Part II Dynamic Games of Complete Information
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Game Theory: Static and Dynamic Games of Incomplete Information
Game Theory: Static and Dynamic Games of Incomplete Information Branislav L. Slantchev Department of Political Science, University of California – San Diego June 3, 2005 Contents. 1 Static Bayesian Games 2 1.1BuildingaPlant......................................... 2 1.1.1Solution:TheStrategicForm............................ 4 1.1.2Solution:BestResponses.............................. 5 1.2Interimvs.ExAntePredictions............................... 6 1.3BayesianNashEquilibrium.................................. 6 1.4 The Battle of the Sexes . 10 1.5AnExceedinglySimpleGame................................ 12 1.6TheLover-HaterGame.................................... 13 1.7TheMarketforLemons.................................... 16 2 Dynamic Bayesian Games 18 2.1ATwo-PeriodReputationGame............................... 19 2.2Spence’sEducationGame.................................. 22 3 Computing Perfect Bayesian Equilibria 24 3.1TheYildizGame........................................ 24 3.2TheMyerson-RosenthalGame................................ 25 3.3One-PeriodSequentialBargaining............................. 29 3.4AThree-PlayerGame..................................... 29 3.5RationalistExplanationforWar............................... 31 Thus far, we have only discussed games where players knew about each other’s utility func- tions. These games of complete information can be usefully viewed as rough approximations in a limited number of cases. Generally, players may not possess full information about their opponents. In particular, players may possess -
Minimax TD-Learning with Neural Nets in a Markov Game
Minimax TD-Learning with Neural Nets in a Markov Game Fredrik A. Dahl and Ole Martin Halck Norwegian Defence Research Establishment (FFI) P.O. Box 25, NO-2027 Kjeller, Norway {Fredrik-A.Dahl, Ole-Martin.Halck}@ffi.no Abstract. A minimax version of temporal difference learning (minimax TD- learning) is given, similar to minimax Q-learning. The algorithm is used to train a neural net to play Campaign, a two-player zero-sum game with imperfect information of the Markov game class. Two different evaluation criteria for evaluating game-playing agents are used, and their relation to game theory is shown. Also practical aspects of linear programming and fictitious play used for solving matrix games are discussed. 1 Introduction An important challenge to artificial intelligence (AI) in general, and machine learning in particular, is the development of agents that handle uncertainty in a rational way. This is particularly true when the uncertainty is connected with the behavior of other agents. Game theory is the branch of mathematics that deals with these problems, and indeed games have always been an important arena for testing and developing AI. However, almost all of this effort has gone into deterministic games like chess, go, othello and checkers. Although these are complex problem domains, uncertainty is not their major challenge. With the successful application of temporal difference learning, as defined by Sutton [1], to the dice game of backgammon by Tesauro [2], random games were included as a standard testing ground. But even backgammon features perfect information, which implies that both players always have the same information about the state of the game. -
Game Theory with Sparsity-Based Bounded Rationality∗
Game Theory with Sparsity-Based Bounded Rationality Xavier Gabaix NYU Stern, CEPR and NBER March 14, 2011 Extremely preliminary and incomplete. Please do not circulate. Abstract This paper proposes a way to enrich traditional game theory via an element of bounded rationality. It builds on previous work by the author in the context of one-agent deci- sion problems, but extends it to several players and a generalization of the concept of Nash equilibrium. Each decision maker builds a simplified representation of the world, which includes only a partial understanding of the equilibrium play of the other agents. The agents’desire for simplificity is modeled as “sparsity” in a tractable manner. The paper formulates the model in the concept of one-shot games, and extends it to sequen- tial games and Bayesian games. It applies the model to a variety of exemplary games: p beauty contests, centipede, traveler’sdilemma, and dollar auction games. The model’s predictions are congruent with salient facts of the experimental evidence. Compared to previous succesful proposals (e.g., models with different levels of thinking or noisy decision making), the model is particularly tractable and yields simple closed forms where none were available. A sparsity-based approach gives a portable and parsimonious way to inject bounded rationality into game theory. Keywords: inattention, boundedly rational game theory, behavioral modeling, sparse repre- sentations, `1 minimization. [email protected]. For very good research assistance I am grateful to Tingting Ding and Farzad Saidi. For helpful comments I thank seminar participants at NYU. I also thank the NSF for support under grant DMS-0938185. -
Bayesian Games Professors Greenwald 2018-01-31
Bayesian Games Professors Greenwald 2018-01-31 We describe incomplete-information, or Bayesian, normal-form games (formally; no examples), and corresponding equilibrium concepts. 1 A Bayesian Model of Interaction A Bayesian, or incomplete information, game is a generalization of a complete-information game. Recall that in a complete-information game, the game is assumed to be common knowledge. In a Bayesian game, in addition to what is common knowledge, players may have private information. This private information is captured by the notion of an epistemic type, which describes a player’s knowledge. The Bayesian-game formalism makes two simplifying assumptions: • Any information that is privy to any of the players pertains only to utilities. In all realizations of a Bayesian game, the number of players and their actions are identical. • Players maintain beliefs about the game (i.e., about utilities) in the form of a probability distribution over types. Prior to receiving any private information, this probability distribution is common knowledge: i.e., it is a common prior. After receiving private information, players conditional on this information to update their beliefs. As a consequence of the com- mon prior assumption, any differences in beliefs can be attributed entirely to differences in information. Rational players are again assumed to maximize their utility. But further, they are assumed to update their beliefs when they obtain new information via Bayes’ rule. Thus, in a Bayesian game, in addition to players, actions, and utilities, there is a type space T = ∏i2[n] Ti, where Ti is the type space of player i.There is also a common prior F, which is a probability distribution over type profiles. -
Outline for Static Games of Complete Information I
Outline for Static Games of Complete Information I. Definition of a game II. Examples III. Definition of Nash equilibrium IV. Examples, continued V. Iterated elimination of dominated strategies VI. Mixed strategies VII. Existence theorem on Nash equilibria VIII. The Hotelling model and extensions Copyright © 2004 by Lawrence M. Ausubel Definition: An n-player, static game of complete information consists of an n-tuple of strategy sets and an n-tuple of payoff functions, denoted by G = {S1, … , Sn; u1, … , un} Si, the strategy set of player i, is the set of all permissible moves for player i. We write si ∈ Si for one of player i’s strategies. ui, the payoff function of player i, is the utility, profit, etc. for player i, and depends on the strategies chosen by all the players: ui(s1, … , sn). Example: Prisoners’ Dilemma Prisoner II Remain Confess Silent Remain –1 , –1 –5 , 0 Prisoner Silent I Confess 0, –5 –4 ,–4 Example: Battle of the Sexes F Boxing Ballet Boxing 2, 1 0, 0 M Ballet 0, 0 1, 2 Definition: A Nash equilibrium of G (in pure strategies) consists of a strategy for every player with the property that no player can improve her payoff by unilaterally deviating: (s1*, … , sn*) with the property that, for every player i: ui (s1*, … , si–1*, si*, si+1*, … , sn*) ≥ ui (s1*, … , si–1*, si, si+1*, … , sn*) for all si ∈ Si. Equivalently, a Nash equilibrium is a mutual best response. That is, for every player i, si* is a solution to: ***** susssssiiiiin∈arg max{ (111 , ... ,−+ , , , ... , )} sSii∈ Example: Prisoners’ Dilemma Prisoner II -
Efficiency and Welfare in Economies with Incomplete Information∗
Efficiency and Welfare in Economies with Incomplete Information∗ George-Marios Angeletos Alessandro Pavan MIT, NBER, and FRB of Minneapolis Northwestern University September 2005. Abstract This paper examines a class of economies with externalities, strategic complementarity or substitutability, and incomplete information. We first characterize efficient allocations and com- pare them to equilibrium. We show how the optimal degree of coordination and the efficient use of information depend on the primitives of the environment, and how this relates to the welfare losses due to volatility and heterogeneity. We next examine the social value of informa- tion in equilibrium. When the equilibrium is efficient, welfare increases with the transparency of information if and only if agents’ actions are strategic complements. When the equilibrium is inefficient, additional effects are introduced by the interaction of externalities and informa- tion. We conclude with a few applications, including investment complementarities, inefficient fluctuations, and market competition. Keywords: Social value of information, coordination, higher-order beliefs, externalities, transparency. ∗An earlier version of this paper was entitled “Social Value of Information and Coordination.” For useful com- ments we thank Daron Acemoglu, Gadi Barlevi, Olivier Blanchard, Marco Bassetto, Christian Hellwig, Kiminori Matsuyama, Stephen Morris, Thomas Sargent, and especially Ivàn Werning. Email Addresses: [email protected], [email protected]. 1Introduction Is coordination socially desirable? Do markets use available information efficiently? Is the private collection of information good from a social perspective? What are the welfare effects of the public information disseminated by prices, market experts, or the media? Should central banks and policy makers disclose the information they collect and the forecasts they make about the economy in a transparent and timely fashion, or is there room for “constructive ambiguity”? These questions are non-trivial. -
Learning About Bayesian Games
Learning to Play Bayesian Games1 Eddie Dekel, Drew Fudenberg and David K. Levine2 First draft: December 23, 1996 Current revision: July 13, 2002 Abstract This paper discusses the implications of learning theory for the analysis of games with a move by Nature. One goal is to illuminate the issues that arise when modeling situations where players are learning about the distribution of Nature’s move as well as learning about the opponents’ strategies. A second goal is to argue that quite restrictive assumptions are necessary to justify the concept of Nash equilibrium without a common prior as a steady state of a learning process. 1 This work is supported by the National Science Foundation under Grants 99-86170, 97-30181, and 97- 30493. We are grateful to Pierpaolo Battigalli, Dan Hojman and Adam Szeidl for helpful comments. 2 Departments of Economics: Northwestern and Tel Aviv Universities, Harvard University and UCLA. 2 1. Introduction This paper discusses the implications of learning theory for the analysis of games with a move by Nature.3 One of our goals is to illuminate some of the issues involved in modeling players’ learning about opponents’ strategies when the distribution of Nature’s moves is also unknown. A more specific goal is to investigate the concept of Nash equilibrium without a common prior, in which players have correct and hence common beliefs about one another’s strategies, but disagree about the distribution over Nature’s moves. This solution is worth considering given the recent popularity of papers that apply it, such as Banerjee and Somanathan [2001], Piketty [1995], and Spector [2000].4 We suggest that Nash equilibrium without a common prior is difficult to justify as the long- run result of a learning process, because it takes very special assumptions for the set of such equilibria to coincide with the set of steady states that could arise from learning. -
Heiko Rauhut: Game Theory. In: "The Oxford Handbook on Offender
Game theory Contribution to “The Oxford Handbook on Offender Decision Making”, edited by Wim Bernasco, Henk Elffers and Jean-Louis van Gelder —accepted and forthcoming— Heiko Rauhut University of Zurich, Institute of Sociology Andreasstrasse 15, 8050 Zurich, Switzerland [email protected] November 8, 2015 Abstract Game theory analyzes strategic decision making of multiple interde- pendent actors and has become influential in economics, political science and sociology. It provides novel insights in criminology, because it is a universal language for the unification of the social and behavioral sciences and allows deriving new hypotheses from fundamental assumptions about decision making. The first part of the chapter reviews foundations and assumptions of game theory, basic concepts and definitions. This includes applications of game theory to offender decision making in different strate- gic interaction settings: simultaneous and sequential games and signaling games. The second part illustrates the benefits (and problems) of game theoretical models for the analysis of crime and punishment by going in- depth through the “inspection game”. The formal analytics are described, point predictions are derived and hypotheses are tested by laboratory experiments. The article concludes with a discussion of theoretical and practical implications of results from the inspection game. 1 Foundations of game theory Most research on crime acknowledges that offender decision making does not take place in a vacuum. Nevertheless, most analytically oriented research applies decision theory to understand offenders. Tsebelis (1989) illustrates why this is a problem by using two examples: the decision to stay at home when rain is 1 probable and the decision to speed when you are in a hurry. -
Lecture 3 1 Introduction to Game Theory 2 Games of Complete
CSCI699: Topics in Learning and Game Theory Lecture 3 Lecturer: Shaddin Dughmi Scribes: Brendan Avent, Cheng Cheng 1 Introduction to Game Theory Game theory is the mathematical study of interaction among rational decision makers. The goal of game theory is to predict how agents behave in a game. For instance, poker, chess, and rock-paper-scissors are all forms of widely studied games. To formally define the concepts in game theory, we use Bayesian Decision Theory. Explicitly: • Ω is the set of future states. For example, in rock-paper-scissors, the future states can be 0 for a tie, 1 for a win, and -1 for a loss. • A is the set of possible actions. For example, the hand form of rock, paper, scissors in the game of rock-paper-scissors. • For each a ∈ A, there is a distribution x(a) ∈ Ω for which an agent believes he will receive ω ∼ x(a) if he takes action a. • A rational agent will choose an action according to Expected Utility theory; that is, each agent has their own utility function u :Ω → R, and chooses an action a∗ ∈ A that maximizes the expected utility . ∗ – Formally, a ∈ arg max E [u(ω)]. a∈A ω∼x(a) – If there are multiple actions that yield the same maximized expected utility, the agent may randomly choose among them. 2 Games of Complete Information 2.1 Normal Form Games In games of complete information, players act simultaneously and each player’s utility is determined by his actions as well as other players’ actions. The payoff structure of 1 2 GAMES OF COMPLETE INFORMATION 2 the game (i.e., the map from action profiles to utility vectors) is common knowledge to all players in the game. -
Minimax AI Within Non-Deterministic Environments
Minimax AI within Non-Deterministic Environments by Joe Dan Davenport, B.S. A Thesis In Computer Science Submitted to the Graduate Faculty Of Texas Tech University in Partial Fulfillment of The Requirements for The Degree of MASTER OF SCIENCE Approved Dr. Nelson J. Rushton Committee Chair Dr. Richard Watson Dr. Yuanlin Zhang Mark Sheridan, Ph.D. Dean of the Graduate School August, 2016 Copyright 2016, Joe Dan Davenport Texas Tech University, Joe Dan Davenport, August 2016 Acknowledgements There are a number of people who I owe my sincerest thanks for arriving at this point in my academic career: First and foremost I wish to thank God, without whom many doors that were opened to me would have remained closed. Dr. Nelson Rushton for his guidance both as a scientist and as a person, but even more so for believing in me enough to allow me to pursue my Master’s Degree when my grades didn’t represent that I should. I would also like to thank Dr. Richard Watson who originally enabled me to come to Texas Tech University and guided my path. I would also like to thank Joshua Archer, and Bryant Nelson, not only for their companionship, but also for their patience and willingness to listen to my random thoughts; as well as my manger, Eric Hybner, for bearing with me through this. Finally I would like to thank some very important people in my life. To my Mother, I thank you for telling me I could be whatever I wanted to be, and meaning it. To Israel Perez for always being there for all of us. -
Bayesian Games with Intentions
Bayesian Games with Intentions Adam Bjorndahl Joseph Y. Halpern Rafael Pass Carnegie Mellon University Cornell University Cornell University Philosophy Computer Science Computer Science Pittsburgh, PA 15213 Ithaca, NY 14853 Ithaca, NY 14853 [email protected] [email protected] [email protected] We show that standard Bayesian games cannot represent the full spectrum of belief-dependentprefer- ences. However, by introducing a fundamental distinction between intended and actual strategies, we remove this limitation. We define Bayesian games with intentions, generalizing both Bayesian games and psychological games [5], and prove that Nash equilibria in psychological games correspond to a special class of equilibria as defined in our setting. 1 Introduction Type spaces were introduced by John Harsanyi [6] as a formal mechanism for modeling games of in- complete information where there is uncertainty about players’ payoff functions. Broadly speaking, types are taken to encode payoff-relevant information, a typical example being how each participant val- ues the items in an auction. An important feature of this formalism is that types also encode beliefs about types. Thus, a type encodes not only a player’s beliefs about other players’ payoff functions, but a whole belief hierarchy: a player’s beliefs about other players’ beliefs, their beliefs about other players’ beliefs about other players’ beliefs, and so on. This latter point has been enthusiastically embraced by the epistemic game theory community, where type spaces have been co-opted for the analysis of games of complete information. In this context, types encode beliefs about the strategies used by other players in the game as well as their types. -
6 Dynamic Games with Incomplete Information
22 November 2005. Eric Rasmusen, [email protected]. Http://www.rasmusen.org. 6 Dynamic Games with Incomplete Information 6.1 Perfect Bayesian Equilibrium: Entry Deterrence II and III Asymmetric information, and, in particular, incomplete information, is enormously impor- tant in game theory. This is particularly true for dynamic games, since when the players have several moves in sequence, their earlier moves may convey private information that is relevant to the decisions of players moving later on. Revealing and concealing information are the basis of much of strategic behavior and are especially useful as ways of explaining actions that would be irrational in a nonstrategic world. Chapter 4 showed that even if there is symmetric information in a dynamic game, Nash equilibrium may need to be refined using subgame perfectness if the modeller is to make sensible predictions. Asymmetric information requires a somewhat different refine- ment to capture the idea of sunk costs and credible threats, and Section 6.1 sets out the standard refinement of perfect bayesian equilibrium. Section 6.2 shows that even this may not be enough refinement to guarantee uniqueness and discusses further refinements based on out-of- equilibrium beliefs. Section 6.3 uses the idea to show that a player’s ignorance may work to his advantage, and to explain how even when all players know something, lack of common knowledge still affects the game. Section 6.4 introduces incomplete information into the repeated prisoner’s dilemma and shows the Gang of Four solution to the Chain- store Paradox of Chapter 5. Section 6.5 describes the celebrated Axelrod tournament, an experimental approach to the same paradox.