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Equilibrium routing under uncertainty

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Abstract

We provide a brief introduction to the basic models used to describe traffic on congested networks, both in urban transport and telecommunications. We discuss traffic equilibrium models, covering atomic and non-atomic routing games, with emphasis on situations where the travel times are subject to random fluctuations. We use convex optimization to present the models in a unified framework that stresses the common underlying structures. As a prototypical example of traffic equilibrium with elastic demands, we discuss some models for routing and congestion control in telecommunications. We also describe a class of stochastic dynamics that model the adaptive behavior of agents and which provides a plausible micro-foundation for the equilibrium. Finally we present some recent ideas on how risk-averse behavior might be incorporated in the equilibrium models.

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Notes

  1. We thank Michael Patriksson for pointing out the earlier paper [105] which already formulated an equivalent optimization problem for equilibrium.

  2. The \(\epsilon _a^k\)’s are allowed to depend on k, which may capture differences in travel time perceptions for different classes of agents. However, they are not required to be independent and may be taken the same for all k, allowing even for correlations among arcs.

  3. Since \(\tau _{d_k}^k \equiv 0\) we adopt the convention \(\varphi _{d_k}^k(\cdot )\equiv 0\). Note that \(\varphi _i^k\) is a smooth and concave lower approximation of the polyhedral function \(\bar{\varphi }_i^k(z^k)=\min _{a\in A_i^+}z_a^k\), so that (4) is a smoothed version of the standard shortest path equations \(\bar{\tau }_i^k=\min _{a\in A_i^+}t_a+\bar{\tau }_{j_a}^k\).

  4. Note that the common usage in telecommunications is to state the model as the maximization of a network utility function. Here we follow the convention in traffic equilibrium by stating the model in the form of a convex minimization problem. This choice also facilitates the use of the convex duality theory.

  5. This corresponds to the case of 2 parallel links with linear costs and identical players.

  6. Not all preference relations can be represented in this form, though this is not too restrictive (see e.g. [47, Proposition 2.13]).

  7. The limit cases \(\beta \rightarrow \pm \infty \) can also be considered as extreme attitudes toward risk with \(\rho ^{ent}_{\infty }(X)=\text {ess sup\,}X\) and \(\rho ^{ent}_{-\infty }(X)=\text {ess inf\,}X\). In this paper we only consider finite \(\beta \)’s.

  8. As a technical remark, we note that in order to properly define a random variable \(X^h\) from its distribution, we need \((\varOmega ,{\mathcal {F}},{\mathbb {P}})\) to be a standard non-atomic probability space. However the explicit integral formula for \(\rho ^h(X)\) does not require this.

  9. A simple entropic risk measure corresponds to the case of a Dirac distribution F.

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Acknowledgments

A significant portion of the material covered in this review comes from four papers written in collaboration with my co-authors Jean-Bernard Baillon, Cristóbal Guzmán, Emerson Melo, Sylvain Sorin and Alfredo Torrico. Our synthesis also benefited greatly from my collaborations with Mario Bravo, Luis Briceño, Manuel Cepeda, José Correa, Cristián Cortés, Mike Florian, Omar Larré, Francisco Martínez, and Nicolas Stier-Moses. I am deeply indebted to all of them for sharing their own views on the subject. I also express my gratitude to Neil Walton for his comments as well as to the five anonymous referees for they careful and detailed revision of the first draft of this paper. Their comments and suggestions were extremely useful to improve the paper presentation. Naturally, I am fully responsible for any inaccuracies that may remain. Along the years my work has received support from different sources. I especially thank the Chilean tax payers for their support through the Grants FONDECYT 1100046 and 1130564, the Núcleo Milenio Información y Coordinación en Redes (ICM/FIC P10-024F), and the Instituto de Sistemas Complejos de Ingeniería (ICM/FIC P05-004-F and Proyecto Basal FBO/16).

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    Roberto Cominetti

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Cominetti, R. Equilibrium routing under uncertainty. Math. Program. 151, 117–151 (2015). https://doi.org/10.1007/s10107-015-0889-y

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