Compensation-based Game for Spectrum Sharing in the Gaussian Interference Channel

This paper considers an optimization problem of sum-rate in the Gaussian frequency-selective channel. We construct a competitive game with an asymptotically optimal compensation to approximate the optimization problem of sum-rate. Once the game achieves the Nash equilibrium, all users in the game will operate at the optimal sum-rate boundary. The contributions of this paper are twofold. On the one hand, a distributed power allocation algorithm called iterative multiple waterlevels water-filling algorithm is proposed to efficiently achieve the Nash equilibrium of the game. On the other hand, we derive some sufficient conditions on the convergence of iterative multiple waterlevels water-filling algorithm in this paper. Through simulation, the proposed algorithm has a significant improvement of the performance over iterative water filling algorithm and achieves the close-to-optimal performance

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

Game theoretic aspects of distributed spectral coordination with application to DSL networks

In this paper we use game theoretic techniques to study the value of cooperation in distributed spectrum management problems. We show that the celebrated iterative water-filling algorithm is subject to the prisoner's dilemma and therefore can lead to severe degradation of the achievable rate region in an interference channel environment. We also provide thorough analysis of a simple two bands near-far situation where we are able to provide closed form tight bounds on the rate region of both fixed margin iterative water filling (FM-IWF) and dynamic frequency division multiplexing (DFDM) methods. This is the only case where such analytic expressions are known and all previous studies included only simulated results of the rate region. We then propose an alternative algorithm that alleviates some of the drawbacks of the IWF algorithm in near-far scenarios relevant to DSL access networks. We also provide experimental analysis based on measured DSL channels of both algorithms as well as the centralized optimum spectrum management

The Variable-Processor Cup Game

The problem of scheduling tasks on $p$ processors so that no task ever gets too far behind is often described as a game with cups and water. In the $p$-processor cup game on $n$ cups, there are two players, a filler and an emptier, that take turns adding and removing water from a set of $n$ cups. In each turn, the filler adds $p$ units of water to the cups, placing at most $1$ unit of water in each cup, and then the emptier selects $p$ cups to remove up to $1$ unit of water from. The emptier's goal is to minimize the backlog, which is the height of the fullest cup. The $p$-processor cup game has been studied in many different settings, dating back to the late 1960's. All of the past work shares one common assumption: that $p$ is fixed. This paper initiates the study of what happens when the number of available processors $p$ varies over time, resulting in what we call the \emph{variable-processor cup game}. Remarkably, the optimal bounds for the variable-processor cup game differ dramatically from its classical counterpart. Whereas the $p$-processor cup has optimal backlog $\Theta(\log n)$, the variable-processor game has optimal backlog $\Theta(n)$. Moreover, there is an efficient filling strategy that yields backlog $\Omega(n^{1 - \epsilon})$ in quasi-polynomial time against any deterministic emptying strategy. We additionally show that straightforward uses of randomization cannot be used to help the emptier. In particular, for any positive constant $\Delta$, and any $\Delta$-greedy-like randomized emptying algorithm $\mathcal{A}$, there is a filling strategy that achieves backlog $\Omega(n^{1 - \epsilon})$ against $\mathcal{A}$ in quasi-polynomial time

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