  Consider a Bayesian binary decision-making problem in star networks, where
local agents make selfish decisions independently, and a fusion agent makes a
final decision based on aggregated decisions and its own private signal. In
particular, we assume all agents have private beliefs for the true prior
probability, based on which they perform Bayesian decision making. We focus on
the Bayes risk of the fusion agent and counterintuitively find that incorrect
beliefs could achieve a smaller risk than that when agents know the true prior.
It is of independent interest for sociotechnical system design that the optimal
beliefs of local agents resemble human probability reweighting models from
cumulative prospect theory.
  We also consider asymptotic characterization of the optimal beliefs and
fusion agent's risk in the number of local agents. We find that the optimal
risk of the fusion agent converges to zero exponentially fast as the number of
local agents grows. Furthermore, having an identical constant belief is
asymptotically optimal in the sense of the risk exponent. For additive Gaussian
noise, the optimal belief turns out to be a simple function of only error costs
and the risk exponent can be explicitly characterized.
