Can stake shorten a challenge window when proposers can be bribed?

What it found

In optimistic verification a fraud pays G unless a challenge is included within w blocks; the attacker controls each proposer w.p. ρ and otherwise pays b to censor. The exact attacker recursion V(n) = ρV(n−1) + (1−ρ)max(−S, V(n−1)−b) gives a closed-form deterrence window (0 mismatches in 5000 instances; Monte Carlo agrees) with a gamble-first, bribe-last optimal attack. A luck-only analysis underestimates the window 100–170×; with an exogenous bribe price the window is 63 blocks for every stake from 10 to 1600; if the bounty is paid as a priority fee the price becomes θS and S·w ≈ G/((1−ρ)θ). Stylised.

A figure from its own output

Paper, code and results

• README and method
• White paper
• Experiments and results
• Code
• Tests

More in Verification games & slashing

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• Do free retries break hash-derived spot checks?Hash-derived audit challenges under free retries: cheat profit exactly G − c/p, deterrence needs ln(G/c)·T/k opened steps vs ln2·T/k for a beacon (10×–60×), stake is irrelevant, and the proof-of-work optimum is x* = v/λ_p − c0.
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• Are reward pools identity-proof?Identity splitting vs merging under stake-weighted pools: split-invariant iff the weight is linear in stake, per-head optimum √(Rn/c)−n, sharp deterrence fee Rn/((n+1)(n+2)), wagering split-neutral.
• Can a bond tell reliable hardware from unreliable?Can a bond tell reliable hardware from unreliable? Single crossing runs the wrong way: reliable rent (θ_H−θ_L)(c+(1+ρ)s_H)/(1−θ_H) is minimised at zero stake, a flat wage is optimal, and a deterrence stake scales rent by (c+(1+ρ)s)/c.

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Other research topics

Proper scoring & elicitation28Effort & peer prediction9Verification games & slashing19Detection & tolerances14Markets as routers13Prediction-market design9Paying contributors8Decentralised training systems23Digital twins & sim-to-real164LLM chip design loops40Robot manufacturing loops53Ambient AI assistant46Self-replicating manufacturing34Simulation-first design64Engineered clothing45Task-specific AI accelerators40Algorithmic chip-design loops42Open-source purpose-built robots44The software-engineering loop43Adversarial human imitation41Open-source residential construction41Autonomous multi-angle capture34Sensing, imaging and 3D31Energy supply and dense energy storage39Robot societies among people31Agentic co-design of low-cost robots27Laundry-hanging robots25Machine learning on qualitative data23Robots on the building site27Civilizations & group psychology1History & 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.