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

We study a dynamic Bayesian persuasion model called Markovian persuasion, illustrated here with two players: the sender (he) and the receiver (she). In such a model, the belief of the receiver regarding the current state of a Markov chain ${(X_n)}_{n\ge 1}$, over a finite state space K, is controlled through signals she obtains from a sender, who observes ${(X_n)}_{n\ge 1}$ in real time. At each stage $n\ge 1$, 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 ${(X_n)}_{n\ge 1}$, the trajectory of the $\delta$-discounted value is monotone decreasing in $\delta$. By combining this result with the opposite increasing monotone trajectories found in Lehrer and Shaiderman [Lehrer E, Shaiderman D (2025) Markovian persuasion with stochastic revelations. Games Econom. Behav. 154:411–439], 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 ${(X_n)}_{n\ge 1}$ is ergodic. The results for the Markovian persuasion model are then extended to the Markov chain games model of Renault (2006).