Extreme Points in Multi-Dimensional Screening

A common explanation for the difficulty with multi-dimensional screening is that binding incentive constraints depend on the choice of mechanism, making it a priori unclear which constraints will be binding in an optimal mechanism; our results corroborate this explanation. The perturbation described in the previous paragraph modifies the binding incentive constraints of an exhaustive mechanism for an arbitrarily small set of types. Thus, since exhaustive mechanisms are defined only in terms of binding feasibility constraints, the qualitative properties that distinguish extreme points from other mechanisms are essentially only properties of binding feasibility constraints. In contrast, for all one-dimensional problems, properties of binding incentive constraints impose significant restrictions on the structure of the extreme points, e.g., by limiting their menu size to no more than a few allocations in typical applications.

A potential concern is that our results characterize the structure of optimal mechanisms across all instances of the principal’s problem, i.e., for arbitrary utility functions and beliefs about the agent’s type, while in some applications, the principal’s utility function is known. For example, when a monopolist maximizes revenue, certain extreme points are suboptimal for every belief of the monopolist about the agent’s valuations. Our main insights remain the same in sample applications where the principal’s utility is fixed and state-independent, such as in the monopoly problem. In particular, with multi-dimensional types, we show that the extreme points that are (uniquely) optimal for some belief of the principal are again virtually as rich as the set of all IC mechanisms.

Our results offer some insights into the capabilities and limitations of the classical mechanism design paradigm. An important pillar for the success of the theory is that, in many applications, it makes predictions for optimal mechanisms that are independent of the specific details of the environment. We confirm that such predictions are obtainable for all one-dimensional linear screening problems, whereas they are largely unattainable for all multi-dimensional linear screening problems. When the structure of the optimal mechanism depends too finely on the model parameters, it is difficult to derive tangible practical guidance and testable implications from the theory since parameters such as type distributions may be unknown or unobservable in practice.

We emphasize that we do not provide a full solution to multi-dimensional linear screening in that we do not identify the optimal mechanism for each instance of the principal’s problem and show how this mechanism varies across instances. However, given the overwhelming complexity of the structure of extreme points, it seems implausible that such comparative statics exercises are feasible in full generality.

We obtain our results by establishing a connection between extreme points of the set of IC mechanisms and extremal elements of certain spaces of convex sets. Instead of studying the set of IC mechanisms or the agent’s associated indirect utility functions,⁷⁷7For the indirect-utility approach, see e.g. [103], [104], [80, 81], and [36]. we study the space of all menus that the principal could offer the agent. By the well-known taxation principle, any IC mechanism is the agent’s choice function from some menu of allocations and vice versa. Since preferences are linear, offering the agent a menu is payoff-equivalent to offering the agent the menu’s convex hull. Thus, we can establish a bijection between payoff-equivalence classes of IC mechanisms and certain convex sets contained in the allocation space. We show that this bijection preserves convex combinations (in the sense of Minkowski) and therefore preserves extreme points. Analogous bijections hold onto the set of indirect utility functions.

The extremal elements of the space of compact convex sets in Euclidean space are relatively well understood in the mathematical literature and are referred to as indecomposable convex bodies, first studied by [46]. Most of our results are derived from translating these mathematical insights into economic insights via the connection between IC mechanisms and menus in the form of convex sets. Two kinds of complications arise in this translation. First, feasibility requires that menus are contained in the space of allocations; these constraints are not generally considered in the literature on indecomposability. Second, certain menus are equivalent from the agent’s perspective when the type space is restricted, i.e., when the agent’s preferences are constrained to a subset of all linear preferences.

Indecomposable convex bodies in the plane are points, line segments, and triangles, but they are so plentiful and complex in higher dimensions that a complete description has not been obtained and is not to be expected.⁸⁸8[115] writes (p. 166): “Most [(in the sense of topological genericity)] convex bodies in $\mathbb{R}^{d}$, $d\geq 3$, are smooth, strictly convex and indecomposable. It appears that no concrete example of such a body is explicitly known. This is not too surprising, since it is hard to imagine how such a body should be described.” We note that algebraic characterizations of indecomposable polytopes are known; see [92, 88, 118]. We provide a characterization along these lines in Appendix B. However, what is known in the mathematical literature is enough to obtain the relevant economic insights we present in this paper. The complexity of indecomposable convex bodies in two- versus higher dimensions mirrors the dichotomy between one- and multi-dimensional screening problems since, with linear utility and up to redundancies, an allocation space of a given dimension always corresponds to a type space of one dimension less. (Transfers would here be counted as an allocation dimension of its own.)

Section 2 introduces relevant notation and mathematical definitions. Section 3 introduces the model. Section 4 gives a characterization of extreme points in terms of mechanisms that make an inclusion-wise maximal set of incentive and feasibility constraints binding. Section 5 clarifies the role of feasibility constraints by defining and characterizing exhaustive mechanisms. Section 6 presents our core results for one- versus multi-dimensional problems, along with several supporting results. Section 7 introduces the relevant mathematical tools and sketches the proof of our core results in the context of a delegation problem among lotteries over finitely many alternatives, with an emphasis on the special role of the three-alternative case.⁹⁹9Problems with three alternatives have been considered as the simplest departure from the two-alternative case often studied in the literature on mechanism design without transfers; see [25]. Section 8 discusses applications to monopolistic selling and veto bargaining, including essentially complete characterizations of undominated mechanisms in the sense of [81] for these settings. Section 9 provides an extensive discussion of the related literature, including multi-dimensional screening, extreme points in mechanism design, delegation and veto bargaining, and the mathematical foundations underlying this paper. Section 10 concludes.

Appendix A collects several auxiliary results, including the translation between the set of IC mechanisms and a certain space of convex sets. Appendix B deals with the geometry of the set of finite-menu mechanisms and provides an algebraic characterization of finite-menu extreme points (which generalizes the main result in [81]). Appendix C provides a complete characterization of extreme points for one-dimensional problems omitted from the main text for brevity. Appendix D contains the proofs for all results in the main text.

There is a principal and an agent. The principal chooses an allocation $a\in A\subset\mathbb{R}^{d}$, where $A$ is a $d$-dimensional polytope. The principal’s preferences over allocations depend on the agent’s private information, their type $\theta\in\Theta\subset\mathbb{R}^{d}\setminus\{0\}$, where the set $\{\lambda\theta\mid\theta\in\Theta,\,\lambda\in\mathbb{R}_{+}\}$ of all rays through the type space $\Theta$ is a $d$-dimensional polyhedral cone. We say that the type space is unrestricted if $\operatorname{cone}\Theta=\mathbb{R}^{d}$. An agent of type $\theta\in\Theta$ derives utility $a\cdot\theta$ from allocation $a\in A$. Given the agent’s type $\theta\in\Theta$, the principal derives utility $a\cdot v(\theta)$ from allocation $a\in A$, where $v:\Theta\to\mathbb{R}^{d}$ is a bounded objective function that captures the conflict of interest between both parties. There may be a veto allocation $\underaccent{\bar}{a}\in\operatorname{ext}A$ that the agent can enforce unilaterally.

Since utility is linear, we can identify types on the same ray from the origin because they have the same preferences over the allocations in $A$. We select normalized types in the unit sphere $\mathbb{S}^{d-1}=\{y\in\mathbb{R}^{d}:\>||y||=1\}$ as canonical representatives, i.e., $\Theta\subseteq\mathbb{S}^{d-1}$. In applications, we occasionally make other selections, e.g., when considering transferable utility. Thus, in our model, a $d$-dimensional allocation space $A$ always corresponds to a $(d-1)$-dimensional type space $\Theta$.¹¹¹¹11Contrary to other notions of one-dimensionality in the mechanism design literature (see e.g. [24, Chapter 5.6]), a one-dimensional type space need here not imply a linear order on the underlying preferences. For example, $\Theta=\mathbb{S}^{1}$ may be a circle.

The principal designs a (direct and measurable) mechanism $x:\Theta\to A$ to screen the agent.¹²¹²12It is without loss of generality to consider deterministic mechanisms: every randomized allocation in $\Delta(A)$ can be replaced with its barycenter since both principal and agent have linear utility. In applications, we may think of the allocation space $A$ as a set of lotteries over an underlying finite set of alternatives. In this case, a mechanism can be interpreted as a stochastic mechanism. A mechanism asks the agent to report their type $\theta$ and then implements an allocation $x(\theta)$. By the revelation principle, it is without loss of generality for the principal to focus on mechanisms that are incentive-compatible (IC) and individually rational (IR):

IC means that the agent has no incentive to misreport their type. IR means the agent has no incentive to veto the principal’s choice. To simplify the analysis, we assume that there exists a type $\underaccent{\bar}{\theta}\in\Theta$ for whom the veto allocation is one of their favorite allocations, i.e., $\underaccent{\bar}{a}\in\operatorname*{arg\,max}_{a\in A}a\cdot\underaccent{\bar}{\theta}$. If no veto allocation exists, IR is satisfied by convention.

An optimal mechanism is any solution to the principal’s problem

where $\mu\in\Delta(\Theta)$ is the principal’s belief about the agent’s type. We assume that $\mu$ admits a bounded probability density, i.e., is absolutely continuous.

We say that a set of (IC) and (IR) mechanisms is a candidate set for optimality if it contains an optimal mechanism for every objective function $v$ and belief $\mu$ of the principal.

Instead of designing a mechanism, the principal can equivalently offer the agent a menu (or delegation set) $M\subseteq A$, with $\underaccent{\bar}{a}\in M$, from which the agent may choose their favorite allocation. That is,

defines an IC and IR mechanism $x:\Theta\to A$ (if maximizers exist). The value function $U(\theta)=\theta\cdot x(\theta)$ is the agent’s indirect utility function associated with the mechanism $x$.

Mechanisms defined by the same menu are payoff-equivalent, i.e., the associated indirect utility functions are the same. For IC mechanisms, it can be shown that payoff-equivalence is equivalent to equality almost everywhere (Corollary A.5).¹³¹³13 Almost everywhere equality is with respect to the spherical measure (since $\Theta\subseteq\mathbb{S}^{d-1}$). For a Borel subset $B\subseteq\mathbb{S}^{d-1}$, the spherical measure is proportional to the Lebesgue measure of the set $\{\lambda\theta\mid\theta\in B,\,\lambda\in[0,1]\}$. Thus, payoff-equivalent mechanisms yield the principal the same expected utility since the belief $\mu$ is absolutely continuous.

We define the (essential) menu

associated with an IC and IR mechanism as (the closure of) the set of allocations that are commonly made by all mechanisms in its payoff-equivalence class. For example, if the menu size $|\operatorname{menu}(x)|$ is finite, then the menu simply consists of the allocations that are made by the mechanism with strictly positive probability (cf. [36, Definition 7]).

We henceforth identify payoff-equivalent mechanisms, i.e., $x=x^{\prime}$ if $x(\theta)=x^{\prime}(\theta)$ for almost every $\theta\in\Theta$, and write $\mathcal{X}$ for the set of payoff-equivalence classes of IC and IR mechanisms.¹⁴¹⁴14See Section A.1 for a brief discussion of tie-breaking. In Section A.2, we show that $\mathcal{X}$ is $L^{1}$-compact and convex. Therefore, a solution to (OPT) exists and can be found among the extreme points of $\mathcal{X}$ (Bauer’s maximum principle).

In the remainder of this section, we present additional results for the multi-dimensional case that further strengthen Theorem 6.2. We separately discuss extreme points of finite and infinite menu size as well as uniquely optimal mechanisms. All proofs are in Section D.3.

We first provide a genericity condition under which an exhaustive mechanism of finite menu size is an extreme point. For this, we say that a set of points $M\subseteq A$ is in general position if every hyperplane in $\mathbb{R}^{d}$ intersects $M$ in at most $d$ points.

That is, every exhaustive mechanism with a finite menu can be transformed into an extreme point by perturbing its menu into general position. By carrying out such perturbations, we obtain the following genericity result:

Thus, extreme points remain prevalent among exhaustive mechanisms even when restricting attention to mechanisms that make only a limited number of allocations.

It is easy to show that mechanisms with a finite menu size are dense in the set of all mechanisms. Consequently, we have:

We next turn to mechanisms of infinite menu size.

Exhaustive mechanisms can also be approximated by mechanisms that are uniquely optimal for some objective and prior of the principal. That is, even the most parsimonious candidate sets are dense in the set of exhaustive mechanisms. The formal result is a consequence of a theorem due to Straszewicz and Klee ([67]), which asserts that the exposed points of a norm-compact convex set are dense in its extreme points.

In Section 8, we show that the gist of our results continues to hold if we only consider those extreme points that are unique maximizers for specific objectives of the principal such as revenue-maximization. That is, candidate sets remain complex even if the principal’s objective is a priori known and fixed and only their belief is considered a free parameter.

We proceed in the context of the linear delegation problem and discuss the necessary adjustments for other problems at the end of this section:

• •

  $A=\{a\in\mathbb{R}^{d}_{+}\mid\sum_{i=1}^{d}a_{i}\leq 1\}$ is the unit simplex, i.e., the allocation space when considering lotteries over $m=d+1$ alternatives or when dividing time or a budget across the alternatives ($a$ lists the probabilities or shares of the first $d$ alternatives);

• •

  $\Theta=\mathbb{S}^{d-1}$ is the unrestricted type space, i.e., the agent can have all possible von Neumann-Morgenstern preferences over $A$;

• •

  the principal’s objective function $v:\Theta\to\mathbb{R}^{d}$ is an arbitrary bounded function, i.e., the principal relies on the agent’s information in order to make an informed decision;

• •

  there is no veto alternative $\underaccent{\bar}{a}$ for the agent.

The linear delegation problem features multi-dimensional types whenever there are $m\geq 4$ alternatives and thus differs from classical formulations of delegation problems à la [59, 60], which assume one-dimensional allocation and type spaces and single-peaked preferences; see Section 9 for further discussion.

Next to being a natural application of our model, there are two systematic reasons for considering the linear delegation problem:

1. (1)

   In the linear delegation problem, incentive constraints are completely independent from feasibility constraints in the sense that every mechanism that makes an inclusion-wise maximal set of incentive constraints binding is an extreme point up to positive homothety (Lemma 7.1). This independence simplifies our arguments and renders the connection between extremal menus and indecomposable convex bodies most transparent.

2. (2)

   Every linear screening problem is linear delegation with a restricted type space (modulo IR constraints). This is because every linear screening problem can be represented with the unit simplex as its allocation space through an appropriate type space restriction.²¹²¹21Consider a problem with allocation space $A$ and type space $\Theta$. Any allocation polytope $A\subset\mathbb{R}^{d}$ is the image of a higher-dimensional simplex $S\subset\mathbb{R}^{n}$ under a linear map $f:\mathbb{R}^{n}\to\mathbb{R}^{d}$ [49, Chapter 5.1]. An appropriate type space in $\mathbb{R}^{n}$ corresponding to the simplex is given by $f^{T}(\Theta)$, where $f^{T}$ is the transpose of $f$. With such reformulations, however, $\operatorname{cone}\Theta$ is no longer full-dimensional, and because of this additional complexity, we do not use reformulations to linear delegation in our general proofs.

So far, we have followed the literature in that we have stated our results in terms of direct mechanisms. However, IC mechanisms can equivalently be understood as the agent’s choice functions from different (closed) menus $M\subseteq A$.

We call a closed set $M\subseteq A$ an extended menu if every allocation in $A\setminus M$ is strictly preferred by at least one type $\theta\in\Theta$ to every allocation in $M$. In other words, if $M$ is an extended menu, then there is no allocation that can be added to $M$ without necessarily changing the agent’s choice function. Since the agent has linear utility and we are considering the unrestricted type space in this section, every menu $M\subseteq A$ is extended by passing to its convex hull $\operatorname{conv}(M)$, which is a convex body in allocation space.

It is straightforward to show that the map which assigns to every mechanism $x\in\mathcal{X}$ the extended menu $\operatorname{conv}(\operatorname{menu}(x))\subseteq A$ is a bijection between the space $\mathcal{X}$ of (payoff-equivalence classes of) IC mechanisms and the space $\mathcal{M}$ of convex bodies in allocation space.

In Appendix A (Theorem A.2), we show that the bijection between $\mathcal{X}$ and $\mathcal{M}$ commutes with convex combinations and, therefore, preserves the linear structure of the underlying spaces. For convex bodies $M,M^{\prime}\subset\mathbb{R}^{d}$, this linear structure is given by Minkowski addition and positive scalar multiplication, defined as

where $\lambda,\rho\in\mathbb{R}_{+}$. In particular, extreme points of one space map to extreme points of the other space.

We also show in Appendix A that convergence of extended menus with respect to the Hausdorff distance implies convergence of the corresponding mechanisms in $L^{1}$ (Lemma A.7). Thus, any statement about compactness or denseness in the former space carries over to the latter.

We next explain how extreme points of the set of extended menus $\mathcal{M}$ can be understood in terms of the notion of indecomposability from the mathematical literature and the notion of exhaustiveness defined in Section 5.

A menu $M\in\mathcal{M}$ is an extreme point of $\mathcal{M}$ if and only if it does not admit either of the following decompositions:

1. (1)

   $M=\lambda M^{\prime}+(1-\lambda)M^{\prime\prime}$ for $\lambda\in(0,1)$ and $M^{\prime},M^{\prime\prime}\in\mathcal{M}$ homothetic to $M$;

2. (2)

   $M=\lambda M^{\prime}+(1-\lambda)M^{\prime\prime}$ for $\lambda\in(0,1)$ and $M^{\prime},M^{\prime\prime}\in\mathcal{M}$ not homothetic to $M$.²²²²22The two cases are mutually exclusive: if one of $M^{\prime}$ or $M^{\prime\prime}$ is homothetic to $M$, then so is the other.

We call (1) a homothetic decomposition and (2) a non-homothetic decomposition.

Lemma D.2 in Appendix D shows that a mechanism $x\in\mathcal{X}$ is exhaustive if and only if the associated extended menu $M\in\mathcal{M}$ admits no homothetic decomposition. We can straightforwardly extend the definition of exhaustiveness to extended menus because exhaustiveness is solely a property of the feasibility constraints of the allocation space that are intersected by the menu of a mechanism.

Non-homothetic decompositions are closely related to the notion of decomposability from the mathematical literature. A convex body $K\subset\mathbb{R}^{d}$ is decomposable if there exist convex bodies $K^{\prime},K^{\prime\prime}\subset\mathbb{R}^{d}$ not homothetic to $M$ such that $M=K^{\prime}+K^{\prime\prime}$. By scaling the summands, decomposability is equivalent to the existence of convex bodies $K^{\prime},K^{\prime\prime}\subset\mathbb{R}^{d}$ not homothetic to $K$ such that $K=\lambda K^{\prime}+(1-\lambda)K^{\prime\prime}$ with $\lambda\in(0,1)$. A convex body that is not decomposable is indecomposable.

If an extended menu $M\in\mathcal{M}$ is indecomposable, then $M$ has no non-homothetic decomposition. The converse does not generally hold because the summands $\lambda M^{\prime}$ and $(1-\lambda)M^{\prime\prime}$ of a non-homothetic decomposition in our model are required to be subsets of $A$, i.e., feasible extended menus.²³²³23In the absence of feasibility constraints, every convex body trivially has homothetic decompositions, e.g. through translations into opposite directions. This is why homothetic decompositions are ruled out in the definition of indecomposability. However, when the allocation space is a simplex, indecomposability is necessary and sufficient for the absence of non-homothetic decompositions.

Before proceeding with a characterization of the indecomposable convex bodies, we briefly discuss the economic meaning of indecomposability. Recall that an extreme point of finite menu size is determined by its binding incentive and feasibility constraints (Theorem 4.1). Indecomposability of the associated extended menu ensures that (up to payoff-equivalence) there is no other, non-constant mechanism that makes an inclusion-wise larger set of incentive constraints binding; exhaustiveness ensures the same for the feasibility constraints. Thus, by Lemma 7.1 and in the linear delegation problem, the role of incentive and feasibility constraints in whether or not a mechanism is an extreme point can be completely separated. Indeed, in other linear screening problems, extreme points need not make inclusion-wise maximal sets of incentive constraints binding. (Nevertheless, it is helpful to analyze feasibility constraints separately from the incentive constraints, as we have done in Section 5.)

Given Lemma 7.1, it remains to characterize indecomposable and exhaustive extended menus. Indecomposability has been characterized in the mathematical literature.

Figure 1 depicts the proof idea for convex polygons. The figure shows a quadrilateral and two deformations of the quadrilateral that translate the right-most, vertical facet-defining line either to the left or to the right. The resulting deformed quadrilaterals yield a non-homothetic decomposition of the original quadrilateral. Similar deformations can be found for any polygon, but triangles are the only polygons for which these deformations yield homotheties of the triangle. Thus, (degenerate) triangles are the only indecomposable convex polygons. The extension to all plane convex bodies requires a more involved argument.

Shephard identifies a large class of indecomposable polytopes, with the simplest being the simplicial polytopes, i.e., polytopes of which every proper face is a simplex. Roughly speaking, a simplicial polytope $S$ is indecomposable because each two-dimensional face of $S$ is a triangle and any decomposition of $S$ into non-homothetic polytopes would also have to decompose every face of $S$ individually, which is impossible because triangles are indecomposable. Simplicial polytopes are Hausdorff-dense in the space of all convex bodies. First, every convex body is arbitrarily close to a polytope. (Take the convex hull of a finite set of points on the body’s boundary that is $\varepsilon$-dense in the boundary.) Second, every polytope can be transformed into a simplicial polytope by perturbing its vertices into general position; Figure 4 illustrates.

Exhaustiveness admits a simple economic characterization in the linear delegation problem, which follows immediately from Theorem 5.3 (recall Example 5.4). We state the characterization in terms of mechanisms, but it can equivalently be stated in terms of the associated extended menus:

• •

  A constant mechanism $x$ is exhaustive if and only if it dictates an alternative: there exists $a\in\operatorname{ext}A$ such that $\operatorname{menu}(x)=\{a\}$.

• •

  A non-constant mechanism $x\in X$ is exhaustive if and only if it grants a strike: for every alternative $k=1,\ldots,m$ there exists a lottery $a\in\operatorname{menu}(x)$ in which alternative $k$ is chosen with probability 0. That is, the agent is given the option to strike out any one of the alternatives. Geometrically speaking, this means that $\operatorname{menu}(x)$ touches all facets of the allocation simplex.

The following characterization result follows at once from the previous arguments and the bijection between the set of mechanisms $\mathcal{X}$ and the set of extended menus $\mathcal{M}$.

Thus, the theory predicts simple solutions for linear delegation problems with three alternatives, but with four or more alternatives and up to approximation, the only distinguishing property of extreme points is that they dictate or grant a strike.

We remark that optimality in the linear delegation problem, even when there are only three alternatives, may require the use of stochastic mechanisms that offer the agent lotteries over the alternatives.²⁴²⁴24See [75] and [70] for a discussion about the optimality of stochastic mechanisms in the classical one-dimensional delegation model. Lotteries can be interpreted as risky courses of action or as budget or time shares. Optimality may even require lotteries with full support, i.e., interior points of the simplex. Intuitively, lotteries give the principal more leeway in screening the agent and make it more difficult for the agent to align the allocation with their own preferences.

We finally discuss the necessary adjustments to our approach when considering (IR) constraints, allocation spaces different from the simplex, and restricted type spaces.

In the context of the linear delegation problem, IR would mean that the menu of a mechanism must contain the veto alternative $\underaccent{\bar}{a}\in\operatorname{ext}A$. Any decomposition of a given convex body that contains $\underaccent{\bar}{a}$ must also contain $\underaccent{\bar}{a}$. Thus, introducing IR constraints simply amounts to considering extreme points of the set of IC mechanisms that also satisfy IR. The same conclusion obtains in other linear screening problems; see Section A.3.

Suppose the allocation space $A$ differs from the simplex. If an extended menu $M\in\mathcal{M}$ is indecomposable, then it does not admit a non-homothetic decomposition. However, the converse is no longer true. This is inconsequential for the denseness results for multi-dimensional problems since we only get additional, extremal but decomposable extended menus. For one-dimensional type spaces, these additional extreme points drive the bound on the menu size from three up to the number of feasibility constraints of the allocation space (Theorem 6.1). We provide a complete characterization of extremal extended menus for one-dimensional type spaces and arbitrary allocation spaces $A$ (Theorem C.1 in Appendix C). This characterization builds on a mathematical result due to [93].

Suppose the type space is restricted, i.e. $\operatorname{cone}\Theta\neq\mathbb{R}^{d}$. Extending a menu now entails more than taking the convex hull because there are certain directions in the allocation space along which all types are worse off. Geometrically speaking, these directions form the polar cone of the type space. To prove our result, it is a technical convenience to extend menus beyond the boundaries of the allocation space and work with closed convex sets that share the polar cone as a common recession cone. Indecomposability for closed convex sets with a common recession cone is analogous to indecomposability for convex bodies and has been discussed in [118].

For multi-dimensional problems, we have identified exhaustive mechanisms as a reference set in which the extreme points lie dense. However, with a fixed objective $v$, not every extreme point remains relevant for optimality. For example, an extreme point might minimize expected revenue for some type distribution $\mu$. The appropriate reference set now becomes the set of undominated mechanisms, originally defined for the multi-good monopoly problem by [81].

[81] show for the monopoly problem that every undominated mechanism is optimal for some belief about the agent’s type. Their benchmark result can be extended from revenue maximization to arbitrary objectives:

Conversely, every mechanism that is optimal for some fully supported type distribution $\mu\in\Delta(\Theta)$ must clearly be undominated.

A priori, not every undominated mechanism is a necessary candidate for optimality. (Undominated mechanisms need not be extreme or exposed points). However, in the following applications and as long as types are multi-dimensional, we show that every undominated mechanism is arbitrarily close to a mechanism that is uniquely optimal for some type distribution, i.e., arbitrarily close to a mechanism that is a necessary candidate for optimality.

The multi-good monopoly problem is the following linear screening problem:

• •

  $A=[0,1]^{m}\times[0,\kappa]$, where the first $m$ allocation dimensions are the probabilities with which good $i=1,\ldots,m$ is sold to the agent, and the last allocation dimension is the payment by the agent (and $\kappa$ is some sufficiently large constant, which is without loss of generality whenever valuations are bounded);

• •

  $\Theta=[0,1]^{m}\times\{-1\}$, i.e., the consumer has valuations in $[0,1]$ for each good $i=1,\ldots,m$ and money is the numeraire;²⁵²⁵25Due to linear utility, we implicitly assume that the goods are neither substitutes nor complements for the agent. This assumption is made in most papers on the multi-good monopoly problem. We could incorporate substitutes and complements by allowing the agent to have one valuation for each possible bundle $B\subseteq{\{1,\ldots,m\}}$. The allocation space is then the unit simplex over $2^{m}$ deterministic allocations, i.e., all possible bundles, plus an extra dimension representing money as before. Free disposal, i.e., the agent being willing to pay weakly more for inclusion-wise larger bundles, and a fixed marginal utility for money can be modeled as a family of affine restrictions on the type space.

• •

  $\underaccent{\bar}{a}=(0,\ldots,0,0)$, i.e., the consumer can leave without paying anything;

• •

  $v(\theta)=\bar{v}=\{0,\ldots,0,1\}$ for all $\theta\in\Theta$, i.e., the principal maximizes expected revenue (and goods can be produced at zero cost).²⁶²⁶26The literature makes the zero-cost assumption for simplicity. It can easily be relaxed to a constant marginal cost for each good. With decreasing marginal costs, extreme points also remain the relevant candidates for optimality (see the discussion in [81]). With increasing marginal costs, one has to follow the approach taken by [104].

In line with standard terminology in mechanism design with transfers, we abuse our language by referring to $a\in[0,1]^{m}$ as an allocation and $t\in[0,\kappa]$ as the transfer. Instead of probabilities, allocations can also be interpreted as quantities or as quality-differentiated goods with multiple attributes (for which the consumer has unit demand).

We next show that a large class of mechanisms in the monopoly problem is undominated. A pricing function is a continuous convex function $p:[0,1]^{m}\to\mathbb{R}_{+}$ such that $p(0)=0$ that assigns a price to each possible allocation.²⁷²⁷27Convexity and continuity are without loss of generality because the agent has linear utility. $p(0)=0$ reflects the (IR) constraint. See also [56, Appendix A.2 ]. The marginal price for good $i=1,\ldots,d$ at allocation $a\in[0,1]^{m}$ with $a_{i}<1$ is the directional derivative $\nabla_{e_{i}}p(a)$ of $p$ at $a$ in the coordinate direction $e_{i}$ (which exists by the convexity and continuity of $p$). The mechanism $x\in\mathcal{X}$ obtained from a pricing function is the agent’s choice function from the menu $M=\{(a,p(a))\mid a\in[0,1]^{m}\}$.

In plain words, a mechanism that, on the margin, prevents low-valuation types from buying additional quantity while enabling high-valuation types to buy additional quantity is undominated. Such a mechanism features “no-distortion at the top” (the highest type receives the efficient allocation) and “exclusion at the bottom” (the lowest type receives nothing), which are well-known properties of optimal mechanisms in screening problems with transfers. In particular, such a mechanism features these two properties separately in each allocation dimension. Not all undominated mechanisms have marginal prices bounded away from zero and one, but the gap to the mechanisms that do admit this bound is negligible.²⁸²⁸28For an example, see the mechanism depicted in Figure 2 in [81]. In the bottom-right “market segment,” the marginal price for good one is 1.

For a rough intuition for the richness of undominated mechanisms, consider the following trade-off. When the principal increases the price for some allocations, revenue increases from those types who continue to choose these allocations. However, some types that have previously chosen an allocation at the lower price may now opt for a cheaper allocation, decreasing revenue from the types that switch. This trade-off rules out a dominance relationship between many mechanisms.

Given the characterization of undominated mechanisms, the same arguments as in Section 7 can be applied to conclude that extreme points are dense in the set of undominated mechanisms and, therefore, in the set of all mechanisms by Corollary 8.4. In the following result, the first part is well-known (see, for example, [81, Lemma 4 ]).

The second part says that any incentive-compatible and individually rational mechanism can be turned into a mechanism that is uniquely optimal for some belief of the seller by applying an arbitrarily small perturbation. The claim about uniquely optimal mechanisms is not an application of Straszewicz’ theorem upon showing denseness of the extreme points in the set of undominated mechanisms. While Straszewicz’ theorem guarantees that exposed points are arbitrarily close to extreme points, these points may be exposed by linear functionals unrelated to revenue maximization. Our proof modifies the theorem to obtain the desired result.

We now discuss the following linear veto bargaining problem:

• •

  $A=\{a\in\mathbb{R}^{d}_{+}\mid\sum_{i=1}^{d}a_{i}\leq 1\}$ is the unit simplex, i.e., the allocation space when considering lotteries over $m=d+1$ alternatives or when dividing time or a budget across the alternatives ($a$ lists the probabilities or shares of the first $d$ alternatives);

• •

  $\Theta=\mathbb{S}^{d-1}$ is the unrestricted domain, i.e., the agent can have all possible von Neumann-Morgenstern preferences over $A$;

• •

  there is a veto alternative $\underaccent{\bar}{a}\in\operatorname{ext}A$ for the agent (e.g., the status quo in a political context), and we set $\underaccent{\bar}{a}=(0,\ldots,0)$ without loss of generality;

• •

  the principal’s preferences are given by a Bernoulli utility vector $\bar{v}$ independently of the agent’s information, i.e., $v(\theta)=\bar{v}$ for all $\theta\in\Theta$, and we assume for simplicity that (1) $\bar{v}\in\mathbb{R}_{++}$, i.e., the veto alternative is the principal’s least preferred alternative, and (2) $\operatorname*{arg\,max}_{i}\bar{v}_{i}$ is a singleton, i.e., the principal has a unique favorite alternative.

The problem can be seen as a delegation problem with a state-independent objective and an IR constraint. Therefore, the extreme points of linear veto bargaining are exactly the extreme points of linear delegation that satisfy IR (Lemma A.9). Linear veto bargaining can also be seen as a no-transfers analogue of the monopoly problem since both problems feature state-independent objectives with an IR constraint.

The richness of undominated mechanisms in the veto bargaining problem comes from a trade-off similar to that in the monopoly problem. By adding an alternative to the menu of a mechanism, some types prefer the new alternative over their previous choice. Among those who switch, some types will do so in the principal’s favor, i.e., switch away from alternatives that the principal likes less than the new alternative. Other types will not switch in the principal’s favor, i.e., switch away from alternatives that the principal likes more than the new alternative. A similar trade-off arises when removing an alternative from the menu. These trade-offs prevent a dominance relationship between mechanisms that allocate the principal’s most preferred alternative.

As before, given the characterization of undominated mechanism above, the same arguments as in Section 7 can be applied to conclude that the extreme points are dense in the set of undominated mechanisms whenever there are four or more alternatives. The claim about uniquely optimal mechanisms again requires additional arguments.

In the context of the multi-good monopoly problem, MV provide the first—and, prior to this paper, only—analysis of extreme points in multi-dimensional mechanism design, with two main contributions. First, they provide an algebraic characterization of finite-menu extreme points in terms of whether or not a certain linear system associated with a given mechanism has a unique solution. This characterization is based on auxiliary results about the facial structure of the set of incentive-compatible mechanisms. Second, they define the notion of undominated mechanisms and show that every undominated mechanism maximizes expected revenue for some distribution of types.

In comparison to MV, we consider arbitrary linear screening problems with or without transfers and subsume the multi-good monopoly problem as a special case. We contribute explicit, non-algebraic extreme-point characterizations (Section 6). These characterizations reveal the precise structure of the set of extreme points and, therefore, the structure of the possible solutions to linear screening problems. Along the way, we obtain generalizations of MVs results in our more general framework; see Appendix B and Theorem 8.2.

In comparison to MV, we also characterize undominated extreme points and uniquely optimal mechanisms. While MV show for the monopoly problem that all undominated mechanisms are potentially optimal, it has not been known which undominated mechanisms are necessary candidates for optimality, i.e., which extreme points are undominated and uniquely optimal for some type distribution. A priori, one might conjecture that parsimonious candidate sets are significantly smaller than the set of undominated mechanisms. We show that this is not the case: the relevant exposed points are dense in the set of undominated mechanisms. Moreover, we provide new results about undominated mechanisms, showing that these mechanisms are themselves virtually as rich as the set of all IC and IR mechanisms.

Finally, we note that MV have shown for the monopoly problem, modulo minor details, that the extreme points of menu size $k\in\mathbb{N}$ are relatively open and dense in the IC mechanisms of menu size $k$, provided $k$ is smaller than the number of goods for sale plus one (their Remark 25). We show that this substantial qualifier on $k$ is not necessary and that the result holds for arbitrary multi-dimensional screening problems with linear utility (Theorem 6.4).

The literature on multi-dimensional screening—and on the multi-good monopoly problem in particular—is much too large to be summarized here in detail. We focus on recent developments and point to a survey by [105] for work up to the early 2000s.²⁹²⁹29A sample of important early work includes [1], [114], [87], [122], [8], and [104].

Recent work focuses mostly on the multi-good monopoly problem and can be classified into several approaches for gaining insights into multi-dimensional screening problems or for circumventing the severe difficulties associated with their classical formulations:

• •

  provide conditions for the optimality of common mechanisms such as separate sales or bundling ([87]; [80]; [41]; [100]; [36]; [90]; [19]; [53]; [47]; [124]);

• •

  provide duality results that can be used to certify the optimality of a given mechanism ([36]; [69]; [27]; [73]; [68]);

• •

  identify specific structural properties of optimal mechanisms (e.g., subadditive pricing, monotonicity, no randomization) and show when such structure arises ([87]; [80]; [56]; [13]; [17]; [21]);

• •

  quantify the worst-case performance (approximation ratio) of common mechanisms or classes of mechanisms ([54]; [55]; [77]; [11]; [109]; [57]; [12]; [17]);

• •

  identify mechanisms with the optimal worst-case performance for a mechanism designer with Knightian uncertainty over the set of type distributions ([28]; [37]; [33]);

• •

  derive asymptotic optimality results for a large number of i.i.d. goods ([9]; [14]) or for the speed of convergence to first-best as the principal gains increasingly precise information about the agent’s type ([44]).

Our paper is orthogonal to these developments. We do not focus on specific properties and classes of mechanisms or attempt to escape intractabilities. Instead, we shed light on where these intractabilities originate and identify the limits of the qualitative predictions that can be drawn within the standard Bayesian framework. Moreover, to the best of our knowledge, multi-dimensional screening without transfers has not been studied, with the exception of [68], whose duality approach to a multi-dimensional delegation problem is complementary to our extreme points approach.

Besides the implications for optimal mechanism design, we contribute to the literature on implementability with multi-dimensional type spaces (e.g., [103]; [110]; [20]) by characterizing extreme points of the set of incentive-compatible mechanisms. By Choquet’s theorem, every non-extreme point can be represented as a mixture over extreme points.

In general, little is known about optimal multi-dimensional mechanism design with multiple agents ([99]; [62]; [30]; [61]; [73]); see the conclusion for further discussion.

A number of papers have approached mechanism design problems by studying the extreme points of the set of incentive-compatible mechanisms. However, aside from the previously discussed work by [81], this approach has only been applied to one-dimensional problems. For instance, [23] uses extreme points—hierarchical allocations—in a characterization of the set of feasible interim allocation rules. Building on Border’s insights, [82] demonstrate the equivalence of Bayesian and dominant strategy incentive-compatibility in standard auction problems. A similar approach is discussed in [121, Chapter 6 ].

[70] present characterizations of the extreme points of certain majorization sets and show how these majorization sets naturally arise as feasible sets in many economic design problems. In the context of mechanism design, their results immediately imply a characterization of the extreme points of the set of feasible and incentive-compatible interim allocation rules in one-dimensional symmetric allocation problems, providing a new perspective on Border’s theorem as well as BIC-DIC equivalence. Their approach is tailored to one-dimensional problems, elegantly handling both the IC constraints (monotonicity for one-dimensional types) and the Maskin-Riley-Matthews-Border feasibility constraints (majorization with respect to the efficient allocation rule).

In subsequent work, [71] characterize certain extreme points of the set of measures defined on a compact convex subset of $\mathbb{R}^{d}$ that are dominated in the convex order by a given measure. Their result is a multi-dimensional analogue of results obtained in [70] about the set of monotone functions that majorize a given monotone function (see also [7]). These results apply to information design but have no obvious applications to mechanism design.

[96, 97] builds on the majorization approach, allowing for additional constraints on the majorization sets. These constraints may, for example, correspond to fairness or efficiency constraints in a revenue-maximization problem. [125] provide a complementary analysis to [70] based on characterizations of extreme points of sets of distributions characterized by first-order stochastic dominance conditions rather than second-order stochastic dominance conditions (majorization).

Extreme point approaches have also been used in mechanism design without transfers (e.g., [18]; [95]), and several other mechanism design papers use extreme points as a technical tool (e.g., [34]).

Much of the literature on optimal delegation has focused on one-dimensional allocation (action) and type spaces with single-peaked preferences; see [59, 60], [89], [84], [4], [5], [74], and [70].

The applications of our results to delegation differ from the classical literature in two ways. First, allocations in our delegation problem are lotteries over finitely many alternatives.³⁰³⁰30Delegation over a finite set of alternatives is also studied in the project selection literature; see [10], [98], [31], and [52]. Second, both the principal and agent have arbitrary vNM preferences over these alternatives; that is, our problem features an unrestricted rather than single-peaked preference domain and therefore multi-dimensional types (and allocations). We can allow more general allocation spaces, provided the agent’s utility remains linear.

A small number of papers consider multi-dimensional type or allocation spaces. [72] study optimal delegation in a setting with a one-dimensional type space and two allocation dimensions across which the principal and the agent have separable quadratic preferences. [42] links multiple independent, one-dimensional delegation problems. Frankel shows that “halfspace delegation,” i.e., imposing a quota on the weighted average of actions across problems, is optimal for normally distributed states and approximately optimal for general distributions as the number of linked problems goes to infinity. See also [43] for a robust mechanism design approach. [68] studies optimal delegation with both multi-dimensional type and allocation spaces. Kleiner’s duality-based approach is complementary to our extreme-point approach.

Veto bargaining is a classical problem in political science, originally studied in [108]. The case with incomplete information about the agent’s (vetoer’s) preferences has only recently been studied using a mechanism design approach by [64].³¹³¹31See also [2]. Their model features one-dimensional private information. [6] and [113] study related one-dimensional delegation problems with IR constraints where the principal does not necessarily have state-independent preferences.³²³²32See also the ”balanced” delegation problem in [74]. Similarly, our model can nest linear delegation problems with IR constraints.

[46] introduced the notion of an indecomposable convex body and announced the first results about indecomposability. Gale’s results were later proven and published in [116], [91]/[117], and [112]. These and other papers have provided many novel results that go beyond Gale’s original presentation. [88, 92, 118] provide algebraic characterizations of indecomposable polytopes. [118] discusses indecomposable polyhedra. Related results characterize extremal convex bodies within a given compact convex set in the plane ([93, 50]); see Theorems 6.1 and C.1 for the application in our paper. Decomposability is related to deformations of polytopes, which we briefly use in Appendix B; [29, Section 2 ] provide a concise treatment. Textbook references on indecomposability include [115, Chapter 3.2 ], [101, Chapter 6 ], and [49, Chapter 15 ].

Characterizations of indecomposable convex bodies can alternatively be seen, via support function duality, as characterizations of the extremal rays of the cone of sublinear (i.e. convex and homogeneous) functions. A subset of the results known in the literature on indecomposable convex bodies have been independently obtained in studies of the extremal rays of the cone of convex functions by [63] (for two-dimensional domains) and [26] (for $d$-dimensional domains).³³³³33We thank Andreas Kleiner for pointing us to these references.

We finally mention a result due to [65, Proposition 2.1, Theorem 2.2 ], which shows that for most (in the sense of topological genericity) compact convex subsets of an infinite-dimensional Banach space, the extreme points of the set are dense in the set itself. This follows since such sets have an empty interior, support points are dense in the boundary, hence in the set itself, and since most such sets are strictly convex, so that every support point is an extreme point. However, the set of IC mechanisms is a specific compact convex subset of an infinite-dimensional Banach space, which, in particular, is not strictly convex. The content of our results is that whenever the type space is multi-dimensional, the extreme points are nevertheless dense in a certain part of the set.