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Feasible Conditional Belief Distributions

Itai Arieli, Yakov Babichenko, Fedor Sandomirskiy

Abstract

Agents receive private signals about an unknown state. The resulting joint belief distributions are complex and lack a simple characterization. Our key insight is that, when conditioned on the state, the structure of belief distributions simplifies: feasibility constrains only the marginal distributions of individual agents across states, with no joint constraints within a state. We apply this insight to multi-receiver persuasion, identifying new tractable cases and introducing optimal transportation and duality tools.

Feasible Conditional Belief Distributions

Abstract

Agents receive private signals about an unknown state. The resulting joint belief distributions are complex and lack a simple characterization. Our key insight is that, when conditioned on the state, the structure of belief distributions simplifies: feasibility constrains only the marginal distributions of individual agents across states, with no joint constraints within a state. We apply this insight to multi-receiver persuasion, identifying new tractable cases and introducing optimal transportation and duality tools.
Paper Structure (26 sections, 11 theorems, 105 equations, 5 figures)

This paper contains 26 sections, 11 theorems, 105 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Related literature.
  4. Feasible Belief Distributions
  5. Implications for Persuasion
  6. Comparison to the single-receiver duality.
  7. Proposition \ref{['prop_dual value representation']} as an analog of the concavification formula.
  8. Comparison to the Kantorovich duality.
  9. Applications
  10. One-state persuasion
  11. Supermodular Persuasion
  12. Constructing Solutions via Duality
  13. Conclusion
  14. Infinite state spaces and uncommon priors.
  15. Posterior means.
  16. ...and 11 more sections

Key Result

Theorem 1

Distributions $(\mu^\omega)_{\omega\in \Omega}$ are feasible conditional distributions if and only if the one-receiver marginals $(\mu_i^\omega)_{\omega\in \Omega}$ are feasible in a one-receiver problem for each receiver $i$.

Figures (5)

  • Figure 1: Conditional and unconditional belief distributions for Example \ref{['ex_conditional_vs_unconditional']} placing as much weight on disagreement outcomes as permitted by feasibility. For accuracy $r$ close to $1$, the marginals force $\mu^\ell$ to put almost all weight on $(r,r)$ and $\mu^h$, on $(1-r,1-r)$. As a result, beliefs become almost perfectly correlated under $\mu$. Red/blue colors correspond to $\omega=\ell$ and $\omega=h$.
  • Figure 2: The joint distribution of beliefs for Example \ref{['ex_one_state']}. Prior is $1/2$. The numbers inside the square indicate the probabilities of each outcome, and red/blue colors correspond to $\omega=\ell$ and $\omega=h$, respectively.
  • Figure 3: The joint distribution of beliefs for Example \ref{['ex_olygop']} with pollution cost $C(q_1)=1-\exp(-2q_1)$. The numbers inside the square indicate the probabilities of each outcome, and red/blue colors correspond to $\omega=\ell$ and $\omega=h$, respectively.
  • Figure 4: The construction of $\alpha$ from \ref{['eq_alpha_h']} for $v(x_1,x_2)=|x_1-x_2|^3$; see Example \ref{['ex_x_to_third']}.
  • Figure 5: Construction of $\alpha_p$ from \ref{['eq_alpha_p']}. Left:$v(x_1,x_2)=|x_1-x_2|$ with $p=1/3$; full-information/no-information is optimal and thus $b_p=c_p=p$ (Example \ref{['retailer']}) Right:$v(x_1,x_2)=|x_1-x_2|\cdot \left|x_1-{1}/{2}\right|\cdot \left|x_2-{1}/{2}\right|$ with $p=1/2$; full-information/partial information with beliefs $b_p=(3-\sqrt{3})/6$ and $c_p=1-b_p$ of the partially-informed receiver is optimal (Example \ref{['ex_partial']}).

Theorems & Definitions (30)

  • Definition 1
  • Theorem 1: conditional feasibility for $n$ receivers
  • proof
  • Example 1: conditional vs. unconditional feasibility
  • Corollary 1: primal value representation
  • Proposition 1: dual value representation
  • Lemma 1
  • Lemma 2
  • Example 2: supporting group morale in bad states
  • Example 3
  • ...and 20 more