# Coalition Formation Game for Cost-Efficient Multiparty Payment Channel in Payment Channel Networks
## 2 Related Works  
The payment channel imbalance and depletion problem is a well-known problem and has been studied for the past several years. In this section, we introduce details of these studies and compare them as in Table 1. Since the channel depletion is caused by imbalanced payment flows in the PCN, many researchers solved it by equalizing the PBT flows. As a seminal work, Revive [35] proposed an artificial opposite PBT flow to achieve balance between bi-directional PBT flows within a payment channel. For this, a user has to find a circular path for the rebalancing, in which each user probably suffers from imbalanced flows. Revive has a critical limitation in that all users along the circular path should trust each other; otherwise, there is no mechanism to receive payback when a certain party denies pay forwarding. For this, Circle [36] was recently proposed, which creates a smart contract to prevent the misbehavior of these participants in the circular payment.  
In [38], a multiparty channel was proposed for the first time for the Bitcoin micropayment channel. The authors created a middle layer between the Bitcoin and micropayment channel layers, called a channel factory, that replaces a transaction with a shared lock on the channel deposit whenever a rebalance among the micro-payment channels is required. However, it is still costly for relocation transactions due to the nature of the Bitcoin network and relies on complicated data structures of timelocked transactions for state consensus.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Name** & **Cryptocurrency** & **Layer** & **Main Idea** \\ \hline Revive [35] & Any & L2 & Circular PBT flow \\ \hline Channel Factory [38] & Bitcoin & L1.5 & Create timelock sub-transactions \\ \hline Perun [39; 40] & Ethereum & L2 & Create virtual channel using smart contract \\ \hline Perun extend [42; 43] & Bitcoin & L2 & Create virtual channel using timelocks and multi-signatures \\ \hline Circle [36] & Ethereum & L2 & Circular PBT flow by smart contract \\ \hline MPC [37] & Ethereum & L3 & Scheduled sub-PCs in the fixed MPC \\ \hline Proposed MPC & Ethereum & L2 & Partially/fully synchronized MPC with dynamic topology \\ \hline \hline \end{tabular}
\end{table}
Table 1: Previous works for imbalanced deposit in PCN.  
Perun, as a virtual payment channel over the L2 protocol, was proposed in [39; 40]. Perun is a payment hub that has direct channels with several parties without intermediary users. The authors extended Perun for a general multiparty channel [41]. Perun can be easily realized by smart contracts, compared to previous studies. However, it also assumes that the intermediary users along the virtual channel are trusted or can be punished for misbehavior by the smart contract. Accordingly, Perun can reuse the deposit for the virtual channel but it requires smart contracting with a delay. Meanwhile, [42; 43] extended Perun to the Bitcoin network as a lightweight virtual payment channel, only requiring timelocks and multi-signatures.  
Recently, the MPC concept based on multiple PCs was introduced in [37], which is designed only for a fixed PCN topology, in contrast to our proposal. As the MPC demands consensus on every transaction to all parties and schedules its individual PCs proportionally to balance, the transaction latency can increase exponentially according to the number of users. However, MPC formation is not discussed for such a scalability problem.