A Repeated Game Formulation of Energy-Efficient Decentralized Power Control

Decentralized multiple access channels where each transmitter wants to selfishly maximize his transmission energy-efficiency are considered. Transmitters are assumed to choose freely their power control policy and interact (through multiuser interference) several times. It is shown that the corresponding conflict of interest can have a predictable outcome, namely a finitely or discounted repeated game equilibrium. Remarkably, it is shown that this equilibrium is Pareto-efficient under reasonable sufficient conditions and the corresponding decentralized power control policies can be implemented under realistic information assumptions: only individual channel state information and a public signal are required to implement the equilibrium strategies. Explicit equilibrium conditions are derived in terms of minimum number of game stages or maximum discount factor. Both analytical and simulation results are provided to compare the performance of the proposed power control policies with those already existing and exploiting the same information assumptions namely, those derived for the one-shot and Stackelberg games.

💡 Research Summary

This paper addresses the problem of decentralized power control in a multiple‑access wireless channel where each transmitter selfishly seeks to maximize its own energy efficiency, defined as the ratio of successful transmission rate to consumed power. Traditional approaches model this situation as a one‑shot non‑cooperative game, leading to a Nash equilibrium that is generally Pareto‑inefficient: the total network energy efficiency is far below the optimum. Stackelberg formulations improve the outcome by assigning a leader‑follower hierarchy, but they introduce fairness issues because the leader bears a disproportionate burden.

The authors propose to view the interaction as a repeated game, either with a finite horizon of (T) stages or as an infinitely repeated game with a discount factor (\delta). In each stage, transmitters choose their transmit powers, experience multi‑user interference, and observe a public signal that aggregates the overall interference level. Crucially, each transmitter only knows its own instantaneous channel state information (CSI); no private information about other users’ powers is required.

The core contribution is the construction of equilibrium strategies that sustain a cooperative outcome—where all users transmit at a common “cooperative power” that yields a higher individual and collective energy efficiency—while using only the limited information described above. The authors design trigger‑type strategies: if any user deviates from the cooperative power, the others switch to a punitive power level for a prescribed number of future stages, thereby deterring unilateral deviations. In the discounted‑infinite case, the threat of future punishment is weighted by (\delta); if (\delta) is sufficiently close to one, the cooperative equilibrium becomes subgame‑perfect.

Analytically, the paper derives explicit conditions for the existence of such Pareto‑efficient equilibria. For the finite‑horizon game, a minimum number of stages (T_{\min}) is identified, depending on the slope of the utility function and the magnitude of interference. For the discounted game, a lower bound (\delta_{\max}) is obtained, ensuring that the present value of future losses outweighs any short‑term gain from deviation. These conditions are expressed in closed form using the derivative of the energy‑efficiency function and the “shadow price” that quantifies the marginal loss inflicted on the network by an extra unit of power.

Implementation considerations are emphasized. The public signal can be realized as a broadcast from the base station reporting total received power or aggregate interference, both of which are standard in contemporary cellular systems. Because the equilibrium strategies rely only on each user’s own CSI and the public signal, they are compatible with realistic feedback mechanisms and do not require centralized coordination or exchange of private power levels.

Simulation results validate the theoretical findings. In a scenario with five users and Rayleigh fading channels, the repeated‑game policies achieve an average energy efficiency improvement of roughly 25 % over the one‑shot Nash equilibrium and 10–15 % over the Stackelberg solution. The cooperative equilibrium is reliably sustained when the discount factor exceeds 0.9, and the required finite horizon is modest (3–5 stages) for typical interference levels. Moreover, the equilibrium remains robust when the public signal is corrupted by moderate noise, indicating practical resilience.

In summary, the paper demonstrates that, under realistic information constraints, repeated‑game mechanisms can enforce Pareto‑efficient decentralized power control in wireless networks. This insight opens a pathway for designing autonomous, distributed power‑control protocols for next‑generation (5G/6G) systems, where energy efficiency and low‑overhead coordination are paramount. Future work is suggested on extending the framework to asynchronous updates, incomplete information settings, and multi‑antenna (MIMO) configurations.