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Coalitions in Cooperative Wireless Networks
Suhas Mathur, Lalitha Sankar, Narayan B. Mandayam
TL;DR
The paper asks when rational users in a wireless interference channel will remain in the grand coalition despite incentives to form smaller groups. It analyzes receiver and transmitter cooperation under different gain-apportionment schemes. The results show that stability depends on the cooperation strategy, detector, channel gains, and jamming conditions.
Problem
For rational users, it is not clear when the grand coalition is stable, since coalition members may prefer to leave for higher rates.
Source: §I · p. 1Method
The paper uses coalitional game theory to study rate-based utility in an interference channel with receiver and transmitter cooperation.
Source: §I · p. 2Results
Jointly decoding receivers have a stable, sum-rate-maximizing grand coalition, while stability in other cooperation models depends on the detector, channel gains, or jamming conditions.
Source: §I · pp. 1–2, §VI · p. 11Takeaways & Limitations
The grand coalition's sum-rate optimality does not guarantee stability; whether users stay together depends on the cooperation and rate-apportionment model.
Source: §I · p. 1, §VI · p. 11
Abstract
from arXiv · showhide
Cooperation between rational users in wireless networks is studied using coalitional game theory. Using the rate achieved by a user as its utility, it is shown that the stable coalition structure, i.e., set of coalitions from which users have no incentives to defect, depends on the manner in which the rate gains are apportioned among the cooperating users. Specifically, the stability of the grand coalition (GC), i.e., the coalition of all users, is studied. Transmitter and receiver cooperation in an interference channel (IC) are studied as illustrative cooperative models to determine the stable coalitions for both flexible (transferable) and fixed (non-transferable) apportioning schemes. It is shown that the stable sum-rate optimal coalition when only receivers cooperate by jointly decoding (transferable) is the GC. The stability of the GC depends on the detector when receivers cooperate using linear multiuser detectors (non-transferable). Transmitter cooperation is studied assuming that all receivers cooperate perfectly and that users outside a coalition act as jammers. The stability of the GC is studied for both the case of perfectly cooperating transmitters (transferrable) and under a partial decode-and-forward strategy (non-transferable). In both cases, the stability is shown to depend on the channel gains and the transmitter jamming strengths.
I. INTRODUCTION
The paper asks when rational wireless users will stay in the grand coalition (GC), since cooperation costs and unequal rate sharing can make smaller coalitions more attractive. It uses coalitional game theory to study receiver and transmitter cooperation, showing that GC stability depends on the cooperation model and how gains are apportioned.
I. INTRODUCTION
- Cooperation may impose costs on users who do not receive the greatest immediate benefits, while gain-apportionment rules can discourage participation.Users may instead form coalitions closed to outsiders.Source: §I · p. 1
- With equal division of sum-rate, the GC is desirable only when users have similar SNRs; stable coalitions can instead exclude a weak interferer.Although the three-user MAC achieves maximum sum-rate when all users cooperate, that fact alone does not establish GC stability.Source: §I · p. 1
- The paper studies rate as user utility and distinguishes transferable from non-transferable gain apportionment in a K-link interference channel.The sections proceed from coalitional-game theory and system models to receiver and transmitter cooperation.Source: §I · pp. 1–2
- Joint decoding by cooperating receivers makes the GC the stable, sum-rate-maximizing coalition structure.The introduction also reports that the GC is always stable with an MMSE detector, but is stable with a decorrelating detector only in the high-SNR regime.Source: §I · pp. 1–2
- With perfect transmitter cooperation, the GC maximizes sum-rate but is not always stable; stability depends on relative channel strengths.The model treats outside coalitions as worst-case jammers, and the GC is the only viable stable structure when it is cohesive.Source: §I · p. 2
- Under partial decode-and-forward, clustered users can have incentives to leave when jamming is weak, and arbitrary inter-user channels may produce non-cohesive games.For clustered channels, the achievable rate region is maximized when transmitters in a coalition decode one another's messages.Source: §I · p. 2
II. COALITIONAL GAME THEORY FOR RECEIVER AND TRANSMITTER COOPERATION
This section introduces coalitional-game concepts for evaluating stable cooperation. It distinguishes games where coalition members' utilities depend on outsiders from characteristic-function games, then defines transferable and non-transferable utility and core-based stability.
II. COALITIONAL GAME THEORY FOR RECEIVER AND TRANSMITTER COOPERATION
- A characteristic-function game assumes coalition utilities are unaffected by outsiders, but transmitter-cooperation rates depend on outsiders' strategies.The paper models those outsiders as worst-case jammers; receiver-only cooperation is in characteristic-function form.Source: §II · p. 2
- Transferable-utility games allow arbitrary feasible division of coalition value, whereas non-transferable games constrain the attainable payoff vectors.A payoff vector records members' shares of coalition rate.Source: §II · pp. 2–3
- Coalition structures grow exponentially in number with the users, and finding the sum-rate-maximizing structure is NP-complete.Cohesiveness simplifies the search by making the grand coalition the only possible stable structure.Source: §II · pp. 2–3
- A cohesive transferable-utility game gives the grand coalition at least the value of every partition, while superadditivity requires this property for every pair of disjoint coalitions.Cohesiveness makes the grand coalition optimal, whereas superadditivity applies the comparison to any two disjoint groups and their union.Source: §II · p. 3
- For cohesive games, the core contains feasible grand-coalition payoffs that no coalition can improve upon for all its members.For transferable utility, core non-emptiness can be tested as feasibility of a linear program.Source: §II · p. 3
- A non-transferable-utility game represents each coalition by a set of attainable rate vectors, and its core excludes payoffs that another coalition can improve for every member.The paper defines cohesive NTU games and focuses on GC stability for that class.Source: §II · p. 3
A. Channel Model
The channel model consists of K transmitter–receiver links with fixed gains and additive white Gaussian noise. Transmitters use Gaussian signaling subject to power constraints.
A. Channel Model
- The network has K transmitter–receiver pairs, indexed by the set K, with each link modeled as an additive white Gaussian noise channel.The received signal is specified at receiver m.Source: A. Channel Model · pp. 3–4
- Noise samples at the receivers are independent, identically distributed, proper complex Gaussian variables with zero mean and unit variance.Source: A. Channel Model · p. 4
- Transmitters use Gaussian signaling subject to their power constraints; co-located receivers reduce the model to a multiple-access channel.The paper uses user and transmitter interchangeably and denotes the set of signals for a coalition S by X_S.Source: A. Channel Model · p. 4
B. Cooperation Models
The paper considers receiver cooperation through joint decoding or linear multiuser detectors, and transmitter cooperation through perfect cooperation or partial decode-and-forward. The transmitter models assume joint decoding at receivers and worst-case jamming by competing coalitions.
B. Cooperation Models
- Cooperating receivers communicate over noiseless links, treat outside transmitters as interference, and jointly decode as a SIMO-MAC.The Gaussian input signaling achieves the known capacity region for this model.Source: B. Cooperation Models · p. 4
- The linear-detector model uses a co-located-receiver, single-antenna MAC with synchronized BPSK CDMA, no power control, and common signature correlation ρ.A bank of matched filters produces the received signal vector.Source: B. Cooperation Models · p. 4
- Transmitter cooperation assumes joint decoding by all receivers and worst-case jamming from competing coalitions.Perfect cooperation models each coalition as a multi-input, K-output MIMO channel maximizing sum-capacity under per-antenna power constraints.Source: B. Cooperation Models · p. 4
- Partial decode-and-forward lets coalition members decode and jointly forward part of their message streams in a full-duplex cooperative MAC.The model assumes co-located receivers.Source: B. Cooperation Models · p. 4
- Clustered channels have stronger inter-user links than links to the destination, a setting associated with users overcoming relatively poor direct channels.Source: B. Cooperation Models · p. 4
IV. RECEIVER COOPERATION
This section presents stable-coalition results for receiver cooperation in an interference channel; the cooperation models are described in Section III-B.
IV. RECEIVER COOPERATION
- The section reports stable coalitions for receiver cooperation in an interference channel.Source: §IV · p. 4
- The receiver-cooperation models are described in Section III-B.Source: §IV · p. 4
- The section presents the results for those models.Source: §IV · p. 4
A. Receiver Cooperation via Joint Decoding (TU game)
For joint-decoding receiver cooperation, each coalition’s value is its maximum information-theoretic sum-rate. The game is superadditive, and the grand coalition is the stable structure maximizing spectrum utilization.
A. Receiver Cooperation via Joint Decoding (TU game)
- A coalition’s value v(S) is the maximum information-theoretic sum-rate achieved by its links.The value can be apportioned among coalition members in any manner, and users may leave if another coalition offers a greater rate.Source: A. Receiver Cooperation via Joint… · p. 4
- Figure 3 depicts receiver coalitions in a K-link interference channel with joint decoding and noncooperating transmitters.Source: Figure 3 · p. 5
- The grand coalition is the stable coalition structure that maximizes spectrum utilization with jointly decoding cooperating receivers.The game is superadditive, so the grand coalition maximizes the sum-rate, which is equivalent to spectrum utilization.Source: A. Receiver Cooperation via Joint… · pp. 4–5
- Every feasible payoff profile on the capacity region’s dominant face lies in the core, which is nonempty and generally non-unique.The dominant face comprises rate profiles maximizing the sum-rate of all transmitters.Source: A. Receiver Cooperation via Joint… · p. 5
B. Receiver Cooperation using Multiuser Detectors (NTU game)
For receiver cooperation using linear multiuser detectors, grand-coalition stability depends on the detector: it holds at every SNR for MMSE, but only at high SNR for decorrelating detection.
Detector model
- Coalitions use linear multiuser detectors to process matched-filter signals from random-signature transmitters, yielding non-transferable utility games.Each link's rate is assumed to increase monotonically with its receiver SINR.Source: B. Receiver Cooperation using… · pp. 5–6
MMSE detector
- The grand coalition is always the stable, sum-rate-maximizing coalition with an MMSE detector.The linear MMSE receiver minimizes both noise and interference for links in a coalition.Source: B. Receiver Cooperation using… · p. 6
Decorrelating detector
- With a decorrelating detector, the grand coalition is stable and sum-rate maximizing only in the high-SNR regime.At high SNR, every link achieves its largest SINR and therefore rate in the grand coalition.Source: B. Receiver Cooperation using… · p. 6
A. Transmitter Cooperation: Perfect Transmit Side-Information
Perfect transmitter cooperation forms a cohesive game, making the grand coalition the only possible stable structure, but numerical analysis shows its core can be empty. The analysis also conjectures stability when channel gains and powers are comparable.
Transmitter cooperation model
- Under any coalition partition, each coalition acts as a multi-antenna transmitter in a MIMO-MAC; the grand coalition reduces to a MIMO point-to-point channel.The channel has per-antenna power constraints.Source: A. Transmitter Cooperation: Perfect… · p. 7
Game properties
- The transmitter cooperation jamming game is cohesive, and each coalition's sum-rate is evaluated under worst-case interference from users outside it.The coalition's rate can be arbitrarily apportioned among its members, making the game transferable-utility.Source: A. Transmitter Cooperation: Perfect… · p. 7
Cohesiveness and stability
- Cohesiveness makes the grand coalition the only possible stable structure, but does not guarantee a stable payoff profile when the core is empty.The analysis says an analytical core proof is intractable because it requires comparing channel- and power-dependent rate regions.Source: A. Transmitter Cooperation: Perfect… · p. 8
Stability of the grand coalition
- Numerical evaluation finds an empty core, so the grand coalition is unstable; breakaway incentives can instead produce oscillatory departures, with no stable alternative.The authors conjecture that the core is non-empty when channel gains and transmitter powers are comparable.Source: A. Transmitter Cooperation: Perfect… · p. 8
B. Transmitter Cooperation: Partial Decode-and-Forward (PDF)
This section studies clustered transmitter cooperation under partial decode-and-forward (PDF), with users outside each coalition treated as worst-case jammers. It identifies a power allocation that maximizes the PDF rate region, but shows that the grand coalition need not be cohesive.
B. Transmitter Cooperation: Partial Decode-and-Forward (PDF)
- PDF gives each cooperating user two new message streams and a cooperative message; cooperating users decode one stream from one another, and the destination decodes all streams.The users outside a coalition are modeled as worst-case jammers.Source: B. Transmitter Cooperation: Partial… · p. 8
- The PDF jamming game is non-transferable, with each coalition valued by its achievable rate region under worst-case jamming from outsiders.The coalition value is a rate region rather than a freely transferable payoff.Source: B. Transmitter Cooperation: Partial… · pp. 8–9
- For clustered users, the PDF rate region is maximized by setting each cooperating user's direct-message power p_k,d to zero.The resulting rate bounds are concave, simplifying rate-region evaluation.Source: B. Transmitter Cooperation: Partial… · pp. 9–10
- Setting p_k,d to zero makes the rate bounds concave, so evaluating a coalition's PDF rate region no longer requires a convex-hull operation.The concavity follows from the maximizing allocation established for clustered coalitions.Source: B. Transmitter Cooperation: Partial… · pp. 9–10
- Checking cohesiveness requires verifying rate-region conditions whose complexity grows exponentially with the number of users K.The conditions depend on channel parameters and transmit powers.Source: B. Transmitter Cooperation: Partial… · p. 11
- In Example 25, the {1, 2} rate region contains the GC's R1-R2 projection, so the GC is not cohesive or stable in that example.The distant user 3's decoding requirement tightens the GC bounds, while user 3 acts as a weak jammer outside {1, 2}.Source: B. Transmitter Cooperation: Partial… · pp. 10–11, Figure 7 · p. 10
VI. CONCLUDING REMARKS
The concluding remarks show that grand-coalition stability depends on the cooperation scheme and its conditions. In particular, detector choice, channel gains, jamming, and noisy inter-user links can affect stability.
VI. CONCLUDING REMARKS
- With receiver cooperation by joint decoding, the GC is both stable and sum-rate optimal.Source: §VI · p. 11
- With linear multiuser detectors, GC stability depends on the detector and SNR; decorrelator noise enhancement can disincentivize GC membership at low SNR.Source: §VI · p. 11
- With noise-free transmitter cooperation, the GC is sum-rate optimal but may not be stable.Source: §VI · p. 11
- For clustered transmitters using PDF, GC optimality and stability depend on network geometry and users' jamming potential; noisy inter-user channels can also affect GC stability.Source: §VI · p. 11