Three variations of Heads or Tails Game for Bitcoin
Cyril Grunspan, Ricardo Perez-Marco
TL;DR
The paper uses simple Heads or Tails coin-toss models with chip economies to analyze incentive effects in Bitcoin's proof-of-work, focusing on temporary forks, connectivity, and orphan blocks. It derives precise thresholds for when a miner would deviate from honesty, notably around $32.94\%$ in zero-connectivity settings and about $42.9\%$ in some temporary-fork scenarios, highlighting how orphan blocks influence profitability. It also proposes a modified difficulty adjustment framework based on the progression $D=H+U$ to rectify incentive misalignments and presents a third variant, JM_3, which yields an unbiased outcome where honesty is optimal. The results clarify vulnerabilities in the current difficulty adjustment and offer a straightforward, implementable approach aligned with prior literature to strengthen Bitcoin's consensus security.
Abstract
We present three very simple variants of the classic Heads or Tails game using chips, each of which contributes to our understanding of the Bitcoin protocol. The first variant addresses the issue of temporary Bitcoin forks, which occur when two miners discover blocks simultaneously. We determine the threshold at which an honest but temporarily ``Byzantine'' miner persists in mining on their fork to save his orphaned blocks. The second variant of Heads or Tails game is biased in favor of the player and helps to explain why the difficulty adjustment formula is vulnerable to attacks of Nakamoto's consensus. We derive directly and in a simple way, without relying on a Markov decision solver as was the case until now, the threshold beyond which a miner without connectivity finds it advantageous to adopt a deviant mining strategy on Bitcoin. The third variant of Heads or Tails game is unbiased and demonstrates that this issue in the Difficulty Adjustment formula can be fully rectified. Our results are in agreement with the existing literature that we clarify both qualitatively and quantitatively using very simple models and scripts that are easy to implement.