Robust Equilibria in Additively Coupled Games in Communications Networks

We obtain the robust Nash equilibrium (RNE) for a wide range of multi-user communications networks under uncertainty by utilizing the robust optimization theory for the worst-case uncertainties. To do so, we consider the uncertainty as a distance between the estimated and the actual values of the system parameters as a general norm function, and utilize the finite-dimensions variational inequalities (VI) to derive the conditions for existence and uniqueness of RNE. Two effects of uncertainty on the performance of the system are investigated: the difference between the achieved social utility at the RNE and the Nash equilibrium (NE) of the nominal game, and the distance between the deployed strategies of users at the RNE and at the NE. We quantify these two effects for the cases of unique NE and multiple NEs, and show that when the NE is unique, the achieved social utility at the RNE is always less than that of the NE. Interestingly, the worst-case robustness approach may lead to a higher social utility at the RNE in the multiple NEs scenario. Considering uncertainty at RNE introduces coupling between users, and hence, developing distributed algorithms for reaching RNE is more challenging as compared to the NE in the nominal game. However, for some special forms of utilities and norm functions, we propose simultaneous and sequential distributed algorithms; and investigate the performance of the robust game for power control in interference channels, and for flow control in Jackson networks.