Competitive Information Disclosure with Multiple Receivers

This paper analyzes a model of competition in Bayesian persuasion in which two symmetric senders vie for the patronage of multiple receivers by disclosing information about the qualities (i.e., binary state -- high or low) of their respective proposals. Each sender is allowed to commit to a signaling policy where he sends a private (possibly correlated) signal to every receiver. The sender's utility is a monotone set function of receivers who make a patron to this sender. We characterize the equilibrium structure and show that the equilibrium is not unique (even for simple utility functions). We then focus on the price of stability (PoS) in the game of two senders -- the ratio between the best of senders' welfare (i.e., the sum of two senders' utilities) in one of its equilibria and that of an optimal outcome. When senders' utility function is anonymous submodular or anonymous supermodular, we analyze the relation between PoS with the ex ante qualities $\lambda$ (i.e., the probability of high quality) and submodularity or supermodularity of utility functions. In particular, in both families of utility function, we show that $\text{PoS} = 1$ when the ex ante quality $\lambda$ is weakly smaller than $1/2$, that is, there exists equilibrium that can achieve welfare in the optimal outcome. On the other side, we also prove that $\text{PoS} > 1$ when the ex ante quality $\lambda$ is larger than $1/2$, that is, there exists no equilibrium that can achieve the welfare in the optimal outcome. We also derive the upper bound of $\text{PoS}$ as a function of $\lambda$ and the properties of the value function. Our analysis indicates that the upper bound becomes worse as the ex ante quality $\lambda$ increases or the utility function becomes more supermodular (resp.\ submodular).