  Information exchange systems differ in many ways, but all share a common
vulnerability to selfish behavior and free-riding. In this paper, we build
incentives schemes based on social norms. Social norms prescribe a social
strategy for the users in the system to follow and deploy reputation schemes to
reward or penalize users depending on their behaviors. Because users in these
systems often have only limited capability to observe the global system
information, e.g. the reputation distribution of the users participating in the
system, their beliefs about the reputation distribution are heterogeneous and
biased. Such belief heterogeneity causes a positive fraction of users to not
follow the social strategy. In such practical scenarios, the standard
equilibrium analysis deployed in the economics literature is no longer directly
applicable and hence, the system design needs to consider these differences. To
investigate how the system designs need to change when the participating users
have only limited observations, we focus on a simple social norm with binary
reputation labels but allow adjusting the punishment severity through
randomization. First, we model the belief heterogeneity using a suitable
Bayesian belief function. Next, we formalize the users' optimal decision
problems and derive in which scenarios they follow the prescribed social
strategy. With this result, we then study the system dynamics and formally
define equilibrium in the sense that the system is stable when users
strategically optimize their decisions. By rigorously studying two specific
cases where users' belief distribution is constant or is linearly influenced by
the true reputation distribution, we prove that the optimal reputation update
rule is to choose the mildest possible punishment. This result is further
confirmed for higher order beliefs in simulations.
