On the Monotonicity and Rate of Convergence of the Markovian Persuasion Value

TL;DR

This paper analyzes the properties of the Markovian persuasion model, showing how the discounted value behaves as the discount factor varies and providing bounds on its convergence rate in ergodic cases.

Contribution

It establishes the monotonicity of the discounted value trajectory and derives an upper bound on the convergence rate for ergodic Markov chains in persuasion games.

Findings

The discounted value decreases monotonically with the discount factor.

An upper bound on the convergence rate as the discount factor approaches 1.

Extension of results to Markov chain games.

Abstract

We study a dynamic Bayesian persuasion model called Markovian persuasion. In such a model, the belief of the receiver regarding the current state of a Markov chain $(X_n)_{n\geq 1}$, over a finite state space $K$, is controlled through signals she obtains from a sender, who observes $(X_n)_{n\geq 1}$ in real time. At each stage $n\geq 1$, the receiver takes an action based on his current belief, which together with the realized state of $X_n$, determines the $n$'th stage payoff of the sender. The sender's goal in a Markovian persuasion game is to find a signaling policy that maximizes her expected $\delta$-discounted sum of stage payoffs for a discount factor $\delta \in [0,1)$. We show that starting from any invariant distribution $(X_n)_{n\geq 1}$ the trajectory of the $\delta$-discounted value is a monotone decreasing in $\delta$. By combining this result with the opposite increasing monotone trajectories found in Lehrer and S.\ (2025, GEB), we are able to derive an upper bound on the rate of convergence of the $\delta$-discounted values (as $\delta \to 1^-$) in the case where $(X_n)_{n\geq 1}$ is ergodic. The results for the Markovian persuasion model are then extended to the Markov chain games model of Renault (2006, MOR).