Repeated rounds of the Robust Midpoint protocol reduce the span of honest values, leading to approximate agreement.

Byzantine Approximate Agreement: From One Round to a Complete Protocol https://decentralizedthoughts.github.io/2026-10-17-approximate-agreement-complete-protocol/ The previous post showed that one round of the Robust Midpoint protocol keeps every honest value inside the old honest range and cuts the span of those values in half. Repeating this round gives approximate agreement. We do this formally in this post by specifying how many rounds to run and when parties decide. The update is the

Byzantine Approximate Agreement: Grade, Detect, Average https://decentralizedthoughts.github.io/2026-10-18-byzantine-approximate-agreement-grade-detect-average/ The previous post gave a complete approximate agreement protocol: send the current value, remove the $f$ largest and $f$ smallest values, and take the midpoint. Each round halves the span of the honest values. Starting in $[0,1]$, reaching span less than $1/n$ takes $O(log n)$ rounds. Ben-Or, Dolev, and Hoch improve this bound to