The Inefficiency of Nash and Subgame Perfect Equilibria for Network Routing

This paper provides new bounds on the quality of equilibria in finite congestion games with affine cost functions, specifically for atomic network routing games. It is well known that the price of anarchy equals exactly 5/2 in general. For symmetric network routing games, it is at most (5n−2)/(2n+ 1), where n is the number of players. The paper answers to two open questions for congestion games. First, we show that the price of anarchy bound (5n−2)/(2n+ 1) is tight for symmetric network routing games, thereby answering a decade-old open question. Secondly, we ask if sequential play and subgame perfection allows to evade worst-case Nash equilibria, and thereby reduces the price of anarchy. This is motivated by positive results for congestion games with a small number of players, as well as recent results for other resource allocation problems. Our main result is the perhaps surprising proof that subgame perfect equilibria of sequential symmetric network routing games with linear cost functions can have an unbounded price of anarchy. We complete the picture by analyzing the case with two players: We show that the sequential price of anarchy equals 7/5, and that computing the outcome of a subgame perfect equilibrium is NP-hard

The curse of ties in congestion games with limited lookahead

We introduce a novel framework to model limited lookahead in congestion games. Intuitively, the players enter the game sequentially and choose an optimal action under the assumption that the k - 1 subsequent players play subgame-perfectly. Our model naturally interpolates between outcomes of greedy best-response (k = 1) and subgame-perfect outcomes (k = n, the number of players). We study the impact of limited lookahead (parameterized by k) on the stability and inefficiency of the resulting outcomes. As our results reveal, increased lookahead does not necessarily lead to better outcomes; in fact, its effect crucially depends on the existence of ties and the type of game under consideration

Network Investment Games with Wardrop Followers

We study a two-sided network investment game consisting of two sets of players, called providers and users. The game is set in two stages. In the first stage, providers aim to maximize their profit by investing in bandwidth of cloud computing services. The investments of the providers yield a set of usable services for the users. In the second stage, each user wants to process a task and therefore selects a bundle of services so as to minimize the total processing time. We assume the total processing time to be separable over the chosen services and the processing time of each service to depend on the utilization of the service and the installed bandwidth. We provide insights on how competition between providers affects the total costs of the users and show that every game on a series-parallel graph can be reduced to an equivalent single edge game when analyzing the set of subgame perfect Nash equilibria

Exact Price of Anarchy for Weighted Congestion Games with Two Players

This paper gives a complete analysis of worst-case equilibria for various versions of weighted congestion games with two players and affine cost functions. The results are exact price of anarchy bounds which are parametric in the weights of the two players, and establish exactly how the primitives of the game enter into the quality of equilibria. Interestingly, some of the worst-cases are attained when the players' weights only differ slightly. Our findings also show that sequential play improves the price of anarchy in all cases, however, this effect vanishes with an increasing difference in the players' weights. Methodologically, we obtain exact price of anarchy bounds by a duality based proof mechanism, based on a compact linear programming formulation that computes worst-case instances. This mechanism yields duality-based optimality certificates which can eventually be turned into purely algebraic proofs.Comment: 17 pages, 9 figures, 4 table

Selfishness Level of Strategic Games

We introduce a new measure of the discrepancy in strategic games between the social welfare in a Nash equilibrium and in a social optimum, that we call selfishness level. It is the smallest fraction of the social welfare that needs to be offered to each player to achieve that a social optimum is realized in a pure Nash equilibrium. The selfishness level is unrelated to the price of stability and the price of anarchy and is invariant under positive linear transformations of the payoff functions. Also, it naturally applies to other solution concepts and other forms of games. We study the selfishness level of several well-known strategic games. This allows us to quantify the implicit tension within a game between players' individual interests and the impact of their decisions on the society as a whole. Our analyses reveal that the selfishness level often provides a deeper understanding of the characteristics of the underlying game that influence the players' willingness to cooperate. In particular, the selfishness level of finite ordinal potential games is finite, while that of weakly acyclic games can be infinite. We derive explicit bounds on the selfishness level of fair cost sharing games and linear congestion games, which depend on specific parameters of the underlying game but are independent of the number of players. Further, we show that the selfishness level of the $n$-players Prisoner's Dilemma is $c/(b(n-1)-c)$, where $b$ and $c$ are the benefit and cost for cooperation, respectively, that of the $n$-players public goods game is $(1-\frac{c}{n})/(c-1)$, where $c$ is the public good multiplier, and that of the Traveler's Dilemma game is $\frac{1}{2}(b-1)$, where $b$ is the bonus. Finally, the selfishness level of Cournot competition (an example of an infinite ordinal potential game, Tragedy of the Commons, and Bertrand competition is infinite.Comment: 34 page

Análise de problemas de transporte sob a perspectiva da teoria de jogos

Orientadores: Flávio Keidi Miyazawa, Rafael Crivellari Saliba SchoueryDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Problemas relacionados com meios de transporte são comumente encontrados na área de Otimização Combinatória, como, por exemplo, o Problema do Caixeiro Viajante e o Problema de Roteamento de Veículos. Nesta dissertação, consideramos um problema de transporte sob a perspectiva da teoria de jogos algorítmica onde todos os jogadores querem ser transportados a um destino em comum o mais rápido possível, e para isso eles devem escolher um dentre os ônibus disponíveis. Revisamos alguns resultados quanto à existência e à ineficiência de equilíbrios puros de Nash em relação a duas funções sociais. Então, apresentamos limitantes para o Preço de Anarquia para uma nova função social, chamada de função utilitária. Consideramos também o jogo na forma extensiva, o qual chamamos de jogos de transporte sequenciais e apresentamos limitantes para o Preço da Anarquia Sequencial considerando três funções sociais, para instâncias métricas e não-métricasAbstract: Problems related to transportation, such as the Traveling Salesman Problem and the Vehicle Routing Problem, commonly appear in the Combinatorial Optimization area. In this master¿s thesis, we present a game-theoretic analysis of a transportation game where all players want to be transported to a common destination as quickly as possible and, in order to achieve this goal, they have to choose one of the available buses. We review some results concerned with the existence and inefficiency of Pure Nash Equilibria in relation with two social functions. Then, we give bounds on the Price of Anarchy to a new social function, called the utilitarian function. Furthermore, we consider the game in its extensive form, which we call sequential transportation games, and then we provide bounds for the Sequential Price of Anarchy considering three social functions in both metric and non-metric instancesMestradoCiência da ComputaçãoMestre em Ciência da Computação2017/05223-9148027/2016-4FAPESPCNP

LNCS

In resource allocation games, selfish players share resources that are needed in order to fulfill their objectives. The cost of using a resource depends on the load on it. In the traditional setting, the players make their choices concurrently and in one-shot. That is, a strategy for a player is a subset of the resources. We introduce and study dynamic resource allocation games. In this setting, the game proceeds in phases. In each phase each player chooses one resource. A scheduler dictates the order in which the players proceed in a phase, possibly scheduling several players to proceed concurrently. The game ends when each player has collected a set of resources that fulfills his objective. The cost for each player then depends on this set as well as on the load on the resources in it – we consider both congestion and cost-sharing games. We argue that the dynamic setting is the suitable setting for many applications in practice. We study the stability of dynamic resource allocation games, where the appropriate notion of stability is that of subgame perfect equilibrium, study the inefficiency incurred due to selfish behavior, and also study problems that are particular to the dynamic setting, like constraints on the order in which resources can be chosen or the problem of finding a scheduler that achieves stability

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