Optimal RANDAO Manipulation in Ethereum

It is well-known that RANDAO manipulation is possible in Ethereum if an adversary controls the proposers assigned to the last slots in an epoch. We provide a methodology to compute, for any fraction $\alpha$ of stake owned by an adversary, the maximum fraction $f(\alpha)$ of rounds that a strategic adversary can propose. We further implement our methodology and compute $f(\cdot)$ for all $\alpha$. For example, we conclude that an optimal strategic participant with $5\%$ of the stake can propose a $5.048\%$ fraction of rounds, $10\%$ of the stake can propose a $10.19\%$ fraction of rounds, and $20\%$ of the stake can propose a $20.68\%$ fraction of rounds.