Persuasion with limited communication capacity

We consider a Bayesian persuasion problem where the persuader and the decision maker communicate through an imperfect channel which has a fixed and limited number of messages and is subject to exogenous noise. Imperfect communication entails a loss of payoff for the persuader. We establish an upper bound on the payoffs the persuader can secure by communicating through the channel. We also show that the bound is tight: if the persuasion problem consists of a large number of independent copies of the same base problem, then the persuader can achieve this bound arbitrarily closely by using strategies which tie all the problems together. We characterize this optimal payoff as a function of the information-theoretic capacity of the communication channel. JEL Classification Numbers: C72, D82, D83. The authors thank James Best, Olivier Gossner, Frédéric Koessler, Marie Laclau, Daniel Martin, Ludovic Renou, Thomas Rivera, Jakub Steiner, Colin Stewart for stimulating discussions and comments. We also thank participants of: the 6th workshop on Stochastic Methods in Game Theory, Erice May 2017; the 13th European Meeting on Game Theory (SING13), Paris July 2017; the XXVI Colloque Gretsi, Juan-Les-Pins, September 2017; the 10th Transatlantic Theory Workshop, Paris September 2017; the 55th Allerton Conference, Monticello, Illinois, October 2017. We thank the Institute Henri Poincaré for hosting numerous research meetings. ETIS UMR 8051, Université Paris Seine, Université Cergy-Pontoise, ENSEA, CNRS, F-95000, Cergy, France; [email protected]; sites.google.com/site/maelletreust/. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01). HEC Paris and GREGHEC, 1 rue de la Libération, 78351 Jouy-en-Josas, France; [email protected]; studies2.hec.fr/jahia/Jahia/tomala. Tristan Tomala gratefully acknowledges the support the HEC foundation and ANR/Investissements d’Avenir under grant ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX0047.