Convergence in Models with Bounded Expected Relative Hazard Rates

TL;DR

This paper introduces a broad framework for analyzing the convergence of stochastic sequences in various learning models, showing conditions under which they reliably reach optimal actions.

Contribution

It develops a unified theoretical approach to study convergence in models with bounded expected relative hazard rates, applicable across economics, computer science, and marketing.

Findings

Sequences with small step-sizes converge to 1 with high probability.

Sequences with shrinking step-sizes converge almost surely to 1.

Provides conditions for learning models to select payoff-maximizing actions long-term.

Abstract

We provide a general framework to study stochastic sequences related to individual learning in economics, learning automata in computer sciences, social learning in marketing, and other applications. More precisely, we study the asymptotic properties of a class of stochastic sequences that take values in $[0,1]$ and satisfy a property called "bounded expected relative hazard rates." Sequences that satisfy this property and feature "small step-size" or "shrinking step-size" converge to 1 with high probability or almost surely, respectively. These convergence results yield conditions for the learning models in B\"orgers, Morales, and Sarin (2004), Erev and Roth (1998), and Schlag (1998) to choose expected payoff maximizing actions with probability one in the long run.