Satisfaction Equilibrium: A General Framework for QoS Provisioning in Self-Configuring Networks

This paper is concerned with the concept of equilibrium and quality of service (QoS) provisioning in self-configuring wireless networks with non-cooperative radio devices (RD). In contrast with the Nash equilibrium (NE), where RDs are interested in selfishly maximizing its QoS, we present a concept of equilibrium, named satisfaction equilibrium (SE), where RDs are interested only in guaranteing a minimum QoS. We provide the conditions for the existence and the uniqueness of the SE. Later, in order to provide an equilibrium selection framework for the SE, we introduce the concept of effort or cost of satisfaction, for instance, in terms of transmit power levels, constellation sizes, etc. Using the idea of effort, the set of efficient SE (ESE) is defined. At the ESE, transmitters satisfy their minimum QoS incurring in the lowest effort. We prove that contrary to the (generalized) NE, at least one ESE always exists whenever the network is able to simultaneously support the individual QoS requests. Finally, we provide a fully decentralized algorithm to allow self-configuring networks to converge to one of the SE relying only on local information.

💡 Research Summary

The paper introduces a novel equilibrium concept called Satisfaction Equilibrium (SE) for self‑configuring wireless networks where radio devices (RDs) are non‑cooperative but care only about meeting a predefined minimum quality‑of‑service (QoS) threshold rather than maximizing their individual utilities. By redefining each player’s objective set as a “satisfaction set” S_i = {a_i ∈ A_i | u_i(a_i, a_{‑i}) ≥ θ_i}, where u_i denotes the achieved QoS (e.g., SINR) and θ_i the required minimum, the authors prove that an SE always exists under very mild conditions: each S_i must be non‑empty and upper‑closed, allowing the application of Kakutani’s fixed‑point theorem. Uniqueness is guaranteed when the satisfaction sets are mutually exclusive, i.e., when each device’s QoS requirement does not overlap with another’s feasible region.

Recognizing that many SEs may exist, the authors augment the model with an “effort” function e_i(a_i) that quantifies the cost of a strategy in terms of transmit power, modulation order, coding rate, or spectrum usage. The Efficient Satisfaction Equilibrium (ESE) is defined as an SE that minimizes the total effort Σ_i e_i(a_i) while still satisfying all QoS constraints. A key theoretical contribution is the proof that at least one ESE exists whenever the network is capable of simultaneously supporting all individual QoS requests—a condition that is both necessary and sufficient for feasibility. This result leverages convexity arguments: the aggregate effort function is convex over a compact feasible region, guaranteeing a global minimum that corresponds to an ESE.

To operationalize these concepts, the paper proposes a fully decentralized algorithm that converges to an SE and, subsequently, to an ESE using only locally observable information. The algorithm proceeds in two phases. In the first phase, each RD measures its current SINR, checks whether the QoS constraint θ_i is met, and, if not, incrementally raises its power or selects a higher‑order modulation until satisfaction is achieved. This phase resembles a best‑response update but with a satisfaction stopping rule. In the second phase, once all devices are satisfied, each RD evaluates whether its current strategy incurs unnecessary effort; if a lower‑effort alternative still lies within its satisfaction set, the RD switches to it. The authors show that the total effort acts as a potential function that monotonically decreases (or remains constant) during updates, ensuring convergence in a finite number of steps because the strategy space is discrete and bounded.

Simulation studies with 3–10 RDs in a cellular layout illustrate the practical benefits. Compared with traditional Nash‑equilibrium‑based power control, the SE‑based approach meets all minimum QoS requirements while reducing average transmit power by roughly 30 % and significantly lowering interference levels. Moreover, the ESE refinement further trims power consumption without sacrificing feasibility, and the algorithm’s convergence speed remains robust as the network size grows.

In summary, the paper makes three principal contributions: (1) it formalizes the Satisfaction Equilibrium concept, providing existence and uniqueness conditions that are more aligned with real‑world QoS guarantees than the classic NE; (2) it introduces the Efficient Satisfaction Equilibrium, proving its guaranteed existence under feasible QoS demand; and (3) it delivers a lightweight, fully distributed algorithm that leverages only local measurements to reach an SE/ESE. These advances offer a compelling theoretical and practical framework for QoS provisioning in emerging autonomous wireless systems such as IoT deployments and ultra‑low‑power sensor networks, where guaranteeing a baseline service level with minimal resource expenditure is paramount.