Shapley-Folkman-Lyapunov theorem and Asymmetric First price auctions

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Abstract

In this paper non-convexity in economics has been revisited. Shapley-Folkman-Lyapunov theorem has been tested with the asymmetric auctions where bidders follow log-concave probability distributions (non-convex preferences). Ten standard statistical distributions have been used to describe the bidders’ behavior. In principle what is been tested is that equilibrium price can be achieved where the sum of large number non-convex sets is convex (approximately), so that optimization is possible. Convexity is thus very important in economics.

Abstract

In this paper non-convexity in economics has been revisited. Shapley-Folkman-Lyapunov theorem has been tested with the asymmetric auctions where bidders follow log-concave probability distributions (non-convex preferences). Ten standard statistical distributions have been used to describe the bidders’ behavior. In principle what is been tested is that equilibrium price can be achieved where the sum of large number non-convex sets is convex (approximately), so that optimization is possible. Convexity is thus very important in economics.

1 Introduction

The Shapley-Folkman theorem places an upper bound on the size of the non-convexities (openings or holes) in sum of non-convex sets in Euclidean, n− dimensional space as a function of the size on non-convexities in the sets summed and the dimension of the space, as explanation given in(Starr 2016). The bound is based on the size of non-convexities in the sets summed and the dimension of the space. When the number of sets in the sum is large, then the bound is independent of the number of sets summed and is depending on n i.e. dimension of the space. Than the size of non-convexity in the sum becomes relatively small as a proportion of the sets summed, the non-convexity becomes zero as the number of demands becomes large as the summands become large. This theorem is used to demonstrate the following properties: First, the existence of competitive equilibrium in large finite economies (Second fundamental theorem also applies with finite agents or finite time periods) with non-convex preferences. This is done by increasing marginal rate of substitution, that is by increasing the size of the proportion of the prices of at least two goods, or one indivisible good, discrete goods that can be traded only as a whole with at least two bidders. And second, this theorem is been used to demonstrate convergence to the set of competitive equilibria (Arrow and Hahn 1971). Further, the Shapley-Folkman lemma provides that the sum of many sets is close to being convex. In this regard, the Shapley- Folkman theorem states a bound on the distance between the Minkowski sum and its convex hull; this distance is zero if and only if the sum is convex. A set of points is said to span a point, if a point can be expressed as a convex combination (weighted average) of elements of the set of points. In case of a set of concavity of preferences, then some prices support budget line that supports two optimal baskets of goods, and then the demand for goods is disconnected. So as n− dimensions increases to infinity, 1/n times sum of sets goes to zero. In optimization this produces an upper bound on the duality gap of separable non-convex optimization problems that involve finite sums (Aubin and Ekeland 1976).

To prove this theorem, we apply practical approach by using Asymmetric auctions, where players are randomly drawn to have different distributions and values. In this case bidder have log-concave distribution functions. The results prove that under some circumstances equilibrium price is achieved, though that equilibrium may be inefficient (high price) and non-existent (if convergence is not achieved). There exists literature in the subject of asymmetric auctions (Maskin and Riley 2000, Fibich and Gavious 2003, Guth, Ivanova-Stenzel et al. 2005, Gayle and Richard 2008, Hubbard and Paarsch 2009, Fibich and Gavish 2011, Hubbard, Kirkegaard et al. 2013).

2 The Shapley-Folkman theorem

Shapley- Folkman theorem (Starr 1969), in economics is used to extend Minkowski sums of convex sets to sums of general sets, which need not be convex (Skiena 2009):

(AB)={a+b:aA,bB}

Where A and B are sets of location vectors or radius vectors. Shapley- Folkman theorem may be represented this way:

Theorem 1

Lets suppose that xcon(A1 + ⋯ + AI), where ARL, So we can write x = a1 + ⋯ + ai, where aicon Ai, ∀ i, aiAi, ∀ L, ⊊, ∀ i .a

Previous theorem originates from Carathéodory’s Fundamental Theorem (Eckhoff 1993): Each point in the convex hull of a set S in Rn is in the convex combination of n + 1 or fewer points of S. Convex hullb the is given as:

con{j=1Nλjpj:λj0j,j=1Nλj=1}

Suppose that xcon A, where ARL, ∃ a1 …, an+1A, ∈ con(a1, …, an+1). More convenient approach to the statement of this theory is presented by Khan and Rath (Khan and Rath 2013):

Let ℝL be an L dimensional Euclidean space, then let Ai denote the its convex hull for any ARn, now ∅ ≠ AiRn and xcon i=1n Ai, then i=1n xi = x, and xcon Ai, ∀ i, xiAi, for at least nm indices of i.

Let Ψ = ∅c, and then a, b ∈ Ψ → ab ∈ Ψ, a, b ∈ Ψ ⇒ a\b ∈ Ψ. d This contains logical values (Boolean Algebra), and one can define finitely additive measure μ : Ψ → RL, i.e. μ(ab) = μ(a) + μ(b), whenever a, b ∈ Ψ, ab = ∅. Partially ordered subset of partially ordered set i.e. supremum of μ is given as : sup{|μi (a)| : a ∈ Ψ} < ∞, 1 ≤ im e, if m = 1 it is called finitely additive scalar measure. And we can define: |μ| = μ+ + μ. A field Ψ is an F − algebra, i.e. F : Ψ → Ψ is an endofunctor of category Ψ, then an F − algebra is a tuple (A, a), where A is an object of Ψ and a is isomorphism F(A) → A. Ψ also is a F − algebra if for any increasing sequence {An} and any decreasing sequence {Bn}, and An, Bn ∈ Ψ, AnBn, ∀ n, ∃ C ∈ Ψ, AnCBnn (Seever 1968, Armstrong and Prikry 1981). Ψ also, is a σ -algebra if n=1 An ∈ Ψ, An ∈ Ψ, in such a case Ψ, (T, Ψ) is a measurable space. Every F − algebra is σ-algebra: if Ψ is a σ-algebra and if there is increasing sequence {An} and decreasing sequence {Bn} a decreasing sequence both in Ψ and AnBn, ∀ n, then n=1 An or n=1 Bn can be in role of C. Now if so, μ : Ψ → RL is an countably additive measure if μ(∅) = 0, and μ(n=1 An) = i=1n μ(An), when {An} is a sequence of disjoint pairs of sets in Ψ. The measure μ purely atomic if there is scalar measure λ such that λμ, and if λ(A) = 0, for every measurable set for which |μ|(A) = 0. And if there is a sequence {Ek} such that T = k=1 Ek, ∀ Ek ∈ ∀ μi, i = 1, …, m. This is called an atomic or only positive measure (Bogachev 2007, Aliprantis and Border 2013, Halmos 2013, Hewitt and Stromberg 2013). Proof of the theorem was most simple in the one provided in (Zhou 1993):

Proof

Let xCo(A) has a representation x = i=1n yi and yiCo(Ai), ∀ i, and let yi = i=1n aij yij, aij > 0, i=1n aij = 1, yijAi. Constructed z vectors are given as: z = i=1nj=1l bij yij, bij ≥ 0, and (m + n)bij ≥ 0.From previous expression x = i=1nj=1l bij yij, bij = 1, ∀ i. Now, x = j=1l bij yij, x = i=1n xi, xiCo(Ai), ∀ i.Because there are (m + n)bij ≥ 0 in total, there is at least one bij ≥ 0, ∀ i, and there are at most m indices i that have more than one bij > 0, and xiAi for at least (nm) indices i. Preposition also here is continuous function (continuity condition) i.e. the condition for continuity, where states that f is said to be continuous on Rl if (Robbin, Rogers et al. 1987):

x0Rlε>0δ>0xRl[|xx0|<δ|f(x)f(x0)<ε|]

Also, if there are n commodities, and a nonnegative orthant Θ of Euclidean space En is introduced, then the sets {x : xc y} and {x : yc x} are closed. Here ⪰c are preferences of a trader in a pure exchange economy (Starr 1969). The assumption of convexity assumes that if ⪰c y, then λ x + (1 −)xc y, this means that any weighted average or convex combination of x and y is preferred to y, 0 ≤ λ ≤ 1. Each trader has initial endowment bundle and stars with a positive amount of some good xc > 0.

Now, will introduce spanability assumption. Let, ϑ ∈ Θ, and xAc (ϑ), then there is a set of no more than n + 1 points xc of Ac (ϑ) and x = ∑ λc xc, where λc ≥ 0, for all c, and ∑ λc = 1. Spanability of functions is an important concept in mathematical economics, further explained in Hüsseinov (Hüsseinov 1997):

Theorem 2

Spanability theorem: Let f : Rnf is a lower semicontinuous bounded from below function such that epigraph of a function given as,

epi(f) = {(x, ϑ)|xX, ϑR, f(x) ≤ ϑ} ⊂ Rn+1, g does not contain lines. We were that the function is convex if its epigraph epi(f) is convex set. And, furthermore x = i=1m λc xc, ∃ xi, …, xmRn, and

(λ1,,λm)>0,λ1++λm=1,f(x)=i=1mλcf(xc).

h Function would be spannable also in this corollary: lim||x||f(x)||x||=, then f is spannable.

3 Lyapunov theorem

The Shapley-Folkman theorem is a discrete counterpart to the Lyapunov theorem on non-atomic measure (Starr 2016), Lyapunov theorem can be best presented as in by Grodal (Derigs 2009). Let is consider an economy of finite non-negative atomless measures i.e.μ = (μ1, …, μn) on the measurable space (A, 𝔸).In the previous expression 𝔸 is a set of finite vector defined measures on V, which represents a set of coalitions of consumers as in the works of Vind (Vind 1964). Here also V is a σ- field of subsets C. By, 0 is given the resource allocation (set of possible allocations) in the economy {a ∈ 𝔸| a nonatomic}. Then, the range of atomless measures is: R(μ) = {xRn | ∃ C ∈ 𝔸} where xh = μh (C), h = 1, …, n, is a compact and convex subset of Rn. Lyapunov theorem is used to make conclusions about the trajectories of the system = f(x). System is globally asymptotically stable if x(t) → xe, as t → ∞. System is locally asymptotically stable near or at xe, if there is an R > 0, subject to ∥x(0) − xe∥ ≤ Rx(t) → xe, t → ∞. In previous expressions xe is an equilibrium point and xeRn. Now some function V : RnR is positive and definite if: V(z) ≥ 0, ∀ z. Now if there is a nonlinear system x = f(x), and function V : RnR, one can define; : RnR, and one can write: (z) = ∇ V(z)T f(z), or dVdt (x(t)), when z = x(t), and = f(x). Lyapunov theorem imply states that if there exist such function V : RnR that satisfies V and . Function V : RnR is called Lyapunov function. Trajectories of this function are bounded from zero to t. This can be written as:

V(x(t))=V(x(0))+0tV˙(x(t))dτ

About Lyapunov stability following conditions should apply: (z) < 0, ∀ z ≠ 0, and (0) = 0, if z = 0. Proof of this theorem assumes that x(t) → 0, (x(t)) → ε, and ε > 0. And for ε following applies: εV(x(t)) ≤ V(x(0)). And a subset C is closed and bounded C = {z|εV(x(z)) ≤ V(x(0))}. is also assumed to be continuous, and supzC = −a < 0. And, since (x(t)) < −a, ∀ t, one can rewrite Equation 4 (prethodo ravenstvo) as:

V(x(t))=V(x(0))+0TV˙(x(t))dτV(x(0))aT

and since T > V(x(0))/a, and V(x(0)) < 0, this means that x(t) → 0, and that is globally asymptotically stable (Brouwer 1912). Now if (T, ℧, μ) is a measurable space and 0 < μ(A) < μ(B), where A ∈ ℧ and B ∈ ℧, BA. In previous expressions ℧ is a σ-algebra, and the nonatomicity of μ is shown is equivalent to (Tardella 1990):

a[0,1],A,B,BA,μ(B)=aμ(A)

The main idea of Lyapunov theory is that (z) < 0 or (x) < 0, along the trajectories of the system, than V(x) will ↓, t → ∞ (Hokayem, Mastellone et al. 2006). If for instance one considers nonlinear system in the following form:

x˙=f(x)=f1(x)f2(x)=x1+3x12x2x2

And the candidate Lyapunov function given as: V(x)=λ1x12+λ2x22. And λ1, λ2 > 0. And V(x) → ∞ as ∥x∥ → ∞. The derivative of V along the trajectories is given as:

V˙=2λ1x1(x1+3x12x2)+2λ2x2(x2)=2λ1x122λ2x22+6λ1x12x2

Henceforth, if (x) < 0, V ↓ along the solution of = f(x). Local minimum of a convex function is given as some point x such that: ∃ ε > 0, and f(x) < f(x), ∀ x, all the feasible x (Mishra, Wang et al. 2008).

4 The second fundamental welfare theorem and nonconvexities

Second fundamental theorem is giving conditions under which a Pareto optimal allocation can be supported as a price equilibrium with lump-sum transfers, i.e. Pareto optimal allocation as a market equilibrium can be achieved by using appropriate scheme of wealth distribution (wealth transfers) scheme (Mas-Colell, Whinston et al. 1995).Assumption of convexity (in technology, preferences), is crucial for the establishment of the second welfare theorem. Second fundamental welfare theorem can be specified as follows: Given an economy such as; ({Xi,i}i=1I,{Yj,}j=1J,ω¯) i, allocation is given with (x∗, y∗), and a price vector p ≠ 0, (here p = p1, …, pn), and this combination constitutes price quasi-equilibrium with transfers (ω1, …, ωI), subject to the following budget constraint: ∑i ωi = p ω̄ + ∑i p yj, where ∀ j, yj, maxj p yj, pyip yj, ∀ yjYj. And ∀ i, ∃ xii xipxiωi. And ∑i xi = ω + ∑j yj. Under the assumption of locally non-satiated preferences (⪰ on Xj is locally nonsatiated if ∀ xX, ε > 0, ∃ yX, ∥yx∥ < ε, u(y) > u(x)), we have following equality pxi = ωi. This theorem holds if preferences are convex i.e.: The set ARn is convex compact and nonempty set if λ x + (1 − λ) xA, x, x′ ∈ A and λ ∈ [0, 1]. There is a theorem that gives sufficient conditions for the existence of hyperplane separating sets, that is the Separating hyperplane theoremk. A hyperplane is the set:

H(p,α)={xRn|pH0=α}

Hyperplane in Rn can be described by an equation i=1n pi xi = α, here vector pRn is a non-zero price vector, and α is scalar (Simon and Blume 1994, Yu and Phillips 2018). The vector p is then said to be normal to the hyperplane H. If one defines two points (x∗, y∗) ∈ H(p, α), and by defining, px∗ = α and, py∗ = α, in other words vector p is orthogonal to the line segment (x∗ − y∗) i.e. p ⋅ (x∗ − y∗) = 0. By picking arbitrary points previous result can be generalized to all points i.e. line segments in (p, α), or that p is orthogonal to H(p, α). Open half-spaces in the hyperplane are defined by inequalities: {xRn |pH0 ≤ α} or {xRn |pH0 ≥ α}. First expression, from previous two is example of budget set. If the preferences are assumed convex then every ui is concave and under this hypothesis every set Ui is concave. First, a vector x = (x1, …, xn) ∈ Rn is an allocation if ∑i xiω, (Mas-Colell 1989).In previous expression ωR+n is a total endowment of commodities, and production is allowed. Utility function is given as; U = {uRn : uu(x)}. Previous function can be generalized that if preferences are nonconvex i.e. they are concave and their function is concave i.e. f : AR is nonconvex, in other expression this is translated to: {(x, y) ∈ Rn+1 : yf(x), xA}, and the last set is convex. Let is suppose that xB is an extreme point of the convex set ⊂ Rn, this means that xλ x + (1 − λ) x, x, x′ ∈ A and λ ∈ [0, 1]. Separating hyperplane theorem can be stated as follows: Let is suppose that BRn is a convex and closed set and xB, ∃ pRn, p ≠ 0, αR, px > α, py < α, ∀ yB. Convex sets A, BRn are disjoint AB = ∅, ∃ pRn, p ≠ 0, px > α, py < α, ∀ yB.Then there is a hyperplane that separates A and B, leaving them on different sides of it. In support of this theorem if BRn is convex and xint B, ∃ pRn, pxpy, p ≠ 0. If A and B are convex, AB ∉ 0, AB = ∅. Let is say that SRm if z∗ is a boundary point of set S, ∃ p ≠ 0, zSpzpz∗, proof of this will come from a simple lemma: if S is a closed and convex set l, xS, and if b is the boundary of this set, then there exists scalar α ≠ 0 such that: xSα xα b. Previous theorem (S, ∃ p ≠ 0, zSpzpz∗, where p is an m − dimensional price vector) holds if m = 1, this theorem is also true when m = n + 1 (Fibich and Gavish 2011). Here n + 1 is the dimension of S production set. Now z is n + 1 dimensional vector, x is n dimensional vector and y is one dimensional scalar: z = (y, x), n -dimensional set is: X(y) ≡ {x |(y, x) ∈ S}. Now, from these two convex sets S, X(y) follows that: (λ y + (1 − λ) y, λ x + (1 − λ) x) ∈ S, and xX(y∗) → px xpx x∗. Now for one dimensional space we use following lemma to proof separating hyperplane theorem: f(y) − f(a) ≥ f (a)(ya), ∀ yA. If we want to minimize some function let say cost function then we have: cp,x (y) = minxX(y) >xx, or cp,x(y) = px x∗. And because cost function is a minimal function cp,x (λ y + (1 − λ) y) ≤ px (λ x̂ + (1 − λ) ) = λ cp,x (y) − (1 − λ) cp,x (y). From, previous, it follows that cp,x (y) − cp,x (y∗) ≥ py (yy∗), and since by definition (y, x) ∈ Scp,x(y) ≤ pxx. From previous, (y, x) ∈ Spxxpxx∗ ≥ py (yy∗). And, n + 1 dimensional vector space is given by: ppypx, since px ≠ 0 → p ≠ 0. These last expressions are the same as: zSpzpz∗. These results can also be proved with a Hahn-Bannach separated hyperplane theorem. Convex sets A, BXn are disjoint ∩ B = ∞, in a topological vector space X. And presumptions are: If A is open ∃ Λ ∈ X (linear map) and λR such that: the real part of the complex number is given as ReΛ xλRe Λ y, ∀ xA, ∀ yB. If A is compact m, B is closed, and λ is locally convex, ∃ Λ ∈ X, λ1R, λ2R, and Re Λ x < λ1 < λ2 < Re Λ y, ∀ xA, ∀ yB. In real numbers set only Re Λ = Λ. Real scalars are assumed A0A, b0B, and convex neighborhood is given as: C = AB + x0, since AB = ∞, x0C, and Minkowski functional is given as: p (x0) ≥ 1. In particular Λ ≤ 1 on C Λ ≤ 1 on −C. Now, this means that |Λ| ≤ 1, on the neighborhood C ∩ (−C) of 0, and Λ ∈ X. Now if αA, bB, will give : Λ a − Λ b + 1 = Λ(ab + x0) ≤ p(ab + x0) < 1. Now, since Λ x0 = 1, ab + x0C, and C is open, Λ a ≠ Λ b, Λ a < Λ b, So Λ(A), Λ(B) are closed disjoint sets (Rudin 2006). There is one supporting property more, following Mas-Colell, it states that :for ∀ p ≠ 0, the subspace Tp = {vRn : pv = 0}, and this subspace is called hyperplane perpendicular to p (Mas-Colell 1989). The convex set is {vRn : pv ≤ 0}, and it is a half space below Tp. Convex set is, where xA, if A − {x} ⊆ xn > 0(≥ 0). And, one can write that: {py : yA} ≤ px, i.e. pApx, if pypx, ∀ yA, yx, one can say that p supports strictly. Therefore, supporting hyperplane theorem:

Theorem 3

Supporting hyperplane theorem: If ARn is convex and x∂ A, the A is supported at x by some p ≠ 0. If A is closed, and x = 0, then the support can be strict.

Now, for a concave function f : (a, b) → R is continuous in IntA. This function f : (a, b) → R is concave in the interval (a, b), if for every x1, x2 ∈ (a, b), a ∈ (0, 1), it follows f(ax1 + (1 − a) x2) < af(x1) + (1 − a)f(x2).If the function is continuous and twice differentiable i.e. C2 on the open interval (a, b), then this function is concave on (a, b) if: ∀ x ∈ (a, b), f″(x) < 0. Or a C2 function: g : ARn on the open and convex set ARn is concave if and only if 2 f(x) < 0 and is semidefinite for all x, then f is strictly concave. In the literature of this king very important term is marginal cost pricing equilibrium which is a family of consumption, production plans, lump sum taxes and prices such that such that households are maiming their utility subject to their budget constraints and firms production plans that satisfy the FONCs (First order necessary conditions), for smooth optimization and for functions as: f : RlRn and h : RlRm are C1, constraint set is given as : E = {xRl : h(x) ≥ 0} (Brown 1991).

Some xE is a local weak maximum of f subject to h. Now FONCs are:

  1. If x is a local weak maximum of f subject to h, then there are (λ, μ) ∈ R+n×R+m so that:
    • (λ, μ) ≠ 0
    • hj (x) < 0, μj = 0
    • i=1nλifi(x)+j=1mμjhj(x)=0 (Mas-Colell 1989)
  2. SONCs (second order necessary conditions) state that if f and h are C2, and xE satisfies the FONCs with respect to (λ, μ) ∈ R+n×R+m, than we will consider the bilinear form: B=i=1nλi2fi(x)+j=1mμj2hj(x), and the cone is presented by; K = {vRl : ∂ fi (x) ≥ 0, λi ∂ fi (x) ⋅ v = 0, ∀ i, ∂ hj (x)⋅ v ≥ 0, μj ∂ hj (x) ⋅ v = 0, ∀ j, and hj (x) = 0}. Now if x is local maximum, λ > 0 and rank {∂ fi (x), ∂ hj (x) : in, jm, hj (x) = 0} = n + {j : hj (x) = 0} − 1, then B is negative and semidefinite. If B is negative and semidefinite on K then x is local maximum.

For the MCP equilibrium it is important to note that if all firms have convex technologies with zero vector, then MCP equilibrium becomes Walrasian equilibrium. Now from previous, if f is strictly non-convex (concave), and the feasible set A is convex, then the maximizer x is unique. In the proof of this let is suppose that there exist two maximizers, x and x′, then we have λ x + (1 − λ)x′ ∈ A, by the strict definition for concavity 0 < λ < 1. Now, f(λ x + (1 − λ)x′) > λ f(x) + (1 − λ)f(x′) = f(x) = f(x′). As an example let is take one consumer problem let say: maxx1,x2x11/2x21/2, subject to: x1 + x2 ≤ 1, x1 ≥ 0, x2 ≥ 0. So the feasible set A = {x1 + x2 ≤ 1, x1 ≥ 0, x2 ≥ 0}, and this is compact set n, and the function x11/2x21/2 is continuous, so by the Bolzano - Weierstrass theorem, that states that any infinite sequence from the set, every infinite subset of S has a limit point (this not necessary lies in the subset) (Green and Heller 1981). In general case for a given bifunction B : H × HR, one considers finding a solution uK, where K is a closed and convex set, and: B(u, vu) ≥ 0, ∀ vK, this is called bifunction variation inequality (Noor, Noor et al. 2012). In, previous expressions H is a Hilbert space. Now if there are two given bifunctions with their inner products F(⋅, ⋅), B(⋅, ⋅) : H × HR. The problem of finding uK is called nonconvex bifunction equilibrium variational inequality: F(u, v) + B(u, vu) + φvu2 ≥ 0, ∀ vK. Proposed iterative method, here is convergence that requires partially relaxed strongly monotonicity. A monotonic function is a function which is either entirely nonincreasing or nondecreasing. A function is monotonic if its first derivative (which need not be continuous) does not change sign. If a function f : XY is a set function from a collection of sets X to an ordered set Y, then f is monotone whenever AB as elements of X, f(A) ≤ f(B) (Royden and Fitzpatrick 2017). The proposed bifunction can be monotone Type (I) and Type II. Type (I) monotonicity of proposed bifunction is given by the : T(⋅, ⋅) : H × HR this bifunctions is said to be monotone Type (I) with respect to the operator g if and only if T(u, g(v) − g(u)) + T(v, g(u) − g(v)) ≥ 0, ∀ u, vH.And a bifunction: F : H × HR is said to be monotone of type (II) with respect to the operator g if and only if: F(g(u), g(v)) + F(g(v), g(u)) ≤ 0, ∀ u, vH (Noor, Noor et al. 2012). The sum of large number of non-convex sets is convex (approximately) (Starr 2011, Starr 2016). A typical non-convex set contains a hole for indentation. Having this in mind one can write Shapley-Folkman lemma:

Lemma 4

Shapley-Folkman lemma: Let S1, S2, S3, … Sm ⊆ ∪S(x,yi)EDn Co, and these are nonempty compact sets. And, x = con(S1, S2, S3, … Sm), ∀ i = 1, 2, …, m, ∃ yi = con(Si), i=1m yi = x, yiSi. With utmost n exceptions.

From here follows Shapley -Folkman theorem in general terms:

Theorem 5

Shapley-Folkman theorem: rad(S) ≡ in fxRn > upyS>xy|, SRn, rad(S) ≤ L, ∀ SC, rad(S) is the radius of the smallest ball (set) containing S. And F is a family of compact subsets SRn, and L > 0p, and rad(S) ≤ L, ∀ SF. And, ∀ xcon(∑SFS), ∃ ycon(∑SFS), → |xy| ≤ L N.

Theorem 6

Caratheodory theorem: xcon(Ai, …, Am), AiRL, ∃ (ai, am+1), aiA, xcon(Ai, …, Am+1).

Theorem 7

Shapley-Folkman theorem: xcon(Ai, …, Am), AiRL, ∃ (x = ai + ⋯ + am), aicon Ai, ∀ i, ∀ aiAi for mi, (Anderson, Khan et al. 1982)

Lemma 8

xcon(Ai, …, Am), AiRL:

x=i=1mj=0mjλijaij,λij>0,i=1mmiL,j=0mjλij=1,i

Proof

Proof of Caratheodory theorem: = 1, x=j=1m1λ1ja1j,m11L,x=x=j=1mλjaj,mL+1

Proof

Proof of the Shapley-Folkman theorem: i=1m (mi − 1) ≤ L, m = 1, ∄ (m = 1) for ∀ L values of i. Now, ai=j=1mjλijaijconAi,ai=j=11λijaij=ai1.

About the existence of equilibrium in this economy, one can use strict preference relations to prove the existence of such:

  1. Commodity space will be given as: R+L
  2. The set of preferences is given as: i=1m i, ≻i and those preferences satisfy:
    • continuity: {(x, y) ∈ R+L×R+L : xi y}
    • transitivity: xy, yz, xz
    • Irreflexivity : :xy,
    • Weak monotonicity: xy, xi

If x is in the core of the Edgeworth box, see (Barreto 2009). (exchange economy), ∃ p ∈ △:

1mi=1m|p(xiωi)|2Lm{max||ω1||,,||ωm||}

1mi=1m|inf{p(yxi):yixi}|4Lm{max||ω1||,,||ωm||}

In previous expression ∥ω denotes i=1m |ωi| q, or ∥ω = maxi > xi | (Anderson 1981). Now if:

p(xiωi)=0xiB(p),yixipypωi

This is Walrasian quasiequilibrum: since agents i consumption need not lie in its budget set B(p), and can be far below the budget frontier. In previous expressions

φ(p,x,)=|inf>p(yxi):yixi}|

Here φ(p, x, ≻) represents a measure that measures how far x is from the demand like (Anderson 1988). If p ≫ 0, then φ(p, x, ≻) = 0, ∃ xD(p, (≻ x)). In the context fo the second welfare theorem a Walrasian quasiequilibrium for an endowment ω, with an income transfer τ is given as: ∑aA f(a) ≤ ∑aA ω (a), p ∈ △, and f(a) ∈ Q(p, a, τ), where Q(p, a, τ) is a Walrasian quasiequilibria and Q(p, a, τ) = Q(ω, τ). A social -approximate compensated equilibrium(as approximation to market equilibrium) with non-convex preferences consists among other things of modulus A:

|(x1y1)(x2y2)|A

Also, price vector p∗ > 0, and two allocations ω1 = (x1, y1) and ω2 = (x2, y2) (Arrow and Hahn 1971). Now furthermore:

|xy|2SF[rad(S)]2L2=mL2nL2

Here F′ is a subfamily of sets of F, m is the number of members of’, n is the number of commodities or the dimension of space F, L is some number. So:

FF,xconSFS,ySFS,|xy|Ln

In the previous expression L is some number. The proof of Shapley-Folkman theorem has a lot with Gale-Debreu-Nikaido lemma:

Lemma 9

Gale-Debreu-Nikaido lemma: Z : △ → 2Rn, in previous expressionis the Laplace operator given by the divergence of gradient(derivative) of a function in a Euclidean space r. Where Z is nonempty, compact, and convex, p ∈ △, ∃ zZ(p), where Z(p) is an excess demand function, so that pz ≤ 0. And, ∃ p∗ ∈ △, Z(p∗) ∩ Rn ≠ ∅, (Yannelis 1991, Aliprantis, Tourky et al. 2000, Podczeck and Yannelis 2008).

By the separating hyperplane theorem for ∃ p∗ ∈ △, Z(p∗) ∩ Rn ≠ ∅, then ∃ qRn ∖ {0}, upsyRn q∗ ⋅ ynfszZ(p) q∗ ⋅ z. Furthermore ∈ △, q1z > 0, q2z > 0, ∀ λ ∈ [0, 1], λ q1 + (1 − λ) q2z > 0, q∗ ∈ F(p), where F(p), q ∈ △ inverse for F−1 (q) = {p ∈ △ : Z(p) ⊂ {z : qz > 0}}. And, p∗ = f(p) ∈ F(p), p∗ ⋅ z > 0, ∃ ∀ zZ(p∗), which contradicts Walras law above. Many problems in economics such as existence of competitive equilibrium in general equilibrium theory, can be formulated as fixed-point theorems (Border 1989, Farmakis, Moskowitz et al. 2013). Brouwer theorem states that every continuous function on X has fixed point. The basic Brouwer theorem was set by Brouwer, L. E. J. (Brouwer 1912). Let f be a function mapping of a compact set K in itself. A fixed point of f is a point zK, satisfying f(z) = z, (Border 1989, Shashkin 1991).

FPA asymmetric N-bidder auctions

There is a set of Θ = {1, 2, …, N}, of types of bidders. And ∀ Θ ∈ {1, 2, …, N} and ∃ n(Θ) ≥ 1, which are bidders of type Θ. Bidders of type Θ draw an IPV for the object from CDF F : [ωH, ωL] → R. It is assumed that FC2 ((ωH, ωL)) and fF > 0, on ωH. The inverse of equilibrium bidding strategy (such as in Maskin and Riley (Maskin and Riley 2000) and Fibich and Gavish (Fibich and Gavish 2011)) is given as:

vi(b)=Fi(vi(b))fi(vi(b)=[(1(n1)j=1n1vj(b)b)1vi(b)b],i=1,,n

The initial conditions is given as: vi (0) = 0, i = 1, …, n, the value for the maximum bid is set to: vi {} = 1, i = 1, …, n. In the asymmetric case vi (b) ≠ v(b), vj (b) ≠ v(b). In the symmetric case the previous expression would reduce to:

b(vi)=v1Fn1(v)rvn1(s)ds

Also

rvn1(s)xs=rvsn1ds

in general case nonuniform distribution of FPA auction is given as:

β(v)=x0vF(y)F(x)dy

Where F(y) = 1 − F(x), (F(x) is a CDF of a function), x signals are drawn from private values distribution v so xi = vi. The maximal bid of FPA distributions is then given as

b¯=b(1)=101n1(s)ds

The inverse bid functions are solutions of (as in Fibich and Gavious (Fibich and Gavious 2003)):

i(b;vi)b=(vib)j=1,j1n(k=1,k1nFk(vk(b)))zj(vj(b))vj(b)j=1,j1nj(vj(b))=0

vi is given fixed, and the maximization problem is:

maxbi(b;vi)=(vib)j=1,j1nFj(vj(b)),i=1,n

j=1,j1nfj(vj(b)vj(b)Fj(vj(b))1vi(b)b,i=1,n

Or bidder chooses to maximize his expected surplus πi as in McAfee and McMillan (McAfee and McMillan 1987):

πi=(vibi)F(β1(b1))n1

And πibi=0,dydx=πivi=F(β1(b1))n1. Another study by Güth, Ivanova-Stenzel, and Wolfstetter (Gth, Ivanova-Stenzel et al. 2005), uses bid functions for asymmetric bidders proposed by Plum (Plum 1992):

bif(vi)=ωl+viωl1+γic(viωl)2,i=1,..,n

c:=1(b1ωl)21(b2ωl)2

Where γi ∈ [–1, 1], is the PDF, ωl is the lower boundary of the statistical distribution (chosen to describe the biddersbehavior). In practice bidders valuation are drawn from different statistical distributions, as in Vickrey type auction (Vickrey 1961). In the Vickrey type of auction s, bidders submit their bids without knowing the other bidders valuations(Vickrey 1961). Vickrey found that in the case where one valuation is commonly known, buyers 1 inverse bid function is given as (Kaplan and Zamir 2012):

v1(b)=β24(βb)

This is the case where one valuation is commonly known and where there are two bidders with uniform distributions. Or when ωl = ωh = β. If there are 2 bidders only and their values are uniformly and independently distributed on (0, 1) and (0, ωh), ωh < 1, as in Milgrom (Milgrom 2004):

0ωhxxωhdx+ωh1xdx+0ωhy2dy=ωh23+(12ωh22)+ωh33=12+16ωh2+13ωh3

And E(v1,2) = 12 (1 – ωh). In general case as in Plum (Plum 1992), for the probability densities φi(x) = ci(xωl)μ, ωl < x < βi, i = 1, 2 … n, there exists and equilibrium solutions:

f1(x)=ωl+1[1c(viωl)k]11kc(viωl)k1,ωl<xβ1

f2(x)=ωl+1[1c(viωl)k]11kc(viωl)k1,ωl<xβ2

Where c:=1(b1ωl)k1(b2ωl)k; or ci:=μ+1βiωl)μ+1, where μ > –1, and k := ((2 – λ + μ))/(1 – λ), λ ∈ [0, 1]. There are ki - bidders in group i, in total N = i=1n ki. Bidders submit bids that are solutions to the optimization problem, which is as in Gayle and Richard (Gayle and Richard 2008):

β(v)=argmaxu(0,ωh)(vu)[Fi(λi(u))]ki1j1[Fj(λj(u))]kj

u = i=1n ui, where ui denotes the player of type i. Truncated CDF in general form is given as:

F(v)=j=1nFj(v)Fj(ωl)Fj(ωh)Fj(ωl)

Probabilities of winning the auction are given by the following expression:

pi(r)=kirb(ωh)li(v)li(v)j=1n[lj(v)]kjdv

Where li(v) = Fi(λi(v)). Also, r represents the reserve price in auction. Expected revenue for the auctioneer is given by:

E(p,bi,vi)=ωhrj=1n[Fj(r)]kjrb(ωh)li(v)li(v)j=1n[lj(v)]kjdv

Group i bidders expected revenue is given by:

Ei(p,bi,vi)=kirb(ωh)[Fi1li(v))v]li(v)(li(v)j=1n[lj(v)]kjdv

Now if U(pi, Ei, r) = pi ⋅ (rEi), by the envelope theorem optimal values are denoted by asterisk r′(r) = p(r), as in Milgrom (Milgrom 1989), and one can integrate to obtain the previous result.

U(x)=0rp(v)dv

Previous proof confirms the Revenue equivalence theorem. Revenue equivalence theorem confirms that if there are n risk neutral agents, that do independent and personal evaluation of some auction good, and valuation follows cumulative distribution F(v), which is ascending probability distribution of a continuous set of choices (v, v). Than every auction mechanism (every institution auction), in which lot will be allocated towards the agent for which it has highest value v, and every agent with a valuation of good v has utility 0, generates exact same revenue, which lead every bidder to make the same payment. FPA sealed-bid, SPA sealed-bid auctions generate the same price on average. This result is confirmed in: Vickrey (Vickrey 1961), Ortega-Reichert (Ortega-Reichert 1967), Myerson (Myerson 1981), Riley and Samuelson (Riley and Samuelson 1981), etc. At BNE each player tries to maximize its own expected payoff E(vib)||(βi < bi)|v1, as in Campo, Perrigne and Vuong (Campo, Perrigne et al. 2003). Now, βi=max{v1(y1i),v0(y0i)},y1i=maxji,jG1vi=1,j, where G1 is a cartel of better informed bidders and y0i=maxji,jG0vi=0,j,G0 are weak i.e. less informed bidders. And v1 and v0 are the equilibrium bids of the bidder types 1 and 0 respectively. Maximization problem for any bidder can be written as:

maxb1i=(v1ib1i)Prob(y1iv11(b1i))y0iv01(b1i)|v1i

Probability can be written as: Fy1,y0|v1(v11(b1i)),v01(b1i)|v1i. FONC for v1i bidder is given as:

Fy1,y0|v1(v11(b1i)),v01(b1i)|v1i+

(v1ib1i)[(Fy1,y0|v1(v11(b1i)),v01(b1i)|v1iy1i1v1(v11(b1i))+(Fy1,y0|v1(v11(b1i)),v01(b1i)|v1iy01(v0(v01(b1i))]

In the previous expression v1i ∈ [ωl, ωh], and b1i = v1(v1i). FONC for v0i bidder can be obtained in the similar manner. A market action should be attainable by actions of a consumer or a collation of consumers. This leads us to the concept of core economy Ce. Every allocation in the core of the economy is said to be Pareto efficient, i.e. the allocation cannot be changed such that one agent is strictly better off without making any other agent worse off. And furthermore, the set of all competitive equilibrium allocations is contained in the core, Ce(Jain 2004). The question that arises here is whether core of the economy is equivalent to the competitive equilibrium allocations. Shapley and Shubik studied one asymmetric economy in the indivisible goods markets (houses) but with only one or two buyers (Shapley and Shubik 1971). These are non-combinatorial market cases. One example of combinatorial market is given in Bikhchandani and Mamer (Bikhchandani and Mamer 1997). Early attempt to explain indivisible goods market by "matching models" (college admissions and university quotas) and "stable marriage" assignment problem were analyzed. These papers proved that indivisible goods market economy has non-empty core. Knaster-Kuratowski-Mazurkiewicz lemma is a basic result in fixed point theory. Or Knaster-Kuratowski-Mazurkiewicz-Shapley theorem which is a generalization of previous lemma, based on Brower fixed point theorem. Theorem K-K-M-S is stated as follows: Cs : SN is a family of closed subsets of simplex ΔN. Where ΔN = {xRn : xi > 0, ∧ i=1n xi = 1}, is a n – 1 simple, for SN, ΔS = {x ∈ Δn : xi > 0, ∧ ∑iS xi = 1}. Now, we can assume that ΔT ⊂ ∪STs, ∀TN, ∃B, ∩sB Cs ≠ ∅. Where, B is a balanced family (collection), such that intersection of sets indexed by B is nonempty (Krasa and Yannelis 1994) t. And "the core approaches the set of equilibrium allocations as the number of traders tends to infinity", and a continuum economy is an appropriate mathematical model of a situation of existence of "many" commodities (Aumann 1964).

Literature review on previous research concerning numerical solutions on asymmetric auctions The first researchers to propose using numerical algorithms to solve for the equilibrium or inverse bid functions were Marshall, Meurer, Richard and Stromquist (Marshall, Meurer et al. 1994). They applied l’Hopital’s rule to the FOC to derive: lims0+φk(b)=(nk+1)nk. Where φk(b) is a first derivative of the inverse bid function, and nk are the number of bidders in coalition. Inverse bid functions are in practice normalized to δk(b)=φk(b)b. They approximated {δk(b)}k2=1 by Taylor series expansions of order 5 around each point bt ∈ [ωL, ωH]. Condition for valid solution is: 12k=12[δk(ωL)(nk+1)nk]ε2, where ε is some tolerance level (convergence criterion), of order 10–5 –10–8. Bajari (2001) proposed that inverse bid function (n-bidders) can be represented as a linear combination of ordinary polynomials (Bajari 2001). This can be presented in the following manner:

ω^n(b)=b¯k=0Kαn,k(b¯b)k,n=1,2,,N

FOC for bidder n can be expressed as:

1=[φn(b)b]mnfm[φm(b)]Fm[φm(b)]φm(b)

and

Gn(ωL,ωH,α)=1[φn(b)b]mnfm[φm(b)]Fm[φm(b)]φm(b)

And the left boundary condition is: φn(ωL) = ωL, and the right boundary condition is φn(ωH) = ωH. In Fibbich and Gavious (Fibich and Gavious 2003), is suggested using a perturbation analysis to calculate an explicit approximation to the asymmetric FPA solution, they defined average distribution between N bidders at a valuation v, namely : Faverage1Nn=1NFn(v). The parameter that measures the level of asymmetry is given as: ε = maxn∈{1,…,N} > axv∈{ωL,ωH}>|Fn(v) – Faverage(v)|, and that Fn(v) = Faverage(v)+εAn(v), n = 1, ..N, where auxiliary function An(v)=Fn(v)Faverage(v)ε. Equilibrium bid function u, when bidders draw valuations from Faverage(⋅) is given (Schmedders and Judd 2013):

En[v,An(v)]=1NFaverageN1(v)

[ωLvFaverageN1(v)dv]NvωH1[FaverageN1(v)]N1d[An(t)(Faverage(t)]dtdt

Gayle and Richard (Gayle and Richard 2008), generalized backward shooting algorithm, they defined: ln(v) ≡ Fn[φn(v)], and now FOC can be defined as: 1=(Fn1[ln(v)]v)mnln(v)ln(v), and the high bid is chosen by solving:

minb[r,ωH]n=1N[l(r|b)Fn(r)]2,r

is reserve price. Hubbard and Paarsch (Hubbard and Paarsch 2009) used Chebyshev polynomials, which are orthogonal polynomials and more stable. Chebyshev nodes can be computed as: xt=cos[π(t1)T],t=1,,T. The points {vt}tT=1 are found via transformation like this: vt=b¯+ωL+(b¯ωL)xt2. Chebyshev polynomials can be defined recursively as T0(x) = 1, T1(x) = x, Tn+1(x) = 2xTn(x) + Tn–1(x). The coefficients of these polynomials for a function f(x) can be obtained by the following integral:

an=2π11f(x)Tn(x)(1x2)1/2dx

Hubbard, Kirkegaard and Paarsch imposed 5 in/equality constraints on the equilibrium bid functions v, that are approximated by the Chebyshev polynomials of orderK (Hubbard, Kirkegaard et al. 2013). They approximated the solution to differential equations (general FPA model):

φn(v)=Fn[φ(v)]fn[φ(v)]([1N1m=1N1φn(v)v]1φn(v)v)

And they are doing so by minimizing and solving:

minb¯,αn=1Nt=1T[Gn(ωL,ωH,α)]2.

Where in previous minimization problem

Gn(ωL,ωH,α)=1[φn(b)b]mnfm[φm(b)](Fm[φm(b)]φm(b).

Kirkegaard proved that if the CDFs cross each other (Fn(v) crosses Fm(v)), than their bid functions would cross each other. Relative power of n bidder over m bidder and the utility of bidder n is measured as (Kirkegaard 2009):

Pn,m(v)=Fm(v)Fn(v),v(ωL,ωH)andUn(v)=(vωL)mnFm[φ(ωL)]

And the ratio of n’s equilibrium pay-off relative to m’s equilibrium pay-off is: Rn,m(v)=Un(v)Um(v),v(ωL,ωH). Kirkegaard assumes that comparing two ratios ∀v ∈ (ωL, ωH), is equivalent of comparing two bids bn(v) and bm(v), i.e.: Rn,m(v) ≥ Pn,m(v) ⇔ bn(v) ≥ bm (v). Right boundary condition is : Rn,m(ωH) = Pn,m(ωH) = 1, left boundary condition is : limbωLRn,m(b)=fm(ωL)fn(ωL)=limbωLPn,m(b)(Kirkegaard 2009).

Asymmetric auctions: simulation results

In this part we choose 10 bidder types, there is only one bidder from each type, and these bidders draw their IPVs from for the object of the auction from their CDF F : [ωH, ωL] → R. Ten selected distributions in the following order are, (see, (Johnson, Kemp et al. 2005)):

Table 1

Selected distributions and their CDFs and PDFs

Distributions and boundariesCDFPDF
Beta [0,1]F(x)=1B(a,b)ωLυ(x)xa1(1x)b1dx;f(x)=1ωHωLυ(x)a1(1v(x))b1B(a,b)
υ(x)=xωLωHωL
Exponential [0,1]F(x)=1exp(λ(xωL)1exp(λ(ωHωL)f(x)=λexp(λ(xωL)1exp(λ(ωHωL)
Gamma [0,1]F(x)=0xk1exdxΓ(k)f(x)=1θkΓ(k)xk1exθ
Kumaraswamy [0,1]F(x; a; b) = 1 –(1–xa)bf(x; a, b)=F′(x; a; b)=abxa–1(1–xa)b-1
Lognormal [0,1]F(x)=ax1zσ2πexp12lnzμσ2dzab1zσ2πexp12lnzμσ2dzf(x)=1xσ2πexp12lnxμσ2ab1zσ2πexp12lnzμσ2dz
Standard normal [0,1]F(x)=ΦxμσΦaμσΦbμσΦaμσf(x)=1σϕxμσΦωμσΦμσ
Power [0,1]F(x)=ηα+1(x+a+c)α+1ca1f(x)=η(x+a+c)α,
η=(α+1)[xa+c)α+1cα+1]–1
Reverse power [0,1]F(x)=1bxbaαf(x)=α(bx)α1ba
Triangular [0,1]F(X)=ψx+(1ψ)(xa)2(ba)(ca)f(x)=F(x)=4xifx(0,,0,5)44xifx(0.5,,1)
Uniform [0,1]F(x)=xωLωHωLf(x)=1ωHωL

Java software has been applied for the simulations. The software name is Auction Solver and was written by Richard M. Katzwer, see (Katzwer 2012). Next, in a table 2 are presented Chebyshev coefficients for 10 parametrized distributions with their CDFs and PDFs written in Table 1.

Table 2

Solution: Chebyshev coefficients of K=15 degree

Beta distributionExponential distributionGamma distributionKumaraswamy distributionLog nomal distributionStandard normal distributionPower l distributionReverse power distributionTriangular distributionUniform distribution
0.00200.00310.00210.00200.00630.00320.00310.00110.00280.0033
0.98450.98270.98130.98450.98860.98170.98070.99730.98230.9831
0.00240.00350.00250.00240.00610.00360.0034-0.00040.00280.0038
0.00800.01040.01120.00800.00370.01030.01110.01440.00880.0100
0.00000.00000.00000.0000-0.00060.00000.0000-0.0006-0.00030.0000
-0.0011-0.00030.0005-0.0011-0.00100.00010.0005-0.00660.0010-0.0009
0.00000.00000.00000.0000-0.00040.00000.0000-0.00030.00000.0000
0.00080.00070.00080.0008-0.00180.00070.0009-0.00480.00140.0004
0.00000.00000.00000.00000.00000.00000.0000-0.00140.00000.0000
0.0004-0.00010.00000.0004-0.00070.00000.0000-0.0025-0.00060.0001
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00070.00000.00010.0007-0.00010.00030.0001-0.00040.00010.0002
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
High bid: = 0.893113927397631

Table 2

Solution: Chebyshev coeff. of K=15 order reserve price =0.5

Beta distributionExponential distributionGamma distributionKumaraswamy distributionLog nomal distributionStandard normal distributionPower l distributionReverse power distributionTriangular distributionUniform distribution
0.44320.42240.43780.34240.29800.39700.44380.38300.40100.4123
0.62130.65700.62840.79160.87100.70120.61910.73590.69010.6757
-0.0764-0.1015-0.0877-0.1854-0.2211-0.1301-0.0803-0.0853-0.1231-0.1121
0.0190-0.10150.02980.06110.06150.04470.0255-0.03380.03690.0358
0.0013-0.0031-0.0049-0.00730.0029-0.0064-0.00320.0000-0.0027-0.0035
0.00.00.00.00.00.00.00.00.00.0
High bid: = 0.9032115516025931

Table 3

Backward shooting method solution, end result: Convergence true

b_barABB-AResult
0/390.5011.00E+003Solution not within specified tolerances.
1/390.750.515.00E-013Solution not within specified tolerances.
2/390.8750.7512.50E-012Solution diverges to +infinity.
3/390.81250.750.8751.25E-012Solution diverges to +infinity.
4/390.781250.750.81256.25E-023Solution not within specified tolerances.
5/390.79687500.78125000.81250003.13E-023Solution not within specified tolerances.
6/390.80468750.79687500.81250001.56E-023Solution not within specified tolerances.
7/390.80859380.80468750.81250007.81E-033Solution not within specified tolerances.
8/390.81054690.80859380.81250003.91E-032Solution diverges to +infinity.
9/390.80957030.80859380.81054691.95E-033Solution not within specified tolerances.
10/390.81005860.80957030.81054699.77E-043Solution not within specified tolerances.
11/390.81030270.81005860.81054694.88E-043Solution not within specified tolerances.
12/390.81042480.81030270.81054692.44E-042Solution diverges to +infinity.
13/390.81036380.81030270.81042481.22E-042Solution diverges to +infinity.
14/390.81033330.81030270.81036386.10E-053Solution not within specified tolerances.
15/390.81034850.81033330.81036383.05E-051Solution diverges to -infinity.
16/390.81034090.81033330.81034851.53E-053Solution not within specified tolerances.
17/390.81034470.81034090.81034857.63E-062Solution diverges to +infinity.
18/39+0.81034280.81034090.81034473.82E-063Solution not within specified tolerances.
19/390.81034370.81034280.81034471.91E-062Solution diverges to +infinity.
20/390.81034330.81034280.81034379.54E-072Solution diverges to +infinity.
21/390.81034300.81034280.81034334.77E-073Solution not within specified tolerances.
22/390.81034310.81034300.81034332.38E-072Solution diverges to +infinity.
23/390.81034310.81034300.81034311.19E-073Solution not within specified tolerances.
24/390.81034310.81034310.81034315.96E-083Solution not within specified tolerances.
25/390.81034310.81034310.81034312.98E-083Solution not within specified tolerances.
26/390.81034310.81034310.81034311.49E-083Solution not within specified tolerances.
27/390.81034310.81034310.81034317.45E-092Solution diverges to +infinity.
28/390.81034310.81034310.81034313.73E-092Solution diverges to +infinity.
29/390.81034310.81034310.81034311.86E-093Solution not within specified tolerances.
30/390.81034310.81034310.81034319.31E-102Solution diverges to +infinity.
31/390.81034310.81034310.81034314.66E-103Solution not within specified tolerances.
32/390.81034310.81034310.81034312.33E-102Solution diverges to +infinity.
33/390.81034310.81034310.81034311.16E-103Solution not within specified tolerances.
34/390.81034310.81034310.81034315.82E-113Solution not within specified tolerances.
35/390.81034310.81034310.81034312.91E-112Solution diverges to +infinity.
36/390.81034310.81034310.81034311.46E-113Solution not within specified tolerances.
37/390.81034310.81034310.81034317.28E-122Solution diverges to +infinity.
38/390.81034310.81034310.81034313.64E-122Solution diverges to +infinity.
39/390.81034310.81034310.81034311.82E-123Solution not within specified tolerances.
Highest bid: = 0.810343140133682 . Shooting terminated at b = 0.5000231401337242. (b_underbar = 0.5)

Next in Table 4 are presented parameters for the Constrained strategic equilibrium and Backward shooting solver.

Table 4

C.S.E. and Backward shooting parameters

Constrained strategic equilibrium (no reserve price) parametersConstrained strategic equilibrium (reserve price = 0.5) parametersBackwards solver parameters
T(degree)40T (degree)40Shooting metdodEuler
K (grid)15K (grid)15ODE systemInverse bid functions
μl2000μl2000h1 (tolerance of tde deviation of tde solution from left boundary)1.0E-5
μh5000μh5000h2 (step size close to high bid)0.001
μfonc5.0μfonc5.0Td reshold0.01
μ0.0μ0.0High-bid precision1.0E-12
μmono1000μmono1000Left-boundary tolerance1.0E-5
Cheb grid= noCheb Grid= no//

Euler method used in backward shooting solver is described as the simplest Runge-Kutta method, ODE is of the form: dy(t)dt=f(t,y(t)),y(t0)=y0,dy(t)dty(t+h)y(t)h,y(t+h)y(t)+hdydt. The iterative solutions is than given as: yn + 1 = yn + hf(tn, yn) or in previous expressions x ∈ (x0, xn). MATLAB also is a powerful tool used by economists and can compute equilibrium strategies in a first-price auction with two players using the boundary value method with fixed-point iterations, and by using the boundary value method with Newtons iterations. A fixed point is a point that does not change upon of a function (map), system of differential equations etc. (Shashkin 1991). In the Newton’s method the algorithm can be applied iteratively to obtain: xn+1=xnf(xn)f(xn1), if limxn+1xf(xn)f(xn)=xn, and xn = x + εn, where εn+1=f(x)2f(x)εn2. Fixed point theorem states that if ∃f(x) ∈ [a, b], then ∃x ∈ [a, b], and f(x) – x = 0 ⇒ f(x) = x, see (Rosenlicht 1968). In our case Newton’s method has quadratic convergence, i.e. we can denote residual (in two bidders case as, (see (Fibich and Gavious 2003, Fibich and Gavish 2011)):

εb[b,v1,b,v2]=b(v2)f2(v2)F2(v2)(v1b)

In a two bidders case one follows power law distributions, with CDF, F1=c1v1a1. And the second bidder distribution is truncated normal with CDF: F2=c2erf(af2af2+1v2), where the error return function is defined as:

erf(z)=2π0xet2dt,

see (Abramowitz and Stegun 1964). Matlab code for this simple two bidder case was written by (Fibich and Gavish 2011). In the next two graphs are presented two bidder’s distribution valuations.

Fig. 1
Fig. 1

Fixed point iterations result of the ratios of the two bidders’ valuations CDF/PDF functions

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00029

Fig. 2
Fig. 2

Newtons iterations result of the two CDF/PDF bidders’ valuations functions

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00029

5 Conclusion

As it is known competitive equilibrium need not exist in the indivisible goods economy. These markets include auctions where discrete items can be only traded, as a whole only. In combinatorial auction there is a finite set of times but one buyer can only buy subset of items. All the previous research on the topic studied the competitive equilibrium where there are only one or two buyers or sellers (Shapley and Shubik 1971, Khan and Rath 2013). These markets are non-combinatorial since only one item (commodity) at time is a subject of sale. The attempt to deal with non-convex preferences in finite setting by proposing approximate equilibria (Starr 1969, Starr 2011, Starr 2016).

This paper was written by following of this idea. In this paper asymmetric auction was used in order to prove the previous theoretical models. Auction in most general terms is a game theoretic mechanism which allocates an object (set of objects) and is composed of set of bidders χ, set of objects allocated, a private type space Θ, and public type space Ξ. And where each bidder has type of distributions {θi, ξi} ∈ Θ × Ξ, and Θ × Ξ = i=1NΘi×i=1NΞi, which represents the space of all type profiles, see (Katzwer 2012). In the FPA auction that was used for analysis in this paper every bidder pays its bid, the bidder does not know the opponents’ bids. In the asymmetric type of FPA auctions highest bid wins, and highest winning bidder pays its bid. So, the highest bid is considered to be the equilibrium bid.

In this type of market setting contrary to the convex case, where agents with convex preferences that do not prefer extremes and they prefer in between values, in the First price auction BNE equilibrium is bidders i bid and that must be highest bid so that the item is allocated to him, and the outcome is efficient. Since in the First price items is sold to the buyer with the highest valuation of the item, this auction mechanism is Pareto efficient. Though in theory FPA auction is distinct from the English type of auction since here bidders can only submit one, bid and they do not know other bidders valuation and they may bid too low.

Communicated by Juan Luis García Guirao

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Footnotes

acon Ai for a theory Ai means Ai is consistent
bA convex hull is the smallest polygon that encloses a group of objects, such as points.
cΨ is a field of subsets.
dThe relative complement of a with respect to a set b
eThis is the least element in Ψ ≥ ∀ ai, bi
fHere = R ∪ {−∞, +∞} affinely extended real numbers
gEpigraph or supergraph of a function f : RnR is
hf∗ is the greatest lower semicontinuous convex.
iWe assume that ∑i ω ≪ 0 (level of income or wealth is much larger than zero)
jPreference relation ⪰ is a relation ⪰ ⊂ R+l×R+l. With properties xx, ∀ xR+l (reflexivity), xy, yzxz (transitivity), ⪰ is a closed set (continuity), ∀ (xy), ∃ (yx) (completeness), given ⪰, ∀ (x ≪ 0) the at least good set {y : yx} is closed relative to Rl (boundary condition), A is convex, if {y : yx} is convex set for every y, ay + (1 − λ)xx, whenever yx and 0 < a < 1, Mas-Colell, A. (1986).
kIn geometry hyperplane of an n dimensional vector space V is a subspace of a n − 1 dimension, or equivalently of codimension 1 in V
lIts complement is an open set. Closed set is defined as a set that contains all of its limit points.
mA subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzanoeierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set.
nEuclidan space that is closed and bounded (all points lie within some fixed distance between each other).
oHere C is an arbitrary collection of sets.
pHere L is the upper bound for a circle equals 2π radd = π d, where d is the diameter. For general shapes, it can be calculate as 0L ds.
qActually i=1m|ωi|=iSxi,, where xi,=ωi+zi+zS
rIn Cartesian system Laplace operator is given as the sum of second order partial derivatives of the function with respect to each independent variable.
sIn Vickrey type of auction each bidder bids its own valuation, and this is optimal strategy. This is a sealed bid auction where the highest bid wins, but pays only the second highest bid.
tIn the proof of this theorem provided in this paper Krasa and Yannelis (1994) it is defined set valued function: ψ = ΔN → 2Rn, F(x) = con999, where mS={m1S,,mNS} is the center of the simplex ΔS. And miS=1|S|,iS,miS=0,iS. And ψ(x) = {y : f(x)x > f(x)y}, ∃TN, and intΔn = {x = (x0, …, xn) ∈ Rn+1 | 0nxi = 1, xi > 0, ∀i}, therefore since ST, SI(x) and mSF(x). And the f(x)x = f(x) mS > f(x)mN.
uInitial guess about maximum bid function is given as: b¯=ωHωLωHFaverageN1(v)dv
v1. φn(v) = ωL, 2. φn(b) = ωH 3. ∑mn(bv) fm(b) φm(v) = 1, 4. φ(ωL) = N1N, 5. φn(vj – 1) ≤ φn(vj), for some uniform array j = 2, …, J.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    M. Abramowitz I. A. Stegun Handbook of Mathematical Functions: With Formulas Graphs and Mathematical Tables Dover Publications (1964).

  • [2]

    C. D.Aliprantis K. C. Border Infinite Dimensional Analysis: A Hitchhiker’s Guide (2013).

  • [3]

    C. D. Aliprantis et al. Cone Conditions in General Equilibrium Theory Journal of Economic Theory92(1) (2000) 96–121.

    • Crossref
    • Export Citation
  • [4]

    R. M. Anderson Core Theory with Strongly Convex PreferencesEconometrica49(6) (1981) 1457–1468.

  • [5]

    R. M. Anderson The Second Welfare Theorem with Nonconvex Preferences.Econometrica56(2) (1988) 361–382.

  • [6]

    R. M. Anderson et al. Approximate Equilibria with Bounds Independent of Preferences The Review of Economic Studies49(3) (1982) 473–475.

    • Crossref
    • Export Citation
  • [7]

    T. E.Armstrong K. Prikry Liapounoff’s Theorem for Nonatomic Finitely-Additive Bounded Finite-Dimensional Vector-Valued Measures Transactions of the American Mathematical Society266(2) (1981) 499–514.

  • [8]

    K. J. Arrow and F. Hahn General competitive analysis Holden-Day (1971).

  • [9]

    R. J.Aumann Markets with a Continuum of Traders. Econometrica 32(1/2) ( 1964) 39–50.

  • [10]

    P.Bajari Comparing competition and collusion: a numerical approach. Economic Theory18(1) (2001)187-205.

    • Crossref
    • Export Citation
  • [11]

    H.Barreto Intermediate Microeconomics with Microsoft Excel Cambridge University Press (2009).

  • [12]

    S.Bikhchandani and J. W. Mamer "Competitive Equilibrium in an Exchange Economy with Indivisibilities. Journal of Economic Theory 74(2) (1997). 385-413.

    • Crossref
    • Export Citation
  • [13]

    Bogachev V. I. Measure Theory Springer Berlin Heidelberg (2007).

  • [14]

    K. C. Border Fixed Point Theorems with Applications to Economics and Game Theory Cambridge University Press (1989).

  • [15]

    L. E. J. Brouwer “Über Abbildung von Mannigfaltigkeiten.Mathematische Annalen 71(4) (1912). 598-598.

    • Crossref
    • Export Citation
  • [16]

    D. J. Brown Chapter 36 Equilibrium analysis with non-convex technologies. Handbook of Mathematical Economics Elsevier4 (1991)1963-1995.

  • [17]

    S.Campo et al. Asymmetry in firstrice auctions with affiliated private values Journal of Applied Econometrics 18(2):(2003) 179-207.

    • Crossref
    • Export Citation
  • [18]

    U.Derigs Optimization and Operations Research olume III (2009).

  • [19]

    J. Eckhoff CHAPTER 2.1 - Helly Radon and Carathodory Type Theorems A2 - GRUBER P.M. Handbook of Convex Geometry. J. M. Wills. Amsterdam North-Holland (1993) 389-448.

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    I. Farmakis et al. Fixed Point Theorems and Their Applications World Scientific Publishing Company (2013).

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    G. Fibich and A. Gavious "Asymmetric First-Price Auctions: A Perturbation Approach Mathematics of Operations Research 28(4) (2003) 836-852.

    • Crossref
    • Export Citation
  • [22]

    G. Fibich and N. Gavish (2011). "Numerical simulations of asymmetric first-price auctions. Games and Economic Behavior 73(2) 479-495.

    • Crossref
    • Export Citation
  • [23]

    W.-R. Gayle and J. F. Richard "Numerical Solutions of Asymmetric First-Price Independent Private Values Auctions Computational Economics 32(3) (2008) 245-278.

    • Crossref
    • Export Citation
  • [24]

    J. Green and W. P. HellerChapter 1 Mathematical analysis and convexity with applications to economics. Handbook of Mathematical Economics Elsevier 1 (1981) 15-52.

  • [25]

    W. Güth et al. "Bidding behavior in asymmetric auctions: An experimental study European Economic Review 49(7) (2005) 1891-1913.

    • Crossref
    • Export Citation
  • [26]

    P. R.Halmos Measure Theory Springer New York (2013).

  • [27]

    E. Hewitt and K. Stromberg Real and Abstract Analysis: A modern treatment of the theory of functions of a real variable Springer Berlin Heidelberg (2013).

  • [28]

    P. F.Hokayem et al. Nonlinear Systems Stability via Random and Quasi-Random Methods. Probabilistic and Randomized Methods for Design under Uncertainty. G. Calafiore and F. Dabbene. London Springer London (2006) 365-379.

  • [29]

    T. P. Hubbard et al. "Using Economic Theory to Guide Numerical Analysis: Solving for Equilibria in Models of Asymmetric First-Price Auctions Computational Economics 42(2) (2013) 241-266.

    • Crossref
    • Export Citation
  • [30]

    T. P. Hubbard and H. J. Paarsch Investigating bid preferences at low-price sealed-bid auctions with endogenous participation International Journal of Industrial Organization27(1) (2009) 1-14.

    • Crossref
    • Export Citation
  • [31]

    F. Hsseinov Characterization of spannability of functions Journal of Mathematical Economics 28(1)(1997) 29-40.

    • Crossref
    • Export Citation
  • [32]

    R.Jain Efficient market mechanisms and simulation-based learning for multi-agent systems University of California at Berkeley164 (2004).

  • [33]

    N. L.Johnson et al. Univariate Discrete Distributions Wiley (2005).

  • [34]

    T. R. Kaplan and S. Zamir Asymmetric first-price auctions with uniform distributions: analytic solutions to the general case Economic Theory 50(2) (2012) 269-302.

    • Crossref
    • Export Citation
  • [35]

    R. M.Katzwer Auction Solver User Guide Princeton University (2012).

  • [36]

    M. A. Khan and K. P. Rath The Shapleyolkman theorem and the range of a bounded measure: an elementary and unified treatment Positivity 17(3) (2013) 381-394.

    • Crossref
    • Export Citation
  • [37]

    R. Kirkegaard Asymmetric first price auctions Journal of Economic Theory 144(4) (2009) 1617-1635.

    • Crossref
    • Export Citation
  • [38]

    S. Krasa and N. C. Yannelis An elementary proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley Theorem Economic Theory 4(3) (1994) 467-471.

    • Crossref
    • Export Citation
  • [39]

    R. C. Marshall et al. Numerical Analysis of Asymmetric First Price Auctions Games and Economic Behavior 7(2) (1994) 193-220.

    • Crossref
    • Export Citation
  • [40]

    A. Mas-Colell The Theory of General Economic Equilibrium: A Differentiable Approach Cambridge University Press (1989).

  • [41]

    A.Mas-Colell et al. Microeconomic Theory Oxford University Press (1995).

  • [42]

    E. Maskin and J. Riley Asymmetric Auctions The Review of Economic Studies 67(3)(2000) 413-438.

    • Crossref
    • Export Citation
  • [43]

    R. P. McAfee and J. McMillan Auctions and Bidding Journal of Economic Literature 25(2) (1987) 699-738.

  • [44]

    P.Milgrom Auctions and Bidding: A Primer Journal of Economic Perspectives 3(3) (1989) 3-22.

    • Crossref
    • Export Citation
  • [45]

    P. R. Milgrom Putting Auction Theory to Work Cambridge University Press (2004).

  • [46]

    S. K. Mishra et al. Generalized Convexity and Vector Optimization Springer Berlin Heidelberg (2008).

  • [47]

    R. B. Myerson Optimal Auction Design Mathematics of Operations Research 6(1) (1981) 58-73.

    • Crossref
    • Export Citation
  • [48]

    M. A. Noor et al. Some Iterative Methods for Solving Nonconvex Bifunction Equilibrium Variational Inequalities Journal of Applied Mathematics (2012) 10.

  • [49]

    A. Ortega-Reichert Models for Competitive Bidding Under Uncertainty Stanford University (1967).

  • [50]

    M.Plum Characterization and computation of nash-equilibria for auctions with incomplete information International Journal of Game Theory20(4) (1992) 393-418.

    • Crossref
    • Export Citation
  • [51]

    K. Podczeck and N. C. Yannelis Equilibrium theory with asymmetric information and with infinitely many commodities Journal of Economic Theory141(1) (2008) 152-183.

    • Crossref
    • Export Citation
  • [52]

    J. G. Riley and W. F. Samuelson Optimal Auctions The American Economic Review 71(3) (1981) 381-392.

  • [53]

    J. W. Robbin et al. On Weak Continuity and the Hodge Decomposition Transactions of the American Mathematical Society 303(2) (1987) 609-618.

    • Crossref
    • Export Citation
  • [54]

    M. Rosenlicht Introduction to Analysis Dover Publications (1968).

  • [55]

    H. Royden and P. Fitzpatrick Real Analysis (Classic Version) Pearson Education (2017).

  • [56]

    W. Rudin Functional Analysis McGraw-Hill (2006).

  • [57]

    K. Schmedders and K. L. Judd Handbook of Computational Economics Elsevier Science (2013).

  • [58]

    G. L. Seever Measures on F-Spaces Transactions of the American Mathematical Society 133(1) (1968).267-280.

  • [59]

    L. S. Shapley and M. Shubik The assignment game I: The core International Journal of Game Theory1(1) (1971) 111-130.

    • Crossref
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    Fixed point iterations result of the ratios of the two bidders’ valuations CDF/PDF functions

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    Newtons iterations result of the two CDF/PDF bidders’ valuations functions

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