Slack and Budget Breaking in Threshold Team Production

Abstract

A threshold system completes a public task only after $κ$ verifiable shares are publicly committed. If the honest schedule creates \( \Nstar=κ+Δ\) share opportunities by deadline $t^\star$, then $Δ$ shares are slack such that a coalition delays completion if and only if it withholds at least $Δ+1$ shares. The incentive problem is therefore to price the cheapest sabotage set. Agents receive a direct fee $f$ per committed share. A delaying coalition may also obtain delay value at most $L$, and may earn additional fee revenue during recovery after the deadline. Let $R_1^+$ be a pathwise upper bound on the coalition's incremental fee revenue in a recovery slot that completes the task, including any same-slot overshoot. The principal can post a nonnegative completion bounty that depends only on committed shares, uses no deposits or punishments, and expires if completion is late. The optimal rule is uniform, as if completion occurs by $t^\star$, every admissible horizon share receives $B/\Nstar$, otherwise no bounty is paid. Full participation is ex-post strongly delay proof exactly when \( (Δ+1)f+\frac{Δ+1}{\Nstar}B \ge L+R_1^+ . \) Equivalently, the exact worst-case budget is \( B^\star = \frac{\Nstar}{Δ+1} \bigl(L+R_1^+-(Δ+1)f\bigr)^+ . \) The bound is tight for every nonnegative completion measurable bounty, among the $\Nstar$ horizon shares, some $Δ+1$ receive total bounty at most $(Δ+1)B/\Nstar$, and withholding precisely those shares delays completion. The result applies to threshold signatures, data availability certification, coded dissemination, and generic $k$-of-$n$ completion tasks. We also isolate a separate limit, no transfer rule based only on completed shares can remove a final slot race in which a coalition has already observed enough pre-completion shares to act.