a point Xˆ,Qˆbelongs to the boundary ofF
The integral is maximized pointwise. Its any maximizer is a linear disclosure (15) with coefficients α=λ, α0 =λ0.
13This step might fail if there are infinitely many attributes,|J|=∞. IfF has infinite dimensions then it might have some boundary points that cannot be supported by a hyperplane. A sufficient condition for the existence of a supporting hyperplane is thatF has a non-empty interior.
Proof of Proposition 4. For an arbitrary trade expectation Y ∈ X, the problem of finding the maximal probability experiment can be written as:
q:Xmax→[0,1]
Z
q(x)dG(x) s.t.
R xq(x)dG(x)
R q(x)dG(x) =Y.
Equivalently, it can be written in terms of the attribute surpluses and trade probabilities as:
(Xmax,Q)∈FQ s.t. X − QY = 0.
As F is compact, the solution to this problem exists and belongs to the boundary of F. Thus, by Proposition 3, it can be achieved (only) by a linear disclosure.
Proof of Proposition 5. The seller’s problem can be written in terms of attribute sur-pluses and trade probabilities as:
{r(θ),X(θ),Q(θ),p(θ)}max
Z
θ∈Θ(r(θ) +Q(θ)p(θ)) dF (θ) (43) subject to incentive-compatibility constraints: ∀θ, θ′ ∈Θ,
θ· X (θ)− Q(θ)p(θ)−r(θ)≥θ· X (θ′)− Q(θ′)p(θ′)−r(θ′), (44) θ· X (θ)− Q(θ)p(θ)−r(θ)≥θ·(E[x]− X(θ′))−(1− Q(θ′))p(θ′)−r(θ′), (45) θ· X (θ)− Q(θ)p(θ)−r(θ)≥θ·E[x]−p(θ′)−r(θ′), (46)
θ· X (θ)− Q(θ)p(θ)−r(θ)≥ −r(θ′), (47)
the individual-rationality constraints: ∀θ∈Θ,
θ· X (θ)− Q(θ)p(θ)−r(θ)≥0, (48) and the feasibility constraint: ∀θ ∈Θ,
(X(θ),Q(θ))∈ F. (49)
Consider any feasible profile of tariff functions, attribute surpluses, and trade probabilities r(θ), X (θ), Q(θ), p(θ) that satisfy constraints (44), (45), (46), (47), (48). Take any type
θˆsuch that Xθˆ,Qθˆ ∈ int (F). Consider the following perturbation: keeping Xθˆ andpθˆQθˆfixed, minimizeQθˆwithinF. By (3),F is compact so an optimum exists and belongs to the boundary. By construction, Q′θˆ ≤ Qθˆ. The perturbation keeps the objective (43) and the constraints (44), (47), (48) intact. At the same time, it strictly increasespθˆand hence strictly relaxes the constraints (45) and (46) for all types deviating to type ˆθ. Hence, the perturbed profile is implementable and delivers the same allocation as the original profile. As one can do this perturbation to all typesθ ∈Θ, the result follows.
Proof of Theorem 1. The seller’s problem (43) can be seen as a maximization of a continuous function over a compact set. Hence, an optimal menu exists. By Proposition 5, there exists an optimal menu with all allocations located on the boundary of the feasibility set F. By Proposition 3, such allocations are achieved by linear disclosures. The result follows.
Proof of Proposition 6. Introduce auxiliary attributes, as many as there are types.
Define an auxiliary attributex′θ as the valuation of a typeθ,x′θ ,v(θ, x). The new attribute vector, x′, is distributed over a set X′ ⊆ R|Θ|, according to the distribution of original attributes Gand the valuation functionv(θ, x). By construction, the valuation of each type can be defined as the corresponding auxiliary attribute,v′(θ, x′) = x′θ. This is a special case of the formulation (1). Thus, Theorem 1 applies and there exists an optimal menu with every experiment in it being a linear disclosure of auxiliary attributes x′:
q(x′) =
1, if Pθ∈Θαθx′θ > α0, 0, if Pθ∈Θαθx′θ < α0,
for α ∈ R|Θ|, α0 ∈ R, not all zeros. In the original formulation, these are linear forms. The result follows.
Calculations behind Example 1. Consider the linear form (21). If α1+α2 6= 0, then, by rescaling the term α0, the sum can be normalized to equal 1. By rearranging the terms, the linear form can be rewritten as:
q(x) =
1, if −(x−(α1θ1+α2θ2))2 ≷α0′, 0, if −(x−(α1θ1+α2θ2))2 ≶α0′,
with α′0 = −v0 +α0 +α1α2(θ1−θ2)2 and the inequality sign depending on the sign of the original coefficient sum. This is a neighborhood disclosure with ˆθ = α1θ1 +α2θ2 and
α1+α2 = 1.
If α1 = α2 = 0, then the linear form provides no disclosure and, as X is bounded, is equivalent to a neighborhood disclosure for a sufficiently large |α0|.
Finally, if α1 +α2 = 0 and α1 6= 0, then the linear form (21) is a linear disclosure that informs about the direction of types’ difference:
q(x) =
1, if (θ1−θ2)·x≷α′0, 0, if (θ1−θ2)·x≶α′0,
with α′0 = α0/(2α1) + (θ21−θ22)/2 and the inequality sign depending on the sign of α1. However, the proof of Proposition 3 established that the attribute surplus and probability achieved by a linear form with parameters (α1, α2, α0) correspond to a boundary point of the feasibility set F in the auxiliary attributes, supported by the hyperplane orthogonal to the vector (α1, α2, α0). If θ1 6= θ2, the feasibility set has a strict interior. Hence, the set of boundary points supported by hyperplanes with α1+α2 = 0 has a measure zero and, hence, generically does not matter for an optimal mechanism.
Proof of Theorem 2. The argument is given in the text. The only difference from Myerson (1981)’s problem is that X can take values in [0,E[x]], not in [0,1]. However, it does not affect the extremal nature of the solution.
Lemma 2. (Directional Decomposition)
Let (x1, x2, . . . , xJ)be J attributes distributed independently overX ⊆RJ according to prior distributions G1, . . . , GJ. Let E = (S, π), π : X → ∆ (S) be an arbitrary experiment.
Let (µ(s, E),Pr (s, E)) be the belief distribution induced by E, so that µ(s, E) is a dis-tribution over X conditional on s given E. Denote by µj(s, E) the jth marginal distribu-tion of µ(s, E). Then, there exists a collection of experiments {Ej}Jj=1 such that experi-ment Ej provides the same information as E about attribute j and provides no information about other attributes: Ej = (S, πj) induces a belief distribution (µ(s, Ej),Pr (s, Ej)) with µ(s, Ej) = (µj(s, E), G−j) and Pr (s, Ej) = Pr (s, E) for all s∈S.
Proof. The proof is constructive. Introduce dummy variables (x′1, x′2, . . . , x′J) that have the same prior distributions as (x1, x2, . . . , xJ) but drawn independently of them. For a given j, construct Ej as an experiment that informs about the vector xj, x′−j according to the likelihood function of E. By construction, Ej induces the same marginal distribution of beliefs about attribute j. However, as (x1, x′1, x2, x′2. . . , xJ, x′J) are independent, it provides no information about other attributes. The result follows.
Proof of Proposition7. Consider an arbitrary responsive experimentEj(θj). By Lemma 2 there exists a linear disclosure Ej′ (θj) such that the relevant attribute surplus and trade probability remain the same Xj′(θj) = Xj(θj), QEj′(θj)= Q(Ej(θj)), and all other sur-pluses stay at the ex-ante expectations,Xk′(θj) =E[xk] for allk 6=j. Hence, replacingEj(θj) with Ej′ (θj) does not change incentive compatibility within cohort Θj but, by Blackwell’s Theorem, relaxes the incentive-compatibility constraints of other cohorts.
Proof of Theorem3. Consider the seller’s problem (34) and its arbitrary solution. Define p = minθ{p(θ)}. This is the relevant price of the deviations across types. Towards the contradiction, assumep(θ)> p for some type θ.
IfE[xj(θ)]≥p, then the incentive compatibility constraint is binding. Hence,Q(θ)p(θ) = X(θ)−E[xj(θ)] +p. For smallε >0 consider a modified mechanism withX′(θ) = X(θ) +ε, Q′(θ)p′(θ) = X′(θ)−E[xj(θ)] +p. Because Q(X) is continuous, the mechanism remains incentive compatible yet brings higher revenue. Contradiction.
If E[xj(θ)]< p, then the individual-rationality constraint is binding. Hence,Q(θ)p(θ) = X(θ). For smallε >0, consider the modified mechanism withp′(θ) = p(θ)−ε,X′(θ)/Q′(θ) = X(θ)/Q(θ)−ε. The mechanism remains incentive compatible yet brings higher revenue.
Contradiction.
Now, consider optimal disclosure for a given object price. By feasibility and individual rationality, X (θ)/Q(θ) ≥ maxnp,E[xj(θ)]o. If X(θ)/Q(θ) > maxnp,E[xj(θ)]o, then for small ε > 0 the mechanism with X′(θ)/Q′(θ) = X(θ)/Q(θ)−ε is incentive compatible and increases trade probability, Q′(θ)>Q(θ), and consequently, revenue. Contradiction.
Proof of Lemma 1. The seller’s problem can be written as:
{rj(θj),Xmaxj(θj),pj(θj)}
J
X
j=1
f(Θj)
Z
θj∈Θj
(rj(θj) +Qj(θj)pj(θj)) dFj(θj) s.t. θjXj(θj)−pj(θj)Qj(θj)−rj(θj)≥θjXj
θ′j−pj
θ′jQj
θj′−rj
θj′, ∀j, θj, θj′ ∈Θj, θjXj(θj)−pj(θj)Qj(θj)−rj(θj)≥θjE[xj]−pk(θk)−rk(θk), ∀j, k, θj ∈Θj, θk∈Θk, θjXj(θj)−pj(θj)Qj(θj)−rj(θj)≥0,
Xj(θj)≥ Qj(θj)E[xj], Qj(θj) = Qj(Xj(θj)), ∀j, θj ∈Θj. Define the expected transfer function:
Tj(θj),Qj(θj)pj(θj).
I can use standard one-dimension arguments within each cohort to establish the connection between the attribute surplus and the expected transfer function. Incentive compatibility requires the slope of the indirect utility function be equal toXj(θ) almost everywhere. Hence, the indirect utility function is convex and, by the Envelope Theorem, the optimal transfers can be recovered to be
Tj(θj) =θjXj(θj)−
Z θj
0 Xj(z) dz.
Individual rationality and incentive compatibility within each cohort are satisfied by con-struction. However, incentive compatibility between different cohorts imposes one additional constraint,
where p is the minimal object price in the menu determined by {Xj}Jj=1. The deviations from all other types θj ∈Θ follow because the indirect utility function is convex and grows slower than θjE[xj]. Applying double integration to the objective function completes the derivation.
Proof of Theorem 4. The argument in the text establishes that all high types are offered the minimal price. The optimal mechanism should then solve the problem (38) with the additional constraints that all high types are offered the same fixed pricep∗, and are served the fixed attribute surplusXj∗θj
. These constraints can be written as:
Z θj
I can then consider a relaxed problem with the original integral constraints and the mono-tonicity constraints dropped. In this relaxed problem, byLuenberger (1969), there exist the Lagrange multipliers {λj}such that optimal Xj(θj) maximize the Lagrange function
L ∼
i. If all type distributions are log-concave, then the integrand is increasing in θj. Hence, the optimal Xj(θj) are bang-bang: Xj(θj) = 0 for θj < θ∗j,Xj(θj) = X∗θj
forθj > θ∗j. This relaxed solution corresponds to a single item per each attribute cohort. Hence, the relaxed constraints are satisfied and the relaxed solution solves the original problem as well.
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