Conjectural Equilibrium in Water-filling Games

TL;DR

This paper introduces the concept of conjectural equilibrium in water-filling games, showing how users can learn and adapt their power allocations to improve their rates without prior knowledge of opponents.

Contribution

It models interference as a function of power allocation, proves the existence of conjectural equilibrium, and develops algorithms for belief formation and strategy optimization.

Findings

Foresighted users can learn interference functions effectively.

Conjectural equilibrium generalizes Nash and Stackelberg equilibria.

Numerical results show improved rates for all users.

Abstract

This paper considers a non-cooperative 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 multi-user interaction, the conjectural equilibrium is introduced, and the existence of this equilibrium for the investigated water-filling game is proved. Interestingly, both the Nash equilibrium and the Stackelberg equilibrium are shown to be special cases of the generalization of conjectural equilibrium. We develop practical algorithms to form accurate beliefs and search desirable power allocation strategies. Numerical simulations indicate that a foresighted user without any a priori knowledge of its competitors' private information can effectively learn the required information, 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.