Heterogeneous quantal response equilibrium and cognitive hierarchies

Abstract

We explore an equilibrium model of games where players' choice behavior is given by logit response functions, but their payoff responsiveness is heterogeneous. We extend the definition of quantal response equilibrium to this setting, calling it heterogeneous quantal response equilibrium (HQRE), and prove existence under weak conditions. We generalize HQRE to allow for limited insight, in which players can only imagine others with low responsiveness. We identify a formal connection between this new equilibrium concept, called truncated quantal response equilibrium (TQRE), and the Cognitive Hierarchy (CH) model. We show that CH can be approximated arbitrarily closely by TQRE. We report a series of experiments comparing the performance of QRE, HQRE, TQRE and CH. A surprise is that the fit of the models are quite close across a variety of matrix and dominance-solvable asymmetric information betting games. The key link is that in the QRE approaches, strategies with higher expected payoffs are chosen more often than strategies with lower expected payoff. In CH this property is not built into the model, but generally holds true in the experimental data.JEL classification numbers: 024, 026

Uniform Sampling of the Infinite Noncooperative Game on Unit Hypercube and Reshaping Ultimately Multidimensional Matrices of Player’s Payoff Values

“…This might be useful inasmuch as multidimensional matrices of players' payoff values in the hypercubic lattice form are preferred to other forms [10]. However, the main issue is to make problems of the type (26) effectively computable for accelerating the consistency checks in (28) and (32).…”

“…However, NE-solutions render a lot of the refined or modified principles of optimality, allowing to smooth differences in utility and equity [2], [10], [11]. Mainly, they are principles of Pareto equilibrium [2], [6], [8], [10], [13], [14], Mertens-stable equilibrium [15], trembling hand perfect equilibrium [16], proper equilibrium [17], [18], correlated equilibrium [19], sequential equilibrium [20], [21], quasi-perfect equilibrium [18], [22], [23], perfect Bayesian equilibrium [18], [20], [24], [25], quantal response equilibrium [26], [27], self-confirming equilibrium [28], [29], strong Nash equilibrium [30], [31], Markov perfect equilibrium [32], [33]. The question is only to find NEsolutions as fast as possible.…”

Measuring Attention and Strategic Behavior in Games with Private Information

“…Male subjects, subjects who have been exposed to Game Theory, subjects with experience in Poker or Bridge, and subjects who answer correctly all three CRT questions seem to perform slightly better than their counterparts. 40 A variety of two-sample Wilcoxon rank-sum tests show that the distributions of the probabilities of playing Nash conditional on looking at MIN are indeed statistically different in some cases. In particular in D1, there are gender, game theory, and CRT effects.…”

Exploration vs Exploitation, Impulse Balance Equilibrium, and a Specification Test for the El Farol Bar Problem

“…Various answers have been proposed, as in e.g., Armantier and Treich (2009), Rogers, Camerer and Palfrey (2009) and Camerer, Nunnari and Palfrey (2016). 2 Although making these modifications shows that assuming heterogeneity in agents' traits considerably improves the model's goodness-of-fit, it does not address the question of when one should indeed forego the assumption of homogeneous agents.…”

“…Unlike Golman's normative approach that is suited to QRE and that generates most useful predictions for the QRE analysis of binary-choice 2 See McKelvey, Palfrey and Weber (2000) and Weizsäcker (2003) for earlier investigations of QRE models with non-homogenous agents. Rogers et al (2009) also consider a QRE model with subjective beliefs, i.e., where the distributions of others' traits are not common knowledge but each player believes that the others' traits are i.i.d. from the same distribution as her/his (which is private information).…”