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Self-Fulfilling Signal of an Endogenous State in Network Congestion Games

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Abstract

We consider the problem of coordination via signaling in network congestion games to improve social welfare deteriorated by incomplete information about traffic flow. Traditional studies on signaling, which focus on exogenous factors of congestion and ignore congestion externalities, fail to discuss the oscillations of traffic flow. To address this gap, we formulate a problem of designing a coordination signal on endogenous information about traffic flow and introduce a self-fulfilling characteristic of a signal that guarantees an outcome flow consistent with the signal itself without causing the unwanted oscillation. An instance of the self-fulfilling signal is shown in the case of a Gaussian signal distribution. In addition, we show simple numerical examples. The results reveal how a self-fulfilling signal suppresses the oscillation and simultaneously improves social welfare through improved network efficiency.

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Correspondence to Tatsuya Iwase.

Appendix: Bayesian Inference for Linear Gaussian Model

Appendix: Bayesian Inference for Linear Gaussian Model

Here, we summarize the general results of Bayesian inference for a linear Gaussian model (Bishop 2006). Let prior Gaussian distribution of x be denoted by

$$ p(\boldsymbol{x})=N(\boldsymbol{x};\boldsymbol{\mu},\boldsymbol{\Lambda}^{-1}). $$
(46)

In linear Gaussian model, the conditional distribution of y given x has a mean that is a linear function of x, such as

$$ p(\boldsymbol{y}|\boldsymbol{x})=N(\boldsymbol{y};\boldsymbol{A}\boldsymbol{x}+\boldsymbol{b},\boldsymbol{L}^{-1}). $$
(47)

The posterior distribution of x given y is denoted by

$$ p(\boldsymbol{x}|\boldsymbol{y})=N(\boldsymbol{x};\boldsymbol{\Sigma}\{\boldsymbol{A}^{T}\boldsymbol{L} (\boldsymbol{y}-\boldsymbol{b})+\boldsymbol{\Lambda}\boldsymbol{\mu}\},\boldsymbol{\Sigma}), $$
(48)

where

$$ \boldsymbol{\Sigma}=(\boldsymbol{\Lambda}+\boldsymbol{A}^{T}\boldsymbol{L}\boldsymbol{A})^{-1}. $$
(49)

The marginal distribution of y is denoted by

$$ p(\boldsymbol{y})=N(\boldsymbol{y};\boldsymbol{A}\boldsymbol{\mu}+\boldsymbol{b},\boldsymbol{L}^{-1}+\boldsymbol{A} \boldsymbol{\Lambda}^{-1}\boldsymbol{A}^{T}). $$
(50)

If covariance matrices are not full rank, it is impossible to calculate their inverse matrices. In this case, there exists a linear transformation that reduces the dimension of vector x and makes its covariance Σ full rank as follows

$$\begin{array}{@{}rcl@{}} \tilde{\boldsymbol{x}}&=&\boldsymbol{C}(\boldsymbol{x}-\boldsymbol{\mu}) \\ \tilde{\boldsymbol{\Sigma}}&=&\boldsymbol{C}\boldsymbol{\Sigma}\boldsymbol{C}^{T}. \end{array} $$
(51)

This transformation matrix C is obtained by singular value decomposition. The calculation including inverse matrices can be processed in this reduced vector space. Let \(\tilde {\boldsymbol {x}}_{2}\) be the result. This is converted into the original vector space by the following reverse transformation

$$ \boldsymbol{x}_{2}=\boldsymbol{C}^{T}\tilde{\boldsymbol{x}}_{2}+\boldsymbol{\mu}. $$
(52)

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Iwase, T., Tadokoro, Y. & Fukuda, D. Self-Fulfilling Signal of an Endogenous State in Network Congestion Games. Netw Spat Econ 17, 889–909 (2017). https://doi.org/10.1007/s11067-017-9351-4

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