Equivalences of Extensive Forms with Perfect Recall
|
|
- Cynthia Tate
- 5 years ago
- Views:
Transcription
1 Equivalences of Extensive Forms with Perfect Recall Carlos Alós-Ferrer and Klaus Ritzberger University of Cologne and Royal Holloway, University of London, 1 and VGSF 1 as of Aug. 1, 2016
2 1 Introduction Extensive form is the basic representation of a game... needed to verify that the rules are complete. Yet, extensive forms may differ even though the game is the same. When are details of the extensive form strategically relevant? Solutions are often defined in the normal form (as strategy profiles). I Some refinement concepts are defined purely for the extensive form, e.g., subgame perfection, perfect Bayesian, sequential equilibrium. I But these are sensitive to inessential details of the extensive form. I Therefore some have argued that strategically stable solutions should only depend on the normal form... or even only on the reduced normal form (Kohlberg & Mertens 1986, Mertens 1989, 1991, 1992,...). 2
3 1.1 Which Details are inessential? One approach: I Transformations of the extensive form that do not affect the semireduced normal form (Thompson 1952, Elmes & Reny 1994). Thompson: Interchange of moves, Coalescing of moves, Addition of a superfluous move, Inflation/Deflation. Elmes & Reny: Interchange of moves, Coalescing of moves, Modified addition of a superfluous move to preserve perfect recall. F F Inflation/Deflation and Addition can destroy perfect recall! I Thompson-transformations take payoffs! Thompson s Theorem: Two extensive form games have the same semireduced normal form game if and only if one emerges from the other by iterative application of the four transformations or their inverses. 3
4 Here are the four Thompson-transformations: 4
5 from this paper 5
6 Complementary approach: I Which information is lost/preserved in the transition from the extensive to the normal form? I But now without payoffs in a pure representation of the rules. I No transformations, but identify what is essential. I What are the structures in the extensive form that can be used for solution concepts that are as robust as normal form concepts? Related (because also without payoffs) but different is... Battigalli s Conjecture: Two extensive forms have the same reduced 2 normal form if and only if one emerges from the other by iterative application of the first two Thompson transformations (Interchange & Coalescing) or their inverses (Leonetti 2015). 2 without payoffs = semi-reduced 6
7 2 Definitions A game tree =( ) is a collection of nonempty subsets (the nodes) of an underlying set (of plays) partially ordered by set inclusion such that, { } for all, and (GT1) is a chain if and only if :, (GT2) every chain in the set = \{{ }} of moves has a maximum and either an infimum in the set = {{ }} of terminal nodes or a minimum. A node is finite if the set \{ } = { } has a minimum ( ), otherwise it is infinite. If is a union of nodes, define = { } and ( ) ={ : = \ } (1) as the nodes where 2 is available. 7
8 A discrete extensive form (DEF) with player set is a pair ( ), where =( ) isagametreewithset of plays and =( ) is a system consisting of collections (the set of s choices) of nonempty unions of nodes such that (DEF1) if ( ) ( 0 ) 6= and 6= 0,then ( ) = ( 0 ) and 0 =, for all 0 and n all, and (DEF2) 1 ( ) = ( o ( ) ) ( ) ( ) ( ) for all, where ( ) ={ ( )} are the choices available to at and ( ) ={ ( ) 6= } 6= are the decision makers at. The set of pure strategies of player is the set of all functions : = { ( )} that satisfy 1 ( ) = ( ) for all ( ) (2) and = is the set of all pure strategy combinations. 8
9 I For every DEF there is a surjection : that assigns to every strategy combination the play that it induces (AR 2008, Th. 4 & 6). I For let ( ( )) = ( ) denote the plays passing through information set ( ). I Assume w.l.o.g. that ( ( )) for all and all. I For a DEF no-absent-mindedness holds (AR 2005, Prop. 13), but not necessarily perfect recall. ADEF( ) satisfies perfect recall (Kuhn 1953) if and only if if 0 6= then either 0 or 0 (3) for all 0 { ( ( )) } and all (AR 2016, Th. 2). 9
10 Want more details? 10
11 2.1 Normal Form The normal form of a DEF ( ) is the triplet ( ), where = are the strategy combinations and : assigns the induced plays. Two strategies 0 ofthesameplayerinadef( ) are strategically equivalent if ( )= ( 0 ) for all 6=. Proposition 1 Two strategies 0 are strategically equivalent if and only if, for all, ( ) 6= 0 ( ) ( ( ) ( 0 )) =. Denote the quotient space w.r.t. strategic equivalence for by and = ( ). By definition : induces a surjection :,forall [ ], by ([ ]) = ( ). ³ The triplet is the reduced normal form of the DEF ( ). 11
12 2.1.1 Comparing Normal Forms Let ( ) and be the normal forms of two DEFs, ( ) and ( 0 0 ). They are isomorphic if there are bijections : 0, : 0, : 0,and : 0 for all such that 0 ( ( )) = ( ( )) for all (4) and ( ) ( ) = ( ) for all and all. Example 1 The following two normal forms are isomorphic. a b A 1 1 B 2 3 C 2 4 X Y Z
13 2.2 Decision System Beginwithasingleplayer,thedecision maker (DM). A (sequential) decision problem (with perfect recall) for a DM on a set is a pair ( ) where is a collection of nonempty subsets of and is a partition of such that, for all 0, if 0 and 0 6= then = 0 (5) and, for all 0 { ( ) }, if 0 6= then either 0 or 0. (6) Denote ( ) = and ( ) ={ ( ) }. Decision problems can be partially ordered. Say that decision problem ( 0 0 ) is smaller than decision problem ( ) if 0 and ( 0 ) ( ). 13
14 Decision problems may contain redundancies, though. E.g., = { } for and. Say that for a decision problem ( ) the pair ( ) is redundant if = ( ). Proposition 2 For every decision problem ( ) there is a unique largest decision problem that is smaller than ( ) and contains no redundant pairs. Call this the reduced decision problem of ( ). Remark 1 In epistemics an information structure is a function Π : W such that Π ( ) and 0 Π ( ) Π ( 0 ) Π ( ) for all. Such a function can be associated to a decision problem by Π ( ) =min{ ( ) ( ) } provided it is the reduced decision problem. 14
15 Now return to multi-player games. The decision system associated with a DEF ( ) is the collection ( ) of decision problems where, for each, the partition of is induced by, for all 0, 0 ( ) = ( 0 ). Each is a choice and each is an information set. If in a decision system the decision problems ( ) arereplacedbytheir reduced decision problems, the resulting collection is called the reduced decision system of the DEF. With the decision problem ( ) of each player comes the space of s choice functions. Θ = { : ( ), } (7) 15
16 Choice functions and strategies are bijective and preserve. Lemma 1 For each there is a bijection : Θ such that, with = ( ) Θ = Θ and = ( ) : Θ, { ( ) ( ) } = ( ( )) Comparing Decision Systems The decision system ( ) of a DEF ( ) and the decision system ( 0 0 ) 0 of a DEF ( 0 0 ) are isomorphic if there are bijections : 0 and : 0 such that 0 ( ) = { ( ) } and 0 ( ) = {{ ( ) } } for all, where ( ) ={ ( ) } for all 2. 16
17 Example: Let = { } be the plays and for two players, = {1 2}, 1 = {{1} {2 3 4} {3} {4}} with 1 = {{{1} {2 3 4}} {{3} {4}}} and 2 = {{2} {3 4}} with 2 = {{{2} {3 4}}}. 17
18 3 First Results I For what follows let ( ) and ( 0 0 ) be two DEFs with normal forms ( ) and ³ respectively, 0 reduced normal forms and ³ 0 0, decision systems ( ) and ( 0 0 ) respectively, and 0 reduced decision systems and 0 0 I All DEFs involved satisfy perfect recall. Theorem 1 Two DEFs have isomorphic normal forms if and only if their decision systems are also isomorphic. Conjecture: This corresponds to Thompson s Interchange of moves." Choices and information sets stay constant, only the tree changes
19 Idea of proof for Theorem 1: if: From the isomorphism between decision systems construct bijections between the choice functions. Use Lemma 1 to construct an isomorphism between normal forms. only if: Exploits perfect recall on every path to an information set of player this player has to take the same choices. Now take the chain of image choices (under the isomorphism of normal forms), intersect them, and obtain the images of choices. Show that this results in an isomorphism between decision systems. (This takes DEF1 and DEF2.) I Important: No reduction step needed! And multiple strategically equivalent strategies still show up. I Hence, the only difference between the two DEFs concerns the trees but not choices and information sets. 19
20 Theorem 2 Two DEFs have isomorphic reduced normal forms if and only if their reduced decision systems are also isomorphic. Conjecture: This corresponds to the first two Thompson-transformations, Interchange of moves and Coalescing of moves. Idea of proof for Theorem 2: Similar to proof of Theorem 1, but now reduction step is involved. DEF 7 decision system 7 reduced decision system 7 new DEF Involves an algorithm that reconstructs a DEF from a (possibly reduced) decision system. Let 0 = 0 = { }. For each =1 2 and each 1 1 \{{ } } let ( ) ={ : ( )} and 1 ( ) = ( ) 6= ( ) ( ) ª and set = 1 1 ( ) and = \{{ } }. 20
21 4 What s that Saying? I If the conjectures are correct, then Battigalli s conjecture is correct, but canberefined: Two extensive forms have the same normal form if and only if one emerges from the other by iterative application of the first Thompson transformation (Interchange) or its inverse. Two extensive forms have the same reduced normal form if and only if one emerges from the other by iterative application of the first two Thompson transformations (Interchange & Coalescing) or their inverses. F Beware: This applies to forms not to games. E.g., the following is not the normal form of a DEF with = { }
22 4.1 Implications for Solutions I Robust solutions for extensive form games cannot depend on the tree! For instance, Myerson & Reny s attempts to generalize sequential equilibrium to large games will have to work without the notion of beliefs. For, beliefs are definedonthe nodes of the tree and the tree changes both under Theorem 1 and 2. I Will robust solutions have to work without a sequential structure? No! The decision system captures all the relevant sequential structure, because (for a given player) choices together with information sets are like a tree. But the appropriate domain for probabilistic assessments is the set of plays not the nodes. I Why not work with the (reduced) normal form right away? The normal form does not identify subgames and information sets (recall, though, Mailath, Samuelson, and Swinkels 1993, 1994). Hence, backwards induction intuition can only be captured indirectly, e.g., by proper equilibrium. But (proper) strategy perturbations in large games are bound to depend on the topology and for large games there is no natural topology on strategy space. 22
23 4.2 ABitofSpeculation I The existing results show that the decision system captures precisely the same information as the normal form, and the reduced decision system the same as the reduced normal form. I If the normal form (resp. reduced normal form) contains all strategically relevant information, then so does the decision system (resp. reduced decision system). This raises the following question: Is it possible to represent a game purely by its decision system a game in decision form, as it were? One would need to find conditions on a decision system such that the algorithm produces a DEF that has the original decision system (preferably also for the reduced version). 23
24 5 Conclusions Working with extensive forms and normal forms without payoffs, but with perfect recall, we characterize equivalence classes of DEFs that have isomorphic normal forms that have isomorphic reduced normal forms by the property that they have isomorphic decision systems they have isomorphic reduced decision systems. This is most likely related to the first two Thompson transformations (Interchange & Coalescing) and to Battigalli s conjecture. 24
25 Thank you for your attention! 25
4: Dynamic games. Concordia February 6, 2017
INSE6441 Jia Yuan Yu 4: Dynamic games Concordia February 6, 2017 We introduce dynamic game with non-simultaneous moves. Example 0.1 (Ultimatum game). Divide class into two groups at random: Proposers,
More informationHigher Order Beliefs in Dynamic Environments
University of Pennsylvania Department of Economics June 22, 2008 Introduction: Higher Order Beliefs Global Games (Carlsson and Van Damme, 1993): A B A 0, 0 0, θ 2 B θ 2, 0 θ, θ Dominance Regions: A if
More informationRESEARCH PAPER NO AXIOMATIC THEORY OF EQUILIBRIUM SELECTION IN SIGNALING GAMES WITH GENERIC PAYOFFS. Srihari Govindan.
RESEARCH PAPER NO. 2000 AXIOMATIC THEORY OF EQUILIBRIUM SELECTION IN SIGNALING GAMES WITH GENERIC PAYOFFS Srihari Govindan Robert Wilson September 2008 This work was partially funded by a National Science
More informationCHARACTERIZATIONS OF STABILITY
CHARACTERIZATIONS OF STABILITY Srihari Govindan and Robert Wilson University of Iowa and Stanford University 1 Motive for Equilibrium Selection The original Nash definition allows 1. Multiple equilibria
More informationON FORWARD INDUCTION
Econometrica, Submission #6956, revised ON FORWARD INDUCTION SRIHARI GOVINDAN AND ROBERT WILSON Abstract. A player s pure strategy is called relevant for an outcome of a game in extensive form with perfect
More informationSuper Weak Isomorphism of Extensive Games
Super Weak Isomorphism of Extensive Games André Casajus Accepted for publication in Mathematical Social Sciences as of July 6 2005. (March 2005, this version: July 10, 2005, 12:13) Abstract It is well-known
More informationEconomics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Extensive games with perfect information OR6and7,FT3,4and11
Economics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Extensive games with perfect information OR6and7,FT3,4and11 Perfect information A finite extensive game with perfect information
More informationAxiomatic Equilibrium Selection for Generic Two-Player Games
Stanford University From the SelectedWorks of Robert B Wilson May, 2009 Axiomatic Equilibrium Selection for Generic Two-Player Games Srihari Govindan Robert B Wilson Available at: https://works.bepress.com/wilson_robert/16/
More informationSequential Equilibria of Multi-Stage Games with Infinite Sets of Types and Actions
Sequential Equilibria of Multi-Stage Games with Infinite Sets of Types and Actions by Roger B. Myerson and Philip J. Reny* Draft notes October 2011 http://home.uchicago.edu/~preny/papers/bigseqm.pdf Abstract:
More informationStrategic Independence and Perfect Bayesian Equilibria
journal of economic theory 70, 201234 (1996) article no. 0082 Strategic Independence and Perfect Bayesian Equilibria Pierpaolo Battigalli* Department of Economics, Princeton University, Princeton, New
More informationOpen Sequential Equilibria of Multi-Stage Games with Infinite Sets of Types and Actions
Open Sequential Equilibria of Multi-Stage Games with Infinite Sets of Types and Actions By Roger B. Myerson and Philip J. Reny Department of Economics University of Chicago Paper can be found at https://sites.google.com/site/philipjreny/home/research
More informationEconomics 3012 Strategic Behavior Andy McLennan October 20, 2006
Economics 301 Strategic Behavior Andy McLennan October 0, 006 Lecture 11 Topics Problem Set 9 Extensive Games of Imperfect Information An Example General Description Strategies and Nash Equilibrium Beliefs
More informationGame Theory with Information: Introducing the Witsenhausen Intrinsic Model
Game Theory with Information: Introducing the Witsenhausen Intrinsic Model Michel De Lara and Benjamin Heymann Cermics, École des Ponts ParisTech France École des Ponts ParisTech March 15, 2017 Information
More informationInteractive epistemology in games with payoff uncertainty
Research in Economics 61 (2007) 165 184 www.elsevier.com/locate/rie Interactive epistemology in games with payoff uncertainty Pierpaolo Battigalli a,, Marciano Siniscalchi b,1 a Università Bocconi, IEP
More informationExtensive Form Games I
Extensive Form Games I Definition of Extensive Form Game a finite game tree X with nodes x X nodes are partially ordered and have a single root (minimal element) terminal nodes are z Z (maximal elements)
More informationThe Index of Nash Equilibria
Equilibria in Games, Santiago, Chile January 10, 2017 Finite Normal-Form Games We consider in these lectures, the set of finite games with fixed strategy sets and parametrized by the payoff functions.
More informationPayoff Continuity in Incomplete Information Games
journal of economic theory 82, 267276 (1998) article no. ET982418 Payoff Continuity in Incomplete Information Games Atsushi Kajii* Institute of Policy and Planning Sciences, University of Tsukuba, 1-1-1
More informationSUFFICIENT CONDITIONS FOR STABLE EQUILIBRIA. 1. Introduction
SUFFICIENT CONDITIONS FOR STABLE EQUILIBRIA SRIHARI GOVINDAN AND ROBERT WILSON Abstract. A refinement of the set of Nash equilibria that satisfies two assumptions is shown to select a subset that is stable
More informationRobust Knowledge and Rationality
Robust Knowledge and Rationality Sergei Artemov The CUNY Graduate Center 365 Fifth Avenue, 4319 New York City, NY 10016, USA sartemov@gc.cuny.edu November 22, 2010 Abstract In 1995, Aumann proved that
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 informationEconomics 201A Economic Theory (Fall 2009) Extensive Games with Perfect and Imperfect Information
Economics 201A Economic Theory (Fall 2009) Extensive Games with Perfect and Imperfect Information Topics: perfect information (OR 6.1), subgame perfection (OR 6.2), forward induction (OR 6.6), imperfect
More informationBACKWARD INDUCTION IN GAMES WITHOUT PERFECT RECALL
BACKWARD INDUCTION IN GAMES WITHOUT PERFECT RECALL JOHN HILLAS AND DMITRIY KVASOV Abstract. The equilibrium concepts that we now think of as various forms of backwards induction, namely subgame perfect
More informationHIERARCHIES OF BELIEF AND INTERIM RATIONALIZABILITY
HIERARCHIES OF BELIEF AND INTERIM RATIONALIZABILITY JEFFREY C. ELY AND MARCIN PESKI Abstract. In games with incomplete information, conventional hierarchies of belief are incomplete as descriptions of
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 information1 The General Definition
MS&E 336 Lecture 1: Dynamic games Ramesh Johari April 4, 2007 1 The General Definition A dynamic game (or extensive game, or game in extensive form) consists of: A set of players N; A set H of sequences
More informationOn the Notion of Perfect Bayesian Equilibrium
On the Notion of Perfect Bayesian Equilibrium Julio González-íaz epartamento de Estadística e Investigación Operativa, niversidad de Santiago de Compostela Miguel A. Meléndez-Jiménez epartamento de Teoría
More information6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2
6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2 Daron Acemoglu and Asu Ozdaglar MIT October 14, 2009 1 Introduction Outline Mixed Strategies Existence of Mixed Strategy Nash Equilibrium
More informationIterated Dominance Revisited
Iterated Dominance Revisited Amanda Friedenberg H. Jerome Keisler August 14, 2011 1 Introduction Iterated deletion of strongly dominated strategies has a long tradition in game theory, going back at least
More informationMicroeconomics. 2. Game Theory
Microeconomics 2. Game Theory Alex Gershkov http://www.econ2.uni-bonn.de/gershkov/gershkov.htm 18. November 2008 1 / 36 Dynamic games Time permitting we will cover 2.a Describing a game in extensive form
More informationExtensive games (with perfect information)
Extensive games (with perfect information) (also referred to as extensive-form games or dynamic games) DEFINITION An extensive game with perfect information has the following components A set N (the set
More information6.207/14.15: Networks Lecture 11: Introduction to Game Theory 3
6.207/14.15: Networks Lecture 11: Introduction to Game Theory 3 Daron Acemoglu and Asu Ozdaglar MIT October 19, 2009 1 Introduction Outline Existence of Nash Equilibrium in Infinite Games Extensive Form
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 informationPerfect Conditional -Equilibria of Multi-Stage Games with Infinite Sets of Signals and Actions (Preliminary and Incomplete)
Perfect Conditional -Equilibria of Multi-Stage Games with Infinite Sets of Signals and Actions (Preliminary and Incomplete) Roger B. Myerson and Philip J. Reny Department of Economics University of Chicago
More informationSubjective expected utility in games
Theoretical Economics 3 (2008), 287 323 1555-7561/20080287 Subjective expected utility in games ALFREDO DI TILLIO Department of Economics and IGIER, Università Bocconi This paper extends Savage s subjective
More informationEntropic Selection of Nash Equilibrium
Entropic Selection of Nash Equilibrium Zeynel Harun Alioğulları Mehmet Barlo February, 2012 Abstract This study argues that Nash equilibria with less variations in players best responses are more appealing.
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 informationDEDUCTIVE REASONING IN EXTENSIVE GAMES
DEDUCTIVE REASONING IN EXTENSIVE GAMES GEIR B. ASHEIM AND MARTIN DUFWENBERG Abstract. We justify the application to extensive games of the concept of fully permissible sets, which corresponds to choice
More informationWhat is Backward Induction?
What is ackward duction? dam randenburger y manda Friedenberg z September 9 bstract Which solution concepts satisfy backward induction (I)? We de ne a property we call it Di erence which relates the behavior
More informationRealization Plans for Extensive Form Games without Perfect Recall
Realization Plans for Extensive Form Games without Perfect Recall Richard E. Stearns Department of Computer Science University at Albany - SUNY Albany, NY 12222 April 13, 2015 Abstract Given a game in
More informationSelf-Confirming Games: Unawareness, Discovery, and Equilibrium
Self-Confirming Games: Unawareness, Discovery, and Equilibrium Burkhard C. Schipper Preliminary: April 6, 07 Abstract Equilibrium notions for games with unawareness in the literature cannot be interpreted
More informationStructural Rationality in Dynamic Games
Structural Rationality in Dynamic Games Marciano Siniscalchi May 3, 2016 Abstract The analysis of dynamic games hinges on assumptions about players actions and beliefs at information sets that are not
More informationINFORMATIONAL ROBUSTNESS AND SOLUTION CONCEPTS. Dirk Bergemann and Stephen Morris. December 2014 COWLES FOUNDATION DISCUSSION PAPER NO.
INFORMATIONAL ROBUSTNESS AND SOLUTION CONCEPTS By Dirk Bergemann and Stephen Morris December 2014 COWLES FOUNDATION DISCUSSION PAPER NO. 1973 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY
More informationLevels of Knowledge and Belief Computational Social Choice Seminar
Levels of Knowledge and Belief Computational Social Choice Seminar Eric Pacuit Tilburg University ai.stanford.edu/~epacuit November 13, 2009 Eric Pacuit 1 Introduction and Motivation Informal Definition:
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 informationFUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets
FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be
More informationINVERSE LIMITS AND PROFINITE GROUPS
INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological
More informationWEAKLY DOMINATED STRATEGIES: A MYSTERY CRACKED
WEAKLY DOMINATED STRATEGIES: A MYSTERY CRACKED DOV SAMET Abstract. An informal argument shows that common knowledge of rationality implies the iterative elimination of strongly dominated strategies. Rationality
More informationGuilt aversion in trust games
Guilt aversion in trust games Elena Manzoni Università degli Studi di Milano Bicocca MMRG, 5 May 01 E. Manzoni Bicocca 5/5 1 / 31 Plan of the talk Trust games Experimental results Psychological games Attanasi
More informationOn minimal models of the Region Connection Calculus
Fundamenta Informaticae 69 (2006) 1 20 1 IOS Press On minimal models of the Region Connection Calculus Lirong Xia State Key Laboratory of Intelligent Technology and Systems Department of Computer Science
More informationConsistent Beliefs in Extensive Form Games
Games 2010, 1, 415-421; doi:10.3390/g1040415 OPEN ACCESS games ISSN 2073-4336 www.mdpi.com/journal/games Article Consistent Beliefs in Extensive Form Games Paulo Barelli 1,2 1 Department of Economics,
More informationEconS Advanced Microeconomics II Handout on Subgame Perfect Equilibrium (SPNE)
EconS 3 - Advanced Microeconomics II Handout on Subgame Perfect Equilibrium (SPNE). Based on MWG 9.B.3 Consider the three-player nite game of perfect information depicted in gure. L R Player 3 l r a b
More informationEconomics 703: Microeconomics II Modelling Strategic Behavior 1
Economics 703: Microeconomics II Modelling Strategic Behavior 1 George J. Mailath Department of Economics University of Pennsylvania June 29, 2017 1 Copyright June 29, 2017 by George J. Mailath. Contents
More informationENDOGENOUS REPUTATION IN REPEATED GAMES
ENDOGENOUS REPUTATION IN REPEATED GAMES PRISCILLA T. Y. MAN Abstract. Reputation is often modelled by a small but positive prior probability that a player is a behavioral type in repeated games. This paper
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 informationTree sets. Reinhard Diestel
1 Tree sets Reinhard Diestel Abstract We study an abstract notion of tree structure which generalizes treedecompositions of graphs and matroids. Unlike tree-decompositions, which are too closely linked
More informationGame Theory. Wolfgang Frimmel. Perfect Bayesian Equilibrium
Game Theory Wolfgang Frimmel Perfect Bayesian Equilibrium / 22 Bayesian Nash equilibrium and dynamic games L M R 3 2 L R L R 2 2 L R L 2,, M,2, R,3,3 2 NE and 2 SPNE (only subgame!) 2 / 22 Non-credible
More informationLearning Equilibrium as a Generalization of Learning to Optimize
Learning Equilibrium as a Generalization of Learning to Optimize Dov Monderer and Moshe Tennenholtz Faculty of Industrial Engineering and Management Technion Israel Institute of Technology Haifa 32000,
More informationIterative Weak Dominance and Interval-Dominance Supermodular Games
Iterative Weak Dominance and Interval-Dominance Supermodular Games Joel Sobel April 4, 2016 Abstract This paper extends Milgrom and Robert s treatment of supermodular games in two ways. It points out that
More informationRationalization and Incomplete Information
Rationalization and Incomplete Information Pierpaolo Battigalli Bocconi University and IGIER pierpaolo.battigalli@uni-bocconi.it Marciano Siniscalchi Northwestern University and Princeton University marciano@northwestern.edu
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 informationIntroduction to Topology
Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about
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 informationNash-solvable bidirected cyclic two-person game forms
DIMACS Technical Report 2008-13 November 2008 Nash-solvable bidirected cyclic two-person game forms by Endre Boros 1 RUTCOR, Rutgers University 640 Bartholomew Road, Piscataway NJ 08854-8003 boros@rutcor.rutgers.edu
More informationGuilt in Games. P. Battigalli and M. Dufwenberg (AER, 2007) Presented by Luca Ferocino. March 21 st,2014
Guilt in Games P. Battigalli and M. Dufwenberg (AER, 2007) Presented by Luca Ferocino March 21 st,2014 P. Battigalli and M. Dufwenberg (AER, 2007) Guilt in Games 1 / 29 ADefinitionofGuilt Guilt is a cognitive
More informationIntroduction to Game Theory
COMP323 Introduction to Computational Game Theory Introduction to Game Theory Paul G. Spirakis Department of Computer Science University of Liverpool Paul G. Spirakis (U. Liverpool) Introduction to Game
More informationNTU IO (I) : Classnote 03 Meng-Yu Liang March, 2009
NTU IO (I) : Classnote 03 Meng-Yu Liang March, 2009 Kohlberg and Mertens (Econometrica 1986) We will use the term (game) tree for the extensive form of a game with perfect recall (i.e., where every player
More informationDefinitions and Proofs
Giving Advice vs. Making Decisions: Transparency, Information, and Delegation Online Appendix A Definitions and Proofs A. The Informational Environment The set of states of nature is denoted by = [, ],
More informationRefinements - change set of equilibria to find "better" set of equilibria by eliminating some that are less plausible
efinements efinements - change set of equilibria to find "better" set of equilibria by eliminating some that are less plausible Strategic Form Eliminate Weakly Dominated Strategies - Purpose - throwing
More informationAn axiomatic characterization of the position value for network situations
An axiomatic characterization of the position value for network situations Anne van den Nouweland Marco Slikker September 22, 2011 Abstract Network situations as introduced by Jackson and Wolinsky (1996)
More informationAlgorithms for cautious reasoning in games
Algorithms for cautious reasoning in games Geir B. Asheim a Andrés Perea b October 16, 2017 Abstract We provide comparable algorithms for the Dekel-Fudenberg procedure, iterated admissibility, proper rationalizability
More informationPlayer-Compatible Equilibrium
Player-Compatible Equilibrium Drew Fudenberg Kevin He arxiv:1712.08954v3 [q-fin.ec] 20 Aug 2018 First version: September 23, 2017 This version: August 20, 2018 Abstract Player-Compatible Equilibrium (PCE)
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 informationRefined best-response correspondence and dynamics
Refined best-response correspondence and dynamics Dieter Balkenborg, Josef Hofbauer, and Christoph Kuzmics February 18, 2007 Abstract We study a natural (and, in a well-defined sense, minimal) refinement
More informationThe computational complexity of trembling hand perfection and other equilibrium refinements
The computational complexity of trembling hand perfection and other equilibrium refinements Kristoffer Arnsfelt Hansen 1, Peter Bro Miltersen 1, and Troels Bjerre Sørensen 2 1 {arnsfelt,bromille}@cs.au.dk
More informationCRITICAL TYPES. 1. Introduction
CRITICAL TYPES JEFFREY C. ELY AND MARCIN PESKI Abstract. Economic models employ assumptions about agents infinite hierarchies of belief. We might hope to achieve reasonable approximations by specifying
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationBounded Reasoning and Higher-Order Uncertainty
Bounded Reasoning and Higher-Order Uncertainty Willemien Kets JOB MARKET PAPER November 15, 2010 Abstract Harsanyi type structures, the device traditionally used to model players beliefs in games, generate
More informationExploring the Gap between Perfect Bayesian Equilibrium and Sequential Equilibrium
Article Exploring the Gap between Perfect Bayesian Equilibrium and Sequential Equilibrium Giacomo Bonanno Department of Economics, University of California, Davis, CA 9566-8578, USA; gfbonanno@ucdavis.edu
More informationConfronting Theory with Experimental Data and vice versa. European University Institute. May-Jun Lectures 7-8: Equilibrium
Confronting Theory with Experimental Data and vice versa European University Institute May-Jun 2008 Lectures 7-8: Equilibrium Theory cannot provide clear guesses about with equilibrium will occur in games
More informationWeak Robust (Virtual) Implementation
Weak Robust (Virtual) Implementation Chih-Chun Yang Institute of Economics, Academia Sinica, Taipei 115, Taiwan April 2016 Abstract We provide a characterization of (virtual) implementation in iterated
More informationCW complexes. Soren Hansen. This note is meant to give a short introduction to CW complexes.
CW complexes Soren Hansen This note is meant to give a short introduction to CW complexes. 1. Notation and conventions In the following a space is a topological space and a map f : X Y between topological
More informationMATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X.
MATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X. Notation 2 A set can be described using set-builder notation. That is, a set can be described
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 informationAppendix B for The Evolution of Strategic Sophistication (Intended for Online Publication)
Appendix B for The Evolution of Strategic Sophistication (Intended for Online Publication) Nikolaus Robalino and Arthur Robson Appendix B: Proof of Theorem 2 This appendix contains the proof of Theorem
More informationThe Local Best Response Criterion: An Epistemic Approach to Equilibrium Refinement. Herbert Gintis
The Local est Response Criterion: An Epistemic Approach to Equilibrium Refinement Herbert Gintis February 6, 2009 Abstract The standard refinement criteria for extensive form games, including subgame perfect,
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 informationConservative Belief and Rationality
Conservative Belief and Rationality Joseph Y. Halpern and Rafael Pass Department of Computer Science Cornell University Ithaca, NY, 14853, U.S.A. e-mail: halpern@cs.cornell.edu, rafael@cs.cornell.edu January
More informationPerfect Bayesian Equilibrium
Perfect Bayesian Equilibrium For an important class of extensive games, a solution concept is available that is simpler than sequential equilibrium, but with similar properties. In a Bayesian extensive
More informationGame Theory. Kuhn s Theorem. Bernhard Nebel, Robert Mattmüller, Stefan Wölfl, Christian Becker-Asano
Game Theory Albert-Ludwigs-Universität Freiburg Bernhard Nebel, Robert Mattmüller, Stefan Wölfl, Christian Becker-Asano Research Group Foundations of Artificial Intelligence June 17, 2013 June 17, 2013
More informationEconomics 201B Economic Theory (Spring 2017) Bargaining. Topics: the axiomatic approach (OR 15) and the strategic approach (OR 7).
Economics 201B Economic Theory (Spring 2017) Bargaining Topics: the axiomatic approach (OR 15) and the strategic approach (OR 7). The axiomatic approach (OR 15) Nash s (1950) work is the starting point
More informationEconomics 209B Behavioral / Experimental Game Theory (Spring 2008) Lecture 3: Equilibrium refinements and selection
Economics 209B Behavioral / Experimental Game Theory (Spring 2008) Lecture 3: Equilibrium refinements and selection Theory cannot provide clear guesses about with equilibrium will occur in games with multiple
More informationColumbia University. Department of Economics Discussion Paper Series
Columbia University Department of Economics Discussion Paper Series Equivalence of Public Mixed-Strategies and Private Behavior Strategies in Games with Public Monitoring Massimiliano Amarante Discussion
More informationSolving Extensive Form Games
Chapter 8 Solving Extensive Form Games 8.1 The Extensive Form of a Game The extensive form of a game contains the following information: (1) the set of players (2) the order of moves (that is, who moves
More information2. Prime and Maximal Ideals
18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let
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 informationThe Folk Theorem for Finitely Repeated Games with Mixed Strategies
The Folk Theorem for Finitely Repeated Games with Mixed Strategies Olivier Gossner February 1994 Revised Version Abstract This paper proves a Folk Theorem for finitely repeated games with mixed strategies.
More informationEquational Logic. Chapter Syntax Terms and Term Algebras
Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from
More informationConditional equilibria of multi-stage games with infinite sets of signals and actions
Conditional equilibria of multi-stage games with infinite sets of signals and actions by Roger B. Myerson and Philip J. Reny Department of Economics, University of Chicago Abstract: We develop concepts
More informationGame Theory and Social Psychology
Game Theory and Social Psychology cf. Osborne, ch 4.8 Kitty Genovese: attacked in NY in front of 38 witnesses no one intervened or called the police Why not? \Indierence to one's neighbour and his troubles
More informationRationalizable Partition-Confirmed Equilibrium
Rationalizable Partition-Confirmed Equilibrium Drew Fudenberg and Yuichiro Kamada First Version: January 29, 2011; This Version: July 30, 2014 Abstract Rationalizable partition-confirmed equilibrium (RPCE)
More information