  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).
