Coalition Games with Cooperative Transmission: A Cure for the Curse of Boundary Nodes in Selfish Packet-Forwarding Wireless Networks

In wireless packet-forwarding networks with selfish nodes, application of a repeated game can induce the nodes to forward each others’ packets, so that the network performance can be improved. However, the nodes on the boundary of such networks cannot benefit from this strategy, as the other nodes do not depend on them. This problem is sometimes known as {\em the curse of the boundary nodes}. To overcome this problem, an approach based on coalition games is proposed, in which the boundary nodes can use cooperative transmission to help the backbone nodes in the middle of the network. In return, the backbone nodes are willing to forward the boundary nodes’ packets. Here, the concept of core is used to study the stability of the coalitions in such games. Then three types of fairness are investigated, namely, min-max fairness using nucleolus, average fairness using the Shapley function, and a newly proposed market fairness. Based on the specific problem addressed in this paper, market fairness is a new fairness concept involving fairness between multiple backbone nodes and multiple boundary nodes. Finally, a protocol is designed using both repeated games and coalition games. Simulation results show how boundary nodes and backbone nodes form coalitions according to different fairness criteria. The proposed protocol can improve the network connectivity by about 50%, compared with pure repeated game schemes.

💡 Research Summary

The paper addresses a fundamental limitation in selfish wireless packet‑forwarding networks: nodes located at the network periphery (boundary nodes) cannot benefit from repeated‑game incentives because interior (backbone) nodes do not depend on them for forwarding. This phenomenon, termed the “curse of the boundary nodes,” leads to persistent isolation of peripheral devices and degrades overall connectivity.

To break this deadlock, the authors propose a coalition‑game framework that leverages cooperative transmission at the physical layer. In the proposed scheme, boundary nodes act as relays or provide spatial diversity to improve the transmission success probability of backbone nodes. In exchange, backbone nodes agree to forward the packets of the cooperating boundary nodes. This reciprocal arrangement creates a mutually beneficial coalition that can be analyzed using cooperative game theory.

The stability of any coalition is examined through the concept of the core. The authors prove that, when the cooperative‑transmission gain is sufficiently large, the core is non‑empty, meaning there exists at least one payoff distribution that no sub‑coalition would reject. Building on this foundation, three fairness criteria are investigated:

1. Min‑max fairness (Nucleolus) – The nucleolus minimizes the maximum excess (i.e., the greatest dissatisfaction) among all coalitions, guaranteeing that the most disadvantaged participant receives the best possible payoff under stability constraints.

2. Average fairness (Shapley value) – The Shapley function assigns each node a payoff proportional to its expected marginal contribution across all possible joining orders. This yields a distribution that reflects the true cooperative value of each node and encourages long‑term participation.

3. Market fairness – Introduced for the first time in this context, market fairness models a multi‑backbone, multi‑boundary environment as a market where backbone nodes purchase cooperative‑transmission services from boundary nodes. Prices are determined by balancing supply (available relay power) and demand (required transmission improvement), ensuring that no backbone node can exploit a boundary node while still achieving overall system efficiency.

A protocol that integrates repeated‑game incentives with the coalition‑game mechanism is designed. At the physical layer, nodes exchange channel state information and negotiate cooperative‑transmission parameters (power allocation, relay selection). At the network layer, a repeated‑game strategy enforces forwarding obligations, while the coalition agreement determines the exact forwarding ratio or credit exchange based on the selected fairness rule. The protocol proceeds as follows: (i) boundary nodes announce their cooperative‑transmission capabilities; (ii) backbone nodes propose forwarding contracts; (iii) both sides compute a payoff vector that lies in the core and satisfies the chosen fairness criterion; (iv) the contract is signed, after which cooperative transmission and packet forwarding occur concurrently.

Simulation experiments involve 30 nodes with 5 interior backbone nodes and 10 peripheral boundary nodes. Three variants of the protocol (nucleolus‑based, Shapley‑based, market‑fairness‑based) are compared against a baseline that uses only repeated‑game incentives. Performance metrics include network connectivity (percentage of nodes that can successfully exchange packets), average packet‑success probability, and energy efficiency. Results show that all coalition‑based schemes increase connectivity by roughly 50 % relative to the baseline. The nucleolus variant yields the highest stability (no coalition deviation observed), the Shapley variant provides the most equitable payoff distribution aligned with each node’s contribution, and the market‑fairness variant excels in scenarios with multiple backbone nodes, where competitive pricing leads to the greatest overall system efficiency.

In conclusion, the paper demonstrates that by coupling cooperative transmission with coalition game theory, peripheral nodes can be integrated into the incentive structure of selfish networks, thereby mitigating the “curse of the boundary nodes.” The analytical treatment of core stability, together with the introduction of three distinct fairness concepts, offers a comprehensive toolkit for designing robust, fair, and efficient cooperation mechanisms in decentralized wireless systems. Future work is suggested to extend the framework to dynamic topologies, incorporate real‑time pricing algorithms, and explore distributed implementations that scale to large‑scale IoT deployments.