Introducing Hierarchy in Energy Games

In this paper, we consider a decentralized multiple access channel (MAC). By definition [2], the MAC consists of a network of several transmitters and one receiver. The network is said to be decentralized in the sense that the receiver does not dictate to the users their transmit power level. Indeed, from the sole knowledge of his own uplink channel, each user can choose freely his power control policy in order to selfishly maximize a certain individual performance criterion, called utility (or payoff) in the context of game theoretic studies. The The material in this paper has been presented in part at the 2nd ACM-ICST International Workshop on Game Theory in Communication Networks (GAMECOMM), Athens, Greece, 20 Oct. 2008 [1]. use of game theoretic tools is at the heart of the design of the recently advocated mobile flexible networks [3], which intend to break the spectral efficiency barrier through the use of intelligence. The selfish behavior enables to reduce the signaling overhead, especially for highly mobile terminals where topological information (channel state information -CSI-is one aspect) on the network can not be centralized. In this paper, unlike many works concerning this problem, the chosen users' utility is not the transmission rate (e.g., [4], [5], [6]) but the energy-efficiency of their communication. The latter approach, which consists in maximizing the ratio of the net number of information bits that are transmitted without error per time unit to the transmit power level, has been introduced in [7] for flat fading channels and recently re-used by [8] for multi-carrier CDMA (code division multiple access) systems and linear receivers, motivated by the facts that mobile terminals have a limited battery lifetime and in some applications (e.g., a sensor network measuring a temperature field) the main concern is not the transmission rate.

As mentioned in [7] the Nash equilibrium (NE) in such games can be energy inefficient. The NE of this power control game is shown to be Pareto inefficient. This is why [9] proposed, for MACs with flat fading links and single-user decoding (SUD), a pricing mechanism to obtain improvements in the users’ utilities with respect to the case with no pricing. To our knowledge, since the release of [9], no alternative way of tackling this problem in the context of energy-efficient power control games has been proposed. In this paper we propose an alternative approach to [9] for improving the network efficiency by introducing a certain degree of hierarchy between the users. We propose two schemes. For the first scheme, we propose a Stackelberg formulation of the problem when SUD is assumed at the receiver. For the second scheme, we consider an a priori more efficient (and non-linear) receiver namely successive interference cancellation (SIC). Technically, our approach not only aims at improving the network equilibrium efficiency but has also two nice features:

1. It allows one to analyze the equilibrium uniqueness issue rigorously. Note that even in the simple case of linear pricing analyzed in [9], only simulations are provided to justify uniqueness; 2. Implementing pricing in real wireless networks is still an open issue for many well-used types of utilities (the problem being to know how to measure the modified utility) whereas energy-efficiency can be physically measured (for this purpose each terminal can evaluate its frame error rate over a certain period of time from a feedback mechanism and store the corresponding sequence of power levels); 3. Only individual CSI is needed at each transmitter in the regime of non-saturated equilibria, which is not case with pricing. More generally, our approach contributes to designing networks where intelligence is split between the base station (BS) and mobile stations (MSs) in order to find a desired trade-off between the global network performance reached at the equilibrium and the amount of signaling needed to make it work. As we will see, in both hierarchical approaches proposed the receiver only broadcasts common messages and the corresponding amount of additional signaling is reasonable. Note that the Stackelberg formulation arises naturally in some contexts of practical interest. For example, hierarchy is naturally present in contexts where there are primary (licensed) users and secondary (unlicensed) users who can sense their environment because there are equipped with a cognitive radio [12], [13], [14]. It is also natural if the users have access to the medium in an asynchronous manner. Note that there have been many works on Stackelberg games in the context of wireless communications [15], but they do not consider energy-efficiency for the individual utility as defined in [7], [8], [16]. Rather, they consider transmission rate-type utilities (see e.g., [5], [17], [18]). This paper is structured as follows. The general signal model is provided in Sec. II-A. Sec. II-B reviews the main results of [8] for the non-c

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