Distributed energy resources (DERs) offer an opportunity to regulate voltage profiles in distribution networks through reactive power (VAR) support. The problem can be cast as to achieve a desired voltage profile by optimally coordinating the limited VAR resources. Limited by the communication capability in current distribution networks, the optimal voltage-VAR control results based on centralized or distributed optimization/control algorithms may be unachievable. Hence, it is important to understand the communication-optimality tradeoff in voltage-VAR control to facilitate communication resource deployment. To this end, we first characterize the voltage-VAR control problem with scarce communication links from a game theoretic perspective. The property of the Nash equilibrium is analyzed and the conditions for the uniqueness are investigated. We also propose a distributed voltage regulation algorithm to achieve the NE under limited communications. To further understand the value of communication links, we propose a bi-level optimization problem for communication link placement and provide some preliminary numerical results for this NP-hard problem.