A Variant of the Logistic Quantal Response Equilibrium to Select a Perfect Equilibrium

Abstract

The concept of perfect equilibrium, formulated by Selten (Int J Game Theory 4:25–55, 1975), serves as an effective characterization of rationality in strategy perturbation. In our study, we propose a modified version of perfect equilibrium that incorporates perturbation control parameters. To match the beliefs with the equilibrium choice probabilities, the logistic quantal response equilibrium (logistic QRE) was established by McKelvey and Palfrey (Games Econ Behav 10:6–38, 1995), which is only able to select a Nash equilibrium. By introducing a linear combination between a mixed strategy profile and a given vector with positive elements, this paper develops a variant of the logistic QRE for the selection of the special version of perfect equilibrium. Expanding upon this variant, we construct an equilibrium system that incorporates an exponential function of an extra variable. Through rigorous error-bound analysis, we demonstrate that the solution set of this equilibrium system leads to a perfect equilibrium as the extra variable approaches zero. Consequently, we establish the existence of a smooth path to a perfect equilibrium and employ an exponential transformation of variables to ensure numerical stability. To make a numerical comparison, we capitalize on a variant of the square-root QRE, which yields another smooth path to a perfect equilibrium. Numerical results further verify the effectiveness and efficiency of the proposed differentiable path-following methods.

Appendix

This appendix proves that the Jacobian matrix $Dp(x,\lambda ,\mu ,t;\alpha )$ of $p(x,\lambda ,\mu ,t;\alpha )$ is of full-row rank for any $(x,\lambda ,\mu ,t;\alpha )\in \mathbb {R}^m\times \mathbb {R}^m\times \mathbb {R}^n\times (0,1]\times \mathbb {R}^m$. This property is used in the proof of Theorem 3.4.

We first consider the case where $t\in (0,1)$. Let $g(x,\lambda ,\mu ,t; \alpha )$ denote the left-hand side of the first group of equations in the system (9). The Jacobian matrix $Dp(x,\lambda ,\mu ,t;\alpha )$ is given by

$$\begin{aligned} \left( \begin{array}{ccccc} D_{x}g &{} D_{\lambda }g &{} D_{\mu }g &{} D_{t}g &{} -\beta (t)(1-\beta (t))I\\ A &{} 0 &{} 0 &{} 0 &{} 0\\ B &{} C &{} 0 &{} D &{} 0 \end{array} \right) , \end{aligned}$$

where $D_{x}g \in \mathbb {R}^{m\times m}$, $D_{\lambda }g \in \mathbb {R}^{m\times m}$, $D_\mu g \in \mathbb {R}^{m\times n}$, $D_t g \in \mathbb {R}^{m\times 1}$, I is an identity matrix of size m, $A=\begin{pmatrix} 1 &{} \cdots &{} 1\\ {} &{}&{}&{} 1 &{} \cdots &{} 1\\ {} &{}&{}&{}&{}&{}&{} \ddots \\ {} &{}&{}&{}&{}&{}&{}&{} 1 &{} \cdots &{} 1\end{pmatrix}\in \mathbb {R}^{n\times m}$, B is a diagonal matrix with its elements equal to $\lambda ^i_j+\beta (t)(\ln x^i_j+1)$, C is a diagonal matrix with its elements equal to $x^i_j$, and $D\in \mathbb {R}^{m\times 1}$. Clearly, I, A, B and C are of full-row rank. Hence, the Jacobian matrix $Dp(x,\lambda ,\mu ,t;\alpha )$ is of full-row rank for any $t\in (0,1)$.

Consider the case where $t=1$. The Jacobian matrix $Dp(x,\lambda ,\mu ,1;\alpha )$ is given by

$$\begin{aligned} \left( \begin{array}{ccccc} 0 &{} I &{} E &{} D_{t}g &{} 0\\ A &{} 0 &{} 0 &{} 0 &{} 0\\ F &{} C &{} 0 &{} D &{} 0 \end{array} \right) , \end{aligned}$$

where $E=A^\top$, and F is a diagonal matrix with its elements equal to $\lambda ^i_j+\ln x^i_j+1$. Since I, A, F and C are of full-row rank, $Dp(x,\lambda ,\mu ,1;\alpha )$ is of full-row rank.