On Learning with Finite Memory

TL;DR

This paper investigates the conditions under which a sequence of agents with limited memory and private signals can learn the true state of the world, revealing fundamental limitations and possibilities for convergence.

Contribution

It establishes impossibility results for almost sure learning, identifies conditions for learning in probability with different memory lengths, and analyzes strategic agents' impact on learning.

Findings

Almost sure learning is impossible for any memory length K.

Learning in probability is impossible for K=1, but possible for K≥2 with appropriate rules.

Strategic, forward-looking agents fail to learn in probability regardless of K.

Abstract

We consider an infinite collection of agents who make decisions, sequentially, about an unknown underlying binary state of the world. Each agent, prior to making a decision, receives an independent private signal whose distribution depends on the state of the world. Moreover, each agent also observes the decisions of its last K immediate predecessors. We study conditions under which the agent decisions converge to the correct value of the underlying state. We focus on the case where the private signals have bounded information content and investigate whether learning is possible, that is, whether there exist decision rules for the different agents that result in the convergence of their sequence of individual decisions to the correct state of the world. We first consider learning in the almost sure sense and show that it is impossible, for any value of K. We then explore the possibility of convergence in probability of the decisions to the correct state. Here, a distinction arises: if K equals 1, learning in probability is impossible under any decision rule, while for K greater or equal to 2, we design a decision rule that achieves it. We finally consider a new model, involving forward looking strategic agents, each of which maximizes the discounted sum (over all agents) of the probabilities of a correct decision. (The case, studied in previous literature, of myopic agents who maximize the probability of their own decision being correct is an extreme special case.) We show that for any value of K, for any equilibrium of the associated Bayesian game, and under the assumption that each private signal has bounded information content, learning in probability fails to obtain.