Costly Multidimensional Screening

This paper studies when a principal can improve upon simple one-dimensional mechanisms by also deploying costly nonprice screening instruments — actions that are socially wasteful yet potentially informative about the agent’s private type.

The model features a principal and an agent with quasilinear, additively separable preferences across two components: (i) a productive component, where allocations lie in a one-dimensional compact space X and generate genuine surplus, and (ii) a costly component, where any allocation y in an arbitrary measurable space Y satisfies sB(y, θB) ≤ 0 — it destroys or at best does not create social surplus. The agent’s private type is multidimensional, θ = (θA, θB), drawn from a commonly known distribution. Both components allow for nonlinear valuations and, on the principal’s side, interdependent preferences.

The central result (Theorem 1) establishes that if the agent’s preferences between the productive and costly components are positively correlated — meaning that a higher θA implies a stochastically higher θB — then there exists an optimal mechanism that involves no costly screening. Moreover, if instruments are strictly costly, every optimal mechanism involves no costly screening almost everywhere. Positive correlation is defined in terms of stochastic dominance: θB | θA is stochastically nondecreasing in θA. A sufficient but not necessary condition is affiliation in the sense of Milgrom and Weber (1982).

The intuition centers on two observations. First, under positive correlation, costly instruments can only help relax upward incentive constraints (deterring lower types from mimicking higher types). Second, under the surplus condition — a single-crossing condition on the surplus function sA(x, θA) requiring that if x generates more surplus than x’ at some type, it continues to do so at all higher types — the principal can safely ignore upward incentive constraints at the optimum. The Downward Sufficiency Theorem (Theorem 2) formalizes the second observation: in any one-dimensional screening problem satisfying the surplus condition, there exists an optimal solution to the relaxed program (with only downward IC constraints) that also satisfies all upward IC constraints. Because monetary transfers fully substitute for costly instruments in relaxing downward constraints without destroying surplus, the costly instruments add no value under positive correlation.

The proof proceeds via a monotone path decomposition of the multidimensional type space, exploiting a measurable monotone coupling (Lemma 1) to write θ = (θA, h(θA; ε)) where ε is independent of θA and h is nondecreasing. This reduces the problem to a family of one-dimensional paths, on each of which the Reconstruction Lemma (Lemma 2) shows that any costly mechanism can be weakly improved upon by one with no costly screening that satisfies all downward IC constraints.

A partial converse (Proposition 1) shows that under negative correlation — when some dimension of θB is stochastically nonincreasing in θA — there exist utility functions satisfying the surplus condition for which any mechanism screening only the productive component is strictly dominated.

The paper derives three applications. In monopoly pricing with costly signals (waiting in line, climbing stairs, collecting coupons), profit-maximizing mechanisms require no costly signals when higher-willingness-to-pay consumers also face weakly lower signal costs (Proposition 2). In monopsonistic labor market screening, the firm need not make offers contingent on costly credentials when higher-ability workers find credentialing easier — in contrast to the competitive Spence (1973) model where all screening must occur through costly effort because wages are pinned down by expected output (Proposition 3). In multiproduct pricing, the paper reinterprets bundle components as costly instruments for screening grand-bundle values, recovering Haghpanah and Hartline’s (2021) pure bundling optimality result and extending it to nested bundling (Proposition 4), under conditions that the incremental value of adding items to nested bundles is strictly increasing in type while the value of any non-nested bundle is nonincreasing relative to some nested superset.

Q: What is the paper’s central research question? A: The paper asks whether a principal can improve upon simple one-dimensional mechanisms by also deploying costly nonprice screening instruments when the agent has multidimensional private information. The goal is to characterize conditions under which augmenting a standard price menu with surplus-destroying actions — such as waiting in line, climbing stairs, or obtaining credentials — is or is not beneficial for the principal.

Q: What does “positively correlated preferences” mean precisely in this model? A: Positive correlation means that θB is stochastically nondecreasing in θA: for any θA < θ̂A, the conditional distribution of θB given θA first-order stochastically dominates that given θ̂A — i.e., θB | θA ≤_st θB | θ̂A. Observing a high θA conveys good news about θB in the stochastic dominance sense. A sufficient but not necessary condition is affiliation in the sense of Milgrom and Weber (1982). The condition is asymmetric and does not require full independence or monotone dependence in a deterministic sense.

Q: What is the surplus condition and why does it matter? A: The surplus condition is a single-crossing condition on the productive surplus function: for any x < x̂ and θA < θ̂A, if sA(x̂, θA) > sA(x, θA) then sA(x̂, θ̂A) > sA(x, θ̂A). It says that if a higher allocation generates more total surplus at some type, it continues to do so at all higher types. This condition ensures the existence of a monotone efficient allocation rule, and it is the key enabling condition for the Downward Sufficiency Theorem. It is automatically satisfied when the principal has no interdependent preferences and the agent satisfies increasing differences, and also when sA is strictly increasing in x or has nonnegative cross partial derivative.

Q: What is the Downward Sufficiency Theorem and why is it the key technical result? A: Theorem 2 states that in any one-dimensional screening problem satisfying the surplus condition, there exists an optimal solution to the relaxed program — which ignores all upward IC constraints — that also satisfies all upward IC constraints. This means the principal can solve the easier downward-IC-only problem and the solution is fully incentive compatible. The result is novel and uncovers a general property of one-dimensional screening problems beyond the standard monotone allocation rule setting. It is key because, combined with the observation that costly instruments under positive correlation can only relax upward constraints, it implies there is no benefit to using costly screening.

Q: How does the proof handle the case of multidimensional types? A: The proof uses a monotone path decomposition. By Lemma 1 (measurable monotone coupling), under positive correlation there exists a random variable ε independent of θA and a nondecreasing measurable function h such that θ =^d (θA, h(θA; ε)). This writes the joint type distribution as a family of monotone paths indexed by ε. On each path ε = e, the types are ordered by θA alone, reducing the problem to a one-dimensional screening problem. The Reconstruction Lemma (Lemma 2) then shows that on each such path, any mechanism involving costly screening can be replaced by one without costly screening that weakly improves principal payoff and satisfies all downward IC constraints.

Q: What does the partial converse (Proposition 1) establish? A: Proposition 1 shows that when some dimension i of the costly component satisfies that θi is stochastically nonincreasing in θA (negative correlation), and the type distribution has a density with |X| > 1 and |Y| > 1, then there exist utility functions satisfying the surplus condition for which any mechanism screening only the productive component is strictly dominated by one involving costly screening. This is not a full converse — it establishes existence of cases where costly screening is strictly beneficial, not that it is always beneficial under negative correlation.

Q: How does the insurance example illustrate the two correlation cases? A: In Example 1 (negative correlation), a low-risk type (θA = 0) values insurance at 2, a high-risk type (θA = 1) values it at 3; costs are 0 and 5/2 respectively; and the high-risk type also has higher disutility for the costly action. Without costly screening, the optimal mechanism sells full insurance at price 2 to both types for a profit of 3/4. With costly screening (e.g., requiring the agent to climb stairs to get full insurance), only the low-risk type purchases, yielding profit of 1 > 3/4. In Example 2 (positive correlation), the high-risk type has lower disutility for the costly action; any mechanism using the costly instrument is strictly dominated by simply selling full insurance at price 2 to both types.

Q: How does the labor market application differ from Spence (1973)? A: In Spence (1973), wages are competitive and pinned down by expected output, leaving no room to screen workers via monetary payments, so all screening must occur through costly credentials. In Yang’s model, the monopsonistic firm sets wages and all types face the same outside option, so monetary transfers can screen types. Proposition 3 says that when θB is stochastically nondecreasing in θA — higher-ability workers find credentials easier — no credential is needed in the optimal mechanism. The paper thus shows that costly screening is a feature of competitive, not monopsonistic, labor markets, under positive correlation of preferences.

Q: What is the bundling application and what new results does it yield? A: The paper reinterprets the multiproduct pricing problem by treating the grand bundle as the productive component and sub-bundles as costly instruments (since selling a sub-bundle instead of the grand bundle destroys social surplus relative to selling the grand bundle). Proposition 4 (nested bundling) establishes that a nested menu B of bundles is optimal among deterministic mechanisms if: (i) the incremental value of adding items to move from bundle b to b’ ⊃ b in B is strictly increasing in θ, and (ii) for any bundle b not in B, there exists a nested superset b’ ∈ B such that the value of b relative to b’ is nonincreasing in θ. This extends and complements Haghpanah and Hartline (2021), which is recovered as the special case of pure bundling (Proposition 5).

Q: What are the key scope conditions that delimit when Theorem 1 applies? A: Theorem 1 requires: (i) additive separability of preferences across productive and costly components; (ii) the surplus condition on sA (single-crossing of total surplus in the productive component); (iii) the positive correlation condition (stochastic monotonicity of θB in θA); and (iv) the costly instruments satisfy sB(y, θB) ≤ 0 for all y, θB. The productive allocation space X must be compact and one-dimensional; Y can be any measurable space. The agent’s type space can be multidimensional. The result holds for both private values and interdependent valuations on the principal’s side.

Q: Under what conditions does costly screening arise in practice, according to the model? A: The model predicts that if costly screening instruments are observed in practice, the consumers or agents with higher willingness to pay (or ability) for the productive good must tend to face higher costs for the screening action. For instance, higher-willingness-to-pay consumers who find waiting in line more costly (positively correlated preferences) would not be subjected to waiting as a screening device. If a firm uses waiting in line, it must be because higher-willingness-to-pay consumers find waiting less costly — consistent with negative correlation.

Costly Instruments: Allocations in the space Y such that the ex post social surplus sB(y, θB) = uB(y, θB) + vB(y, θB) ≤ 0 for all y and all θB. These include actions like waiting in line, collecting coupons, or obtaining credentials that destroy social surplus but may convey private information useful for screening.

Productive Component: The one-dimensional allocation dimension X in which both principal and agent derive non-negative surplus, representing the intrinsically valuable output of the mechanism (e.g., insurance coverage, job placement, bundle of goods).

Positive Correlation (Stochastic Monotonicity): The condition that θB is stochastically nondecreasing in θA: for any θA < θ̂A, the conditional distribution of θB given θA first-order stochastically dominates that given θ̂A. Equivalently, observing a higher θA conveys good news about θB. A sufficient condition is affiliation (Milgrom-Weber), but positive correlation is strictly weaker.

Surplus Condition: A single-crossing condition on the total surplus function sA(x, θA) for the productive component: for any x < x̂ and θA < θ̂A, if x̂ generates strictly more surplus than x at type θA, it continues to do so at θ̂A. This ensures a monotone efficient allocation rule exists and is the enabling condition for the Downward Sufficiency Theorem.

Downward Sufficiency Theorem (Theorem 2): The result that in any one-dimensional screening problem satisfying the surplus condition, there exists an optimal solution to the relaxed program (which ignores upward IC constraints) that also satisfies all upward IC constraints. This implies the principal need only enforce downward incentive constraints at the optimum.

Monotone Path Decomposition: A proof technique that writes the multidimensional type distribution as θ =^d (θA, h(θA; ε)) where ε ⊥ θA and h is nondecreasing in θA. Borrowed from dynamic mechanism design (Eso-Szentes, Pavan-Segal-Toikka), it reduces multidimensional IC problems to families of one-dimensional paths indexed by the independent residual ε.

Nested Bundling: A menu B of product bundles that can be totally ordered by set inclusion (b1 ⊂ b2 ⊂ … ⊂ bK). The paper shows that nested bundling is optimal under conditions that the incremental value of nesting is strictly increasing in type for bundles within B, and nonincreasing relative to any nested superset for bundles outside B.