Equilibria in Repeated Games with Countably Many Players and Tail
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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 .................................. -
Cooperative Strategies in Groups of Strangers: an Experiment
KRANNERT SCHOOL OF MANAGEMENT Purdue University West Lafayette, Indiana Cooperative Strategies in Groups of Strangers: An Experiment By Gabriele Camera Marco Casari Maria Bigoni Paper No. 1237 Date: June, 2010 Institute for Research in the Behavioral, Economic, and Management Sciences Cooperative strategies in groups of strangers: an experiment Gabriele Camera, Marco Casari, and Maria Bigoni* 15 June 2010 * Camera: Purdue University; Casari: University of Bologna, Piazza Scaravilli 2, 40126 Bologna, Italy, Phone: +39- 051-209-8662, Fax: +39-051-209-8493, [email protected]; Bigoni: University of Bologna, Piazza Scaravilli 2, 40126 Bologna, Italy, Phone: +39-051-2098890, Fax: +39-051-0544522, [email protected]. We thank Jim Engle-Warnick, Tim Cason, Guillaume Fréchette, Hans Theo Normann, and seminar participants at the ESA in Innsbruck, IMEBE in Bilbao, University of Frankfurt, University of Bologna, University of Siena, and NYU for comments on earlier versions of the paper. This paper was completed while G. Camera was visiting the University of Siena as a Fulbright scholar. Financial support for the experiments was partially provided by Purdue’s CIBER and by the Einaudi Institute for Economics and Finance. Abstract We study cooperation in four-person economies of indefinite duration. Subjects interact anonymously playing a prisoner’s dilemma. We identify and characterize the strategies employed at the aggregate and at the individual level. We find that (i) grim trigger well describes aggregate play, but not individual play; (ii) individual behavior is persistently heterogeneous; (iii) coordination on cooperative strategies does not improve with experience; (iv) systematic defection does not crowd-out systematic cooperation. -
Collusion Constrained Equilibrium
Theoretical Economics 13 (2018), 307–340 1555-7561/20180307 Collusion constrained equilibrium Rohan Dutta Department of Economics, McGill University David K. Levine Department of Economics, European University Institute and Department of Economics, Washington University in Saint Louis Salvatore Modica Department of Economics, Università di Palermo We study collusion within groups in noncooperative games. The primitives are the preferences of the players, their assignment to nonoverlapping groups, and the goals of the groups. Our notion of collusion is that a group coordinates the play of its members among different incentive compatible plans to best achieve its goals. Unfortunately, equilibria that meet this requirement need not exist. We instead introduce the weaker notion of collusion constrained equilibrium. This al- lows groups to put positive probability on alternatives that are suboptimal for the group in certain razor’s edge cases where the set of incentive compatible plans changes discontinuously. These collusion constrained equilibria exist and are a subset of the correlated equilibria of the underlying game. We examine four per- turbations of the underlying game. In each case,we show that equilibria in which groups choose the best alternative exist and that limits of these equilibria lead to collusion constrained equilibria. We also show that for a sufficiently broad class of perturbations, every collusion constrained equilibrium arises as such a limit. We give an application to a voter participation game that shows how collusion constraints may be socially costly. Keywords. Collusion, organization, group. JEL classification. C72, D70. 1. Introduction As the literature on collective action (for example, Olson 1965) emphasizes, groups often behave collusively while the preferences of individual group members limit the possi- Rohan Dutta: [email protected] David K. -
Correlated Equilibria and Communication in Games Françoise Forges
Correlated equilibria and communication in games Françoise Forges To cite this version: Françoise Forges. Correlated equilibria and communication in games. Computational Complex- ity. Theory, Techniques, and Applications, pp.295-704, 2012, 10.1007/978-1-4614-1800-9_45. hal- 01519895 HAL Id: hal-01519895 https://hal.archives-ouvertes.fr/hal-01519895 Submitted on 9 May 2017 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. Correlated Equilibrium and Communication in Games Françoise Forges, CEREMADE, Université Paris-Dauphine Article Outline Glossary I. De…nition of the Subject and its Importance II. Introduction III. Correlated Equilibrium: De…nition and Basic Properties IV. Correlated Equilibrium and Communication V. Correlated Equilibrium in Bayesian Games VI. Related Topics and Future Directions VII. Bibliography Acknowledgements The author wishes to thank Elchanan Ben-Porath, Frédéric Koessler, R. Vijay Krishna, Ehud Lehrer, Bob Nau, Indra Ray, Jérôme Renault, Eilon Solan, Sylvain Sorin, Bernhard von Stengel, Tristan Tomala, Amparo Ur- bano, Yannick Viossat and, especially, Olivier Gossner and Péter Vida, for useful comments and suggestions. Glossary Bayesian game: an interactive decision problem consisting of a set of n players, a set of types for every player, a probability distribution which ac- counts for the players’ beliefs over each others’ types, a set of actions for every player and a von Neumann-Morgenstern utility function de…ned over n-tuples of types and actions for every player. -
Characterizing Solution Concepts in Terms of Common Knowledge Of
Characterizing Solution Concepts in Terms of Common Knowledge of Rationality Joseph Y. Halpern∗ Computer Science Department Cornell University, U.S.A. e-mail: [email protected] Yoram Moses† Department of Electrical Engineering Technion—Israel Institute of Technology 32000 Haifa, Israel email: [email protected] March 15, 2018 Abstract Characterizations of Nash equilibrium, correlated equilibrium, and rationaliz- ability in terms of common knowledge of rationality are well known (Aumann 1987; arXiv:1605.01236v1 [cs.GT] 4 May 2016 Brandenburger and Dekel 1987). Analogous characterizations of sequential equi- librium, (trembling hand) perfect equilibrium, and quasi-perfect equilibrium in n-player games are obtained here, using results of Halpern (2009, 2013). ∗Supported in part by NSF under grants CTC-0208535, ITR-0325453, and IIS-0534064, by ONR un- der grant N00014-02-1-0455, by the DoD Multidisciplinary University Research Initiative (MURI) pro- gram administered by the ONR under grants N00014-01-1-0795 and N00014-04-1-0725, and by AFOSR under grants F49620-02-1-0101 and FA9550-05-1-0055. †The Israel Pollak academic chair at the Technion; work supported in part by Israel Science Foun- dation under grant 1520/11. 1 Introduction Arguably, the major goal of epistemic game theory is to characterize solution concepts epistemically. Characterizations of the solution concepts that are most commonly used in strategic-form games, namely, Nash equilibrium, correlated equilibrium, and rational- izability, in terms of common knowledge of rationality are well known (Aumann 1987; Brandenburger and Dekel 1987). We show how to get analogous characterizations of sequential equilibrium (Kreps and Wilson 1982), (trembling hand) perfect equilibrium (Selten 1975), and quasi-perfect equilibrium (van Damme 1984) for arbitrary n-player games, using results of Halpern (2009, 2013). -
Games with Delays. a Frankenstein Approach
Games with Delays – A Frankenstein Approach Dietmar Berwanger and Marie van den Bogaard LSV, CNRS & ENS Cachan, France Abstract We investigate infinite games on finite graphs where the information flow is perturbed by non- deterministic signalling delays. It is known that such perturbations make synthesis problems virtually unsolvable, in the general case. On the classical model where signals are attached to states, tractable cases are rare and difficult to identify. Here, we propose a model where signals are detached from control states, and we identify a subclass on which equilibrium outcomes can be preserved, even if signals are delivered with a delay that is finitely bounded. To offset the perturbation, our solution procedure combines responses from a collection of virtual plays following an equilibrium strategy in the instant- signalling game to synthesise, in a Frankenstein manner, an equivalent equilibrium strategy for the delayed-signalling game. 1998 ACM Subject Classification F.1.2 Modes of Computation; D.4.7 Organization and Design Keywords and phrases infinite games on graphs, imperfect information, delayed monitoring, distributed synthesis 1 Introduction Appropriate behaviour of an interactive system component often depends on events gener- ated by other components. The ideal situation, in which perfect information is available across components, occurs rarely in practice – typically a component only receives signals more or less correlated with the actual events. Apart of imperfect signals generated by the system components, there are multiple other sources of uncertainty, due to actions of the system environment or to unreliable behaviour of the infrastructure connecting the compo- nents: For instance, communication channels may delay or lose signals, or deliver them in a different order than they were emitted. -
Correlated Equilibria 1 the Chicken-Dare Game 2 Correlated
MS&E 334: Computation of Equilibria Lecture 6 - 05/12/2009 Correlated Equilibria Lecturer: Amin Saberi Scribe: Alex Shkolnik 1 The Chicken-Dare Game The chicken-dare game can be throught of as two drivers racing towards an intersection. A player can chose to dare (d) and pass through the intersection or chicken out (c) and stop. The game results in a draw when both players chicken out and the worst possible outcome if they both dare. A player wins when he dares while the other chickens out. The game has one possible payoff matrix given by d c d 0; 0 4; 1 c 1; 4 3; 3 with two pure strategy Nash equilibria (d; c) and (c; d) and one mixed equilibrium where each player mixes the pure strategies with probability 1=2 each. Now suppose that prior to playing the game the players performed the following experiment. The players draw a ball labeled with a strategy, either (c) or (d) from a bag containing three balls labelled c; c; d. The players then agree to follow the strategy suggested by the ball. It can be verified that there is no incentive to deviate from such an agreement since the suggested strategy is best in expectation. This experiment is equivalent to having the following strategy profile chosen for the players by some third party, a correlation device. d c d 0 1=3 c 1=3 1=3 This matrix above is not of rank one and so is not a Nash profile. And, the social welfare in this scenario is 16=3 which is greater than that of any Nash equilibrium. -
Repeated Games
Repeated games Felix Munoz-Garcia Strategy and Game Theory - Washington State University Repeated games are very usual in real life: 1 Treasury bill auctions (some of them are organized monthly, but some are even weekly), 2 Cournot competition is repeated over time by the same group of firms (firms simultaneously and independently decide how much to produce in every period). 3 OPEC cartel is also repeated over time. In addition, players’ interaction in a repeated game can help us rationalize cooperation... in settings where such cooperation could not be sustained should players interact only once. We will therefore show that, when the game is repeated, we can sustain: 1 Players’ cooperation in the Prisoner’s Dilemma game, 2 Firms’ collusion: 1 Setting high prices in the Bertrand game, or 2 Reducing individual production in the Cournot game. 3 But let’s start with a more "unusual" example in which cooperation also emerged: Trench warfare in World War I. Harrington, Ch. 13 −! Trench warfare in World War I Trench warfare in World War I Despite all the killing during that war, peace would occasionally flare up as the soldiers in opposing tenches would achieve a truce. Examples: The hour of 8:00-9:00am was regarded as consecrated to "private business," No shooting during meals, No firing artillery at the enemy’s supply lines. One account in Harrington: After some shooting a German soldier shouted out "We are very sorry about that; we hope no one was hurt. It is not our fault, it is that dammed Prussian artillery" But... how was that cooperation achieved? Trench warfare in World War I We can assume that each soldier values killing the enemy, but places a greater value on not getting killed. -
Correlated Equilibrium Is a Distribution D Over Action Profiles a Such That for Every ∗ Player I, and Every Action Ai
NETS 412: Algorithmic Game Theory February 9, 2017 Lecture 8 Lecturer: Aaron Roth Scribe: Aaron Roth Correlated Equilibria Consider the following two player traffic light game that will be familiar to those of you who can drive: STOP GO STOP (0,0) (0,1) GO (1,0) (-100,-100) This game has two pure strategy Nash equilibria: (GO,STOP), and (STOP,GO) { but these are clearly not ideal because there is one player who never gets any utility. There is also a mixed strategy Nash equilibrium: Suppose player 1 plays (p; 1 − p). If the equilibrium is to be fully mixed, player 2 must be indifferent between his two actions { i.e.: 0 = p − 100(1 − p) , 101p = 100 , p = 100=101 So in the mixed strategy Nash equilibrium, both players play STOP with probability p = 100=101, and play GO with probability (1 − p) = 1=101. This is even worse! Now both players get payoff 0 in expectation (rather than just one of them), and risk a horrific negative utility. The four possible action profiles have roughly the following probabilities under this equilibrium: STOP GO STOP 98% <1% GO <1% ≈ 0.01% A far better outcome would be the following, which is fair, has social welfare 1, and doesn't risk death: STOP GO STOP 0% 50% GO 50% 0% But there is a problem: there is no set of mixed strategies that creates this distribution over action profiles. Therefore, fundamentally, this can never result from Nash equilibrium play. The reason however is not that this play is not rational { it is! The issue is that we have defined Nash equilibria as profiles of mixed strategies, that require that players randomize independently, without any communication. -
Role Assignment for Game-Theoretic Cooperation ⋆
Role Assignment for Game-Theoretic Cooperation ? Catherine Moon and Vincent Conitzer Duke University, Durham, NC 27708, USA Abstract. In multiagent systems, often agents need to be assigned to different roles. Multiple aspects should be taken into account for this, such as agents' skills and constraints posed by existing assignments. In this paper, we focus on another aspect: when the agents are self- interested, careful role assignment is necessary to make cooperative be- havior an equilibrium of the repeated game. We formalize this problem and provide an easy-to-check necessary and sufficient condition for a given role assignment to induce cooperation. However, we show that finding whether such a role assignment exists is in general NP-hard. Nevertheless, we give two algorithms for solving the problem. The first is based on a mixed-integer linear program formulation. The second is based on a dynamic program, and runs in pseudopolynomial time if the number of agents is constant. However, the first algorithm is much, much faster in our experiments. 1 Introduction Role assignment is an important problem in the design of multiagent systems. When multiple agents come together to execute a plan, there is generally a natural set of roles in the plan to which the agents need to be assigned. There are, of course, many aspects to take into account in such role assignment. It may be impossible to assign certain combinations of roles to the same agent, for example due to resource constraints. Some agents may be more skilled at a given role than others. In this paper, we assume agents are interchangeable and instead consider another aspect: if the agents are self-interested, then the assignment of roles has certain game-theoretic ramifications. -
Interactively Solving Repeated Games. a Toolbox
Interactively Solving Repeated Games: A Toolbox Version 0.2 Sebastian Kranz* March 27, 2012 University of Cologne Abstract This paper shows how to use my free software toolbox repgames for R to analyze infinitely repeated games. Based on the results of Gold- luecke & Kranz (2012), the toolbox allows to compute the set of pure strategy public perfect equilibrium payoffs and to find optimal equilib- rium strategies for all discount factors in repeated games with perfect or imperfect public monitoring and monetary transfers. This paper explores various examples, including variants of repeated public goods games and different variants of repeated oligopolies, like oligopolies with many firms, multi-market contact or imperfect monitoring of sales. The paper also explores repeated games with infrequent external auditing and games were players can secretly manipulate signals. On the one hand the examples shall illustrate how the toolbox can be used and how our algorithms work in practice. On the other hand, the exam- ples may in themselves provide some interesting economic and game theoretic insights. *[email protected]. I want to thank the Deutsche Forschungsgemeinschaft (DFG) through SFB-TR 15 for financial support. 1 CONTENTS 2 Contents 1 Overview 4 2 Installation of the software 5 2.1 Installing R . .5 2.2 Installing the package under Windows 32 bit . .6 2.3 Installing the package for a different operating system . .6 2.4 Running Examples . .6 2.5 Installing a text editor for R files . .7 2.6 About RStudio . .7 I Perfect Monitoring 8 3 A Simple Cournot Game 8 3.1 Initializing the game . -
New Perspectives on Time Inconsistency
Behavioural Macroeconomic Policy: New perspectives on time inconsistency Michelle Baddeley July 19, 2019 Abstract This paper brings together divergent approaches to time inconsistency from macroeconomic policy and behavioural economics. Developing Barro and Gordon’s models of rules, discretion and reputation, behavioural discount functions from behavioural microeconomics are embedded into Barro and Gordon’s game-theoretic analysis of temptation versus enforce- ment to construct an encompassing model, nesting combinations of time consistent and time inconsistent preferences. The analysis presented in this paper shows that, with hyperbolic/quasi-hyperbolic discounting, the enforceable range of inflation targets is narrowed. This suggests limits to the effectiveness of monetary targets, under certain conditions. The pa- per concludes with a discussion of monetary policy implications, explored specifically in the light of current macroeconomic policy debates. JEL codes: D03 E03 E52 E61 Keywords: time-inconsistency behavioural macroeconomics macroeconomic policy 1 Introduction Contrasting analyses of time inconsistency are presented in macroeconomic pol- icy and behavioural economics - the first from a rational expectations perspec- tive [12, 6, 7], and the second building on insights from behavioural economics arXiv:1907.07858v1 [econ.TH] 18 Jul 2019 about bounded rationality and its implications in terms of present bias and hyperbolic/quasi-hyperbolic discounting [13, 11, 8, 9, 18]. Given the divergent assumptions about rational choice across these two approaches, it is not so sur- prising that there have been few attempts to reconcile these different analyses of time inconsistency. This paper fills the gap by exploring the distinction between inter-personal time inconsistency (rational expectations models) and intra-personal time incon- sistency (behavioural economic models).