DOCSLIB.ORG

Draft, December 2013 CONTINUOUS TIME GAME THEORY: AN

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Draft, December 2013 CONTINUOUS TIME GAME THEORY: AN Draft, December 2013 CONTINUOUS TIME GAME THEORY: AN INFINITESIMAL APPROACH MAXWELL B. STINCHCOMBE 1 Contents 1. Introduction and Overview . 6 1.1. Equilibrium Refinement via Infinitesimals. 6 1.2. Compact and Continuous Normal Form Games . 7 1.2.1. Limit games and limit equilibria . 8 1.2.2. The need for exhaustiveness . 8 1.2.3. Respecting weak dominance . 8 1.3. Extensive Form Games with Infinite Action Sets . 8 1.4. Control Problems and Differential Games . 8 1.4.1. An Easy Control Problem . 8 1.4.2. A Less Easy Control Problem. 9 1.4.3. Differential Games. 10 1.5. Continuous Time Stochastic Processes and Diffuse Monitoring . 10 1.5.1. Brownian monitoring . 10 1.5.2. Poisson monitoring . 10 1.5.3. Levy processes and monitoring . 11 1.6. Continuous Time Game Theory with Sharp Monitoring . 11 1.7. Large Population Games. 11 2. Constructing Infinitesimals and the Tools to Use Them . 12 2.1. A Purely Finitely Additive Point Mass. 12 2.2. The Equivalence Classes . 13 2.3. Normal Form Equilibrium Refinement . 13 2.3.1. Notation . 13 2.3.2. Perfection . 14 2.3.3. Properness . 14 2.3.4. p-Stability . 14 2.4. The Basic Results . 14 2.5. Compact and Continuous Games . 15 3. Extensive Form Games . 15 3.1. Decision Theory with Full Support Probabilities . 15 3.2. Dynamic Decisions, the Basic Model . 15 3.3. Bridge Crossing . 16 3.4. Heirarchies of Beliefs . 16 3.5. Extensive Form Equilibrium Refinement . 17 3.5.1. Perfect, proper, and stable equilibrium outcomes. 17 3.5.2. Iterative dominance arguments . 17 4. Continuous Time Control and Games . 17 4.1. Control Theory . 17 4.1.1. When it Works Wonderfully . 17 4.1.2. When it Works Less Well . 17 4.2. Games with Instantaneous Monitoring . 18 4.2.1. (R; ≤) is Totally Ordered but Not Well-Ordered . 18 4.2.2. Implications . 18 4.3. Games Played on a Near Interval . 19 2 4.3.1. Actions, Histories, Strategies, and Outcomes . 20 4.3.2. Safety in Continuously Repeated Games . 21 4.3.3. Some Examples. 22 4.3.4. Revisiting Cournot vs. Bertrand . 23 4.3.5. Preemption Games . 23 4.3.6. Wars of Attrition . 25 4.4. Brownian Monitoring . 28 4.5. Poisson Monitoring . 28 4.6. Continuous Time Martingales. 28 4.7. It^o's Lemma . 28 5. Standard and Nonstandard Superstructures . 28 5.1. Purely Finitely Additive Point Masses . 28 ∗ 5.2. The equivalence relation ∼µ and X .................................. 29 5.3. Superstructures. 29 5.4. Defining V (∗S) inductively . 29 5.5. Internal Sets for Stochastic Processes . 30 5.6. Some External Sets . 30 5.7. Statements and the Transfer Principle . 31 6. Some Real Analysis. 32 6.1. Closed Sets and Closure . 32 6.1.1. The standard part mapping. 32 6.1.2. Closedness of Refined Sets of Equilibria. 32 6.2. Continuity and Uniform Continuity . 32 6.2.1. C(X; R), X compact . 32 6.2.2. Near continuity. 32 6.2.3. The Riemann-Stieltjes Integral. 32 6.2.4. Some near interval control theory . 32 6.3. Theorem of the Maximum . 32 6.3.1. In Control Theory . 32 6.3.2. Single person problems . 32 6.3.3. Limit games and limit equilibria . 32 6.4. Compactness . 32 6.4.1. Existence of optima . 32 6.4.2. Existence of equilibria . 32 6.4.3. Existence of extened equilibrium outcomes. 33 6.4.4. Compact sets of probabilities on R................................ 33 6.5. Probabilities on Metric Spaces . 33 6.5.1. Loeb Measures . ..
Load more

Recommended publications
  • Equilibrium Refinements
    Equilibrium Refinements Mihai Manea MIT Sequential Equilibrium I In many games information is imperfect and the only subgame is the original game. subgame perfect equilibrium = Nash equilibrium I Play starting at an information set can be analyzed as a separate subgame if we specify players’ beliefs about at which node they are. I Based on the beliefs, we can test whether continuation strategies form a Nash equilibrium. I Sequential equilibrium (Kreps and Wilson 1982): way to derive plausible beliefs at every information set. Mihai Manea (MIT) Equilibrium Refinements April 13, 2016 2 / 38 An Example with Incomplete Information Spence’s (1973) job market signaling game I The worker knows her ability (productivity) and chooses a level of education. I Education is more costly for low ability types. I Firm observes the worker’s education, but not her ability. I The firm decides what wage to offer her. In the spirit of subgame perfection, the optimal wage should depend on the firm’s beliefs about the worker’s ability given the observed education. An equilibrium needs to specify contingent actions and beliefs. Beliefs should follow Bayes’ rule on the equilibrium path. What about off-path beliefs? Mihai Manea (MIT) Equilibrium Refinements April 13, 2016 3 / 38 An Example with Imperfect Information Courtesy of The MIT Press. Used with permission. Figure: (L; A) is a subgame perfect equilibrium. Is it plausible that 2 plays A? Mihai Manea (MIT) Equilibrium Refinements April 13, 2016 4 / 38 Assessments and Sequential Rationality Focus on extensive-form games of perfect recall with finitely many nodes. An assessment is a pair (σ; µ) I σ: (behavior) strategy profile I µ = (µ(h) 2 ∆(h))h2H: system of beliefs ui(σjh; µ(h)): i’s payoff when play begins at a node in h randomly selected according to µ(h), and subsequent play specified by σ.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • Fall Back Proper Equilibrium
    TOP DOI 10.1007/s11750-017-0447-2 REGULAR PAPER Fall back proper equilibrium John Kleppe1 · Peter Borm1 · Ruud Hendrickx1 Received: 26 August 2015 / Accepted: 26 April 2017 © The Author(s) 2017. This article is an open access publication Abstract Proper equilibrium plays a prominent role in the literature on non- cooperative games. The underlying thought experiment in which the players play a passive role is, however, unsatisfying, as it gives no justification for its fundamen- tal idea that severe mistakes are made with a significantly smaller probability than innocuous ones. In this paper, we introduce a more active role for the players, leading to the refinement of fall back proper equilibrium. Keywords Proper equilibrium · Fall back proper equilibrium Mathematics Subject Classification 91A10 1 Introduction: proper equilibrium and its thought experiment In this paper, we reconsider the concept of proper equilibrium (Myerson 1978)in mixed extensions of finite strategic games, from now on just abbreviated to games. To adequately state our purposes and ideas, we first recall the underlying framework = ( , { } , {πi } ) and basic notation and definitions. A game is given by G N Mi i∈N i∈N , ={ ,..., } ∈ with N 1 n the player set, Mi the mixed strategy space of player i N, i ={ ,..., i } πi : → R with M 1 m the set of pure strategies, and j∈N M j the von Neumann Morgenstern expected payoff function of player i. A pure strategy k ∈ Mi i i of player i is alternatively denoted by ek, a typical element of Mi by x . We denote B Ruud Hendrickx [email protected] 1 CentER and Department of Econometrics and Operations Research, Tilburg University, PO Box 90153, 5000 LE Tilburg, The Netherlands 123 J.
    [Show full text]
  • Refinement1 Outline A. Subgame Perfection Revisited B
    Economics 703 Advanced Microeconomics Prof. Peter Cramton Lecture Note 4: Refinement1 Outline A. Subgame Perfection Revisited B. Sequential Equilibrium 1. Sequential Rationality 2. Consistency 3. Structural Consistency C. Trembling-Hand Perfection D. Strategic Stability 1. Backward Induction and Invariance 2. Admissibility and Iterated Dominance 3. Stable Equilibrium A. Subgame Perfection Revisited The refinements of the Nash equilibrium concept surveyed in this section are all attempts to extend the spirit of Selten's [1965] definition of subgame perfection to games with imperfect information. To begin, we review the effect of subgame perfection in games of perfect information and the problems that arise with imperfect information. Game 1: Normal Form Extensive Form 1 2 l r L R 2 L 1, 1 1, 1 1 1 1 l r R 2, 0 -1,-1 1: 2 -1 2: 0 -1 The normal form reveals that both (L,r) and (R,l) are Nash equilibria, but the former relies on an incredible threat by player 2: 2 threatens to play r if player 1 gives 2 the move; this would be bad for 1, so 1 plays L, but it also would be bad for 2. Given the opportunity to move, 2 prefers to play l, so (R,l) is the only sensible Nash equilibrium. 1These notes are based without restraint on notes by Robert Gibbons, MIT. Recall that a subgame consists of the game tree following a singleton information set, provided the resulting subtree does not cut any information sets. A Nash equilibrium in the original game is subgame perfect if it specifies Nash equilibrium strategies in every subgame.
    [Show full text]
  • A Active Radar, 1103 Adaptive Adversary, 1259 Adaptive Dynamics
    Index A G-atom, 297 Active radar, 1103 Attack strategy, 1188, 1207, 1214 Adaptive adversary, 1259 Attenuation level, 434, 437, 444 Adaptive dynamics, 477–484, 489, 490, Aubin, J.-P., 971 502, 503 Auction, 1187, 1197, 1204, 1206 asymmetric game, 502 Augmented proportional navigation, 1081 multi-dimensional, 489 Automatic generation control (AGC), 1189, Adaptive utility, 1260 1207, 1216, 1217, 1220 Additive reward and additive transitions Auxiliary problem, 438 (ARAT) game, 295 Availability, 1207 Adjoint equation, 645 Average Stackelberg cost, 43 Adjoint variable, 381 Averaging dynamics, 911–913 Admissible control, 163, 195 Avoidance, 1001 Advertising games, 869–875 Aggregative game, 589 Agreement, 637, 647–649 B Aircraft Ba¸sar, T., 1003 control, 957 Backward constructions, 953 landing, 957 Backward induction, 498 take-off, 957 Backward stochastic differential Airport security, 1250–1251 equations, 405 AK model, 180 Bankruptcy games, 829, 837–845 Algebraic Riccati equation, 1095 Barrier, see Barrier line Algebraic sum, see Minkowski sum Barrier line, 968, 970, 973, 977, 981, 986, 987, Altruism 990, 993, 995, 997 non-paternalistic, 327 Battery degradation, 1200 paternalistic, 323 Battle of the sexes, 1054 Amazon Mechanical Turk (AMT), 1257 Bayesian games, 55 Angular frequency, 1216, 1217 Bayesian Nash equilibrium, 56 Anonymous sequential games, 351 Bayesian Stackelberg game, 1229, Anti-Bourgeois, 1056 1250, 1254 Approachable set, 267, 268 Behavioral models, 1244, 1257, 1259, 1266 Approximating sets, 756 Behavioral Stackelberg equilibrium, 35 ARMOR, 1225, 1228, 1236, 1251 Behavior strategy, 750 ASPEN, 1237–1239 Belief-free equilibrium, 754 Asymmetric games, 1053 Bellman equations, 173, 176, 196, 203, Asymptotically stable, 1049 207, 646 Asymptotic uniform value, 271 Ben-Asher, J.Z., 1029 Asymptotic value, 255, 266 Bequest games, 742, 761 Asynchronous model, 921 Bernhard, P., 953, 956, 992 © Springer International Publishing AG, part of Springer Nature 2018 1271 T.
    [Show full text]
  • Equilibrium Computation — Dagstuhl Seminar —
    07471 Abstracts Collection Equilibrium Computation | Dagstuhl Seminar | Jean-Jacques Herings1, Marcin Jurdzinski2, Peter Bro Miltersen3, Eva Tardos4 and Bernhard von Stengel5 1 Maastricht, [email protected] 2 Warwick, [email protected] 3 Aarhus, [email protected] 4 Cornell, [email protected] 5 LSE, [email protected] Abstract. From 18 to 23 November 2007, the Dagstuhl Seminar 07471 \Equilibrium Computation" was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, sev- eral participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given dur- ing the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available. Keywords. Equilibrium, algorithm, polynomial time, game theory, eco- nomics The purpose of this Dagstuhl seminar was to bring together researchers from different disciplines working on algorithmic problems of finding equilibria. As explained in the Motivation for this seminar, the different topics were (with talks given by) { Parity Games, Andersson (junior researcher), Gimbert (junior researcher), Gr¨adel(survey talk), Svensson (junior researcher), Zwick (survey talk), { Complexity of Finding Equilibria, Elkind (junior researcher), Etessami (sur- vey talk), Goldberg (introductory survey talk on the class PPAD, not listed below), Fabrikant (junior researcher), Hoefer (junior researcher), Monien, V¨ocking (survey talk), { Mathematical Programming, Halman, Morris, Theobald, { Economic Equilibria, Heydenreich (junior researcher), Jain, Peeters (survey talk), { Game Theory, Balthasar (junior researcher), Goldberg, Jiang (junior re- searcher), Sørensen (junior researcher), Turocy (survey talk), Vermeulen, von Stengel.
    [Show full text]
  • Equilibrium Selection in Multi-Player Games with Auction Applications∗
    Equilibrium Selection in Multi-Player Games with Auction Applications∗ Paul Milgrom Joshua Mollner May 23, 2014 Abstract We introduce two new equilibrium refinements for finite normal form games, both of which incorporate the intuitive idea that a costless deviation by one player is more likely than a costly deviation by the same or another player. These refinements lead to new restrictions in games with three or more players and select interesting equilibria in some well-known auction games. 1 Introduction This paper introduces two new refinements of Nash equilibrium. Both in- corporate the restriction that a deviation by one player to a best response is more likely than a deviation by the same or another player to a strategy that is not a best response. One incorporates the additional restriction that the same deviation is also more likely than any combination of deviations by multiple players. Neither restriction is implied by Myerson's (1978) proper equilibrium in games with three or more players, and these restrictions can be consequential. In two well known auction games { the menu auction and the generalized second-price auction { some of the main conclusions that were originally derived using idiosyncratic equilibrium selection criteria can instead be derived using our new refinements. To motivate our equilibrium refinements, consider the three-player game in Figure 1. Each player has two strategies. The \Geo" player picks the pay- off matrix { East or West. Geo's payoff is always zero, regardless of what ∗For helpful comments, we thank Gabriel Carroll, Piotr Dworczak, Michael Ostrovsky, Bernhard von Stengel, Alex Wolitzky, and participants in seminars at Stanford.
    [Show full text]
  • The Price of Anarchy for Network Formation in an Adversary Model
    Games 2011, 2, 302-332; doi:10.3390/g2030302 OPEN ACCESS games ISSN 2073-4336 www.mdpi.com/journal/games Article The Price of Anarchy for Network Formation in an Adversary Model Lasse Kliemann Department of Computer Science, Christian-Albrechts-Universitat¨ zu Kiel, Christian-Albrechts-Platz 4, Kiel 24098, Germany; E-Mail: [email protected]; Tel.: +49-431-880-7454; Fax: +49-431-880-1725 Received: 27 May 2011; in revised form: 18 July 2011 / Accepted: 8 August 2011 / Published: 23 August 2011 Abstract: We study network formation with n players and link cost α > 0. After the network is built, an adversary randomly deletes one link according to a certain probability distribution. Cost for player v incorporates the expected number of players to which v will become disconnected. We focus on unilateral link formation and Nash equilibrium. We show existence of Nash equilibria and a price of stability of 1 + o(1) under moderate assumptions on the adversary and n ≥ 9. We prove bounds on the price of anarchy for two special adversaries: one removes a link chosen uniformly at random, while the other removes a link that causes a maximum number of player pairs to be separated. We show an O(1) bound on the price of anarchy for both adversaries, the constant being bounded by 15 + o(1) and 9 + o(1), respectively. Keywords: network formation; equilibrium; price of anarchy; unilateral link formation; adversary model; network robustness 1. Network Formation In network formation, a multitude of individuals, called players, form a network in such a way that each player decides for herself to which other players she would like to connect.
    [Show full text]
  • An Epistemic Approach to Equilibrium Refinement Herbert Gintis
    The Local Best Response Criterion: An Epistemic Approach to Equilibrium Refinement Herbert Gintis February 6, 2009 Abstract The standard refinement criteria for extensive form games, including sub- game perfect, perfect, perfect Bayesian, sequential, and proper, reject im- portant classes of reasonable Nash equilibria and accept many unreasonable Nash equilibria. This paper develops a new refinement criterion, based on epistemic game theory, that captures the concept of a Nash equilibriumthat is plausible when players are rational. I call this the local best response (LBR) criterion. This criterion is conceptually simpler than the standard refinement criteria because it does not depend on out-of-equilibrium, counterfactual, or passage to the limit arguments. The LBR is also informationally richer because it clarifies the epistemic conditions that render a Nash equilibrium reasonable. The LBR criterion appears to render the traditional refinement criteria superfluous. 1 Introduction A Nash equilibrium refinement of an extensive form game is a criterion that ap- plies to all Nash equilibria that are deemed reasonable, but fails to apply to other Nash equilibria that are deemed unreasonable, based on our informal understand- ing of how rational individuals might play the game. A voluminous literature has developed in search of an adequate equilibrium refinement criterion. A number of criteria have been proposed, including subgame perfect, perfect, perfect Bayesian, sequential, and proper equilibrium (Harsanyi, 1967; Myerson, 1978; Selten, 1980, Kreps and Wilson 1982; Kohlberg and Mertens, 1986), that introduce player error, Santa Fe Institute and Central European University. I would like to thank Robert Aumann, Jeffrey Ely, Larry Samuelson, and Fernando Vega-Redondo for their help, and the European Science Foundation for financial support.
    [Show full text]
  • Tommy C. Tan Sergio R.D.C. Werlang Graduate School of Business IMPA & University of Chicago Getulio Vargas Foundation Chicag
    A GUIDE TO KNOWLEDGE AND GAMES Tommy C. Tan Sergio R.D.C. Werlang Graduate School of Business IMPA & University of Chicago Getulio Vargas Foundation Chicago, Illinois Rio de Janeiro, Brazil ASTRACT This paper serves as an informal introduction to some of the main concepts in non-cooperative games as well as a guide to where more careful treatments can be found. We pay particular attention to the explicit and implicit role of knowledge and information in games. No attempt is made to be formal and exhaustive. The interested reader should refer to the excellent book of Luce and Raiffa (195) which covers the classical material exhaustively and to the almost encyclopaedic lecture note of Van Damme (1983) which covers the vast literature on refinements of solution concepts. Games of incomplete information games are dealt with in Harsanyi (1967-68) and Myerson (1985a) at a foundational level. Tan and Werlang (1984, 1986) provides an environment in which games and the knowledge of players in games are explicitly modelled and several solution concepts are derived from axioms placed on the knowledge of players. As this note was prepared in a short time, some of the notation used may appear to be cryptic and unexplained. The intention is to go through these notes at the conference to clarify the points. The notation however, is more or less standard in literature on games. 163 164 Tutorial Session The Nash Equilibrium There are n players in a game identified by the index i. A normal form game (see Luce and Raiffa or Van Damme) is given by {xi, Ai} for i=l ....
    [Show full text]