Another convenient way to view, compare, and solve Cheap Talk vs. Bayesian Persuasion.
Linear-Programming Formulation: Cheap Talk vs. Bayesian Persuasion
Consider an information environment $\langle \Omega, A, M, \mu_0, u, v \rangle$ with state space $\Omega$, action space $A$, signal space $M$, prior $\mu_0$, receiver utility $u(\omega,a)$, and sender utility $v(\omega,a)$. (Let every set be finite).
Consider direct mechanisms. In general, the sender’s strategy $\sigma:\Omega \to \Delta(A)$ maps each state to a distribution over recommended actions, and the receiver’s strategy $\rho:A \to \Delta(A)$ specifies how she responds to each recommendation.
But note that we can absorb the mixed-strategy randomness into the sender’s stagey — assume that in equilibrium the receiver follows the sender’s recommendation, so $\rho(a)$ is degenerate at $a$. Let $$ x_{a \omega} = \mu_0(\omega)\sum_{m\in M}\sigma(m|\omega) \rho(a\mid m) $$ denote the (ex-ante) joint probability of state $\omega$ and induced action $a$ under $\sigma$. Now under this framework we can write Bayesian Persuasion and Cheap Talk in a unified optimization framework — say we’re maximizing sender’s payoff:
Bayesian Persuasion
$$ \begin{align*} \max_{x \ge 0} \quad & \sum_{a \omega} x_{a \omega}v(\omega,a) \cr \text{s.t.} \quad & \sum_a x_{a \omega} = \mu_0(\omega) && \forall \omega\cr & \sum_\omega x_{a \omega}[u(\omega,a) - u(\omega,a’)] \ge 0 && \forall a,a’ . \end{align*} $$
Cheap Talk
$$ \begin{align*} \max_{x \ge 0} \quad & \sum_{a, \omega} x_{a \omega}v(\omega,a) \cr \text{s.t.} \quad & \sum_a x_{a \omega} = \mu_0(\omega) && \forall \omega \cr & \sum_\omega x_{a \omega}[u(\omega,a) - u(\omega,a’)] \ge 0 && \forall a,a’ \cr & \sum_a x_{a \omega}[v(\omega,a) - \mathbf 1_{\lbrace{\sum_{\omega’}x_{a’ \omega’}>0}\rbrace}v(\omega,a’)] \ge 0 && \forall \omega,a’ . \end{align*} $$
Note it’s interesting and perhaps nontrivial to verify that these are indeed equivalent to the OG cheap talk/Bayesian persuasion definitions. But happy Friday :)