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Closed form solutions for symmetric water filling games
We study power control in optimization and game frameworks. In the
optimization framework there is a single decision maker who assigns network
resources and in the game framework users share the network resources according
to Nash equilibrium. The solution of these problems is based on so-called
water-filling technique, which in turn uses bisection method for solution of
non-linear equations for Lagrange multiplies. Here we provide a closed form
solution to the water-filling problem, which allows us to solve it in a finite
number of operations. Also, we produce a closed form solution for the Nash
equilibrium in symmetric Gaussian interference game with an arbitrary number of
users. Even though the game is symmetric, there is an intrinsic hierarchical
structure induced by the quantity of the resources available to the users. We
use this hierarchical structure to perform a successive reduction of the game.
In addition, to its mathematical beauty, the explicit solution allows one to
study limiting cases when the crosstalk coefficient is either small or large.
We provide an alternative simple proof of the convergence of the Iterative
Water Filling Algorithm. Furthermore, it turns out that the convergence of
Iterative Water Filling Algorithm slows down when the crosstalk coefficient is
large. Using the closed form solution, we can avoid this problem. Finally, we
compare the non-cooperative approach with the cooperative approach and show
that the non-cooperative approach results in a more fair resource distribution
Closed form solutions for symmetric water filling games
We study power control in optimization and game frameworks. In the optimization framework there is a single decision maker who assigns network resources and in the game framework users share the network resources according to Nash equilibrium. The solution of these problems is based on so-called water-filling technique, which in turn uses bisection method for solution of non-linear equations for Lagrange multiplies. Here we provide a closed form solution to the water-filling problem, which allows us to solve it in a finite number of operations. Also, we produce a closed form solution for the Nash equilibrium in symmetric Gaussian interference game with an arbitrary number of users. Even though the game is symmetric, there is an intrinsic hierarchical structure induced by the quantity of the resources available to the users. We use this hierarchical structure to perform a successive reduction of the game. In addition, to its mathematical beauty, the explicit solution allows one to study limiting cases when the crosstalk coefficient is either small or large. We provide an alternative simple proof of the convergence of the Iterative Water Filling Algorithm. Furthermore, it turns out that the convergence of Iterative Water Filling Algorithm slows down when the crosstalk coefficient is large. Using the closed form solution, we can avoid this problem. Finally, we compare the non-cooperative approach with the cooperative approach and show that the non-cooperative approach results in a more fair resource distribution
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