# Coalition Formation Game for Cost-Efficient Multiparty Payment Channel in Payment Channel Networks
## 6 Experimental Results  
We generate multi-PBT flows in SNS-driven PCNs and build MPC formation using the proposed CFG algorithm. Random graphs such as a power-law clustering or partitioning model with a default of 20 PSPs are used for our experiment, as shown in Figure 8. The power-law clustering has a degree of a PSP as a vertex of 3 and clique probability of 0.7 as a default. The revenue and cost rate are configured as \(\epsilon_{r}=0.2\) and \(\epsilon_{c}\)= 0.1. Default \(\mathbb{E}\Delta\) is 0.1. User demand is given randomly by 50 or 100. We derive the average throughput with 100 randomly generated PCN topologies. First, we investigate the convergence of MPC partitioning via our distributed coalition algorithm with two preference rules. Figure 9 shows that the split and merge algorithm allows the MPCs to converge to a Nash-stable partition regarding the overall utility saturation of the PCN. For each Pareto and selfish preference order, the MPC partitions converge successfully to the equilibrium partition in very limited switch operations, 75 and 30, respectively. The selfish order constitutes an aggressive MPC, as Figure 9a shows a drastic decrement in MPCs compared to the Pareto in Figure 9b. The overall utility of the Pareto order increases monotonically. In contrast, the selfish order shows variations in several iterations because the Pareto rule prevents the payoff loss of MPCs caused by new partitions. For the Pareto test, we configure \(\epsilon_{c}=0.2\), the same as \(\epsilon_{r}\), for an SPC to leave an MPC without harming the payoff of other members. Figure 8: SNS-like PCN models. (**a**) Power-law cluster with 20 PSPs. (**b**) Ring of 5 cliques with 4 PSPs each. Figure 9: MPC partition convergence and utility. (**a**) Selfish order. (**b**) Pareto order. Figure 10a shows the average number of MPCs with varying PCN sizes regarding the number of PSPs where the selfish order was applied. The average number of MPCs almost linearly increases with more PSPs, but each size is at saturated around 8 as the degree of each PSP is limited, as in the graph topology. Figure 10b shows that the MPC formation is rather affected by the SPC turnaround delay cost, \(\mathbb{E}\Delta\). The number of MPCs increases with the higher \(\mathbb{E}\Delta\) values; the increment is notably between 0.3 and 0.4. With 0.1, 40 PSPs only have 27 MPCs, while almost 120 MPCs have 0.5. The higher turnaround delay limits the MPC size and causes more partitions in the PCN. In addition to the MPC delay, \(\mathbb{E}\Delta\), collateral cost rate \(\epsilon_{c}\) is critical for MPC formation at a given revenue rate \(\epsilon_{r}=0.5\), as in Figure 11. Figure 11a shows the average payoff of each MPC with different \(\epsilon_{c}\), which decreases exponentially with the increasing delay cost and is saturated between 0.15 and 0.2 of \(\mathbb{E}\Delta\) since there is no major change in partitions. The average utility of an individual MPC is higher with a lower cost rate \(\epsilon_{c}\)--for instance, 100 vs. 400 for \(\epsilon_{c}=\)0.5 and 0.1, respectively, at \(\mathbb{E}\Delta\) = 0.05. Contrarily, the total marginal payoff of MPCs as the sum of all coalitional gains, \(\sum_{k}v(S_{k})-\sum_{ij}v(\{(i,j)\})\), is higher with the cost rate \(\epsilon_{c}=0.5\) than others, owing to more deposit reuse. For instance, the marginal gain increases from 500 to 2500 with \(\epsilon_{c}=\)0.1 and 0.5, respectively, in Figure 11b when \(\mathbb{E}\Delta\) is 0.01. However, the \(\mathbb{E}\Delta\) limits such a coalition gain. Figure 11c shows the average MPC ratio (i.e., number of MPCs/number of initial partitions) increasing with higher cost rates, discouraging PSPs from joining the MPCs. The higher collateral cost rate aggravates the coalition gain and the turnaround delay. Accordingly, the higher \(\epsilon_{c}\) generates more MPCs with small sizes (the minimum MPC size is 2 for a single SPC).