The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities

Discrete Applied Mathematics xxx (xxxx) xxx

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The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities Koji Yokote Waseda Institute for Advanced Study, Waseda University, 1-6-1, Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan

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Article history: Received 11 May 2018 Received in revised form 1 August 2019 Accepted 28 August 2019 Available online xxxx Keywords: Discrete convex analysis Separation theorem Hall’s theorem Auction Walrasian taˆ tonnement

a b s t r a c t The separation theorem in discrete convex analysis states that two disjoint discrete convex sets can be separated by a hyperplane with a 0–1 normal vector. We apply this theorem to markets with heterogeneous commodities and uncover the mathematical structure behind price adjustment processes. When p is not an equilibrium price vector, i.e., when aggregate demand and aggregate supply are disjoint, the separation theorem indicates the existence of overdemanded/underdemanded items. This observation yields a generalization of Hall’s (1935) theorem and a characterization of equilibrium price vectors by Gul and Stacchetti (2000). Building on this characterization, we show that adjusting the prices of overdemanded/underdemanded items corresponds to Ausbel’s (2006) auction. We further extend our approach to markets with continuous commodities and uncover a striking connection between auctions and classical taˆ tonnement processes. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The separation theorem for disjoint convex sets is a powerful tool in theoretical economics. This theorem has been used to prove fundamental theorems, e.g., the Kuhn–Tucker theorem, the second welfare theorem in market theory, and the min–max theorem in game theory (see von Neumann and Morgenstern [16]). The discrete version of the separation theorem, called the discrete separation theorem, is introduced in discrete convex analysis [13], but has not yet been applied to economic models. The purpose of the present paper is to apply the discrete separation theorem to markets with heterogeneous commodities and uncover the mathematical structure behind price adjustment processes. We explain three contributions. First, we show that an equilibrium price vector in discrete markets is characterized by the notions of excess demand/supply. This result is derived based on a geometrical interpretation of a Walrasian equilibrium. In an auction model with the gross-substitutes condition, the aggregate demand of buyers forms a discrete convex set called the M♮ -convex set. Walrasian equilibrium can be described as a situation where the aggregate demand and aggregate supply intersect. Put differently, p is not an equilibrium price vector if and only if the aggregate demand and aggregate supply are disjoint. Applying the discrete separation theorem to the two sets, the separation normal vector turns out to describe overdemanded/underdemanded items. This observation yields a generalization of Hall’s [8] theorem and a characterization of equilibrium price vectors by Gul and Stacchetti [7]. Second, we connect the first result with auctions. Based on an economic intuition of auctions, we consider adjusting the prices of overdemanded/underdemanded items induced by the discrete separation theorem. It turns out that this process coincides with Ausubel’s [3] auction. Our result indicates that Ausubel’s [3] auction can be implemented using the information of the agent’s demand, which is easier to gather than that of their utility functions. Our result relies on a new E-mail address: [email protected]. https://doi.org/10.1016/j.dam.2019.08.022 0166-218X/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: K. Yokote, The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.022.

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Fig. 1. Comparison with existing results.

mathematical finding that the decreased amount of the Lyapunov function is equal to the overdemanded/underdemanded quantity. The above-mentioned results together imply that the price adjustment directions of Ausubel’s [3] auction coincide with the separation normal vectors. Our final result extends this finding to markets with continuous commodities. We prove that existing continuous-time price adjustment processes (known as taˆ tonnement processes; see Arrow and Hahn [2]) adjust prices in the direction of a separation normal vector induced from the standard separation theorem. This result uncovers a striking connection between classical taˆ tonnement processes and auctions. Related literature The extensive literature on iterative auctions has provided a variety of algorithms that find an equilibrium price vector. To formulate a price adjustment algorithm, it is essential to find overdemanded/underdemanded items from submitted demand information. In unit-demand auctions, Hall’s [8] theorem in graph theory has been a powerful tool for finding such items. Indeed, the Vickrey–English auction by Demange et al. [5] proceeds by increasing the prices of overdemanded items induced by Hall’s theorem.1 In a recent study, the English–Vickrey–Dutch auction by Andersson and Erlanson [1], which is a generalization of the Vickrey–English auction and the Vickrey–Dutch auction [11], adjusts the prices of overdemanded/underdemanded items induced by Hall’s theorem.2 However, no multi-demand counterparts to Hall’s theorem have been proposed to date. Although Gul and Stacchetti [7] formulate the notion of excess demand, their formulation is ‘‘one-sided’’ in the sense that excess supply is not discussed. Our first result remedies this deficiency. We characterize a Walrasian equilibrium in multi-demand settings as a situation in which no excess demand/supply exists. We compare our result in discrete markets with that of Murota et al. [15], who also apply discrete convex analysis to iterative auctions (see Fig. 1). Using the techniques in discrete convex analysis, they prove that Ausubel’s [3] auction is a generalization of the above-mentioned unit-demand auctions.3 However, they do not touch the underlying mathematical tool that induces overdemanded/underdemanded items. As represented by the solid lines in Fig. 1, this paper generalizes Hall’s theorem to the discrete separation theorem, from which we derive Ausubel’s auction. This study is part of the recent literature on an application of discrete convex analysis to economic models with indivisibility. Other applications include assignment markets [20], matching problems under constraints [10], and trading networks [4]. The rest of the paper is organized as follows. Section 2 presents basic concepts in discrete convex analysis. Section 3 applies the discrete separation theorem to discrete markets and presents the first two results. Section 4 extends the results in Section 3 to continuous markets and presents the third result. Section 5 concludes. 2. Discrete convex analysis Let R be the set of real numbers, R+ be the set of nonnegative real numbers including 0, Z be the set of integers, and Z+ be the set of nonnegative integers including 0. Let K be an arbitrary finite set. For x ∈ ZK , we define supp+ x = {k ∈ K : xk > 0}, supp− x = {k ∈ K : xk < 0}. 1 See also Section 8.3 of Roth and Sotomayor [18] for a detailed discussion on this auction. 2 Precisely speaking, the definitions of overdemanded/underdemanded items originate from the characterization of equilibrium price vectors by Mishra and Talman [12]. Their characterization is proved by using Hall’s theorem. 3 In addition to providing this generalization result, Murota et al. [15] analyze the number of iterations in iterative auctions. Please cite this article as: K. Yokote, The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.022.

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Fig. 2. Discrete separation theorem for K = {k, ℓ}.

For each A ⊆ K , let 1lA ∈ {0, 1}K denote the characteristic vector of A, i.e., 1lAk

1

if k ∈ A,

0

otherwise.

{ =

For a singleton set {k} ⊆ K , we write 1lk for 1l{k} . We say that a function v : {0, 1}K → R is an M♮ -concave function if, for any x, y ∈ {0, 1}K and k ∈ supp+ (x − y), we have (i) v (x) + v (y) ≤ v (x − 1lk ) + v (y + 1lk ), or (ii) there exists ℓ ∈ supp− (x − y) such that v (x) + v (y) ≤ v (x − 1lk + 1lℓ ) + v (y + 1lk − 1lℓ ). For an interpretation of M♮ -concavity, see Section 3 of Kojima et al. [10]. We say that X ⊆ ZK with X ̸ = ∅ is an M♮ -convex set if, for any x, y ∈ X and k ∈ supp+ (x − y), we have (i) x − 1lk ∈ X , y + 1lk ∈ X , or (ii) there exists ℓ ∈ supp− (x − y) such that x − 1lk + 1lℓ ∈ X , y + 1lk − 1lℓ ∈ X . The following is a discrete analogue of the separation theorem. Theorem 1.

Let X1 , X2 ⊆ ZK be M♮ -convex sets. If X1 ∩ X2 = ∅, there exists α ∈ {0, 1}K ∪ {0, −1}K such that

sup α · x < inf α · x. x∈X1

x∈X2

Proof. See Appendix A. □ Remark 1. The separation theorem in discrete settings is already proved for two sets satisfying M-convexity, which is stronger than (but essentially equivalent to) M♮ -convexity (see Murota [13], Theorem 4.21). Theorem 1 immediately follows from the polyhedral description of the convex hull of an M♮ -convex set (known as the g-polymatroid). In Appendix A, we provide a formal proof for its fundamental importance. An earlier version [21] contains another proof which does not rely on the polyhedral description. ■ As is the case for continuous settings, this theorem states that two ‘‘convex’’ sets can be separated by a hyperplane. The key difference is that the normal vector α can be taken as a characteristic vector (i.e., a vector with 0–1 coordinates). Fig. 2 shows an example of the discrete separation theorem for K = {k, ℓ}. Note that each ‘‘edge’’ of X1 and X2 is parallel to 1lk − 1lℓ or 1lk or 1lℓ .4 The normal vector of the separating hyperplane is taken as α = (1, 1). For two sets X1 , X2 ⊆ ZK , we define the Minkowski sum X1 + X2 by X1 + X2 = {x1 + x2 : x1 ∈ X1 , x2 ∈ X2 }. 4 An M♮ -convex set in a general n-dimensional space can be characterized in terms of the direction of edges; see Murota [13], p. 119. Please cite this article as: K. Yokote, The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.022.

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3. Application to discrete markets We show that the discrete separation theorem is a fundamental mathematical tool in analyzing auctions. Let N be a finite set of agents and K be a finite set of items. Each agent i has a valuation function vi : {0, 1}K → Z; we identify a subset of items A ⊆ K with 1lA . For each i ∈ N, we define the demand correspondence Di : RK+ → {0, 1}K by Di (p) = x ∈ {0, 1}K : vi [p](x) ≥ vi [p](y) for all y ∈ {0, 1}K

{

}

for all p ∈ RK+ ,

where vi [p](x) = vi (x) − p · x. We say that vi (·) is monotonic if for any A, B ⊆ K with A ⊆ B, we have vi (1lA ) ≤ vi (1lB ). We say that vi (·) satisfies the gross substitutes condition (Kelso and Crawford [9]) if for any p, q ∈ RK+ with p ≤ q and x ∈ Di (p), there exists y ∈ Di (q) such that xk ≤ yk if pk = qk . Throughout this section, we assume that vi (·) is monotonic and satisfies the gross substitutes condition for all i ∈ N. As proven by Fujishige and Yang [6], under monotonicity of vi (·), the gross substitutes condition is equivalent to M♮ -concavity. Hence, vi (·) is M♮ -concave for all i ∈ N. An allocation is a set of bundles (xi )i∈N satisfying xi ∈ {0, 1}K for all i ∈ N ,

∑

xi = 1lK ,

i∈N K

where 1l is intended to mean that each ( item has ) a single unit. A Walrasian equilibrium is a pair p∗ , (x∗i )i∈N , where p∗ ∈ RK+ , x∗ is an allocation, and x∗i ∈ (Di (p∗ ) for all ) i ∈ N. We say that p∗ is a (Walrasian) equilibrium price vector if there exists an allocation (x∗i )i∈N such that p∗ , (x∗i )i∈N is a Walrasian equilibrium. For each p ∈ RK+ , we define the aggregate demand D(p) by D(p) =

∑

Di (p).

i∈N

The following lemma immediately follows from the definition of an equilibrium price vector. Lemma 1.

A price vector p ∈ RK+ is an equilibrium price vector if and only if 1lK ∈ D(p).

Equivalently, p is not an equilibrium price vector if and only if 1lK ∈ / D(p).

(1)

It is known that M♮ -concavity of vi (·) implies M♮ -convexity of Di (p) for any p ∈ RK+ . (see Murota [13], Theorem 6.30). Since M♮ -convexity is preserved under the Minkowski sum (see Murota [13], Theorem 4.23), D(p) is an M♮ -convex set for any p ∈ RK+ . Regarding 1lK as a singleton set, {1lK } is an M♮ -convex set. Hence (1) refers to two disjoint M♮ -convex sets. By Theorem 1, there exists α ∈ {0, 1}K ∪ {0, −1}K such that

α · 1lK < min α · x.

(2)

x∈D(p)

Assume α ∈ {0, 1}K and let A ⊆ K be such that α = 1lA . Then (2) is equivalent to

|A| < =

x∈

∑ i∈N

=

∑

∑min

∑ i∈N

i∈N Di (p)

min

xi ∈Di (p)

xk

k∈A

∑

(xi )k

k∈A

min ⏐{k ∈ A : (xi )k = 1}⏐.

⏐

⏐

(3)

xi ∈Di (p)

For each i ∈ N, Gul and Stacchetti [7] called the above minimum value the requirement function and interpreted it as follows: ‘‘the minimal number of objects in A that she would need to construct any of her optimal consumption bundles’’. In the above inequality, the sum of the minimal values among agents is greater than the number of items in A, which means excess demand. When α ∈ {0, −1}K , letting A ⊆ K be such that α = −1lA , (2) is equivalent to

|A| >

∑ i∈N

max ⏐{k ∈ A : (xi )k = 1}⏐,

⏐

⏐

xi ∈Di (p)

which means excess supply. For p ∈ RK+ and A ⊆ K , we define Rmin (p, A) = min ⏐{k ∈ A : (xi )k = 1}⏐, Rmin (p, A) = i

⏐

xi ∈Di (p)

⏐

∑

Rmin (A, p), i

i∈N

Please cite this article as: K. Yokote, The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.022.

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∑ ⏐ ⏐ Rmax (p, A) = max ⏐{k ∈ A : (xi )k = 1}⏐, Rmax (p, A) = Rmin (A, p). i i xi ∈Di (p)

i∈N

We summarize the above discussion as a theorem. Theorem 2.

5

p ∈ RK+ is an equilibrium price vector if and only if

Rmin (p, A) ≤ |A| ≤ Rmax (p, A) for all A ⊆ K . This theorem has an intuitive interpretation that p is an equilibrium price vector if and only if overdemanded/ underdemanded items do not exist. The above argument enables a refined analysis of a characterization of equilibrium price vectors by Gul and Stacchetti [7]. Indeed, their characterization immediately follows from the discrete separation theorem. Let [0, 1lK ] denote the integer interval between 0 and 1lK , i.e., [0, 1lK ] = {x ∈ ZK : 0 ≤ x ≤ 1lK }. We say that p ∈ RK+ is a quasi equilibrium price vector if [0, 1lK ] ∩ D(p) ̸ = ∅. Namely, p is a quasi equilibrium price vector if all the items are consumed by at most one agent. Corollary 1 (Gul and Stacchetti [7], Corollary). p ∈ RK+ is a quasi equilibrium price vector if and only if Rmin (p, A) ≤ |A| for all A ⊆ K . Proof. The only if part immediately follows from the definition of a quasi equilibrium price vector. We prove the contrapositive of the if part. If p is not a quasi equilibrium price vector, then [0, 1lk ] ∩ D(p) = ∅. Since any integer interval is an M♮ -convex set, by Theorem 1, there exists α ∈ {0, 1}K ∪ {0, −1}K such that max α · x < min α · x.

(4)

x∈D(p)

x∈[0,1lK ]

If α ∈ {0, −1}K , then we derive max α · x = 0 ≥ min α · x, a contradiction to (4). Hence, α ∈ {0, 1}K . Since x∈D(p)

x∈[0,1lK ]

max α · x = α · 1l , by choosing A ⊆ K with α = 1lA , the same transformation as (3) yields the desired condition. □ K

x∈[0,1lK ]

Remark 2. As discussed by Gul and Stacchetti [7], Corollary 1 is a generalization of Hall’s [8] theorem. Hence, Hall’s [8] theorem can also be proved by using the discrete separation theorem. ■ Next, we connect Theorem 2 with auctions. For each i ∈ N, we define the indirect utility function Vi : RK+ → R by Vi (p) = max vi [p](x) for all p ∈ RK+ x∈{0,1}K

Ausubel [3] proves that p is an equilibrium price vector if and only if p is a minimizer of the Lyapunov function L : ZK+ → Z defined by6 L(p) =

∑

Vi (p) + p · 1lK for all p ∈ ZK+ .

i∈N

Starting from an arbitrary price vector p, Ausubel’s [3] auction adjusts p in a way to decrease the value of L(·). Together with the economic interpretation of auctions, decreasing the value of L(·) seems to be related to increasing (res. decreasing) the prices of overdemanded (res. underdemanded) items. The following theorem supports this guess and states an even stronger result: the decreased amount of L(·) is equal to the overdemanded/underdemanded quantity. Theorem 3.

Let p ∈ ZK+ and A ⊆ K . Then,

(i) L(p + 1lA ) − L(p) = |A| − Rmin (A, p). In particular, L(p) > L(p + 1lA ) if and only if Rmin (A, p) > |A|. (ii) L(p − 1lA ) − L(p) = Rmax (A, p) − |A|. In particular, L(p) > L(p − 1lA ) if and only if |A| > Rmax (A, p). To prove this theorem, we introduce some preliminaries. For i ∈ N and a ∈ Z+ , we define Dai (p) = x ∈ {0, 1}K : vi [p](x) ≥ Vi (p) − a .

{

}

Note that D0i (p) = Di (p). 5 The above discussion proves the contrapositive of the if part of Theorem 2. The only if part immediately follows from the definition of an equilibrium price vector. 6 Under the assumption that v (·) are integer-valued, the existence of an integer equilibrium price vector is guaranteed. Hence we restrict our i

attention to the integer domain ZK+ . Note that L(p) is integer-valued if p is an integer vector and vi (·) are integer-valued.

Please cite this article as: K. Yokote, The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.022.

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Claim 1.

Let i ∈ N. Then for any a ∈ Z+ , we have

(A, p) − a. x ∈ Dai (p) implies x · 1lA ≥ Rmin i Proof. For each a ∈ Z+ , set (A, p) − a}. E a = {x ∈ {0, 1}K : x ∈ Dai (p) and x · 1lA < Rmin i It suffices to prove that E a = ∅ for all a ∈ Z+ . We proceed by induction on a. If a = 0, the result follows from the definition of Rmin (A, p). Suppose that the claim holds for a ≥ 0 and we prove the claim for a + 1. i Suppose by way of contradiction that there exists x ∈ E a+1 . Let x¯ ∈ Di (p) be such that |supp+ (x¯ − x) ∩ A| ≤ |supp+ (y − x) ∩ A| for all y ∈ Di (p). By x¯ · 1lA ≥ Rmin (A, p), it holds that x¯ · 1lA > x · 1lA . This means that there exists i + ¯ k ∈ supp (x − x) ∩ A. By Theorem 6.15 of Murota [13], vi [p](·) is an M♮ -concave function. With the notation 1l0 = (0, . . . , 0), by M♮ -concavity, there exists ℓ ∈ supp− (x¯ − x) ∪ {0} such that

vi [p](x¯ ) + vi [p](x) ≤ vi [p](x¯ − 1lk + 1lℓ ) + vi [p](x + 1lk − 1lℓ ). ℓ

(5) ℓ

Since (x + 1l − 1l ) · 1l ≤ x · 1l + 1 < , p) − a, by the induction hypothesis, x + 1l − 1l ∈ / Dai (p). Together with x ∈ Dai +1 (p) and the fact that vi [p](·) takes integer values, we obtain vi [p](x) ≥ vi [p](x + 1lk − 1lℓ ). This inequality and (5) imply vi [p](x¯ ) ≤ vi [p](x¯ − 1lk + 1lℓ ). Namely, k

A

A

Rmin (A i

k

x¯ − 1lk + 1lℓ ∈ Di (p) and |supp+ ((x¯ − 1lk + 1lℓ ) − x) ∩ A| = |supp+ (x¯ − x) ∩ A| − 1, a contradiction to the choice of x¯ .

□

Proof of Theorem 3. Let A ⊆ K and p ∈ ZK+ . We prove (i) and omit the proof of (ii) which can be obtained analogously. For each i ∈ N, let x¯ i ∈ Di (p) be such that x¯ i · 1lA = Rmin (A, p). Then, for any i ∈ N and xi ∈ {0, 1}K , by letting i a(xi ) := Vi (p) − vi [p](xi ),

vi [p + 1lA ](xi ) = vi [p](xi ) − xi · 1lA = Vi (p) − a(xi ) − xi · 1lA ≤ Vi (p) − a(xi ) − Rmin (A, p) + a(xi ) i = Vi (p) − Rmin (A, p) i = vi [p](x¯ i ) − x¯ i · 1lA = vi [p + 1lA ](x¯ i ), where the inequality follows from Claim 1. This inequality implies x¯ i ∈ Di (p + 1lA ) for all i ∈ N .

(6)

We obtain L(p + 1lA ) =

∑

=

∑

Vi (p + 1lA ) + p · 1lK + |A|

i∈N

=

vi [p + 1lA ](x¯ i ) + p · 1lK + |A|

i∈N ∑{

} vi [p](x¯ i ) − x¯ i · 1lA + p · 1lK + |A|

i∈N

=

∑

Vi (p) −

∑

i∈N

= L(p) + |A| −

Rmin (A, p) + p · 1lK + |A| i

i∈N

∑

Rmin (A, p), i

i∈N

where the second equality follows from (6). □ As a byproduct of Theorem 3, we relate the descent direction of the Lyapunov function with the set in excess demand7 and prove their lattice structure. For each p ∈ ZK+ , we define O(p) = {A ⊆ K : A is a minimizer of |A| − Rmin (A, p)},

˜ O(p) = {A ⊆ K : A is a minimizer of L(p + 1lA ) − L(p)}.

7 This relationship is already known for unit-demand settings (see Lemma 5.6 of Murota et al. [15]). Here we generalize it to multi-demand settings. Please cite this article as: K. Yokote, The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.022.

K. Yokote / Discrete Applied Mathematics xxx (xxxx) xxx

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Corollary 2. For any p ∈ ZK+ , we have O(p) = ˜ O(p). Moreover, each of these sets forms a lattice. Proof. The first part immediately follows from Theorem 3(i). As proven in Section 8 of Murota [14], L(·) satisfies a notion of discrete convexity called L♮ -convexity. L♮ -convexity implies that ˜ O(p) is a lattice, which proves the latter part of the statement. □ Remark 3. Gul and Stacchetti [7] prove that O(p) is a lattice for any p ∈ ZK+ . The latter part of Corollary 2 provides a shorter proof of this result by relating the set to L♮ -convexity. We derive Corollary 2 from Theorem 3(i). Analogously we can derive a corollary from Theorem 3 (ii). ■ 4. Extension to continuous markets In this section we extend the approach in Section 3 to markets with continuous commodities. Let N and K denote the set of agents and commodities. We assume that the commodity space is continuous, i.e., each agent i consumes a point in [0, 1]K . We assume that i’s valuation function vi : [0, 1]K → R is monotonic,8 continuous and concave. We define i’s demand correspondence Di : RK+ → [0, 1]K and i’s indirect utility function Vi : RK+ → R in the same way as in Section 3. Under the concavity assumption on vi (·), by the conjugacy argument, Vi (·) is convex. Allocations and Walrasian equilibria are defined analogously, where the aggregate supply is represented by 1lK . We remark that a Walrasian equilibrium always exists ∑ (see Ausubel [3]). For each p ∈ RK+ , let D(p) = i∈N Di (p). The following is a continuous analogue of Lemma 1. Lemma 2.

A price vector p ∈ RK+ is an equilibrium price vector if and only if 1lK ∈ D(p).

Since vi (·) are concave, D(p) is convex for all p ∈ RK+ . By the standard separation theorem in continuous settings, there exists α ∈ RK such that9

α · 1lK < min α · x = x∈D(p)

x∈

∑min

i∈N Di (p)

∑

α·x=

min α · xi .

xi ∈Di (p)

i∈N

(7)

Remark 4. Regarding the separation normal vector α , the following claim holds: Claim 2. Let p ∈ RK+ . Then, there exists α ∈ RK that satisfies (7) and the following condition: for any k ∈ K , pk = 0 implies αk ≥ 0.

(8)

Proof. See Appendix B. □ By Claim 2, for any p ∈ RK+ , we can confine our attention to separation normal vectors α ∈ RK satisfying (8). In what follows, when we refer to two vectors α ∈ RK and p ∈ RK+ , we implicitly assume that they satisfy (8). ■ For each i ∈ N, α ∈ RK and p ∈ RK+ , we define Rˆ min (α, p) = min α · xi , Rˆ min (α, p) = i xi ∈Di (p)

∑

Rˆ min (α, p). i

i∈N

We define the Lyapunov function L : RK+ → R by L(p) =

∑

Vi (p) + p · 1lK for all p ∈ RK+ .

i∈N

The notion of a Lyapunov function was originally introduced in the literature on dynamical systems. Consider a continuous-time price adjustment process given by a differential equation p˙ (t) = z(p(t)) for some function z(·).10 A remarkable result is that, if a function satisfying certain properties (which is called a Lyapunov function) exists, then the process converges to an equilibrium price vector; see Varian [19], p. 102 for a more precise discussion on this point. This mathematical result has been utilized by economists who aim to theoretically analyze a price adjustment process in continuous markets. The process is called a taˆ tonnement (see Arrow and Hahn [2], p. 264 for its formal definition). Previous studies find a suitable Lyapunov function in economic models and interpret a taˆ tonnement as a minimization process of this function.11 8 v (·) is monotonic if v (x) ≤ v (y) whenever x ≤ y. i i i 9 The minimum values in the transformation below always exist because D (p) are compact. i 10 In a market model, z(·) typically represents excess demands. 11 For example, Arrow and Hahn [2] consider an economy consisting of a single household, where a Lyapunov function is given by V (·), and write: ‘‘What the auctioneer is doing in this example is following an iterative procedure of minimizing V (p). He does this by making sure that he is always changing p in such a way as to be going ‘downhill’ · · ·’’ (Arrow and Hahn [2], p. 277). Ausubel’s [3] auction for continuous markets also proceeds by minimizing a Lyapunov function (see the proof of Lemma 3). Please cite this article as: K. Yokote, The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.022.

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We are in a position to state our final result which, together with a parallel result in discrete markets (Theorem 3), connects auctions and taˆ tonnement processes. They both adjust prices in the direction of a separation normal vector. Theorem 4.

Let α ∈ RK and p ∈ RK+ . Then, there exists ε > 0 such that L(p) > L(p + εα ) if and only if Rˆ min (α, p) > α · 1lK .

To prove this theorem, we introduce some preliminaries. Let α ∈ RK and p ∈ RK+ . We define the directional derivative of Vi (·) at p in the direction α by12 Vi′ (p; α ) = lim

Vi (p + εα ) − Vi (p)

.

ε

ε↓0

This limit always exists and is finite as Vi (·) is convex (see Rockafellar [17], Theorem 23.1). We define the subdifferential of Vi (·) at p by

∂ Vi (p) = {x ∈ RK : Vi (q) − Vi (p) ≥ x · (q − p) for all q ∈ RK }. For X ⊆ RK , we define −X = {x ∈ RK : −x ∈ X }. Proof of Theorem 4. If: Suppose Rˆ min (α, p) > α · 1lK . Then,

∑ i∈N

min α · xi > α · 1lK ⇐⇒

∑

⇐⇒

∑

⇐⇒

∑

xi ∈Di (p)

i∈N

i∈N

i∈N

⇐⇒

∑ i∈N

⇐⇒

∑

⇐⇒

∑

min α · (−xi ) > α · 1lK

xi ∈−Di (p)

− max α · xi > α · 1lK xi ∈−Di (p)

max α · xi < −α · 1lK

xi ∈−Di (p)

max α · xi < −α · 1lK

xi ∈∂ Vi (p)

Vi′ (p; α ) < −α · 1lK

i∈N

Vi′ (p; α ) + α · 1lK < 0

i∈N

⇐⇒ L′ (p; α ) < 0, where the fourth equivalence follows from ∂ Vi (p) = −Di (p) (see Rockafellar [17], Theorem 23.5) and the fifth equivalence follows from Vi′ (p; α ) = max{α · x : x ∈ ∂ Vi (p)} (see Rockafellar [17], Theorem 23.4). The last inequality implies the desired condition. Only if: We prove the contrapositive. Suppose there exist xi ∈ Di (p) for i ∈ N such that

α · 1lK ≥ α ·

∑

xi .

(9)

i∈N

Then, for any ε > 0, L(p) =

∑

Vi (p) + p · 1lK

i∈N

=

∑ {vi (xi ) − p · xi } + p · 1lK

≤

∑

i∈N

vi (xi ) − p ·

i∈N

∑

(

xi + p · 1lK + ε α · 1lK − α ·

i∈N

=

∑

=

∑{

≤

∑

vi (xi ) − (p + εα ) ·

i∈N

∑ ) xi

i∈N

∑

xi + (p + εα ) · 1lK

i∈N

} vi (xi ) − (p + εα ) · xi + (p + εα ) · 1lK

i∈N

Vi (p + εα ) + (p + εα ) · 1lK

i∈N

= L(p + εα ), 12 We implicitly assume that ε is sufficiently small so that p + εα ∈ RK . This is possible as α and p satisfy (8) (see Remark 4). + Please cite this article as: K. Yokote, The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.022.

K. Yokote / Discrete Applied Mathematics xxx (xxxx) xxx

9

where the first inequality follows from (9) and the second inequality follows from the definition of the indirect utility function. Since ε > 0 is arbitrarily chosen, the proof is completed. □ 5. Conclusion In the economics literature, discrete and continuous markets have been studied using different mathematical tools. With the aid of discrete convex analysis, we have succeeded in analyzing them in a parallel manner. The key difference between the two markets is that, in discrete markets, the separation normal vectors reduce to 0–1 vectors. Hence the price adjustment direction is clear, and a finite-step algorithm can be formulated. This explains why finding an equilibrium price vector is easy in discrete markets compared to continuous markets. Acknowledgments The author is grateful to the four referees of the journal, as well as Yukihiko Funaki, Yoko Kawada, Fuhito Kojima, Kazuo Murota, Yuta Nakamura, and Noriaki Okamoto for their valuable comments. This work was supported by JSPS, Japan Grant-in-Aid for Research Activity Start-up (Grant number: 17H07179) and Waseda University, Japan Grant for Special Research Projects (Project number: 2017S-006). Appendix A We prove Theorem 1. To this end, we prove two claims. Claim 3.

Let X ⊆ ZK be an M♮ -convex set and x ∈ / X . Then, there exists α ∈ {0, 1}K ∪ {0, −1}K such that

α · x < inf α · y.

(10)

y∈X

Proof. Let X¯ denote the convex hull of X . By (4.34) of Murota [13] (known as convexity in intersection), x ∈ / X¯ . By (4.36) of Murota [13], there exist a pair (ρ, µ) of a submodular function ρ : 2K → Z ∪ {+∞} and a supermodular function µ : 2K → Z ∪ {+∞} such that13 X¯ = {x ∈ RK : µ(A) ≤ x · 1lA ≤ ρ (X ) for all A ⊆ K }. By x ∈ / X¯ , (i) there exists A ⊆ K such that µ(A) > x · 1lA , or (ii) there exists A ⊆ K such that ρ (X ) < x · 1lA . By setting α = 1lA in the former case and α = −1lA in the latter case, (10) holds. □ Claim 4.

Let X ⊆ ZK be an M♮ -convex set. Then, −X is also an M♮ -convex set.

Proof. This claim immediately follows from the definition of M♮ -convexity. □ Proof of Theorem 1. As mentioned in Section 3, M♮ -convexity is preserved under the Minkowski sum (see Murota [13], Theorem 4.23). Together with Claim 4, −X1 + X2 is an M♮ -convex set. By X1 ∩ X2 = ∅, 0 ∈ / −X1 + X2 . Applying Claim 3, there exists α ∈ {0, 1}K ∪ {0, −1}K such that 0<

inf

x∈−X1 +X2

α · x,

0 < inf α · (−x1 ) + inf α · x2 , x1 ∈X1

x2 ∈X2

0 < − sup α · x1 + inf α · x2 , x2 ∈X2

x1 ∈X1

sup α · x1 < inf α · x2 , x2 ∈X2

x1 ∈X1

which completes the proof. □ Appendix B We prove Claim 2. Let p ∈ RK+ and we define A := α ∈ RK : α · 1lK < min α · x .

}

{

x∈D(p)

13 µ and ρ satisfy a system of inequalities; see (4.37) of Murota [13]. Please cite this article as: K. Yokote, The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.022.

10

K. Yokote / Discrete Applied Mathematics xxx (xxxx) xxx

By the separation theorem, A ̸ = ∅. For each α ∈ A, we define I(α ) := |{k ∈ K : pk = 0 and αk < 0}|. Let α¯ ∈ A be such that I(α¯ ) ≤ I(α ) for all α ∈ A. It suffices to prove that I(α¯ ) = 0. Suppose to the contrary that I(α¯ ) ≥ 1. Then, there exists ℓ ∈ K such that pℓ = 0 and ′ α¯ ℓ < 0. We define α¯∑ ∈ RK by α¯ ℓ′ = 0 and α¯ k′ = α¯ k for all k ̸= ℓ. Since D(p) is compact, there exists yi ∈ Di (p) for each i ∈ N such that α¯ ′ · i∈N yi = minx∈D(p) α¯ ′ · x. For each i ∈ N, we define y′i ∈ [0, 1]K by (y′i )ℓ = 1 and (y′i )k = (yi )k for all k ̸ = ℓ. By monotonicity of vi (·) and pℓ = 0, we have y′i ∈ Di (p) for all i ∈ N. Hence, min α¯ · x ≤ α¯ ·

x∈D(p)

∑ i∈N

y′i = α¯ ′ ·

∑

y′i + |N | · α¯ ℓ = α¯ ′ ·

∑

i∈N

yi + |N | · α¯ ℓ = min α¯ ′ · x + |N | · α¯ ℓ .

i∈N

x∈D(p)

Then, α¯ ℓ < 0 and the above inequality imply

α¯ · 1lK − α¯ ′ · 1lK = α¯ ℓ ≥ |N | · α¯ ℓ ≥ min α¯ · x − min α¯ ′ · x. x∈D(p)

x∈D(p)

(11)

Since α¯ ∈ A, we have

α¯ · 1lK < min α¯ · x. x∈D(p)

(12)

By (11) and (12), we obtain α¯ ′ · 1lK < minx∈D(p) α¯ ′ · x. Namely, α¯ ′ ∈ A and I(α¯ ′ ) = I(α¯ ) − 1, a contradiction to the choice of

α¯ . □

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Please cite this article as: K. Yokote, The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.022.