# Stochastic Modelling of Selfish Mining in Proof-of-Work Protocols +
###### Abstract  
In blockchain-based systems whose consensus mechanisms resort to Proof-of-Work (PoW), it is expected that a miner's share of total block revenue is proportional to their share of hashing power with respect to the rest of the network. The protocol relies on the immediate broadcast of blocks by miners, to earn precedence in peers' local blockchains. However, a deviation from this strategy named _selfish mining_ (SM), may lead miners to earn more than their \"fair share\". In this paper, we introduce an agent-based model to simulate the dynamics of SM behaviour by a single miner as well as mining pools to understand the influence of (a) mining power distribution, (b) overlay network topology, (c) positioning of the selfish nodes within the peer to peer network. Our minimalistic model allows us to find that in high levels of latency, SM is always a more profitable strategy; our results are very robust to different network topologies and mining nodes' centrality in the network. Moreover, the power-law distribution of the miners' hashing power can make it harder for a selfish miner to be profitable. In addition, we analyze the effect of SM on system global efficiency and fairness. Our analysis confirms that SM is always more profitable for hashing powers representing more than one-third of the total computing power. Further, it also confirms that SM behaviour could cause a statistically significant high probability of continuously mined blocks opening the door for empirical verification of the phenomenon. blockchain; Proof-of-Work; selfish mining; agent-based model; hashing power; network latency +
Footnote †: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/py/4.0/](https://creativecommons.org/licenses/py/4.0/)]([https://creativecommons.org/licenses/py/4.0/](https://creativecommons.org/licenses/py/4.0/))). Footnote †: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:creativecommons.org/licenses/py/4.0/). Footnote †: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/py/4.0/](https://creativecommons.org/licenses/py/4.0/)]([https://creativecommons.org/licenses/py/4.0/](https://creativecommons.org/licenses/py/4.0/))). Footnote †: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons. a crucial assumption is that a majority of network participants act honestly and are not controlled by a single colluding entity. In 2014, [PERSON] and [PERSON] [2] introduced an attack vector on the Bitcoin protocol. They suggest that a miner that controls more than one-third of the network's computational resources may break Bitcoin's incentive compatibility. They find that a miner strategically deviating from the Bitcoin protocol may lead to an increase in their relative revenue. They call this strategy selfish mining. The main idea of selfish mining is to make honest miners waste computational resources on blocks that are not going to be part of the main chain [3], i.e., get no block reward. A selfish miner may achieve this by keeping newly mined blocks private, rather than broadcasting them immediately. This effectively causes a chain split, although honest miners do not know about it initially. Honest miners continue to mine on the _public chain_, whereas the selfish miner mines on their _private chain_. The selfish miner only broadcasts their private blocks as honest miners catch up to the private chain's height, with the intention to invalidate honest blocks. If a selfish miner is relatively more profitable they can invest more resources into computational hardware, further increasing their share of the network's computational resources.