Feasible Joint Posterior Beliefs

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(joint with Itai Arieli, Yakov Babichenko, and Omer Tamuz)
 
Abstract: Distribution of posterior belief of an agent is known to satisfy the martingale condition: the expected posterior is equal to the prior. For one agent, the martingale condition is both necessary and sufficient: for any distribution satisfying this condition, there is a signaling policy that induces it. This result, sometimes dubbed ''the Splitting lemma'', is the key technical tool in the theory of Bayesian persuasion with one receiver and the theory of games with incomplete information on one side. However, no extension of the Splitting lemma to several agents was known.
 
We characterise the set of feasible joint distributions of beliefs that can be induced by private signals sent to several agents. Along the way, we discover links to optimal transportation and maximal flows. Results are applied to multi-receiver Bayesian persuasion.
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