Stability of mixed-strategy-based iterative logit quantal response dynamics in game theory.

Qian Zhuang,Zengru Di,Jinshan Wu,Luo-Luo Jiang

doi:10.1371/journal.pone.0105391

Abstract

Using the Logit quantal response form as the response function in each step, the original definition of static quantal response equilibrium (QRE) is extended into an iterative evolution process. QREs remain as the fixed points of the dynamic process. However, depending on whether such fixed points are the long-term solutions of the dynamic process, they can be classified into stable (SQREs) and unstable (USQREs) equilibriums. This extension resembles the extension from static Nash equilibriums (NEs) to evolutionary stable solutions in the framework of evolutionary game theory. The relation between SQREs and other solution concepts of games, including NEs and QREs, is discussed. Using experimental data from other published papers, we perform a preliminary comparison between SQREs, NEs, QREs and the observed behavioral outcomes of those experiments. For certain games, we determine that SQREs have better predictive power than QREs and NEs.

Highlights

Game theory has become a powerful and popular tool in many sociological studies
We observed the following: first, quantal response equilibrium (QRE) exist for all of the games discussed above, and QREs cover all of the Nash equilibriums (NEs) in the limit of b??; second, for games with a preferred NE, the stable QREs (SQREs) can be used as a refinement of the NEs; and mixed QREs become unstable for large enough b values; the SQRE can be regarded as a refinement of the QREs
Our dynamic process differs from the so-called Logit response dynamics, which generally results in correlated equilibrium, even for non-cooperative games

Summary

Introduction

Game theory has become a powerful and popular tool in many sociological studies. several studies have questioned predictive power of the Nash equilibrium (NE) [1,2], it has been used as a primary game solution since its initial proposition [3,4]. We refer to such an NE as the preferred NE In this case, a proposed evolutionary model is a good theory if the model predicts that long-term solutions of the corresponding dynamic processes converge to the refined NE. Our goal of proposing this new dynamic process is solely to capture the preferred NE with longterm stable solutions of ILQRD, which we denote as stable QREs (SQREs) This space includes the correlated strategy, whereas D is the set of only independent strategies In this notation, a general possibly correlated equilibrium [19] can be defined as r1ce2[DðSÞ such that for every player i, Vri[Di,. Using the previously described matrix-based notation, for a 2player game our iterative Logit quantal response dynamics (ILQRD) is defined as follows: ðtz1Þ~

ZR1 ðtÞ ebHR1 ðr2

Conclusion and Discussion