# Stochastic Modelling of Selfish Mining in Proof-of-Work Protocols +
## 5 Results
### Consensus Block Rate  
Selfish mining could cause a lot of orphan blocks, by intentionally splitting the chain. Most orphan blocks are honest blocks that are part of stale chains. However, most orphan  
Figure 5: Relative selfish revenue as a function of: (**a**) latency (\(\lambda_{nd}\)) and (**b**) selfish hashing power (\(\alpha\)). Selfish miners are located according to various centrality measures. Figure 6: Orphan block rate as a function of: (**a**) latency (\(\lambda_{nd}\)) and (**b**) selfish hashing power (\(\alpha\)). blocks do not split the main chain themselves, but are merely attached to it directly or indirectly via a chain of more orphan blocks. We are interested to reduce this \"noise\" of orphan blocks and only consider blocks that split the main chain. This will allow us to better understand the relationship between selfish mining and its impact on the blockchain forks. We call a block a _main chain no-sibling block_ or a _consensus block_, if the main chain block does not share the same parent block with any other block. To compute the _main chain no-sibling rate_, we divide the number of main chain no-sibling blocks by the total number of main chain blocks. The dynamics of the consensus block rate (as shown in Figure 7) are relatively complex. Considering Figure 7b: First, without selfish mining (\(\alpha=0\)), the rate of consensus blocks converges to one as the block diffusion rate become very fast. Conversely, the sibling rate of main chain blocks is high for high latency. Second, when \(\alpha=0.5\) the selfish miner effectively controls the blockchain. Consequently, honest miners do not manage to build on top of the main chain as they continue to mine on stale chains. Therefore, main chain splits can barely take place and thus the sibling rate for main chain blocks is close to zero. Third, consider the graph for fast block diffusion, \(\lambda_{nd}=79.43\). Initially, the share of siblings grows as \(\alpha\) increases, but beyond some inflexion point, the rate of siblings decreases again. This point of inflexion corresponds roughly to the threshold for which selfish mining becomes relatively more profitable than honest mining (for \(\lambda_{nd}=79.43\) this corresponds to \(\alpha\approx\frac{1}{3}\) as per Figure 2b). This makes intuitive sense, because while selfish mining is not relatively more profitable, selfish miners continuously and intentionally split the main chain, without managing to build up an advantage of multiple blocks. Hence, there are many main chain splits, or few main chain no-sibling blocks. Once selfish mining becomes relatively more profitable, the intentional chain splits are outweighed by the relatively frequent advantages the selfish miners can now build up, thus leading to fewer main chain sibling blocks. The results in Figure 7a are very similar. First, when there is no selfish mining, high latency implies more main chain splits. Before being equivalent to the block creation rate, the faster block diffusion will cause more sibling blocks. After exceeding the block creation rate by far, the faster block diffusion brings the no-sibling rate to one. On the other hand, when there is selfish mining, for very high latency, the selfish miner more effectively controls the main chain and so the rate of main chain siblings converges to zero. As latency decreases, selfish mining is less efficient in splitting the main chain, meaning that they tend to lose more block races against honest miners. Thus, the main chain no-sibling rate decreases until a point of inflexion. This point of inflexion is driven by the selfish mining strategy becoming relatively less profitable for higher \(\lambda_{nd}\). For high \(\lambda_{nd}\) the rate of siblings decreases again as the dynamics are now dominated by honest miners, who observe fewer accidental chain splits. Intuitively, the points of inflexion show that the efficiency of selfish mining is more resistant to higher selfish mining power levels.