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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7141813b616ed4363675fa73f14e945bf1c57b75 | subsection | 97 | 127 | Complementary results for estimating the parameter | We suppose that the covariance function r_z of Z is known.
We can estimate the density f_U by \widehat{f}_U(t)=f_V(\widehat{\tau }_{n}t), where \widehat{\tau }_{n} is a consistent estimator of \tau , the common value of the eigenvalues of matrix A under the isotropic case and where V has the same law as that of U, wher... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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-0.011436873115599155,... | |
edc370ab41562d525eaa8b22440d020b4e7018b6 | subsection | 98 | 127 | Complementary results for estimating the parameter | By Theorem REF we already know that if
0 < \lambda < 1 and -\frac{\pi }{2} < \theta _{o} < \frac{\pi }{2},2n \left({
\widehat{\lambda }_{n} - \lambda ; \,\widehat{\theta }_{o,n} - \theta _{o}
}\right)
\xrightarrow[n \rightarrow +\infty ]{Law} \mathcal {N}(0; \mathit {\Sigma }_{\lambda , \theta _{o}}(u)).So let us look ... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... | |
c422d114ef07882ca6b568dbe825d930626b7a04 | subsection | 99 | 127 | Complementary results for estimating the parameter | In this aim, remember that
\mathit {\Sigma }_{\lambda , \frac{\pi }{2}}(u)=C(\lambda , \frac{\pi }{2}) \times \mathit {\Sigma }^{(\star )}(u) \times C^{t}(\lambda , \frac{\pi }{2})
withC(\lambda , \frac{\pi }{2})= \frac{1}{J_F(\lambda , \frac{\pi }{2})}
\begin{pmatrix}
\frac{\partial F_2}{\partial \theta _o}(\lambda ... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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-0.015473... | |
4683f59b36ffdcb7c00ab0cee240eaeb83cf290d | subsection | 100 | 127 | Complementary results for estimating the parameter | Now we consider the second part taking \lambda =1 (and -\frac{\pi }{2} < \theta _{o} \leqslant \frac{\pi }{2}).The decomposition obtained in (REF ) gives2n\left({X_n -\tfrac{2}{\pi }}\right) \left({X_n +\tfrac{2}{\pi }}\right) \times \\2n \left[{ I^4(\widehat{\lambda }_{n}) \,(X_n^2+Y_n^2) +(\tfrac{2}{\pi })^2 \,I^4(\w... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0... | |
79811e487356b784e7f2138d07e1faa4454b4f60 | subsection | 101 | 127 | Complementary results for estimating the parameter | Indeed, by the first part of Theorem REF and the latter almost sure convergence result, we deduce thatZ_n=I^4(\widehat{\lambda }_{n}) \,(X_n^2+Y_n^2) +(\frac{2}{\pi })^2 \,I^4(\widehat{\lambda }_{n}) -I^2(\widehat{\lambda }_{n})\,(\widehat{\lambda }_{n}^2+1) \xrightarrow[n\rightarrow +\infty ]{a.s.} 0,thus we have to r... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.030970878899097443,
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-0.012510403990745544,
... | |
b11d2493edf215856452c6bf5ba8f7d4319411dd | subsection | 102 | 127 | Complementary results for estimating the parameter | This ends the proof of this theorem.
\BoxProof of Remark REF .
The density f_U of the positive random variable U can be expressed as f_U(t)=f_V(t \tau ^2), where \tau is the common eigenvalue of matrix A under the isotropic case and V is defined as U, substituting Q \mathit {\Sigma }_{\star }(u, 1, I_2) Q^t to \mathit... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.028029995039105415,
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-0.0035171005874872208,
-0.002355541801080107... | |
9cd3a9a34d760387fb5f0f4ab560f517ac8b66bf | subsection | 103 | 127 | Testing the isotropy | We testH_0: \lambda =1\qquad \mbox{against} \qquad H_1: \lambda <1.We still obtain a way to detect the possible isotropy of the process via the following corollaries.
For -\frac{\pi }{2} < \theta _{o} \leqslant \frac{\pi }{2}, under the hypothesis H_0, the following convergence holds:\frac{J_{f^{\star }}^{(n)}(u)}{J_... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.06603100895881653,
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0.017423154786229134,
0.0... | |
b3ec7e59bf317bcf0aa52688e7ccada3f8769165 | subsection | 104 | 127 | Testing the isotropy | Appling Proposition REF , under H_1, the test statistic T_{f^{\star }}^{(n)}(u) converges in law to a Gaussian random variable with asymptotically mean equivalent to 2n\left({
\displaystyle \frac{{\rm E}_{{}}\hspace{-2.0pt}\left[{J_{f^{\star }}^{(n)}(u)}\right]}{{\rm E}_{{}}\hspace{-2.0pt}\left[{J_{\bf 1}^{(n)}(u)}\rig... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.05135539546608925,
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0.030880369246006012,
0.0031048196833580732,
-0.00... | |
37ea97d2ca929b0b3bb6c28c1d2b11a938389a70 | subsection | 105 | 127 | Testing the isotropy | In fact when \lambda < 1, \frac{1}{(2n)^2} \Xi _{f^{\star }}^{(n)}(u) converges in probability to b >0, and this implies that \Xi _{f^{\star }}^{(n)}(u) converges in probability to +\infty .Proof of Theorem REF and Remark REF .
As we have already pointed out in Remark REF , the matrix \mathit {\Sigma }^{\star }(u, \ta... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.05596980080008507,
-0.004035990219563246,
-0.0119782704859972,
0.016220256686210632,
-0.0015049132052809,
-0.006126465741544962,
-0.0008864491828717291,
0.018173400312662125,
0.004840899724513292,
0.014763027429580688,
-0.04342694953083992,
0.014679103158414364,
-0.0032062854152172804,
... | |
70cbb052d80fafb0ebf0fa25dd264561822c3825 | subsection | 106 | 127 | Testing the isotropy | LetS_{f^{\star }}^{(n)}(u)= \widehat{\tau }_n\, \mathit {\Gamma }_{\star }^{-\frac{1}{2}} R^t T_{f^{\star }}^{(n)}(u).Since \widehat{\tau }_n is a consistent estimator of \tau under hypothesis H_0, by Corollary REF , asymptotically this random vector is a standard Gaussian one and Theorem REF ensues.To achieve the proo... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.058322031050920486,
0.025254569947719574,
-0.0020962818525731564,
0.00469994405284524,
0.03302168473601341,
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0.008095195516943932,
0.046297501772642136,
-0.03784370422363281,
0.015267188660800457,
-0.014755993150174618,
0... | |
eb47fbdadbb4b5e0f63b536202975a860144c740 | subsection | 107 | 127 | Testing the isotropy | \Box | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.028452035039663315,
0.006941381376236677,
-0.05391393601894379,
0.0135242510586977,
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-0.03826150298118591,
0.03856661915779114,
0.007917751558125019,
-0.03009966015815735,
0.021... | |
44b0704cc528057053133fabd60a2cabe32101d5 | subsection | 108 | 127 | Appendix | Proof of Lemma REF .
We have(1)&=\int _{0}^{n-1} \int _{0}^{n-1} \int \limits _{\textrm {C}_{]t\,, \,t+1[\times ]s\, ,\,s +1[\,,\, X}(u)} f(\nu _{X}(x)) {\rm \,d}\sigma _{1}(x)\, {\rm \,d}t {\rm \,d}s \\
&\quad - \int _{0}^{1}\int _{0}^{1} \int _{\textrm {C}_{]0\,,\,t[\times ]0\,,\,s[\,, \,X}(u)} f(\nu _{X}(x)) {\rm \... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
0.0024167781230062246,
0.04779383912682533,
-0.034731410443782806,
-0.002706715138629079,
0.004711477551609278,
-0.021455343812704086,
0.004761071875691414,
0.01890694908797741,
0.010697152465581894,
0.01261226274073124,
-0.008164017461240292,
-0.01672479137778282,
-0.015374294482171535,
-... | |
2cb21897e9ab5fde6b271450ae5c265952023cee | subsection | 109 | 127 | Appendix | To obtain the second inequality, arguing as previously we obtain the following lower bound(2)=\int _{0}^{n+1} \int _{0}^{n+1} \int _{\textrm {C}_{]t-1\,,\,t[\times ]s-1\,,\,s[\,,\, X}(u)} f(\nu _{X}(x)) {\rm \,d}\sigma _{1}(x)\, {\rm \,d}t {\rm \,d}s \\
=\int _{n}^{n+1} \int _{n}^{n+1} H(t, s) {\rm \,d}t {\rm \,d}s -\i... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.0380929596722126,
0.03928336128592491,
-0.027028296142816544,
0.013254701159894466,
0.021335719153285027,
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-0.0005670638638548553,
-0.028935996815562248,
-0.0037753386422991753,
-0.024159114807844162,
... | |
7e8d5b61a646740a354ddaa6a0ee744c61d77693 | subsection | 110 | 127 | Appendix | X_n >0.X_n=\Big \langle \frac{J_{f^{\star }}^{(n)}(u)}{J_{\bf 1}^{(n)}(u)} , v^{\star } \Big \rangle = \frac{1}{J_{\bf 1}^{(n)}(u)} \int _{\textrm {C}_{T,\, X}(u)} \left| { \langle \nu _{X}(t) , v^{\star } \rangle } \right| d\sigma _1(t) \geqslant 0,and in a manner similar to that previously used it is proved that, a.s... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.02074883133172989,
0.014569951221346855,
-0.05394696071743965,
-0.007948633283376694,
0.02526475302875042,
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0.007963890209794044,
0.0007413702551275492,
0.0007013219292275608,
-0.016507526859641075,
... | |
92200ed908b53b2f892a67bbd4744411b129e9b1 | subsection | 111 | 127 | Appendix | For 0 < \lambda \leqslant 1 and -\frac{\pi }{2} < \theta _o \leqslant \frac{\pi }{2}, one has{\left\lbrace \begin{array}{ll}
\dfrac{\partial F_1}{\partial \lambda }(\lambda , \theta _o)= \displaystyle \frac{1}{I^2(\lambda )}\, \frac{\left[{\lambda \sin ^2(\theta _o)I(\lambda )-(\cos ^2(\theta _o)+\lambda ^2 \sin ^2(\th... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.07184984534978867,
0.01774880290031433,
-0.0042426204308867455,
0.03214013949036598,
0.020678957924246788,
-0.0010129637084901333,
0.018450820818543434,
0.04450172930955887,
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0.03583335503935814,
-0.059427205473184586,
0.010438676923513412,
-0.012926257215440273,
0... | |
a9eedf0e2d206128a21f7f6ec02cdf4a0c9c4b30 | subsection | 112 | 127 | Appendix | \end{array}\right.}From straightforward calculations we deduce that the Jacobian J_F of the transformation F can be written asJ_F(\lambda , \theta _o)= \displaystyle \frac{\left({1-\lambda ^2}\right)}{I^3(\lambda )}\,\frac{\left[{\lambda \sin ^2(\theta _o)I(\lambda )+(\cos ^2(\theta _o)-\lambda ^2 \sin ^2(\theta _o))I^... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.03699580207467079,
0.011591712944209576,
-0.03449278697371483,
-0.012133524753153324,
0.02235926315188408,
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0.00970681942999363,
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0.01285848394036293,
-0.0013926844112575054,
-0... | |
4c9e087711156812158290ce95eaadcc56134366 | subsection | 113 | 127 | Appendix | \exists \, C >0, \exists \, B>0, \underset{v \in S^1}{\inf } \int _{\left\Vert {t} \right\Vert _{2} < B}
\langle t , v \rangle ^2 \,f_x(t) {\rm \,d}t \geqslant C.Proof of Lemma .
Let us prove it by contradiction supposing that for all B>0, \underset{v \in S^1}{\inf } \int _{\left\Vert {t} \right\Vert _{2} < B}
\langl... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.0491475947201252,
0.037089645862579346,
-0.035593848675489426,
0.040752820670604706,
0.01898745633661747,
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-0.022299576550722122,
0.004101992584764957,
... | |
7c06d1153646ac4df92b5107db46d244e0b36537 | subsection | 114 | 127 | Appendix | Now by using the following inequality, \exists \, D>0, such that if \left| {y} \right| \leqslant 1, then 1- \cos (y) \geqslant D y^2 and (REF ), we get the serie of inequalitiesr_x(0)-r_x(\tau ) = \int _{\mathbb {R}^2} (1- \cos \langle t , \tau \rangle f_x(t)\, {\rm \,d}t
\geqslant \int _{\left\Vert {t} \right\Vert _{2... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.021882768720388412,
0.01336771622300148,
-0.022691546007990837,
-0.01095664408057928,
0.026659132912755013,
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0.0381193533539772,
-0.046970125287771225,
0.013840774074196815,
-0.004543649964034557,
-... | |
3a3ff7c1a7a91a1388c531a387c71e05f9384881 | subsection | 115 | 127 | Appendix | 269) a straightforward calculation on Gaussian characteristic functions gives{\rm E}_{{}}\hspace{-2.0pt}\left[{\prod _{i=1}^{6} \exp (t_i X_i - \tfrac{1}{2} t_i^2)}\right]= \exp \Big (\sum _{i=1}^{3} \sum _{j=1}^{3} \rho _{ij} \,t_i t_{j+3}\Big )\\
= \sum _{r=0}^{\infty } \frac{1}{r!}\, \Big ({\sum _{i=1}^{3} \sum _{j=... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.05930190160870552,
0.03180266544222832,
-0.02296689711511135,
-0.03354234993457794,
0.000900363374967128,
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0.009949778206646442,
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0.0030463566072285175,
-0.015779249370098114,
0.... | |
175fcba32f259b63bf3441ee6fd572876f5e8a61 | subsection | 116 | 127 | Appendix | In the case where \left| {{{k}}} \right| = \left| {{{m}}} \right|, one gets{\rm E}_{{}}\hspace{-2.0pt}\left[{\widetilde{H}_{{{k}}}(X) \widetilde{H}_{{{m}}}(Y)}\right]=\sum _{
{
d_{ij} \geqslant 0 \\ \sum _{j}d_{ij}=k_i \\ \sum _{i}d_{ij}=m_j
}
} {{k}}! {{m}}! \prod _{1 \leqslant i, j \leqslant 3} \frac{\rho _{ij}^{d_{i... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.02671171724796295,
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0.029335597530007362,
-0.015361906960606575,
0.... | |
b87f267befdcb0fc29b18aead8b1ae46fcbbc779 | subsection | 117 | 127 | Appendix | The coefficients a_{f_{1}^{\star }} are given by: for m, \ell \in \mathbb {N}\begin{pmatrix}
a_{f_1^{\star }}(2m, 2\ell )& a_{f_1^{\star }}(2m+1, 2\ell +1) & a_{f_1^{\star }}(2m, 2\ell +1)& a_{f_1^{\star }}(2m+1, 2\ell )\\
a_{f_2^{\star }}(2m, 2\ell )& a_{f_2^{\star }}(2m+1, 2\ell +1) & a_{f_2^{\star }}(2m, 2\ell +1)& ... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... | |
663f71db08ee4d62320ec95761378e2310b625eb | subsection | 118 | 127 | Appendix | First, let us compute a_{f_1^{\star }}(k_1, k_2), for k_1, k_2 \in \mathbb {N}. Let P=(a_{i, j})_{1\leqslant i, j \leqslant 2}.
By definition of coefficient a_{f_1^{\star }}(k_1, k_2), we havea_{f_{1}^{\star }}(k_1, k_2) = \frac{\sqrt{\mu }}{k_1!k_2!} \int _{\mathbb {R}^2} (a_{11}\,\lambda _1\,y_1+a_{12}\,\lambda _2\,y... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
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0... | |
93a387540dcef25e162c2f0c3b12698d70524eaf | subsection | 119 | 127 | Appendix | In that way and using that polynomial H_{2n} is even, one hasa_{f_1^{\star }}(2m, 2\ell ) = \tfrac{2\sqrt{\mu }}{(2m)!(2\ell )!} \int _{\mathbb {R}^2} (a_{11}\,\lambda _1\,y_1+a_{12}\,\lambda _2\,y_2) \,\times \\
1_{\big \lbrace \big \langle {\scriptsize \begin{pmatrix}
\lambda _1\,y_1\\
\lambda _2\,y_2
\end{pmatrix}},... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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d4b5521d3dc7bc754c714e3eeeda4bc1e9703e6c | subsection | 120 | 127 | Appendix | If \omega ^{\star }_1 >0, A can be written asA= \int _{\mathbb {R}} H_{2\ell }(y_2) \,\phi (y_2) \left[{\int _{-\frac{\lambda _2\, \omega ^{\star }_2}{\lambda _1\, \omega ^{\star }_1}y_2}^{+\infty }
y_1\,
H_{2m}(y_1) \,\phi (y_1){\rm \,d}y_1}\right]{\rm \,d}y_2.For real x and m \in \mathbb {N}, the polynomial form of H... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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7684481dce0a8fcc52bda509e5530adf34223391 | subsection | 121 | 127 | Appendix | \frac{(-2)^{n-\ell }}{(2n)!\, (\ell -n)!} \,G_{n+p-k}\left({
\frac{\lambda _2 \omega ^{\star }_2}{\lambda _1 \omega ^{\star }_1}
}\right),where for q \in \mathbb {N} and x \in \mathbb {R}, we definedG_q(x)=\int _{-\infty }^{+\infty } y^{2q}\, \phi (y)\, \phi (xy)\, dy=\frac{1}{\sqrt{2\pi }} \,\frac{1}{(1+x^2)^{q+\frac{... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
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29154d0e4d1f9faf609a15cb113bd82e856e3b28 | subsection | 122 | 127 | Appendix | \sum _{p=0}^{\ell } \frac{(-2)^{p-\ell }}{(2p+1)!\, (\ell -p)!} \,x^{2p+1}.To conclude the proof of Lemma , just remark that a_{f_2^{\star }}(2m+1, 2\ell )=a_{f_2^{\star }}(2m, 2\ell +1)=0 and that a_{f_2^{\star }}(2m, 2\ell ) (resp.
a_{f_2^{\star }}(2m+1, 2\ell +1)) would be computed replacing a_{11} by a_{21} and a_{... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
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d738786148401a02daab915542b80dc9059dcbaf | subsection | 123 | 127 | Appendix | \ell !} \sum _{p=0}^{\ell } \sum _{q=0}^{m}
{{\ell }{p}}
{{m}{q}}
(-1)^{p+q} \lambda _{2} \left({
\tfrac{\lambda _2}{\lambda _1}
}\right)^{2q+1} \times \\
\sum _{n=0}^{\infty }
\frac{
{{q+n}{q}}
{{2q+2n}{q+n}}
}{{{2q}{q}}
}
\left({
\tfrac{1}{2}
}\right)^{2n}
\frac{1}{\beta (p+q+n+1; \frac{1}{2})}
\left({
1- \left({
\tf... | {
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"raw": "Marie F. Kratz and José R. León, Central limit theorems for level functionals of stationary gaussian processes and fields, Journal of Theoretical Probability 14 (2001), no. 3, 639–672. MR 1860517",
... | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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3a1cbdebced2fae16b1e999a1ed350d8a1f1cfd0 | subsection | 124 | 127 | Appendix | (m-p)!(\ell -q)!} \,\lambda ^{2p}\, \times \\
\int _{0}^{\frac{\pi }{2}} \int _{0}^{+\infty } r^{2p+2q+2} \cos ^{2q}(\theta ) \sin ^{2p}(\theta )
e^{-\frac{1}{2}r^2\left[{\cos ^2(\theta )+\lambda ^2 \sin ^2(\theta )
}\right]}\, {\rm \,d}r {\rm \,d}\theta .Now making the change of variable r^2=\frac{2v}{\cos ^2(\theta )... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
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2382b50959b17525d88ace6a54d184325e8189b2 | subsection | 125 | 127 | Appendix | We have the following decomposition of \det \left({\mathit {\Sigma }_{(f_1, f_2)}(u)}\right).{\det \left({\mathit {\Sigma }_{(f_1, f_2)}(u)}\right)}\\
&=\mathit {\Sigma }_{f_1, f_1}(u) \mathit {\Sigma }_{f_2, f_2}(u) -\left({\mathit {\Sigma }_{f_1, f_2}(u)}\right)^2\\
&=\sum _{q=1}^{\infty } \sum _{q^{\prime }=1}^{\inf... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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932428768a91adf9f16eeba3f48c4c04f2b6c897 | subsection | 126 | 127 | Appendix | This argument yields inequality (REF ) and Lemma REF .
\BoxProof of Lemma REF .
Using that B_n^2= A_n, \lim _n A_n=A=B^2 and that \lim _n {\rm tr}(B_n)= {\rm tr}(B) >0, we obviously deduce that \lim _n B_n=B.
\Box | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
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f3c12144b0f884541f64b648bc3f8992604d1f9c | abstract | 0 | 216 | Abstract | We study a routing game in an environment with multiple heterogeneous
information systems and an uncertain state that affects edge costs of a
congested network. Each information system sends a noisy signal about the state
to its subscribed traveler population. Travelers make route choices based on
their private beliefs... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
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88da6a9ee3c5ffcff74e80fec9b214f2c0a4515b | subsection | 1 | 216 | Introduction | Travelers are increasingly relying on traffic navigation services to make their route choice decisions. In the past decade, numerous services have come to the forefront, including Waze/Google maps, Apple maps, INRIX, etc. These Traffic Information Systems (TISs) provide their subscribers with costless information about... | {
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"Manxi Wu",
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e45a62964a5b82d89dd92a498e32cac29f95aaf8 | subsection | 2 | 216 | Our Model and Contributions | We model the traffic routing problem in an asymmetric and incomplete information environment as a Bayesian routing game. We consider a general traffic network with an uncertain state that is realized from a finite set according to a prior probability distribution. The cost function (travel time) of each edge in the net... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
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7109b8881faad0f96fa6b4a6326fa1a902afc014 | subsection | 3 | 216 | Our Model and Contributions | Among the two perturbed populations, we say that one population is the “minor population” if its size is smaller than a certain (population-specific) threshold. Then, the corresponding IIC is tight in equilibrium, i.e. the impact of information on the minor population is fully attained. These population-specific thresh... | {
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39753e7d69e653c6a40995908436db5f25aeb364 | subsection | 4 | 216 | Our Model and Contributions | Intuitively, if it is possible for the non-users of a particular TIS to receive a greater value by adopting it, then they may continue to do so until there is no longer a positive relative value in adopting that TIS. Theorem REF shows that the set of equilibrium adoption rates (i.e. fraction of travelers choosing each ... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
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d3a9a65aa22bb7a7ffaed22bd8cad6b63e8b9a41 | subsection | 5 | 216 | Related Work | Congestion games. Well-known results in classical congestion games include their equivalence with potential games (, , and ), analysis of network formation games as congestion games (, and ), and equilibrium inefficiency (, , , , and ). has studied congestion games with player-specific cost functions, which can model b... | {
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"Manxi Wu",
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ff671fa185f70aed41a48d45a0714f0ed6b826c0 | subsection | 6 | 216 | Motivating Example | In this section, we motivate our analysis using a simple game of two asymmetrically informed traveler populations routing over a network of two parallel routes, denoted _1 and _2. The network state s belongs to the set \mathcal {S}=\lbrace \mathbf {a}, \mathbf {n}\rbrace , where the state \mathbf {a} represents an inci... | {
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496a32f0c47cca66d998ef28262efdfe9b2af47b | subsection | 7 | 216 | Motivating Example | Each population, given the signal it receives, can either assign all its demand on one of the two routes, or splits on both routes. Thus, there are 3^3=27 possible cases. Our results can be used to study how the equilibrium strategies and route flows change as the size of a population varies from 0 to 1. Detailed analy... | {
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aa26aef364d413ebe399610a929df55c8140ab7b | subsection | 8 | 216 | Motivating Example | The induced equilibrium flow on _1 is given by f_1^{*}=q^{2*}_1 if ^1=\mathbf {a}, and f_1^{*}=\lambda ^1+q^{2*}_1 if ^1=\mathbf {n}.On the other hand, in the second regime, i.e. when \lambda ^1 \in [\underline{\lambda }^1, 1], the equilibrium set may not be singleton, and can be represented as follows: q^{1*}_1(\mathb... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
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"Asuman E. Ozdaglar"
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bb997c3632e704631e8dc4c13ab117dcdd50b368 | subsection | 9 | 216 | Motivating Example | If \lambda ^1 \in [0, \underline{\lambda }^1), it is easy to check that C^{2*}(\lambda )-C^{1*}(\lambda )>0, i.e. when the state information is only available to a small fraction of travelers, the informed travelers have an advantage over the uninformed ones. On the other hand, if the size of informed population exceed... | {
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"Manxi Wu",
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29c0129ec1c0952e9466f319ef1b27b03e1dcd8d | subsection | 10 | 216 | Environment | To generalize the simple routing game in Section , we consider a transportation network modeled as a directed graph. For ease of exposition, we assume that the network has a single origin-destination pair. All our results apply to networks with multiple origin-destination pairs; see Sec . Let \mathcal {E} denote the se... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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3dca7a9a59699df04233d2253bf8b35ae5add232 | subsection | 11 | 216 | Environment | In our modeling environment, the signals of different TIS can be correlated, conditional on the state.
Each population i generates a belief about the state s and the other populations^{\prime } types t^{-i} based on the signal received from the information system i\in I. We denote the population i^{\prime }s belief as ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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3bee017adab4d99ce4aeb7db2f641045f1975ce4 | subsection | 12 | 216 | Bayesian Routing Game | The Bayesian routing game for a fixed size vector \lambda can be defined as\Gamma (\lambda )\stackrel{\Delta }{=}\left(I, \mathcal {S}, \mathcal {Q}(\lambda ), \mathcal {C}, \beta \right),where-
I: Set of populations, I= \lbrace 1,2, \dots , I\rbrace
-
\mathcal {S}: Set of states with prior distribution \theta \in \De... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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fc4151e95eeee1fc71002b7b8b91f9846a844f50 | subsection | 13 | 216 | Bayesian Routing Game | \begin{equation}
f_{}()=\sum _{i\in I} q_{}^{i}(t^i), \quad \forall \in \mathcal {R}, \quad \forall \in
\end{equation}
Note that the dependence of f on q is implicit and is dropped for notational convenience.
}Again, for the strategy profile qQ(), we denote the induced edge load as w =(we())eE, , where we() is the agg... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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b10966edad071ff5e4f8f4a98aae5175b08c25ca | subsection | 14 | 216 | Bayesian Routing Game | We define the equilibrium population cost, denoted C^{i*}(\lambda ), as the expected cost incurred by a traveler of a given population across all types and network states in equilibrium: C^{i*}(\lambda )\stackrel{\Delta }{=}\frac{1}{\lambda ^{i} \sum _{t^i\in {i}} \mathrm {Pr}(t^i) \sum _{\in \mathcal {R}} \mathbb {E}[... | {
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{
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"doi": "10.1006/jeth.2000.2696",
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"raw": "William H Sandholm. Potential games with continuous player sets. Journal of Economic theory, 97(1):81–108, 2001.",
"source_ref_id": "3f9f... | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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868a62bfb6f5537519e33daa6aa2c1ffa7171873 | subsection | 15 | 216 | Bayesian Routing Game | Lemma 1
Game \Gamma (\lambda ) is a weighted potential game with the potential function \Phi as follows:
\Phi \left(q\right) \stackrel{\Delta }{=}\sum _{s\in \mathcal {S}} \sum _{e\in \mathcal {E}} \sum _{\in \pi \left(s, \right) \int _{0}^{\sum _{\ni e} \sum _{i\in I} q_{}^{i}(t^i)} c_{e}^{s}(z) dz,
}
and the positi... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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06b05c1be371f73e1c45f72bbc630e4d4e2f7bb7 | subsection | 16 | 216 | Bayesian Routing Game | Importantly, since the equilibrium edge load w^{*}(\lambda ) is unique, the equilibrium population cost C^{i*}(\lambda ) for each population i\in I in () must also be unique for any \lambda . Thus, the equilibria of \Gamma (\lambda ) can be viewed as essentially unique. We denote the optimal value of (), i.e. the value... | {
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{
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"doi": "10.1016/j.orl.2012.11.009",
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"raw": "Gerd Wachsmuth. On LICQ and the uniqueness of Lagrange multipliers. Operations Research Letters, 41(1):78–80, 2013.",
"source_ref_id":... | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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49c236ac17bcb16ad79ad2e7d714b3c4cd48b5d7 | subsection | 17 | 216 | Bayesian Routing Game | Equilibrium Route Flows
Our main question of interest is how the set of BWE \mathcal {\mathcal {Q}}^{*}(\lambda ), i.e. optimal solution set of (), and more importantly, the equilibrium edge load w^{*}(\lambda ), change with the perturbations in the size vector \lambda . However, characterizing the effect of \lambda di... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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a6a31ab19df2bc0fb74bfdc5828fd5eb5fecb671 | subsection | 18 | 216 | Bayesian Routing Game | The constraints (REF )-() do not depend on the size vector \lambda and can be understood as follows: (REF ) captures the fact that the change in the flow through any route resulting from change in the type of population i\in I does not depend on the particular types of the remaining populations; () ensures that all the... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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b5ff01cfde7aacb5e9e4013603a05ab6e0e28dfe | subsection | 19 | 216 | Bayesian Routing Game | We will refer to them as information impact constraints (IIC). We use (\ref {prime:popu_i}) and (\ref {sub:popu_i}) interchangeably, and refer the constraint in (\ref {prime:popu_i}) corresponding to population iI as (\ref {prime:popu_i}i). Also, it is easy to see that for each iI, (\ref {prime:popu_i}i) can be written... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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3f299af3be92ad88a97088981d0683f76482f765 | subsection | 20 | 216 | Bayesian Routing Game | In Step III, we prove that if f satisfies (REF )-(), then we can indeed construct a vector \chi that satisfies (), and the corresponding q in (REF ) is a feasible strategy profile that induces f. By combining Steps II and III, we can conclude that any f satisfying (REF )-() can be induced by at least one feasible strat... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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1039083e34cabcf43de6d98fb6c87ac6747b5954 | subsection | 21 | 216 | Bayesian Routing Game | The total size of the remaining populations is |\lambda ^{-ij}| \stackrel{\Delta }{=}\sum _{k\in I\setminus \lbrace i, j\rbrace } \lambda ^{k}. For pairwise comparison, we only consider the case when the sizes of both populations are strictly positive so that |\lambda ^{-ij}| < 1, and the range of the perturbations in ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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b967442dbd20e3e10e7065ac454bd9137c3edffa | subsection | 22 | 216 | Bayesian Routing Game | \end{split}Before proceeding further, we need to define two thresholds for the size of one of the two perturbed populations (say, population i):
\begin{align}
\underline{\lambda }^i&\stackrel{\Delta }{=}\frac{1}{ \min _{f^{ij, \dagger }\in \mathcal {F}^{ij, \dagger }}\left\lbrace \widehat{J}^{i}(f^{ij, \dagger })\right... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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c2f536f3303a6b0dd19239b1a14cd1273774be4d | subsection | 23 | 216 | Bayesian Routing Game | These regimes are defined by the following sets:
\begin{}
\begin{align}
\Lambda _1^{ij}&\stackrel{\Delta }{=}\lbrace \left(\lambda ^{i}, \lambda ^{j}, \lambda ^{-ij}\right) \left|\lambda ^i\in (0, \underline{\lambda }^i) \right.\rbrace , \\
\Lambda _2^{ij}&\stackrel{\Delta }{=}\lbrace \left(\lambda ^{i}, \lambda ^{j}, ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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11541268e20fc8aa1a9dcd5a3c108cfba6dbae1c | subsection | 24 | 216 | Bayesian Routing Game | \begin{}
For any two populations i, j\in I, and any given \lambda ^{-ij}\in \Lambda ^{-ij}, the set of equilibrium route flows \mathcal {F}^{*}(\lambda ) when \lambda is in regime \Lambda _1^{ij} or regime \Lambda _3^{ij} can be expressed as follows:
\begin{align}
\mathcal {F}^{*}(\lambda )=\left\lbrace \operatornamewi... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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7980e41448b3e8d4bf5acde75453b3c3f51b5838 | subsection | 25 | 216 | Bayesian Routing Game | We can replace the constraints (_i) and (_j) in the optimization problem (REF ) by (REF ) without changing its optimal value, i.e. the optimal value of (REF ) is equal to \Psi (\lambda ). However, since the set \mathcal {F}^{ij, \dagger } (as defined in (REF )) contains all route flows that attain the optimal value \Ps... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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f25f13d548d5c758f3227e76f92900e63f9fbecd | subsection | 26 | 216 | Bayesian Routing Game | Therefore, \Psi (\lambda )={(w^{ij, \dagger }), which does not change when \lambda is perturbed in the direction z^{ij}.
}The necessary and sufficient condition for the invariance of w*() under relative perturbations in the sizes of any two populations in Proposition \ref {bathtub} is a direct consequence of the monoto... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/springerreference_72673",
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"raw": "Anthony V Fiacco. Sensitivity and stability in NLP: Continuity and differential stability. Encyclopedia of Optimization, pages 3467–34... | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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19f2ca96a9841c1ba60f9ec042c3ccd1d7a60432 | subsection | 27 | 216 | Bayesian Routing Game | Next, we can check that () satisfies the following conditions: (1) The potential function \Phi (q) is continuously differentiable in q, and constraints (REF )-() are linear in q and \lambda ; (2) The optimal solution set \mathcal {\mathcal {Q}}^{*}(\lambda ) is non-empty and bounded (Theorem ). The Lagrange multipliers... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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58b8328462c5d75cb70384f2e62a424da748003a | subsection | 28 | 216 | Bayesian Routing Game | From Lemma REF , since both \mu ^{*} and \nu ^{*} are unique in equilibrium, \nabla _{z^{ij}} \Psi (\lambda ) can be simplified:
\nabla _{z^{ij}} \Psi (\lambda )&=\left(\sum _{t^i\in {i}} \mu ^{*t^i}-\sum _{^j\in j} \mu ^{*^j}\right)
&\stackrel{(\ref {define_al})}{=}\left(\sum _{t^i\in {i}} \min _{\in \mathcal {R}}\... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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f7822242f1729786e83a6b53ddc07107dbad03ae | subsection | 29 | 216 | Bayesian Routing Game | As a result, in equilibrium, the travelers in the minor population do not choose the routes with a high expected cost based on the signal they receive from their TIS; however, the travelers in the other population may still choose these routes. On the other hand, in the middle regime, neither population has an advantag... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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ce3c33c62d3019c42cc2864577f7f3f7c42a0459 | subsection | 30 | 216 | Bayesian Routing Game | Therefore, the uninformed population has no further information besides the common knowledge. We show that the equilibrium cost of the uninformed travelers is no less than the cost of any other population.
Proposition 4
Consider the game \Gamma (\lambda ) in which population j is uninformed. Then, for any size vector ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.026364823803305626,
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0.034695375710725784,
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-0.0038353342097252607,
0.0076859258115291595,
-0.014204659499228,
0.003839148674160242,
-0.04516196623444557... | |
5db53dc353e04d0cc0b5de5f819e37e526f52b9d | subsection | 31 | 216 | Bayesian Routing Game | In Fig. \ref {fig:compare}, we illustrate the equilibrium population costs in two cases: (i) Types 1 and 2 are perfectly correlated, i.e. 1=2; (ii) Types 1 and 2 are independent conditional on the state, i.e. Pr(1, 2|s)=Pr(1|s) Pr(2|s).
This example illustrates how the correlation among received signals (or lack thereo... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.006005838979035616,
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... | |
de8c8ea106990fc96903a5fc1cdaf798a61a5f7f | subsection | 32 | 216 | Bayesian Routing Game | We now explicitly characterize the set of size vectors, denoted \Lambda ^{\dagger }, for which the edge load is size-independent. Since (REF ) is a convex optimization problem, and () are the only size-dependent constraints, we can equivalently view \Lambda ^{\dagger } as the set of size vectors for which all the IICs ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.04644924774765968,
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0.037293680012226105,
-0.029023151844739914,
-... | |
bf8a75a95e1f1828925b089bfb059cf55151b667 | subsection | 33 | 216 | Bayesian Routing Game | Therefore, for each \lambda \in \Lambda ^{\dagger }, there must exist a f^{\dagger }\in \mathcal {F}^{\dagger } that is an equilibrium route flow, i.e. at least one f^{\dagger }\in \mathcal {F}^{\dagger } satisfies the (\ref {prime:popu_i}) constraints corresponding to \lambda :
\begin{align}
\Lambda ^{\dagger }\stackr... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.008863200433552265,
-0.0517757311463356,
-0.042348094284534454,
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0.020472314208745956,
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0.019679049029946327,
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0.0051524024456739426,
0.053423281759023666,
-0.012295592576265335,
-0.0017896112985908985,
-0.0122803375124931... | |
697b3e7f7af313f2660b4b075d8c85b30a16f08d | subsection | 34 | 216 | Bayesian Routing Game | Therefore, no traveler has the incentive to change her TIS subscription if and only if:
\lambda ^{i}>0 \quad \Rightarrow \quad C^{i*}(\lambda )=\min _{j\in I} C^{j*}(\lambda ), \quad \forall i\in I.
Such population size vector \lambda can be viewed as the vector of equilibrium adoption rates, one for each TIS. For an... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.017869295552372932,
-0.0241563580930233,
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-0.002428225241601467,
-0.004421544261276722,
-0.01800663396716118,
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0.011689052917063236,
0.04172045737504959,
-0.004463508725166321,
-0.004890785086899996,
-0.022706670686602592,... | |
6568b17ddb24c9df24336ac5ccad53c8c6c9d6c5 | subsection | 35 | 216 | Bayesian Routing Game | Thus, C^{i*}(\lambda )\ge C^{j*}(\lambda ). The first and the second steps together show that any \lambda \in \Lambda ^{\dagger } satisfies (REF ), and hence is a vector of equilibrium adoption rates.
Finally, we show that for any feasible \lambda \notin \Lambda ^{\dagger }, (REF ) is not satisfied. Since \Lambda ^{\da... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.030863754451274872,
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0.04839339107275009,
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0.012723861262202263,
-0.006491610314697027,
... | |
8799b424ed40266f01eb0aa4412f1f3e51f9865b | subsection | 36 | 216 | Bayesian Routing Game | \lambda ^{i\dagger }_{max}=max_{\lambda ^{\dagger } \in \Lambda ^{\dagger }} \lambda ^{i\dagger }) is the minimum (resp. maximum) equilibrium adoption rate. Furthermore, since the set \Lambda ^{\dagger } is determined by the heterogeneous information environment created by all TIS, the equilibrium adoption rate of each... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.005086353048682213,
-0.04233921319246292,
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0.014018950052559376,
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0.027457911521196365,
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0.028724730014801025,
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0.004983328748494387,
-0.03290675953030586,
... | |
ba292296929268c829e268a0f6b19bb19f447f53 | subsection | 37 | 216 | Bayesian Routing Game | One future research question of interest is to analyze how the travelers' expected cost and TIS adoption rates change when one or more TIS providers make technological changes to their service (for example, improving accuracy levels), or when a new TIS is introduced. Addressing this problem would involve applying our r... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1257/aer.101.6.2590",
"end": 957,
"openalex_id": "https://openalex.org/W4255572092",
"raw": "Emir Kamenica and Matthew Gentzkow. Bayesian persuasion. American Economic Review, 101(6):2590–2615, 2011.",
"source_ref_id": "438d74054c3... | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.05417797714471817,
-0.00633348198607564,
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0.017901625484228134,
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0.008164850063621998,
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0.02461664192378521,
0.08967600017786026,
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0.017489567399024963,
-0.022861581295728683,
... | |
b624f4cb8b166ebf764111f9541f6cbb2e0ee42e | subsection | 38 | 216 | Bayesian Routing Game | The first order partial derivative of {(w) with respect to w_{e}() can be written as: \frac{\partial {(w)}{\partial w_{e}()}= \sum _{s\in \mathcal {S}} \pi (s, ) c_{e}^{s}\left(w_{e}\left(\right)\right) for any e\in \mathcal {E}, and any \in . Also, the second order derivative of {(w) can be written as follows:
\begin{... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.020834140479564667,
0.03861565887928009,
0.006101427134126425,
-0.015705736353993416,
0.010668911971151829,
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0.01668257638812065,
0.011363383382558823,
0.01442363578826189,
0.06270084530115128,
-0.029915688559412956,
0.018987305462360382,
0.012447063811123371,
-0.0... | |
2b9fddd37fa99905e05a781269da7b1190fb4908 | subsection | 39 | 216 | Bayesian Routing Game | For any optimal solution q, there must exist \mu and \nu such that \left(q, \mu , \nu \right) satisfies the following Karush-Kuhn-Tucker (KKT) conditions:
{\begin{}
\begin{align}
\frac{\partial \mathcal {L}}{\partial q_{}^{i}(t^i)} &= \frac{\partial \Phi }{\partial q_{}^{i}(t^i)}-\mu ^{t^i}- \nu _{r}^{t^i}=0, \quad &\f... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.030267251655459404,
0.034617308527231216,
-0.037120502442121506,
-0.03388466686010361,
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0.008181163109838963,
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-0.0015139349270612001,
0.05626076087355614,
-0.008288007229566574,
0.010920937173068523,
-0.0237803217023611... | |
572cdddd734e9331586dcebed3050dd937ad6494 | subsection | 40 | 216 | Bayesian Routing Game | Thus, for any \in \mathcal {R}, and t^i\in {i}, i\in I:
\begin{equation*}
\begin{split}
q_{}^{i}(t^i)>0 \quad \Rightarrow \quad \mathrm {Pr}(t^i) \mathbb {E}[c_{}({q})|t^i]=\mu ^{t^i}\le \mu ^{t^i}+\nu _{r^{^{\prime }}}^{t^i} =\mathrm {Pr}(t^i) \mathbb {E}[c_{^{^{\prime }}}({q})|t^i], \quad \forall ^{^{\prime }} \in \m... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.019772808998823166,
0.012091011740267277,
-0.04607553035020828,
-0.02950664609670639,
-0.005355136003345251,
-0.009260876104235649,
0.027004532516002655,
-0.00891759805381298,
0.021405287086963654,
0.031856194138526917,
-0.02897265926003456,
0.031017068773508072,
0.011465483345091343,
0... | |
2437c958d6981d52d2c3a670ef16c4f9bf4902e4 | subsection | 41 | 216 | Bayesian Routing Game | Noting that \Phi (q) \equiv {(w), where the induced edge load w is linear in q (see (\ref {eq:q_w})), and that {(w) is strictly convex in w (Lemma \ref {lemma:potential_convex}), we conclude that \Phi (q) is a convex function of q. Furthermore, since \mathcal {Q}(\lambda ) is a convex polytope, (\ref {eq:potential_opt}... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.0292403232306242,
0.027515815570950508,
-0.02229650877416134,
-0.05139948055148125,
0.006947628688067198,
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0.012384406290948391,
0.005574127193540335,
0.025699740275740623,
0.04553920775651932,
-0.018618576228618622,
0.03711506351828575,
-0.01941215619444847,
0.0442... | |
1f19e75b223157aec3cb671549876e5b53b7182d | subsection | 42 | 216 | Bayesian Routing Game | However, this violates the equality constraint in (\ref {sub:demand}) as R qi*(i)=i0, and we arrive at a contradiction.
Since LICQ holds, for any equilibrium strategy profile q^{*}\in \mathcal {\mathcal {Q}}^{*}(\lambda ), the corresponding \mu ^{*} and \nu ^{*} must be unique. Following the proof of Theorem , we concl... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.0481351763010025,
0.009996366687119007,
-0.01913808099925518,
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0.011095203459262848,
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0.036536335945129395,
0.011095203459262848,
-0.022938227280974388,
-0.01191170047968626,
... | |
053abd4fd296faa3331aa43f7dd2863a1ff1bd2d | subsection | 43 | 216 | Bayesian Routing Game | Additionally,
&\sum _{\in \mathcal {R}}\min _{t^i\in {i}}f_{}(t^i, t^{-i}) \stackrel{(\ref {eq:load})}{=}\sum _{\in \mathcal {R}} \sum _{j\in I\setminus \lbrace i\rbrace }q^{j}_(^j)-\sum _{\in \mathcal {R}}\min _
{t^i\in {i}} q_{}^{i}(t^i)\\ \stackrel{(\ref {sub:demand})}{=}& \sum _{j\in I\setminus \lbrace i\rbrace }\... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.004937349818646908,
0.022221889346837997,
-0.038827259093523026,
-0.03369912877678871,
0.011751960963010788,
0.03553060442209244,
0.0003791724448092282,
0.041696567088365555,
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0.023397086188197136,
-0.01869630068540573,
-0.008699503727257252,
-0.005326538346707821,
... | |
e52fe602e77cc5d1175b805ec74b3f448ec5f8dc | subsection | 44 | 216 | Bayesian Routing Game | Following (REF ), we can write:
&\sum _{i\in I} f_{}(t^i, \widehat{}^{-i})-(|I|-1)f_{}(\widehat{}) =f_{}(^1, \widehat{}^{-1})+f_{}(^2, \widehat{}^{-2}) +\sum _{i=3}^{|I|} f_{}(t^i, \widehat{}^{-i})-(|I|-1)f_{}(\widehat{})\\ \stackrel{(\ref {sub:balance})}{=}&f_{}(^1, ^2, \widehat{}^{-1-2})+f_{}(\widehat{})+\sum _{i=3}... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.029075805097818375,
0.01938387006521225,
-0.02678637206554413,
-0.03217417374253273,
0.012530832551419735,
0.02388642355799675,
0.01576656475663185,
0.04804757982492447,
0.007047639694064856,
0.0137442322447896,
-0.021062787622213364,
-0.022787494584918022,
0.015888668596744537,
0.00475... | |
5a7535caec9d43703fe7f1be5a4117511214ecf3 | subsection | 45 | 216 | Bayesian Routing Game | Since q satisfies (REF ), we obtain that \lambda ^{i}{(\ref {sub:demand})}{=} \sum _{\in \mathcal {R}} q^{i}_(t^i)\stackrel{(\ref {eq_rep})}{=} \sum _{\in \mathcal {R}} \left(f_{}(t^i, \widehat{}^{-i})-f_{}(\widehat{}^{i}, \widehat{}^{-i})+\chi _^i\right)\stackrel{\text{(\ref {sub:ldemand})}}{=} \sum _{\in \mathcal {R}... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.010438686236739159,
-0.006566147319972515,
-0.04004548490047455,
-0.03336106240749359,
-0.009858759120106697,
0.0035005463287234306,
0.029484709724783897,
-0.010988090187311172,
0.0003088015946559608,
0.005558905657380819,
0.004170132800936699,
0.010118200443685055,
-0.013414626941084862,... | |
1e2fc3d29a455521cbaeea96a087fdf6b17554e4 | subsection | 46 | 216 | Bayesian Routing Game | Consider any f\in \mathcal {F}(\lambda ), we explicitly construct the following \chi , and show that such \chi satisfies ():
\chi _^i=\gamma _\cdot \left(\lambda ^{i}\sum _{\in \mathcal {R}} \max _{t^i\in {i}} \left(f_{}(\widehat{})-f_{}(t^i, \widehat{}^{-i})\right)\right) + \max _{t^i\in {i}} \left(f_{}(\widehat{})-f... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.006763333920389414,
0.012916327454149723,
-0.03439268097281456,
-0.013793691992759705,
-0.0041541289538145065,
-0.016265571117401123,
0.016723325476050377,
0.03787161782383919,
-0.004310528747737408,
0.011825344525277615,
-0.014884675852954388,
0.010391044430434704,
-0.017898231744766235,
... | |
766617a3dff1cb2e824106cfa31c4e00c8168aa8 | subsection | 47 | 216 | Bayesian Routing Game | Next, \lambda ^{i}\sum _{\in \mathcal {R}} \max _{t^i\in {i}} \left(f_{}(\widehat{})-f_{}(t^i, \widehat{}^{-i})\right) \stackrel{(\ref {sub:ldemand})}{=}\lambda ^{i}\left(\sum _{\in \mathcal {R}} \min _{t^i\in {i}}f_{}(t^i, \widehat{}^{-i})\right) \stackrel{(\ref {sub:popu_i})}{\ge } 0. Using the above inequalities, we... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.007057350128889084,
0.006084580440074205,
-0.030564049258828163,
-0.012970265001058578,
-0.004024596884846687,
0.050538256764411926,
0.002224019030109048,
0.0009908900829032063,
-0.0159610565751791,
0.0004060360661242157,
-0.050751883536577225,
0.010689024813473225,
-0.006931462325155735,... | |
a6f424a3d2965f232ea22ed6c34a639cac6f93cf | subsection | 48 | 216 | Bayesian Routing Game | If \sum _{\in \mathcal {R}} \left[f_{}(\widehat{})-\sum _{i\in I} \max _{t^i\in {i}} \left(f_{}(\widehat{})-f_{}(t^i, \widehat{}^{-i})\right)\right] >0, then:
\begin{align*}
\sum _{\in \mathcal {R}} \chi _^i&=\sum _{\in \mathcal {R}} \gamma _\cdot \left(\lambda ^{i}\sum _{\in \mathcal {R}} \max _{t^i\in {i}} \left(f_{}... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.02489805407822132,
0.0091384407132864,
-0.026820331811904907,
-0.02558458223938942,
-0.007158953696489334,
0.01307453028857708,
-0.03316689282655716,
0.04003216698765755,
-0.01251005195081234,
0.013135554268956184,
-0.04110009968280792,
0.023280901834368706,
-0.03139717876911163,
-0.0146... | |
88a906700f9605738a7797705e59f96adb655145 | subsection | 49 | 216 | Bayesian Routing Game | Since in this case, \gamma _=0, we can conclude that \sum _{\in \mathcal {R}}\chi _^i= \sum _{\in \mathcal {R}}\max _{t^i\in {i}} \left(f_{}(\widehat{})-f_{}(t^i, \widehat{}^{-i})\right)=\lambda ^{i}, i.e. \chi satisfies (\ref {sub:x_sum}).
\end{align}Finally, i also satisfies (\ref {sub:sum_demand}). If R [f()-iI tii ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
0.006308401469141245,
0.011762460693717003,
-0.03936076536774635,
-0.012654943391680717,
-0.01971089467406273,
0.014813682995736599,
-0.004706509876996279,
0.03435676172375679,
-0.01777336746454239,
0.0038235625252127647,
-0.03090887889266014,
0.013837291859090328,
-0.016354549676179886,
0... | |
7b9c3ca022195f8ae64b6f990f1ee955523a4d72 | subsection | 50 | 216 | Bayesian Routing Game | \end{align*}
If R [f()-iI tii (f()-f(ti, -i))] =0, then we have
0=\sum _{\in \mathcal {R}} \left[f_{}(\widehat{})-\sum _{i\in I} \max _{t^i\in {i}} \left(f_{}(\widehat{})-f_{}(t^i, \widehat{}^{-i})\right)\right] \stackrel{(\ref {min_t})}{=}\sum _{\in \mathcal {R}} \min _{\in f_{}() \ge 0, which implies that for any \in... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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0.024799268692731857,
... | |
1527771d4b336dc6e9a64d0ae1b3d20b06870f22 | subsection | 51 | 216 | Bayesian Routing Game | \square
Supplementary Material for Section
Lemma 9.1
The route flows f^{ij, \dagger }\in \mathcal {F}^{ij, \dagger } induce a unique edge load w^{ij, \dagger }.
Proof of Lemma REF
Following () and (REF ), any edge load w^{ij, \dagger } induced by route flows in \mathcal {F}^{ij, \dagger } (which we defined as opti... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
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0.0... | |
5c92f2d6e3cea91f94bbe5a1e664718ddb25e0ab | subsection | 52 | 216 | Bayesian Routing Game | Since \underline{\lambda }^i is attainable on the set \mathcal {F}^{ij, \dagger }, there exists \tilde{f}^{ij, \dagger }\in \mathcal {F}^{ij, \dagger } such that:
\begin{split}
\underline{\lambda }^i= \frac{1}{\widehat{J}^{i}(\tilde{f}^{ij, \dagger }) \stackrel{\text{(\ref {widehatj})}}{=}\frac{1}{\left(\sum _{\in \ma... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.022110112011432648,
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0.0... | |
bd9d1f8a796880d13a78185c795cd89bf3ca0833 | subsection | 53 | 216 | Bayesian Routing Game | Therefore, (\ref {linear_program_lambli}) is a linear programming. Analogously, the threshold \bar{\lambda }^i is the optimal value of the following linear program:
\begin{equation}
\begin{split}
\max \quad &y \\
s.t. \quad & -|\lambda ^{-ij}| \sum _{\in \mathcal {R}} f_{}(^j_, \widehat{}^{-j}) \ge y \cdot \quad \foral... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.02347475290298462,
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0.05284108966588974,
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0.016942117363214493,
-0.009432638064026833,
0.03... | |
6b1a96962cd9b302067d130d0e7e5234addba0f4 | subsection | 54 | 216 | Bayesian Routing Game | Rearranging, we obtain: \frac{1}{\widehat{J}^{j}(\tilde{f}^{ij, \dagger }) < 1-|\lambda ^{-ij}| -\lambda ^{i}=\lambda ^{j}, and so such \tilde{f}^{ij, \dagger } also satisfies (\ref {prime:popu_i}_{j}). Since \tilde{f}^{ij, \dagger } is an optimal solution of (\ref {drop_i_j}), which minimizes the same objective functi... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.010167873464524746,
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0.0549202486872673,
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0.0006283410475589335,
-0.03536253795027733,
0... | |
9d0369627e8070e2b2ecda0b62a6d13bffeaa06e | subsection | 55 | 216 | Bayesian Routing Game | Hence, (\ref {prime:popu_i}_{j}) can be dropped in (\ref {opt_l}) without changing the optimal solution set.
}\left[\text{Regime } \Lambda _3^{ij}\right]: Analogous to the proof given for regime \Lambda _1^{ij}, we can argue that constraint (\ref {prime:popu_i}_{j}) is tight in any equilibrium for any \lambda in regime... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.03748438507318497,
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0.017994336783885956,
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0.00578062329441309,
-0.04325355961918831,
0.... | |
903d46c9905f109c0870764ee67ae108997228b9 | subsection | 56 | 216 | Bayesian Routing Game | Such \tilde{f}^{ij, \dagger } also satisfies constraint (\ref {prime:popu_i}_{j}). Therefore, \tilde{f}^{ij, \dagger } satisfies all the constraints in (\ref {eq:Lprime}), and minimizes \widehat{\Phi }(f). So \tilde{f}^{ij, \dagger } is an equilibrium route flow, which implies that \mathcal {F}^{*}(\lambda )\cap \mathc... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.058207444846630096,
-0.004311379976570606,
-0.028783230111002922,
-0.04337324574589729,
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0.044288937002420425,
-0.03491836041212082,
0.004883687011897564,
-0.02544095739722252,
... | |
e07da857cc5dce9ca87012f1be7d01dd773891b3 | subsection | 57 | 216 | Bayesian Routing Game | If \underline{\underline{\lambda }}^{i}< \bar{\bar{\lambda }}^{i}, for any \lambda ^{i}\in (\underline{\underline{\lambda }}^{i}, \bar{\bar{\lambda }}^{i}), we can check that any f^{ij, \dagger }\in \mathcal {F}^{ij, \dagger } satisfies the constraint (\ref {prime:popu_i}_i): \frac{1}{\widehat{J}^{i}(f^{ij, \dagger }) ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.03362688794732094,
-0.027493489906191826,
-0.04159114882349968,
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0.044062819331884384,
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-0.01135136280208826,
-0.018064534291625023... | |
ecf47459ff5442963e6f3a220d5a515db7b1fbb4 | subsection | 58 | 216 | Bayesian Routing Game | Moreover, if there are two populations, then the equilibrium strategy profile is unique in regime \Lambda _1^{12} or \Lambda _3^{12}, and can be written as follows:
\begin{}
\begin{align}
\text{In regime $\Lambda _1^{ij}$: }\quad q^{1*}_(^1)&=f^{*}_{}(^1, \widehat{}^2)-\min _{\widehat{}^1\in \mathcal {T}^1} f^{*}_{}(\w... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.036988984793424606,
0.004650327377021313,
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0.04312329739332199,
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-0.012100769206881523,
0.0006957370205782354,
... | |
6207a5e53930db9f75ec5bcb682590e0f009c2a4 | subsection | 59 | 216 | Bayesian Routing Game | Therefore, from (\ref {sub:x_sum}) and (\ref {sub:x_bound}), we obtain:
\begin{align*}
\lambda ^1 {(\ref {sub:x_sum})}{=} \sum _{\in \mathcal {R}} \chi _^1 \stackrel{(\ref {sub:x_bound})}{\ge } \sum _{\in \mathcal {R}}\max _{^1\in \mathcal {T}^1} \left(f_r^{*}(\widehat{}^1, \widehat{}^2)-f_r^{*}(^1, \widehat{}^2)\right... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.01596120372414589,
0.01141393929719925,
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0.013931719586253166,
0.03025914542376995,
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0.010467863641679287,
-0.02957247756421566,
0.020264320075511932,
-0.0042802272364497185,
-0.... | |
56716908cb01d7c4d43ed815f01a8d6f83bf6b40 | subsection | 60 | 216 | Bayesian Routing Game | We know from Theorem \ref {l_behavior} that constraint (\ref {prime:popu_i}_i) is tight in equilibrium, and thus f^{*}(\lambda ) and f^{*}(\lambda ^{^{\prime }}) satisfy: \frac{1}{\widehat{J}^{i}(f^{*}(\lambda )) = \lambda ^{i}< \lambda ^{i^{\prime }} =\frac{1}{\widehat{J}^{i}(f^{*}(\lambda ^{^{\prime }})). Consequentl... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.01422121375799179,
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0.04943855479359627,
-0.03491216525435448,
-0.006755839101970196,
-0.02195743098855... | |
279f1eea82b04e9ba442ab68699fb7bcda63730d | subsection | 61 | 216 | Bayesian Routing Game | Thus, \Psi (\lambda ) as well as w^{*}(\lambda ) remain fixed in regime \Lambda _2^{ij}.
}}\left[\text{Regime $\Lambda _3^{ij}$}\right]: Following similar argument in regime \Lambda _1^{ij}, we conclude that \Psi (\lambda ) monotonically increases in the direction z^{ij} in regime \Lambda _3^{ij}. As a result, w^{*}(\l... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf00938592",
"end": 642,
"openalex_id": "https://openalex.org/W2076370241",
"raw": "Anthony V Fiacco and Jerzy Kyparisis. Convexity and concavity properties of the optimal value function in parametric nonlinear programming. Journal ... | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.04476289823651314,
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0.02650073543190956,
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... | |
dcbc35398931abade9c9c902108ba350f344c1c9 | subsection | 62 | 216 | Bayesian Routing Game | From Theorem , we know that the constraint (_j) must be tight in equilibrium when \lambda is in regime \Lambda _3^{ij}. However, since \widehat{J}^{j}(f^{*})=0 for any \lambda , the constraint (_j) is tight only when \lambda ^{j}=0, i.e. \lambda ^{i}=1-|\lambda ^{-ij}|. This implies that the regime \Lambda _3^{ij} is i... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.04302571713924408,
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0.021116167306900024,
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-0.010710657574236393,
-0.012083818204700947,
... | |
f81e78b0a068546c9c452c1a1e0129d61e16f37d | subsection | 63 | 216 | Bayesian Routing Game | REF .
[Figure: Effects of varying population sizes for Example : (a) Equilibrium route flows on r_1; (b) Equilibrium population costs.]Supplementary material for Section
Proof of Proposition REF
Firstly, we prove that for any \lambda \in \Lambda ^{\dagger }, \mathcal {F}^{*}(\lambda ) \subseteq \mathcal {F}^{\dagger }... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.034615110605955124,
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0.025686608627438545,
-0.010492895729839802,
0... | |
61255929c3ab5ef52e561c24a9433ece70a66552 | subsection | 64 | 216 | Bayesian Routing Game | Additionally, for any \lambda \in \operatornamewithlimits{arg\,min}_{\lambda } \Psi (\lambda ), we have \Psi (\lambda )=\min _{\lambda } \Psi (\lambda )=\widehat{\Phi }(f^{\dagger }). Since \mathcal {F}^{\dagger } includes all route flows that satisfy (REF )-() and attain the minimum value of \Psi (\lambda ), any equil... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.0265129953622818,
-0.013798046857118607,
-0.03307259827852249,
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0.003603967372328043,
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0.03655071184039116,
0.0016570622101426125,
... | |
ecffc7de5f17b65d01c7d7d6e0f9d6089085fb8b | subsection | 65 | 216 | Bayesian Routing Game | The information environment – state, TISs, signals and common prior –is the same as that introduced in Sec. REF .
The fraction of travelers between o-d pair k\in \mathcal {K} who subscribe to TIS i \in I is \lambda _k^i. A feasible size vector \lambda = (\lambda _k^i)_{k \in \mathcal {K}, i \in I} satisfies \lambda _k^... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.03432202339172363,
-0.0011945969890803099,
-0.039935577660799026,
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0.003388346405699849,
0.057599980384111404,
0.004862286616116762,
0.0016512706642970443,
0.010838131420314312,... | |
efeec9f794bcc2b053cc7ef47f7028b1c9d65519 | subsection | 66 | 216 | Bayesian Routing Game | For any feasible strategy profile q, the induced route flow vector is f=(f_{r, k}(t))_{r \in \mathcal {R}_k, k \in \mathcal {K}, t \in , where f_{r, k}(t) is the flow on route r induced by travelers between o-d pair k when the type profile is t:
\begin{align}
f_{r, k}(t)= \sum _{i \in I}q_{r,k}^{i}(t^i), \quad \forall ... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.03739476203918457,
-0.02651212364435196,
-0.02599317580461502,
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0.03357896953821182,
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0.05619898810982704,
0.020040540024638176,
0.013431588187813759,
0.037089500576257706,
-0.0... | |
076223acc496e4823aedd4ebbf6864f2cfa05302 | subsection | 67 | 216 | Bayesian Routing Game | Firstly, we can check that the following function of q is a weighted potential function of the Bayesian congestion game with K o-d pairs:
\begin{align*}
\Phi (q)= \sum _{e \in \mathcal {E}}\sum _{s \in \mathcal {S}} \sum _{t \in \pi (s, t) \int _{0}^{\sum _{i \in I}\sum _{k \in \mathcal {K}}\sum _{r \in \left\lbrace \m... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.007625954691320658,
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0.005459100008010864,
0.05133309215307236,
-0.018479302525520325,
0.017472172155976295,
0.01153621170669794,... | |
d2b7c949612c17a3143f9ddaff7871a332ba9eed | subsection | 68 | 216 | Bayesian Routing Game | We can show that Theorem (\ref {l_behavior}) holds: three regimes (one or two may be empty) can be distinguished by whether or not the information impact all the travelers between o-d pair k who subscribe to TIS i (resp. j), i.e. whether or not (\ref {IIC}) is tight at the optimum of (\ref {opt_l}).
\end{align*}Fourthl... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.039527855813503265,
-0.006642663851380348,
-0.05900181084871292,
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0.03665865212678909,
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0.0038192453794181347,
-0.021351145580410957,
... | |
313b4787e0afe79f6fa60ad98d65945a5ba5bfb5 | subsection | 69 | 216 | Equilibrium Characterization | In this section, we show that the game \Gamma (\lambda ) is a weighted potential game. This property enables us to express the sets of equilibrium strategy profiles and route flows as optimal solution sets of certain convex optimization problems. | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
] | 2,018 | en | Computer Science | [
-0.017290949821472168,
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0.016558412462472916,
-0.010728630237281322,
0.06623364984989166,
-0.00393739202991128,
... |
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