# Coalition Formation Game for Cost-Efficient Multiparty Payment Channel in Payment Channel Networks
## 1 Introduction
In this study, we focus on the two major challenges of the PCN as below. Unbalanced collateral: The PC user balance varies dynamically with payment flows. Accordingly, asymmetric payment flows cause unbalanced collateral and balance depletion, which stops payment for upcoming PBTs. Collateral cost: A PSP must lock the collateral amount for each PBT to prevent double spending. This collateral cost increases with the number of PBTs and their delays. First, the exhaustion of the balance due to the biased payment direction has been solved by an artificial opposite PBT flow. Revive [35] and Circle [36] search the circular payment flows to rebalance the channel. However, this approach assumes mutual trust among users on the circular path and has difficulty in finding the circular path without causing an imbalance in other channels. [PERSON] et al. [37] introduced a MPC for rebalance that provides the order of transactions among multiple PCs within a group. This work does not consider utility for MPC formation; accordingly, a given MPC topology limits rebalance. Second, there are several L3 approaches to rebalance deposits among existing L2 PCs. [PERSON] et al. [38] introduced the channel factory concept that reassigns deposit tokens to a specific pairwise channel dynamically based on agreement by \(n\)-party users. [PERSON] et al. [39; 40] proposed the virtual channel concept that creates a direct virtual channel between two users linked by multiple PCs. Following this, more studies [41; 42] improved upon the virtual channel concept. However, the security of these concepts relies on the behavior of underlying PSPs. This study proposes a multiparty payment channel (MPC) only using the L2 protocol that reduces the collateral cost of the PSP and the probability of deposit imbalance. For instance, if four users with pairwise channels are grouped into an MPC, the collateral cost decreases by one third, while the probability of incoming PBT flows increases by three. The MPC adopts slotted payment for sequential updates of an off-chain state. However, the utility of a grand coalition does not increase additively due to the delay cost in underlying networks. Because the turnaround delay for individual payment slots increases proportionally to the MPC size, multiple MPC coalitions must be created regionally to maximize their utility in the PCN. We propose a distributed algorithm based on the switching operation for each PSP in such a coalition formation game, which permits the PCN to converge to the Nash equilibrium of MPC partitions. We simulate the user payment flows in random social networks and investigate the performance of the MPC formation algorithm with varying turnaround delays and cost parameters. Experimental results show that the proposed algorithm forms MPC coalitions stably and improves the PCN utility consistently by limiting the individual MPC size according to the incidental cost. In summary, our key contributions in this study are highlighted below. * A multiparty payment channel is designed above Layer 2 in the blockchain network. * Expected threats to the MPC are discussed, such as double spending and the Sybil attack. * A mathematical model is established to derive the optimal MPC size. * A distributed MPC formation algorithm is proposed for this coalition formation game. The rest of this study is organized as follows. First, the background of the PC and network is briefly reviewed in Section 3. We introduce the MPC algorithm and its protocol design with smart contracts in Section 4. Section 5 describes a distributed MPC formation algorithm that maximizes PSP utility in the PCN. In Section 6, we evaluate the performance of the coalition formation algorithm and discuss the experimental results. We conclude this paper in Section 7.