Optimal RANDAO Manipulation in Ethereum

TL;DR

A methodology to compute the maximum fraction of stake owned by an adversary, the maximum fraction of rounds that a strategic adversary can propose, for any fraction $\alpha$ of stake owned by an adversary is provided.

Abstract

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 $α$ of stake owned by an adversary, the maximum fraction $f(α)$ of rounds that a strategic adversary can propose. We further implement our methodology and compute $f(\cdot)$ for all $α$. 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.

Figures (7)

• Figure 1: Average tail length attained for each $\alpha$ when running $\textsc{Tail-max}$. The adversary controls a larger tail value as $\alpha$ rises as expected. There is a quick jump when approaching $\alpha = 50\%$, indicating that the adversary can propose almost all blocks.
• Figure 2: Average reward of the optimal policy and $\textsc{Tail-max}$ for $\ell = 32$. The figure on the left shows the $0 < \alpha \leq 0.3$ range, the figure on the right show the entire range of $0 < \alpha < 1$.
• Figure 3: Percentage improvement of the optimal policy and $\textsc{Tail-max}$ over the honest policy for $\ell = 32$. Improvement is defined as $(\text{policy average reward})/(\text{honest average reward}) - 1$. We also analyze the strategy Value-max here which we define as the strategy that maximizes the reward in the next epoch (chooses the pair that maximizes $c_{i,j} + t_{i,j} - i$).
• Figure 4: Percentage improvement over honest for $\ell \in \{16,32,64,128\}$. Improvement is defined as $(\text{optimal average reward})/(\text{honest average reward}) - 1$.
• Figure 5: Block miss rate by slot index from epoch $146876$ to $272341$. The average block miss rate is $0.9482\%$.

Theorems & Definitions (18)

• Proposition 1: puterman2014markov
• Proposition 2: puterman2014markov
• Lemma 3: howard1960dynamic
• Theorem 4: Bellman optimality equation for average-reward MDPs, howard1960dynamic
• Definition 5: RANDAO Manipulation Game v1
• Definition 6: RANDAO Manipulation Game v2
• Definition 7: RANDAO MDP $M_G$
• Definition 8: RANDAO MDP $M'_{G}$
• Proposition 9
• Definition 10: $\textsc{Tree}\,(\cdot,\cdot)$