  We examine the behavior of multi-agent networks where information-sharing is
subject to a positive communications cost over the edges linking the agents. We
consider a general mean-square-error formulation where all agents are
interested in estimating the same target vector. We first show that, in the
absence of any incentives to cooperate, the optimal strategy for the agents is
to behave in a selfish manner with each agent seeking the optimal solution
independently of the other agents. Pareto inefficiency arises as a result of
the fact that agents are not using historical data to predict the behavior of
their neighbors and to know whether they will reciprocate and participate in
sharing information. Motivated by this observation, we develop a reputation
protocol to summarize the opponent's past actions into a reputation score,
which can then be used to form a belief about the opponent's subsequent
actions. The reputation protocol entices agents to cooperate and turns their
optimal strategy into an action-choosing strategy that enhances the overall
social benefit of the network. In particular, we show that when the
communications cost becomes large, the expected social benefit of the proposed
protocol outperforms the social benefit that is obtained by cooperative agents
that always share data. We perform a detailed mean-square-error analysis of the
evolution of the network over three domains: far field, near-field, and
middle-field, and show that the network behavior is stable for sufficiently
small step-sizes. The various theoretical results are illustrated by numerical
simulations.
