History-Independent Distributed Multi-agent Learning

Abstract

How should we evaluate a rumor? We address this question in a setting where multiple agents seek an estimate of the probability, b, of some future binary event. A common uniform prior on b is assumed. A rumor about b meanders through the network, evolving over time. The rumor evolves, not because of ill will or noise, but because agents incorporate private signals about b before passing on the (modified) rumor. The loss to an agent is the (realized) square error of her opinion.

Our setting introduces strategic behavior based on evidence regarding an exogenous event to current models of rumor/influence propagation in social networks.

We study a simple Exponential Moving Average (EMA) for combining experience evidence and trusted advice (rumor), quantifying its resulting performance and comparing it to the optimal achievable using Bayes posterior having access to the agents private signals.

We study the quality of $p_T$, the prediction of the last agent along a chain of T rumor-mongering agents. The prediction $p_T$ can be viewed as an aggregate estimator of b that depends on the private signals of T agents. We show that

• When agents know their position in the rumor-mongering sequence, the expected mean square error of the aggregate estimator is $\varTheta (\frac{1}{T})$. Moreover, with probability $1-\delta$, the aggregate estimator’s deviation from b is $\varTheta \left( \sqrt{\frac{\ln (1/\delta )}{T}}\right)$.

• If the position information is not available, and agents act strategically, the aggregate estimator has a mean square error of $O(\frac{1}{\sqrt{T}})$. Furthermore, with probability $1~-~\delta$, the aggregate estimator’s deviation from b is $\widetilde{O}\left( \sqrt{\frac{\ln (1/\delta )}{\sqrt{T}}}\right)$.

“Rumor is not always wrong” De vita et moribus Iulii Agricolae — Publius Cornelius TACITUS (56 - 117)