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Shapley-Folkman-Lyapunov theorem and Asymmetric First price auctions

Dushko Josheski
Dushko Josheski
, 
Elena Karamazova
Elena Karamazova
 and 
Mico Apostolov
Mico Apostolov
   | Aug 23, 2019
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Published Online: Aug 23, 2019
Page range: 331 - 350
Received: Mar 05, 2019
Accepted: Apr 25, 2019
DOI: https://doi.org/10.2478/AMNS.2019.2.00029
Keywords
© 2019 F. B. Benli et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 Public License.

Fig. 1

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

Fig. 2

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

C.S.E. and Backward shooting parameters

Constrained strategic equilibrium (no reserve price) parameters Constrained strategic equilibrium (reserve price = 0.5) parameters Backwards solver parameters
T(degree) 40 T (degree) 40 Shooting metdod Euler
K (grid) 15 K (grid) 15 ODE system Inverse bid functions
μl 2000 μl 2000 h1 (tolerance of tde deviation of tde solution from left boundary) 1.0E-5
μh 5000 μh 5000 h2 (step size close to high bid) 0.001
μfonc 5.0 μfonc 5.0 Td reshold 0.01
μb̄ 0.0 μb̄ 0.0 High-bid precision 1.0E-12
μmono 1000 μmono 1000 Left-boundary tolerance 1.0E-5
Cheb grid = no Cheb Grid = no / /

Backward shooting method solution, end result: Convergence true

b_bar A B B-A Result
0/39 0.5 0 1 1.00E+00 3 Solution not within specified tolerances.
1/39 0.75 0.5 1 5.00E-01 3 Solution not within specified tolerances.
2/39 0.875 0.75 1 2.50E-01 2 Solution diverges to +infinity.
3/39 0.8125 0.75 0.875 1.25E-01 2 Solution diverges to +infinity.
4/39 0.78125 0.75 0.8125 6.25E-02 3 Solution not within specified tolerances.
5/39 0.7968750 0.7812500 0.8125000 3.13E-02 3 Solution not within specified tolerances.
6/39 0.8046875 0.7968750 0.8125000 1.56E-02 3 Solution not within specified tolerances.
7/39 0.8085938 0.8046875 0.8125000 7.81E-03 3 Solution not within specified tolerances.
8/39 0.8105469 0.8085938 0.8125000 3.91E-03 2 Solution diverges to +infinity.
9/39 0.8095703 0.8085938 0.8105469 1.95E-03 3 Solution not within specified tolerances.
10/39 0.8100586 0.8095703 0.8105469 9.77E-04 3 Solution not within specified tolerances.
11/39 0.8103027 0.8100586 0.8105469 4.88E-04 3 Solution not within specified tolerances.
12/39 0.8104248 0.8103027 0.8105469 2.44E-04 2 Solution diverges to +infinity.
13/39 0.8103638 0.8103027 0.8104248 1.22E-04 2 Solution diverges to +infinity.
14/39 0.8103333 0.8103027 0.8103638 6.10E-05 3 Solution not within specified tolerances.
15/39 0.8103485 0.8103333 0.8103638 3.05E-05 1 Solution diverges to -infinity.
16/39 0.8103409 0.8103333 0.8103485 1.53E-05 3 Solution not within specified tolerances.
17/39 0.8103447 0.8103409 0.8103485 7.63E-06 2 Solution diverges to +infinity.
18/39+ 0.8103428 0.8103409 0.8103447 3.82E-06 3 Solution not within specified tolerances.
19/39 0.8103437 0.8103428 0.8103447 1.91E-06 2 Solution diverges to +infinity.
20/39 0.8103433 0.8103428 0.8103437 9.54E-07 2 Solution diverges to +infinity.
21/39 0.8103430 0.8103428 0.8103433 4.77E-07 3 Solution not within specified tolerances.
22/39 0.8103431 0.8103430 0.8103433 2.38E-07 2 Solution diverges to +infinity.
23/39 0.8103431 0.8103430 0.8103431 1.19E-07 3 Solution not within specified tolerances.
24/39 0.8103431 0.8103431 0.8103431 5.96E-08 3 Solution not within specified tolerances.
25/39 0.8103431 0.8103431 0.8103431 2.98E-08 3 Solution not within specified tolerances.
26/39 0.8103431 0.8103431 0.8103431 1.49E-08 3 Solution not within specified tolerances.
27/39 0.8103431 0.8103431 0.8103431 7.45E-09 2 Solution diverges to +infinity.
28/39 0.8103431 0.8103431 0.8103431 3.73E-09 2 Solution diverges to +infinity.
29/39 0.8103431 0.8103431 0.8103431 1.86E-09 3 Solution not within specified tolerances.
30/39 0.8103431 0.8103431 0.8103431 9.31E-10 2 Solution diverges to +infinity.
31/39 0.8103431 0.8103431 0.8103431 4.66E-10 3 Solution not within specified tolerances.
32/39 0.8103431 0.8103431 0.8103431 2.33E-10 2 Solution diverges to +infinity.
33/39 0.8103431 0.8103431 0.8103431 1.16E-10 3 Solution not within specified tolerances.
34/39 0.8103431 0.8103431 0.8103431 5.82E-11 3 Solution not within specified tolerances.
35/39 0.8103431 0.8103431 0.8103431 2.91E-11 2 Solution diverges to +infinity.
36/39 0.8103431 0.8103431 0.8103431 1.46E-11 3 Solution not within specified tolerances.
37/39 0.8103431 0.8103431 0.8103431 7.28E-12 2 Solution diverges to +infinity.
38/39 0.8103431 0.8103431 0.8103431 3.64E-12 2 Solution diverges to +infinity.
39/39 0.8103431 0.8103431 0.8103431 1.82E-12 3 Solution not within specified tolerances.

Selected distributions and their CDFs and PDFs

Distributions and boundaries CDF PDF
Beta [0,1] F(x)=1B(a,b)∫ωLυ(x)xa−1∗(1−x)b−1dx; $\begin{array}{} F(x)=\frac{1}{B(a,b)}\int\limits_{\omega_{L}}^{\upsilon(x)}x^{a-1}* (1-x)^{b-1}dx; \end{array}$ f(x)=1ωH−ωLυ(x)a−1(1−v(x))b−1B(a,b) $\begin{array}{} \displaystyle f(x)=\frac{1}{\omega_{H}-\omega_{L}}\frac{\upsilon(x)^{a-1}(1-v(x))^{b-1}}{B(a,b)} \end{array}$
υ(x)=x−ωLωH−ωL $\begin{array}{} \displaystyle \upsilon(x)=\frac{x-\omega_{L}}{\omega_{H}-\omega_{L}} \end{array}$
Exponential [0,1] F(x)=1−exp⁡(−λ(x−ωL)1−exp⁡(−λ(ωH−ωL) $\begin{array}{} \displaystyle F(x)=\frac{1-\exp (-\lambda(x-\omega_{L})}{1-\exp (-\lambda(\omega_{H}-\omega_{L})} \end{array}$ f(x)=λexp⁡(−λ(x−ωL)1−exp⁡(−λ(ωH−ωL) $\begin{array}{} \displaystyle f(x)=\frac{\lambda \exp (-\lambda(x-\omega_{L})}{1-\exp (-\lambda(\omega_{H}-\omega_{L})} \end{array}$
Gamma [0,1] F(x)=∫0∞xk−1e−xdxΓ(k) $\begin{array}{} \displaystyle F(x)=\frac{\int\nolimits_{0}^{\infty}x^{k-1}e^{-x}dx}{\Gamma(k)} \end{array}$ f(x)=1θkΓ(k)xk−1e−xθ $\begin{array}{} \displaystyle f(x)=\frac{1}{\theta^{k} \Gamma(k)}x^{k-1}e^{-\frac{x}{\theta}} \end{array}$
Kumaraswamy [0,1] F(x; a; b) = 1 –(1–xa)b f(x; a, b)=F′(x; a; b)=abxa–1(1–xa)b-1
Lognormal [0,1] F(x)=∫ax1zσ2πexp−12lnz−μσ2dz∫ab1zσ2πexp−12lnz−μσ2dz $\begin{array}{} \displaystyle F(x)=\frac{\int_{a}^{x}\displaystyle\frac{1}{z\sigma\sqrt{2\pi}}exp\left[-\frac{1}{2}\left(\frac{lnz-\mu}{\sigma} \right)^{2} \right]dz}{\int_{a}^{b}\displaystyle\frac{1}{z\sigma\sqrt{2\pi}}exp\left[-\displaystyle\frac{1}{2}\left(\frac{lnz-\mu}{\sigma} \right)^{2} \right]dz} \end{array}$ f(x)=1xσ2πexp−12lnx−μσ2∫ab1zσ2πexp−12lnz−μσ2dz $\begin{array}{} \displaystyle f(x)=\frac{\displaystyle\frac{1}{x\sigma\sqrt{2\pi}}exp\left[-\frac{1}{2}\left(\frac{lnx-\mu}{\sigma} \right)^{2} \right]}{\int_{a}^{b}\displaystyle\frac{1}{z\sigma\sqrt{2\pi}}exp\left[-\displaystyle\frac{1}{2}\left(\frac{lnz-\mu}{\sigma} \right)^{2} \right]dz} \end{array}$
Standard normal [0,1] F(x)=Φx−μσ−Φa−μσΦb−μσ−Φa−μσ $\begin{array}{} \displaystyle \rm{F(x)=\frac{\displaystyle\Phi\left(\frac{x-\mu}{\sigma} \right)-\Phi\left(\frac{ a-\mu}{\sigma}\right)}{\displaystyle\Phi\left(\frac{ b-\mu}{\sigma} \right)-\Phi\left(\frac{ a-\mu}{\sigma}\right)}} \end{array}$ f(x)=1σϕx−μσΦω−μσΦ−μσ $\begin{array}{} \displaystyle f(x)={\rm\frac{\displaystyle\frac{1}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right)}{\displaystyle\Phi\left(\frac{\omega-\mu}{\sigma}\right)\Phi\left(\frac{-\mu}{\sigma}\right)}} \end{array}$
Power [0,1] F(x)=ηα+1(x+a+c)α+1−ca∓1 $\begin{array}{} \displaystyle F(x)=\frac{\eta}{\alpha+1}\left[(x+a+c)^{\alpha+1}-c^{a \mp 1}\right] \end{array}$ f(x)=η(x+a+c)α,
η=(α+1)[x–a+c)α+1–cα+1]–1
Reverse power [0,1] F(x)=1−b−xb−aα $\begin{array}{} \displaystyle F(x)=1-\left(\frac{b-x}{b-a}\right)^{\alpha} \end{array}$ f(x)=α(b−x)α−1b−a $\begin{array}{} \displaystyle f(x)=\frac{\alpha(b-x)^{\alpha-1}}{b-a} \end{array}$
Triangular [0,1] F(X)=ψx+(1−ψ)(x−a)2(b−a)(c−a) $\begin{array}{} \displaystyle F(X)=\psi x+(1-\psi)\frac{(x-a)^{2}}{(b-a)(c-a)} \end{array}$ f(x)=F′(x)=4x if x∈(0,…,0,5)4−4xif x∈(0.5,…,1) $\begin{array}{} \displaystyle f(x)=F^{\prime}(x)=\left\{\begin{array}{c}4x ~if~ x \in (0,\ldots,0,5)\\ 4-4x \,if~x \in (0.5,\ldots,1) \end{array}\right. \end{array}$
Uniform [0,1] F(x)=x−ωLωH−ωL $\begin{array}{} \displaystyle F(x)=\frac{x-\omega_{L}}{\omega_{H}-\omega_{L}} \end{array}$ f(x)=1ωH−ωL $\begin{array}{} \displaystyle f(x)=\frac{1}{\omega_{H}-\omega_{L}} \end{array}$

Solution: Chebyshev coefficients of K=15 degree

Beta distribution Exponential distribution Gamma distribution Kumaraswamy distribution Log nomal distribution Standard normal distribution Power l distribution Reverse power distribution Triangular distribution Uniform distribution
0.0020 0.0031 0.0021 0.0020 0.0063 0.0032 0.0031 0.0011 0.0028 0.0033
0.9845 0.9827 0.9813 0.9845 0.9886 0.9817 0.9807 0.9973 0.9823 0.9831
0.0024 0.0035 0.0025 0.0024 0.0061 0.0036 0.0034 -0.0004 0.0028 0.0038
0.0080 0.0104 0.0112 0.0080 0.0037 0.0103 0.0111 0.0144 0.0088 0.0100
0.0000 0.0000 0.0000 0.0000 -0.0006 0.0000 0.0000 -0.0006 -0.0003 0.0000
-0.0011 -0.0003 0.0005 -0.0011 -0.0010 0.0001 0.0005 -0.0066 0.0010 -0.0009
0.0000 0.0000 0.0000 0.0000 -0.0004 0.0000 0.0000 -0.0003 0.0000 0.0000
0.0008 0.0007 0.0008 0.0008 -0.0018 0.0007 0.0009 -0.0048 0.0014 0.0004
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0014 0.0000 0.0000
0.0004 -0.0001 0.0000 0.0004 -0.0007 0.0000 0.0000 -0.0025 -0.0006 0.0001
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0007 0.0000 0.0001 0.0007 -0.0001 0.0003 0.0001 -0.0004 0.0001 0.0002
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

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

Beta distribution Exponential distribution Gamma distribution Kumaraswamy distribution Log nomal distribution Standard normal distribution Power l distribution Reverse power distribution Triangular distribution Uniform distribution
0.4432 0.4224 0.4378 0.3424 0.2980 0.3970 0.4438 0.3830 0.4010 0.4123
0.6213 0.6570 0.6284 0.7916 0.8710 0.7012 0.6191 0.7359 0.6901 0.6757
-0.0764 -0.1015 -0.0877 -0.1854 -0.2211 -0.1301 -0.0803 -0.0853 -0.1231 -0.1121
0.0190 -0.1015 0.0298 0.0611 0.0615 0.0447 0.0255 -0.0338 0.0369 0.0358
0.0013 -0.0031 -0.0049 -0.0073 0.0029 -0.0064 -0.0032 0.0000 -0.0027 -0.0035
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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