# Stochastic Modelling of Selfish Mining in Proof-of-Work Protocols +
## 6 Selfish Mining Pool  
All the previous simulations and results are based on the assumption that there is only a single selfish miner in the network. However, in reality, miners often cooperate in mining pools to share the larger computational power, and to ensure a more stable reward payout. Therefore, we want to further consider the situation in which there is a selfish mining pool.  
We assume there is a selfish mining pool in which members share the information with their pool peers (i.e., each selfish node is connected to all the other selfish nodes in a peer-to-peer (p2p) network). That means, if one selfish mining pool peer finds a block, it is shared within the selfish pool, but not necessarily with honest nodes. As a result, selfish nodes will mine on top of the latest selfish block, but honest nodes will not unless it is shared with them (when selfish miners need to win a block race against honest miners). They do not act as individual selfish nodes, but they operate like a mining pool (in practice they would share the profits proportionally too, which is not relevant for the model as we only care about the aggregate performance). For the simulation, we randomly assign each selfish node a hashing power and normalise the values so that the sum of all selfish miners' hash rates equals \(\alpha\).  
Figure 10 shows mainly two results for multiple selfish miners operating within a pool. First, the relationship between the share of main chain blocks and the share of hashing power is very linear within types of miners. This implies that within honest miners, rewards are distributed fairly, as well as within the selfish mining pool. Here fairness refers to fairness _within_ groups of miners that are of the same type (selfish or honest), not across types. Trivially, selfish mining causes a bias in block reward distribution across types of miners. Second, latency can cause different dynamics, if the selfish share of the network's total hashing power (\(\alpha\)) is shared among multiple miners. Figure 9(c) shows that even for \(\alpha=0.4\) selfish mining is relatively less profitable for high latency (\(\lambda_{nd}=0.63\)). This is due to the fact that we assume block latency within the selfish mining pool to be the same as  
Figure 9: \(MSB\) index as a function of (a) latency (\(\lambda_{nd}\)) and (b) selfish hashing power (\(\alpha\)).  
that outside of the pool. If the selfish mining power is shared across many miners, the pool suffers from high block latency, because by the time selfish nodes find out about blocks from selfish peers, the honest nodes are likely to have found a block themselves already. Finally, for \(\alpha=0.4\) and a relatively fast block propagation rate (\(\lambda_{nd}=7.14\)) selfish mining is relatively more profitable than honest mining.  
We have shown that there is no bias in the distribution of block rewards within miner types. Further, we have shown a glimpse into the dynamics of the system with a selfish mining pool. However, it is worth noting that we currently assume that the communication among selfish and honest miners follows the same distribution. In reality, it would be sensible for selfish nodes to set up very fast communication channels with their selfish mining pool peers which are not necessarily using the default p2p layer. Therefore, a variation in block propagation speeds for communication between pool members and non-members remains an important dynamic to analyze.