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This chapter lays out the general theory of quantal response equilibrium (QRE) for normal-form games. It starts with the reduced-form approach to QR, based on the direct specification of “regular” quantal or smoothed best-response functions required to satisfy four intuitive axioms of stochastic choice. A simple asymmetric matching pennies game is used to illustrate these ideas and show that QRE imposes strong restrictions on the data, even without parametric assumptions on the quantal response functions. Particular attention is given to the logit QRE, since it is the most commonly used approach taken when QRE is applied to experimental or other data. The discussion includes the topological and limiting properties of logit QRE and connections with refinement concepts. QRE is also related to several other equilibrium models of imperfectly rational behavior in games, including a game-theoretic equilibrium version of Luce's (1959) model of individual choice, Rosenthal's (1989) linear response model, and Van Damme's (1987) control cost model; these connections are explained in the chapter.

This chapter considers a set of closely related binary-choice games that have been applied to model questions in political science. It starts with an analysis of “participation games,” which are n-player games where each player has only two possible actions: to participate or not. The payoff for either decision depends on the number of other players who make that decision. In some cases, a threshold level of participation is required for the group benefit to be obtained. The first example is the “volunteer's dilemma,” which pertains to the special case where the threshold is 1, that is, only a single volunteer is needed. The chapter ends with an analysis of bargaining situations, including an application of agent quantal response equilibrium to bilateral “crisis bargaining” and of Markov QRE to the Baron–Ferejohn model of multilateral “legislative bargaining.”

This introductory chapter provides an overview of the book's main themes. This book focuses on a class of general models, quantal response equilibrium (QRE) models, and its applications to economics, political science, and pure game theory. It aims to lay out for an informed reader the broad array of theoretical and experimental results based on QRE. It contains some genuinely new material, offering new directions and open issues that have not been explored in depth yet, as well as delving into a range of peripheral issues, tangencies, and more detailed data analysis than the typical journal-article format allows for. There is also a “how-to” aspect to the book, as it provides details and sample programs to show readers how to compute QRE and how to take it to the data.

This chapter lays out the general theory of quantal response equilibrium (QRE) for extensive-form games. The formulation of the model is necessarily more complicated because timing and information now play a direct role in the decision maker's choice. This can have interesting and unanticipated consequences. It first describes four possible ways to define QRE in extensive-form games, depending on how the games are represented. It then turns to the structural agent quantal response equilibrium (AQRE) extensive-form games. This is followed by a discussion of the logit AQRE model, which implies a unique selection from the set of Nash equilibria. This selection is defined by the connected component of the logit AQRE correspondence. The final section presents an AQRE analysis of the centipede game.

Players have different skills, which has implications for the degree to which they make errors. Low-skill hitters in baseball often swing at bad pitches, beginning skiers frequently fall for no apparent reason, and children often lose at tic-tac-toe. At the other extreme, there are brilliant chess players, bargainers, and litigators who seem to know exactly what move to make or offer to decline. From a quantal response equilibrium (QRE) perspective, these skill levels can be modeled in terms of variation in error rates or in responsiveness of quantal response functions. This chapter explores issues related to individual heterogeneity with respect to player error rates. It also describes some extensions of QRE that relax the assumption that player expectations about the choice behavior of other players are correct. For example, in games that are played only once, players are not able to learn from others' prior decisions, and expectations must be based on introspection. The chapter develops the implications of noisy introspection embedded in a model of iterated thinking.

This chapter explores questions related to learning and dynamics. The first part explores dynamic quantal response equilibrium models where strategies are conditioned on observed histories of past decisions and outcomes of stage games. The second part considers models in which players are learning about others' behavior via a process in which they may update and respond to current beliefs in a noisy (quantal) manner. The final section explores learning models that involve quantal responses to beliefs formed by processing information from finite (but possibly long) histories of prior or observed action profiles. The formulation permits consideration of a wide variety of exogenous or even endogenous (e.g., least squares) learning rules.

This chapter discusses the use of maximum-likelihood methods to estimate the precision and other parameters of interest, for example, those that capture risk aversion, inequity aversion, altruism, etc. It also considers “noisy Nash” and other (nonequilibrium) error specifications. This estimation can be done with any standard statistical package, for example, GAUSS, Mathematica, R, and MATLAB. The sample programs contained in this chapter are specified in a manner that is intended to clarify the underlying structure of the quantal response equilibrium (QRE) analysis and to illustrate how to use numerical methods to calculate QRE. The chapter is organized around a series of applications, each based on a laboratory experiment that presented interesting estimation issues.

This chapter explores several applications of quantal response equilibrium (QRE) to specific games in order to illustrate and expand on the wide range of game-theoretic principles and phenomena associated with QRE that have been highlighted in the previous chapters. The first application considered belongs to the class of continuous games. With a continuum of decisions, QRE predicts a choice distribution that is not merely a (possibly asymmetric) spread to each side of a Nash equilibrium, since “feedback effects” from deviations by one player alter others' expected payoff profiles, which would induce further changes. The second application is a symmetric game with binary actions where players have continuously distributed private information about an unknown state of the world that affects both players' payoffs. The remainder of the chapter looks at three applications to extensive-form games, all of which are games of incomplete information.

This chapter explores whether the equilibrium effects of noisy behavior can cause large deviations from standard predictions in economically relevant situations. It considers a simple price-competition game, which is also partly motivated by the possibility of changing a payoff parameter that has no effect on the unique Nash equilibrium, but which may be expected to affect quantal response equilibrium. In the minimum-effort coordination game studied, any common effort in the range of feasible effort levels is a Nash equilibrium, but one would expect that an increase in the cost of individual effort or an increase in the number of players who are trying to coordinate would reduce the effort levels observed in an experiment. The chapter presents an analysis of the logit equilibrium and rent dissipation for a rent-seeking contest that is modeled as an “all-pay auction.” The final two applications in this chapter deal with auctions with private information.

This chapter begins by underscoring the basic philosophy behind the quantal response equilibrium (QRE) framework. First, and perhaps foremost, is that QRE is an equilibrium theory. The second key aspect of the QRE framework is that it is stochastic, and does not rule out any action a priori. The third aspect of the QRE framework is the idea that choice probabilities are monotone in payoffs, but this is a restriction that is not imposed on the original general formulation of structural QRE. The chapter then speculates on the direction of QRE research in the next two decades. The discussion is organized in three sections: “Theory,” “Applications,” and “Methodology.”Each section briefly describes a few examples of areas that offer interesting opportunities for future research in the QRE framework, or bringing QRE analysis to bear on specific problems of interest to social scientists and game theorists.