Sampling Logit Equilibrium and Endogenous Payoff Distortion

Abstract

We introduce the sampling logit equilibrium (SLE), a stationary concept for population games in which agents evaluate actions using a finite sample of opponents' plays and respond according to a logit choice rule. This framework combines informational frictions from finite sampling with stochastic choice. When the sample size is large, SLE is well approximated by a logit equilibrium of a virtual game whose payoffs incorporate explicit distortion terms generated by sampling noise. Examples illustrate how finite sampling can systematically shift equilibrium behavior and generate equilibrium selection effects.

Figures (7)

• Figure 1: Choice probability of action $1$ in the game \ref{['eq:A.2x2.coord']} under different rules.
• Figure 2: Sampling logit equilibria for $k \in \{1,2,3,5,20\}$ in the game \ref{['eq:A.2x2.coord']}.
• Figure 3: Phase diagrams of the four dynamics in Young's game \ref{['eq:A.Young']}. Arrows show sample trajectories, and background contours depict the speed of adjustment: warmer colors indicate faster adjustment, whereas cooler colors indicate slower adjustment. This figure and the next were generated with the Dynamo software Franchetti-Sandholm-BT2013.
• Figure 4: Illustration of \ref{['cor:suboptimal']} by the plots of $\widehat{\sigma}_i(x) = 2 k \eta^2 \;\widehat{v}_i(x)$ for the case $A = 2001$ with different $\eta$. The interior Nash equilibrium of the game is $x_1^* \equiv \frac{1}{3}$, and $\operatorname{br}(x) = \{2\}$ if $x < x_1^*$ and $\operatorname{br}(x) = \{1\}$ if $x_1 > x_1^*$.
• Figure 5: Illustration of \ref{['prop:separable.2']} based on graphs of $2k\eta^2(\widehat{v} + \widehat{q}) = \widehat{\sigma} + \eta\widehat{q}$ under different values of $\eta$ in a separable two-action congestion game. $\operatorname{br}(x) = \{1\}$ if $x_1 < \frac{1}{2}$ and $\operatorname{br}(x) = \{2\}$ if $x > \frac{1}{2}$. For comparison, the dashed curves show only $\widehat{\sigma}$.

Theorems & Definitions (12)

• proposition 1
• proposition 2
• theorem 1: Uniform approximation
• theorem 2: Virtual payoff representations
• proposition 3
• corollary 1: Virtual preference for the suboptimal
• proposition 4: Analytical approximation for the interior equilibrium
• corollary 2
• proposition 5
• proposition A