Equilibrium Refinements
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Game Theory 2: Extensive-Form Games and Subgame Perfection
Game Theory 2: Extensive-Form Games and Subgame Perfection 1 / 26 Dynamics in Games How should we think of strategic interactions that occur in sequence? Who moves when? And what can they do at different points in time? How do people react to different histories? 2 / 26 Modeling Games with Dynamics Players Player function I Who moves when Terminal histories I Possible paths through the game Preferences over terminal histories 3 / 26 Strategies A strategy is a complete contingent plan Player i's strategy specifies her action choice at each point at which she could be called on to make a choice 4 / 26 An Example: International Crises Two countries (A and B) are competing over a piece of land that B occupies Country A decides whether to make a demand If Country A makes a demand, B can either acquiesce or fight a war If A does not make a demand, B keeps land (game ends) A's best outcome is Demand followed by Acquiesce, worst outcome is Demand and War B's best outcome is No Demand and worst outcome is Demand and War 5 / 26 An Example: International Crises A can choose: Demand (D) or No Demand (ND) B can choose: Fight a war (W ) or Acquiesce (A) Preferences uA(D; A) = 3 > uA(ND; A) = uA(ND; W ) = 2 > uA(D; W ) = 1 uB(ND; A) = uB(ND; W ) = 3 > uB(D; A) = 2 > uB(D; W ) = 1 How can we represent this scenario as a game (in strategic form)? 6 / 26 International Crisis Game: NE Country B WA D 1; 1 3X; 2X Country A ND 2X; 3X 2; 3X I Is there something funny here? I Is there something funny here? I Specifically, (ND; W )? I Is there something funny here? -
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 .................................. -
Recent Advances in Understanding Competitive Markets
Recent Advances in Understanding Competitive Markets Chris Shannon Department of Economics University of California, Berkeley March 4, 2002 1 Introduction The Arrow-Debreu model of competitive markets has proven to be a remark- ably rich and °exible foundation for studying an array of important economic problems, ranging from social security reform to the determinants of long-run growth. Central to the study of such questions are many of the extensions and modi¯cations of the classic Arrow-Debreu framework that have been ad- vanced over the last 40 years. These include dynamic models of exchange over time, as in overlapping generations or continuous time, and under un- certainty, as in incomplete ¯nancial markets. Many of the most interesting and important recent advances have been spurred by questions about the role of markets in allocating resources and providing opportunities for risk- sharing arising in ¯nance and macroeconomics. This article is intended to give an overview of some of these more recent developments in theoretical work involving competitive markets, and highlight some promising areas for future work. I focus on three broad topics. The ¯rst is the role of asymmetric informa- tion in competitive markets. Classic work in information economics suggests that competitive markets break down in the face of informational asymme- 0 tries. Recently these issues have been reinvestigated from the vantage point of more general models of the market structure, resulting in some surpris- ing results overturning these conclusions and clarifying the conditions under which perfectly competitive markets can incorporate informational asymme- tries. The second concentrates on the testable implications of competitive markets. -
Subgame Perfect (-)Equilibrium in Perfect Information Games
Subgame perfect (-)equilibrium in perfect information games J´anosFlesch∗ April 15, 2016 Abstract We discuss recent results on the existence and characterization of subgame perfect ({)equilibrium in perfect information games. The game. We consider games with perfect information and deterministic transitions. Such games can be given by a directed tree.1 In this tree, each node is associated with a player, who controls this node. The outgoing arcs at this node represent the actions available to this player at this node. We assume that each node has at least one successor, rather than having terminal nodes.2 Play of the game starts at the root. At any node z that play visits, the player who controls z has to choose one of the actions at z, which brings play to a next node. This induces an infinite path in the tree from the root, which we call a play. Depending on this play, each player receives a payoff. Note that these payoffs are fairly general. This setup encompasses the case when the actions induce instantaneous rewards which are then aggregated into a payoff, possibly by taking the total discounted sum or the long-term average. It also includes payoff functions considered in the literature of computer science (reachability games, etc.). Subgame-perfect (-)equilibrium. We focus on pure strategies for the players. A central solution concept in such games is subgame-perfect equilibrium, which is a strategy profile that induces a Nash equilibrium in every subgame, i.e. when starting at any node in the tree, no player has an incentive to deviate individually from his continuation strategy. -
Subgame-Perfect Equilibria in Mean-Payoff Games
Subgame-perfect Equilibria in Mean-payoff Games Léonard Brice ! Université Gustave Eiffel, France Jean-François Raskin ! Université Libre de Bruxelles, Belgium Marie van den Bogaard ! Université Gustave Eiffel, France Abstract In this paper, we provide an effective characterization of all the subgame-perfect equilibria in infinite duration games played on finite graphs with mean-payoff objectives. To this end, we introduce the notion of requirement, and the notion of negotiation function. We establish that the plays that are supported by SPEs are exactly those that are consistent with the least fixed point of the negotiation function. Finally, we show that the negotiation function is piecewise linear, and can be analyzed using the linear algebraic tool box. As a corollary, we prove the decidability of the SPE constrained existence problem, whose status was left open in the literature. 2012 ACM Subject Classification Software and its engineering: Formal methods; Theory of compu- tation: Logic and verification; Theory of computation: Solution concepts in game theory. Keywords and phrases Games on graphs, subgame-perfect equilibria, mean-payoff objectives. Digital Object Identifier 10.4230/LIPIcs... 1 Introduction The notion of Nash equilibrium (NE) is one of the most important and most studied solution concepts in game theory. A profile of strategies is an NE when no rational player has an incentive to change their strategy unilaterally, i.e. while the other players keep their strategies. Thus an NE models a stable situation. Unfortunately, it is well known that, in sequential games, NEs suffer from the problem of non-credible threats, see e.g. [18]. In those games, some NE only exists when some players do not play rationally in subgames and so use non-credible threats to force the NE. -
(501B) Problem Set 5. Bayesian Games Suggested Solutions by Tibor Heumann
Dirk Bergemann Department of Economics Yale University Microeconomic Theory (501b) Problem Set 5. Bayesian Games Suggested Solutions by Tibor Heumann 1. (Market for Lemons) Here I ask that you work out some of the details in perhaps the most famous of all information economics models. By contrast to discussion in class, we give a complete formulation of the game. A seller is privately informed of the value v of the good that she sells to a buyer. The buyer's prior belief on v is uniformly distributed on [x; y] with 3 0 < x < y: The good is worth 2 v to the buyer. (a) Suppose the buyer proposes a price p and the seller either accepts or rejects p: If she accepts, the seller gets payoff p−v; and the buyer gets 3 2 v − p: If she rejects, the seller gets v; and the buyer gets nothing. Find the optimal offer that the buyer can make as a function of x and y: (b) Show that if the buyer and the seller are symmetrically informed (i.e. either both know v or neither party knows v), then trade takes place with probability 1. (c) Consider a simultaneous acceptance game in the model with private information as in part a, where a price p is announced and then the buyer and the seller simultaneously accept or reject trade at price p: The payoffs are as in part a. Find the p that maximizes the probability of trade. [SOLUTION] (a) We look for the subgame perfect equilibrium, thus we solve by back- ward induction. -
14.12 Game Theory Lecture Notes∗ Lectures 7-9
14.12 Game Theory Lecture Notes∗ Lectures 7-9 Muhamet Yildiz Intheselecturesweanalyzedynamicgames(withcompleteinformation).Wefirst analyze the perfect information games, where each information set is singleton, and develop the notion of backwards induction. Then, considering more general dynamic games, we will introduce the concept of the subgame perfection. We explain these concepts on economic problems, most of which can be found in Gibbons. 1 Backwards induction The concept of backwards induction corresponds to the assumption that it is common knowledge that each player will act rationally at each node where he moves — even if his rationality would imply that such a node will not be reached.1 Mechanically, it is computed as follows. Consider a finite horizon perfect information game. Consider any node that comes just before terminal nodes, that is, after each move stemming from this node, the game ends. If the player who moves at this node acts rationally, he will choose the best move for himself. Hence, we select one of the moves that give this player the highest payoff. Assigning the payoff vector associated with this move to the node at hand, we delete all the moves stemming from this node so that we have a shorter game, where our node is a terminal node. Repeat this procedure until we reach the origin. ∗These notes do not include all the topics that will be covered in the class. See the slides for a more complete picture. 1 More precisely: at each node i the player is certain that all the players will act rationally at all nodes j that follow node i; and at each node i the player is certain that at each node j that follows node i the player who moves at j will be certain that all the players will act rationally at all nodes k that follow node j,...ad infinitum. -
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). -
How Proper Is the Dominance-Solvable Outcome? Yukio Koriyama, Matias Nunez
How proper is the dominance-solvable outcome? Yukio Koriyama, Matias Nunez To cite this version: Yukio Koriyama, Matias Nunez. How proper is the dominance-solvable outcome?. 2014. hal- 01074178 HAL Id: hal-01074178 https://hal.archives-ouvertes.fr/hal-01074178 Preprint submitted on 13 Oct 2014 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. ECOLE POLYTECHNIQUE CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE HOW PROPER IS THE DOMINANCE-SOLVABLE OUTCOME? Yukio KORIYAMA Matías NÚÑEZ September 2014 Cahier n° 2014-24 DEPARTEMENT D'ECONOMIE Route de Saclay 91128 PALAISEAU CEDEX (33) 1 69333033 http://www.economie.polytechnique.edu/ mailto:[email protected] How proper is the dominance-solvable outcome?∗ Yukio Koriyamay and Mat´ıas Nu´ nez˜ z September 2014 Abstract We examine the conditions under which iterative elimination of weakly dominated strategies refines the set of proper outcomes of a normal-form game. We say that the proper inclusion holds in terms of outcome if the set of outcomes of all proper equilibria in the reduced game is included in the set of all proper outcomes of the original game. We show by examples that neither dominance solvability nor the transference of decision-maker indiffer- ence condition (TDI∗ of Marx and Swinkels [1997]) implies proper inclusion. -
Political Game Theory Nolan Mccarty Adam Meirowitz
Political Game Theory Nolan McCarty Adam Meirowitz To Liz, Janis, Lachlan, and Delaney. Contents Acknowledgements vii Chapter 1. Introduction 1 1. Organization of the Book 2 Chapter 2. The Theory of Choice 5 1. Finite Sets of Actions and Outcomes 6 2. Continuous Outcome Spaces* 10 3. Utility Theory 17 4. Utility representations on Continuous Outcome Spaces* 18 5. Spatial Preferences 19 6. Exercises 21 Chapter 3. Choice Under Uncertainty 23 1. TheFiniteCase 23 2. Risk Preferences 32 3. Learning 37 4. Critiques of Expected Utility Theory 41 5. Time Preferences 46 6. Exercises 50 Chapter 4. Social Choice Theory 53 1. The Open Search 53 2. Preference Aggregation Rules 55 3. Collective Choice 61 4. Manipulation of Choice Functions 66 5. Exercises 69 Chapter 5. Games in the Normal Form 71 1. The Normal Form 73 2. Solutions to Normal Form Games 76 3. Application: The Hotelling Model of Political Competition 83 4. Existence of Nash Equilibria 86 5. Pure Strategy Nash Equilibria in Non-Finite Games* 93 6. Application: Interest Group Contributions 95 7. Application: International Externalities 96 iii iv CONTENTS 8. Computing Equilibria with Constrained Optimization* 97 9. Proving the Existence of Nash Equilibria** 98 10. Strategic Complementarity 102 11. Supermodularity and Monotone Comparative Statics* 103 12. Refining Nash Equilibria 108 13. Application: Private Provision of Public Goods 109 14. Exercises 113 Chapter 6. Bayesian Games in the Normal Form 115 1. Formal Definitions 117 2. Application: Trade restrictions 119 3. Application: Jury Voting 121 4. Application: Jury Voting with a Continuum of Signals* 123 5. Application: Public Goods and Incomplete Information 126 6. -
2.- Game Theory Overview Game Theory Studies Situations in Which
2.- Game Theory Overview Game Theory studies situations in which the outcome of the decisions of an indi- vidual depends on the decisions taken by other individuals as well as on his own - Strategic Interdependence - 2.- Game Theory Overview Plan 2.1 Introduction 2.2 Static Games of Complete Information: Nash Equilibrium 2.3 Dynamic Games of Complete Information: Backwards Induction and Subgame Perfection 2.4 Static Games of Incomplete Information:Bayesian Equilibrium 2.5 Dynamic Games of Incomplete Information: Perfect Bayesian Equilibrium and Sequential Equilibrium 2.- Game Theory Overview 2.1 Introduction Any (non cooperative) game has the following elements 1. The Players Who is to take decisions 2 The Rules Who moves first, who moves next, what can be done and what can not, ... 3 The Strategies What actions are available to the players 4 The Outcomes What is the consequence of each combination of actions by the players 5 The Payoffs What each player obtains in each possible outcome 2.- Game Theory Overview 2.1 Introduction Example: Rock - Scissors - Paper 1. The Players Player 1 and Player 2 2 The Rules Simultaneously, both players show either rocks, scissors or paper 3 The Strategies Each player can choose among showing rocks, scissors or paper 4 The Outcomes Rock beats scissors, which beats paper, which beats rock 5 The Payoffs The winning player receives 1 from the loser (thus getting -1) 2.- Game Theory Overview 2.1 Introduction Game Scenarios Complete Information Incomplete Information Static Players have all relevant information Not all players have all the information and move simultaneously and move simultaneously Dynamic Players have all relevant information Not all players have all the information and move in turns and move in turns Examples Complete Information Incomplete Information Static Rock, Scissors, Paper Sealed-bid Auction Dynamic Chess English Auction 2.- Game Theory Overview 2.2 Static Games of Complete Information Players take decisions without knowing the actions taken by other players Basic Assumptions 1. -
TESIS DOCTORAL Three Essays on Game Theory
UNIVERSIDAD CARLOS III DE MADRID TESIS DOCTORAL Three Essays on Game Theory Autor: José Carlos González Pimienta Directores: Luis Carlos Corchón Francesco De Sinopoli DEPARTAMENTO DE ECONOMÍA Getafe, Julio del 2007 Three Essays on Game Theory Carlos Gonzalez´ Pimienta To my parents Contents List of Figures iii Acknowledgments 1 Chapter 1. Introduction 3 Chapter 2. Conditions for Equivalence Between Sequentiality and Subgame Perfection 5 2.1. Introduction 5 2.2. Notation and Terminology 7 2.3. Definitions 9 2.4. Results 12 2.5. Examples 24 2.6. Appendix: Notation and Terminology 26 Chapter 3. Undominated (and) Perfect Equilibria in Poisson Games 29 3.1. Introduction 29 3.2. Preliminaries 31 3.3. Dominated Strategies 34 3.4. Perfection 42 3.5. Undominated Perfect Equilibria 51 Chapter 4. Generic Determinacy of Nash Equilibrium in Network Formation Games 57 4.1. Introduction 57 4.2. Preliminaries 59 i ii CONTENTS 4.3. An Example 62 4.4. The Result 64 4.5. Remarks 66 4.6. Appendix: Proof of Theorem 4.1 70 Bibliography 73 List of Figures 2.1 Notation and terminology of finite extensive games with perfect recall 8 2.2 Extensive form where no information set is avoidable. 11 2.3 Extensive form where no information set is avoidable in its minimal subform. 12 2.4 Example of the use of the algorithm contained in the proof of Proposition 2.1 to generate a game where SPE(Γ) = SQE(Γ). 14 6 2.5 Selten’s horse. An example of the use of the algorithm contained in the proof of proposition 2.1 to generate a game where SPE(Γ) = SQE(Γ).