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2.
Experience from a Course in Game Theory: Pre and Post-Class Problem Sets as a Didactic Device
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...]
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3.
Game Theory
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...]
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4.
Game Theory
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...]
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5.
Game Theory Course: 1. Strategic-Form Games
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...]
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6.
Game Theory Course: 2.1 Strategic Dominance
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...]
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7.
Game Theory Course: 2.2 Iterated Dominance and Rationalizability
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...]
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8.
Game Theory Course: 3.1 Nash Equilibrium
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...]
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9.
Game Theory Course: 3.2 Computing Mixed-Strategy Nash Equilibria of 2 x 2 Strategic-Form Games
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...]
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10.
Game Theory Course: 4.1 Introduction to Extensive-Form Games
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...]
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13.
Game Theory Course: 5.1 Introduction to Repeated Games
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...]
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14.
Game Theory Course: 5.2 Infinitely Repeated Games with Discounting
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...]
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15.
Game Theory Course: 5.3 A Folk Theorem Sampler
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...]
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16.
Game Theory Course: 6.1 Static Games of Incomplete Information
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...]
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18.
Game Theory Course: 6.3 Perfect Bayesian Equilibria of Extensive-Form Games
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...]
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