Scaling of optimal RANDAO manipulation with epoch length ℓ

Determine whether the percentage improvement over the honest policy achieved by the optimal RANDAO manipulation strategy in the reduced Markov decision process M'_G scales with the epoch length ℓ in the manner indicated by the truncated evaluation plot for ℓ ∈ {16, 32, 64, 128}. Specifically, ascertain if the empirical scaling behavior shown in the figure “Percentage improvement over honest for ℓ ∈ {16,32,64,128}” accurately reflects the true dependence on ℓ despite floating‑point numerical instability in the evaluation for ℓ > 32.

Background

The paper extends its methodology beyond Ethereum’s current epoch length ℓ = 32 to explore how optimal RANDAO manipulation varies with different epoch lengths. For ℓ > 32, the authors note that numerical instability in 64‑bit floating‑point arithmetic affects the evaluation of a key probability expression, leading them to introduce a mitigation that truncates inner sums close to 1 to improve stability.

Using this mitigation, they present a plot of percentage improvement over the honest policy for ℓ ∈ {16, 32, 64, 128}. However, they caution that these extended results are not provably accurate due to the numerical instability, and explicitly conjecture that the observed plot is representative of the true scaling behavior with ℓ. This leaves open the task of rigorously confirming (or refuting) the conjectured scaling, potentially by analyzing numerical errors or devising provably stable evaluation methods for larger ℓ.