Geometry of gross substitutes valuations

Discrete Applied Mathematics xxx (xxxx) xxx

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Geometry of gross substitutes valuations Sven de Vries a , Ulf Friedrich b , Stephen Raach a , a b

∗

Mathematics, Trier University, 54296 Trier, Germany Chair of Operations Research, Department of Mathematics, Technical University of Munich, Arcisstr. 21, 80333 Munich, Germany

article

info

Article history: Received 29 September 2017 Received in revised form 26 October 2018 Accepted 5 November 2018 Available online xxxx Keywords: Combinatorial auctions Gross substitutes valuations Polyhedral dimension Johnson graph

a b s t r a c t We consider the set of gross substitutes valuations which is one of the most important classes of valuation functions in combinatorial auctions. This is due the fact that the existence of a Walrasian equilibrium is guaranteed if all bidders have gross substitutes valuations. We interpret normalized valuation functions v : 2N → R with v (∅) = 0 on n N := {1, . . . , n} as points of R2 −1 . The set of gross substitutes valuations turns out to be the union of finitely many polyhedral cones. It has been proved that the set of gross n substitutes valuations has Lebesgue-measure zero in R2 −1 , but the question whether the dimension of each of these cones is polynomial in the number of items n is open. We answer this question negatively by showing that the dimension of each cone is actually exponential 1 · (2n − n − 2)⌉ + 2n − 1 for their dimension. By in n and provide a lower bound of ⌈ n+ 1 explicit calculation we verify that our lower bound matches the dimension in the case of n ≤ 3, but becomes loose for larger n. © 2018 Elsevier B.V. All rights reserved.

1. Introduction The class of gross substitutes valuations is important in the theory of combinatorial auctions, see, e.g., de Vries and Vohra [2]. This is due to the nice properties that combinatorial auctions with bidders possessing gross substitutes valuations have as, e.g., the problem of maximizing the social welfare can be solved in polynomial time and the existence of a Walrasian equilibrium is guaranteed (Bikhchandani and Mamer [1]). The gross substitutes condition has been introduced by Kelso and Crawford [7] as a sufficient condition under which a salary adjustment process converges to an equilibrium in a market consisting of firms and workers. Combinatorial auctions with gross substitutes are widely studied by Gul and Stacchetti and by Milgrom [6,10]. Gul and Stacchetti [5] show the importance of this class by proving that the gross substitutes condition is necessary to ensure existence of a Walrasian equilibrium in exchange economies. A first price-free characterization is given by Reijnierse et al. [12]. Important steps towards understanding the class of gross substitutes valuations are made by Murota [11], linking them to discrete convex analysis. As a consequence of this connection another price-free characterization of gross substitutes is given by Fujishige and Yang [3]. For a finite set N let 2N denote the power set of N. We consider normalized valuation functions v : 2N → R+ with v (∅) = 0 in which n ∈ N is the number of items and N := {1, . . . , n} is the corresponding set of items to be auctioned in a combinatorial N n auction. We interpret each function v as an element of R2 \{∅} ≃ R2 −1 . Lehmann et al. [9] prove that the set of submodular 2N \{∅} valuations is full-dimensional in R and show that the set of gross substitutes valuations v : 2N → R+ consists of the union of finitely many polyhedral cones. They pose the question whether the dimension of these cones is polynomial in n. Our main result is that to the contrary this dimension is exponential in n. We use a variation of the characterization by Reijnierse et al. [12], refine the construction of Lehmann [9] for submodular valuations and apply results from graph theory ∗ Corresponding author. E-mail addresses: [email protected] (S. de Vries), [email protected] (U. Friedrich), [email protected] (S. Raach). https://doi.org/10.1016/j.dam.2018.11.003 0166-218X/© 2018 Elsevier B.V. All rights reserved.

Please cite this article as: S. de Vries, U. Friedrich and S. Raach, Geometry of gross substitutes valuations, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.003.

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1 · (2n − n − 2)⌉ + 2n − 1 dimensional polyhedral cone contained in the set of gross substitutes to construct an at least ⌈ n+ 1 valuations. In the next section, we review necessary definitions and basic results on valuations. In Section 3, we use the double maximum condition and a construction on hypergraphs to deduce a bound on the dimension of the cones describing the gross substitutes valuations. This section constitutes the main theoretical construction for our results. In Section 4, we refine the bound of the previous section with the help of a graph-theoretic argument. We compare this improved bound with the actual dimension by illustrating two example cases in Section 5 and conclude with a discussion.

2. Preliminaries 2.1. Definitions We focus on a special class of valuation functions that, e.g., agents in a combinatorial auction have. We begin with the definition of a valuation and use the abbreviations S ∪ a := S ∪ {a}, S ∪ ab := S ∪ {a} ∪ {b} and v (a) := v ({a}). Definition 1. Let n ∈ N and N := {1, . . . , n}. A valuation is a function v : 2N → R+ that is nondecreasing (v (A) ≤ v (B) if A ⊆ B) and normalized (v (∅) = 0). Definition 2. For a valuation v : 2N → R+ and a price-vector p ∈ Rn+ with p(S) := correspondence Dv (p) is defined by

∑

i∈S

pi for S ⊆ N the demand

Dv (p) := arg max(v (S) − p(S)) S⊆N

= {S ⊆ N | v (S) − p(S) ≥ v (A) − p(A) for all A ⊆ N } as the set of utility maximizing sets. Definition 3. Given n ∈ N, a valuation function v : 2N → R+ is called a gross substitutes valuation (GS) if for all p, q ∈ RN+ with q ≥ p (component wise) and all A ∈ Dv (p) there exists B ∈ Dv (q) with {a ∈ A | pa = qa } ⊆ B. Definition 4. A function f : 2N → R is submodular, if for all S ⊂ T ⊂ N , a ∈ N \ T .

f (T ∪ a) − f (T ) ≤ f (S ∪ a) − f (S)

The fact that gross substitutability implies submodularity is shown by Gul and Stacchetti [5]. Reijnierse et al. [12] give a first price-free condition for a submodular valuation to be gross substitutes, which we state next. Theorem 5 (Reijnierse et al. [12]). For a finite set N a valuation v : 2N → R+ is (GS) if and only if v is submodular and for all S ⊂ N, a, b, c ∈ N \ S it holds

v (S ∪ ab) + v (S ∪ c) ≤ max{v (S ∪ ac) + v (S ∪ b), v (S ∪ bc) + v (S ∪ a)}. A trivial consequence of this theorem is the following reformulation which we will frequently use. A submodular valuation is (GS) if and only if for all S ⊂ N, a, b, c ∈ N \ S the triple

(

) v (S ∪ ab) + v (S ∪ c), v (S ∪ ac) + v (S ∪ b), v (S ∪ bc) + v (S ∪ a)

(1)

has a double maximum, i.e., two terms of (1) have the same value that is at least as large as the third term. We call this property the triple condition. n −1

2.2. Interpreting valuations as points of R2

With the idea of analyzing the structure of certain classes of valuation functions, Lehman et al. [9] interpret valuation n n functions v : 2N → R+ on n items as points of R2 −1 . This means labeling the axes of the real valued vector space R2 with N the elements of 2 . Since valuation functions are normalized, the value of the coordinate induced by ∅ is always zero and we omit this axis. As usual, the dimension dim(P) of a polyhedral cone P is the cardinality of the largest linearly independent subset of P. n Using this identification with R2 −1 , Lehman et al. [9] show that the polyhedral cone of the submodular valuations Psub is full dimensional by showing that a subcone Vsub , defined below, is full dimensional. Theorem 6 (Lehman et al. [9]). Let 2 ≤ n ∈ N, g: 2N → R+ , g(S) := 1 −

{ H :=

h : 2N → R | h(∅) = 0, |h(S)| ≤

( 1 )n+2 2

( )|S | 1 2

and

}

for all S ⊆ N .

Please cite this article as: S. de Vries, U. Friedrich and S. Raach, Geometry of gross substitutes valuations, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.003.

S. de Vries, U. Friedrich and S. Raach / Discrete Applied Mathematics xxx (xxxx) xxx

3 n −1

Then Vsub := {v | v = g + h, h ∈ H } has, as a translation of the hypercube H, positive Lebesgue-measure in R2 of the submodular valuations.

and is a subset

We call a valuation symmetric for k if v (S) = v (T ) for all S , T ⊆N with |S | = |T | = k. A valuation that is symmetric for n all k is called symmetric. We use GSn ⊆ R2 −1 for the set of gross substitutes valuations on n items. 3. A simple lower bound for the dimension of the cones describing GSn In the first step, we show that GSn actually consists of a union of polyhedral cones. This statement was implicitly already given by Lehman et al. [9]. 3.1. GSn is the union of finitely many polyhedral cones We define the sets L := {(S , {abc }) | S ⊂ N , a, b, c ∈ N \ S }, F := {(S , {ab}, c) | (S , {abc }) ∈ L} and M := {M ⊆ F | (S , {ab}, c) ∈ M or (S , {ac }, b) ∈ M or (S , {bc }, a) ∈ M

for all (S , {abc }) ∈ L}. We want to interpret each (S , {ab}, c) ∈ F as description of the two terms in the triple condition (1) for which the maximum is attained. Therefore, for (S , {ab}, c) ∈ F we define the cone n

P(S ,{ab},c) := {v ∈ R2+ −1 | vSab + vSc ≤ vSac + vSb = vSbc + vSa } that contains the valuations fulfilling the triple condition with joint maximum vSac + vSb = vSbc + vSa , where vj denotes as n usual the jth component of the vector v in the space R2 −1 . For M ⊆ F we set PM :=

⋂

P(S ,{ab},c) .

(2)

(S ,{a,b},c)∈M

The cone PM enforces that a double maximums is attained at vSac + vSb = vSbc + vSa for all (S , {ab}, c) ∈ M. On the one hand, it is a straightforward consequence of Theorem 5 that for every v ∈ GSn there exists an M ∈ M with v ∈ PM . It obviously holds that GSn ⊆ Psub and hence GSn ⊆ ∪M ∈M (Psub ∩ PM ). On the other hand, Psub ∩ PM ⊆ GSn for all M ∈ M and hence ∪M ∈M (Psub ∩ PM ) ⊆ GSn . Because M is finite, this shows that GSn is the union of finitely many polyhedral cones. Notice that for M ′ ⊇ M ∈ M it holds PM ′ ⊆ PM . 3.2. A simple bound We want to find a set M ∈ M for which PM has dimension exponential in n. Towards this let us recapitulate: Definition 7. A hypergraph is a pair (X , E ) in which X is a finite set of vertices and E ⊆ 2X is a set of hyperedges. Definition 8. Let G = (V , E) be an undirected graph with set of vertices V and set of edges E. A subset S ⊆ V is independent if {u, v} ∈ / E for all u, v ∈ S with u ̸= v . We denote by

(N ) k

the set of all subsets A ⊆ N of cardinality exactly k.

Definition 9. Let n, k ∈ N with k ≤ n. The graph (V , E) with V := Johnson Graph J(n, k).

(N ) k

and E := {{A, B} | A, B ∈ V , |A ∩ B| = k − 1} is called

We study large independent sets of the Johnson Graph to show the existence of a subcone of GSn with dimension exponential in n. Lemma 10 (Graham and Sloane [4]). The Johnson Graph has an independent set with at least ⌈ 1n ·

(n) k

⌉ elements.

The proof uses a coset argument that we will adapt in our proof of Theorem 14. The next lemma is related to Stirling’s formula. Lemma 11. Let n ∈ N, then

(2n) n

>

4n √ . 2· π ·n

Source: page 228 in [8]

Please cite this article as: S. de Vries, U. Friedrich and S. Raach, Geometry of gross substitutes valuations, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.003.

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S. de Vries, U. Friedrich and S. Raach / Discrete Applied Mathematics xxx (xxxx) xxx 2N \{∅}

N

We search for an M ′ ∈ M such that PM ′ ∩ Psub has dimension exponential in n. Let R|S |=k be the subspace of R2 \{∅} which is given by all coordinates indexed by the sets of cardinality k. If PM ′ ∩ Psub has dimension exponential in n, then there 2N \{∅}

exists a k ∈ N such that the dimension of PM ′ ∩ Psub ∩ R|S |=k is also exponential in n. 2N \{∅}

2N \{∅} n . Focusing on the elements of R|S |= n 2 2 n PM and S with S 1 2n 2 we 2 double maximum). For all PM there is

Assuming that n is even, we start with k such that dim R|S |=k is maximal, hence k = yields the idea to define M := {(S , {ab}, c) ∈ F : |S | ∈ / { 2n − 1, 2n − 2}}, i.e., for all v ∈ assume that the triple condition is satisfied by a triple maximum (and not only a 2N \{∅}

| |∈ /{ − , − } v∈

no restriction to vT for any T ∈ R|S |= n . In view of Theorem 12, we observe that by (2) for M defined above it holds 2

Usym n

:= {v : 2 → R+ | v is symmetric for k ≤ n, k ̸= N

n 2

} ⊆ PM .

We show that there exists an M ′ ⊃ M such that M ′ ∈ M and PM ′ ∩ Psub ∩Usym is a subcone of GSn with dimension n exponential in n. Theorem 12. Let n ∈ N be even, then there exists an M ′ ∈ M such that PM ′ ∩ Psub ∩ Usym has a dimension that is exponential n in n. Proof. For v ∈ Usym / { 2n − 1, 2n − 2} and a, b, c ∈ N \ S because n , the triple condition is fulfilled for all S with |S | ∈ (vSab + vSc , vSac + vSb , vSbc + vSa ) has a triple maximum. For |S | = 2n − 1 and a, b, c ∈ N \ S the triple condition simplifies to (vSc , vSb , vSa ) having a double maximum. Analogously, for |S | = 2n − 2 and a, b, c ∈ N \ S the triple condition simplifies to (vSab , vSac , vSbc ) having a double maximum. (N ) We encode these double maximum conditions in a hypergraph H = (X , E ) by interpreting every T ∈ k as a vertex. The set of hyperedges is given by every triple of k-sets that has to fulfill a triple condition. In an attempt to obtain a set with few variables restricted by equations we choose vertices representing variables that are not constrained by equations. Each selection of such a free variable makes others unfree. From each hyperedge {R, S , Q } we can choose at most one element Q and set the element’s value vQ arbitrarily within submodularity restrictions. We fix vR = vS to some value for the unselected elements R, S that are connected to Q by an arbitrary hyperedge. In this construction the triple condition remains fulfilled. (N ) We now transform the condition on the hypergraph into a condition on a graph G = (V , E) with vertex set k . Two vertices T1 and T2 in G are adjacent if there is a hyperedge H containing both. A set of vertices of G is independent if no two vertices in the set belong to a common hyperedge of H. We illustrate this idea for the case n = 4 and k = 2. The triple condition for S = ∅ and |S | = 1 requires that the following triples have a double maximum:

|S | = 1 (v12 , v13 , v14 ) (v12 , v23 , v24 ) (v13 , v23 , v34 ) (v14 , v24 , v34 )

S=∅ (v12 , v13 , v23 ) (v12 , v14 , v24 ) (v13 , v14 , v34 ) (v23 , v24 , v34 )

The associated two hypergraphs and the graph G constructed from them are shown in Fig. 1. As in the example setting, the graph G is exactly the Johnson Graph J(n, 2n ). By combining Lemmas 10 and 11, it follows that the Johnson Graph J(n, 2n ) contains an independent set of cardinality at least ⌈ 1n · 2n−1 · (π 2n )−1/2 ⌉. Therefore, there exists an in n exponential set of vertices that do not have a hyperedge in common. Choosing a bundle T contained in an independent set I means choosing all of the cones P(S ,{ab},c) for S ⊂ N and a, b, c ∈ N \ S so that T ∈ / {Sc , Sab}. That means that the value of vT is not restricted by any equation. Let PT be the intersection of all these cones and PM ′ := ∩T ∈I PT . Then, it holds true M ′ = {(S , {ab}, c) ∈ F | T ∈ / {Sab, Sc } for all T ∈ I }. It is easy to see that M ′ ∈ M and hence for v ∈ PM ′ all triple conditions hold. The cone PM ′ ∩ Usym has dimension exponential in n if and only if |I | is exponential in n. Fix I as a maximal n independent set of the Johnson Graph J(n, 2n ), then |I | is exponential in n. It remains to show that PM ′ ∩ Usym intersected with the submodular cone remains a cone with dimension exponential n in n. Making use of the construction in Theorem 6, this can be seen from the following subcone Vsym of PM ′ ∩ Usym that has n dimension exponential in n.

{

(

Vsym := v : 2N → R+ | v (S) = λ· 1 −

( 1 )|S | 2

−αS

( 1 )n+2 ) 2

λ ≥ 0, αS ∈ [0, 1] if S ∈ I and αS = 0 if S ∈ N \ I n−1

2 has dimension |I | ≥ ⌈ 1n · √

π · 2n

with

}

⌉ and is submodular. □

Analogously, we can use this construction to show that for every odd n there exists PM ′ ⊆ GSn with dimension that is 1 exponential in n. This follows by restricting GSn to the subset in which v is symmetric for all k ≤ n, k ̸ = n− and then using 2 n−1 a large independent set of J(n, 2 ). Please cite this article as: S. de Vries, U. Friedrich and S. Raach, Geometry of gross substitutes valuations, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.003.

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Fig. 1. The hypergraph on the upper right side is the graph which arises for the triple condition with S = ∅ while the hypergraph on the upper left arises for the triple condition with |S | = 1. Every hyperedge is marked by lines of one style. The graph below is the resulting graph J(4, 2) which connects all vertices that have at least one common hyperedge in the above hypergraphs.

3.3. Improving the bound We assume for simplicity that n is even in this section. We recapitulate the proof of the previous section: For fulfilling the triple condition, the bundles containing 2n elements interact only with bundles containing 2n − 1 and 2n + 1 elements. Generalizing this observation leads to the idea of setting v symmetric for every second l ≤ n, say for every odd l. Then, for S ⊂ N with |S | odd and a, b, c ∈ N \ S the triple condition reduces to the triples (vSc , vSb , vSa ). Analogously, for S ⊂ N with |S | even and a, b, c ∈ N \ S, the triple condition reduces to (vSab , vSac , vSbc ). We can use the result of the previous section to construct maximal independent sets Ik of the Johnson Graph J(n, k) for every even k. The values vS for the bundles S that are contained in these maximal independent sets Ik can be chosen freely within the restrictions of submodularity. This results

∑n

∑n

(n)

1 2 in at least k2=1 |I2k | ≥ k=1 n 2k unconstrained components. ∑n (n) k n−k We can bound the latter sum by making use of the binomial theorem (x + y)n = . Choosing x = 1 and k=0 k · x · y y = −1 yields

∑n

∑ 2n ( n )

∑ 2n (

, n n ( ) ( n ) ∑ ∑ ∑ 2n ( n ) 1 n−1 n n 1 2 2 n − 1. and therefore k=0 2k = k=1 2k−1 . As 2 = k=0 2k + k=1 2k−1 it follows directly that n k=1 2k = n · 2 ⋃⌊ 2n ⌋ Analogously to Vsym , we can construct a polyhedral cone with I := k=1 I2k by defining { ( ( 1 ) |S | ( 1 )n+2 ) Veven := v : 2N → R+ | v (S) = λ· 1 − −αS with 2 2 } λ ≥ 0, αS ∈ [0, 1] if S ∈ I and αS = 0 if S ∈ N \ I . 0=

k=0

(n) k

· 1k · (−1)n−k =

∑ 2n ( n )

∑ 2n (

k=0

2k

−

k=1

n 2k−1

)

)

This results in a submodular cone of dimension at least ⌈ 1n · 2n−1 − 1⌉ in GSn . We show in the next section how to further improve this result. Please cite this article as: S. de Vries, U. Friedrich and S. Raach, Geometry of gross substitutes valuations, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.003.

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4. A good bound for the cones describing GSn We start this section with an observation about sums of (GS) and additive valuations, ∑which we use to improve the lower bound on the cones’ dimensions. A valuation v : 2N → R+ is called additive if v (S) = i∈S v (i) for all S ⊆ N. Lemma 13. The sum of a (GS) valuation and an additive valuation is a (GS) valuation. Proof. Let n ∈ N, v : 2N → R+ be a (GS) valuation and f : 2N → R+ an additive valuation. For S ⊂ N and a, b, c ∈ N \ S it holds f (S ∪ ab) + f (S ∪ c) = f (S ∪ ac) + f (S ∪ b) = f (S ∪ bc) + f (S ∪ a) =: q, which yields for v + f the triple

( ) v (S ∪ ab) +v (S ∪ c) + q, v (S ∪ ac) +v (S ∪ b) + q, v (S ∪ bc) +v (S ∪ a) + q that has a double maximum if and only if

( ) v (S ∪ ab) + v (S ∪ c), v (S ∪ ac) + v (S ∪ b), v (S ∪ bc) + v (S ∪ a) has a double maximum. The claim follows since the sum of an additive and a submodular function is submodular. □ In the next section we determine the exact dimensions of GS2 to be 3 and the dimension of GS3 to be 6. Comparison to the bound from the previous section reveals that there is still some room for improvement. We improve the bound by recapitulating the idea of finding a system of sets A ⊂ 2N \ {∅}, so that the cardinality of A is large, but T ∈ A is not constrained by equations in any triple condition. Again, the triple condition is encoded in a condition on a hypergraph H = (X , E ) by interpreting every T ∈ 2N \ {∅} as a vertex. This time we define the set of hyperedges E as the set of those triples of vertices which appear in different terms of some triple in (1). Note the difference to the previous section, where we considered a reduced triple condition and the hyperedges linked exclusively sets of the same cardinality. Now, any hyperedge is constructed by choosing the index sets of one of the two summands of every entry of (1), e.g., (S ∪ ab, S ∪ ac , S ∪ a) for S ⊂ N , a, b, c ∈ N \ S). This results in 23 = 8 hyperedges for every pair (S , {abc }) with S ⊂ N and pairwise different a, b, c ∈ N \ S. Choosing the value of an arbitrary vertex T appropriately, we force all vertices which are connected to T by some hyperedge to be constrained by equations. We derive a graph G = (V , E) with V = 2N \ {∅} from the hypergraph H: Two vertices T1 , T2 are adjacent if they have a common hyperedge in H. The set of edges E is therefore given by E := {{A, B} | A, B ∈ V \ {N }: (|A| = |B| and |A ∩ B| = n − 1) or (|A| + 1 = |B| and A ⊂ B)}. Note that for two bundles that occur in the same term of (1) no edge is added. For calculating a good lower bound on the maximal independent set of G we use a coset argument. 1 · (2n − 1)⌉. Theorem 14. The graph G = (V , E) described above has an independent set of cardinality at least ⌈ n+ 1

Proof. For two bundles A, B of equal cardinality and connected by an edge, it holds that

∑ ∑ | i− i| < n i∈B

(3)

i∈A

since A and B coincide in all except one element. For two bundles A, B with A ⊂ B and c = |B \ A| it holds that

∑ ∑ ∑ ∑ | i− i| = | i+c− i| = c ≤ n i∈B

i∈A

i∈A

(4)

i∈A

as c ∈ N. Therefore two bundles of V connected by an edge have a difference smaller than n + 1 with respect to the sum of the elements contained in the bundles. We define f : V → {1, . . . , n + 1}, f (S) :=

∑

i

mod (n + 1).

i∈S

It follows from (3) and (4) that two elements connected by an edge cannot be in the same coset and, as a direct consequence, each coset is independent. Since all the cosets are disjoint and every S ∈ 2N \ {∅} belongs to a coset, f yields a disjoint 1 partition of 2N \ {∅}. Hence, at least one coset contains at least ⌈ n+ · (2n − 1)⌉ bundles and hence (V , E) has an independent 1 set I with at least that number of elements. □ Please cite this article as: S. de Vries, U. Friedrich and S. Raach, Geometry of gross substitutes valuations, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.003.

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1 · (2n − 1)⌉, analogously We are now ready to provide a polyhedral cone satisfying the triple condition with dimension ⌈ n+ 1 to the construction in the previous section. Exactly as in the proof of Theorem 12, this can be done by choosing the cones corresponding to the elements of the independent set I. However, it is possible to improve this bound further. We observe that for every n ∈ N the value v (N) can be chosen freely within the restrictions of submodularity. This follows directly by Theorem 5 since v (N) is not involved in any triple in (1) and hence v (N) only has to fulfill the submodularity conditions. Furthermore, at most one singleton bundle is contained in every coset. In contrast, it is a consequence of Lemma 13 that the values of all n singletons can be chosen arbitrarily: If some arbitrary gross substitutes valuation v : 2N → R+ is given and for all S such that i ∈ S for some fixed i ∈ N we add the same ϵ > 0 to v (S), then the resulting valuation function remains gross substitutes. Hence, there are n degrees of freedom for the values of the singletons. This leads to distributing just 2n − 2 − n elements to n + 1 disjoint cosets and choosing the mentioned values independently. Finally, we observe that for every k ∈ {2, . . . , n − 1} we can pick an arbitrary element with cardinality k not contained in the independent set I and vary its value by ϵ . All triple conditions remain fulfilled when adding this ϵ to the value of every set with cardinality k. Hence there are n − 2 additional degrees of freedom. 1 In total, this gives us at least ⌈ n+ · (2n − 2 − n)⌉ + n + (n − 2) + 1 values that can be chosen freely without violating any 1 triple condition. It remains to show that intersecting with the cone of submodular valuations does not reduce the dimension. We do this again by providing a subcone. 1 Theorem 15. There is an at least ⌈ n+ · (2n − 2 − n)⌉ + 2n − 1 dimensional cone contained in GSn . 1

Proof. Let G = (V , E) with V = {T ∈ 2N | |T | ∈ / {0, 1, N }} and E := {{A, B} | A, B ∈ V :(|A| = |B| and |A ∩ B| = n − 1) or (|A| + 1 = |B| and A ⊂ B)}. Let I be a maximal independent set of G, then

{

(

W := v : 2N → R+ | v (S) = λ· 1 −

( 1 )|S |

−αS

( 1 )n+2

+β|S |

( 1 )n+2

2 2 2 with λ ≥ 0, αS ∈ [0, 1] if S ∈ I and αS = 0 if S ∈ N \ I ,

+u

)

β|S | ∈ [0, 1] if |S | ∈ {2, . . . , n} and β|S | = 0 if |S | ∈ {0, 1}, } u: 2N → R+ additive valuation fulfills the statement. For v ∈ W , the submodularity is guaranteed by Theorem 6 and the triple condition (1) is satisfied by construction. □ The bound provided by Theorem 15 is best possible up to O(n) because the actual dimension cannot be greater than 2n − 1. 5. Comparing the bounds with the dimensions of the cones We abbreviate v ({1, 2}) with v (12) et cetera. For n = 1 and n = 2 our bound matches the dimension. In the case n = 2, it follows by Theorem 5 that the set of submodular valuations coincides with the set of gross substitutes valuations. For n = 3 it is provided by Theorem 5 that for being a (GS) valuation v : 2{1,2,3} \ {∅} → R+ has to be submodular and the triple

(

v (12) + v (3), v (23) + v (1), v (13) + v (2)

)

has to have a double maximum. Without loss of generality let

v (12) + v (3) = v (23) + v (1) ≥ v (13) + v (2). It is easy to verify that the dimension of the cone is 6. This coincides with the lower bound. We remark that GS3 consists of three isomorphic polyhedral cones. For n = 4 we have to consider a few more triples to check if a valuation v : 2{1,2,3,4} \ {∅} → R+ fulfills the (GS) condition. For S = ∅ the following triples must have a double maximum:

( ) v (23) + v (1), v (13) + v (2), v (12) + v (3) , ( ) v (13) + v (4), v (14) + v (3), v (34) + v (1) , ( ) v (24) + v (1), v (14) + v (2), v (12) + v (4) , ( ) v (23) + v (4), v (24) + v (3), v (34) + v (2) .

(5) (6) (7) (8)

Choosing S as singleton yields the following four triples:

( ) v (124) + v (13), v (134) + v (12), v (123) + v (14) ,

for S = {1},

(9)

Please cite this article as: S. de Vries, U. Friedrich and S. Raach, Geometry of gross substitutes valuations, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.003.

8

S. de Vries, U. Friedrich and S. Raach / Discrete Applied Mathematics xxx (xxxx) xxx

( ) v (234) + v (12), v (124) + v (23), v (123) + v (24) , ( ) v (234) + v (13), v (134) + v (23), v (123) + v (34) , ( ) v (234) + v (14), v (134) + v (24), v (124) + v (34) ,

for S = {2},

(10)

for S = {3},

(11)

for S = {4}.

(12)

In all triples we choose the first and the second term to be the double maximum. Considering the resulting (in)equalities and the submodularity inequalities, we calculated the resulting polyhedral dimension to be 10. Explicit calculations show that the dimensions of the other cones describing GS4 are not greater than 10, regardless of which terms we choose to be the double maximum. Our bound has a deficit of one here. In the triples (5)–(8) the bundles 12 and 34 are not involved in any equality. So we can choose their values within some bounds freely. The same occurs to the bundle 123. We can fulfill the double maximum in (9)–(12) without involving the bundle 123. On the contrary, the coset construction of the last section does not allow this. If for instance 12 and 34 are elements (4) of the maximal independent set I, no superset with cardinality 3 is allowed to be in I. It turns out that all elements of 3 are supersets of 12 or 34. 6. Conclusion We have interpreted the set of gross substitutes valuations as a set in a real vector space and discussed its geometry. It has been shown that this set is the union of finitely many polyhedral cones and we analyzed the dimension of these cones. We started with a simple bound that shows that these cones have dimension exponential in n. After improving this bound step by step, we obtained a new bound that is best possible within O(n). A consequence of our result is that an arbitrary gross substitutes valuation can generally not be encoded in space polynomial in n as the valuation may take in n exponentially many different values. An immediate open problem is given by determining the exact dimension of the cones of GSn for n ≥ 5. Acknowledgments The authors like to thank the anonymous reviewers for their very helpful comments. All authors were supported by the Research Training Group 2126 Algorithmic Optimization (ALOP), funded by the German Research Foundation DFG. The second author was supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research (BMBF). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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Please cite this article as: S. de Vries, U. Friedrich and S. Raach, Geometry of gross substitutes valuations, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.003.