Belief-based Learning
|
|
- Charleen Robertson
- 6 years ago
- Views:
Transcription
1 Belief-based Learning Algorithmic Game Theory Marcello Restelli
2 Lecture Outline Introdutcion to multi-agent learning Belief-based learning Cournot adjustment Fictitious play Bayesian learning Equilibrium learners Targeted learning Learning to cooperate Evolutionary game theory
3 Cournot Adjustment The simplest approach to model other agents Assumes that other agents will repeat the same action this stage as they did the previous time BR 1 For every θ The agent generates a best response to 2 the line states this model Each agent has a prior model of the other agent θ 2 θ t = (θ t 1, θ t2 ) the BR of player 1 against it. The value for player 1 is the height at point θ 2 θ t 2 θ t+1 Can you convince yourself this point is a Nash? BR 2 θ t 1 New BR if 2 plays θ t 2 θ 1
4 Cournot Dynamics A movement between profiles such that θ t+1 = f(θ t ), f i (θ t ) = BR i (θ t -i ) A steady state is θ s so that θ s = f(θ s ) Once θ t =θ s the system remains there Claim: θ s is a Nash equilibrium Proof: by definition for every player θ s =BR(θ -i ), so players don't want to move So: Every steady state is a Nash equilibrium
5 Best Response and Adjustment Fighters and Bombers problem Suppose that the fighter believes that the bomber will look up every time We can compute the expected values E(Sun) = 0.95*1 + 1*0 = 0.95 E(Bottom) = 1*1 + 0*0 = 1 best response What happens if the bomber looks down? Fighter/Bomber Look Up Look Down Sun Attack Bottom Attack 1 0 The best response is to attack from sun Sometimes Cournot dynamics find the Nash equilibrium (PD) Other times Cournot dynamics cycles...
6 Cycles Rock-Paper-Scissors paper Prior beliefs P row (Rock)=1, P row (Paper)=0, P row (Scissors)=0 P row (Rock)=1, P row (Paper)=0, P row (Scissors)=0 Next round Row/Column Rock Paper Scissors Rock (0,0) (-1,1) (1,-1) Paper (1,-1) (0,0) (-1,1) Scissors (-1,1) (1,-1) (0,0) P row (Rock)=0, P row (Paper)=1, P row (Scissors)=0 P row (Rock)=0, P row (Paper)=1, P row (Scissors)=0 The cycle continues indefinitely
7 Fictitious Play Agents behave assuming that they are facing stationary opponents The learner explicitly mantains a belief about the opponent's strategy At each iteration Play a best response to the assessed opponent's strategy Observe the opponent's action and update beliefs The opponents' payoffs are unknown The agent knows his own payoff matrix in the stage game
8 Two-Player Fictitious Play Player i starts with a weight function Each time an opponent strategy is played, its weight is update by adding 1 The belief about player i playing s -i at time t is
9 Two-Player Fictitious Play A FP is any rule that is a best response to the estimated opponent strategy Many best responses are possible Several tie-breaking rules can be used Example: prefer pure strategies over mixed strategies Which rule is used is not much important FP is very sensitive to players' initial beliefs The prior beliefs cannot be (0,...,0) If the initial weights are positive, there is no finite sample to which the beliefs assign probability zero
10 Example: Matching Pennies As the number of rounds tends to infinity, the empirical distribution of each player will converge to (0.5,0.5)
11 Convergence: Pure Strategies Does FP converge? Steady state: a strategy profile is a steady state of FP whenever it is played in every period after some finite time T If a pure-strategy profile is a strict Nash equilibrium of a stage game, then it is a steady state of FP in the repeated game If a pure-strategy profile is a steady state of FP in the repeated game, then it is a (possibly weak) Nash equilibrium in the stage game
12 Convergence: Mixed Strategies What happens if there are no pure Nash? If all the equilibria of the stage game are mixed, FP cannot converge to a pure profile In matching pennies FP cycles and never converges to Nash, but the empirical distribution over player i's strategy converges to (1/2,1/2) So the product of the two empirical marginal distributions is the mixed-strategy equilibrium of the game If the empirical distribution of each player's strategies converges in fictitious play, then it converges to a Nash equilibrium
13 Convergence: More Results Under FP the empirical distributions converge if the stage game is: Generic payoff and 2x2 Zero sum Identical interest Solvable by iterated strict dominance When empirical distribution converge, what about the outcome? Empirical distributions need not converge
14 Example: Anti-Coordination Two pure Nash equilibria Each player earns 1 One mixed Nash equilibrium Each player earns 0.5 Using FP each player converge to the mixed strategy, but... each player earns 0
15 Example: Shapley's RPS The empirical play of this game never converges to any fixed distribution
16 Generalizations of FP Mainly generalize the update rule for the belief Adaptive forecasting E.g., exponentially weigthed If opponent follows a fixed mixed-strategy, then the assessments do not converge If β goes to 1 then the rule is asymptotically empirical, and the standard FP results hold
17 Rational Learning Also known as Bayesian learning Allows to consider a richer set of beliefs about opponents' strategies than FP Repeated-game strategies, e.g. Tit-for-Tat Each player begins with some prior beliefs Beliefs are updated using Bayesian updating
18 Example: Iterated Prisoner's Dilemma Suppose that the support of the prior belief is g 1,g 2,...,g g : trigger strategy g T : trigger strategy before T, defect later Suppose that the best response is selected from among g 0,g 1,g 2,...,g After each round, each player performs a Bayesian update after history h t Assuming that the other player has always cooperate up to t
19 Convergence Analysis Formal analysis focuses on self-play Under some conditions, self-play rational learning results in correct beliefs about opponent's strategy Under some conditions, self-play rational learning converges toward a Nash equilibrium with high probability The main condition is an absumption of absolute continuity Let X be a set and let μ, μ' ϵ Π(X) be probability distributions over X. Then the distribution μ is said to be absolutely continuous with respect to the distribution μ' iff for x ⅽ X that is measurable it is the case that if μ(x) > 0 then μ'(x) > 0
20 Belief Accuracy Player's beliefs will eventually converge if Use Bayesian updating Play best respone The play predicted by the other players' real strategies is absoutely continuous with respect to that predicted by his belief If the above conditions are met, then, for every ε and for every history, there is a time T such that for all t > T the distribution induced by the player's beliefs is ε-close to the actual strategy profile Prediction is accurate only for the on-path portion
21 Nash Equilibrium Player's strategies will eventually converge to ε-nash if Use Bayesian updating Play best respone The play predicted by the other players' real strategies is absoutely continuous with respect to that predicted by his belief The space of repeated-games equilibria is huge: which equilibrium? Optimism can lead to high rewards (Tit-for-Tat) Pessimism can lead to low rewards (defect, grim-trigger)
Game Theory, Evolutionary Dynamics, and Multi-Agent Learning. Prof. Nicola Gatti
Game Theory, Evolutionary Dynamics, and Multi-Agent Learning Prof. Nicola Gatti (nicola.gatti@polimi.it) Game theory Game theory: basics Normal form Players Actions Outcomes Utilities Strategies Solutions
More informationFirst Prev Next Last Go Back Full Screen Close Quit. Game Theory. Giorgio Fagiolo
Game Theory Giorgio Fagiolo giorgio.fagiolo@univr.it https://mail.sssup.it/ fagiolo/welcome.html Academic Year 2005-2006 University of Verona Summary 1. Why Game Theory? 2. Cooperative vs. Noncooperative
More informationIterated Strict Dominance in Pure Strategies
Iterated Strict Dominance in Pure Strategies We know that no rational player ever plays strictly dominated strategies. As each player knows that each player is rational, each player knows that his opponents
More informationEvolution & Learning in Games
1 / 27 Evolution & Learning in Games Econ 243B Jean-Paul Carvalho Lecture 2. Foundations of Evolution & Learning in Games II 2 / 27 Outline In this lecture, we shall: Take a first look at local stability.
More informationGame-Theoretic Learning:
Game-Theoretic Learning: Regret Minimization vs. Utility Maximization Amy Greenwald with David Gondek, Amir Jafari, and Casey Marks Brown University University of Pennsylvania November 17, 2004 Background
More informationComputing Minmax; Dominance
Computing Minmax; Dominance CPSC 532A Lecture 5 Computing Minmax; Dominance CPSC 532A Lecture 5, Slide 1 Lecture Overview 1 Recap 2 Linear Programming 3 Computational Problems Involving Maxmin 4 Domination
More informationPrisoner s Dilemma. Veronica Ciocanel. February 25, 2013
n-person February 25, 2013 n-person Table of contents 1 Equations 5.4, 5.6 2 3 Types of dilemmas 4 n-person n-person GRIM, GRIM, ALLD Useful to think of equations 5.4 and 5.6 in terms of cooperation and
More informationExponential Moving Average Based Multiagent Reinforcement Learning Algorithms
Exponential Moving Average Based Multiagent Reinforcement Learning Algorithms Mostafa D. Awheda Department of Systems and Computer Engineering Carleton University Ottawa, Canada KS 5B6 Email: mawheda@sce.carleton.ca
More informationA (Brief) Introduction to Game Theory
A (Brief) Introduction to Game Theory Johanne Cohen PRiSM/CNRS, Versailles, France. Goal Goal is a Nash equilibrium. Today The game of Chicken Definitions Nash Equilibrium Rock-paper-scissors Game Mixed
More informationBELIEFS & EVOLUTIONARY GAME THEORY
1 / 32 BELIEFS & EVOLUTIONARY GAME THEORY Heinrich H. Nax hnax@ethz.ch & Bary S. R. Pradelski bpradelski@ethz.ch May 15, 217: Lecture 1 2 / 32 Plan Normal form games Equilibrium invariance Equilibrium
More informationNormal-form games. Vincent Conitzer
Normal-form games Vincent Conitzer conitzer@cs.duke.edu 2/3 of the average game Everyone writes down a number between 0 and 100 Person closest to 2/3 of the average wins Example: A says 50 B says 10 C
More information6.254 : Game Theory with Engineering Applications Lecture 8: Supermodular and Potential Games
6.254 : Game Theory with Engineering Applications Lecture 8: Supermodular and Asu Ozdaglar MIT March 2, 2010 1 Introduction Outline Review of Supermodular Games Reading: Fudenberg and Tirole, Section 12.3.
More informationLectures Road Map
Lectures 0 - Repeated Games 4. Game Theory Muhamet Yildiz Road Map. Forward Induction Examples. Finitely Repeated Games with observable actions. Entry-Deterrence/Chain-store paradox. Repeated Prisoners
More informationEvolutionary Game Theory
Evolutionary Game Theory ISI 330 Lecture 18 1 ISI 330 Lecture 18 Outline A bit about historical origins of Evolutionary Game Theory Main (competing) theories about how cooperation evolves P and other social
More informationOnline Learning Class 12, 20 March 2006 Andrea Caponnetto, Sanmay Das
Online Learning 9.520 Class 12, 20 March 2006 Andrea Caponnetto, Sanmay Das About this class Goal To introduce the general setting of online learning. To describe an online version of the RLS algorithm
More information6.891 Games, Decision, and Computation February 5, Lecture 2
6.891 Games, Decision, and Computation February 5, 2015 Lecture 2 Lecturer: Constantinos Daskalakis Scribe: Constantinos Daskalakis We formally define games and the solution concepts overviewed in Lecture
More informationComputing Minmax; Dominance
Computing Minmax; Dominance CPSC 532A Lecture 5 Computing Minmax; Dominance CPSC 532A Lecture 5, Slide 1 Lecture Overview 1 Recap 2 Linear Programming 3 Computational Problems Involving Maxmin 4 Domination
More informationMS&E 246: Lecture 12 Static games of incomplete information. Ramesh Johari
MS&E 246: Lecture 12 Static games of incomplete information Ramesh Johari Incomplete information Complete information means the entire structure of the game is common knowledge Incomplete information means
More informationA Polynomial-time Nash Equilibrium Algorithm for Repeated Games
A Polynomial-time Nash Equilibrium Algorithm for Repeated Games Michael L. Littman mlittman@cs.rutgers.edu Rutgers University Peter Stone pstone@cs.utexas.edu The University of Texas at Austin Main Result
More informationSatisfaction Equilibrium: Achieving Cooperation in Incomplete Information Games
Satisfaction Equilibrium: Achieving Cooperation in Incomplete Information Games Stéphane Ross and Brahim Chaib-draa Department of Computer Science and Software Engineering Laval University, Québec (Qc),
More informationNotes on Coursera s Game Theory
Notes on Coursera s Game Theory Manoel Horta Ribeiro Week 01: Introduction and Overview Game theory is about self interested agents interacting within a specific set of rules. Self-Interested Agents have
More informationMS&E 246: Lecture 4 Mixed strategies. Ramesh Johari January 18, 2007
MS&E 246: Lecture 4 Mixed strategies Ramesh Johari January 18, 2007 Outline Mixed strategies Mixed strategy Nash equilibrium Existence of Nash equilibrium Examples Discussion of Nash equilibrium Mixed
More informationBounded Rationality, Strategy Simplification, and Equilibrium
Bounded Rationality, Strategy Simplification, and Equilibrium UPV/EHU & Ikerbasque Donostia, Spain BCAM Workshop on Interactions, September 2014 Bounded Rationality Frequently raised criticism of game
More information4. Opponent Forecasting in Repeated Games
4. Opponent Forecasting in Repeated Games Julian and Mohamed / 2 Learning in Games Literature examines limiting behavior of interacting players. One approach is to have players compute forecasts for opponents
More informationIndustrial Organization Lecture 3: Game Theory
Industrial Organization Lecture 3: Game Theory Nicolas Schutz Nicolas Schutz Game Theory 1 / 43 Introduction Why game theory? In the introductory lecture, we defined Industrial Organization as the economics
More informationA Generic Bound on Cycles in Two-Player Games
A Generic Bound on Cycles in Two-Player Games David S. Ahn February 006 Abstract We provide a bound on the size of simultaneous best response cycles for generic finite two-player games. The bound shows
More informationNOTE. A 2 2 Game without the Fictitious Play Property
GAMES AND ECONOMIC BEHAVIOR 14, 144 148 1996 ARTICLE NO. 0045 NOTE A Game without the Fictitious Play Property Dov Monderer and Aner Sela Faculty of Industrial Engineering and Management, The Technion,
More informationBounded Rationality Lecture 2. Full (Substantive, Economic) Rationality
Bounded Rationality Lecture 2 Full (Substantive, Economic) Rationality Mikhail Anufriev EDG, Faculty of Business, University of Technology Sydney (UTS) European University at St.Petersburg Faculty of Economics
More informationWeak Dominance and Never Best Responses
Chapter 4 Weak Dominance and Never Best Responses Let us return now to our analysis of an arbitrary strategic game G := (S 1,...,S n, p 1,...,p n ). Let s i, s i be strategies of player i. We say that
More informationGame Theory. Professor Peter Cramton Economics 300
Game Theory Professor Peter Cramton Economics 300 Definition Game theory is the study of mathematical models of conflict and cooperation between intelligent and rational decision makers. Rational: each
More informationGame Theory and Rationality
April 6, 2015 Notation for Strategic Form Games Definition A strategic form game (or normal form game) is defined by 1 The set of players i = {1,..., N} 2 The (usually finite) set of actions A i for each
More informationModels of Reputation with Bayesian Updating
Models of Reputation with Bayesian Updating Jia Chen 1 The Tariff Game (Downs and Rocke 1996) 1.1 Basic Setting Two states, A and B, are setting the tariffs for trade. The basic setting of the game resembles
More informationGames A game is a tuple = (I; (S i ;r i ) i2i) where ffl I is a set of players (i 2 I) ffl S i is a set of (pure) strategies (s i 2 S i ) Q ffl r i :
On the Connection between No-Regret Learning, Fictitious Play, & Nash Equilibrium Amy Greenwald Brown University Gunes Ercal, David Gondek, Amir Jafari July, Games A game is a tuple = (I; (S i ;r i ) i2i)
More informationKlaus Kultti Hannu Salonen Demented Prisoners. Aboa Centre for Economics
Klaus Kultti Hannu Salonen Demented Prisoners Aboa Centre for Economics Discussion Paper No. 43 Turku 2009 Copyright Author(s) ISSN 1796-3133 Printed in Uniprint Turku 2009 Klaus Kultti Hannu Salonen Demented
More informationBasic Game Theory. Kate Larson. January 7, University of Waterloo. Kate Larson. What is Game Theory? Normal Form Games. Computing Equilibria
Basic Game Theory University of Waterloo January 7, 2013 Outline 1 2 3 What is game theory? The study of games! Bluffing in poker What move to make in chess How to play Rock-Scissors-Paper Also study of
More informationNon-zero-sum Game and Nash Equilibarium
Non-zero-sum Game and Nash Equilibarium Team nogg December 21, 2016 Overview Prisoner s Dilemma Prisoner s Dilemma: Alice Deny Alice Confess Bob Deny (-1,-1) (-9,0) Bob Confess (0,-9) (-6,-6) Prisoner
More informationGeneral-sum games. I.e., pretend that the opponent is only trying to hurt you. If Column was trying to hurt Row, Column would play Left, so
General-sum games You could still play a minimax strategy in general- sum games I.e., pretend that the opponent is only trying to hurt you But this is not rational: 0, 0 3, 1 1, 0 2, 1 If Column was trying
More informationExponential Moving Average Based Multiagent Reinforcement Learning Algorithms
Artificial Intelligence Review manuscript No. (will be inserted by the editor) Exponential Moving Average Based Multiagent Reinforcement Learning Algorithms Mostafa D. Awheda Howard M. Schwartz Received:
More informationUC Berkeley Haas School of Business Game Theory (EMBA 296 & EWMBA 211) Summer 2016
UC Berkeley Haas School of Business Game Theory (EMBA 296 & EWMBA 211) Summer 2016 More on strategic games and extensive games with perfect information Block 2 Jun 12, 2016 Food for thought LUPI Many players
More informationQuantum Games. Quantum Strategies in Classical Games. Presented by Yaniv Carmeli
Quantum Games Quantum Strategies in Classical Games Presented by Yaniv Carmeli 1 Talk Outline Introduction Game Theory Why quantum games? PQ Games PQ penny flip 2x2 Games Quantum strategies 2 Game Theory
More informationConvergence and No-Regret in Multiagent Learning
Convergence and No-Regret in Multiagent Learning Michael Bowling Department of Computing Science University of Alberta Edmonton, Alberta Canada T6G 2E8 bowling@cs.ualberta.ca Abstract Learning in a multiagent
More informationPlayers as Serial or Parallel Random Access Machines. Timothy Van Zandt. INSEAD (France)
Timothy Van Zandt Players as Serial or Parallel Random Access Machines DIMACS 31 January 2005 1 Players as Serial or Parallel Random Access Machines (EXPLORATORY REMARKS) Timothy Van Zandt tvz@insead.edu
More informationBiology as Information Dynamics
Biology as Information Dynamics John Baez Stanford Complexity Group April 20, 2017 What is life? Self-replicating information! Information about what? How to self-replicate! It is clear that biology has
More informationMultiagent Learning Using a Variable Learning Rate
Multiagent Learning Using a Variable Learning Rate Michael Bowling, Manuela Veloso Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213-3890 Abstract Learning to act in a multiagent
More informationOn the Rate of Convergence of Fictitious Play
Preprint submitted to Theory of Computing Systems June 7, 2011 On the Rate of Convergence of Fictitious Play Felix Brandt Felix Fischer Paul Harrenstein Abstract Fictitious play is a simple learning algorithm
More informationMechanism Design: Implementation. Game Theory Course: Jackson, Leyton-Brown & Shoham
Game Theory Course: Jackson, Leyton-Brown & Shoham Bayesian Game Setting Extend the social choice setting to a new setting where agents can t be relied upon to disclose their preferences honestly Start
More informationGame Theory for Linguists
Fritz Hamm, Roland Mühlenbernd 4. Mai 2016 Overview Overview 1. Exercises 2. Contribution to a Public Good 3. Dominated Actions Exercises Exercise I Exercise Find the player s best response functions in
More informationSolving Zero-Sum Extensive-Form Games. Branislav Bošanský AE4M36MAS, Fall 2013, Lecture 6
Solving Zero-Sum Extensive-Form Games ranislav ošanský E4M36MS, Fall 2013, Lecture 6 Imperfect Information EFGs States Players 1 2 Information Set ctions Utility Solving II Zero-Sum EFG with perfect recall
More informationGame Theory and Algorithms Lecture 2: Nash Equilibria and Examples
Game Theory and Algorithms Lecture 2: Nash Equilibria and Examples February 24, 2011 Summary: We introduce the Nash Equilibrium: an outcome (action profile) which is stable in the sense that no player
More informationEquilibrium Computation
Equilibrium Computation Ruta Mehta AGT Mentoring Workshop 18 th June, 2018 Q: What outcome to expect? Multiple self-interested agents interacting in the same environment Deciding what to do. Q: What to
More informationObservations on Cooperation
Introduction Observations on Cooperation Yuval Heller (Bar Ilan) and Erik Mohlin (Lund) PhD Workshop, BIU, January, 2018 Heller & Mohlin Observations on Cooperation 1 / 20 Introduction Motivating Example
More informationCommunities and Populations
ommunities and Populations Two models of population change The logistic map The Lotke-Volterra equations for oscillations in populations Prisoner s dilemma Single play Iterated play ommunity-wide play
More informationStatic (or Simultaneous- Move) Games of Complete Information
Static (or Simultaneous- Move) Games of Complete Information Introduction to Games Normal (or Strategic) Form Representation Teoria dos Jogos - Filomena Garcia 1 Outline of Static Games of Complete Information
More informationBargaining Efficiency and the Repeated Prisoners Dilemma. Bhaskar Chakravorti* and John Conley**
Bargaining Efficiency and the Repeated Prisoners Dilemma Bhaskar Chakravorti* and John Conley** Published as: Bhaskar Chakravorti and John P. Conley (2004) Bargaining Efficiency and the repeated Prisoners
More informationExtending Fictitious Play with Pattern Recognition
Extending Fictitious Play with Pattern Recognition November 13, 2013 Author: Ronald Chu (info@ronaldchu.nl) Supervisor: dr. G.A.W. Vreeswijk Abstract Fictitious play, an algorithm to predict the opponents
More informationBrown s Original Fictitious Play
manuscript No. Brown s Original Fictitious Play Ulrich Berger Vienna University of Economics, Department VW5 Augasse 2-6, A-1090 Vienna, Austria e-mail: ulrich.berger@wu-wien.ac.at March 2005 Abstract
More informationCSC304 Lecture 5. Game Theory : Zero-Sum Games, The Minimax Theorem. CSC304 - Nisarg Shah 1
CSC304 Lecture 5 Game Theory : Zero-Sum Games, The Minimax Theorem CSC304 - Nisarg Shah 1 Recap Last lecture Cost-sharing games o Price of anarchy (PoA) can be n o Price of stability (PoS) is O(log n)
More informationREPEATED GAMES. Jörgen Weibull. April 13, 2010
REPEATED GAMES Jörgen Weibull April 13, 2010 Q1: Can repetition induce cooperation? Peace and war Oligopolistic collusion Cooperation in the tragedy of the commons Q2: Can a game be repeated? Game protocols
More informationLecture 1. Evolution of Market Concentration
Lecture 1 Evolution of Market Concentration Take a look at : Doraszelski and Pakes, A Framework for Applied Dynamic Analysis in IO, Handbook of I.O. Chapter. (see link at syllabus). Matt Shum s notes are
More informationBasics of Game Theory
Basics of Game Theory Giacomo Bacci and Luca Sanguinetti Department of Information Engineering University of Pisa, Pisa, Italy {giacomo.bacci,luca.sanguinetti}@iet.unipi.it April - May, 2010 G. Bacci and
More informationEC3224 Autumn Lecture #04 Mixed-Strategy Equilibrium
Reading EC3224 Autumn Lecture #04 Mixed-Strategy Equilibrium Osborne Chapter 4.1 to 4.10 By the end of this week you should be able to: find a mixed strategy Nash Equilibrium of a game explain why mixed
More informationBiology as Information Dynamics
Biology as Information Dynamics John Baez Biological Complexity: Can It Be Quantified? Beyond Center February 2, 2017 IT S ALL RELATIVE EVEN INFORMATION! When you learn something, how much information
More informationTheory Field Examination Game Theory (209A) Jan Question 1 (duopoly games with imperfect information)
Theory Field Examination Game Theory (209A) Jan 200 Good luck!!! Question (duopoly games with imperfect information) Consider a duopoly game in which the inverse demand function is linear where it is positive
More informationGame Theory Lecture 10+11: Knowledge
Game Theory Lecture 10+11: Knowledge Christoph Schottmüller University of Copenhagen November 13 and 20, 2014 1 / 36 Outline 1 (Common) Knowledge The hat game A model of knowledge Common knowledge Agree
More informationLecture Notes on Game Theory
Lecture Notes on Game Theory Levent Koçkesen Strategic Form Games In this part we will analyze games in which the players choose their actions simultaneously (or without the knowledge of other players
More informationA Dynamic Level-k Model in Games
Dynamic Level-k Model in Games Teck Ho and Xuanming Su UC erkeley March, 2010 Teck Hua Ho 1 4-stage Centipede Game 4 2 16 8 1 8 4 32 1 2 3 4 64 16 5 Outcome Round 1 2 3 4 5 1 5 6.2% 30.3% 35.9% 20.0% 7.6%
More informationEVOLUTIONARY STABILITY FOR TWO-STAGE HAWK-DOVE GAMES
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS olume 25, Number 1, Winter 1995 EOLUTIONARY STABILITY FOR TWO-STAGE HAWK-DOE GAMES R. CRESSMAN ABSTRACT. Although two individuals in a biological species often interact
More informationGame Theory. 2.1 Zero Sum Games (Part 2) George Mason University, Spring 2018
Game Theory 2.1 Zero Sum Games (Part 2) George Mason University, Spring 2018 The Nash equilibria of two-player, zero-sum games have various nice properties. Minimax Condition A pair of strategies is in
More informationBayesian-Nash Equilibrium
Bayesian-Nash Equilibrium A Bayesian game models uncertainty over types of opponents a player faces The game was defined in terms of players, their types, their available actions A player s beliefs about
More informationSONDERFORSCHUNGSBEREICH 504
SONDERFORSCHUNGSBEREICH 504 Rationalitätskonzepte, Entscheidungsverhalten und ökonomische Modellierung No. 02-03 Two-Speed Evolution of Strategies and Preferences in Symmetric Games Possajennikov, Alex
More informationEquilibrium Refinements
Equilibrium Refinements Mihai Manea MIT Sequential Equilibrium In many games information is imperfect and the only subgame is the original game... subgame perfect equilibrium = Nash equilibrium Play starting
More informationSmall Sample of Related Literature
UCLA IPAM July 2015 Learning in (infinitely) repeated games with n players. Prediction and stability in one-shot large (many players) games. Prediction and stability in large repeated games (big games).
More informationEvolutionary Dynamics and Extensive Form Games by Ross Cressman. Reviewed by William H. Sandholm *
Evolutionary Dynamics and Extensive Form Games by Ross Cressman Reviewed by William H. Sandholm * Noncooperative game theory is one of a handful of fundamental frameworks used for economic modeling. It
More informationComputational Evolutionary Game Theory and why I m never using PowerPoint for another presentation involving maths ever again
Computational Evolutionary Game Theory and why I m never using PowerPoint for another presentation involving maths ever again Enoch Lau 5 September 2007 Outline What is evolutionary game theory? Why evolutionary
More informationTechnical Report Documentation Page
1. Report No. SWUTC/13/600451-00065-1 4. Title and Subtitle GAME THEORY AND TRAFFIC ASSIGNMENT Technical Report Documentation Page 2. Government Accession No. 3. Recipient's Catalog No. 5. Report Date
More informationErgodicity and Non-Ergodicity in Economics
Abstract An stochastic system is called ergodic if it tends in probability to a limiting form that is independent of the initial conditions. Breakdown of ergodicity gives rise to path dependence. We illustrate
More informationGame Theory. Monika Köppl-Turyna. Winter 2017/2018. Institute for Analytical Economics Vienna University of Economics and Business
Monika Köppl-Turyna Institute for Analytical Economics Vienna University of Economics and Business Winter 2017/2018 Static Games of Incomplete Information Introduction So far we assumed that payoff functions
More informationLearning by (limited) forward looking players
Learning by (limited) forward looking players Friederike Mengel Maastricht University April 2009 Abstract We present a model of adaptive economic agents who are k periods forward looking. Agents in our
More informationGovernment 2005: Formal Political Theory I
Government 2005: Formal Political Theory I Lecture 11 Instructor: Tommaso Nannicini Teaching Fellow: Jeremy Bowles Harvard University November 9, 2017 Overview * Today s lecture Dynamic games of incomplete
More informationIncremental Policy Learning: An Equilibrium Selection Algorithm for Reinforcement Learning Agents with Common Interests
Incremental Policy Learning: An Equilibrium Selection Algorithm for Reinforcement Learning Agents with Common Interests Nancy Fulda and Dan Ventura Department of Computer Science Brigham Young University
More informationComputational Problems Related to Graph Structures in Evolution
BACHELOR THESIS Štěpán Šimsa Computational Problems Related to Graph Structures in Evolution Department of Applied Mathematics Supervisor of the bachelor thesis: Study programme: Study branch: Prof. Krishnendu
More informationRecap Social Choice Fun Game Voting Paradoxes Properties. Social Choice. Lecture 11. Social Choice Lecture 11, Slide 1
Social Choice Lecture 11 Social Choice Lecture 11, Slide 1 Lecture Overview 1 Recap 2 Social Choice 3 Fun Game 4 Voting Paradoxes 5 Properties Social Choice Lecture 11, Slide 2 Formal Definition Definition
More informationGame Theory DR. ÖZGÜR GÜRERK UNIVERSITY OF ERFURT WINTER TERM 2012/13. Strict and nonstrict equilibria
Game Theory 2. Strategic Games contd. DR. ÖZGÜR GÜRERK UNIVERSITY OF ERFURT WINTER TERM 2012/13 Strict and nonstrict equilibria In the examples we have seen so far: A unilaterally deviation from Nash equilibrium
More informationMultiagent Systems Motivation. Multiagent Systems Terminology Basics Shapley value Representation. 10.
Multiagent Systems July 2, 2014 10. Coalition Formation Multiagent Systems 10. Coalition Formation B. Nebel, C. Becker-Asano, S. Wöl Albert-Ludwigs-Universität Freiburg July 2, 2014 10.1 Motivation 10.2
More informationEC319 Economic Theory and Its Applications, Part II: Lecture 7
EC319 Economic Theory and Its Applications, Part II: Lecture 7 Leonardo Felli NAB.2.14 27 February 2014 Signalling Games Consider the following Bayesian game: Set of players: N = {N, S, }, Nature N strategy
More informationThe Time Consistency Problem - Theory and Applications
The Time Consistency Problem - Theory and Applications Nils Adler and Jan Störger Seminar on Dynamic Fiscal Policy Dr. Alexander Ludwig November 30, 2006 Universität Mannheim Outline 1. Introduction 1.1
More informationFor general queries, contact
PART I INTRODUCTION LECTURE Noncooperative Games This lecture uses several examples to introduce the key principles of noncooperative game theory Elements of a Game Cooperative vs Noncooperative Games:
More information6.254 : Game Theory with Engineering Applications Lecture 13: Extensive Form Games
6.254 : Game Theory with Engineering Lecture 13: Extensive Form Games Asu Ozdaglar MIT March 18, 2010 1 Introduction Outline Extensive Form Games with Perfect Information One-stage Deviation Principle
More information6 The Principle of Optimality
6 The Principle of Optimality De nition A T shot deviation from a strategy s i is a strategy bs i such that there exists T such that bs i (h t ) = s i (h t ) for all h t 2 H with t T De nition 2 A one-shot
More informationVII. Cooperation & Competition
VII. Cooperation & Competition A. The Iterated Prisoner s Dilemma Read Flake, ch. 17 4/23/18 1 The Prisoners Dilemma Devised by Melvin Dresher & Merrill Flood in 1950 at RAND Corporation Further developed
More informationGame Theory and its Applications to Networks - Part I: Strict Competition
Game Theory and its Applications to Networks - Part I: Strict Competition Corinne Touati Master ENS Lyon, Fall 200 What is Game Theory and what is it for? Definition (Roger Myerson, Game Theory, Analysis
More informationC 4,4 0,5 0,0 0,0 B 0,0 0,0 1,1 1 2 T 3,3 0,0. Freund and Schapire: Prisoners Dilemma
On No-Regret Learning, Fictitious Play, and Nash Equilibrium Amir Jafari Department of Mathematics, Brown University, Providence, RI 292 Amy Greenwald David Gondek Department of Computer Science, Brown
More informationComputation of Efficient Nash Equilibria for experimental economic games
International Journal of Mathematics and Soft Computing Vol.5, No.2 (2015), 197-212. ISSN Print : 2249-3328 ISSN Online: 2319-5215 Computation of Efficient Nash Equilibria for experimental economic games
More informationהאוניברסיטה העברית בירושלים
האוניברסיטה העברית בירושלים THE HEBREW UNIVERSITY OF JERUSALEM TOWARDS A CHARACTERIZATION OF RATIONAL EXPECTATIONS by ITAI ARIELI Discussion Paper # 475 February 2008 מרכז לחקר הרציונליות CENTER FOR THE
More informationarxiv: v1 [physics.soc-ph] 3 Nov 2008
arxiv:0811.0253v1 [physics.soc-ph] 3 Nov 2008 TI-games : An Exploration of Type Indeterminacy in Strategic Decision-making Jerry Busemeyer, Ariane Lambert-Mogiliansky. February 19, 2009 Abstract In this
More informationComputing Equilibria of Repeated And Dynamic Games
Computing Equilibria of Repeated And Dynamic Games Şevin Yeltekin Carnegie Mellon University ICE 2012 July 2012 1 / 44 Introduction Repeated and dynamic games have been used to model dynamic interactions
More informationMeaning, Evolution and the Structure of Society
Meaning, Evolution and the Structure of Society Roland Mühlenbernd November 7, 2014 OVERVIEW Game Theory and Linguistics Pragm. Reasoning Language Evolution GT in Lang. Use Signaling Games Replicator Dyn.
More informationEquilibria in Games with Weak Payoff Externalities
NUPRI Working Paper 2016-03 Equilibria in Games with Weak Payoff Externalities Takuya Iimura, Toshimasa Maruta, and Takahiro Watanabe October, 2016 Nihon University Population Research Institute http://www.nihon-u.ac.jp/research/institute/population/nupri/en/publications.html
More informationLearning To Cooperate in a Social Dilemma: A Satisficing Approach to Bargaining
Learning To Cooperate in a Social Dilemma: A Satisficing Approach to Bargaining Jeffrey L. Stimpson & ichael A. Goodrich Computer Science Department, Brigham Young University, Provo, UT 84602 jstim,mike@cs.byu.edu
More informationIrrational behavior in the Brown von Neumann Nash dynamics
Irrational behavior in the Brown von Neumann Nash dynamics Ulrich Berger a and Josef Hofbauer b a Vienna University of Economics and Business Administration, Department VW 5, Augasse 2-6, A-1090 Wien,
More information