Bribing the verifiers

research/verifier-bribery

Bribing verifiers in refereed verification: closed-form bribe floor, a sharp collusion-proof stake that falls as 1/m, and jackpots that raise the floor one-for-one.

7 tests

What it found

Closed-form bribe floor for buying verifier silence in refereed verification: a sharp collusion-proof stake (falls as 1/m verifiers), and planted-fault jackpots that lift the floor. Verified against brute force and Monte Carlo. Stylised model.

Paper, code and results

More in Verification games & slashing

Do verifiers and solvers learn the audit equilibrium?No-regret learning in the refereed-verification inspection game: conserved potential, closed-form cycle period, averages converge but per-round audit probability collapses; optimistic Hedge converges.
Many verifiers, one solver: the volunteer's dilemma in refereed verificationM verifiers in the inspection game: detection pinned at s/(s+S), cheating rises with m under split rewards, symmetric equilibrium is a saddle (volunteer's dilemma), bandit learners need an exploration floor.
Are stake and audits substitutes in trace verification?Sampled audits of committed training traces: payoff is convex in corrupted steps, deterrence iff min(F, f·k) ≥ T·g, so stake and audits are complements and stake must scale with job length.
What does secret task assignment buy?Random k-fold replication with slashing against a colluding fraction β: secret assignment makes cheating a coordination game with unique-honesty stake Gβ^{k−1}/(1−β^{k−1}), public assignment leaves β^k corruption at any stake, and assignment leaks leave a λβ^k floor.
How should a scarce audit budget be spread?Scarce audit budget across heterogeneous jobs: deterrence threshold g/(g+S) makes it a knapsack, greedy with a 1/3 guarantee (within 1% on 91% of instances), uniform audits need up to 1.9× the budget, heterogeneity lowers deterrence cost (Jensen), stake substitutes for audits.
How many verifiers does deterrence-by-audit need?Staffing the audit pool that stake-based deterrence needs: exact Erlang-C waiting (matches simulation), square-root staffing R+1.06√R (spare capacity 60% at R=5, 1.1% at 10⁴), stake–headcount optimum G+S=√(wsG/(rd₀)) (cost 24 vs 77 at too little stake), and bursty audit demand delays 1.8× what Allen–Cunneen predicts.

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Proper scoring & elicitation28 Effort & peer prediction9 Verification games & slashing19 Detection & tolerances14 Markets as routers13 Prediction-market design9 Paying contributors8 Decentralised training systems23 Digital twins & sim-to-real164 LLM chip design loops40 Robot manufacturing loops53 Ambient AI assistant46 Self-replicating manufacturing34 Simulation-first design64 Engineered clothing45 Task-specific AI accelerators40 Algorithmic chip-design loops42 Open-source purpose-built robots44 The software-engineering loop43 Adversarial human imitation41 Open-source residential construction41 Autonomous multi-angle capture34 Sensing, imaging and 3D31 Energy supply and dense energy storage39 Robot societies among people31 Agentic co-design of low-cost robots27 Laundry-hanging robots25 Machine learning on qualitative data23 Robots on the building site27 Civilizations & group psychology1 History & human sciences1

Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.