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 Formula: see text, over a finite state space K, is controlled through signals she obtains from a sender, who observes Formula: see text in real time. At each stage Formula: see text, the receiver takes an action based on his current belief, which, together with the realized state of Formula: see text, 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 Formula: see text-discounted sum of stage payoffs for a discount factor Formula: see text. We show that starting from any invariant distribution Formula: see text, the trajectory of the Formula: see text-discounted value is monotone decreasing in Formula: see text. 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 Formula: see text-discounted values (as Formula: see text) in the case where Formula: see text is ergodic. The results for the Markovian persuasion model are then extended to the Markov chain games model of Renault (2006).