New paper reveals optimal bounty to prevent delays in threshold systems

In threshold systems, a public task completes only after κ verifiable shares are committed. Honest participants create Nstar = κ + Δ share opportunities by deadline t*, where Δ is the slack. A coalition can delay completion by withholding at least Δ+1 shares, earning delay value L and potentially extra fee revenue during recovery. The paper introduces a nonnegative completion bounty that depends only on committed shares, expires if late, and uses no deposits or punishments.

The optimal incentive rule is a uniform bounty: each admissible horizon share receives B/Nstar if completion occurs by t*, otherwise zero. Full participation is ex-post strongly delay-proof exactly when (Δ+1)f + (Δ+1)B/Nstar ≥ L + R1+, where R1+ is an upper bound on coalition's incremental fee revenue in a recovery slot. The worst-case budget B* is tight for any nonnegative completion-measurable bounty. The result applies to threshold signatures, data availability certification, coded dissemination, and generic k-of-n tasks. They also show no transfer rule based only on completed shares can eliminate a final slot race.