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4bed6860f3d324aae541b06e0f6eba1b80ca2415 | subsection | 70 | 216 | Equilibrium Strategy Profiles | Following , the game \Gamma (\lambda ) is a weighted potential game if there exists a continuously differentiable function \Phi : \mathcal {Q}(\lambda ) \rightarrow \mathbb {R} and a set of positive, type-specific weights \lbrace \gamma (t^i)\rbrace _{t^i\in {i}, i\in I} such that:\frac{\partial \Phi (q())}{\partial q_... | {
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"Manxi Wu",
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593fc316366a193df2c6f040f0ec437caa1c8404 | subsection | 71 | 216 | Equilibrium Strategy Profiles | In addition, {(w) satisfies the following property:
\begin{}
The function {(w) is twice continuously differentiable and strictly convex in w.
}
\end{}}}}Our first result provides a characterization of the set of equilibrium strategy profiles:
\begin{}
A strategy profile q\in \mathcal {Q}(\lambda ) is a BWE if and only ... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
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25a36e97574f037e5217f2a3531865e9f5ab061d | subsection | 72 | 216 | Equilibrium Strategy Profiles | The next lemma shows that for any BWE q^{*}\in \mathcal {\mathcal {Q}}^{*}(\lambda ), the optimal Lagrange multipliers \mu ^{*} and \nu ^{*} in (REF ) associated with q^{*} are unique.Lemma 2
The Lagrange multipliers \mu ^{*} and \nu ^{*} at the optimum of () are unique:\mu ^{t^i*}&=\min _{\in \mathcal {R}}\mathrm {Pr... | {
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"raw": "Gerd Wachsmuth. On LICQ and the uniqueness of Lagrange multipliers. Operations Research Letters, 41(1):78–80, 2013.",
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"Manxi Wu",
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"Asuman E. Ozdaglar"
] | [
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e0092afd9a0f10aaba02e627e66317ab98511d28 | subsection | 73 | 216 | 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 directly from () is not so... | {
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"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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937c413679c1c2a41b8cf2d9f5911afaf738673a | subsection | 74 | 216 | Equilibrium Route Flows | These results enable us to evaluate how the equilibrium edge load and population costs change with perturbations in \lambda .Let us start by introducing the set of route flows\mathcal {F}(\lambda ) \stackrel{\Delta }{=}\lbrace f\in \mathbb {R}^{|\mathcal {R}| \times |} \left|\text{$f$ satisfies (\ref {sub:balance})-(\r... | {
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"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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0e232d1771230e2853e5a45df865f3d5ee5aae54 | subsection | 75 | 216 | Equilibrium Route Flows | Specifically, for any strategy profile q\in \mathcal {Q}(\lambda ) and population i\in I, we define the impact of information on population i\in I as follows:J^i(q) \stackrel{\Delta }{=}\lambda ^i \sum _{\in \mathcal {R}}\min _{t^i\in {i}} q_{}^{i}(t^i).Using (REF ), we can re-write (REF ) as: J^i(q)=\sum _{\in \mathca... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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cc9af3eec0dbdba94cb7e668d02297e23f43fc37 | subsection | 76 | 216 | Equilibrium Route Flows | \end{align}
Thus, F() is a convex polytope. The following proposition relates the set of feasible strategy profiles and the induced route flows.Proposition 1
The set of feasible route flows is the convex polytope \mathcal {F}(\lambda ). Furthermore, for a given route flow f\in \mathcal {F}(\lambda ), any feasible stra... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
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e26c122393edd671c3b89ce6e6e16af69cf62b5b | subsection | 77 | 216 | Equilibrium Route Flows | \quad f\in \mathcal {F}(\lambda ),
\end{split}where \mathcal {F}(\lambda ) is the set of feasible route flow vectors, which satisfy constraints (REF ) – ().We denote the set of equilibrium route flows f^{*} in the game \Gamma (\lambda ) as \mathcal {F}^{*}(\lambda ). From Theorem , equations () and (REF ), we know that... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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0bd705c110b93b5ca303dfd83a7553d1f2f8eeac | subsection | 78 | 216 | Pairwise Comparison of Populations | In this section, we first analyze the effects of perturbations in the relative sizes of any two populations on the equilibrium structure. Next, we study how the cost difference between any two populations depends on the population sizes.In this section, we first analyze the effects of perturbations in the relative size... | {
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"Manxi Wu",
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"Asuman E. Ozdaglar"
] | [
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f6a4891daf55d4a817c2093a801e41f853a520a6 | subsection | 79 | 216 | Equilibrium Regimes | To study the effects of perturbations in the relative sizes of any two populations, we employ the notion of directional perturbation of size vector \lambda . In particular, for any two populations i and j, we consider the |I|-dimensional direction vector z^{ij}\stackrel{\Delta }{=}(\dots 0 \dots , 1, \dots 0 \dots , -1... | {
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"Manxi Wu",
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"Asuman E. Ozdaglar"
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003d66d84ed6c57b26b86ab01c3c100f269e6cc1 | subsection | 80 | 216 | Equilibrium Regimes | Then, the optimal solution set of (REF ) can be written as the following polytope:\begin{split}
\mathcal {F}^{ij, \dagger }= \left\lbrace f\left|\begin{array}{l}
\text{$f$ satisfies (\ref {sub:balance}), (\ref {sub:ldemand}), (\ref {sub:l_positive}), (\ref {prime:popu_i})$\setminus \lbrace i, j\rbrace $, and (\ref {ext... | {
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"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
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6526553c2e6ec62e5befc3a8f10ab4475d38f5b2 | subsection | 81 | 216 | Equilibrium Regimes | 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"
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1d10ffd3ec6cfc50a3c6b49ce666c17b66e0693d | subsection | 82 | 216 | Equilibrium Regimes | \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"
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43c1304a59616af0bd95175fb834d7c8cb7352c2 | subsection | 83 | 216 | Equilibrium Regimes | 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"
] | [
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956b42bf84481b52db32d37d0e81fa2568bb318f | subsection | 84 | 216 | Equilibrium Regimes | 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... | {
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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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7d9e92ae1a96618012547185bf15f4bfefcc8e90 | subsection | 85 | 216 | Equilibrium Regimes | We denote the set of optimal solutions for (REF ) as \mathcal {F}^{ij, \dagger }. Analogously to Theorem , we can show that any f^{ij, \dagger }\in \mathcal {F}^{ij, \dagger } induces a unique edge load w^{ij, \dagger }, which can be obtained by (); see Lemma REF . Then, the optimal solution set of (REF ) can be writte... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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a358b06a9f91f63c31a88decbe44a843ea5a5ca3 | subsection | 86 | 216 | Equilibrium Regimes | These two thresholds play a crucial role in our subsequent analysis.
}We are now ready to introduce the equilibrium regimes that are induced by the relative change in the sizes of populations i and j with fixed sizes of other populations \lambda ^{-ij}\in \Lambda ^{-ij}. These regimes are defined by the following sets:... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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92d1b3ae64e697b2bad3e125f640d0fab255c322 | subsection | 87 | 216 | Equilibrium Regimes | \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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6988b5dd9624bee8ad93c77b5a705359cbb19669 | subsection | 88 | 216 | Equilibrium Regimes | 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"
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c35cc52b6ac7e5d89b2ae73f4c5995f7c2e4c4d6 | subsection | 89 | 216 | Equilibrium Regimes | 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": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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b3752fec9aed3757b9876cfdb419b2f0ff908b67 | subsection | 90 | 216 | Relative Value of Information | We now study the difference between the equilibrium costs of any two populations under perturbations in their relative sizes. For any two populations i, j\in I and size vector \lambda , we define the relative value of information, denoted V^{ij*}(\lambda ), as the expected travel cost saving that a traveler in populati... | {
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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"
] | 2,018 | en | Computer Science | [
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9e5c74db95c434e3fcdc43dc2d1258dfd2768417 | subsection | 91 | 216 | Relative Value of Information | Therefore, from Lemma REF , we know that \Psi (\lambda ) is differentiable in the direction z^{ij}, and \nabla _{z^{ij}} \Psi (\lambda ) can be expressed as:\nabla _{z^{ij}} \Psi (\lambda )&=\min _{q^{*}\in \mathcal {\mathcal {Q}}^{*}(\lambda )} \max _{{
(\mu ^{*}, \nu ^{*})\\
\in \left(M(q^{*}), N(q^{*})\right)}} \nab... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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249f6e4334beb19de90fba3dce18fa09efe183ec | subsection | 92 | 216 | Relative Value of Information | 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}}\mat... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
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b3e84ddc57adc0e7893d14f8bd2c16c05d0c0661 | subsection | 93 | 216 | Relative Value of Information | 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... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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10e949826b7f05b49e38d55e43eeec9fbd74797b | subsection | 94 | 216 | Relative Value of Information | 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 \... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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20cee0f9d15e5da5b6e3c777faa38e0fb4c1c7ff | subsection | 95 | 216 | Relative Value of Information | 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... | {
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} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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5222dcb47096bf509605a5a0f09626a2763e7d42 | subsection | 96 | 216 | Relative Value of Information | We say TIS i is relatively more valuable (resp. less valuable) than TIS j if V^{ij*}(\lambda )>0 (resp. V^{ij*}(\lambda )<0). Similarly, if V^{ij*}(\lambda )=0, TIS i is said to be as valuable as TIS j.It turns out that, for any given size vector \lambda , V^{ij*}(\lambda ) is closely related to the sensitivity of \Psi... | {
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"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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42c00a3657cdba1a18fb17198cedb357ea4cb5c4 | subsection | 97 | 216 | Relative Value of Information | Therefore, from Lemma REF , we know that \Psi (\lambda ) is differentiable in the direction z^{ij}, and \nabla _{z^{ij}} \Psi (\lambda ) can be expressed as:\nabla _{z^{ij}} \Psi (\lambda )&=\min _{q^{*}\in \mathcal {\mathcal {Q}}^{*}(\lambda )} \max _{{
(\mu ^{*}, \nu ^{*})\\
\in \left(M(q^{*}), N(q^{*})\right)}} \nab... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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b3b2ba317bee0b3059a9bd22e97d83c3888bcdd6 | subsection | 98 | 216 | Relative Value of Information | 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}}\mat... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
"cs.GT"
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c32fa707383e882e085ff9f62c8fce60da1b0188 | subsection | 99 | 216 | Relative Value of Information | 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"
] | [
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472f36aa463a5bd1a770ec101db2b1a537b04298 | subsection | 100 | 216 | Relative Value of Information | 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 | [
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791a63c5776d695ca7cef73b2905093ee9c06d48 | subsection | 101 | 216 | Relative Value of Information | 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 | [
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edd078ae8b58bae39a336a64de402f07faae7128 | subsection | 102 | 216 | General Properties of Equilibrium Outcome | In this section, we first extend our approach of pairwise comparison of populations to study how the equilibrium outcome depends on population sizes in general. Then, we analyze the TIS adoption rates in situations where travelers can choose information subscription.In this section, we first extend our approach of pair... | {
"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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59dfe9bcd1e7df7ab4f300a4fbf1df19211ac71a | subsection | 103 | 216 | Size-Independence of Edge Load Vector | Our analysis in Section showed that if perturbations in the relative sizes of any two populations i, j\in I induce a middle regime \Lambda _2^{ij}, then the equilibrium outcome in this regime is independent of the sizes of the perturbed populations i and j. A natural question to ask is whether this result can be genera... | {
"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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561b4208e8fb1a14c675379bc323f061c894b445 | subsection | 104 | 216 | Size-Independence of Edge Load Vector | 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 | [
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49b6edfc6b912a261c2533f4282d4afa95ca642a | subsection | 105 | 216 | Size-Independence of Edge Load Vector | \end{split}Let us denote the optimal solution set of (REF ) as \mathcal {F}^{\dagger }. Analogous to Theorem , one can argue that any optimal solution f^{\dagger }\in \mathcal {F}^{\dagger } induces a unique edge load w^{\dagger }, obtained from (). Thus, \mathcal {F}^{\dagger } can be written as the convex polytope:\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 | [
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7c6b1b00e4f5a09518436b59db834d8682024315 | subsection | 106 | 216 | Size-Independence of Edge Load Vector | The equilibrium edge load vector w^{*}(\lambda ) is size-independent, and is equal to w^{\dagger } if and only if \lambda \in \Lambda ^{\dagger }.This result shows that some of the properties of \Psi (\lambda ) and the change of equilibrium edge load vector under pairwise perturbation (Proposition REF ) also hold for t... | {
"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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fcd24163db6bfe433e8111d0800a98952ae28ae5 | subsection | 107 | 216 | Size-Independence of Edge Load Vector | 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 | [
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57f4178611dbcecd86f6de1f0b7b1ac04d4d455a | subsection | 108 | 216 | Adoption Rates under Choice of TIS | Our analysis so far has focused on the equilibrium properties with fixed population sizes. We now extend our results on the relative value of information (Section ) and the size independence of the equilibrium edge load vector (Section REF ) to analyze travelers' choice of TIS subscription when they can choose to subsc... | {
"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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a12aea745bd5110b9e20269165ff2a9535ac2343 | subsection | 109 | 216 | Adoption Rates under Choice of TIS | Therefore, any two populations with positive size have identical costs in equilibrium.If any \lambda \in \Lambda ^{\dagger } satisfies \lambda ^{i}>0 for all i\in I, then the first step in our proof is sufficient to show that (REF ) is satiesfied. Otherwise, for any \lambda \in \Lambda ^{\dagger }, and any degenerate p... | {
"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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dc7deba7f9b4d559121971308a5af3b268b9bd15 | subsection | 110 | 216 | Adoption Rates under Choice of TIS | Therefore, C^{j*}(\lambda )>C^{i*}(\lambda ), which implies that travelers in population j has incentive to change subscription to TIS i. To sum up, in either case, \lambda \notin \Lambda ^{\dagger } cannot be a vector of equilibrium adoption rates.
\squareNote that the set \Lambda ^{\dagger } is not a singleton set 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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188aa08426e6453193f55f645519fe50f04aedea | subsection | 111 | 216 | Adoption Rates under Choice of TIS | We now extend our results on the relative value of information (Section ) and the size independence of the equilibrium edge load vector (Section REF ) to analyze travelers' choice of TIS subscription when they can choose to subscribe to any TIS in the set I.We model travelers' choice of TIS and the choice of routes as ... | {
"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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bd9349ca558527ddb96b9caec223dd69683b5a27 | subsection | 112 | 216 | Adoption Rates under Choice of TIS | Therefore, any two populations with positive size have identical costs in equilibrium.If any \lambda \in \Lambda ^{\dagger } satisfies \lambda ^{i}>0 for all i\in I, then the first step in our proof is sufficient to show that (REF ) is satiesfied. Otherwise, for any \lambda \in \Lambda ^{\dagger }, and any degenerate p... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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a67244eeeec847cb4680399ee8667cee2e6eb3b0 | subsection | 113 | 216 | Adoption Rates under Choice of TIS | Therefore, C^{j*}(\lambda )>C^{i*}(\lambda ), which implies that travelers in population j has incentive to change subscription to TIS i. To sum up, in either case, \lambda \notin \Lambda ^{\dagger } cannot be a vector of equilibrium adoption rates.
\squareNote that the set \Lambda ^{\dagger } is not a singleton set in... | {
"cite_spans": []
} | 1808.10590 | Value of Information in Bayesian Routing Games | [
"Manxi Wu",
"Saurabh Amin",
"Asuman E. Ozdaglar"
] | [
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722e60a3d8ee962983e7d5902fced790b5c7333b | subsection | 114 | 216 | Adoption Rates under Choice of TIS | We now extend our results on the relative value of information (Section ) and the size independence of the equilibrium edge load vector (Section REF ) to analyze travelers' choice of TIS subscription when they can choose to subscribe to any TIS in the set I.We model travelers' choice of TIS and the choice of routes as ... | {
"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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3eda234e779c2b61a17c930539dcd5dae3227397 | subsection | 115 | 216 | Adoption Rates under Choice of TIS | Therefore, any two populations with positive size have identical costs in equilibrium.If any \lambda \in \Lambda ^{\dagger } satisfies \lambda ^{i}>0 for all i\in I, then the first step in our proof is sufficient to show that (REF ) is satiesfied. Otherwise, for any \lambda \in \Lambda ^{\dagger }, and any degenerate p... | {
"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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... | |
a250104cb5a5d928447bcc765103bdcf2e438883 | subsection | 116 | 216 | Adoption Rates under Choice of TIS | Therefore, C^{j*}(\lambda )>C^{i*}(\lambda ), which implies that travelers in population j has incentive to change subscription to TIS i. To sum up, in either case, \lambda \notin \Lambda ^{\dagger } cannot be a vector of equilibrium adoption rates.
\squareNote that the set \Lambda ^{\dagger } is not a singleton set 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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59218082f2895c79b90b3a8322595776e1fb0dd2 | subsection | 117 | 216 | Concluding Remarks | In this article, we study the equilibrium route choices and costs in a heterogeneous information environment, in which each population receives a private signal from their traffic information system (TIS). Each population maintains a belief about the unknown network state and about the signals received by other travele... | {
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{
"arxiv_id": "",
"doi": "10.1257/aer.101.6.2590",
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"raw": "Emir Kamenica and Matthew Gentzkow. Bayesian persuasion. American Economic Review, 101(6):2590–2615, 2011.",
"source_ref_id": "438d74054c... | 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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b3f369be02e6f480cc2ee5e0f80c2383a8213e67 | subsection | 118 | 216 | Concluding Remarks | However, the characterization of regime thresholds in this case is more complicated from a computational viewpoint due to the non-uniqueness of edge load vector.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 techn... | {
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{
"arxiv_id": "",
"doi": "10.1257/aer.101.6.2590",
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"raw": "Emir Kamenica and Matthew Gentzkow. Bayesian persuasion. American Economic Review, 101(6):2590–2615, 2011.",
"source_ref_id": "438d74054c... | 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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b51e1abb7a99f319623c9094cdffb6e287b7410a | subsection | 119 | 216 | Supplementary Material for Sec. | Proof of Lemma REF .
First note that \Phi (q), as defined in (REF ), is a continuous and differentiable function of the strategy profile q. To show that \Phi (q) is a weighted potential function of \Gamma (\lambda ), we write the first order derivative of \Phi (q) with respect to q_{}^{i}(t^i):\frac{\partial \Phi (q)}{... | {
"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... | |
ceafcb970a735cbd51e819eb36ddc023a3b899ea | subsection | 120 | 216 | Supplementary Material for Sec. | \quad \forall e, e^{^{\prime }} \in \mathcal {E}, \quad \forall , ^{^{\prime }} \in
}{}{equation*}
Since for any e\in \mathcal {E} and any s\in \mathcal {S}, c_{e}^{s}(w_{e}) is increasing in w_{e}, \sum _{s\in \mathcal {S}} \pi \left(s, \right) \frac{d c_{e}^{s}\left(w_{e}\left(\right)\right)}{d w_{e}\left(\right)}>0... | {
"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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0.028793999925255775,... | |
36c7b1ba556470981f5667764d1996d637ec0a52 | subsection | 121 | 216 | Supplementary Material for Sec. | \end{align}
\end{}}Using (\ref {potential_prove}) and (\ref {first_order}), we have \frac{\partial \Phi (q) }{\partial q_{}^{i}(t^i)} = \mathrm {Pr}(t^i) \mathbb {E}[c_{}({q})|t^i]= \mu ^{t^i}+\nu _{r}^{t^i} for any \in \mathcal {R}, and t^i\in {i}, i\in I. From (\ref {com_slack}), we see that for any \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 | [
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0... | |
9e98b24b37914d1ab68775916525a7ad2a070950 | subsection | 122 | 216 | Supplementary Material for Sec. | We can easily check that (\ref {first_order}) and (\ref {beta_positive}) are satisfied by (q^{*}, \bar{\mu }, \bar{\nu }). Since q^{*} is a BWE, we know from (\ref {eq:BWE_fun}) that for a route \in \mathcal {R}, and t^i\in {i}, i\in I, if q^{i*}_{}(t^i)>0, then \mathbb {E}[c_{}({q^{*}})|t^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"
] | 2,018 | en | Computer Science | [
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0... | |
41499e7d3355b8e18e1ab9421dfe01b9819f560c | subsection | 123 | 216 | Supplementary Material for Sec. | \hfill \square }\end{equation}}}}{}}\begin{}{(Theorem 2 in \cite {wachsmuth2013licq})}
The Lagrange multiplies \mu ^{*} and \nu ^{*} associated with any q^{*}\in \mathcal {\mathcal {Q}}^{*}(\lambda ) at the optimum of (\ref {eq:potential_opt}) are unique if and only if the LICQ condition is satisfied in that the gradie... | {
"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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b36c7a2a689a30a52ed66dec725c8ba5356ea80e | subsection | 124 | 216 | Supplementary Material for Sec. | From (), we obtain that for any t^i, \tilde{t}^{i}\in {i}, any t^{-i}, \tilde{}^{-i}\in \mathcal {T}^{-i}, and any i\in I, f satisfies (REF ):\begin{split}
&f_{}(t^i, t^{-i})-f_{}(\tilde{t}^{i}, t^{-i})= q_{}^{i}(t^i)+\sum _{j\in I\setminus \lbrace i\rbrace } q^{j}_(^j)-q^i_(\tilde{t}^{i})-\sum _{j\in I\setminus \lbrac... | {
"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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b14bf918d4c38db87b9bb5916aef6bc31f8dd220 | subsection | 125 | 216 | Supplementary Material for Sec. | For any route \in \mathcal {R}, the linear system of equations () has \prod _{i\in I} |{i}| equations in \sum _{i\in I} |{i}| variables. Note that for any given \widehat{}=\left(\widehat{}^{i}\right)_{i\in I} \in , the following equations are linearly independent:
\begin{equation}
\begin{split}
\sum _{i\in I} q^i_(\wid... | {
"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 | [
-0.00306727085262537,
0.024477127939462662,
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0.010155566036701202,
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0.005192233249545097,
-0.... | |
e6205059589c44ddd0e6ddd33ce83738f88740ca | subsection | 126 | 216 | Supplementary Material for Sec. | We apply the same procedure iteratively for another |I|-2 times:\sum _{i\in I} f_{}(t^i, \widehat{}^{-i})-(|I|-1)f_{}(\widehat{})
&=f_{}(), \quad \forall \inNow for any \in \mathcal {R} and \in , we can write iI qi(ti)=iI (qi(ti)+jI{i} qj(tj))- (|I|-1) iI qi(i)()=iI f(ti, -i)-(|I|-1)f() (REF )=f().
Thus, for any R, 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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618600294b6d0820af2a47f4f4c9008744c040ca | subsection | 127 | 216 | Supplementary Material for Sec. | Thus, \chi _^{i} \ge \max _{t^i\in {i}} \left\lbrace f_{}(\widehat{}^{i}, \widehat{}^{-i})-f_{}(t^i, \widehat{}^{-i})\right\rbrace , i.e. \chi satisfies ().Step III: Finally, we show that the set of \chi satisfying () is non-empty, i.e., any f\in \mathcal {F}(\lambda ) can be induced by at least one feasible strategy p... | {
"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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-0.006398949772119522,... | |
8447c3ca16d0e6be09c52ddedf6658cee8314a73 | subsection | 128 | 216 | Supplementary Material for Sec. | Thus, for any R, we obtain:
\begin{align}
f_{}(\widehat{})-\sum _{i\in I} \max _{t^i\in {i}} \left(f_{}(\widehat{})-f_{}(t^i, \widehat{}^{-i})\right) &=\min _{\in \sum _{i\in I} f_{}(t^i, \widehat{}^{-i})- (|I|-1)f_{}(\widehat{})=\min _{\in f_{}()\ge 0.
}
Hence, we can conclude that \gamma _\ge 0. Next, \lambda ^{i}\s... | {
"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 | [
-0.00027801477699540555,
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0.006931771524250507,
-0.031893014907836914... | |
4d6cafa788706e63c383c8940a0a794f049c3fae | subsection | 129 | 216 | Supplementary Material for Sec. | 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,
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0.013135554268956184,
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0.023280901834368706,
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-0.0146... | |
bd478af02b0bae7b10d4eb3ea22ac670c7844315 | subsection | 130 | 216 | Supplementary Material for Sec. | 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 | [
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-0.03090887889266014,
0.013837291859090328,
-0.016354549676179886,
0... | |
77beefc96c01b071ed81afd5d97cc914a3310177 | subsection | 131 | 216 | Supplementary Material for Sec. | \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 | [
-0.01908961869776249,
-0.012932416051626205,
-0.028184274211525917,
-0.015686754137277603,
0.015450230799615383,
0.032014403492212296,
0.023652205243706703,
0.03119039349257946,
0.0009956816211342812,
0.03296049311757088,
-0.01753315143287182,
0.012573817744851112,
0.026551509276032448,
-0... | |
089fcefe78cde97b7ecf65e1d1bad0b26d44c045 | subsection | 132 | 216 | Supplementary Material for Sec. | \squareProof of Lemma REF .
First note that \Phi (q), as defined in (REF ), is a continuous and differentiable function of the strategy profile q. To show that \Phi (q) is a weighted potential function of \Gamma (\lambda ), we write the first order derivative of \Phi (q) with respect to q_{}^{i}(t^i):\frac{\partial \Ph... | {
"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 | [
-0.07141795009374619,
0.037723325192928314,
-0.010239624418318272,
-0.021196480840444565,
0.004784086719155312,
-0.03671615198254585,
0.022463081404566765,
0.013093290850520134,
0.03851686045527458,
0.042545564472675323,
0.003929513040930033,
0.02640022523701191,
-0.021990014240145683,
0.0... | |
d643c30981313627e65f7986749b95a0e27e2811 | subsection | 133 | 216 | Supplementary Material for Sec. | \quad \forall e, e^{^{\prime }} \in \mathcal {E}, \quad \forall , ^{^{\prime }} \in
}{}{equation*}
Since for any e\in \mathcal {E} and any s\in \mathcal {S}, c_{e}^{s}(w_{e}) is increasing in w_{e}, \sum _{s\in \mathcal {S}} \pi \left(s, \right) \frac{d c_{e}^{s}\left(w_{e}\left(\right)\right)}{d w_{e}\left(\right)}>0... | {
"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 | [
-0.0037156008183956146,
0.022354641929268837,
-0.02726808562874794,
-0.03247145190834999,
0.0018120229942724109,
0.0014458035584539175,
0.011047618463635445,
-0.0050774794071912766,
-0.007492238190025091,
0.07141277939081192,
-0.042664557695388794,
0.022354641929268837,
0.028793999925255775,... | |
49059054a0fb91f70e837f58302b00a2e2b38410 | subsection | 134 | 216 | Supplementary Material for Sec. | \end{align}
\end{}}Using (\ref {potential_prove}) and (\ref {first_order}), we have \frac{\partial \Phi (q) }{\partial q_{}^{i}(t^i)} = \mathrm {Pr}(t^i) \mathbb {E}[c_{}({q})|t^i]= \mu ^{t^i}+\nu _{r}^{t^i} for any \in \mathcal {R}, and t^i\in {i}, i\in I. From (\ref {com_slack}), we see that for any \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 | [
-0.027792664244771004,
0.016794579103589058,
-0.05341926962137222,
-0.03003499284386635,
-0.005403705406934023,
-0.030157024040818214,
0.04344319924712181,
-0.014506489038467407,
0.03813483193516731,
0.03999580815434456,
-0.027029966935515404,
0.044236402958631516,
-0.014613267034292221,
0... | |
51f505a5c4778ed8cf64c56aa50307e6f2f4b08a | subsection | 135 | 216 | Supplementary Material for Sec. | We can easily check that (\ref {first_order}) and (\ref {beta_positive}) are satisfied by (q^{*}, \bar{\mu }, \bar{\nu }). Since q^{*} is a BWE, we know from (\ref {eq:BWE_fun}) that for a route \in \mathcal {R}, and t^i\in {i}, i\in I, if q^{i*}_{}(t^i)>0, then \mathbb {E}[c_{}({q^{*}})|t^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"
] | 2,018 | en | Computer Science | [
-0.03426320105791092,
0.017055325210094452,
-0.012608431279659271,
-0.04067039117217064,
-0.002158612245693803,
-0.03136471286416054,
0.02208954468369484,
0.007345383521169424,
0.027627184987068176,
0.06019705906510353,
-0.0014378034975379705,
0.03127318248152733,
-0.0034629327710717916,
0... | |
b885430796f9484f9dfa4da475214b717579ad92 | subsection | 136 | 216 | Supplementary Material for Sec. | \hfill \square }\end{equation}}}}{}}\begin{}{(Theorem 2 in \cite {wachsmuth2013licq})}
The Lagrange multiplies \mu ^{*} and \nu ^{*} associated with any q^{*}\in \mathcal {\mathcal {Q}}^{*}(\lambda ) at the optimum of (\ref {eq:potential_opt}) are unique if and only if the LICQ condition is satisfied in that the gradie... | {
"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 | [
-0.03107801079750061,
0.01633998565375805,
-0.017209621146321297,
-0.06322399526834488,
-0.01781989075243473,
0.004355803597718477,
0.00034637603675946593,
-0.00039953627856448293,
-0.007277472410351038,
0.023571686819195747,
-0.020245714113116264,
0.000667006301227957,
-0.014043843373656273... | |
365208b714235ce276f370bd0b4cd8a3e9ec94a4 | subsection | 137 | 216 | Supplementary Material for Sec. | From (), we obtain that for any t^i, \tilde{t}^{i}\in {i}, any t^{-i}, \tilde{}^{-i}\in \mathcal {T}^{-i}, and any i\in I, f satisfies (REF ):\begin{split}
&f_{}(t^i, t^{-i})-f_{}(\tilde{t}^{i}, t^{-i})= q_{}^{i}(t^i)+\sum _{j\in I\setminus \lbrace i\rbrace } q^{j}_(^j)-q^i_(\tilde{t}^{i})-\sum _{j\in I\setminus \lbrac... | {
"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 | [
-0.017043352127075195,
0.006267284508794546,
-0.022963514551520348,
-0.005923976190388203,
0.0002593885292299092,
0.025694722309708595,
0.002147584455087781,
-0.00020348171528894454,
-0.0030363716650754213,
0.04220404103398323,
0.005874387454241514,
-0.02372642047703266,
-0.01012378185987472... | |
bd411111957cf757a6c6abef37a1e618ef2a2b3c | subsection | 138 | 216 | Supplementary Material for Sec. | For any route \in \mathcal {R}, the linear system of equations () has \prod _{i\in I} |{i}| equations in \sum _{i\in I} |{i}| variables. Note that for any given \widehat{}=\left(\widehat{}^{i}\right)_{i\in I} \in , the following equations are linearly independent:
\begin{equation}
\begin{split}
\sum _{i\in I} q^i_(\wid... | {
"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 | [
-0.00306727085262537,
0.024477127939462662,
-0.02720867656171322,
-0.043399594724178314,
0.014054509811103344,
0.027346016839146614,
0.011574750766158104,
0.06073501706123352,
-0.013398327864706516,
0.010155566036701202,
-0.03308379650115967,
-0.02655249461531639,
0.005192233249545097,
-0.... | |
cf7bf99051ab44f9436e49de25813dac627be38d | subsection | 139 | 216 | Supplementary Material for Sec. | We apply the same procedure iteratively for another |I|-2 times:\sum _{i\in I} f_{}(t^i, \widehat{}^{-i})-(|I|-1)f_{}(\widehat{})
&=f_{}(), \quad \forall \inNow for any \in \mathcal {R} and \in , we can write iI qi(ti)=iI (qi(ti)+jI{i} qj(tj))- (|I|-1) iI qi(i)()=iI f(ti, -i)-(|I|-1)f() (REF )=f().
Thus, for any R, 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 | [
-0.030669350177049637,
0.009254209697246552,
-0.02082769386470318,
-0.026915788650512695,
0.011710809543728828,
0.02175845392048359,
0.03213415667414665,
0.029860656708478928,
0.00863624457269907,
0.01644853688776493,
-0.015822943300008774,
-0.025893475860357285,
0.02078191749751568,
0.002... | |
734fe688ac2db974f7689719ff357189f2f0fac3 | subsection | 140 | 216 | Supplementary Material for Sec. | Thus, \chi _^{i} \ge \max _{t^i\in {i}} \left\lbrace f_{}(\widehat{}^{i}, \widehat{}^{-i})-f_{}(t^i, \widehat{}^{-i})\right\rbrace , i.e. \chi satisfies ().Step III: Finally, we show that the set of \chi satisfying () is non-empty, i.e., any f\in \mathcal {F}(\lambda ) can be induced by at least one feasible strategy p... | {
"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 | [
-0.012500273995101452,
-0.005540414713323116,
-0.04066023230552673,
-0.03397510573267937,
-0.0011981336865574121,
-0.026526832953095436,
0.03797397017478943,
0.02475634217262268,
0.012118702754378319,
0.021902190521359444,
0.00003589154584915377,
0.0010149795562028885,
-0.006398949772119522,... | |
d3ab955362c826607b0faedfecc92b1d12d0f757 | subsection | 141 | 216 | Supplementary Material for Sec. | Thus, for any R, we obtain:
\begin{align}
f_{}(\widehat{})-\sum _{i\in I} \max _{t^i\in {i}} \left(f_{}(\widehat{})-f_{}(t^i, \widehat{}^{-i})\right) &=\min _{\in \sum _{i\in I} f_{}(t^i, \widehat{}^{-i})- (|I|-1)f_{}(\widehat{})=\min _{\in f_{}()\ge 0.
}
Hence, we can conclude that \gamma _\ge 0. Next, \lambda ^{i}\s... | {
"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 | [
-0.00027801477699540555,
0.013390488922595978,
-0.025087136775255203,
-0.01835755817592144,
-0.006496866699308157,
0.01580916903913021,
-0.0024339405354112387,
0.026750456541776657,
-0.028291698545217514,
0.009979919530451298,
-0.05194441229104996,
0.006931771524250507,
-0.031893014907836914... | |
4b584f3850da9cb305083609bd752d2456029f02 | subsection | 142 | 216 | Supplementary Material for Sec. | 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... | |
203adec0b9eb4d0d4549d633b1ef1e05dbd9f37f | subsection | 143 | 216 | Supplementary Material for Sec. | 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... | |
4f5223051056ea3836d478f0f33091c9f3e61c6c | subsection | 144 | 216 | Supplementary Material for Sec. | \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 | [
-0.01908961869776249,
-0.012932416051626205,
-0.028184274211525917,
-0.015686754137277603,
0.015450230799615383,
0.032014403492212296,
0.023652205243706703,
0.03119039349257946,
0.0009956816211342812,
0.03296049311757088,
-0.01753315143287182,
0.012573817744851112,
0.026551509276032448,
-0... | |
cefd27f76378d5a89726ab5ba0ff227c2d4ec19f | subsection | 145 | 216 | Supplementary Material for Sec. | \squareProof of Lemma REF .
First note that \Phi (q), as defined in (REF ), is a continuous and differentiable function of the strategy profile q. To show that \Phi (q) is a weighted potential function of \Gamma (\lambda ), we write the first order derivative of \Phi (q) with respect to q_{}^{i}(t^i):\frac{\partial \Ph... | {
"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 | [
-0.07141795009374619,
0.037723325192928314,
-0.010239624418318272,
-0.021196480840444565,
0.004784086719155312,
-0.03671615198254585,
0.022463081404566765,
0.013093290850520134,
0.03851686045527458,
0.042545564472675323,
0.003929513040930033,
0.02640022523701191,
-0.021990014240145683,
0.0... | |
656c9741d1c49698a31d99b591f4aba8553f4300 | subsection | 146 | 216 | Supplementary Material for Sec. | \quad \forall e, e^{^{\prime }} \in \mathcal {E}, \quad \forall , ^{^{\prime }} \in
}{}{equation*}
Since for any e\in \mathcal {E} and any s\in \mathcal {S}, c_{e}^{s}(w_{e}) is increasing in w_{e}, \sum _{s\in \mathcal {S}} \pi \left(s, \right) \frac{d c_{e}^{s}\left(w_{e}\left(\right)\right)}{d w_{e}\left(\right)}>0... | {
"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 | [
-0.0037156008183956146,
0.022354641929268837,
-0.02726808562874794,
-0.03247145190834999,
0.0018120229942724109,
0.0014458035584539175,
0.011047618463635445,
-0.0050774794071912766,
-0.007492238190025091,
0.07141277939081192,
-0.042664557695388794,
0.022354641929268837,
0.028793999925255775,... | |
2656aefdf04f50a143c8542fac54387d424ea879 | subsection | 147 | 216 | Supplementary Material for Sec. | \end{align}
\end{}}Using (\ref {potential_prove}) and (\ref {first_order}), we have \frac{\partial \Phi (q) }{\partial q_{}^{i}(t^i)} = \mathrm {Pr}(t^i) \mathbb {E}[c_{}({q})|t^i]= \mu ^{t^i}+\nu _{r}^{t^i} for any \in \mathcal {R}, and t^i\in {i}, i\in I. From (\ref {com_slack}), we see that for any \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 | [
-0.027792664244771004,
0.016794579103589058,
-0.05341926962137222,
-0.03003499284386635,
-0.005403705406934023,
-0.030157024040818214,
0.04344319924712181,
-0.014506489038467407,
0.03813483193516731,
0.03999580815434456,
-0.027029966935515404,
0.044236402958631516,
-0.014613267034292221,
0... | |
8318301b984300e14cbf3cdad871b0e782949d9f | subsection | 148 | 216 | Supplementary Material for Sec. | We can easily check that (\ref {first_order}) and (\ref {beta_positive}) are satisfied by (q^{*}, \bar{\mu }, \bar{\nu }). Since q^{*} is a BWE, we know from (\ref {eq:BWE_fun}) that for a route \in \mathcal {R}, and t^i\in {i}, i\in I, if q^{i*}_{}(t^i)>0, then \mathbb {E}[c_{}({q^{*}})|t^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"
] | 2,018 | en | Computer Science | [
-0.03426320105791092,
0.017055325210094452,
-0.012608431279659271,
-0.04067039117217064,
-0.002158612245693803,
-0.03136471286416054,
0.02208954468369484,
0.007345383521169424,
0.027627184987068176,
0.06019705906510353,
-0.0014378034975379705,
0.03127318248152733,
-0.0034629327710717916,
0... | |
1faa428d60a3dfd1e0834d7c8e693a84d34b3653 | subsection | 149 | 216 | Supplementary Material for Sec. | \hfill \square }\end{equation}}}}{}}\begin{}{(Theorem 2 in \cite {wachsmuth2013licq})}
The Lagrange multiplies \mu ^{*} and \nu ^{*} associated with any q^{*}\in \mathcal {\mathcal {Q}}^{*}(\lambda ) at the optimum of (\ref {eq:potential_opt}) are unique if and only if the LICQ condition is satisfied in that the gradie... | {
"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 | [
-0.03107801079750061,
0.01633998565375805,
-0.017209621146321297,
-0.06322399526834488,
-0.01781989075243473,
0.004355803597718477,
0.00034637603675946593,
-0.00039953627856448293,
-0.007277472410351038,
0.023571686819195747,
-0.020245714113116264,
0.000667006301227957,
-0.014043843373656273... | |
cd9a4ceeee6d717e9ad0b72d2c95de87edac06e9 | subsection | 150 | 216 | Supplementary Material for Sec. | From (), we obtain that for any t^i, \tilde{t}^{i}\in {i}, any t^{-i}, \tilde{}^{-i}\in \mathcal {T}^{-i}, and any i\in I, f satisfies (REF ):\begin{split}
&f_{}(t^i, t^{-i})-f_{}(\tilde{t}^{i}, t^{-i})= q_{}^{i}(t^i)+\sum _{j\in I\setminus \lbrace i\rbrace } q^{j}_(^j)-q^i_(\tilde{t}^{i})-\sum _{j\in I\setminus \lbrac... | {
"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 | [
-0.017043352127075195,
0.006267284508794546,
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0.0002593885292299092,
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0.04220404103398323,
0.005874387454241514,
-0.02372642047703266,
-0.01012378185987472... | |
65b40d68a572b9fedfa0a0cc77f724166124ba55 | subsection | 151 | 216 | Supplementary Material for Sec. | For any route \in \mathcal {R}, the linear system of equations () has \prod _{i\in I} |{i}| equations in \sum _{i\in I} |{i}| variables. Note that for any given \widehat{}=\left(\widehat{}^{i}\right)_{i\in I} \in , the following equations are linearly independent:
\begin{equation}
\begin{split}
\sum _{i\in I} q^i_(\wid... | {
"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 | [
-0.00306727085262537,
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0.011574750766158104,
0.06073501706123352,
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0.010155566036701202,
-0.03308379650115967,
-0.02655249461531639,
0.005192233249545097,
-0.... | |
e5111b096931a3dba7a1660bcedc483e8ab2131b | subsection | 152 | 216 | Supplementary Material for Sec. | We apply the same procedure iteratively for another |I|-2 times:\sum _{i\in I} f_{}(t^i, \widehat{}^{-i})-(|I|-1)f_{}(\widehat{})
&=f_{}(), \quad \forall \inNow for any \in \mathcal {R} and \in , we can write iI qi(ti)=iI (qi(ti)+jI{i} qj(tj))- (|I|-1) iI qi(i)()=iI f(ti, -i)-(|I|-1)f() (REF )=f().
Thus, for any R, 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 | [
-0.030669350177049637,
0.009254209697246552,
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0.01644853688776493,
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-0.025893475860357285,
0.02078191749751568,
0.002... | |
c52928bf5e48bed7d7a96ac10dc0ab8539b78c47 | subsection | 153 | 216 | Supplementary Material for Sec. | Thus, \chi _^{i} \ge \max _{t^i\in {i}} \left\lbrace f_{}(\widehat{}^{i}, \widehat{}^{-i})-f_{}(t^i, \widehat{}^{-i})\right\rbrace , i.e. \chi satisfies ().Step III: Finally, we show that the set of \chi satisfying () is non-empty, i.e., any f\in \mathcal {F}(\lambda ) can be induced by at least one feasible strategy p... | {
"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 | [
-0.012500273995101452,
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0.021902190521359444,
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0.0010149795562028885,
-0.006398949772119522,... | |
3c27b758dc1718c19f2d7316763e961db20a6b52 | subsection | 154 | 216 | Supplementary Material for Sec. | Thus, for any R, we obtain:
\begin{align}
f_{}(\widehat{})-\sum _{i\in I} \max _{t^i\in {i}} \left(f_{}(\widehat{})-f_{}(t^i, \widehat{}^{-i})\right) &=\min _{\in \sum _{i\in I} f_{}(t^i, \widehat{}^{-i})- (|I|-1)f_{}(\widehat{})=\min _{\in f_{}()\ge 0.
}
Hence, we can conclude that \gamma _\ge 0. Next, \lambda ^{i}\s... | {
"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 | [
-0.00027801477699540555,
0.013390488922595978,
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0.009979919530451298,
-0.05194441229104996,
0.006931771524250507,
-0.031893014907836914... | |
7723c1ebaf81e36d5217450916abc49f1c69292b | subsection | 155 | 216 | Supplementary Material for Sec. | 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,
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0.04003216698765755,
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0.013135554268956184,
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0.023280901834368706,
-0.03139717876911163,
-0.0146... | |
1da1e218f9b516baaea48781e97fb2f20dfd8da7 | subsection | 156 | 216 | Supplementary Material for Sec. | 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,
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0.0038235625252127647,
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0.013837291859090328,
-0.016354549676179886,
0... | |
d1c36f24b9b5faaaa43b53e775500b9bff5e4e92 | subsection | 157 | 216 | Supplementary Material for Sec. | \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 | [
-0.017123136669397354,
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0.011522253043949604,
0.02600519172847271,
-... | |
d4fffbeff744719ad2e129172b7719c041545500 | subsection | 158 | 216 | 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 optimal solution set of (REF )) is an optimal soluti... | {
"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 | [
-0.04986855015158653,
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0.04101794958114624,
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0.010559680871665478,
-0.019318722188472748,
... | |
feb5c3945a9c3bf11aa1522a98621dbb0ef13924 | subsection | 159 | 216 | Supplementary Material for Section | 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 \math... | {
"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 | [
-0.02141396515071392,
0.021352913230657578,
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0.02116975747048855,
-0.07130865752696991,
0.024619191884994507,
-0.05039836838841438,
0.0047... | |
9ef235a5dddcb316f5c8011145c6de4de1e45fb3 | subsection | 160 | 216 | Supplementary Material for Section | 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 | [
-0.02334541641175747,
-0.007388900499790907,
-0.048186156898736954,
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0.011527752503752708,
0.010413886047899723,
0.014838834293186665,
0.017654016613960266,
0.0528247244656086,
-0.05239748954772949,
0.018493231385946274,
-0.008361626416444778,
0... | |
a728f6f4d29a3c985ede80cabb18aaa49366cd17 | subsection | 161 | 216 | Supplementary Material for Section | 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,
0.009893273003399372,
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0.011037444695830345,
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0.0549202486872673,
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0.0006283410475589335,
-0.03536253795027733,
0... | |
d0859d373bada69f176f4e7ebf0727e9a08f9210 | subsection | 162 | 216 | Supplementary Material for Section | 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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0.00578062329441309,
-0.04325355961918831,
0.... | |
7393e82c9577ac3ece9ae0f0539d5217973a1298 | subsection | 163 | 216 | Supplementary Material for Section | 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,
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0.004883687011897564,
-0.02544095739722252,
... | |
e6ce458081528eb6fc26f38d22dd6db991efbd72 | subsection | 164 | 216 | Supplementary Material for Section | 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,
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0.044062819331884384,
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-0.018064534291625023... | |
1a6518831c059a2e531565c8a56d25ac549f3c2b | subsection | 165 | 216 | Supplementary Material for Section | 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,
... | |
f9d0796d23c119c34df47ce77e105eb99907b9be | subsection | 166 | 216 | Supplementary Material for Section | 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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0.010467863641679287,
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0.020264320075511932,
-0.0042802272364497185,
-0.... | |
b0a8148b4abe6d2b493e26bafac4d9aa88b986f8 | subsection | 167 | 216 | Supplementary Material for Section | 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,
-0.0025043527130037546,
-0.02877812087535858,
-0.02832035720348358,
0.0000032186558200919535,
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0.04943855479359627,
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-0.02195743098855... | |
2668c36b791cb6a4effc2992c4ffe835f960e25b | subsection | 168 | 216 | Supplementary Material for Section | 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": 641,
"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 | [
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0.026785030961036682,
-0.03278304636478424,
-... | |
8d23b55d5c2526af8ec26a118b938696bba432fa | subsection | 169 | 216 | Supplementary Material for Section | 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 | [
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-0.03238453343510628,
0.019778678193688393,
-0.03442955017089844,
-0.011453624814748764,
-0.009858815930783749,
... |
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