The Gaussian Multiple Access Wiretap Channel with Selfish Transmitters: A Coalitional Game Theory Perspective

TL;DR

This work studies secure communication over the $GMAC-WT$ with selfish transmitters through a coalitional-game lens. It shows that for the degraded channel, forming the grand coalition is in every transmitter’s interest and yields a unique, fair, and core-consistent secrecy-rate allocation; for the non-degraded two-user channel, cooperation may fail to be mutually beneficial, but when beneficial, a unique fair allocation that lies in the core exists. The analysis introduces a rigorous value function based on information-theoretic secrecy guarantees, proves core non-emptiness (or conditions for emptiness), and extends an axiomatic fairness framework (Efficiency, Symmetry, Envy-freeness) to obtain implementable allocations. The results provide fundamental insights into stability and fairness of cooperative protocols in physical-layer security, with open challenges for larger numbers of users. Practical impact lies in guiding protocol design for secure multiuser networks where users pursue individual secrecy rates rather than altruistic global optima.

Abstract

This paper considers the Gaussian multiple access wiretap channel (GMAC-WT) with selfish transmitters, i.e., who are each solely interested in maximizing their individual secrecy rate. The question then arises as to whether selfish transmitters can increase their individual secrecy rate by participating in a collective, i.e., multiple access, protocol instead of operating on their own. If yes, the question arises whether there is a protocol that satisfies all the participating transmitters simultaneously, in the sense that no transmitter has an incentive to deviate from the protocol. Utilizing coalitional game theory, these questions are addressed for the degraded GMAC-WT with an arbitrary number of transmitters and for the non-degraded GMAC-WT with two transmitters. In particular, for the degraded GMAC-WT, cooperation is shown to be in the best interest of all transmitters, and the existence of protocols that incentivize all transmitters to participate is established. Furthermore, a unique, fair, stable, and achievable secrecy rate allocation is determined. For the non-degraded GMAC-WT, depending on the channel parameters, there are cases where cooperation is not in the best interest of all transmitters, and cases where it is. In the latter cases, a unique, fair, stable, and achievable secrecy rate allocation is determined.

Figures (4)

• Figure 1: Degraded GMAC-WT with selfish transmitters in the presence of adversarial jammers, where the transmitters form $q$ coalitions. In the absence of adversarial jammers, set $S^n \leftarrow \emptyset$.
• Figure 2: Representation of a known achievable region tekin2008gaussian for the degraded GMAC-WT, the core $\mathcal{C}(v)$ defined in Remark \ref{['remcore']}, a subset of the core $\mathcal{C}^*(v)$ defined in Theorem \ref{['propcore']}, and the allocation $(R_1^*(v),R_2^*(v))$ defined in Proposition \ref{['prop1']} for two transmitters with power constraints $(\Gamma_1,\Gamma_2)= (2,1.4)$ with $h=0.3$ and $\Lambda =0$.
• Figure 3: Fix $\Gamma_1 = 1$, $\Gamma_2= 0.4$, and $\Lambda =0.1$. $h_1$ and $h_2$ vary between $0$ and $2$ with a step size of $0.1$. A red cross for the pair $(h_1,h_2)$ indicates that cooperation between the two transmitters is beneficial.
• Figure 4: Representation of the known achievable region $\mathcal{R}_a$ from ISIT17, the core $\mathcal{C}(v)$ defined in Definition \ref{['defcore']}, and the unique fair allocation $(R_1^*(v),R_2^*(v))$ defined in Proposition \ref{['Propaxi']} with the parameters $(\Gamma_1,\Gamma_2,h_1,h_2, \Lambda)= (1,0.4,0.6,0.8,0.1)$.

Theorems & Definitions (38)

• Definition 1
• Definition 2
• Theorem 1: chou21
• Theorem 2: chou21
• Remark 1
• Theorem 3
• Definition 3
• proof
• Definition 4: e.g. maschler1979geometric
• Remark 2