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, applications 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 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. The stability of the coalitions is studied using the concept of a core. Then two types of fairness, namely, the min-max fairness using nucleolus and the average fairness using the Shapley function are investigated. Finally, a protocol is designed using both repeated games and coalition games. Simulation results show how boundary nodes and backbone nodes form coalitions together 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 problem in selfish wireless ad‑hoc networks: the “curse of boundary nodes.” In such networks, nodes are reluctant to forward others’ packets because forwarding consumes precious battery energy. Repeated‑game mechanisms can enforce cooperation when nodes mutually depend on each other, but boundary nodes depend only on interior (backbone) nodes while the backbone nodes have no reciprocal dependence. Consequently, backbone nodes have no incentive to forward packets for boundary nodes, and the usual threat of retaliation or reputation loss cannot be applied.

To break this deadlock, the authors propose a novel framework that combines cooperative transmission with coalition game theory. Using an amplify‑and‑forward cooperative relaying scheme, boundary nodes act as relays for backbone nodes. This cooperation reduces the transmit power required by the backbone source, giving the backbone node a tangible benefit. In exchange, the backbone node agrees to forward a certain fraction (α_i) of the boundary node’s packets. The interaction is modeled as a coalition game (N, v), where N is the set of all nodes and v(S) quantifies the total power‑saving (or monetary) value that coalition S can achieve through cooperative transmission. The characteristic function satisfies v(∅)=0 and super‑additivity, ensuring that larger coalitions are never worse than the sum of smaller ones.

Stability of any proposed payoff distribution is examined through the concept of the core. A payoff vector U belongs to the core if (i) the sum of all individual payoffs equals the grand‑coalition value v(N) and (ii) every sub‑coalition S receives at least its worth v(S). If a distribution lies outside the core, some subset of nodes would have an incentive to break away, rendering the coalition unstable.

Two fairness criteria are investigated for allocating the grand‑coalition surplus:

1. Min‑max fairness (nucleolus) – The nucleolus is the allocation that lexicographically minimizes the maximum excess (the amount by which any coalition’s worth exceeds the sum of its members’ payoffs). This yields a distribution that protects the most disadvantaged participant, thereby reducing the risk of coalition breakup.

2. Average fairness (Shapley value) – The Shapley value assigns to each node its expected marginal contribution over all possible joining orders. It reflects each node’s actual contribution to power saving (e.g., channel gains, relay positions) and distributes the surplus proportionally.

The authors embed these game‑theoretic mechanisms into a practical protocol that also retains the reputation‑based enforcement of repeated games. Initially, nodes engage in a repeated‑game phase to discover mutual dependencies. When a boundary node can assist a backbone node via cooperative transmission, it sends a coalition proposal specifying the desired α_i. After both sides accept, the backbone node forwards the agreed fraction of the boundary node’s traffic, and the boundary node supplies the relaying power. The payoff division follows either the nucleolus or the Shapley rule, depending on the chosen fairness policy. Ongoing monitoring via the repeated‑game component ensures that any deviation (e.g., a backbone node refusing to forward) triggers punitive strategies, preserving the coalition’s integrity.

Simulation studies are conducted on a two‑tier topology (one backbone source, multiple boundary relays). The performance metrics include network connectivity (the probability that any two nodes can communicate) and individual utilities. Results show that the proposed coalition‑based scheme improves overall connectivity by roughly 50 % compared with a baseline that relies solely on repeated games. Moreover, the nucleolus‑based allocation yields the highest minimum utility among participants, while the Shapley‑based allocation maximizes the average utility, confirming the theoretical fairness properties.

In summary, the paper makes three key contributions:

• It demonstrates that cooperative transmission can artificially create mutual dependence between boundary and backbone nodes, thereby eliminating the “curse of boundary nodes.”
• It formulates the interaction as a coalition game and rigorously analyzes stability (core) and fairness (nucleolus, Shapley), providing concrete allocation formulas.
• It proposes a hybrid protocol that blends repeated‑game incentives with coalition‑game payoff sharing, achieving substantial empirical gains in connectivity and energy efficiency.

The work bridges game‑theoretic concepts with practical wireless protocol design, offering a viable solution for incentivizing cooperation in selfish, energy‑constrained ad‑hoc networks.