Efficiency of Restricted Tolls in Non-atomic Network Routing Games

An effective means to reduce the inefficiency of Nash flows in non- atomic network routing games is to impose tolls on the arcs of the network. It is a well-known fact that marginal cost tolls induce a Nash flow that corresponds to a minimum cost flow. However, despite their effectiveness, marginal cost tolls suffer from two major drawbacks, namely (i) that potentially every arc of the network is tolled, and (ii) that the imposed tolls can be arbitrarily large. In this paper, we study the restricted network toll problem in which tolls can be imposed on the arcs of the network but are restricted to not exceed a predefined threshold for every arc. We show that optimal restricted tolls can be computed efficiently for parallel-arc networks and affine latency functions. This generalizes a previous work on taxing subnetworks to arbitrary restrictions. Our algorithm is quite simple, but relies on solving several convex programs. The key to our approach is a characterization of the flows that are inducible by restricted tolls for single-commodity networks. We also derive bounds on the efficiency of restricted tolls for multi-commodity networks and polynomial latency functions. These bounds are tight even for parallel-arc networks. Our bounds show that restricted tolls can significantly reduce the price of anarchy if the restrictions imposed on arcs with high-degree polynomials are not too severe. Our proof is constructive. We define tolls respecting the given thresholds and show that these tolls lead to a reduced price of anarchy by using a (\lambda,\mu)-smoothness approach

Uncertainty in Multi-Commodity Routing Networks: When does it help?

We study the equilibrium behavior in a multi-commodity selfish routing game with many types of uncertain users where each user over- or under-estimates their congestion costs by a multiplicative factor. Surprisingly, we find that uncertainties in different directions have qualitatively distinct impacts on equilibria. Namely, contrary to the usual notion that uncertainty increases inefficiencies, network congestion actually decreases when users over-estimate their costs. On the other hand, under-estimation of costs leads to increased congestion. We apply these results to urban transportation networks, where drivers have different estimates about the cost of congestion. In light of the dynamic pricing policies aimed at tackling congestion, our results indicate that users' perception of these prices can significantly impact the policy's efficacy, and "caution in the face of uncertainty" leads to favorable network conditions.Comment: Currently under revie

Jogos de localização de instalações não cooperativos e percepção de custos

Orientadores: Eduardo Candido Xavier, Guido SchäferTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Esta tese de doutorado cobre a interseção entre problemas de localização de instalações e teoria dos jogos algorítmica não cooperativa, com ênfase em alterações da percepção de custos de cada jogador e seu efeito na qualidade de equilíbrios. O problema de localização de instalações é um dos problemas fundamentais em otimização combinatória. Em sua versão clássica, existe um conjunto de terminais e um conjunto de instalações, e cada terminal necessita ser conectado a uma instalação, para que esta providencie bens ou serviços. O objetivo é minimizar o total dos custos associados à abertura das instalações e à conexão dos terminais a essas instalações. Na prática, existem diversos cenários onde é inviável ou não é desejável que uma autoridade central única decida como clientes devem escolher as instalações às quais se conectam. Dessa forma, é importante estudar como a independência desses terminais pode afetar a eficiência social e a complexidade computacional para esses cenários. A teoria dos jogos algorítmica pode ser útil para tais cenários, em particular sua parte não cooperativa. A teoria dos jogos algorítmica preenche uma lacuna entre a ciência da computação teórica e a teoria dos jogos, e está interessada em questões como a complexidade computacional de se encontrar equilíbrios, o quanto o bem-estar social pode ser perdido devido ao egoísmo de jogadores e como desenvolver mecanismos para garantir que o melhor interesse dos jogadores se alinhe com o ótimo social. Nesta tese, estudamos jogos de localização de instalações não cooperativos e algumas de suas variantes. Focamos em responder questões relativas à existência de equilíbrios de Nash puros e sobre as principais medidas de perda de eficiência, o preço da anarquia e preço da estabilidade. Apresentamos uma revisão das descobertas mais importantes para as variantes básicas, com novos resultados nos casos onde nenhum era conhecido. Para a versão capacitada desses jogos, mostramos que, enquanto a simultaneidade pode levar a uma perda de eficiência ilimitada, quando se admite a sequencialidade de jogadores, é possível mostrar que a perda de eficiência tem limites. Também investigamos como mudanças na percepção de custo podem afetar a qualidade de equilíbrios de duas maneiras: através de jogadores altruístas e de esquemas de taxação. No primeiro, adaptamos resultados de jogos de compartilhamento justo de custos e apresentamos novos resultados sobre uma versão sem regras de compartilhamento. No último, propomos um modelo de mudança na percepção de custos, onde os jogadores consideram um pedágio adicional em suas conexões ao calcular seus custos. Apresentamos limitantes para o custo total das taxas no problema de pedágios mínimos, onde o objetivo é encontrar o valor mínimo de pedágio necessário para garantir que um determinado perfil de estratégia socialmente ótimo seja escolhido pelos jogadores. Mostramos algoritmos para encontrar pedágios ótimos para tal problema em casos especiais e relacionamos esse problema a um problema de emparelhamento NP-difícilAbstract: This Ph.D. thesis covers the intersection between facility location problems and non-cooperative algorithmic game theory, with emphasis on possible changes in cost perception and its effects in regards to quality of equilibria. The facility location problem is one of the fundamental problems in the combinatorial optimization field of study. In its classic version, there exists a set of terminals and a set of facilities, and each terminal must be connected to a facility, in order for goods or services to be provided. The objective is to minimize the total costs associated with opening the facilities and connecting all the terminals to these facilities. In practice, there are multiple scenarios where it is either infeasible or not desirable for a single central authority to decide which facilities terminals connect to. Thus, it is important to study how the independence of these terminals may affect social efficiency and computational complexity in these scenarios. For this analysis algorithmic game theory can be of use, in particular its non-cooperative part. Algorithmic game theory bridges a gap between theoretical computer science and game theory, and is interested in questions such as how hard it is computationally to find equilibria, how much social welfare can be lost due to player selfishness and how to develop mechanisms to ensure that players' best interest align with the social optimum. In this thesis we study non-cooperative facility location games and several of its variants. We focus on answering the questions concerning the existence of pure Nash equilibria and the main measures of efficiency loss, the price of anarchy and the price of stability. We present a review of the most important findings for the basic variants and show new results where none were known. For the capacitated version of these games, we show that while simultaneity may lead to unbounded loss of efficiency, when sequentiality is allowed, it is possible to bound the efficiency loss. We also investigate how changes in players' perception of cost can affect the efficiency loss of these games in two ways: through altruistic players and through tolling schemes. In the former we adapt results from fair cost sharing games and present new results concerning a version with no cost sharing rules. In the latter, we propose a model for change in cost perception where players consider an additional toll in their connections when calculating their best responses. We present bounds for total toll cost in the minimum toll problem, where the objective is to find the minimum amount of tolls needed to ensure that a certain socially optimal strategy profile will be chosen by players. We show algorithms for finding optimal tolls for the minimum toll problem in special cases and provide some insight into this problem by connecting it to a matching problem which we prove is NP-hardDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação147141/2016-8CAPESCNP

The dynamic user equilibrium on a transport network: mathematical properties and economic applications

This thesis is focused on dynamic user equilibrium models and their applications to traffic assignment. It aims at providing a mathematically rigorous and general formulation for the dynamic user equilibrium. Particular attention is paid to the representation of transport demand and more specifically to trip scheduling and users with heterogeneous preferences. This is achieved by expressing the dynamic user equilibrium as a Nash game with a continuum of players. This allows for a precise, concise and microeconomically consistent description. This thesis also deals with computational techniques. We solve analytically equilibrium on small networks to get a general intuition of the complex linkage between the demand and supply of transport in dynamic frameworks. The intuition acquired from the resolution is used to elaborate efficient numerical solving methods that can be applied to large size, real life, transport networks. Along the thesis several economic applications are proposed. All of them are dealing with the assessment of congestion pricing policies where are likely to reschedule their trips. In particular, a pricing scheme designed to ease congestion during holiday departure periods is tested. In this scheme a toll varying within the day and from day to day is set on the french motorway network. This form to toll is especially appealing as it enables the operator to influence the departure day as well as the departure time. Indeed it is shown that even moderate variations of the toll with time might have strong impacts on an highly congested interurban network.Cette thèse porte sur les modèles d'équilibres dynamiques sur un réseau de transport et leurs applications à l'affectation de trafic. Elle tente d'en propose une formulation à la fois générale et mathématiquement rigoureuse. Une attention particulière est accordée à la représentation de la demande de transport. Plus spécifiquement, la modélisation de l'hétérogénéité dans les préférences des usagers d'un réseau de transport, ainsi que des stratégies de choix d'horaire dans les déplacements, occupe une place importante dans notre approche. Une caractéristique de ce travail est son fort recours au formalisme mathématique; cela nous permet d'obtenir une formulation concise et micro-économiquement cohérente des réseaux de transport et de la demande de transport dans un contexte dynamique. Cette thèse traite aussi de méthodes de résolution en lien avec les modèles d'équilibres dynamiques. Nous établissons analytiquement des équilibres sur des réseaux de petites tailles afin d'améliorer la connaissance qualitative de l'interaction entre offre et demande dans ce contexte. L'intuition retirée de ces exercices nous permet de concevoir des méthodes numériques de calculs qui peuvent être appliquées à des réseaux de transport de grande taille. Tout au long de la thèse plusieurs applications économiques de ces travaux sont explorées. Toutes traitent des politiques de tarification de la congestion et de leurs évaluation, notamment lorsque les automobilistes sont susceptibles d'ajuster leurs horaires de départ. En particulier une politique tarifaire conçue pour limiter la congestion lors des grands départs de vacances est testée. Elle consiste à mettre en place un péage sur le réseau autoroutier variant selon l'heure de la journée mais aussi de jour en jour. Ce type de péage est particulièrement intéressant pour les exploitants car il leur permet d'influencer à la fois sur l'heure et le jour de départ des vacanciers. Les méthodes développées dans cette thèse permettent d'établir que les gains en termes de réduction de la congestion sont substantiels