This paper considers a problem where multiple users make repeated decisions based on their own observed events. The events and decisions at each time step determine the values of a utility function and a collection of penalty functions. The goal is to make distributed decisions over time to maximize time average utility subject to time average constraints on the penalties. An example is a collection of power constrained sensor nodes that repeatedly report their own observations to a fusion center. Maximum time average utility is fundamentally reduced because users do not know the events observed by others. Optimality is characterized for this distributed context. It is shown that optimality is achieved by correlating user decisions through a commonly known pseudorandom sequence. An optimal algorithm is developed that chooses pure strategies at each time step based on a set of time-varying weights.
The framework of partial knowledge at each user is similar in spirit to a @cite_9 @cite_14 . There, the goal is to design competitive strategies that lead to a Nash equilibrium. This is significantly different from the goal of the current paper. The current paper is not concerned with competition or equilibrium. Rather, there is a single utility function that all users desire to maximize. Distributed algorithms are developed to maximize time average utility subject to time average penalty constraints.
@cite_9 @cite_14
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We study in this paper a noncooperative approach for sharing resources of a common pool among users, wherein each user strives to maximize its own utility. The optimality notion is then a Nash equilibrium. First, we present a general framework of systems wherein a Nash equilibrium is Pareto inefficient, which are similar to the 'tragedy of the commons' in economics. As examples that fit in the above framework, we consider noncooperative flow-control problems in communication networks where each user decides its throughput to optimize its own utility. As such a utility, we first consider the power which is defined as the throughput divided by the expected end-to-end packet delay, and then consider another utility of additive costs. For both utilities, we establish the non-efficiency of the Nash equilibria. We address the problem of spectrum pricing in a cognitive radio network where multiple primary service providers compete with each other to offer spectrum access opportunities to the secondary users. By using an equilibrium pricing scheme, each of the primary service providers aims to maximize its profit under quality of service (QoS) constraint for primary users. We formulate this situation as an oligopoly market consisting of a few firms and a consumer. The QoS degradation of the primary services is considered as the cost in offering spectrum access to the secondary users. For the secondary users, we adopt a utility function to obtain the demand function. With a Bertrand game model, we analyze the impacts of several system parameters such as spectrum substitutability and channel quality on the Nash equilibrium (i.e., equilibrium pricing adopted by the primary services). We present distributed algorithms to obtain the solution for this dynamic game. The stability of the proposed dynamic game algorithms in terms of convergence to the Nash equilibrium is studied. However, the Nash equilibrium is not efficient in the sense that the total profit of the primary service providers is not maximized. An optimal solution to gain the highest total profit can be obtained. A collusion can be established among the primary services so that they gain higher profit than that for the Nash equilibrium. However, since one or more of the primary service providers may deviate from the optimal solution, a punishment mechanism may be applied to the deviating primary service provider. A repeated game among primary service providers is formulated to show that the collusion can be maintained if all of the primary service providers are aware of this punishment mechanism, and therefore, properly weight their profits to be obtained in the future.