Y2K Bibliography of Experimental Economics and Social Science
Quantal Response Equilibrium

Charles A. Holt, cah2k@virginia.edu, suggestions and corrections welcome
(for online and personal use only)

Anderson, Lisa R. (1999) “Payoff Effects in Information Cascade Experiments,” College of William and Mary, Discussion Paper. Keywords: experiments, decisions, information cascades, methodology, incentive effects, logit equilibrium. Abstract: The paper reports the effects of changing payoff levels on behavior in information cascade experiments. Offering a $2 payment for a correct prediction lowers logit error rates estimated with a logit model, below levels obtained with hypothetical incentives. A doubling the payoff to $4 does not produce a further reduction in the error rates. Email Contact: lrande@mail.wm.edu

Anderson, Simon P., Jacob Goeree, and Charles A. Holt (1998) “Control Costs and Equilibria in Games with Bounded Rationality,” University of Virginia, Discussion Paper, presented at the Summer 1998 ESA Meeting. Keywords: experiments, game theory, control costs, quantal response equilibrium. Abstract: Van Damme's notion of control costs is that it is more costly to implement decisions more precisely. This paper derives the relationship between control costs and noisy approaches to equilibrium in games. In two-by-two games, quantal response equilibria are equivalent to Nash equilibria with control costs. Extensions to N-person matrix games are also discussed. Email Contact: jg2n@virginia.edu

Anderson, Simon P., Jacob K. Goeree, and Charles A. Holt (1997) “Stochastic Game Theory: Adjustment and Equilibrium with Bounded Rationality,” University of Virginia, Discussion Paper. Keywords: experiments, game theory, evolution, logit equilibrium, potential games, quantal response. Abstract: This paper specifies an evolutionary model in which agents adjust their decisions in the direction of higher expected payoff, subject to random error (Brownian motion). The process produces a probability distribution of players' decisions that evolves over time according to the Fokker-Planck equation. The evolutionary process is stable for all potential games, and the steady state is a logit equilibrium, so the analysis provides a dynamic justification for the use of this one-parameter generalization of a Nash equilibrium. The evolutionary paths and steady state are sensitive to payoff differences that do not affect the Nash equilibrium, so the stochastic game theory approach can be used to explain intuitive results of laboratory experiments that are "anomalous" when viewed from the perfect rationality perspective of a Nash equilibrium. Email Contact: sa9w@virginia.edu

Anderson, Simon P., Jacob K. Goeree, and Charles A. Holt (1998) “Bounded Rationality in Markets and Complex Systems,” Experimental Economics, forthcoming. Keywords: experiments, markets, bounded rationality, game theory, quantal response equilibrium, learning, logit equilibrium. Abstract: This is a selective survey of the literature on bounded rationality and its implications for experimental economics. Email Contact: holt@virginia.edu

Anderson, Simon P., Jacob K. Goeree, and Charles A. Holt (1998) “A Theoretical Analysis of Altruism and Decision Error in Public Goods Games,” Journal of Public Economics, 70:2 (November), 297-323. Keywords: experiments, public, voluntary contributions, nonlinear public goods games, game theory, logit equilibrium, altruism, quantal response, numbers effects. Abstract: This paper formalizes an equilibrium model in which altruism and decision error parameters determine the distribution of contributions in public goods games. The authors prove existence of a unique symmetric equilibrium density of contributions and show that 1) contributions increase with the marginal value of the public good, 2) total contributions increase with the number of participants if there is altruism, and 3) mean contributions lie between the Nash predictions and half of the endowment. These predictions, which are not implied by a standard Nash analysis, are roughly consistent with laboratory tests. Email Contact: holt@virginia.edu

Anderson, Simon P., Jacob K. Goeree, and Charles A. Holt (1998) “Logit Equilibrium Models of Anomalous Behavior: What to Do when the Nash Equilibrium Says One Thing and the Data Say Something Else,” in Handbook of Experimental Economics Results, edited by C. R. Plott and V. L. Smith, New York: Elsevier Press, forthcoming. Keywords: experiments, game theory, survey, bounded rationality, logit equilibrium, quantal response equilibrium, traveler's dilemma, decision error. Abstract This paper shows how the logit equilibrium can be used to explain anomalous data patterns in a variety of games. Email Contact: sa9w@virginia.edu

Anderson, Simon P., Jacob K. Goeree, and Charles A. Holt (1998) “Rent Seeking with Bounded Rationality: An Analysis of the All Pay Auction,” Journal of Political Economy, 106:4 (August), 828-853. Keywords: rent seeking, experiments, game theory, logit equilibrium, bounded rationality, quantal response. Abstract: A (logit) model of bounded rationality is used to explain why the extent of rent dissipation in an all-pay auction may is increasing in the number of competitors, effort costs, and other factors that have no effect in a Nash analysis. The paper derives existence, uniqueness, and intuitive comparative statics results for the logit equilibrium model. Email Contact: sa9w@virginia.edu

Anderson, Simon P., Jacob K. Goeree, and Charles A. Holt (1999) “Properties of Logit Equilibria in Games with Rank-Based Payoffs,” University of Virginia, Discussion Paper. Keywords: experiments, game theory, logit equilibrium, existence, uniqueness, symmetry, comparative statics, minimum effort coordination, order statistic coordination games, Bertrand price competition, traveler's dilemma, best shot game, all-pay auction. Abstract: The paper characterizes the logit equilibria for a class of games with payoffs that depend on the ranking of players' decisions, including Bertrand price competition, the all-pay auction, Hotelling's location game, the traveler's dilemma, and many variants of coordination games. General existence, uniqueness, and comparative proofs are presented and applied. Email Contact: holt@virginia.edu

Anderson, Simon P., Jacob K. Goeree, and Charles A. Holt (2000) “Minimum Effort Coordination Games: An Equilibrium Analysis of Bounded Rationality,” Games and Economic Behavior, forthcoming. Keywords: experiments, game theory, logit equilibrium, coordination, learning, potential games, stochastic potential, evolution, quantal response. Abstract: This paper considers minimum-effort coordination games with a continuum of Pareto-ranked Nash equilibria. The introduction of noise yields a unique probability distribution of effort decisions that maximizes a stochastic potential function (expected value of the potential of the game plus entrophy). As the noise vanishes, the limiting distribution converges to an outcome that is analogous to the risk-dominant outcome in two-by-two games. In accordance with experimental evidence and economic intuition, the theory predicts that effort distributions decrease with increases in effort cost and the number of players, even though these parameters do not affect the set of Nash equilibria. Email Contact: sa9w@virginia.edu

Baye, Michael R., and John Morgan Morgan (1999) “Bounded Rationality in Homogeneous Product Pricing Games,” Princeton University, Keywords: experiments, markets, Bertrand games, quantal response equilibrium, epsilon equilibrium. Abstract: This paper uses the quantal response equilibrium and the epsilon equilibrium to evaluate data from laboratory Bertrand games. Both of these approaches organize the data significantly better than the standard Nash/Bertrand equilibrium. Email Contact: jmorgan@princeton.edu

Capra, C. Monica, Jacob K. Goeree, Rosario Gomez, and Charles A. Holt (1998) “Learning and Noisy Equilibrium Behavior in an Experimental Study of Imperfect Price Competition,” University of Virginia, Discussion Paper, presented at the Conference of Experimental Economics in Osaka, 1999. Keywords: experiments, game theory, markets, logit equilibrium, simulation, bounded rationality, quantal response. Abstract: The experiments implement a model of imperfect price competition in which the high-price seller is allowed to match the lower price, but the resulting market share is less than one half. The Nash-Bertrand prediction is unaffected by the market share of the high-price seller, but the data respond sharply to changes in this treatment parameter, a response that is consistent with dynamic and equilibrium logit models of noisy behavior. Email Contact: jg2n@virginia.edu

Capra, C. Monica, Jacob K. Goeree, Rosario Gomez, and Charles A. Holt (1999) “Anomalous Behavior in a Traveler's Dilemma?,” American Economic Review, 89:3 (June), 678-690. Keywords: experiments, game theory, traveler's dilemma, logit equilibrium, learning, simulation, bounded rationality, quantal response. Abstract: The observed choices in a traveler's dilemma experiment are moved across the entire set of feasible decisions by changes in a treatment variable that has no effect on the unique Nash prediction. Dynamic patterns are explained by a logit learning model, and the steady state distributions are centered around the predictions of a logit equilibrium that generalizes the Nash equilibrium to allow for noisy behavior. Email Contact: jg2n@virginia.edu

Chen, Hsiao-Chi, James W. Friedman, and Jacques-Francois Thisse (1996) “Boundedly Rational Nash Equilibrium: A Probabilistic Choice Approach,” Games and Economic Behavior, 1832-54. Keywords: experiments, game theory, matrix games, quantal response, bounded rationality, learning. Abstract: This paper uses the Luce probabilistic choice rule to analyze behavior in matrix games.

Goeree, Jacob K., Simon P. Anderson, and Charles A. Holt (1998) “The War of Attrition with Noisy Players,” in Advances in Applied Microeconomics, Volume 7, edited by M. R. Baye, Greenwich, Conn.: JAI Press, 15-29. Keywords: experiments, game theory, bounded rationality, quantal response equilibrium, logit equilibrium, war of attrition. Abstract The authors show that the Nash equilibria for the two-player normal-form war of attrition with asymmetric values involve one player choosing zero effort (conceding immediately). Non-degenerate mixed equilibria are only possible if there is no maximum effort. These equilibria have perverse comparative-statics properties: an increase in one player's value leaves that player's effort distribution unaffected and raises the other player's efforts. In contrast, the logit equilibrium predicts that the player with the higher prize value exerts more effort. Email Contact: jg2n@virginia.edu

Goeree, Jacob K., and Charles A. Holt (1999) “Ten Little Treasures of Game Theory, and Ten Intuitive Contradictions,” University of Virginia, Discussion Paper. Keywords: experiments, game theory, bargaining, matching pennies, traveler's dilemma, coordination, Kreps game, auctions, signaling, extensive form games, backward induction, Nash equilibrium, logit equilibrium, introspection, one shot games. Abstract: The "treasures" are ten static and dynamic games where behavior matches the Nash equilibrium or relevant refinement, and the contradictions are variations of the same game that produces anomalous behavior patterns. In some games, Nash seems to work only by coincidence, e.g. if deviation losses are symmetric or very high. In other games the data are repelled from the Nash prediction and pile up on the opposite side of the set of feasible decisions. Email Contact: jg2n@virginia.edu

Goeree, Jacob K., and Charles A. Holt (1999) “An Experimental Study of Costly Coordination,” University of Virginia, Discussion Paper. Keywords: experiments, game theory, coordination games, logit equilibrium, bounded rationality, learning, simulation, quantal response, stochastic potential. Abstract: The authors present and test a unified view of behavior in coordination games with a continuum of Nash equilibria that are not affected by non-critical changes in effort costs and numbers of players. The theory is a generalization of risk dominance and maximum potential; the maximization of (stochastic) potential explains steady state effort levels in a series of minimum- and median-effort coordination experiments. In both types of coordination games considered, observed effort distributions are inversely related to effort costs, an intuitive experimental result that is not explained by the perfect-rationality Nash model. Email Contact: jg2n@virginia.edu

Goeree, Jacob K., and Charles A. Holt (1999) “Stochastic Game Theory: For Playing Games, Not Just for Doing Theory,” Proceedings of the National Academy of Sciences, 96(September), 10564-10567. Keywords: experiments, game theory, Nash equilibrium, logit equilibrium, quantal response equilibrium, introspection, coordination, traveler's dilemma, evolution, learning. Abstract: This paper argues that noisy models of introspection, learning, and equilibrium can explain the salient behavior patterns in game experiments, patterns that are not predicted by the Nash equilibrium or its refinements. Models of iterated noisy introspection are used to explain initial choices, models of noisy learning and evolution are used to predict dynamic adjustment paths, and logit equilibrium models explain Nash-invariant treatment effects in steady-state distributions of decisions. Email Contact: jg2n@virginia.edu

Goeree, Jacob K., and Charles A. Holt (1999) “Explaining Anomalous Behavior in Binary-Choice Games: Entry, Voting, Public Goods, and the Volunteer's Dilemma,” University of Virginia, Discussion Paper. Keywords: experiments, game theory, quantal response, binary choice games, volunteer's dilemma, step-level public goods, participation games, voting, market entry, coordination. Abstract: This paper characterizes quantal response equilibria for a general class of N-person binary-choice games: participation, entry, voting, step-level public goods, coordination, and volunteer's dilemma games. The quantal response equilibrium tracks deviations from Nash predictions reported in previous laboratory experiments. Email Contact: jg2n@virginia.edu

Goeree, Jacob K., and Charles A. Holt (1999) “Asymmetric Inequality Aversion and Noisy Behavior in Alternating-Offer Bargaining Games,” European Economic Review, forthcoming. Keywords: experiments, bargaining, alternating offer bargaining games, fairness, inequality aversion, logit equilibrium, asymmetric endowments, one-shot games. Abstract: The experiment involves a two-stage, alternating-offer bargaining game in which players receive asymmetric fixed money payments in addition to what they earn by bargaining. The endowments do not affect the perfect positive correlation between the initial subgame perfect offers and the remaining pie size. Endowments, however, are varied to produce a perfect negative relationship between the remaining pie size and the first stage offer that would equalize players' total payoffs. The negative relationship is apparent in the data, which indicates the importance of fairness considerations. An equilibrium model of noisy behavior and inequity aversion is used to provide estimates of utility and logit error parameters. Email Contact: jg2n@virginia.edu

Goeree, Jacob K., and Charles A. Holt (1999) “A Model of Noisy Introspection,” University of Virginia, Discussion Paper. Keywords: experiments, game theory, introspection, logit equilibrium. Abstract: The paper presents a theoretical model of noisy iterated introspection designed to explain behavior in games played only once. The model determines layers of beliefs (about others' decisions, others' beliefs, beliefs about beliefs...), but relaxes the Nash-like requirement that belief distributions coincide with distributions of decisions, i.e. it allows for systematic surprises in the absence of learning and repetition. The introspective solution exists and nests standard Nash and quantal response equilibrium theories, which are rejected on the basis of maximum likelihood estimates of noise and introspection parameters for laboratory data from thirty-seven variations of 2x2 games played once. Email Contact: jg2n@virginia.edu

Goeree, Jacob K., Charles A. Holt, and Susan K. Laury (1998) “Altruism and Error in Two-Person Public Goods Experiments,” University of Virginia, Discussion Paper. Keywords: experiments, public, voluntary contributions, altruism, internal return, external return, MPCR effects, logit equilibrium, one shot games. Abstract: The effects of a contribution to a public good are decomposed into an internal return to the contributor and an external return to each of the others. Contributions in one-shot games are generally increasing in internal returns, external returns, and group size, and a logit model of individual behavior tracks treatment averages well, both for linear and non-linear altruism specifications. Email Contact: jg2n@virginia.edu

Goeree, Jacob K., Charles A. Holt, and Thomas R. Palfrey (1999) “Quantal Response Equilibrium and Overbidding in Private-Value Auctions,” California Institute of Technology, Keywords: experiments, auctions. private values, logit equilibrium, risk aversion. Abstract: This paper reports the results of a private-values auction experiment in which the expected costs of deviating from the Nash equilibrium bidding function are asymmetric, with more upside risk in one treatment and more downside risk in the other. Overbidding relative to Nash is observed in both treatments, but is more severe in the treatment with lower costs of overbidding. A two-parameter model of constant relative risk aversion and noisy (logit equilibrium) behavior provides a tight fit of bid distributions, and estimates are stable across treatments. Email Contact: jg2n@virginia.edu

Lopez, Gladys (1995) “Quantal Response Equilibria for Models of Price Competition,” Ph.D, Dissertation, University of Virginia, Keywords: experiments, game theory, quantal response equilibrium, Bertrand games, markets, bounded rationality, logit equilibrium, probabilistic choice.

McKelvey, Richard, and John Patty (1999) “Quantal Response Voting,” California Institute of Technology, Discussion Paper, presented at the Spring 1999 Public Choice Meetings. Keywords: experiments, game theory, voting, quantal response. Email Contact: rdm@hss.caltech.edu

McKelvey, Richard D., and Thomas R. Palfrey (1995) “Quantal Response Equilibria for Normal Form Games,” Games and Economic Behavior, 10:1 (July), 6-38. Keywords: experiments, game theory, quantal response, bounded rationality, logit equilibrium. Email Contact: rdm@hss.caltech.edu

McKelvey, Richard D., and Thomas R. Palfrey (1996) “A Statistical Theory of Equilibrium in Games,” Japanese Economic Review, 47:2 (June), 186-209. Keywords: experiments, game theory, logit equilibrium, quantal response equilibrium, experiments, bounded rationality. Email Contact: rdm@hss.caltech.edu

McKelvey, Richard D., and Thomas R. Palfrey (1998) “Quantal Response Equilibria for Extensive Form Games,” Experimental Economics, 1:1 9-41. Keywords: experiments, game theory, backward induction, extensive form, quantal response, logit equilibrium, bounded rationality. Email Contact: rdm@hss.caltech.edu

McKelvey, Richard D., and Thomas R. Palfrey (1998) “An Experimental Study of Jury Decisions,” California Institute of Technology, Discussion Paper, presented at the Summer 1998 ESA Meeting. Keywords: experiments, public, political science, law, voting, jury decisions, unanimity rule, majority rule, quantal response equilibrium, swing voter's curse, information. Abstract: The paper develops an a quantal response analysis of group and individual voting behavior in jury deliberations. The focus is on the number of voters and the number of votes needed to convict. Email Contact: rdm@hss.caltech.edu

McKelvey, Richard D., Thomas R. Palfrey, and Roberto Weber (1997) “The Effects of Payoff Magnitude and Heterogeneity on Behavior in 2x2 Games with Unique Mixed Strategy Equilibria,” California Institute of Technology, working paper. Keywords: experiments, game theory, mixed strategies, quantal response equilibrium, logit equilibrium, payoffs, incentives. Email Contact: rdm@hss.caltech.edu

Morgan, John, and Martin Sefton (1998) “An Experimental Investigation of Unprofitable Games,” Princeton University, Discussion Paper, presented at the Fall 1998 ESA Meetings. Keywords: experiments, game theory, matrix games, maxmin behavior, Nash equilibrium, quantal response equilibrium. Abstract: An unprofitable game is one in which maximin strategies do not constitute a Nash equilibrium, yet they guarantee the same payoff as players earn in a Nash equilibrium. In two-person matrix games with three strategies, the data patterns do not correspond to either Nash or maximin play. Email Contact: jmorgan@princeton.edu, martin.sefton@newcastle.ac.uk

Nagel, Rosemarie, and Fang Fang Tang (1998) “Experimental Results on the Centipede Game in Normal Form: An Investigation of Learning,” Journal of Mathematical Psychology, 42356-384. Keywords: experiments, game theory, learning, normal-form centipede game, learning direction theory, quantal response equilibrium. Abstract: The paper specifies and estimates a noisy directional theory of learning in a game where players' payoffs are determined by the minimum decision. Email Contact: nagel@upf.es

Offerman, Theo, Arthur Schram, and Joep Sonnemans (1998) “Quantal Response Models in Step-Level Public Goods Games,” European Journal of Political Economy, 1489-100. Keywords: experiments, public, voluntary contributions, step-level public goods, quantal response, bounded rationality. Email Contact: theoo@fee.uva.nl

Reynolds, Stanley S. (1999) “Sequential Bargaining with Asymmetric Information: The Role of Quantal Response Equilibrium in Explaining Experimental Results,” University of Arizona, Discussion Paper, presented at the Summer 1999 ESA Meetings. Keywords: experiments, bargaining, sequential bargaining, asymmetric information, quantal response equilibrium, logit equilibrium, Nash equilibrium, bounded rationality. Email Contact: sreynolds@bpa.arizona.edu