Shapley-Folkman-Lyapunov theorem and Asymmetric First price auctions

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).

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):

∑<!-- ? -->(A⨁<!-- ? -->B)={a+b:a∈<!-- ? -->A,b∈<!-- ? -->B} $$\begin{array}{} \displaystyle \sum(A\bigoplus B)=\{a+b:a\in A, b\in B\} \end{array}$$(1)

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

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 hull

A convex hull is the smallest polygon that encloses a group of objects, such as points.

con≡<!-- = -->{∑<!-- ? -->j=1Nλ<!-- ? -->jpj:λ<!-- ? -->j≥<!-- = -->0∀<!-- ? -->j,∑<!-- ? -->j=1Nλ<!-- ? -->j=1} $$\begin{array}{} \displaystyle con \equiv \{\sum_{j=1}^{N} \lambda_j p_j :\lambda_j \geq 0 \, \forall j, \sum_{j=1}^{N} \lambda_j=1\} \end{array}$$(2)

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

Let ℝ^L be an L dimensional Euclidean space, then let A_i denote the its convex hull for any A ⊆ Rⁿ, now ∅ ≠ A_i ⊆ Rⁿ and x ∈ con ∑<!-- ? -->i=1n $\begin{array}{} \sum_{i=1}^{n} \end{array}$ A_i, then ∑<!-- ? -->i=1n $\begin{array}{} \sum_{i=1}^{n} \end{array}$ x_i = x, and x ∈ con A_i, ∀ i, x_i ∈ A_i, for at least n − m indices of i.

Let Ψ = ∅

Ψ is a field of subsets.

Also, if there are n commodities, and a nonnegative orthant Θ of Euclidean space Eⁿ is introduced, then the sets {x : x ⪰_c y} and {x : y ⪰_c 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 −)x ⪰_c 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 x_c > 0.

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

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, …, μ_n) 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, Ā 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) = {x ∈ Rⁿ | ∃ C ∈ 𝔸} where x_h = μ_h (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 ẋ = f(x). System is globally asymptotically stable if x(t) → x_e, as t → ∞. System is locally asymptotically stable near or at x_e, if there is an R > 0, subject to ∥x(0) − x_e∥ ≤ R ⇒ x(t) → x_e, t → ∞. In previous expressions x_e is an equilibrium point and x_e ∈ Rⁿ. Now some function V : Rⁿ → R is positive and definite if: V(z) ≥ 0, ∀ z. Now if there is a nonlinear system x = f(x), and function V : Rⁿ → R, one can define; V̇ : Rⁿ → R, and one can write: V̇ (z) = ∇ V(z)^T f(z), or dVdt $\begin{array}{} \frac{dV}{dt} \end{array}$ (x(t)), when z = x(t), and ẋ = f(x). Lyapunov theorem imply states that if there exist such function V : Rⁿ → R that satisfies V and V̇. Function V : Rⁿ → R 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<!-- t --> $$\begin{array}{} \displaystyle V(x(t))=V(x(0))+\int_{0}^{t} \dot{V} (x(t))d\tau \end{array}$$(4)

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

V(x(t))=V(x(0))+∫<!-- ? -->0TV˙<!-- ? -->(x(t))dτ<!-- t -->≤<!-- = -->V(x(0))−<!-- - -->aT $$\begin{array}{} \displaystyle V(x(t))=V(x(0))+\int_{0}^{T} \dot{V} (x(t))d\tau\leq V(x(0))-aT \end{array}$$(5)

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

∀<!-- ? -->a∈<!-- ? -->[0,1],∀<!-- ? -->A∈<!-- ? -->℧<!-- ? -->,∃<!-- ? -->B∈<!-- ? -->℧<!-- ? -->,B⊂<!-- ? -->A,μ<!-- ? -->(B)=aμ<!-- ? -->(A) $$\begin{array}{} \displaystyle \forall a\in [0,1],\forall A\in \mho, \exists B\in \mho,B\subset A,\mu(B)=a\mu(A) \end{array}$$(6)

The main idea of Lyapunov theory is that V̇ (z) < 0 or V̇(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+3x12x2−<!-- - -->x2 $$\begin{array}{} \displaystyle \dot{x}=f(x)=\begin{bmatrix}f_1 (x) \\ f_2 (x)\end{bmatrix}=\begin{bmatrix}-x_1+3x_1^2 x_2 \\-x_2 \end{bmatrix} \end{array}$$(7)

And the candidate Lyapunov function given as: V(x)=λ<!-- ? -->1x12+λ<!-- ? -->2x22. $\begin{array}{} V(x)=\lambda_1 x_1^2+\lambda_2 x_2^2. \end{array}$ And λ₁, λ₂ > 0. And V(x) → ∞ as ∥x∥ → ∞. The derivative of V along the trajectories is given as:

V˙<!-- ? -->=2λ<!-- ? -->1x1(−<!-- - -->x1+3x12x2)+2λ<!-- ? -->2x2(−<!-- - -->x2)=−<!-- - -->2λ<!-- ? -->1x12−<!-- - -->2λ<!-- ? -->2x22+6λ<!-- ? -->1x12x2 $$\begin{array}{} \displaystyle \dot{V} =2\lambda_1 x_1 (-x_1+3x_1^2 x_2 )+2\lambda_2 x_2 (-x_2 )=-2\lambda_1 x_1^2-2\lambda_2 x_2^2+6\lambda_1 x_1^2 x_2 \end{array}$$(8)

Henceforth, if V̇ (x) < 0, V ↓ along the solution of ẋ = 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).

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<!-- ? -->¯<!-- ? -->) $\begin{array}{} (\{X_i,\succeq_i \}_{i=1}^I,\{Y_j,\}_{j=1}^J,\bar{\omega} ) \end{array}$

We assume that ∑_i ω ≪ 0 (level of income or wealth is much larger than zero)

H(p,α<!-- a -->)={x∈<!-- ? -->Rn|pH0=α<!-- a -->} $$\begin{array}{} \displaystyle H(p,\alpha)=\{x\in R^n | p H0=\alpha\} \end{array}$$(9)

Hyperplane in Rⁿ can be described by an equation ∑<!-- ? -->i=1n $\begin{array}{} \sum_{i=1}^{n} \end{array}$ p_i x_i = α, here vector p ∈ Rⁿ 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, p ⋅ x∗ = α and, p ⋅ y∗ = 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: {x ∈ Rⁿ |pH0 ≤ α} or {x ∈ Rⁿ |pH0 ≥ α}. First expression, from previous two is example of budget set. If the preferences are assumed convex then every u_i is concave and under this hypothesis every set U_i is concave. First, a vector x = (x₁, …, x_n) ∈ Rⁿ is an allocation if ∑_i x_i ≤ ω, (Mas-Colell 1989).In previous expression ω ∈ R+n $\begin{array}{} R_+^n \end{array}$ is a total endowment of commodities, and production is allowed. Utility function is given as; U = {u ∈ Rⁿ : u ≤ u(x)}. Previous function can be generalized that if preferences are nonconvex i.e. they are concave and their function is concave i.e. f : A → R is nonconvex, in other expression this is translated to: {(x, y) ∈ Rⁿ⁺¹ : y ≤ f(x), x ∈ A}, and the last set is convex. Let is suppose that x ∈ B is an extreme point of the convex set ⊂ Rⁿ, this means that x ≠ λ x + (1 − λ) x^′, x, x′ ∈ A and λ ∈ [0, 1]. Separating hyperplane theorem can be stated as follows: Let is suppose that B ⊂ Rⁿ is a convex and closed set and x ∉ B, ∃ p ∈ Rⁿ, p ≠ 0, α ∈ R, p ⋅ x > α, p ⋅ y < α, ∀ y ∈ B. Convex sets A, B ⊂ Rⁿ are disjoint A ∩ B = ∅, ∃ p ∈ Rⁿ, p ≠ 0, p ⋅ x > α, p ⋅ y < α, ∀ y ∈ B.Then there is a hyperplane that separates A and B, leaving them on different sides of it. In support of this theorem if B ⊂ Rⁿ is convex and x ∉ int B, ∃ p ∈ Rⁿ, p ⋅ x ≥ p ⋅ y, p ≠ 0. If A and B are convex, A − B ∉ 0, A ∩ B = ∅. Let is say that S ∈ R^m if z∗ is a boundary point of set S, ∃ p ≠ 0, z ∈ S → p ⋅ z ≤ p ⋅ z∗, proof of this will come from a simple lemma: if S is a closed and convex set

Its complement is an open set. Closed set is defined as a set that contains all of its limit points.

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 : A → Rⁿ on the open and convex set A ⊂ Rⁿ 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 : R^l → Rⁿ and h : R^l → R^m are C¹, constraint set is given as : E = {x ∈ R^l : h(x) ≥ 0} (Brown 1991).

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

If x is a local weak maximum of f subject to h, then there are (λ, μ) ∈ R+n×<!-- ? -->R+m $\begin{array}{} R_+^n\times R_+^m \end{array}$ so that:

(λ, μ) ≠ 0

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, $\begin{array}{} max_{x_1,x_2 }x_1^{1/2} x_2^{1/2}, \end{array}$ 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

Euclidan space that is closed and bounded (all points lie within some fixed distance between each other).

From here follows Shapley -Folkman theorem in general terms:

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

Commodity space will be given as: R+L $\begin{array}{} R_+^L \end{array}$

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

1m∑<!-- ? -->i=1m|p⋅<!-- ? -->(xi−<!-- - -->ω<!-- ? -->i)|≤<!-- = -->2Lm{max||ω<!-- ? -->1||∞<!-- 8 -->,…<!-- ? -->,||ω<!-- ? -->m||∞<!-- 8 -->} $$\begin{array}{} \displaystyle \frac{1}{m} \sum_{i=1}^m |p\cdot (x_i-\omega_i )| \leq\frac{2L}{m} \{max ||\omega_1 ||_\infty,\ldots, ||\omega_m ||_\infty \} \end{array}$$(11)

1m∑<!-- ? -->i=1m|inf{p⋅<!-- ? -->(y−<!-- - -->xi):y≻<!-- ? -->ixi}|≤<!-- = -->4Lm{max||ω<!-- ? -->1||∞<!-- 8 -->,…<!-- ? -->,||ω<!-- ? -->m||∞<!-- 8 -->} $$\begin{array}{} \displaystyle \frac{1}{m} \sum_{i=1}^m |inf \{p\cdot (y-x_i ):y\succ_i x_i \} | \leq\frac{4L}{m}\{max ||\omega_1 ||_\infty,\ldots, ||\omega_m ||_\infty \} \end{array}$$(12)

In previous expression ∥ω∥_∞ denotes ∑<!-- ? -->i=1m $\begin{array}{} \sum_{i=1}^{m} \end{array}$ |ωⁱ|

Actually ∑<!-- ? -->i=1m|ω<!-- ? -->i|=∑<!-- ? -->i∈<!-- ? -->Sxi,, $\begin{array}{} \sum_{i=1}^m |\omega^{i} | =\sum_{i\in S} x_i^{,}, \end{array}$ where xi,=ω<!-- ? -->i+zi+zS $\begin{array}{} x_i^{,}=\omega_i+z_i+\frac{z}{S} \end{array}$

p⋅<!-- ? -->(xi−<!-- - -->ω<!-- ? -->i)=0⇒<!-- ? -->xi∈<!-- ? -->B(p),y≻<!-- ? -->ixi⇒<!-- ? -->p⋅<!-- ? -->y≥<!-- = -->p⋅<!-- ? -->ω<!-- ? -->i $$\begin{array}{} \displaystyle p\cdot (x_i-\omega_i )=0 \Rightarrow x_i\in B(p), \, y\succ_i x_i\Rightarrow p\cdot y\geq p\cdot \omega_i \end{array}$$(13)

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<!-- f -->(p,x,≻<!-- ? -->)=|inf><!-- > -->p⋅<!-- ? -->(y−<!-- - -->xi):y≻<!-- ? -->ixi}| $$\begin{array}{} \displaystyle \varphi(p,x,\succ )=|inf\gt p\cdot (y-x_i ):y\succ_i x_i \} | \end{array}$$(14)

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, ∃ x ∈ D(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: ∑_a∈A f(a) ≤ ∑_a∈A 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:

|(x1−<!-- - -->y1)−<!-- - -->(x2−<!-- - -->y2)|≤<!-- = -->A $$\begin{array}{} \displaystyle |(x^1 -y^1 )-(x^2 -y^2 )|\leq A \end{array}$$(15)

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

|x−<!-- - -->y|2≤<!-- = -->∑<!-- ? -->S∈<!-- ? -->F′[rad(S)]2≤<!-- = -->L2=mL2≤<!-- = -->nL2 $$\begin{array}{} \displaystyle |x-y|^2\leq \sum_{S\in F'}[rad(S)]^2\leq L^2=mL^2\leq nL^2 \end{array}$$(16)

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:

F′⊂<!-- ? -->F,x∈<!-- ? -->con∑<!-- ? -->S∈<!-- ? -->F′S,y∈<!-- ? -->∑<!-- ? -->S∈<!-- ? -->F′S,|x−<!-- - -->y|≤<!-- = -->Ln $$\begin{array}{} \displaystyle F^{'}\subset F,x\in con\sum _{S\in F'} S,y\in \sum_{S\in F'} S, |x-y|\leq L\sqrt n \end{array}$$(17)

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

By the separating hyperplane theorem for ∃ p∗ ∈ △, Z(p∗) ∩ R−<!-- - -->n $\begin{array}{} \displaystyle R_-^n \end{array}$ ≠ ∅, then ∃ q^∗ ∈ Rⁿ ∖ {0}, upsy ∈ R−<!-- - -->n $\begin{array}{} \displaystyle R_-^n \end{array}$ q∗ ⋅ y ≤ nfsz ∈ Z(p) q∗ ⋅ z. Furthermore ∈ △, q₁ ⋅ z > 0, q₂ ⋅ z > 0, ∀ λ ∈ [0, 1], λ q₁ + (1 − λ) q₂ ⋅ z > 0, q∗ ∈ F(p), where F(p), q ∈ △ inverse for F⁻¹ (q) = {p ∈ △ : Z(p) ⊂ {z : q ⋅ z > 0}}. And, p∗ = f(p) ∈ F(p), p∗ ⋅ z > 0, ∃ ∀ z ∈ Z(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 z ∈ K, 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 : [ω_H, ω_L] → R. It is assumed that F ∈ C² ((ω_H, ω_L)) and f ≡ F^′ > 0, on ω_H. 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(n−<!-- - -->1)∑<!-- ? -->j=1n1vj(b)−<!-- - -->b)−<!-- - -->1vi(b)−<!-- - -->b],i=1,…<!-- ? -->,n $$\begin{array}{} \displaystyle \small{v'_i (b)=\frac{F_i (v_i (b))}{f_i (v_i (b)}=[(\frac{1}{(n-1)} \sum_{j=1}^n \frac{1}{v_j (b)-b)} -\frac{1}{v_i (b)-b}], i=1,\ldots, n} \end{array}$$(18)

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

b(vi)=v−<!-- - -->1Fn−<!-- - -->1(v)∫<!-- ? -->rvn−<!-- - -->1(s)ds $$\begin{array}{} \displaystyle b(v_i )=v-\frac{1}{F^{n-1} (v)} \int _{r}^{v}{}^{n-1}(s) ds \end{array}$$(19)

Also

∫<!-- ? -->rvn−<!-- - -->1(s)xs=∫<!-- ? -->rvsn−<!-- - -->1ds $$\begin{array}{} \displaystyle \int_{r}^{v}{}^{n-1} (s)xs=\int_{r}^{v}s^{n-1} ds \end{array}$$(20)

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

β<!-- ? -->(v)=x−<!-- - -->∫<!-- ? -->0vF(y)F(x)dy $$\begin{array}{} \displaystyle \beta(v)=x-\int_{0}^{v}\frac{F(y)}{F(x)} dy \end{array}$$(21)

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

b¯<!-- ? -->=b(1)=1−<!-- - -->∫<!-- ? -->01n−<!-- - -->1(s)ds $$\begin{array}{} \displaystyle \bar{b }=b(1)=1-\int_{0}^{1}{}^{n-1}(s) ds \end{array}$$(22)

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

∂<!-- ? -->∪<!-- ? -->i(b;vi)∂<!-- ? -->b=(vi−<!-- - -->b)∑<!-- ? -->j=1,j≠<!-- ? -->1n(∏<!-- ? -->k=1,k≠<!-- ? -->1nFk(vk(b)))zj(vj(b))vj′(b)−<!-- - -->∏<!-- ? -->j=1,j≠<!-- ? -->1nj(vj(b))=0 $$\begin{array}{} \displaystyle \small{\frac{\partial \cup_i (b;v_i )}{\partial b}=(v_i-b) \sum_{j=1,j\neq1 }^{n} ( \prod_{k=1,k\neq1}^nF_k (v_k (b)) ) z_j (v_j (b)) v_j^{'} (b)-\prod_{j=1,j\neq1}^{n}{}_j (v_j (b))=0} \end{array}$$(23)

v_i is given fixed, and the maximization problem is:

maxb∪<!-- ? -->i(b;vi)=(vi−<!-- - -->b)∏<!-- ? -->j=1,j≠<!-- ? -->1nFj(vj(b)),i=1,…<!-- ? -->n $$\begin{array}{} \displaystyle max_{b} \cup_i (b;v_i)= (v_i-b) \prod_{j=1,j\neq1}^{n}F_j (v_j (b)), i=1, \ldots n \end{array}$$(24)

∑<!-- ? -->j=1,j≠<!-- ? -->1nfj(vj(b)vj′(b)Fj(vj(b))−<!-- - -->1vi(b)−<!-- - -->b,i=1,…<!-- ? -->n $$\begin{array}{} \displaystyle \sum_{j=1,j\neq 1 }^{n}\frac{f_j (v_j (b) v_j^{'} (b)}{F_j (v_j (b))}-\frac{1}{v_i (b)-b}, i=1,\ldots n \end{array}$$(25)

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

π<!-- p -->i=(vi−<!-- - -->bi)F(β<!-- ? -->−<!-- - -->1(b1))n−<!-- - -->1 $$\begin{array}{} \displaystyle \pi_i=(v_i-b_i)F(\beta^{-1} (b_1 ))^{n-1 } \end{array}$$(26)

And ∂<!-- ? -->π<!-- p -->i∂<!-- ? -->bi=0,dydx=∂<!-- ? -->π<!-- p -->i∂<!-- ? -->vi=F(β<!-- ? -->−<!-- - -->1(b1))n−<!-- - -->1. $\begin{array}{} \displaystyle \frac{\partial \pi_i}{\partial b_i}=0, \frac{dy}{dx} =\frac{\partial\pi_i}{\partial v_i }=F(\beta^{-1} (b_1 ))^{n-1 }. \end{array}$ 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)=ω<!-- ? -->l+vi−<!-- - -->ω<!-- ? -->l1+γ<!-- ? -->ic(vi−<!-- - -->ω<!-- ? -->l)2,∀<!-- ? -->i=1,..,n $$\begin{array}{} \displaystyle b_i^f (v_i)=\omega_l +\frac{v_i-\omega_l}{\sqrt{1+\gamma_i c(v_i-\omega_l )^2}}, \forall i=1,..,n \end{array}$$(27)

c:=1(b1−<!-- - -->ω<!-- ? -->l)2−<!-- - -->1(b2−<!-- - -->ω<!-- ? -->l)2 $$\begin{array}{} \displaystyle c:=\frac{1}{(b_1-\omega_l )^2} - \frac{1}{(b_2-\omega_l )^2} \end{array}$$(28)

Where γ_i ∈ [–1, 1], is the PDF, ω_l 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

In 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.

v1(b)=β<!-- ? -->24(β<!-- ? -->−<!-- - -->b) $$\begin{array}{} \displaystyle v_1 (b)=\frac{\beta^2}{4(\beta-b) } \end{array}$$(29)

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

∫<!-- ? -->0ω<!-- ? -->hx⋅<!-- ? -->xω<!-- ? -->hdx+∫<!-- ? -->ω<!-- ? -->h1xdx+∫<!-- ? -->0ω<!-- ? -->hy2dy=ω<!-- ? -->h23+(12−<!-- - -->ω<!-- ? -->h22)+ω<!-- ? -->h33=12+16ω<!-- ? -->h2+13ω<!-- ? -->h3 $$\begin{array}{} \displaystyle \int_{0}^{\omega_h} x\cdot \frac{x}{\omega_h } dx+\int_{\omega_h}^1x dx+\int_0^{\omega_h}y^2 dy=\frac{\omega_h^{2}}3+(\frac{1}{2}-\frac{\omega_h^{2}}{2})+\frac{\omega_h^{3}}{3}=\frac{1}{2} +\frac{1}{6} \omega_h^2+\frac{1}{3} \omega_h^3 \end{array}$$(30)

And E(v_1,2) = 12 $\begin{array}{} \displaystyle \frac{1}{2} \end{array}$ (1 – ω_h). In general case as in Plum (Plum 1992), for the probability densities φ_i(x) = c_i(x – ω_l)^μ, ω_l < x < β_i, i = 1, 2 … n, there exists and equilibrium solutions:

f1(x)=ω<!-- ? -->l+1−<!-- - -->[1−<!-- - -->c(vi−<!-- - -->ω<!-- ? -->l)k]1−<!-- - -->1kc(vi−<!-- - -->ω<!-- ? -->l)k−<!-- - -->1,ω<!-- ? -->l<<!-- ? -->x≤<!-- = -->β<!-- ? -->1 $$\begin{array}{} \displaystyle f_1 (x)=\omega_l +\frac{1-[1-c(v_i-\omega_l )^k ]^{1-\frac{1}k}}{c(v_i-\omega_l )^{k-1 }}, \omega_l \lt x\leq \beta_1 \end{array}$$(31)

f2(x)=ω<!-- ? -->l+1−<!-- - -->[1−<!-- - -->c(vi−<!-- - -->ω<!-- ? -->l)k]1−<!-- - -->1kc(vi−<!-- - -->ω<!-- ? -->l)k−<!-- - -->1,ω<!-- ? -->l<<!-- ? -->x≤<!-- = -->β<!-- ? -->2 $$\begin{array}{} \displaystyle f_2 (x)=\omega_l +\frac{1-[1-c(v_i-\omega_l )^k ]^{1-\frac{1}k}}{c(v_i-\omega_l )^{k-1 }}, \omega_l \lt x\leq\beta_2 \end{array}$$(32)

Where c:=1(b1−<!-- - -->ω<!-- ? -->l)k−<!-- - -->1(b2−<!-- - -->ω<!-- ? -->l)k $\begin{array}{} \displaystyle c:=\frac{1}{(b_1-\omega_l )^k} -\frac{1}{(b_2-\omega_l )^k } \end{array}$; or ci:=μ<!-- ? -->+1β<!-- ? -->i−<!-- - -->ω<!-- ? -->l)μ<!-- ? -->+1, $\begin{array}{} \displaystyle c_i :=\frac{\mu+1}{\beta_i-\omega_l )^{\mu+1 }}, \end{array}$ where μ > –1, and k := ((2 – λ + μ))/(1 – λ), λ ∈ [0, 1]. There are k_i - bidders in group i, in total N = ∑<!-- ? -->i=1n $\begin{array}{} \sum_{i=1}^n \end{array}$ k_i. 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,ω<!-- ? -->h)(v−<!-- - -->u)⋅<!-- ? -->[Fi(λ<!-- ? -->i(u))]ki−<!-- - -->1∏<!-- ? -->j≠<!-- ? -->1[Fj(λ<!-- ? -->j(u))]kj $$\begin{array}{} \displaystyle \beta(v)=arg \, max_{u\in(0,\omega_h ) } (v-u)\cdot [F_i (\lambda_i (u))]^{k_i-1} \prod_{j\neq1 } [F_j (\lambda_j (u))]^{k_j } \end{array}$$(33)

∃u = ∑<!-- ? -->i=1n $\begin{array}{} \sum_{i=1}^n \end{array}$ u_i, where u_i denotes the player of type i. Truncated CDF in general form is given as:

F∗<!-- * -->(v)=∏<!-- ? -->j=1nFj(v)−<!-- - -->Fj(ω<!-- ? -->l)Fj(ω<!-- ? -->h)−<!-- - -->Fj(ω<!-- ? -->l) $$\begin{array}{} \displaystyle F^* (v)=\prod_{j=1}^n \frac{ F_j (v)-F_j (\omega_l)}{F_j (\omega_h )-F_j (\omega_l)} \end{array}$$(34)

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

pi(r)=ki∫<!-- ? -->rb(ω<!-- ? -->h)li′(v)li(v)∏<!-- ? -->j=1n[lj(v)]kjdv $$\begin{array}{} \displaystyle p_i (r)=k_i \int_{r}^{b(\omega_h)}\frac{l_i^{' }(v)}{l_i (v)} \prod_{j=1}^{n}[l_{j }(v)]^{k_j } dv \end{array}$$(35)

Where l_i(v) = F_i(λ_i(v)). Also, r represents the reserve price in auction. Expected revenue for the auctioneer is given by:

E(p,bi,vi)=ω<!-- ? -->h−<!-- - -->r∏<!-- ? -->j=1n[Fj(r)]kj−<!-- - -->∫<!-- ? -->rb(ω<!-- ? -->h)li′(v)li(v)∏<!-- ? -->j=1n[lj(v)]kjdv $$\begin{array}{} \displaystyle E(p,b_i,v_i )=\omega_h-r\prod_{j=1}^{n} [F_j (r)]^{k_j } -\int_r^{b(\omega_h)} \frac{l_i^{'} (v)}{l_i (v)} \prod_{j=1}^n [l_j (v)]^{k_j }dv \end{array}$$(36)

Group i bidders expected revenue is given by:

Ei(p,bi,vi)=ki∫<!-- ? -->rb(ω<!-- ? -->h)[Fi−<!-- - -->1li(v))−<!-- - -->v]⋅<!-- ? -->li′(v)(li(v)∏<!-- ? -->j=1n[lj(v)]kjdv $$\begin{array}{} \displaystyle E_i (p,b_i,v_i ) =k_i \int_r^{b(\omega_h)}[F_i^{-1} l_i (v))-v]\cdot \frac{l_i^{'} (v)}{(l_i (v)} \prod_{j=1}^n[l_j (v)]^{k_j } dv \end{array}$$(37)

Now if U(p_i, E_i, r) = p_i ⋅ (r – E_i), 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 $$\begin{array}{} \displaystyle U^{*} (x)=\int_0^rp^{*} (v)dv \end{array}$$(38)

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(v_i – b)||(β_–i < b_i)|v₁, as in Campo, Perrigne and Vuong (Campo, Perrigne et al. 2003). Now, β<!-- ? -->−<!-- - -->i=max{v1(y1i∗<!-- * -->),v0(y0i∗<!-- * -->)},y1i∗<!-- * -->=maxj≠<!-- ? -->i,j∈<!-- ? -->G1vi=1,j, $\begin{array}{} \displaystyle \beta_{-i}=max\{v_1 (y_{1i}^* ),v_0 (y_{0i}^{*} ) \},y_{1i}^{*}=max_{j\neq i,j\in G_1}v_{i=1,j}, \end{array}$ where G₁ is a cartel of better informed bidders and y0i∗<!-- * -->=maxj≠<!-- ? -->i,j∈<!-- ? -->G0vi=0,j,G0 $\begin{array}{} \displaystyle y_{0i}^*=max_{j\neq i,j\in G_0}v_{i=0,j}, G_0 \end{array}$ 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=(v1i−<!-- - -->b1i)Prob(y1i∗<!-- * -->≤<!-- = -->v1−<!-- - -->1(b1i))∧<!-- ? -->y0i≤<!-- = -->v0−<!-- - -->1(b1i)|v1i $$\begin{array}{} \displaystyle max_{b_{1 i}}=(v_{1i}-b_{1i })Prob(y_{1i}^*\leq v_1^{-1} (b_{1i} )) \wedge y_{0i} \leq v_0^{-1} (b_{1i} )|v_{1i} \end{array}$$(39)

Probability can be written as: Fy1∗<!-- * -->,y0|v1(v1−<!-- - -->1(b1i)),v0−<!-- - -->1(b1i)|v1i. $\begin{array}{} \displaystyle F_{y_1^*,y_0 |v_1 } (v_1^{-1} (b_{1i} )),v_0^{-1} (b_{1i} )|v_{1i}. \end{array}$ FONC for v_1i bidder is given as:

−<!-- - -->Fy1∗<!-- * -->,y0|v1(v1−<!-- - -->1(b1i)),v0−<!-- - -->1(b1i)|v1i+ $$\begin{array}{} \displaystyle -F_{y_1^*,y_0 |v_1 } (v_1^{-1} (b_{1i} )),v_0^{-1} (b_{1i} )|v_{1i}+ \end{array}$$

(v1i−<!-- - -->b1i)[(∂<!-- ? -->Fy1∗<!-- * -->,y0|v1(v1−<!-- - -->1(b1i)),v0−<!-- - -->1(b1i)|v1i∂<!-- ? -->y1i∗<!-- * -->⋅<!-- ? -->1v1′(v1−<!-- - -->1(b1i))+(∂<!-- ? -->Fy1∗<!-- * -->,y0|v1(v1−<!-- - -->1(b1i)),v0−<!-- - -->1(b1i)|v1i∂<!-- ? -->y0⋅<!-- ? -->1(v0′(v0−<!-- - -->1(b1i))] $$\begin{array}{c} \displaystyle (v_{1i}-b_{1i} )[ \frac{(\partial F_{y_1^*,y_0 |v_1 } (v_1^{-1} (b_{1i} )),v_0^{-1} (b_{1i} )|v_{1i}}{\partial y_{1i}^* }\cdot \frac{1}{v_1^{'} (v_1^{-1}( b_{1i} )) }\\ \displaystyle+\frac{(\partial F_{y_1^*,y_0 |v_1 } (v_1^{-1} (b_{1i} )),v_0^{-1} (b_{1i} )|v_{1i}}{\partial y_0 }\cdot \frac{1}{(v_0^{'} (v_0^{-1} (b_{1i} )) }] \end{array}$$(40)

In the previous expression v_1i ∈ [ω_l, ω_h], and b_1i = v₁(v_1i). FONC for v_0i 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 Cε is contained in the core, Cε ⊂ 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: C_s : S ⊂ N is a family of closed subsets of simplex Δ^N. Where Δ^N = {x ∈ Rⁿ : x_i > 0, ∧ ∑<!-- ? -->i=1n $\begin{array}{} \sum_{i=1}^n \end{array}$ x_i = 1}, is a n – 1 simple, for S ⊂ N, Δ^S = {x ∈ Δⁿ : x_i > 0, ∧ ∑_i∈S x_i = 1}. Now, we can assume that Δ^T ⊂ ∪_S⊂Ts, ∀T ⊂ N, ∃B, ∩_s∈B C_s ≠ ∅. Where, B is a balanced family (collection), such that intersection of sets indexed by B is nonempty (Krasa and Yannelis 1994)

In the proof of this theorem provided in this paper Krasa and Yannelis (1994) it is defined set valued function: ψ = Δ^N → 2^Rn, F(x) = con999, where mS={m1S,…<!-- ? -->,mNS} $\begin{array}{} \displaystyle m^S=\{m_1^S,\ldots,m_N^S \} \end{array}$ is the center of the simplex Δ^S. And miS=1|S|,i∈<!-- ? -->S,miS=0,i∉<!-- ? -->S. $\begin{array}{} \displaystyle m_i^S=\frac{1}{|S|}, i\in S, m_i^S=0, i \notin S . \end{array}$ And ψ(x) = {y : f(x)x > f(x)y}, ∃T ⊂ N, and intΔⁿ = {x = (x₀, …, x_n) ∈ Rⁿ⁺¹ | ∑<!-- ? -->0nxi $\begin{array}{} \sum_0^n x_i \end{array}$ = 1, x_i > 0, ∀i}, therefore since S ⊂ T, S ∈ I(x^∗) and m^S ∈ F(x^∗). And the f(x^∗)x^∗ = f(x) m^S > f(x^∗)m^N.

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: lims→<!-- ? -->0+→<!-- ? -->φ<!-- f -->k′(b)=(nk+1)nk. $\begin{array}{} \lim_{ s\rightarrow 0^+ } \rightarrow \varphi_k^{'} (b)=\frac{(n_k+1)}{n_k} . \end{array}$ Where φ<!-- f -->k′(b) $\begin{array}{} \displaystyle \varphi_k^{'} (b) \end{array}$ is a first derivative of the inverse bid function, and n_k are the number of bidders in coalition. Inverse bid functions are in practice normalized to δ<!-- d -->k(b)=φ<!-- f -->k(b)b. $\begin{array}{} \displaystyle \delta_k (b)=\frac{\varphi_k (b)}b . \end{array}$ They approximated {δ<!-- d -->k(b)}k2=1 $\begin{array}{} \displaystyle \{ \delta_k (b)\}_k^2=1 \end{array}$ by Taylor series expansions of order 5 around each point b_t ∈ [ω_L, ω_H]. Condition for valid solution is: 12∑<!-- ? -->k=12[δ<!-- d -->k(ω<!-- ? -->L)−<!-- - -->(nk+1)nk]≤<!-- = -->ε<!-- e -->2, $\begin{array}{} \frac{1}2 \sum_{k=1}^2[ \delta_k (\omega_L )-\frac{(n_k+1)}{n_k} ]\leq \varepsilon^2, \end{array}$ 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=0Kα<!-- a -->n,k(b¯<!-- ? -->−<!-- - -->b)k,n=1,2,…<!-- ? -->,N $$\begin{array}{} \displaystyle \widehat{\omega} _n (b)=\overline{b}-\sum_{k=0}^K\alpha_n, _k (\overline{b} -b)^k, n=1,2,\ldots,N \end{array}$$(41)

FOC for bidder n can be expressed as:

1=[φ<!-- f -->n(b)−<!-- - -->b]∑<!-- ? -->m≠<!-- ? -->nfm[φ<!-- f -->m(b)]Fm[φ<!-- f -->m(b)]φ<!-- f -->m′(b) $$\begin{array}{} \displaystyle 1=[\varphi_n (b)-b] \sum_{m\neq n }\frac{f_m [\varphi_m (b)]}{F_m [\varphi_m (b)]} \varphi'_m (b) \end{array}$$(42)

and

Gn(ω<!-- ? -->L,ω<!-- ? -->H,α<!-- a -->)=1−<!-- - -->[φ<!-- f -->n(b)−<!-- - -->b]∑<!-- ? -->m≠<!-- ? -->nfm[φ<!-- f -->m(b)]Fm[φ<!-- f -->m(b)]φ<!-- f -->m(b) $$\begin{array}{} \displaystyle G_n (\omega_L,\omega_H,\alpha)=1-[\varphi_n (b)-b] \sum_{m\neq n }\frac{f_m [\varphi_m (b)]}{F_m [\varphi_m (b)] } \varphi_m (b) \end{array}$$(43)

And the left boundary condition is: φ_n(ω_L) = ω_L, and the right boundary condition is φ_n(ω_H) = ω_H. 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 : Faverage≡<!-- = -->1N∑<!-- ? -->n=1NFn(v). $\begin{array}{} F_{average} \equiv \frac{1}N \sum_{n=1}^N F_n (v). \end{array}$ The parameter that measures the level of asymmetry is given as: ε = max_{n∈{1,…,N}} > ax_{v∈{ωL,ωH}}>|F_n(v) – F_average(v)|, and that F_n(v) = F_average(v)+εA_n(v), n = 1, ..N, where auxiliary function An(v)=Fn(v)−<!-- - -->Faverage(v)ε<!-- e -->. $\begin{array}{} \displaystyle A_n (v)=\frac{F_n (v)-F_{average} (v)}{\varepsilon}. \end{array}$ Equilibrium bid function

Initial guess about maximum bid function is given as: b¯<!-- ? -->=ω<!-- ? -->H−<!-- - -->∫<!-- ? -->ω<!-- ? -->Lω<!-- ? -->HFaverageN−<!-- - -->1(v)dv $\begin{array}{} \overline{b }=\omega_{H}-\int_{\omega_L}^{\omega_H}F_{average}^{N-1} (v)dv \end{array}$

En[v,An(v)]=1−<!-- - -->NFaverageN−<!-- - -->1(v)⋅<!-- ? --> $$\begin{array}{} \displaystyle E_n [v,A_n (v)]=\frac{1-N}{F_{average}^{N-1} (v) }\cdot \end{array}$$

⋅<!-- ? -->[∫<!-- ? -->ω<!-- ? -->LvFaverageN−<!-- - -->1(v)dv]N∫<!-- ? -->vω<!-- ? -->H1[FaverageN−<!-- - -->1(v)]N−<!-- - -->1d[An(t)(Faverage(t)]dtdt $$\begin{array}{} \displaystyle \cdot[\int_{\omega_L}^vF_{average}^{N-1} (v)dv]^N \int_v^{\omega_H}\frac{1}{[F_{average}^{N-1} (v)]^{N-1}} \frac{ d [ \frac{A_n (t)} {(F_{average} (t)}]}{dt}dt \end{array}$$(44)

Gayle and Richard (Gayle and Richard 2008), generalized backward shooting algorithm, they defined: l_n(v) ≡ F_n[φ_n(v)], and now FOC can be defined as: 1=(Fn−<!-- - -->1[ln(v)]−<!-- - -->v)∑<!-- ? -->m≠<!-- ? -->nln′(v)ln(v), $\begin{array}{} 1=(F_n^{-1} [l_{n} (v)]-v) \sum_{m\neq n } \frac{l'_{n} (v)}{l_n (v) }, \end{array}$ and the high bid is chosen by solving:

minb[r,ω<!-- ? -->H]∑<!-- ? -->n=1N[l(r|b−<!-- - -->)−<!-- - -->Fn(r)]2,r $$\begin{array}{} \displaystyle min_{b [r,\omega_H ] } \sum_{n=1}^N [l(r|b^-)-F_n (r)]^2, r \end{array}$$

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<!-- p -->(t−<!-- - -->1)T],t=1,…<!-- ? -->,T. $\begin{array}{} \displaystyle x_t=cos[\frac{\pi(t-1)}{T}], t=1,\ldots, T. \end{array}$ The points {vt}tT=1 $\begin{array}{} \displaystyle \{v_t \}_{t}^T=1 \end{array}$ are found via transformation like this: vt=b¯<!-- ? -->+ω<!-- ? -->L+(b¯<!-- ? -->−<!-- - -->ω<!-- ? -->L)xt2. $\begin{array}{} \displaystyle v_t=\frac{\overline{b} +\omega_L+(\overline{b}-\omega_L ) x_t}2. \end{array}$ Chebyshev polynomials can be defined recursively as T₀(x) = 1, T₁(x) = x, T_n+1(x) = 2xT_n(x) + T_n–1(x). The coefficients of these polynomials for a function f(x) can be obtained by the following integral:

an=2π<!-- p -->∫<!-- ? -->−<!-- - -->11f(x)Tn(x)(1−<!-- - -->x2)1/2dx $$\begin{array}{} \displaystyle a_n=\frac{2}\pi \int_{-1}^1 \frac{f(x) T_{n} (x)}{(1-x^2 )^{1/2}} dx \end{array}$$(45)

Hubbard, Kirkegaard and Paarsch imposed 5 in/equality constraints on the equilibrium bid functions

1. φ_n(v) = ω_L, 2. φ_n(b) = ω_H 3. ∑_m≠n(b – v) f_m(b) φ<!-- f -->m′ $\begin{array}{} \displaystyle \varphi_m^{'} \end{array}$(v) = 1, 4. φ(ω_L) = N−<!-- - -->1N $\begin{array}{} \displaystyle \frac{N-1}{N} \end{array}$, 5. φ_n(v_j – 1) ≤ φ_n(v_j), for some uniform array j = 2, …, J.

φ<!-- f -->n′(v)=Fn[φ<!-- f -->(v)]fn[φ<!-- f -->(v)]([1N−<!-- - -->1∑<!-- ? -->m=1N1φ<!-- f -->n(v)−<!-- - -->v]−<!-- - -->1φ<!-- f -->n(v)−<!-- - -->v) $$\begin{array}{} \displaystyle \varphi_n^{'} (v)=\frac{F_n [\varphi(v)]}{f_n [\varphi(v)] } ([\frac{1}{N-1} \sum_{m=1}^N \frac{1}{\varphi_n (v)-v}]-\frac{1}{\varphi_n (v)-v }) \end{array}$$(46)

And they are doing so by minimizing and solving:

minb¯<!-- ? -->,α<!-- a -->∑<!-- ? -->n=1N∑<!-- ? -->t=1T[Gn(ω<!-- ? -->L,ω<!-- ? -->H,α<!-- a -->)]2. $$\begin{array}{} \displaystyle min_{\overline{b},\alpha} \sum_{n=1}^N \sum_{t=1}^T [G_n (\omega_L,\omega_H,\alpha)]^2 . \end{array}$$