Finite Representation and Greedy Implementation in Markov Persuasion

A sender repeatedly discloses a Markov state to a myopic receiver. We study two simplifications: using a common finite set of posterior destinations and maximizing current persuasion at every date. Neither exact simplification implies the other, even within one open four-action payoff family. When both are available, their combination uses the least dynamic menu. For finite-action payoffs with sender-favorable ties and any fixed finite transition, each positive tolerance admits one finite menu that approximates value uniformly over all priors and discounts below one. At every discount where greedy and unrestricted values agree at all priors, that same menu supports an approximately optimal policy that is exactly greedy at every date. Policies may vary with the discount. A complete binary criterion and a general small-discount theorem identify when current persuasion is dynamically optimal; strictly positive one-period transitions give polynomial menu-size bounds. The complete preprint consists of the main manuscript and its companion supplement. The complete proof of Theorem 7 is in Section S1 of the supplement.