Searchable Menus

Abstract: Multidimensional screening is (in)famously intractable. In this paper, we study optimal screening mechanisms subject to a tractability constraint from the agent's perspective. Specifically, we require that the menu of options offered by the designer can be ordered so that, regardless of her preference type, the agent can find a utility-maximizing option via greedy search: any locally optimal choice must also be globally optimal. In one-dimensional screening with the single-crossing property, this requirement has no bite. In multidimensional environments, however, searchability restricts the set of implementable outcomes. In the multiproduct monopoly problem, the optimal searchable menu is a sparse upgrade menu: higher tiers offer higher allocation probabilities for every good, and the number of tiers is at most the number of goods. In a multidimensional screening problem with money and ordeals, the optimal searchable menu offers the agent a single way to obtain the good. In income taxation with rich multidimensional heterogeneity, a tax schedule is searchable if and only if it is progressive.

• The paper demonstrates that requiring menus to admit a total order under which every agent can find a utility-maximizing option via greedy search simplifies notoriously intractable multidimensional screening problems into ones with simple, distribution-free optimal mechanisms.
• Imposing searchability imposes no additional restriction on implementable outcomes in one-dimensional screening and disfavors interiorly crossed increments in multiproduct monopoly profits.
• Grid-searchability provides a feasible approach for dealing with multidimensional screening problems under separability across dimensions.

This paper introduces the notion of a searchable menu as an agent-side tractability constraint in screening problems (2608.17446). Rather than restricting the designer's mechanism class ex ante or imposing assumptions on the type distribution, the authors require that the menu of options admit a total order under which every agent can locate a utility-maximizing option by greedy search: any locally optimal choice must be globally optimal. The paper delivers a general characterization of searchable menus and applies it to three canonical multidimensional screening environments—multiproduct monopoly pricing, screening with money and ordeals, and optimal income taxation—showing that this single behavioral restriction converts notoriously intractable problems into ones with simple, distribution-free optimal mechanisms.

The concept of searchability

The framework is a standard single-agent screening model with outcome space $\mathcal{X}$, type space $\Theta$, utility $u(x,\theta)$, and type distribution $\mu$. A menu is a subset of outcomes containing the outside option, totally ordered by $\preceq$ and order-embeddable into $\mathbb{R}$. A menu is searchable if for every type, any local maximum of $u(x^s,\theta)$ over the index set $S$ is a global maximum, and the set of global maximizers forms an interval. The interval condition is largely cosmetic—for finite menus it is implied by the local-to-global property under continuity and non-degeneracy.

The definition is given a microfoundation via extensive-form search games: a finite menu is searchable if and only if both ascending and descending search games are one-step simple in the sense of Pycia and Troyan, so that one-step foresight suffices to identify an optimal choice.

Characterization

The central result is a clean set-theoretic characterization. For a finite menu indexed by $\{1,\dots,m\}$, define the incremental better-off sets $$\mathcal{H}^k = \{\theta : u(x^{k-1},\theta) \le u(x^k,\theta)\}$$ and their strict counterparts $\Theta$0. The main theorem states that a menu is searchable if and only if every finite submenu satisfies the incremental nesting condition, $\Theta$1 and $\Theta$2; when the menu itself is finite, checking the full menu suffices. The proof proceeds through two steps: searchability is equivalent to semi-strict quasi-concavity of each type's payoff along the menu order (single-peakedness with flat regions only at global maxima), which in turn reduces to nesting of adjacent comparison sets via a telescoping argument.

Two corollaries frame the scope of the constraint. First, in one-dimensional screening with strict single-crossing preferences, every essential menu ordered by increasing allocation is searchable—searchability imposes no additional restriction on implementable outcomes. This establishes a benchmark: one-dimensional problems are tractable not only for the designer but also for the navigating agent. Second, in regular environments (topological type space, continuity, and a reachability condition ruling out thick indifference sets), only weak nesting of the $\Theta$3 need be checked; in quasilinear payoff-type environments, searchability is equivalent to weak incremental nesting up to outcome-equivalence.

Multiproduct monopoly

The first application considers a seller of $\Theta$4 goods to a buyer with additive values drawn from a full-support continuous distribution. The headline result is striking: regardless of the type distribution, there exists an optimal searchable menu that is a sparse upgrade menu—higher tiers weakly increase allocation probabilities for every good, the top tier is the grand bundle, and the number of priced tiers is at most $\Theta$5. This stands in sharp contrast to the unrestricted problem, where optimal menus may be infinite and irregular, and where recent "anything goes" results show that essentially any IC mechanism is uniquely optimal for some distribution. It also restores revenue monotonicity: unlike the unrestricted bundling problem, where optimal revenue can decrease under first-order stochastic dominance shifts in values, optimal searchable revenue weakly increases.

The proof exploits the geometry of linear preferences: incremental comparison sets are half-spaces bounded by indifference hyperplanes, and searchability forbids interior crossings among them. Any non-upgrade structure can be improved by rotating positively sloped hyperplanes flat, and a final extreme-point argument on an auxiliary linear program—with one capacity constraint per good—yields at most $\Theta$6 nonzero increments. Notably, the same extreme-point machinery that generates complexity results in the unrestricted problem here yields simplicity, because the perturbations preserve searchability. A companion proposition shows the characterization is tight: every upgrade menu identified is undominated and hence optimal for some bounded-density distribution even among all menus, so intermediate tiers with interior probabilities cannot be ruled out without further distributional assumptions.

Screening with money and ordeals

The second application allows a designer allocating a single good to screen using both payments and costly ordeal requirements, maximizing a general weighted welfare objective. Among searchable menus, a posted-requirement menu—a binary option requiring a posted price and completion of all ordeals—is always optimal, irrespective of the number of instruments. When the designer maximizes profit alone, ordeals are never used and the optimum reduces to a posted price. The argument parallels the monopoly case but simplifies further: because ordeal levels are unbounded above, the auxiliary program has a single constraint, so a single nonzero option suffices at an extreme point.

Income taxation

The third application addresses Mirrleesian income taxation with unrestricted heterogeneity, assuming only concavity of utility in consumption and earnings. A key proposition shows that for monotone concave preferences on a convex domain, the incremental nesting condition needs to be tested only against linear preferences. Consequently, if the type space is rich—containing all preferences with constant marginal rate of substitution—a tax schedule is searchable if and only if it is progressive (i.e., $\Theta$7 is convex). Convexity of $\Theta$8 makes disposable income concave in earnings, rendering each type's payoff single-peaked; conversely, any non-progressive schedule admits a linear-preference profile with a locally but not globally optimal earnings choice under any ordering.

The authors are explicit that progressive schedules are generally not optimal in the classical Mirrlees framework except under restrictive parametric assumptions; the contribution is a microfoundation for progressivity as exactly the property guaranteeing greedy-search tractability of the worker's problem. Searchability also has a technical payoff: it validates the first-order approach underlying Saez's perturbation method, since local optimality conditions become sufficient for global optimality.

Grid-searchable menus

Because total ordering may be overly restrictive when preferences are separable across dimensions, the paper extends the analysis to grid-searchability: the designer partitions goods into blocks, each block carries its own ordered component menu, and the final allocation combines choices across components. Local optimality means no improving adjacent move within any single component. Under rich additive-linear preferences and strictly increasing prices, grid-searchability is equivalent to block-additive pricing with each component menu searchable. The necessity direction rests on a $\Theta$9 logic: if a joint upgrade is cheaper than the sum of individual upgrades, some type finds $u(x,\theta)$0 locally but not globally optimal; if more expensive, a different type faces the reverse failure. The general proof proceeds by induction on allocation rank after reducing each component to an effectively one-dimensional problem.

The applications follow directly. In monopoly pricing, optimal grid-searchable menus are block-separable upgrade menus—at most $u(x,\theta)$1 tiers per block—which rules out additive pricing under pure searchability (vertical and horizontal half-space comparison sets cross) but permits it once the partition is chosen. In the ordeal problem, posted-requirement menus remain optimal for any partition, because types place zero value on ordeal-only upgrades. In taxation with multiple income sources (e.g., couples), grid-searchability under the finest partition yields separability of taxes across income sources together with progressivity within each source—an equivalence holding for every income grid.

Limitations and open questions

Several restrictions deserve emphasis. The baseline definition requires a total order, hence a one-dimensional search space; the grid extension accommodates separable preferences but presumes the designer (or salience) fixes the partition, and the appropriate interpretation—designer choice versus exogenous constraint—varies by application. The taxation results rely on richness of the type space and on marginal tax rates strictly below one; the multiproduct results use full-support distributions with densities and finite moments, and the tier bound of $u(x,\theta)$2 is tight only in the sense that stronger predictions require distributional assumptions. The paper leaves open whether searchability would arise endogenously under a worst-case criterion over agents who select any locally optimal option, whether the local-to-global principle extends to tree-structured or partially ordered search spaces, and how the behavioral premise performs experimentally in terms of choice errors and decision times.

Conclusion

By imposing tractability on the agent's choice problem rather than on the mechanism class or the type distribution, this paper identifies a structural property—nested incremental comparison sets—that disciplines multidimensional screening without sacrificing economic content. The resulting optimal mechanisms are sparse upgrade menus in monopoly pricing, posted requirements with ordeals, and progressive tax schedules, each derived from a single characterization theorem. The connection drawn between searchability and single crossing—nesting preserved while the order is endogenized and localized—suggests that agent-side navigability is a productive substitute for classical preference structure in multidimensional environments.