In this paper, we present two completely uncoupled algorithms for utility maximization. In the first part, we present an algorithm that can be applied for general non-concave utilities. We show that this algorithm induces a perturbed (by $epsilon$) Markov chain, whose stochastically stable states are the set of actions that maximize the sum utility. In the second part, we present an approximate sub-gradient algorithm for concave utilities which is considerably faster and requires lesser memory. We study the performance of the sub-gradient algorithm for decreasing and fixed step sizes. We show that, for decreasing step sizes, the Cesaro averages of the utilities converges to a neighbourhood of the optimal sum utility. For constant step size, we show that the time average utility converges to a neighbourhood of the optimal sum utility. Our main contribution is the expansion of the achievable rate region, which has been not considered in the prior literature on completely uncoupled algorithms for utility maximization. This expansion aids in allocating a fair share of resources to the nodes which is important in applications like channel selection, user association and power control.
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