Conjectural Equilibrium in Multiuser Power Control Games

This paper considers a noncooperative game in which competing users sharing a frequency-selective interference channel selfishly optimize their power allocation in order to improve their achievable rates. Previously, it was shown that a user having the knowledge of its opponents' channel state information can make foresighted decisions and substantially improve its performance compared with the case in which it deploys the conventional iterative water-filling algorithm, which does not exploit such knowledge. This paper discusses how a foresighted user can acquire this knowledge by modeling its experienced interference as a function of its own power allocation. To characterize the outcome of the multiuser interaction, the conjectural equilibrium is introduced, and the existence of this equilibrium for the investigated water-filling game is proven. Importantly, we show that both the Nash equilibrium and the Stackelberg equilibrium are special cases of the conjectural equilibrium. We also develop practical algorithms to form accurate beliefs and select desirable power allocation strategies. Numerical simulations indicate that a foresighted user without any a priori knowledge of its competitors' private information can effectively learn how the other users will respond to its actions, and induce the entire system to an operating point that improves both its own achievable rate as well as the rates of the other participants in the water-filling game.