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).
翻译:本文分析巴伊西亚说服中的竞争模式 {, 其中两个对称发送者为多个接收者的利益而争斗, 通过披露关于各自提案的品质( 即二进制状态 -- -- 高或低) 的信息, 每个发送者被允许承诺执行信号政策, 向每个接收者发送私人( 可能相关) 信号。 发送者的效用是让该发送者拥有保护者的单一组合功能。 我们描述平衡结构, 并显示平衡并非独一无二( 即使是简单的工具功能 )。 我们然后关注两个发送者游戏中的稳定价格( POS) -- 发送者福利的最佳比率( 即二进制状态 -- -- 高或低调状态) 。 当发送者功能为匿名子模块或匿名超级模式时, 我们分析POS与美元前质量的关系( e. e. 高质量的可能性) 和 美元下基质( 美元) 的稳定性( 美元) 或超级状态值之间的比率比率值, 也显示我们的一个结果的存在。