# Coalition Formation Game for Cost-Efficient Multiparty Payment Channel in Payment Channel Networks
## 6 Experimental Results
For instance, \(\epsilon_{c}=0.1\) merges the initial coalitions into 20% of them, while \(\epsilon_{c}=0.5\) achieves only 30% in the case of \(\mathbb{E}\Delta\) = 0.1. Moreover, the number of final partitions increases as the MPC delay cost \(\mathbb{E}\Delta\) becomes significant. In Figure 11d, accordingly, each MPC size decreases with the increasing delay cost, especially for the higher cost rate \(\epsilon_{c}\). The MPC ratio and size are comparable between \(\mathbb{E}\Delta\) = 0.15 and 0.2 because the coalitional payoff is limited. Figure 10: MPC formation with varying PSPs and \(\mathbb{E}\Delta\). (a) Average number of MPCs and their size. (b) Number of MPCs with varying \(\mathbb{E}\Delta\) = 0.1\(\sim\)0.5. Figure 11(a) compares the marginal payoff when the degree of a PSP as several PCs is 3 or 4. We use the SPC delay \(\mathbb{E}\Delta=0.05\) to reveal the impact of the degree of a PSP in the power-law cluster PCNs. The total marginal payoff increases as more PSPs participate in the coalitional game and form more MPCs, as shown in Figure 10. The result shows that PSPs with PCs probably have more opportunities to form MPCs and gain payoff. The MPC ratio with the higher degree, 4, is slightly lower than the others because PSPs can be easily merged with more PCs, as in Figure 11(b). Meanwhile, the MPC ratio is almost consistent with the number of PSPs. Figure 11(c) describes the MPC formation performance in the two different PCN topologies shown in Figure 8. More MPCs are established in the power-law cluster graphs rather than the ring of cliques, which has MPCs that are equal to the cliques. For instance, the 40 PSP case has 10 cliques and 9 interconnecting PCs, constituting around 19 MPCs. In the random PSP-clustered network, PC existence between two PSPs increases the probability of forming MPCs, as shown in Figure 11(d). The MPC ratio decreases monotonically with higher PC connectivity probability from 0.5 and 0.7 to 0.9. Moreover, a higher degree of PSP among 3, 5, and 9 helps to build larger MPCs and reduce the MPC ratio. Asymmetric PBT flows of user demands also affect the MPC formation because the revenue of the MPC is proportional to the PBT flow rate. Figure 13 shows the relation of MPC formation and user demand, where we apply \(\mathbb{E}\Delta=0.01\) to reduce the MPC  
Figure 11: MPC formation with varying collateral cost \(\epsilon_{c}=\)0.1, 0.3, and 0.5 and delay cost \(\Delta\) at the revenue rate \(\epsilon_{r}=0.5\). (**a**) Average payoff of an MPC. (**b**) Total marginal payoff of all MPC partitions. (**c**) MPC ratio with varying cost. (**d**) Average MPC size. overhead and highlight the impact of the user demand. Figure 13a shows that the MPC ratio decreases as the PSPs form larger coalitions for more revenue. In Figure 13b, the total marginal payoff from MPCs increases with more user demand. Consequently, more PBT flows increase the MPC utility and form a larger MPC in a profitable situation with a low MPC delay cost. Figure 12: MPC formation with varying PCN topologies. (**a**) Marginal payoff with degree of PCs. (**b**) Clustering MPCs with degree of PCs. (**c**) Power-law clusters vs. ring of cliques. (**d**) MPC ratio with varying connectivity probability. Figure 13: MPC formation with varying user demands. (**a**) MPC coalitions with asymmetric user demands. (**b**) Marginal utility of total partitions.