Quantal Response Equilibrium Model Research Articles

Quantal Response Equilibrium in a Double Auction

This paper establishes existence and uniqueness of Quantal Response Equilibrium (QRE) in a double auction. The concept of QRE has the intuitive property that a deviation from best response is less likely the higher the cost associated with the deviation itself. Thanks to such property, the QRE accommodates stochastic elements in the analysis of the strategic decision-making that arises in the double auction. By providing a theoretical alternative to the Bayesian Nash Equilibrium model, the QRE model offers an appealing tool for analyzing data of double auction experiments.

Quantal response equilibrium in a double auction

This paper establishes existence and uniqueness of Quantal Response Equilibrium (QRE) in a double auction. The concept of QRE has the intuitive property that a deviation from best response is less likely the higher the cost associated with the deviation itself. Thanks to such property, the QRE accommodates stochastic elements in the analysis of the strategic decision-making that arises in the double auction. By providing a theoretical alternative to the Bayesian Nash Equilibrium model, the QRE model offers an appealing tool for analyzing data of double auction experiments.

An Adaptive Learning Model with Foregone Payoff Information

AbstractIn this paper, we provide theoretical predictions on the long-run behavior of an adaptive decision maker with foregone payoff information. In the model, the decision maker assigns a subjective payoff assessment to each action based on his past experience and chooses the action that has the highest assessment. After receiving a payoff, the decision maker updates his assessments of actions in an adaptive manner, using not only the objective payoff information but also the foregone payoff information, which may be distorted. The distortion may arise from “the grass is always greener on the other side” effect, pessimism/optimism or envy/gloating; it depends on how the decision maker views the source of the information. We first provide conditions in which the assessment of each action converges, in that the limit assessment is expressed as an average of the expected objective payoff and the expected distorted payoff of the action. Then, we show that the decision maker chooses the optimal action most frequently in the long run if the expected distorted payoff of the action is greater than the ones of the other actions. We also provide conditions, under which this model coincides with the experience-weighted attraction learning, stochastic fictitious play and quantal response equilibrium models, and thus this model provides theoretical predictions for the models in decision problems.

Truth-telling and trust in sender–receiver games with intervention: an experimental study

Recent experimental studies find excessive truth-telling in sender–receiver games. We show that this phenomenon is robust to the random intervention of a truthful regulator. In addition, intervention significantly increases the excessive trust of receivers while the overall percentage of truthful messages received does not change much with or without intervention. We offer a theoretical explanation for the behavior of senders and receivers, using a logit agent quantal response equilibrium (logit-AQRE) model incorporating a non-monetary lying cost for senders (like Peeters et al. in Scand J Econ 115(2):508–548, 2013). We show that our experimental findings are all consistent with the predictions of this model. Moreover, we find that the lying cost is significantly higher under intervention, implying that truthful intervention is beneficial for receivers and justified as a tool for policy makers acting on behalf of informationally inferior parties.

Social learning in networks: a Quantal Response Equilibrium analysis of experimental data

Individuals living in society are bound together by a social network and, in many social and economic situations, individuals learn by observing the behavior of others in their local environment. This process is called social learning. Learning in incomplete networks, where different individuals have different information, is especially challenging: because of the lack of common knowledge individuals must draw inferences about the actions others have observed, as well as about their private information. This paper reports an experimental investigation of learning in three-person networks and uses the theoretical framework of Gale and Kariv (Games Econ Behav 45:329–346, 2003) to interpret the data generated by the experiments. The family of three-person networks includes several non-trivial architectures, each of which gives rise to its own distinctive learning patterns. To test the usefulness of the theory in interpreting the data, we adapt the Quantal Response Equilibrium (QRE) model of Mckelvey and Palfrey (Games Econ Behav 10:6–38, 1995; Exp Econ 1:9–41, 1998). We find that the theory can account for the behavior observed in the laboratory in a variety of networks and informational settings. This provides important support for the use of QRE to interpret experimental data.

Level-k analysis of experimental centipede games

As one of the best-known examples of the paradox of backward induction, centipede games have prompted a host of studies with various approaches and explanations (McKelvey and Palfrey, 1992; Fey et al., 1996; Nagel and Tang, 1998; Rapoport et al., 2003; Palacios-Huerta and Volij, 2009). Focusing on initial plays observed in experiments, this paper attempts to offer another explanation based on thorough study of level-k models as applied to these games. Borrowing ideas from the cognitive hierarchy model (Camerer et al., 2004), the authors constructed a group of models based on levels of rationality, and also tested for various assumptions on the play of the most naïve player type in these models. It was found that level-k models generally perform better than the agent quantal response equilibrium (AQRE) model and its variant with altruistic player types for increasing-pie centipede games, while the AQRE model with altruistic player types performs better in constant-pie games.

Fairness Ideals in Distribution Channels

Existing research suggests that concerns for fairness may significantly affect the interactions between firms in a distribution channel. We analytically and experimentally evaluate how firms make decisions in a two-stage dyadic channel, in which firms decide on investments in the first stage and then on prices in the second stage. We find that firms’ behaviors differ significantly from the predictions of the standard economic model. We explain the results by allowing the retailer to concern distributive fairness with the manufacturer. Using a Quantal Response Equilibrium (QRE) model, in which both the manufacturer and retailer make noisy best responses, we show significant concerns exist regarding fairness between channel members. Additionally, we propose a new principle of distributive fairnessthe sequence-aligned ideal, that is studied first time in literature, and compare the new fairness ideal with several existing ideals in literature. Surprisingly, the new ideal, according to which the sequence of moving determines the formation of equitable payoff for players, significantly outperforms other fairness ideals, including strict egalitarianism, liberal egalitarianism, and libertarianism.

Comparing Models of Strategic Thinking in Van Huyck, Battalio, and Beil's Coordination Games

This paper compares the leading models of strategic thinking with subjects’ initial responses to Van Huyck, Battalio, and Beil (1990, 1991) coordination games. Among the refined “equilibrium plus noise” models we compare, payoff-dominant equilibrium performs better than risk-dominant or maximin equilibrium. Among the individualistic models we compare, level-k and cognitive hierarchy models usually fit better than logit quantal response equilibrium or noisy introspection models. In Van Huyck, Battalio, and Beil games, payoff-dominant equilibrium usually fits better than level-k or cognitive hierarchy. The data favor versions of the models in which people model others as if they were perfectly correlated over the standard, independent versions. (JEL: C51, C72, C92)

Sequential equilibrium in monotone games: A theory-based analysis of experimental data

A monotone game is an extensive-form game with complete information, simultaneous moves and an irreversibility structure on strategies. It captures a variety of situations in which players make partial commitments and allows us to characterize conditions under which equilibria result in socially desirable outcomes. However, since the game has many equilibrium outcomes, the theory lacks predictive power. To produce stronger predictions, one can restrict attention to the set of sequential equilibria, or Markov equilibria, or symmetric equilibria, or pure-strategy equilibria. This paper explores the relationship between equilibrium behavior in a class of monotone games, namely voluntary contribution games, and the behavior of human subjects in an experimental setting. Several key features of the symmetric Markov perfect equilibrium (SMPE) are consistent with the data. To judge how well the SMPE fits the data, we estimate a model of Quantal Response Equilibrium (QRE) [R. McKelvey, T. Palfrey, Quantal response equilibria for normal form games, Games Econ. Behav. 10 (1995) 6–38; R. McKelvey, T. Palfrey, Quantal response equilibria for extensive form games, Exp. Econ. 1 (1998) 9–41] and find that the decision rules of the QRE model are qualitatively very similar to the empirical choice probabilities.

Strategic Interaction and Interstate Crises: A Bayesian Quantal Response Estimator for Incomplete Information Games

Private information characteristics like resolve and audience costs are powerful influences over strategic international behavior, especially crisis bargaining. As a consequence, states face asymmetric information when interacting with one another and will presumably try to learn about each others' private characteristics by observing each others' behavior. A satisfying statistical treatment would account for the existence of asymmetric information and model the learning process. This study develops a formal and statistical framework for incomplete information games that we term the Bayesian Quantal Response Equilibrium Model (BQRE model). Our BQRE model offers three advantages over existing work: it directly incorporates asymmetric information into the statistical model's structure, estimates the influence of private information characteristics on behavior, and mimics the temporal learning process that we believe takes place in international politics.

Designing Price Contracts for Boundedly Rational Customers: Does the Number of Blocks Matter?

When designing price contracts, one of the major questions confronting managers is how many blocks there should be in the contract. We investigate this question in the setting of a manufacturer-retailer dyad facing a linear deterministic consumer demand. Theoretical marketing models predict that the manufacturer’s profits rise dramatically when the number of blocks in the contract is increased from one to two because both channel efficiency and its share of channel profits increase. However, increasing the number of blocks to three yields no incremental profits.We test these predictions experimentally and find that increasing the number of blocks from one to two raises channel efficiency but not the manufacturer’s share of profits. Surprisingly, having three blocks in the contract increases channel efficiency even further and also gives the manufacturer a slightly higher share of profits. We show that these results can be explained by a quantal response equilibrium model in which the manufacturer accounts for noisy best response due to nonpecuniary payoff components in the retailer’s utility. We also show that the retailer is sensitive to the counterfactual profits it could have earned if it were charged a lower marginal price for earlier blocks in the multiple-block contract.

Comparing Models of Strategic Choice: The Role of Uncertainty and Signaling

Testing the fit of competing equilibrium solutions to extensive form games crucially depends on assumptions about the distribution of player types. To illustrate the importance of these assumptions for differentiating standard statistical models of strategic choice, I draw on a game previously analyzed by Lewis and Schultz (2003). The differences that they highlight between a pair of perfect Bayesian equilibrium and quantal response equilibrium models are not produced by signaling and updating dynamics as claimed, but are instead produced by different assumptions about the distribution of player types. The method of analysis developed and the issues raised are applicable to a broad range of structural models of conflict and bargaining.

Bounded rationality in laboratory bargaining with asymmetric information

Summary. This paper reports an experiment on two-player sequential bargaining with asymmetric information that features some forces present in multi-round monopoly pricing environments. Buyer-seller pairs play a series of bargaining games that last for either one or two rounds of offers. The treatment variable is the probability of continuing into a second round. Equilibrium predictions do a poor job of explaining levels of prices and treatment effects. As an alternative to the conventional equilibrium model, we consider models that allow for bounded rationality of subjects. The quantal response equilibrium model captures some of the important features of the results.

Quantal-response equilibrium models of the ultimatum bargaining game

This paper investigates the implications of quantal response equilibrium (QRE) models [McKelvey and Palfrey, 1995, Games Econ. Behav. 10, 6–38; 1998, Exper. Econ. 1, 9–41] in the ultimatum bargaining game. It is shown that, in a normal-form QRE (NQRE), each bargainer's decision depends critically on the anticipated behavior of the other, and there is a NQRE in which the proposer makes any offer between zero and equal split as a strict best response. The application of NQRE to the experimental data [Slonim and Roth, 1998, Econometrica 66, 569–596] suggests that the history dependence observed in the experiment is a result of the strategic interactions between bargainers.

Ignoring the rationality of others: evidence from experimental normal-form games

Two behavioral models of two-person normal-form game play are presented and estimated, using three experimental data sets. The models are variants of the Quantal Response Equilibrium model defined by McKelvey and Palfrey (1995, Games Econ. Behav. 10, 6–38), but allow a player to hold inaccurate beliefs about the behavior of her opponent. Each model involves two parameters: one captures the player's own level of response rationality, the other the level she attributes to her opponent. In order to allow for type heterogeneity among the subjects in the experiments, parametric distributions of these parameters are assumed. The estimation results indicate that the subjects' choices follow a specific anomalous pattern: on average, subjects play as if they significantly underestimated their opponent's rationality.

A quantal response equilibrium model of order-statistic games

This paper applies McKelvey and Palfrey’s [Games Econ. Behav. 10 (1995) 6] notion of “quantal response equilibrium” to a class of “order-statistic” coordination games closely related to those studied by Van Huyck et al. [Am. Econ. Rev. 80 (1990) 234; Quart. J. Econ. 106 (1991) 885] and Van Huyck et al. [Evidence on Learning in Coordination Games. Manuscript, Texas A&M University, 1996] and those recently analyzed by Anderson et al. [Games Econ. Behav. 34 (2001) 177]. With quadratic effort costs, the limiting QRE as the noise goes to zero in their games is the most efficient equilibrium. This result contrasts with Anderson et al.’s conclusion for order-statistic games with linear effort costs, and allows a fuller assessment of the QRE’s success in describing the limiting outcomes in Van Huyck et al.’s experiments.

Quantal Response Equilibrium and Overbidding in Private-Value Auctions

This paper applies the quantal response equilibrium (QRE) model to study overbidding in private-values auctions. Experimental evidence shows that the prevalence of overbidding depends on the cost of overbidding relative to underbidding, as predicted theoretically. We use QRE as an error structure to estimate parameters of several competing models of overbidding. A QRE model based on risk averse bidders closely tracks the exact distribution of bids. The estimated parameters are significant and consistent across treatments. Journal of Economic Literature Classification Numbers: C72, C92, D44.

Ignoring the Rationality of Others: Evidence from Experimental Normal-form Games

In this paper, two models of two-person normal-form game play behavior are presented and estimated, using three experimental data sets. The models are variants of the Quantal Response Equilibrium model defined by McKelvey and Palfrey (1995, Games and Economic Behavior), and allow a player to hold inaccurate beliefs about the behavior of her opponent. Each model involves two parameters: One captures the player's own level of response rationality, the other the level she attributes to her opponent. In order to allow for type heterogeneity among the subjects in the experiments, parametric distributions of these parameters are assumed. The estimation results indicate that in all three data sets the subjects' choices follow a specific anomalous pattern: On average, subjects play as if they significantly underestimated their opponent's rationality.

The effects of payoff magnitude and heterogeneity on behavior in 2×2 games with unique mixed strategy equilibria

The Logit version of Quantal Response Equilibrium (QRE) predicts that equilibrium behavior in games will vary systematically with payoff magnitudes, if all other factors are held constant (including the Nash equilibria of the game). We explore this in the context of a set of asymmetric 2×2 games with unique totally mixed strategy equilibria. The data provide little support for the payoff magnitude predictions of the Logit Equilibrium model. We extend the theoretical QRE model to allow for heterogeneity, and find that the data fit the heterogeneous version of the theory significantly better.

An experimental study of constant-sum centipede games

In this paper, we report the results of a series of experiments on a version of the centipede game in which the total payoff to the two players is constant. Standard backward induction arguments lead to a unique Nash equilibrium outcome prediction, which is the same as the prediction made by theories of “fair” or “focal” outcomes.We find that subjects frequently fail to select the unique Nash outcome prediction. While this behavior was also observed in McKelvey and Palfrey (1992) in the “growing pie” version of the game they studied, the Nash outcome was not “fair”, and there was the possibility of Pareto improvement by deviating from Nash play. Their findings could therefore be explained by small amounts of altruistic behavior. There are no Pareto improvements available in the constant-sum games we examine. Hence, explanations based on altruism cannot account for these new data.We examine and compare two classes of models to explain these data. The first class consists of non-equilibrium modifications of the standard “Always Take” model. The other class we investigate, the Quantal Response Equilibrium model, describes an equilibrium in which subjects make mistakes in implementing their best replies and assume other players do so as well. One specification of this model fits the experimental data best, among the models we test, and is able to account for all the main features we observe in the data.