Distributed Compression with Selfish Terminals: Correlation Induces Anarchy

We consider the min-cost multicast problem with multiple correlated sources, where each receiver wants to reconstruct all the sources. We initiate the study of the inefficiency brought forth by the selfish behavior of the terminals in this scenario of multicast of multiple sources by modeling it as a noncooperative game among the terminals. The degradation in performance due to the lack of regulation is measured by the Price of Anarchy(POA), which is defined as the ratio between the cost of the worst possible Nash equilibrium and the socially optimum cost. Our main result is that the presence of source correlations can significantly increase the price of anarchy under some reasonable cost-splitting mechanisms as against the case of multicast with independent sources, where for a large class of cost functions, cost-splitting mechanisms can be designed that ensure that the price of anarchy is one. While establishing this result, we derive several interesting properties of the flows and rates at Nash equilibrium and the socially optimum solution and note the striking similarities between them. Further, we show that the Nash equilibrium is indeed a socially optimal solution albeit with a different set of (related) cost functions. This characterization of the Nash equilibrium also allows us to obtain a near tight upper bound on POA. The main techniques in our analysis are Lagrangian duality theory and the usage of the supermodularity of conditional entropy. Finally, all the results in this paper naturally extend to a large class of network information flow problems where the Slepian-Wolf polytope is replaced by any polymatroid, leading to a nice class of succinct multi-player games. We believe that the ideas in this paper allow the investigation of other practical and meaningful scenarios beyond network coding as well.