  Until now, distributed algorithms for rational agents have assumed a-priori
knowledge of $n$, the size of the network. This assumption is challenged here
by proving how much a-priori knowledge is necessary for equilibrium in
different distributed computing problems. Duplication - pretending to be more
than one agent - is the main tool used by agents to deviate and increase their
utility when not enough knowledge about $n$ is given. The a-priori knowledge of
$n$ is formalized as a Bayesian setting where at the beginning of the algorithm
agents only know a prior $\sigma$, a distribution from which they know $n$
originates. We begin by providing new algorithms for the Knowledge Sharing and
Coloring problems when $n$ is a-priori known to all agents. We then prove that
when agents have no a-priori knowledge of $n$, i.e., the support for $\sigma$
is infinite, equilibrium is impossible for the Knowledge Sharing problem.
Finally, we consider priors with finite support and find bounds on the
necessary interval $[\alpha,\beta]$ that contains the support of $\sigma$,
i.e., $\alpha \leq n \leq \beta$, for which we have an equilibrium. When
possible, we extend these bounds to hold for any possible protocol.
