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Game Theory - Article

 A comprehensive chronology spanning ancient and modern times. [Details...]
 This is a revised version of my paper with the same title published in Games and Economic Behavior, 28 (1999), 155-170. The paper summarizes my experience in teaching an undergraduate course in game theory in 1998 and in 1999. Students were required to submit two types of problem sets:pre-class problem sets, which served as experiments, and post-class problem sets, which require the students to study and apply the solution concepts taught in the course. The sharp distinction between the two types of problem sets emphasizes the limited relevance of game theory as a tool for making predictions and giving advice. The paper summarizes the results of 43 experiments which were conducted during the course. It is argued that the crude experimental methods produced results which are not substantially different from those obtained at much higher cost using stricter experimental methods. [Details...]
 Encyclopedia Entry: Game theory is the science of strategy. It attempts to determine mathematically and logically the actions that "players" should take to secure the best outcomes for themselves in a wide array of "games." [Details...]
 Overview of game theory, including the elements of a game, a game theory framework, bimatrix games, extensive form games, strategic form game representation, signaling and threats, and auctions. [Details...]
 We'll look at noncooperative games which are played only once, which involve only a finite number of players, and which give each player only a finite number of actions to choose from. We'll consider what is called the strategic (or normal) form of a game. Although our formal treatment will be more general, our exemplary paradigm will be a two-person, simultaneous-move matrix game. [Details...]
 To begin our analysis we discuss the concept of strategic dominance. Then we will turn to the more precisely relevant concept of "never a best response." These notions will be the foundation for our study of nonequilibrium solution concepts. These concepts are nonequilibrium in the sense that they typically admit outcomes which are not Nash equilibria. If we can make a useful prediction using only nonequilibrium analysis, our conclusion can be much more compelling than if we had achieved the same result using the much stronger (and frequently more dubious) assumptions required by equilibrium analysis. Furthermore, by applying nonequilibrium techniques in our initial analysis of a game we will frequently greatly simplify our subsequent equilibrium analysis. [Details...]
 By assuming that the players' rationality is common knowledge, we can justify an iterative process of outcome rejection--the iterated elimination of strictly dominated strategies--which can often sharpen our predictions. Outcomes which do not survive this process of elimination cannot plausibly be played when the rationality of the players is common knowledge. A similar, and weakly stronger, process--the iterated elimination of strategies which are never best responses--leads to the solution concept of rationalizability. The surviving outcomes of this process constitute the set of rationalizable outcomes. Each such outcome is a plausible result (and these are the only plausible results)when the players' rationality is common knowledge. In two-player games the set of rationalizable outcomes is exactly the set of outcomes which survive the iterated elimination of strictly dominated strategies. In three-or-more-player games, the set of rationalizable outcomes can be strictly smaller than the set of outcomes which survives the iterated elimination of strictly dominated strategies. [Details...]
 When rational players correctly forecast the strategies of their opponents they are not merely playing best responses to their beliefs about their opponents' play; they are playing best responses to the actual play of their opponents. When all players correctly forecast their opponents' strategies, and play best responses to these forecasts, the resulting strategy profile is a Nash equilibrium. [Details...]
 We'll now see explicitly how to find the set of (mixed-strategy) Nash equilibria for general two-player games where each player has a strategy space containing two actions (i.e. a "2x2 matrix game"). We first compute the best-response correspondence for a player. We partition the possibilites into three cases: The player is completely indifferent; she has a dominant strategy; or, most interestingly, she plays strategically (i.e., based upon her beliefs about her opponent's play). [Details...]
 When we model a strategic economic situation we want to capture as much of the relevant detail as tractably possible. A game can have a complex temporal and information structure; and this structure could well be very significant to understanding the way the game will be played. These structures are not acknowledged explicitly in the game's strategic form, so we seek a more inclusive formulation. It would be desirable to include at least the following: 1) the set of players, 2) who moves when and under what circumstances, 3) what actions are available to a player when she is called upon to move, 4) what she knows when she is called upon to move, and 5) what payoff each player receives when the game is played in a particular way. We can incorporate all of these features within an extensive-form description of the game. [Details...]
 We define a strategy for a player in an extensive-form game as a specification for each of her information sets of the (pure or mixed) action she would take at that information set. One such strategy for each player constitutes a strategy profile for the extensive-form game. [Details...]
 We define the solution concept of subgame-perfect equilibrium as a refinement of Nash equilibrium that imposes the desired dynamic consistency. A subgame-perfect equilibrium of an extensive-form game is a behavior-strategy profile whose restriction to each subgame is a Nash equilibrium of that subgame. [Details...]
 One striking feature of many one-shot games we study (e.g., the Prisoners' Dilemma) is that the Nash equilibria are so noncooperative: each player would prefer to fink than to cooperate. Repeated games can incorporate phenomena which we believe are important but which aren't captured when we restrict our attention to static, one-shot games. In particular we can strive to explain how cooperative behavior can be established as a result of rational behavior. We will develop a useful formalism, the semiextensive form, for analyzing repeated games, i.e. those which are repetitions of the same one-shot game (called the stage game). We will describe strategies for such repeated games as sequences of history-dependent stage-game strategies. The payoffs to the players in this repeated game will be functions of the stage-game payoffs. We will define the concept of Nash equilibrium and after identifying the subgames in this formalism the concept of a subgame-perfect equilibrium for a repeated game. [Details...]
 Infinite repetitions of the stage game potentially pose a problem: a player's repeated-game payoff may be infinite. We ensure the finiteness of the repeated-game payoffs by introducing discounting of future payoffs relative to earlier payoffs. Such discounting can be an expression of time preference and/or uncertainty about the length of the game. We introduce the average discounted payoff as a convenience which normalizes the repeated-game payoffs to be "on the same scale" as the stage-game payoffs. [Details...]
 The original motivation for developing a theory of repeated games was to show that cooperative behavior was an equilibrium. The theoreticians were all too clever, for, as we will see, they showed that in many cases a huge multiplicity of even very "noncooperative" stage-game payoffs could be sustained on average as an equilibrium of the repeated game. These findings are made precise in numerous folk theorems. Each folk theorem considers a class of games and identifies a set of payoff vectors each of which can be supported by some equilibrium strategy profile. There are many folk theorems because there are many classes of games and different choices of equilibrium concept. Some folk theorems identify sets of payoff vectors which can be supported by Nash equilibria; of course, of more interest are those folk theorems which identify payoffs supported by subgame-perfect equilibria. [Details...]
 In many economically important situations the game may begin with some player having private information about something relevant to her decision making. These are called games of incomplete information, or Bayesian games. (Incomplete information is not to be confused with imperfect information in which players do not perfectly observe the actions of other players.) Although any given player does not know the private information of an opponent, she will have some beliefs about what the opponent knows, and we will assume that these beliefs are common knowledge. [Details...]
 We consider here the simplest dynamic games of incomplete information: sender-receiver games. There are only two players: a Sender (S) and a Receiver (R). The Sender's action will be to send a message, m, chosen from a message space M to the Receiver. The Receiver will observe this message m and respond to it by choosing an action a from his action space A. [Details...]
 The concept of Perfect Bayesian equilibrium for extensive-form games is defined by four Bayes Requirements. These requirements eliminate the bad subgame-perfect equilibria by requiring players to have beliefs, at each information set, about which node of the information set she has reached, conditional on being informed she is in that information set. [Details...]
 This encyclopedia entry explains how the prisoners' dilemma is the best-known game of strategy in social science. [Details...]

Game Theory - Interactive Tutorial

 Here is a little on-line Javascript utility for game theory (up to five strategies for the row and column player). It is also designed to play against you (using the optimal mixed strategy most of the time...) Notes: This will only work on Netscape or Internet Explorer, version 3 or later. You need only enter the non-zero payoffs. The software will set the others to zero. To play against the computer, enter the payoffs, press "Play" and click on row strategies. (The computer does not know your move...) [Details...]
 This tutorial has been created to allow people to play the game against a variety of computer opponents, and to demonstrate the educational potential of simple interactive Web pages in Javascript. [Details...]
 This interactive game allows you to play against a computer and create different outcomes. [Details...]
 This interactive game allows you choose the strategies you want to include in each round as well as the length of play. The results will then be shown in the window. [Details...]
 The spatial variant of the iterated prisoner's dilemma is a simple yet powerful model for the problem of cooperation versus conflict in groups. The applet here demonstrates the spread of 'altruism' and 'exploitation for personal gain' in an interacting population of individuals learning from each other by experience. [Details...]
 Game theory presents the concept of mixed strategies as a method to determine random action choices. This applet tests whether user-submitted sequences are random. [Details...]
 You play the Prisoner's Dilemma game against five different personalities. You will play each opponent for 25 rounds. A summary screen will appear at the end. [Details...]

Game Theory - Experiment Software

 This program sets up a two-player, two-decision matrix game. Pairings may be fixed (same in all rounds) or randomly reconfigured after each round. You choose the number of rounds and each player's payoffs for each of the four possible outcomes. [Details...]
 This program sets up a multi-person game in which each person chooses a price in a Bertrand game with linear demand and constant marginal cost. The game highlights severe competitive pressures when there are several sellers. [Details...]
 This program sets up a centipede game in which two players make a series of decisions in alternating order. At each stage in the series, the player who has the decision must choose to Stop or Continue. The process continues until one person chooses to stop or until the final stage is reached. Payoffs are determined by the stage in which the process stops. The person who chooses Stop earns more than the other person, but payoffs typically increase with successive stages. [Details...]
 This program provides a graphical interface to create and run extensive form game experiments. [Details...]
 The module is based on a matrix representation of two player strategic form games. Using the game editor, the moderator determines the choices that characterize the set of pure strategies, the naming of the strategies and the two players, as well as the payoffs for each possible outcome of the game. Games are run from the edit mode, too. The moderator can view the history of participation by subjects and their choices, plus some basic statistics describing the games played. Both of these are continuously updated as the session proceeds. The results and the summary statistics are saved in an "html" file that can be imported into a spreadsheet or opened with a browser. [Details...]
 This program sets up a game in which each person chooses an "effort". Players are matched in groups of a specified size, and the payoff for each person is the minimum effort made by people in their group (including themselves). There is a cost of effort, which is a constant amount per unit of effort, and each person must pay the cost of their own effort. Thus the payoff is the minimum effort minus a unit cost times one's own effort. Efforts are required to be greater than or equal to a specified lower bound, and to be less than or equal to an upper bound. [Details...]
 This program sets up one or more markets in which each person is a seller who chooses a production quantity. This is a Cournot game with linear demand and constant marginal cost. The game can be used to motivate discussion of the Nash/Cournot predictions. There may be tacit collusion with few sellers and fixed matchings, and there may be severe competitive pressures when there are several sellers. [Details...]
 This site is based on the perception of game theory as the study of a set of considerations used by individuals in strategic situations. The goal is to deliver a loud and clear message that game theoretic models are not meant to supply predictions of strategic behavior in real life. [Details...]
 This program sets up a two-player, multi-decision matrix game, which allows asymmetries. Pairings may be fixed (in all periods) or randomly reconfigured after each round. [Details...]
 This program sets up a multi-person game in which each person chooses whether or not to enter a market. The payoff for all people who enter is a decreasing function of the number of entrants, and the payoff for not entering is a constant. The incentives are typically such that each person would prefer to enter if the others are unlikely to do so, and would prefer to stay out if the others are likely to enter. [Details...]
 This software allows normal-form game experiments to be conducted over the Internet. Within an experiment, subjects participate in one or more matches. Each match is a series of one or more stage games. Several methods of rematching subjects between matches are offered. [Details...]
 This program runs a set of "Principal/Agent" games. The first mover (employer) makes a contract offer, and the second movers (worker) chooses whether to accept the contract. A worker who accepts a contract then chooses an effort level, which is costly to the worker but which benefits the employer. The possible contracts include fixed wage payments, along with possible ex post bonuses, monitoring, penalties, and/or profit sharing. If the contract only specifies a required fixed wage and an optional bonus, then the Nash equilibrium for selfish preferences in a one-shot game is to offer the minimum possible effort, since the wage is paid irrespective of effort. Efforts may be higher with fixed matchings or if participants are concerned with fairness and reciprocity. A number of contract options based (based on penalties and rewards) are also available. The game highlights issues of contract incentives, reciprocity, and strategy. [Details...]
 This program runs a set of two-person signaling games in which one person (the proposer) observes the true state of nature and makes a decision. The other person (responder) sees the proposer's decision but not the state, and makes a response. [Details...]
 This program sets up a game in which each person chooses a "claim". Players are matched in groups of a specified size, and the payoff for each person is the minimum claim made by people in their group (including themselves). [Details...]
 This program sets up a two-player, two-stage game in which one player moves first and the other follows. You specify the number or rounds, with pairings that may be fixed (same in all periods) or randomly reconfigured after each round. You also choose numbers of decisions, their names for each player, and the players' payoffs for each of the possible outcomes. [Details...]
 This program sets up a multi-person game in which each person chooses whether or not to "volunteer." The payoff for all people in a group is higher if at least one of them volunteers, but a volunteer incurs a cost that cannot be shared. The incentives are such that each person would prefer to volunteer only if it is likely that nobody else will do so. You may specify the cost of volunteering and the individual payoffs when there is at least one volunteer and when there is no volunteer. You also specify the group size and the number of rounds. Groupings may be fixed (same in all periods) or randomly reconfigured after each round. There may be two treatments with differing parameters; but a one-treatment setup is obtained by setting the number of rounds in treatment 2 to be zero. [Details...]

Game Theory - Web Site

 This web site was created by PBS to present the life and achievements of John Nash, as well as various game theory topics. [Details...]
 Game Theory .net provides resource material to educators and students of game theory and its applications to economics, business, political science, computer science, and other disciplines. Primarily, the site is directed at less rigorous presentations of the material, concentrating more on making the lessons of game theory relevant to the student. In aiding class preparation, a list of textbooks, readers, and lecture notes used by other educators is provided. Java applets and online games demonstrate these concepts in a fun, interactive way. Further, links to game-theoretic themes in movies, books, and the popular press serve to reinforce concepts, offer an entertaining diversion in class, and make the material more approachable. Assessment materials are provided both to aid educators in preparing classes and to offer students additional study materials. Educators are requested to submit lecture notes, reviews of textbooks, novel teaching strategies or aids, or other suggestions. Students are encouraged to submit their experiences about learning game theory. [Details...]

Game Theory - Online Book

 The formal theory of bargaining originated with John Nash's work in the early 1950s. In this book we discuss two recent developments in this theory. The first uses the tool of extensive games to construct theories of bargaining in which time is modeled explicitly. The second applies the theory of bargaining to the study of decentralized markets. [Details...]
 Arising out of the author's lifetime fascination with the links between the formal language of mathematical models and natural language, this short book comprises five essays investigating both the economics of language and the language of economics. Ariel Rubinstein touches on the structure imposed on binary relations in daily language, the evolutionary development of the meaning of words, game-theoretical considerations of pragmatics, the language of economic agents, and the rhetoric of game theory. These short essays are full of challenging ideas for social scientists that should help to encourage a fundamental rethinking of many of the underlying assumptions in economic theory and game theory. [Details...]
 The material presented would also be helpful to first-year PhD students learning game theory as part of their microeconomic-theory sequence, as well as to advanced undergraduates learning game theory. I consider the exposition detailed, rigorous, and self-contained. [Details...]
 This is a set of essays covering several game theory concepts. The author states "I have tried to limit these pages to fairly elementary topics and to avoid mathematics other than numerical tables and a very little algebra. The presentation is as intuitive as I have been able to make it, and keep it brief." [Details...]

Game Theory - Software Tool

 This applet allows you to create extensive-form (sequential) games, and have them automatically solved for you. The applet allows up to four players, and up to 14 periods. To use the applet, follow the four steps along the right side of the screen: Pick a prototype game tree. Customize the tree to look like your game Add payoffs Solve To customize the game tree (step two), select a node and drag it, or select "add child" to add a sub-node, or "remove node" to delete it. [Details...]
 Gambit is a library of game theory software and tools for the construction and analysis of finite extensive and normal form games. Gambit is designed to be portable across platforms: it currently is known to run on Linux, FreeBSD, MacOS X, and Windows 98 and later. Gambit provides: 1. A graphical user interface, based upon the wxWidgets (nee wxWindows) library, providing a common look-and-feel across platforms; 2. A Python API for scripting applications (under development); 3. A library of C++ source code for representing games, suitable for use in other applications. [Details...]
 There are correspondingly two types of software available from this page: the Spatial Games software, as the name indicates, simulates a game with strong spatial structure, while the Repeated Prisoner's Dilemma software and the Asymmetric Games software model uncorrelated play. [Details...]
 This applet allows you to create a two-player normal-form (simultaneous move) game with up to four strategies for each player. After you enter the payoffs, the applet solves the game, finding all pure-strategy Nash equilbria (and a unique mixed-strategy equilbrium, if one exists, for two-by-two games). [Details...]

Game Theory - Experiment Software Configuration

 55. garp
 56. Duopoly-HMN This is a configuration for a normal form game that implements an experimental test of the prediction of the Cournot-Nash duopoly model. [Details...]
 A normal-form game representation of King Solomon's Problem. [Details...]
 A normal-form game that highlights the incentives for judges to choose law clerks early. [Details...]
 59. NFG-Duop-4x4
 A normal-form game presenting the prisoner's dilemma in gains. [Details...]
 A normal-form game presenting the prisoner's dilemma in losses. [Details...]
 62. R&D Spending A normal-form game presenting a research and development spending scenario. [Details...]
 This configuration is based on the Selten and Stoecker(1986) experiments. In each supergame the game shown below is repeated ten times. Each subject plays 25 supergames. Each subject faces the same opponent within a supergame, and is rematched for each new supergame. [Details...]
 This experiment setup is designed to go with the EconPort Stackelberg duopoly teaching module. [Details...]
 A normal-form game presenting incentives for hockey players to wear helmets. [Details...]

Game Theory - Non-computerized experiment

 Most models of economic behavior are based on the assumption of rationality of economic agents and common knowledge of rationality. This means that an agent selects a strategy that maximizes his utility believing that all others do the same (are equally rational) and that all agents believe that all others believe that all agents are rational etc. The p-beauty contest game is an appropriate game to test the assumption of this kind of reasoning. In this game a player has to guess what the average choice is going to be and the player will win if his choice is closest to some fraction of the average choice. This experiment can be introduced in many different courses and at all levels of teaching. For example, it can be used in game theory in order to discuss the problems of iterated elimination of (weakly) dominated strategies and the issue of common knowledge of rationality; in macroeconomics to discuss rational expectations; and in microeconomics to discuss strategic interaction between players. [Details...]
 For a number of years, I have been using a simple and brief classroom experiment to illustrate the power of game theory in explaining the behavior of oligopolists. The whole presentation takes about fifteen minutes of class time, and it has worked well in the Principles of Microeconomics course. [Details...]
 We present a simple way of carrying out the Investment Game, introduced by Berg, Dickhaut and McCabe (1995) inside the classroom for instructional purposes. This game is a handy way of illustrating the principle of backward induction in sequential move games. In a slight deviation from the original design we allow each subject to play both as a Sender as well as a Receiver. [Details...]
 Both games are conducted in class and they have a short follow-up assignment that is announced after the game is finished. This assignment is meant to help the students understand what they have been doing and why the two games are different. In the coordination game, the students have a common interest (the equilibria are Pareto ranked, and one is efficient). The problem is aligning expectations (and actions). Generally, the students initially settle on an inefficient equilibrium. Direct communication between students allows students to achieve efficiency and move to the Pareto efficient equilibrium without the need for binding commitments. In contrast, in the motivation game, the players have a personal interest diametrically opposed to the common interest (a sort of multilateral prisoner's dilemma). By playing the game, students come to realize how difficult it can be to achieve cooperation when the benefits to defection are great. Even in the classroom, it seems impossible to get the Pareto optimal equilibrium without some kind of binding agreement. [Details...]
 After discussing perfectly competitive and monopoly market structures, most introductory courses cover oligopolies. I have found that the key to students understanding oligopoly market structures is for them to appreciate interdependence. When I first tell classes that oligopolies are interdependent, they are thrilled to know (after perfect competition and monopoly) that this means no curves (I don't use kinked demand). The thrill is somewhat abated when they realize that it means an alternative treatment is necessary, and it might be worse than the curves were. This is where I think it is important to give the students a sense of the behavioral nature of the models as well as to point out that oligopolies are much more common in the economy. This is a great juncture for introducing a classroom experiment or exercise. [Details...]
 To help the students in my senior level industrial organization class understand predation, I run the experiment printed below, which is a modification of Jung, Kagel, and Levin (1994). The experiment takes about 75 minutes to perform. [Details...]