# Selfish Mining and Dyck Words in Bitcoin and Ethereum Networks

## TL;DR

This paper introduces a combinatorial approach using Dyck words to analyze long-term hashrates of selfish mining strategies in Bitcoin and Ethereum, avoiding complex Markov chain calculations.

## Contribution

It presents a novel elementary probability and combinatorics method for analyzing selfish mining, applicable to Bitcoin and Ethereum, bypassing traditional Markov chain computations.

## Key findings

- Provides explicit formulas for long-term hashrates
- Applies Dyck words to model mining strategies
- Extends analysis to Ethereum network

## Abstract

The main goal of this article is to present a direct approach for the formula giving the long-term apparent hashrates of Selfish Mining strategies using only elementary probabilities and combinatorics, more precisely, Dyck words. We can avoid computing stationary probabilities on Markov chain, nor stopping times for Poisson processes as in previous analysis. We do apply these techniques to other block withholding strategies in Bitcoin, and then, we consider also selfish mining in Ethereum.