FEASIBLE JOINT POSTERIOR BELIEFS BAYESIAN COMMUNICATION N - PowerPoint PPT Presentation

ITAI ARIELI (TECHNION) YAKOV BABICHENKO (TECHNION) FEDOR SANDOMIRSKIY (TECHNION, HSE ST.PETERSBURG) OMER TAMUZ (CALTECH) FEASIBLE JOINT POSTERIOR BELIEFS

BAYESIAN COMMUNICATION N Receivers: POSTERIOR s 1 ∈ S 1 p ′ � 1 = P ( θ = 1 ∣ s 1 ) Random state: … θ = { p 1, prob. … s N ∈ S N 0, prob. 1 − p POSTERIOR p ′ � N = P ( θ = 1 ∣ s N ) signals with joint distribution P ( θ = 1 ∣ s N ) P ( s 1 , s 2 …, s N ∣ θ ) ? ? What joint distributions of posteriors on are feasible [0,1] N KNOWN RESULTS N=1: N>1: ‣ Ziegler (2020) SPLITTING LEMMA (R.AUMANN & M.MASCHLER / D.BLACKWELL) ‣ A necessary condition for feasibility on is feasible satisfies μ [0,1] ⟺ ‣ Mathevet, Perego, and Taneva (2019) ∫ [0,1] ‣ Belief hierarchies x d μ ( x ) = p martingale property NO ANALOG OF SPLITTING LEMMA IS KNOWN!

CHARACTERISATION OF FEASIBILITY FOR N=2 MARTINGALE PROPERTY NEW QUANTITATIVE BOUND ON DISAGREEMENT IS NOT SUFFICIENT ‣ Define δ ( A , B ) = = ∫ A × [0,1] x d μ − ∫ [0,1] × B y d μ ‣ Then Infeasible: ‣ Posteriors are common knowledge ≥ δ ( A , B ) ≥ − μ ( A × B ) μ ( A × B ) ‣ Bayesian-rationals cannot agree to μ for any feasible and disagree Aumann (1976) A , B ⊂ [0,1] . MAIN THEOREM ⟺ A distribution is feasible satisfies ‣ Martingale Property ‣ Quantitative bound on disagreement

APPLICATIONS INDEPENDENT POSTERIORS FEASIBILITY FOR PRODUCT DISTRIBUTIONS ‣ Yes! Is feasible? 1 ϕ Measure on , symmetric around . [0,1] 2 ‣ No! Is feasible? ⟺ ϕ ≥ SOSD Uniform ϕ × ϕ is feasible HOW MANY SIGNALS DO WE NEED? BAYESIAN PERSUASION ‣ Feasible set has extreme points with infinite support Receivers: Informed sender: ‣ Persuasion may require infinite p ′ � 1 number of signals ‣ For N=1, two signals are enough p ′ � EXAMPLE 2 utility E [ u ( p ′ � 2 ) ] ‣ Sender minimises 1 , p ′ � 2 ] = E [ ( p ′ � 2 − 0.5) ] cov[ p ′ � 1 , p ′ � 1 − 0.5)( p ′ � = − 1 ‣ value 32