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TROPICAL GEOMETRY AND MECHANISM DESIGN

ROBERT ALEXANDER CROWELL AND NGOC MAI TRAN

Abstract. We use tropical geometry to analyze finite valued mechanisms. We geometri-cally characterize all mechanisms on arbitrary type spaces, derive geometric and algebraiccriteria for weak monotonicity, incentive compatibility and revenue equivalence. As corollar-ies we obtain various results known in the literature and shed new light on their nature.

Keywords: Mechanism Design, Tropical Geometry, Tropical Convexity, Incentive Compat-ibility, Cyclical Monotonicity, Revenue Equivalence

1. Introduction

Mechanisms are games devised to implement a targeted outcome in an informationally con-strained economy. A well-designed mechanism elicits agents’ private information truthfullygiven their strategic behavior. An important class consists of dominant strategy incentivecompatible (D-IC) mechanisms. With quasi-linear utilities, the classical theorem of Ro-chet [31] states that a mechanism is (D-IC) if and only if all cycles on a certain weightedgraph are nonnegative. This condition, known as cyclical monotonicity [32], is rather difficultto work with in theory. Many papers have been devoted to identifying domains on which sim-pler conditions, such as weak monotonicity, are sufficient for (D-IC), see [3,7,33], and furtherreferences in [37]. In particular, Saks and Yu [33] showed that if the type space is convex,then weak monotonicity implies (D-IC). Ashlagi, Braverman, Hassidim and Monderer [3]showed the converse, if the type space is not convex, then one can construct a mechanismthat is weakly monotone but not (D-IC).

Contributions. Our paper contains three main results. For an arbitrary type space T ,we give a geometric characterization of all possible mechanisms (Theorem 4.2), all possibleweakly monotone mechanisms (Theorem 4.11), and all possible (D-IC) mechanisms (Theorem4.6) that can arise on T . These results make checking and visualizing incentive compatibilityeasy, both theoretically and computationally. Our proofs give a clear understanding of howthe allocation rule and geometry of the type space affect (D-IC) and weak monotonicity. Weprove a generalization of a Saks-Yu in Theorem 6.3, which is closely related to the results ofKushnir and Galichon [22], discovered independently and using different techniques.

Our characterization also allows results about revenue equivalence, such as those in [9,15, 16, 37], to be understood in a unified way. The characterization of revenue equivalencederived here (Corollary 4.7 and Theorem 6.6) emphasizes the role of geometry. Despite beingless general than that of Heydenreich, Muller, Uetz and Vohra [15], these results clarify theinterplay of the allocation rules with the geometry of the type space for revenue equivalenceexplicitly. More importantly, it shows how assumptions pertaining to revenue equivalenceand weak monotonicity can differ, and why. For instance, in Section 5 we construct type

Date: September 11, 2016.1

2 ROBERT ALEXANDER CROWELL AND NGOC MAI TRAN

spaces for which there are different implementable rules, which may or may not be revenueequivalent.

Use of Tropical Geometry. We state and prove our results using tropical geometry. Thisis a young and exciting field with connections to algebraic geometry, combinatorics andoptimization theory. It has found many applications, amongst them phylogenetics [29], meanpay-off games [1], computational complexity [2], discrete events systems [4], chemical reactionnetworks [28] and pairwise ranking [13], to name a few; see the monographs [8, 23] andreferences therein. In particular, under the tropical algebra, deciding (D-IC) becomes aproblem of solving a system of linear tropical equations. Objects in (D-IC) mechanismdesign problems correspond to tropical polytopes and tropical hyperplane arrangements, andeconomic questions about them can thus be answered using tropical convex geometry.

Many problems in mechanism design are questions of network flows [37], which in turncan be cast as problems in tropical linear algebra, as done in [4, 5, 8]. However, there arethree distinct advantages using tropical mathematics. Firstly, tropical geometry providesa convenient language with crisp notation for network flow problems. Secondly, it givesnew geometric insights that would be difficult to obtain from network flows alone. Thirdly,tropical geometry is more than just tropical linear algebra. It has tools from combinatorics,integer programming, and algebraic geometry, which do not have immediate network flowinterpretations. For instance, the tropical Bezout theorem has been used to characterizegeneral equilibrium with indivisibilities, see [5, 6, 36]. These works are the first to applytropical geometry in microeconomic theory. The auctions studied there are a special case ofmechanisms, although the questions addressed in the above papers are distinct from ours.This indicates that there are more applications of tropical geometry to economics.

The results in our paper do not reflect the limit of tropical methods. Instead, they justprovide a starting point for a more systematic development and demonstrate the powerof tropical thinking. On the other hand, we certainly do not claim that all problems inmechanism design can be solved tropically. Due to its idempotency, tropical algebra is lessrich compared to standard algebra.

Organization. Section 2 gives the necessary concepts and results from tropical geometry.Section 3 contains the necessary terminology from mechanism design in tropical language. InSection 4 we present our main results, Theorems 4.2, 4.6 and 4.11. We collect all examplesin Section 5. Section 6 elaborates on the economic interpretation of our results, includingproofs of a generalization of Saks-Yu (cf. Theorem 6.3) and the geometric intuition of theAshlagi-Braverman-Hassidim-Monderer Theorem (cf. Theorem 6.5).

Notation. For an integer n, define [n] = {1, 2, . . . , n}. In general, we use the −i subscript todenote the set or vector excluding its i-th component. For instance, for T = (T1, . . . , Tn) ⊆(Rm)n, define T−i = (T1, . . . , Tj−1, Tj+1, . . . , Tn) ⊆ (Rm)n−1. For t ∈ T , define t−i = (tj : j ∈[n], j 6= i) ∈ T−i. For an m×m matrix L ∈ Rm×m, let G(L) be the directed graph on m nodeswith edge weights L. Let Sm denote the symmetric group on [m]. For sets A,B ⊂ Rm, writedist(A,B) for their distances as measured in the Euclidean norm. As a convention, we shalluse the underline notation, such as ⊕,�,H, . . . to indicate objects defined with arithmeticdone in the min-plus tropical algebra, and the overline notation ⊕,�,H, . . . to indicate thesame objects defined with arithmetic in the max-plus tropical algebra.

TROPICAL GEOMETRY AND MECHANISM DESIGN 3

2. Elements of Tropical Geometry

We will be concerned with two versions of the tropical algebra on R: max-plus and min-plus. The max-plus algebra (R,⊕,�) is defined with tropical addition a⊕b := max(a, b),and tropical multiplication a � b := a + b. The min-plus algebra (R,⊕,�) is defined bya⊕b := min(a, b), a � b := a + b. These algebras are isomorphic via the map x 7→ −x, sotheorems that hold for one have obvious analogues in the other.

In short, tropical mathematics is mathematics done in the min-plus or one of its isomorphicalgebras. The adjective tropical in this context was coined by French mathematicians in honorof the Hungarian-born Brazilian mathematician Imre Simon. There is no deeper meaning ofthe word tropical, ‘it simply stands for the French view of Brazil’, as put by Maclagan andSturmfels [23]. This section is a minimal introduction to tropical linear algebra and tropicalconvex geometry. For a thorough treatment we refer to [4, 8, 18,23].

2.1. Basics. Let L ∈ Rm×m be a matrix, x ∈ Rm a vector and λ ∈ R a scalar. As usual,scalar-vector multiplication is defined element-wise λ�x ∈ Rm, (λ�x)i = λ+xi for i ∈ [m].Matrix-vector multiplication is defined by L�x ∈ Rm, (L�x)i = minj∈[m]{Lij + xj} for i ∈[m]. We say that (x, λ) is an eigenvector-eigenvalue pair of L if

L�x = λ�x,

or explicitly,

minj∈[m]{Lij + xj} = λ+ xi for i ∈ [m].

By Theorem 2.1 of [11], a matrix L ∈ Rm×m has a unique tropical eigenvalue. Thus onecan speak of the tropical eigenvalue of a matrix L, denoted λ(L). The tropical eigenspace ofL ∈ Rm×m is

Eig(L) = {x ∈ Rm : L�x = λ(L)�x}.

Tropical addition is idempotent: a⊕ a = a for a ∈ R. In particular there is no subtraction.This makes tropical linear algebra different from its classical counterpart. For instance, thedeterminant of A equals its permanent, which is

tdet(L) =⊕

σ∈SmL1σ1 � . . .� Lmσm = min

σ∈Sm(L1σ1 + . . .+ Lmσm) .

However, because tropical equations are just a set of classically linear equations and in-equalities, many tropical objects can be computed using linear programming. We give threeexamples relevant for this paper: computing the tropical determinant, tropical eigenvalueand tropical eigenspace.

2.2. Tropical Determinant. Evaluating the tropical determinant means solving the as-signment problem. Imagine that there are m jobs and m workers, and each worker needsto be assigned to exactly one job. Let Lij be the cost of paying worker i to do job j. Thecompany wants to find a least cost assignment. The minimal cost is thus tdet(L).

While there are m! possible matchings, to find the optimal matching, there is no needto evaluate them all. The classic assignment problem above is a linear program over thepermutahedron, and can be solved efficiently with the Hungarian method [21].


2.3. Tropical Eigenvalues.

Theorem 2.1 ( [11]). A matrix L ∈ Rm×m has a unique eigenvalue λ(L), which equals theminimum mean-weight of all simple directed cycles on m vertices.

The mean-weight of a cycle on L in the sum of the edges divided by the number ofedges in the cycle. While there are exponentially many simple cycles, to compute λ(L), itis not necessary to check them all. Computing the min-plus and max-plus eigenvalue arelinear programs over the normalized cycle polytope, for which there is an efficient solution,see [11, 23].

2.4. Tropical Eigenspace. Parallel to classical linear algebra, the tropical eigenspace of anm × m matrix is generated by at most m extreme tropical eigenvectors v1, . . . , vm, in thesense that any x ∈ Eig(L) can be written as

x = a1 � v1⊕ . . .⊕am � vmfor some a1, . . . , am ∈ R. As a set, tropical eigenspaces are polytropes [18], so-called becausesuch a set is both a tropical and an ordinary polytope, see Section 2.8 below. To find Eig(L),one first subtracts λ(L) element-wise from L and reduces to the case λ(L) = 0. In this case,numerically, vij can be found as the value of the shortest path from i to j on G(L). Abstractly,these vectors are the column vectors of the Kleene star of L. In the following definition Idenotes the min-plus identity matrix with zeros on its diagonal and +∞ elsewhere.

Definition 2.2. For L ∈ Rm×m with λ(L) = 0, the Kleene star of L, denoted L∗, is

L∗ = I⊕

(⊕m

i=1L�i

).

A matrix M ∈ Rn×n is called a Kleene star if M∗ = M .

Theorem 2.3. For L ∈ Rm×m, let c1, . . . , cm be the column vectors of (L − λ(L))∗. Then{c1, . . . , cm} is the set of tropical generators of Eig(L).

For this reason, Kleene stars play a fundamental role in tropical spectral theory, andthus have been extensively studied [4, 8, 17, 34]. They are also known as strong transitiveclosures [8, §1.6.2.1], or distance matrix [26]. Kleene stars, and thus the tropical generatorsof a tropical eigenspace, can be computed by multiplication and addition of tropical matrices.

2.5. Tropical Projective Torus. In many economic problems only relative valuations orprices matter, not their absolute values. In tropical terms, this means that valuations andprices are points in the tropical projective torus TPm−1. This shall be our ambient spacewhen speaking about geometry in mechanism design problems.

A set S ⊂ Rm is closed under tropical scalar multiplication if for all a ∈ R we havea� x = (a + x1, . . . , a + xm) ∈ S whenever x ∈ S. Examples include the image of a matrixL ∈ Rm×m

Im(L) = {y ∈ Rm : L�x, x ∈ Rm},or its tropical eigenspace Eig(L). It is sufficient to consider such a set modulo tropical scalarmultiplication. Define an equivalence relation ∼ on Rm via

(1) x ∼ y ⇔ x = a� y for some a ∈ Rm .


The space Rm modulo ∼ is called the tropical projective torus, or tropical affine space TPm−1.Explicitly, it is Rm modulo the line spanned by the all-one vector

TPm−1 ≡ Rm /R ·(1, . . . , 1).

Note that the tropical scalar multiplication does not depend on max or min, so on the samespace TPm−1 one can speak of max-plus as well as min-plus geometric objects such as tropicalpolytopes and tropical hyperplane arrangements.

We follow the convention in [18,23] and identify TPm−1 with Rm−1 by normalizing the firstcoordinate via the following homeomorphism.

(2) TPm−1 → Rm−1, [(x1, . . . , xm)] 7→ (x2 − x1, . . . , xm − x1).

In particular, we use this map to visualize sets in TP2. Of course, one could choose othernormalizations, such as setting the i-th coordinate to zero for some other i ∈ [m], or requirethat the sum of the coordinates is a constant.

Often, checking whether a set S ⊆ Rm is closed under tropical scalar multiplication isstraight forward. In such cases, we shall write S ⊆ TPm−1 without taking explicit notice. Inparticular, we will often write an element x ∈ TPm−1 as a vector in Rm.

2.6. Tropical Polytopes and Hyperplanes. The central objects in tropical convex ge-ometry are tropical polytopes and tropical hyperplanes. A tropical polytope contains alltropical line segments (tropical convex hull) between any two of its points. For each tropicalpolytope there is a finite minimal set of points generating it. Dually a tropical polytope canbe represented as intersections of tropical hyperplanes. Let us elaborate.

Definition 2.4. The tropical min-plus polytope generated by vectors {c1, . . . , cm} ⊂ Rm is

tconv(c1, . . . , cm) = {z1�c1 ⊕ . . .⊕ zm�cm : z ∈ Rm}.

Let L be the matrix whose i-th column is ci. By rewriting, one obtains

im(L) = tconv(c1, . . . , cm).

It is immediate that tconv(c1, . . . , cm) ⊂ TPm−1, and that for constants a1, . . . , am ∈ R,

tconv(a1 � c1, . . . , am � cm) = tconv(c1, . . . , cm).

Thus, we shall work in TPm−1, viewing both {c1, . . . , cm} and their tropical convex hull asa subset of TPm−1. Note that for a particular tropical polytope P , the matrix L such thatIm(L) = P is only defined up to tropical scalings of its columns. Unless stated otherwise, weshall rescale L as to have zeros on its diagonal. This associates P to a unique matrix L.

For the rest of this section, fix a matrix L ∈ Rm×m. Let ci denote its i-th column vec-tor. The tropical polytope Im(L) can also be thought of in terms of its supporting tropicalhyperplanes.

Definition 2.5. For a point p ∈ Rm, j ∈ [m], the min-plus tropical hyperplane with apex−p, denoted H(p), is the set of z ∈ Rm such that the minimum in the tropical inner product

(3) p>�z = min{z1 + p1, . . . , zm + pm}is achieved at least twice. Analogously, the max-plus tropical hyperplane with apex −p,denoted H(p), is the set of z ∈ Rm such that the maximum in

(4) p>�z = max{z1 + p1, . . . , zm + pm}is achieved at least twice.


In classical linear algebra, the complement of a hyperplane is the union of two open half-spaces. In tropical linear algebra, the complement of a tropical hyperplane in TPm−1 is theunion of m open sectors.

Definition 2.6. The j-th open sector of the max-plus tropical hyperplane with apex −p,denoted H◦j(p), is the set of z ∈ Rm such that the maximum (4) is achieved only at j. Thatis,

H◦j(p) := {z ∈ TPm−1 : zj + pj > zk + pk for k 6= j}.Its closure is the j-th closed sector of the max-plus hyperplane with apex −p,

Hj(p) := {z ∈ TPm−1 : zj + pj ≥ zk + pk for k 6= j}.

This is the set of z ∈ Rm such that the maximum (4) is achieved at j and possibly at asecond coordinate.

For a matrix L ∈ Rm×m with zero diagonal, let L1, . . . , Lm ∈ TPm−1 be the m rows of L,viewed as vectors in TPm−1. To simplify notation, write Lj for Hj(Lj), that is,

Lj = Hj(Lj) = {t ∈ TPm−1 : Ljk + tk ≤ tj for k 6= j}.

Write L◦j for the corresponding j-th open sector H◦j(Lj)

L◦j = H◦j(Lj) = {t ∈ TPm−1 : Ljk + tk < tj for k 6= j}.

Write ∂Lj for its boundary Lj\L◦j . Note that the boundary is the union of m − 1 half-

hyperplanes

∂Lj =⋃

k∈[m],k 6=j

Ljk,

where the k-th piece Ljk is the set

Ljk = Hjk(Lj) = {t ∈ TPm−1 : Ljk + tk = tj, Ljk′ + tk′ < tj for k′ 6= j, k}.

As before, we use the underline notation to mean the analogous quantity in min-plus. Forinstance, Lj, the j-th sector of the min-plus hyperplane with apex −Lj in TPm−1, is

Lj = {t ∈ TPm−1 : Ljk + tk ≥ tj for k 6= j}.

Definition 2.7 (Tropical hyperplane arrangement). Let L ∈ Rm×m, L1, . . . , Lm ∈ Rm beits row vectors. Consider the set of tropical hyperplanes {H(Li) : i ∈ [m]} in TPm−1. Theintersections of their various sectors partitions TPm−1 into a polyhedral complex called thetropical hyperplane arrangement H(L).

Definition 2.8 (Dominant arrangement). Let L ∈ Rm×m, L1, . . . , Lm ∈ Rm be its rowvectors. The set of closed sectors {Li : i ∈ [m]} in TPm−1 is called the max-plus domi-nant arrangement of L, denoted D(L). Similarly, {Li : i ∈ [m]} is the min-plus dominantarrangement of L, denoted D(L).

Proposition 2.9 ( [23]). Let L ∈ Rm×m. The tropical hyperplane arrangement H(L) is apolyhedral complex. Furthermore, the union of the bounded cells of H(L) equals Im(L).


2.7. Covectors. Let L ∈ Rm×m. Each hyperplane in H(L) partitions TPm−1 into m sectors.The numbering of the sectors correspond to the coordinate at which the minimum is achievedonce. For a given point p ∈ TPm−1 we can ask for its position relative to the tropicalhyperplanes associated to L. This combinatorial information is encoded in the covectors.

Definition 2.10. The min-plus covector (or combinatorial type) of p ∈ TPm−1 with respectto H(L), denoted coVecL(p), is a matrix in {0, 1}m×m with

coVecL(p)ki = 1⇔ p ∈ Hk(Li).

One can think of a covector as the adjacency matrix of a (m,m) bipartite graph. It hasedge (k, i) iff p is in sector k of the hyperplane with apex at −Li, the i-th row of L. Covectorsare a central concept in tropical convex geometry. This idea was put forward by Develin andSturmfels as combinatorial types [12] and was subsequently further developed by Joswig andLoho [19].

Points in the same open cell of H(L) have the same covector. Thus, we can speak ofthe covector of a cell ν of H(L). Denote this coVecL(ν). Say that coVecL(ν) is invertibleif for each i ∈ [m], there exists some j ∈ [m] such that coVecL(ν)ij = 1. The following is acharacterization of bounded cells in a tropical hyperplane arrangement by their covectors.

Lemma 2.11 ( [12]). Let ν ⊂ TPm−1 be a cell in H(L). Then ν is bounded (as a subset ofTPm−1) if and only if coVecL(ν) is invertible.

2.8. Polytropes and Tropical Eigenspaces.

Definition 2.12. A polytrope P ⊂ TPm−1 is a tropical polytope that is also an ordinarypolytope.

The term polytrope was coined by Joswig and Kulas [18]. There are many equivalentcharacterizations of polytropes. We collect the ones relevant for this paper here. The firstthree results are classical results, see [8,18,23]. The last two statements are characterizationsby Murota [26], who calls polytropes L-convex sets.

Proposition 2.13. Let P ⊂ TPn−1 be a non-empty set. The following are equivalent.

(1) P is a polytrope.(2) There exists a matrix M ∈ Rm×m such that P = Eig(M).

(3) There exists a matrix M ∈ Rm×m such that

P = {y ∈ TPm−1 : yi − yj ≤Mij for all i, j ∈ [m]}.(4) P = Im(M∗) for a unique Kleene star M∗.(5) P is both a min-plus tropical polytope and a max-plus tropical polytope(6) P is both a min-plus convex set and max-plus convex set.

As a set in TPm−1, a polytrope P is a compact convex polytope of dimension k ∈{0, 1, . . . ,m − 1}. Say that P is full-dimensional if it has dimension m − 1. A polytropeof dimension k < m− 1 in TPm−1 is obtained by embedding a k-dimensional polytrope intoa k-dimensional subspace of TPm−1 defined by intersections of hyperplanes of the form

xi − xj = 0

for some i, j ∈ [m]. In matrix terms, this means that if we write a polytrope P as P = Im(L),then as a set in TPm−1, precisely k + 1 of its columns (or rows) are unique. Thus, it is oftensufficient to consider the theory for full-dimensional polytropes.


2.9. Row-Column Duality and Minimal Polytropes. There is a trivial duality betweenthe min and max algebra stemming from the fact that for a, b ∈ R,

min{a, b} = −max{−a,−b}.

This translates to a trivial duality between row and column space of a matrix L ∈ Rm×m asfollows.

Lemma 2.14. Let A ∈ Rm×m. The map x 7→ −x sends H(L) to H(−L>).

There is a more astonishing row and column duality, due to Develin and Sturmfels [12]

Theorem 2.15 ( [12]). Let L ∈ Rm×m. There is an isomorphism between the polyhedralcomplexes Im(L) and Im(L>) obtained by restricting the piecewise linear maps Rm → Rm,z 7→ y := L� (−z), and y 7→ z := (−z)> � L, to Im(L) and Im(L>), respectively.

By part (5) of Proposition 2.13, the isomorphism in Theorem 2.15 reduces to the trivialmap y 7→ −y induced by the map L 7→ −L> if and only if L is a Kleene star, or equivalently,if and only if Im(L) is a polytrope.

When L = −L>, then not only Im(L) is both a max-plus and min-plus tropical polytope, ithas the same set of min-plus and max-plus generators. An example is the standard minimalpolytrope

∆m−1 = conv(0, e1, e1 + e2, e1 + e2 + e3, . . . , e1 + . . .+ em) + R ·(1, . . . , 1),

where conv denotes the classical convex hull, and ei is the i-th standard basis vector in Rm,that is, the vector with 1 in the i-th coordinate and zero elsewhere. Note that for thispolytrope, its set of max-plus tropical generators, min-plus tropical generators and verticesas ordinary polytope all coincide. In a sense, this is the only polytrope with this property.The following theorem is essentially a restatement of [26, Theorem 7.24]. It is key to thecharacterization of weak monotonicity in mechanism design, see Section 6.1.

Definition 2.16. Say that a polytrope P ⊂ TPm−1 of dimension k ∈ {0, 1 . . . ,m − 1} isminimal if as a classical polytope, it has k + 1 vertices.

Theorem 2.17. Let P be a full-dimensional tropical polytope in TPm−1. Then the followingare equivalent

(1) P is a full-dimensional minimal polytrope.(2) P is both a min-plus and max-plus full-dimensional tropical polytope, with the same

set of m generators.(3) Up to permutations, there exists a unique matrix A ∈ Rm×m such that A = −A>,

P = Im(A), and as a subset of TPm−1, the columns of A are unique.(4) The Kleene star A∗ of P has the form

A∗ij = pi − pj + a(A∗∆m−1)ij,

for some vector p ∈ TPm−1 and some scalar a ∈ R, where A∗∆m−1is the Kleene star

of ∆m−1.(5) Up to permutations, there exists a vector p ∈ TPm−1 and a scalar a ∈ R such that

(5) P ≡ p+ a ·∆m−1


Proof. Proposition 2.13 implies (2) ⇔ (3) and (4) ⇔ (5). We first prove (1) ⇔ (5). By [26,Theorem 7.24], any polytrope can be regularly decomposed as unions of smaller polytropes. Itfollows from this proof, using the Lovasz extension of the corresponding L-convex functionsover P , that the minimal polytropes are precisely the convex hull of maximal chains on{0, 1}m. Up to permutation on [m], such a chain is lexicographic. The convex hull of thelexicographic chain is ∆m−1. So up to permutation, scaling and translation, a minimalpolytrope P equals ∆m−1, as needed. Now we prove (5) ⇔ (2). Suppose P is given by (5).Then it is straight-forward to check that it satisfies (2). Conversely, suppose (2). Dueto [26, Theorem 7.24], any L-convex function over P must also be L-concave. Thus, it mustbe a classical hyperplane. So P has a unique Lovasz extension, which implies that up toscaling and translation, it must be a simplex supported on chains of {0, 1}m. Since P isfull-dimensional, the chain is maximal, and thus we have (5). �

2.10. An Example. We illustrate all of the concepts above in the following example. Letm = 3, δ ∈ [0, 1) and consider the following points in TP2: c1 = (0,−1,−4), c2 = (0, δ, 0), c3 =(0,−4,−1). Putting them together as column vectors, we obtain the matrix

A =

0 0 01 −δ 44 0 1

View c1, c2, c3 are vectors in TP2, and renormalize so that the i-th coordinate of the i-thvector is zero, we obtain the new matrix

L =

0 δ −11 0 34 δ 0

.

Let P = Im(L). One can check that P = Im(A). The matrix L is the unique matrix withzero diagonal associated with P .

On the graph G(L), the mean-cycle lengths of two-cycles are 12(1 + δ), 1

2(3 + δ), and 3/2.

The mean-cycle lengths of three cycles are 13(δ+3+4) and 1

3δ. The self-loops have mean-cycle

lengths zero. So for any δ ∈ [0, 1), we have λ(L) = 0.Assume δ ∈ [0, 1). Let us compute the Kleene star of L. Note that

L�2 =

0 δ⊕(1− δ) −11 0 0

1 + δ δ 0

, L�3 =

0 −1 + δ −11 0 0

1 + δ δ 0

So the Kleene star L∗ = L⊕L�2⊕L�3 = L�3. In this case, the tropical eigenspace Eig(L) hasfacet representation

Eig(L) = {x ∈ R3 : −1 ≤ x1 − x2 ≤ −1 + δ,−1− δ ≤ x1 − x3 ≤ −1,−δ ≤ x2 − x3 ≤ 0}.

Figure 1 shows the tropical hyperplane arrangement H(L>) for δ ∈ (0, 1) in panel (A), andδ = 0 in panel (B). This arrangement consists of min-plus tropical hyperplanes whose apicesare the column vectors −L>1 ,−L>2 and −L>3 of L. In the first case, the green triangle in themiddle is the tropical eigenspace Eig(L). In the second case, the tropical eigenspace Eig(L)is the green point. The tropical polytope Im(L) consists of the bounded line segments fromthe vertices of Eig(L) to the points c1, c2, c3 labelled with their coordinates.


−c1 = (0,−1,−4)

−c3 = (0,−4,−1)

−c2 = (0, δ, 0)(0, 0)

a

b

c

(a) The case δ > 0.

−c1 = (0,−1,−4)

−c3 = (0,−4,−1)

−c2 = (0, 0, 0)(0, 0)

(b) The case δ = 0. The tropical eigenspace col-lapses to a point.

Figure 1. The tropical hyperplane arrangement H(L>) for δ ∈ (0, 1) (left),and for δ = 0 (right).

The covectors of the points labelled a though c are given by the following matrices.

coVecL(a) =

0 0 01 1 00 0 1

, coVecL(b) =

1 1 10 0 00 1 0

, coVecL(c) =

1 0 00 1 00 0 1

Note that a and b lie in unbounded cells of the tropical hyperplane arrangement, so theircovectors are not invertible, while c lies in a bounded cell, so its covector is invertible.

3. Mechanism Design in Tropical Notation

We consider an economy with m ∈ N outcomes which is populated by n ∈ N agents withquasi-linear utilities, as in [37, §4]. For i ∈ [n], agent i has a true type ti ∈ Rm, drawn bynature and privately known only to him. The tuple of all possible true types forms the typespace T ⊆ Rm×n. A direct mechanism (g, p) consists of an outcome function g : T → [m]and a payment function p : T → Rn.

We focus on dominant strategy equilibria in the game induced by the mechanism. Astrategy for agent i is a declared type si ∈ Ti for each possible true type. After all agentshave submitted their reports, the game’s outcome is g(s1, . . . , sn) ∈ [m] with payment-vectorp(s1, . . . , sn) ∈ Rn. Agent i chooses his report to maximize his utility, which is

ui((j, p), ti) = tij − pi

for game outcome j ∈ [m] with payment vector p ∈ Rn.We say that an agent tells the truth if his declared type is his true type. A central goal of

mechanism design is to find mechanisms (g, p) under which all agents report truthfully andthe desired outcome is implemented in equilibrium.

Definition 3.1. For a mechanism (g, p), say that truth-telling is dominant strategy incentivecompatible, abbreviated (D-IC), if for each agent i ∈ [n], each t−i ∈ T−i and each t, s ∈ Ti(t−i),

(D-IC) tg((t,t−i)) − pi((t, t−i)) ≥ tg((s,t−i)) − pi((s, t−i)).

That is, no matter what the other agents declare, each agent always maximizes his utility byreporting his true type.


For each agent i and types t−i ∈ T−i, define the matrix Lg,t−i ∈ Rm×m via

(6) Lg,t−i

jk = inft∈g−1(j)

{tj − tk}

For a given outcome function g, the classical theorem of Rochet [31] gives a necessary andsufficient condition for (D-IC) in terms of cycles on the allocation graph G(Lg,t−i), [14, 37].

Theorem 3.2 (Rochet). A mechanism g is (D-IC) if and only if for each i ∈ [n] and eacht−i ∈ T−i, all cycles in G(Lg,t−i) have nonnegative weights.

We can restrict to the single agent case. Fix i ∈ [n] and t−i. Let T = Ti, define g : T → [m]via g(t) := g(t, t−i) and similarly p : T → R via p(t) := pi([t, t−i]). Without loss, assume

g−1(k) = {t ∈ T : g(t) = k} 6= ∅ for all k ∈ [m].

By rewriting (D-IC), we obtain the analogous tropical formulation

(7) Lg,t−i � p = p.

That is, deciding (D-IC) is solving a tropical linear equation. With this view, we restateessential definitions of mechanism design in tropical notation as follows.

Definition 3.3. Say that g is dominant strategy incentive compatible (D-IC) if for eachi ∈ [n] and each t−i ∈ T−i we have that λ(Lg,t−i) = 0. In this case, call the tropicaleigenspace Eig(Lg,t−i) the set of incentive compatible payments.

Definition 3.4. Suppose g is (D-IC). Say that g is revenue equivalent (RE) if for each i ∈ [n]and each t−i ∈ T−i, Eig(Lg,t−i) in TPm−1 consists of exactly one point.

For ease of reference, we also collect the following central definitions here.

Definition 3.5. Fix a type space T ⊂ Rm. Say that a given matrix L ∈ Rm×m is realizableif there exists some outcome function g : T → [m] such that L = Lg,t−i .

Definition 3.6. Let L ∈ Rm×m be a realizable matrix. Say that L is weakly monotone ifL+ L> is element-wise nonnegative.

4. The Geometry of Mechanism Design - Main Results

4.1. Geometric Characterization of all Mechanisms. We fix an agent i and the typesof other agents t−i ∈ T−i. An allocation rule g defines the matrix Lg,t−i . Our characterizationrests on inverting this map. This allows us to study mechanisms on the space of matrices.

It is without loss to assume that T ⊆ TPm−1. In terms of economics, this means that onlyrelative valuation of the agents matter for truthfulness. Though it is common to normalizein equilibrium theory, this is rather uncommon in mechanism design. In our setting it turnsout to be natural and very useful.

Definition 4.1. For j, k ∈ [m], j 6= k, define Ijk = Ljk ∩ T . A (j, k)-witness is a sequence

{sj(r) : r ≥ 1} ⊆ T ∩ Lj◦

such that

limr→∞

dist(sj(r),Ljk) = 0.

We say that L separates T at (j, k) if

dist(T ∩ Lj,Ljk) = 0,

and in addition, whenever Ijk = Ikj = {s} for some s ∈ TPm−1, then there exists a (j, k)-witness or a (k, j)-witness. Say that L separates T if L separates T for all j, k ∈ [m], j 6= k.


The witnessing sequence condition is to exclude strange cases in which two sectors are heldin place by one point. In such a case, the dominant arrangement will move if the shared pointon the boundary is assigned to different outcomes. Thus such a matrix cannot be realizable,see Example 5.2 below. We now state our first main result, which geometrically characterizesall possible mechanisms on arbitrary type-spaces.

Theorem 4.2. Let L ∈ Rm×m with zero diagonal. Then L is realizable if and only if

T ⊆m⋃k=1

Lk

and L separates T .

For L to be realizable, L needs to separate T , as for each i, j ∈ [m], we want

Lij = inft∈g−1(i)

(ti − tj),

so this infimum needs to be achievable by some sequence in g−1(i). The other conditioncomes from the following observation.

Lemma 4.3. Suppose L ∈ Rm×m is realizable with outcome function g : T ⊂ TPm−1 → [m].Then g−1(j) ⊆ Lj for each j ∈ [m].

Proof. For any t ∈ g−1(j) we have that Ljk = infs∈g−1(j)(sj − sk) ≤ tj − tk, so Ljk + tk ≤ tj.Thus maxk∈[m](Ljk + tk) ≤ tj. But by definition for j = k, Ljj = 0 = tj − tj, so Ljj + tj = tjand hence maxk∈[m](Ljk + tk) = Lj�t = tj. It follows that t ∈ Lj. �

Proof of Theorem 4.2. Suppose L is realizable with outcome function g. By Lemma 4.3, fork = 1, 2, . . . ,m, g−1(k) ⊆ Lk. Since g partitions T , we have that,

T =m⋃k=1

g−1(k) ⊆m⋃k=1

Lk.

It remains to show that L separates T . Fix j, k ∈ [m], j 6= k. Since Ljk = inft∈g−1(j){tj − tk},this infimum is achieved either by a (j, k)-witness, or by a point t ∈ Ljk. So dist(T∩Lj,Ljk) =0. Furthermore, if Ijk = Ikj = {s}, then if g(s) = j, the infimum in the definition of Lkj mustbe achieved by a (k, j)-witness, so there must exist a (k, j)-witness. Conversely, if g(s) = k,then there must exist a (j, k)-witness. This shows that L separates T at {j, k}. Since thepair was chosen arbitrarily, L separates T .

For the converse direction, suppose L is a matrix that separates T . Define g : T → [m] as

follows. For a point t ∈ T ∩ L◦j , let

g(t) = j.

The remaining points lie on the boundary T ∩⋃j,k∈[m],j 6=k Ljk =

⋃j,k∈[m],j 6=k Ijk. Assign these

points such that points on Ijk have either outcome j or k, and such that on every non-emptyboundary Ijk there exists a point with outcome j. The only case where this cannot be doneis if Ijk = Ikj = {s}. In this case, if there is a (j, k)-witness, set g(s) = k, else, since Lseparates T , there must exists a (k, j)-witness, so set g(s) = j. We claim that L is realizedby g. Fix j, k ∈ [m], j 6= k. By definition of Lj

infs∈g−1(j)

(sj − sk) ≥ Ljk.


By the above assignment, there must be a point in Ijk or a (j, k)-witness in g−1(j). So theinfimum is achieved, thus g realizes L as claimed. �

It is immediate that when two rows of L coincide in TPm−1, say, Lm ≡ Lm−1, thenLmi = L(m−1)i for all i ∈ [m], i 6= m−1,m. In that case, one can reduce checking realizabilityby projecting onto the subspace xm−1−xm = 0, and deleting the last coordinate. In economicterms, this means that under any incentive compatible payment, outcomes m and m−1 havethe same transfers. We can strengthen this argument to an equivalence by introducing liftabletype spaces. Essentially, this condition requires that the type space has enough points, sothat upon embedding into a spce of one dimension higher, there are enough witnesses.

Definition 4.4. Suppose L separates T . For a pair j, k ∈ [m], k 6= j, let wjk be the numberof (j, k)-witnesses. Let qjk be the number of points in Ljk ∩ T . Now fix r ∈ [m]. Say thatT is liftable at coordinate r with respect to L if for each k ∈ [m], k 6= r, one of the followingcases hold

• wrk + qrk ≥ 3• wrk + qrk = 2, and Lrk ∩ Lkr ∩ T is empty• wrk + qrk = 2, Lrk ∩ Lkr ∩ T is non-empty, and qkr > 0.

We now obtain the following dimension reduction result, which follows straight from The-orem 4.2 by a definition chase.

Corollary 4.5. Let L ∈ Rm×m be a zero diagonal matrix. Suppose that two rows of L arethe same in TPm−1, say Lm−1 ≡ Lm. Let Π : Rm → Rm−1

(8) Π(x) = (x1, . . . , xm−2,min(xm−1, xm))

be the map given by projecting orthogonally onto the subspace xm−1 − xm = 0 and deletingthe m-th entry. View this as a map from TPm−1 to TPm−2. Write Π(L) ∈ R(m−1)×(m−1) forthe matrix obtained by removing the m-th row and column of L. Then L is realizable on typespace T ⊆ TPm−1 if and only if Π(L) is realizable on type space Π(T ) ⊆ TPm−2 and T isliftable at coordinate m− 1 with respect to L.

4.2. Geometric Characterization of Incentive Compatibility. Consider an allocationmatrix L = Lg,t−i . Note that it has zero diagonal, so necessarily λ(L) ≤ 0. To decide whetherit is incentive compatible, we want to know whether λ(L) < 0 or λ(L) = 0.

Theorem 4.6. Let L ∈ Rm×m be a matrix with zero diagonal. Define

Eig0(L) :=m⋂j=1

Lj.

Then λ(L) = 0 if and only if Eig0(L) 6= ∅. In that case, Eig(L) = Eig0(L).

Proof. By definition,

Eig0(L) = {p ∈ TPm−1 : L� p = p}.The theorem follows from uniqueness of the tropical eigenvalue. If λ(L) = 0, then Eig0(L)is the tropical eigenspace of L, which is non-empty. Conversely, suppose Eig0(L) 6= ∅. Thismeans that there exists a tropical eigenvector of L with tropical eigenvalue zero. But thetropical eigenvalue of L is unique, thus, λ(L) = 0. �


This geometric result allows us to visualize and prove incentive compatibility, or its failure,on arbitrary type spaces without calculating cycle lengths. When λ(L) = 0, it gives a geo-metric description for the tropical eigenspace of L, i.e. the tropical semi-module of incentivecompatible payments. We defer the discussion of the economic interpretation to Section 6.3.

Corollary 4.7. Let L ∈ Rm×m be a matrix with zero diagonal. Then Eig0(L) is a singletonif and only if any mechanism g that realizes L is revenue equivalent.

4.3. Verifying Incentive Compatibility Algorithmically. Determining if λ(L) = 0 canbe reduced to computing the tropical determinant. Thus verifying (D-IC) can be solved bythe Hungarian method.

Proposition 4.8. Let L ∈ Rm×m have zero diagonal. The following are equivalent.

(1) λ(L) = 0.(2) Any cycle of G(L) has nonnegative length.(3) tdet(L) = 0(4) The identity assignment e : 1 7→ 1, . . . ,m 7→ m is a minimal matching in the classic

perfect matching problem on a bipartite graph with weight L.

Proof. The assertion (1)⇔ (2) is the Rochet theorem, see [37] for proof using network flows.Since L has zero diagonal, the identity assignment has value 0. So (3) ⇔ (4). Let us prove(2)⇔ (3). Since L has zero diagonal, the identity assignment has value 0. Thus tdet(L) ≤ 0.Suppose (2). Any assignment σ ∈ Sm can be decomposed as a union of simple directed cycleson a disjoint set of vertices. Thus (2) implies tdet(L) ≥ 0, and so we have (3). Now instead,suppose (3). Any simple cycle C can be extended to a permutation by adding self-loops atvertices which are not in C. Thus we have (2). �

Corollary 4.9. For each i ∈ [n] and t−i ∈ T−i checking whether all cycles of G(Lg,t−1) arenonnegative can be done algorithmically using the Hungarian algorithm for perfect matching.

4.4. Characterization of Weak Monotonicity. Weak monotonicity has received wideattention in the literature, it is known to be sufficient on some domains, cf. Theorem 6.4.The following characterizations of weak monotonicity can be traced back to [10].

Lemma 4.10 ( [10]). Suppose L is realizable. For j, k ∈ [m],

Ljk + Lkj > 0⇔ Lj ∩ Lk = ∅,Ljk + Lkj = 0⇔ Lj ∩ Lk on their boundaries, i.e. Lj ∩ Lk = ∂Lj ∩ ∂LkLjk + Lkj < 0⇔ Lj ∩ Lk in their interiors, i.e. L◦j ∩ L◦k 6= ∅.

Proof. Note that Lj and Lk are closed polyhedra, so they either do not intersect, intersecton their boundaries, or intersect in their interiors. So it is sufficient to prove the last twoequivalences in the statements. Suppose there exists t ∈ Lj ∩ Lk. Then by definition of thesectors,

Ljk + Lkj ≤ (tj − tk) + (tk − tj) = 0.

In particular, strict equality holds if and only if t lies on the boundary of Lj ∩Lk, while strictinequality holds if and only if t lies in either of their interiors. In that case, one can pick at′ ∈ L◦j ∩L◦k, then Ljk +Lkj < (t′j− t′k)+(t′k− t′j) = 0. This proves the desired statement. �

Theorem 4.11. L is weakly monotone if and only if it is realizable, and the open setsL◦1, . . . ,L◦m are pairwise disjoint.

Proof. This follows from Lemma 4.10 and Definition 3.6 �


5. Examples

In this section we illustrate our results with various examples of mechanisms with threeoutcomes (m = 3). In particular, these examples show that it is simple to construct mecha-nisms and verify realizability, incentive compatibility (D-IC) and revenue equivalence (RE).

In all figures, for j ∈ {1, 2, 3}, the max-plus sector Lj has apex −Lj, where Lj (up totropical scaling) is the j-th row of L. The boundary of the max-plus sector Lj is shown inblue. The boundary of the min-plus sector Lj is shown in red. Type spaces are shaded ingray. The tropical eigenspace Eig(L) is shaded green.

Example 5.1 (Geometric representation of mechanism and verification of D-IC). Figure 2(a) is the geometric representation of a mechanism g via the dominant arrangement D(L)of its allocation matrix L = Lg,t−i . Note that T lies in the union of the sectors L1, L2,and L3. Furthermore, the sectors are disjoint and T has points on all of the boundariesL12,L13,L21,L23,L31 and L32. So L separates T , as expected from Theorem 4.2. Figure 2(b) verifies that the mechanism defined by Figure 2 (a) is D-IC via the min-plus dominantarrangement D(L) using Theorem 4.6. Indeed, the intersection of the min-plus sectors L1,L2

and L3 is the green region, which is non-empty. This region is the tropical eigenspace Eig0(L)

with eigenvalue zero, which corresponds to incentive compatible payments.

L3

L1 L2

(a) The mechanism g is defined by the max-plus dominant arrangement D(Lg,t−i) of itsallocation matrix Lg,t−i .

L3

L1 L2

(b) The min-plus dominant arrangementD(Lg,t−i) verifies that g is D-IC.

Figure 2. Geometric representation of a mechanism g (left) and verificationthat it is D-IC (right). Figures accompanying Example 5.1.

Example 5.2 (Realizable and non-realizable matrices). On a common type space, Figure 3shows the dominant arrangement D(L) of a realizable matrix L in panel (a), and that of anon-realizable matrix in panel (b). The type space T = {t1, t2, t3, t4} consists of four pointsin TP2. In Figure 3 (a), the mechanism g is defined by g(t2) = g(t3) = 3, g(t1) = 1 andg(t4) = 2. One can readily verify that Lg,t−i = L.

Now consider the matrix L defined via its dominant arrangement D(L) in Figure 3 (b). ByTheorem 4.2, for a mechanism g to realize L as L = Lg,t−i , we must have g(t4) = 2, g(t1) = 1,g(t3) = 3, and g(t2) either 1 or 3. Suppose g(t2) = 1. Then Lg31(t3) = infs∈g−1(3){s3 − s1} =t33− t13, Lg32(t3) = infs∈g−1(3){s3− s2} = t33− t23 and Lg33 = 0. Hence −Lg3 = [(−L31,−L32, 0)] =

[(0,−t23 − t13, t33 − t13)] in TP2 and thus Lg 6= L. So we cannot have g(t2) = 1. Now suppose


instead that g(t2) = 3. Then D(Lg,t−i) must equal the arrangement in Figure 3 (a), soL 6= Lg. Thus L is not realizable. Realizability fails since L does not separate T at {1, 3}.Here, one has I13 = L13 ∩ L31 ∩ T = {t2}, but there is no (1, 3) nor a (3, 1) witnesses.

t4

t3

t1

t2 L3

L1

L2

(a) The dominant arrangement D(L) of a realiz-able matrix L.

t4

t3

t1

t2 L3

L1

L2

(b) The dominant arrangement D(L) of a non-realizable matrix L. Here L does not separate T .

Figure 3. Realizable and non-realizable matrices, defined by their dominantarrangements D(L). Figure accompanies Example 5.2.

Example 5.3 (Realizable matrices with unusual domains). In Figure 4 (a), the domainconsists of the open gray-shaded trapezium and the four-point set {t1, t2, t3, t4}. Indeed,T ⊆

⋃3k=1 Lk, and L separates T . One can define a mechanism that realizes L as follows.

Set g(t1) = g(t2) = 1, g(t4) = 2, g(t3) = 2, and g(t) = 3 for all remaining points t ∈ T . ByTheorem 4.11, in any mechanism g that realizes Figure 4 (a), the two-cycle (2 → 3 → 2) isnegative. Thus g is not D-IC.

t4

t3

t1

t2 L3

L1

L2

(a) The dominant arrangement D(L) of a re-alizable matrix L. Any corresponding mech-anism g is not D-IC.

L3

L1

L2

(b) The max-plus arrangement D(L) in blue,and the min-plus arrangement D(L) in red,of a realizable matrix L. The unique corre-sponding mechanism is RE.

Figure 4. Dominant arrangements D(L) of two realizable matrices L withunusual type spaces T . Using Theorem 4.2, one can write down all the mech-anisms g : T → {1, 2, 3} with allocation matrix L. Figure to Example 5.3.


In Figure 4 (b), the domain consists of three connected pieces: two open circles and a largeopen set cut out by a curve. There is a unique mechanism that realizes L which is RE andD-IC since the three sectors in the min-plus dominant arrangement D(L) intersect at −L3.

L3

L1 L2

(a) The max-plus dominant arrangementD(L).

L3

L1L2

(b) The min-plus dominant arrangementD(L). The green dot is the unique point inthe tropical eigenspace, so g is RE.

Figure 5. A revenue equivalent mechanism on a convex type space. Figureaccompanies Example 5.4.

Example 5.4 (Saks-Yu demonstration). In Figure 5, the type space T is the closed half-spaceshaded gray. Any mechanism g that realizes the arrangement D(L) given in Figure 5 (a)must assign −L3 to outcome 3, −L2 to outcome 2, and assign points in the interior of L1

and L2 to outcomes 1 and 2, respectively. Points on the boundary of any pair Lj and Lkcan take either outcome j or k. By Theorem 4.11, any realizable g has all of its two-cyclesequal to 0. Figure 5 (b) shows the tropical eigenspace Eig(Lg,t−i) as the intersection of themin-plus sectors Lj. In this case the eigenspace consists of a point, so the mechanism is D-ICand RE. This example is an isntance where Theorem 6.4, below applies.

L3

L2

L1

(a) A mechanism that is weaklymonotone but not D-IC.

L3L2

L1

(b) The min-plus arrangementD(L) verifies non-D-IC.

L3

L2L1

(c) On the same type space, an-other mechanism that is RE.

Figure 6. Two different mechanisms on a non-convex type space. Figureaccompanies Example 5.5.


Example 5.5 (Non-convex, not path-connected domain with revenue equivalence). Figure 6depicts a non-convex type space (gray shaded) on which any implementable mechanism mustbe revenue equivalent. Figures 6 (a) and (b) show a weakly monotone mechanism that isnot D-IC. On convex type spaces such a situation cannot occur. Figure 6 (c) depicts a D-ICmechanism that is RE. Note that the closure of the type-space is not path-connected. Alsothe domain is not ‘boundedly grid-wise connected’ cf. [7, Theorem 1 and 4].

6. Economic Interpretation

We present Theorem 6.3, a generalization of a theorem by Saks and Yu [33], on the suf-ficiency of weak monotonicity on convex domains. The converse of Saks-Yu was proved byAshlagi, Braverman, Hassidim and Monderer [3]; we give the geometric intuition of theirproof here. Using the simple geometric characterization of revenue equivalence in Corol-lary 4.7, we obtain a special case of a theorem of Heydenreich-Muller-Uetz-Vohra [15]. Weconclude with a discussion of the economic interpretation of Theorem 4.6 in terms of themin-max duality of linear programming.

6.1. Convexity and Weak Monotonicity. Consider the type graph T g : T ×T → R, withT g(t, s) := tg(t) − tg(s). For a path γst ⊂ T of finite Euclidean length from s to t, let `g(γst)denote the length of γst with respect to T g. Note that such a path has length of the form`g(γst) = Lgi0i1 + Lgi1i2 + . . .+ Lgik−1ik

for some finite sequence i0 → . . .→ ik ⊂ [m], called the

length sequence of γst. Note that i0 = g(s), ik = g(t). Call a path γst simple if its lengthsequence contains no repeated nodes.

Our assumptions on the domain T are

(H1) T is closed(H2) T is simply connected(H3) For any i, j ∈ [m], any point t ∈ T ∩ ∂Li ∩ ∂Lj is the limit of a sequence in the strict

interior T ∩ L◦i and also the limit of a sequence in T ∩ L◦j .(H4) For any i, j ∈ [m], there exist u ∈ T ∩ ∂Lij, v ∈ T ∩ ∂Lji, such that the line segment

[u, v] ⊆ T .

Hypothesis (H3) allows one to reassign t to either outcome i or j leaving the matrix Lunchanged. Thus, we have a family of mechanisms G = {g : T → [m]} such that Lg =Lg′

= L for all g, g′ ∈ G, whose values only differ on boundary points, namely, points in theskeleton T ′ := T ∩

⋃i,j∈[m](Lij ∩ Lji). For a point p ∈ T , write G(p) = {g(p) : g ∈ G} =

{i1, i2, . . . , ik} ⊆ [m] to be the set of possible labels at p.

Lemma 6.1. Suppose T satisfies (H3). If s, t ∈ T are two points in the strict interior ofsectors i and j, respectively, then `g(γst) = `g

′(γst) for all g, g′ ∈ G.

Proof. Consider γst ∩ T ′. As the path γst has finite length and T ′ is piecewise linear, this isa union of points and line segments. Let p ∈ γst ∩ T ′ be an isolated point. Since p 6= s, t,the point p is in the strict interior of the path γst. Let p−, p+ ∈ γst be points immediatelyto the left and right of p, respectively. Since p is isolated, p+, p− /∈ S, one may assume thatG(p+) = {i1}, G(p−) = {i2}. But p ∈ ∩i∈GLi ∩ T , so Li1ir + Liri2 = Li1i2 for all ir ∈ G. Sowe are done. The case of a line segment is similar, as one can define the line segments sothat all points in each line segment have the same set of labels. This proves the claim. �

With the above lemma, we can define `(γst) to be `(γst) := `g(γst) for some g ∈ G.


Proposition 6.2. Suppose T satisfies (H1)-(H3). If s, t ∈ T are two points in the strictinterior of sectors i and j, respectively, then `(γst) = `(γ′st) for all paths γst, γ

′st ⊂ T .

Proof. Note that T is simply connected. Continuously transform γst until the first time thatit becomes a path γ′st with a different length sequence. We want to show that `(γst) = `(γ′st).By choosing intermediate points s′, t′ ∈ T if necessary to break the problem down, one canassume that γst has length sequence i → k → j, and γ′st has length sequence i → i1 →. . . → ir → j with k 6= i1, . . . , ir, and that this length sequence cannot be further reduced.In other words, there exists points p1, . . . , pr ∈ γ′st such that G(p1) ∩ {i, i1, . . . , ir, j} = i1,G(p2) ∩ {i, i1, . . . , ir, j} = i2, and so on. Now, each point of γ′st is the limit of a sequence ofpoints in T ∩ (Li ∪ Lj ∪ Lk). Since T is closed, {i, j, k} ∩ G(q) 6= ∅ for each q = p1, . . . , pr.By assumption, i, j /∈ G(q), thus k ∈ G(q) for each q = p1, . . . , pr. Therefore, Lik + Lkj =Lii1 + Li1i2 + . . .+ Lirj, so `(γst) = `(γ′st), as needed. �

The above Proposition says that all cycles in Lg that are realizable as the length of sequenceof a loop γss ∈ T have length zero. However, if such a cycle involves the edge i → j, say,then one must have Lij + Lji = 0. In general, not all edges satisfy this property: Lg canhave a strictly positive two-cycle. However, with hypothesis (H4), one can be sure that edgesinvolved in a strict positive two-cycle must be large.

Theorem 6.3. Suppose T satisfies (H1)-(H4). If all two-cycles of Lg are nonnegative, theng is D-IC.

Proof. For i, j ∈ [m], take points u, v as in hypothesis (H4). Let i → i1 → . . . → ir → j bethe length sequence of the path [u, v]. We claim a reversed triangle inequality:

ui − uj = Lij ≥ Lii1 + . . .+ Lirj.

Indeed, let w := v − u. There exists a sequence of points p1, . . . , pr ∈ [u, v] correspondingto a sequence of constants 0 ≤ λ1 ≤ . . . ≤ λr ≤ 1 such that u + λ1w ∈ T ∩ Li ∩ Li1 ,u+ λ2w ∈ T ∩Li1 ∩Li2 , and so on. If any of the inequalities are equalities, say, λ1 = 0, thenLii2 = Lii1 +Li1i2 , so one can replace u by p1. Thus, one may assume that all inequalities arestrict. Now, consider the line segment [p1, p2]. Let ei be the i-th standard coordinate vector.Note that

p1i1− p1

i2= 〈u, ei1 − ei2〉+ λ1〈w, ei1 − ei2〉 = Li1i2 ,

andp2i1− p2

i2= 〈u, ei1 − ei2〉+ λ2〈w, ei1 − ei2〉 = −Li2i1 ≤ Li1i2 .

Therefore,

〈w, ei1 − ei2〉 ≤1

λ2 − λ1

(Li1i2 + Li2i1) = 0.

So in particular,ui1 − ui2 = 〈p1, ei1 − ei2〉 − λ1〈w, ei1 − ei2〉 ≥ Li1i2 .

By the same argument with segments [p2, p3], . . . , [pr, v], one concludes that

ui2 − ui3 ≥ Li2i3 , . . . , uir − uij ≥ Lirij .

Therefore,

Lij = ui − uj = (ui − ui1) + (ui1 − ui2) + . . .+ (uir − uj) ≥ Lii1 + . . . Lirj.

Since the line [u, v] goes through adjacent sectors,

Lii1 + . . .+ Lirj = −(Li1i + . . .+ Ljir),


therefore we have

−(Li1i + . . .+ Ljir) ≤ Lij ⇒ Lij + Ljir + . . .+ Li1i ≥ 0.

That is, the cycle j → ir → . . .→ i1 → i→ j has nonnegative length. Reversing the role ofu and v implies that the reverse cycle also has nonnegative length.

Finally, it remains to show that any simple cycle involving j and i can be decomposedthis way. Take an arbitrary cycle i → i1 → . . . → ir → j → i. Each edge is greaterthan or equal to the length of a line segment [ui, vi1 ], [ui1 , vi2 ], [ui2 , vi3 ], . . . , [uj, vi], where thesuperscript indicates the label of the point. Since T is connected, there exists simple pathsγvi1ui1 , γvjuj , γviui . Thus, the path γii := [ui, vi1 ] ∪ γvi1ui1 ∪ [ui1 , vi2 ] ∪ . . . ∪ γvjui is a path inT such that `(γii) = 0, and is less than or equal to the length of the cycle in consideration.This completes the proof. �

We now obtain Saks-Yu as a corollary.

Theorem 6.4 (Saks-Yu). Suppose L is realizable, and all two-cycles of L are nonnegative.If T is convex, then λ(L) = 0.

Proof. Without loss of generality, one can assume that T is closed and for each i ∈ [m],

T ∩ L◦i 6= ∅. Thus, T satisfies (H1)-(H4). The result follows by Theorem 6.3. �

We now consider the converse of Saks-Yu for finite-valued mechanisms. This result wasfirst proved by Ashlagi, Braverman, Hassidim and Monderer [3]

Theorem 6.5 (Ashlagi-Braverman-Hassidim-Monderer). Suppose T = Ti is closed. If T isnot convex, then there exists a mechanism g such that all two-cycles of Lg,t−i are nonnegative,but λ(L) < 0.

Proof. Let us sketch the idea for the case where T is full-dimensional, and conv(T )\T containsan open ball B(p, ε) around some point p. Inside this open ball, fits a full-dimensional minimalpolytrope P . By Theorem 2.17, there exists a matrix L = −L> such that P = Im(L). Thenwith careful analysis, one can choose P such that L separates T . By Theorem 4.2, L isrealizable, that is, L = Lg,t−i for some g. By construction, all the two-cycles of L are zero,and the polytrope P has full-dimension, so λ(L) < 0. �

6.2. Weak Monotonicity and Revenue Equivalence. The literature on revenue equiva-lence is vast [9,15,16,20,25,27,37], see [9,37] for a comprehensive review. Many results giveconditions that ensure revenue equivalence for all implementable allocation rules. As waspointed out in [15], this is a very stringent requirement as there are cases in which revenueequivalent and non-revenue equivalent mechanisms can coexist on the same type-space. Thefollowing corollary of Proposition 2.13 is a special case of the characterization of revenueequivalence in [15], it is the linear algebraic analogue of the geometric characterization inCorollary 4.7.

Theorem 6.6 ( [15], special case for finite outcomes). Let L be a realizable m×m matrix withzeros on its diagonal and with tropical eigenvalue λ(L) = 0. The following are equivalent.

(1) L has a unique eigenvector in TPm−1.(2) L∗ is skew-symmetric.


6.3. An Interpretation via Linear Programming Duality. Loosely formulated we havethe following duality. The rows of Lg,t−i correspond to the agent’s problem (the informedside of the economy), he chooses optimal reports given his type. The columns of (Lg,t−i)∗

correspond to the problem of the mechanism (the uninformed side), it must simultaneouslybalance all incentives of reporting a given type. This observation can be made precise vialinear programing duality and the geometry of the dominant arrangements. A similar inter-pretation in terms of zero sum games [24,35] was given in [30].

Consider the j-th row of −Lg,t−i , its entries

−Lg,t−i

jk = sups∈g−1(j)

{sk − sj} k ∈ [m]

correspond to the most any agent whose truthful statement would entail outcome j couldgain by misreporting to change the outcome to k.

With quasi-linear utilities the set of payments that offset the incentive to misreport is Lj.To see this, note that by definition

p ∈ Lj = {p ∈ TPm−1 : Ljk + pk ≥ pj for k 6= j}.

Thus infg−1(j){tj − tk} + pk ≥ pj and hence tj − pj ≥ tk − pk for all t ∈ g−1(j). However,there is a consistency requirement. Since the mechanism is informationally constrained, itcan only set one payment that must incentivize all possible types simultaneously.

Suppose Lg,t−i is realizable and λ(Lg,t−i) = 0. The min-plus covector of p ∈ Eig0(Lg,t−i)with respect to Lg,t−i systematically lists for which types the incentive constraints hold, i.e.

coVecLg,t−i (p)jj = 1⇔ p ∈ Lj

The intersection of all the sectors Lj for j ∈ [m] is the set of jointly incentive compatiblepayments. Theorem 4.6 identified this set as the the tropical eigenspace, i.e. the tropicalsemi-module of D-IC payments. It is the min-plus image of (Lg,t−i)∗.

Rewriting the eigenvector condition, we obtain

maxk∈[m]

{− Lg,t−i

j,k − pk}

= −pj ∀j ∈ [m].

The terms −Lg,t−i

j,k − pk are utility gains relative to truth-telling net payments. Thus themechanism makes any agent pay the most he could gain.

This can be linked to linear programming duality viz. A reporting strategy of an agentis to announce a preference for some allocation that maximizes his utility, i.e. a probabilitydistribution s ∈ ∆m−1. Recall that types in TPm−1 are normalized, i.e. t = (0, t2−t1, . . . , tm−t1). The mechanism tries to find a vector p ∈ TPm−1 that solves the following family of linearprograms simultaneously for all j ∈ [m].

(P) infµ

sups≥0

{ m∑k=1

sk

((tk − tj)− (pk − pj)

)− µ

( m∑k=1

sk − 1)}

∀t ∈ g−1(j)

The inner problem is the agent’s, he maximizes utility relative truth-telling.Now, suppose p ∈ Eig0(Lg,t−i) and hence coVecLg,t−i (p) has ones on its diagonal. It follows

that for any j ∈ [m] and any t ∈ g−1(j):

tj − pj ≥ tk − pk k ∈ [m], k 6= j


Thus, the inner problem of (P) is maximized by setting sj(t) = 1 and sk(t) = 0 for k 6= j,i.e. truth-telling. Implicitly this is the content of Theorem 4.6. For each j ∈ [m], linearprograming duality lets us then equivalently solve the dual.

(D) sups≥0

infµ

{µ+

m∑r=1

sr

((tr − tj)− (pr − pj)− µ

)}∀t ∈ g−1(j)

Explicitly the problem is to minimize µ subject to µ ≥ 0 and (tk−tj)−(pk−pj) ≤ µ for k 6= j.In economics µ(t) is the shadow-value of a agent with type t from lying. If µ > 0 then theagent has a strict incentive to misreport. For p ∈ Eig0(Lg,t−i) this program is solved by settingµ(t) = 0. Thus (P) is the agent’s problem of maximizing his utility relative to truth-tellingfor given payments. Problem (D) is the mechanism’s problem of minimizing incentives to liegiven optimizing behavior. By flipping sectors at their apices and intersecting the regionsLj, we geometrically characterize payments that induce truth-telling.

7. Acknowledgements

We wish to thank John Weymark for suggesting to explore possible connections of mech-anism design to tropical geometry. Our geometric approach is inspired by his work withcoauthors in [9]. Part of this work was written during the Hausdorff Center Conference“Tropical Geometry and Economics”. We thank the participants for helpful discussions.The Hausdoff Center’s role in catalyzing research in mathematical economics is gratefullyacknowledged. Finally we thank Paul Edelman and John Weymark for pointing out an errorin an earlier version, and Alexey Kushnir for generously sharing his manuscript.

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(Robert Alexander Crowell) Hausdorff Center for Mathematics, Bonn 53115, GermanyE-mail address: [email protected]

(Ngoc Mai Tran) Department of Mathematics, University of Texas at Austin, Texas TX78712, USA and Hausdorff Center for Mathematics, Bonn 53115, Germany

E-mail address: [email protected]

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