Informational Incentives for Congestion Games
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CSC304 Lecture 4 Game Theory
CSC304 Lecture 4 Game Theory (Cost sharing & congestion games, Potential function, Braess’ paradox) CSC304 - Nisarg Shah 1 Recap • Nash equilibria (NE) ➢ No agent wants to change their strategy ➢ Guaranteed to exist if mixed strategies are allowed ➢ Could be multiple • Pure NE through best-response diagrams • Mixed NE through the indifference principle CSC304 - Nisarg Shah 2 Worst and Best Nash Equilibria • What can we say after we identify all Nash equilibria? ➢ Compute how “good” they are in the best/worst case • How do we measure “social good”? ➢ Game with only rewards? Higher total reward of players = more social good ➢ Game with only penalties? Lower total penalty to players = more social good ➢ Game with rewards and penalties? No clear consensus… CSC304 - Nisarg Shah 3 Price of Anarchy and Stability • Price of Anarchy (PoA) • Price of Stability (PoS) “Worst NE vs optimum” “Best NE vs optimum” Max total reward Max total reward Min total reward in any NE Max total reward in any NE or or Max total cost in any NE Min total cost in any NE Min total cost Min total cost PoA ≥ PoS ≥ 1 CSC304 - Nisarg Shah 4 Revisiting Stag-Hunt Hunter 2 Stag Hare Hunter 1 Stag (4 , 4) (0 , 2) Hare (2 , 0) (1 , 1) • Max total reward = 4 + 4 = 8 • Three equilibria ➢ (Stag, Stag) : Total reward = 8 ➢ (Hare, Hare) : Total reward = 2 1 2 1 2 ➢ ( Τ3 Stag – Τ3 Hare, Τ3 Stag – Τ3 Hare) 1 1 1 1 o Total reward = ∗ ∗ 8 + 1 − ∗ ∗ 2 ∈ (2,8) 3 3 3 3 • Price of stability? Price of anarchy? CSC304 - Nisarg Shah 5 Revisiting Prisoner’s Dilemma John Stay Silent Betray Sam -
Potential Games. Congestion Games. Price of Anarchy and Price of Stability
8803 Connections between Learning, Game Theory, and Optimization Maria-Florina Balcan Lecture 13: October 5, 2010 Reading: Algorithmic Game Theory book, Chapters 17, 18 and 19. Price of Anarchy and Price of Staility We assume a (finite) game with n players, where player i's set of possible strategies is Si. We let s = (s1; : : : ; sn) denote the (joint) vector of strategies selected by players in the space S = S1 × · · · × Sn of joint actions. The game assigns utilities ui : S ! R or costs ui : S ! R to any player i at any joint action s 2 S: any player maximizes his utility ui(s) or minimizes his cost ci(s). As we recall from the introductory lectures, any finite game has a mixed Nash equilibrium (NE), but a finite game may or may not have pure Nash equilibria. Today we focus on games with pure NE. Some NE are \better" than others, which we formalize via a social objective function f : S ! R. Two classic social objectives are: P sum social welfare f(s) = i ui(s) measures social welfare { we make sure that the av- erage satisfaction of the population is high maxmin social utility f(s) = mini ui(s) measures the satisfaction of the most unsatisfied player A social objective function quantifies the efficiency of each strategy profile. We can now measure how efficient a Nash equilibrium is in a specific game. Since a game may have many NE we have at least two natural measures, corresponding to the best and the worst NE. We first define the best possible solution in a game Definition 1. -
Lecture Notes
GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies .................................. -
Equilibrium Computation — Dagstuhl Seminar —
07471 Abstracts Collection Equilibrium Computation | Dagstuhl Seminar | Jean-Jacques Herings1, Marcin Jurdzinski2, Peter Bro Miltersen3, Eva Tardos4 and Bernhard von Stengel5 1 Maastricht, [email protected] 2 Warwick, [email protected] 3 Aarhus, [email protected] 4 Cornell, [email protected] 5 LSE, [email protected] Abstract. From 18 to 23 November 2007, the Dagstuhl Seminar 07471 \Equilibrium Computation" was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, sev- eral participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given dur- ing the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available. Keywords. Equilibrium, algorithm, polynomial time, game theory, eco- nomics The purpose of this Dagstuhl seminar was to bring together researchers from different disciplines working on algorithmic problems of finding equilibria. As explained in the Motivation for this seminar, the different topics were (with talks given by) { Parity Games, Andersson (junior researcher), Gimbert (junior researcher), Gr¨adel(survey talk), Svensson (junior researcher), Zwick (survey talk), { Complexity of Finding Equilibria, Elkind (junior researcher), Etessami (sur- vey talk), Goldberg (introductory survey talk on the class PPAD, not listed below), Fabrikant (junior researcher), Hoefer (junior researcher), Monien, V¨ocking (survey talk), { Mathematical Programming, Halman, Morris, Theobald, { Economic Equilibria, Heydenreich (junior researcher), Jain, Peeters (survey talk), { Game Theory, Balthasar (junior researcher), Goldberg, Jiang (junior re- searcher), Sørensen (junior researcher), Turocy (survey talk), Vermeulen, von Stengel. -
Network Formation Games and the Potential Function Method
P1: SBT 9780521872829main CUNY1061-Nisan 0 521 87282 0 May 23, 2007 4:25 CHAPTER 19 Network Formation Games and the Potential Function Method Eva´ Tardos and Tom Wexler Abstract Large computer networks such as the Internet are built, operated, and used by a large number of diverse and competitive entities. In light of these competing forces, it is surprising how efficient these networks are. An exciting challenge in the area of algorithmic game theory is to understand the success of these networks in game theoretic terms: what principles of interaction lead selfish participants to form such efficient networks? In this chapter we present a number of network formation games. We focus on simple games that have been analyzed in terms of the efficiency loss that results from selfishness. We also highlight a fundamental technique used in analyzing inefficiency in many games: the potential function method. 19.1 Introduction The design and operation of many large computer networks, such as the Internet, are carried out by a large number of independent service providers (Autonomous Systems), all of whom seek to selfishly optimize the quality and cost of their own operation. Game theory provides a natural framework for modeling such selfish interests and the networks they generate. These models in turn facilitate a quantitative study of the trade-off between efficiency and stability in network formation. In this chapter, we consider a range of simple network formation games that model distinct ways in which selfish agents might create and evaluate networks. All of the models we present aim to capture two competing issues: players want to minimize the expenses they incur in building a network, but at the same time seek to ensure that this network provides them with a high quality of service. -
Designing Mechanisms for Good Equilibria
Designing Mechanisms for Good Equilibria vorgelegt von Dipl.-Math. Philipp Friedrich Freiherr von Falkenhausen Hamburg Von der Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation Promotionsausschuss Vorsitzender: Prof. Dr. Jochen Blath Berichter: Dr. Tobias Harks (Assistant Professor) Prof. Dr. Rolf H. Möhring Prof. Dr. Britta Peis Tag der wissenschaftlichen Aussprache: 27. Mai 2014 Berlin 2014 D 83 Preface Over the three years that went into writing this thesis, I have received outstanding support. I would like to thank everyone who has had a part in this. Tobias Harks introduced me to algorithmic game theory as my diploma thesis advisor. Our fruitful collaboration motivated me for further research and I am thoroughly glad to still have him as my advisor. I am grateful for his support and guidance and for the chance to work with him. Visits in Maastricht and Berlin brought days of intense mathematical discussions and convivial evenings that I have always looked forward to. The Group for Combinatorial Optimization and Graph Algorithms has been a great home for doing research. I am thankful for Rolf H. Möhring’s generous leadership and the group’s open nature. A special mention goes to Max Klimm and Kai-Simon Goetzmann, with whom I shared an office and many jokes over the last years. Max has also been a great sparring partner for research questions and helped proofreading parts of this thesis. My thanks also go to Britta Peis for taking the third assessment of this thesis. Exposure to the international research community has been a tremendous experience for me. -
Constraint Games: Modeling and Solving Games with Constraints Thi-Van-Anh Nguyen
Constraint Games: Modeling and Solving Games with Constraints Thi-Van-Anh Nguyen To cite this version: Thi-Van-Anh Nguyen. Constraint Games: Modeling and Solving Games with Constraints. Computer Science [cs]. Université de Caen, 2014. English. tel-01138960 HAL Id: tel-01138960 https://hal.archives-ouvertes.fr/tel-01138960 Submitted on 3 Apr 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE´ DE CAEN BASSE NORMANDIE U.F.R. de Sciences ECOLE´ DOCTORALE SIMEM TH ESE` Present´ ee´ par Thi Van Anh NGUYEN soutenue le 12 Decembre´ 2014 en vue de l’obtention du DOCTORAT de l’UNIVERSITE´ de CAEN Specialit´ e´ : Informatique et applications Arretˆ e´ du 07 aoutˆ 2006 Titre Constraint Games: Modeling and Solving Games with Constraints The work presented in this thesis was carried out at GREYC - Universite´ de Caen Basse-Normandie. Jury Francesca ROSSI Professor University of Padova, Italy Reviewer Lakhdar SA¨IS Professor Universite´ d’Artois, France Reviewer Patrice BOIZUMAULT Professor Universite´ de Caen, France Examiner Lucas BORDEAUX Research Engineer Microsoft Research, UK Examiner Philippe JEGOU´ Professor Universite´ d’Aix-Marseille, France Examiner Bruno ZANUTTINI Associate Professor Universite´ de Caen, France Examiner Arnaud LALLOUET Professor Universite´ de Caen, France Supervisor Abstract This thesis presents a topic at the interface of game theory and constraint programming. -
Where Game Theory Meets Network Optimization
Networks' Challenge: Where Game Theory Meets Network Optimization Asu Ozdaglar International Symposium on Information Theory July 2008 Department of Electrical Engineering & Computer Science Massachusetts Institute of Technology, USA 1 Introduction ² Central Question in Today's and Future Networks: { Systematic analysis and design of network architectures and development of network control schemes ² Traditional Network Optimization: Single administrative domain with a single control objective and obedient users. ² New Challenges: { Large-scale with lack of access to centralized information and subject to unexpected disturbances ¤ Implication: Control policies have to be decentralized, scalable, and robust against dynamic changes { Interconnection of heterogeneous autonomous entities, so no central party with enforcement power or accurate information about user needs ¤ Implication: Sel¯sh incentives and private information of users need to be incorporated into the control paradigm { Continuous upgrades and investments in new technologies ¤ Implication: Economic incentives of service and content providers much more paramount 2 Tools and Ideas for Analysis ² These challenges necessitate the analysis of resource allocation and data processing in the presence of decentralized information and heterogeneous sel¯sh users and administrative domains ² Instead of a central control objective, model as a multi-agent decision problem: Game theory and economic market mechanisms ² Game Theory: Understand incentives of sel¯sh autonomous agents and large -
Applications of Game Theory to Networks
6.207/14.15: Networks Lecture 12: Applications of Game Theory to Networks Daron Acemoglu and Asu Ozdaglar MIT October 21, 2009 1 Networks: Lecture 12 Introduction Outline Traffic equilibrium: the Pigou example General formulation with single origin-destination pair Multi-origin-destination traffic equilibria Congestion games and atomic traffic equilibria Potential functions and potential games Network cost-sharing Strategic network formation. Reading: EK, Chapter 8. Jackson, Chapter 6. 2 Networks: Lecture 12 Introduction Motivation Many games are played over networks, in the sense that players interact with others linked to them through a network-like structure. Alternatively, in several important games, the actions of players correspond to a path in a given network. The most important examples are choosing a route in a traffic problem or in a data routing problem. Other examples are cost sharing in network-like structures. Finally, the formation of networks is typically a game-theoretic (strategic) problem. In this lecture, we take a first look at some of these problems, focusing on traffic equilibria and formation of networks. 3 Networks: Lecture 12 Wardrop Equilibria The Pigou Example of Traffic Equilibrium Recall the following simple example from lecture 9, where a unit mass of traffic is to be routed over a network: delay depends on congestion 1 unit of traffic no congestion effects System optimum (minimizing aggregate delay) is to split traffic equally between the two routes, giving S X S S 1 1 3 min Csystem(x ) = li (xi )xi = + = : x1+x2≤1 4 2 4 i Instead, the Nash equilibrium of this large (non-atomic) game, also referred to as Wardrop equilibrium, is x1 = 1 and x2 = 0 (since for any x1 < 1, l1 (x1) < 1 = l2 (1 − x1)), giving an aggregate delay of WE X WE WE 3 Ceq(x ) = li (x )x = 1 + 0 = 1 > : i i 4 i 4 Networks: Lecture 12 Wardrop Equilibria The Wardrop Equilibrium Why the Wardrop equilibrium? It is nothing but a Nash equilibrium in this game, in view of the fact that it is non-atomic|each player is infinitesimal. -
Routing Without Regret: on Convergence to Nash Equilibria of Regret-Minimizing Algorithms in Routing Games∗
THEORY OF COMPUTING, Volume 6 (2010), pp. 179–199 www.theoryofcomputing.org Routing Without Regret: On Convergence to Nash Equilibria of Regret-Minimizing Algorithms in Routing Games∗ Avrim Blum† Eyal Even-Dar Katrina Ligett‡ Received: November 10, 2009; published: September 15, 2010. Abstract: There has been substantial work developing simple, efficient regret-minimizing algorithms for a wide class of repeated decision-making problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on perfor- mance even in adversarially-changing environments. There has also been substantial work on analyzing properties of Nash equilibria in routing games. In this paper, we consider the question: if each player in a routing game uses a regret-minimizing strategy, will behavior converge to a Nash equilibrium? In general games the answer to this question is known to be no in a strong sense, but routing games have substantially more structure. In this paper we show that in the Wardrop setting of multicommodity flow and infinites- imal agents, behavior will approach Nash equilibrium (on most days, the cost of the flow ∗A preliminary version of these results appeared in the Proceedings of the 25th Annual ACM Symposium on Principles of Distributed Computing, July 2006. †Supported in part by the National Science Foundation under grants IIS-0121678, CCR-0122581, and CCF-0514922. ‡This material is based upon work supported in part by an AT&T Labs Graduate Fellowship, an NSF Graduate Research Fellowship, and by the National Science Foundation under Grant #0937060 to the Computing Research Association for the CIFellows program. -
ROUTING WITHOUT REGRET: on CONVERGENCE to NASH EQUILIBRIA of REGRET-MINIMIZING ALGORITHMS in ROUTING GAMES† 1. Introduction. T
ROUTING WITHOUT REGRET: ON CONVERGENCE TO NASH EQUILIBRIA OF REGRET-MINIMIZING ALGORITHMS IN ROUTING GAMES† AVRIM BLUM∗, EYAL EVEN-DAR† , AND KATRINA LIGETT‡ Abstract. There has been substantial work developing simple, efficient no-regret algorithms for a wide class of repeated decision-making problems including online routing. These are adaptive strate- gies an individual can use that give strong guarantees on performance even in adversarially-changing environments. There has also been substantial work on analyzing properties of Nash equilibria in routing games. In this paper, we consider the question: if each player in a routing game uses a no- regret strategy, will behavior converge to a Nash equilibrium? In general games the answer to this question is known to be no in a strong sense, but routing games have substantially more structure. In this paper we show that in the Wardrop setting of multicommodity flow and infinitesimal agents, behavior will approach Nash equilibrium (formally, on most days, the cost of the flow will be close to the cost of the cheapest paths possible given that flow) at a rate that depends polynomially on the players’ regret bounds and the maximum slope of any latency function. We also show that price-of-anarchy results may be applied to these approximate equilibria, and also consider the finite- size (non-infinitesimal) load-balancing model of Azar [2]. Our nonatomic results also apply to a more general class of games known as congestion games. 1. Introduction. There has been substantial work in learning theory and game theory on adaptive no-regret algorithms for problems of repeated decision-making. -
Random Order Congestion Games
Random Order Congestion Games Michal Penn Faculty of Industrial Engineering and Management, Technion, Haifa 32000, Israel email: [email protected] Maria Polukarov School of Electronics and Computer Science at University of Southampton, SO17 1BJ, Southampton, UK email: [email protected] Moshe Tennenholtz Microsoft Israel R&D Center, 13 Shenkar Street, Herzeliya 46725, Israel Faculty of Industrial Engineering and Management, Technion, Haifa 32000, Israel email: [email protected], [email protected] We introduce a new class of games, Random Order Congestion Games [ROCGs]. In an ROCG, each player has a task that can be carried out by any element of a set of resources, and each resource executes its assigned tasks in a random order. Each player’s aim is to minimize his expected cost which is the sum of two terms – the sum of the fixed costs over the set of his utilized resources and the expected cost of his task execution. The cost of a player’s task execution is determined by the earliest time his task is completed, and thus it might be beneficial for him to assign his task to several resources. We prove the existence of pure strategy Nash equilibria in ROCGs. Moreover, we present a polynomial time algorithm for finding such an equilibrium in a given ROCG. Key words: congestion games; asynchronous distributed systems; random order of task execution; pure-strategy Nash equilibrium; algorithms MSC2000 Subject Classification: Primary: 91A10, 91A40; Secondary: 91A06, 91A80 OR/MS subject classification: Primary: Games/group decisions – Noncooperative; Secondary: Comput- ers/computer science – Artificial intelligence 1.