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Accuracy

An unaligned arbitrageur will take one of the two limit orders when the true price is different enough from the limit order price (dispute distance). A disputer earns DD from taking a mispriced limit order, but risks being disputed at some adversarial distance AA, losing A⋅MA \cdot M at the escalated size. The chance they get disputed is (1−Q(A))(1 - Q(A)). Assuming perfect and continuous transaction inclusion, and no swap fees, protocol fees, or gas fees in the oracle game, and the disputer not engaging in a self-dispute strategy, the disputer’s expected value is:
EV=D−(1−Q(A))⋅A⋅MEV = D - (1 - Q(A)) \cdot A \cdot M
Where:
  • DD is the dispute distance (profit from taking the limit order)
  • AA is the adversarial dispute distance (where someone disputes you)
  • Q(A)Q(A) is the probability the price does not reach AA over the settlement time TT
  • MM is the multiplier
We assume a disputer chooses DD such that for any choice of AA maximizing (1−Q(A))⋅A⋅M(1 - Q(A)) \cdot A \cdot M, they remain profitable. In a standard normal distribution, (1−Q(A))⋅A(1 - Q(A)) \cdot A is maximized at A≈0.84A \approx 0.84 standard deviations of settlement time returns. Q(A)Q(A) is the double-no-touch (DNT) probability — the probability the true price stays within ±A\pm A for the entire settlement time. At A=0.84A = 0.84, Q(0.84)≈0.22Q(0.84) \approx 0.22, so the disputer’s EV simplifies to:
EV=D−0.65⋅M⋅σ(T)EV = D - 0.65 \cdot M \cdot \sigma(T)
Therefore, if D>0.65⋅M⋅σ(T)D > 0.65 \cdot M \cdot \sigma(T), the disputer is profitable. We expect the settled oracle price to be within ± 0.65⋅M⋅σ(T)\pm\, 0.65 \cdot M \cdot \sigma(T) of the true price at time of settlement. In practice, honest dispute barriers tend to be tighter than this bound. The 0.65⋅M0.65 \cdot M figure assumes a normal distribution, which appears to often be the worst case. Given a driftless assumption, under leptokurtic and certain jump-diffusion processes, the honest dispute barriers can tighten: fat-tailed (leptokurtic) distributions are characterized by more center mass, and jumpy regimes have a lower diffusion component for fixed total volatility. Skewness alone does not appear to move CC by much: Donsker’s theorem suggests that any driftless iid process converges to Brownian motion in the continuous-time limit, granted this doesn’t apply to jumpy distributions. In terms of bimodals, in our tested 2-component Gaussian-mixture model (fixed total variance), increasing mode separation did not increase CC above 0.65; allocating more variance to directional mean shifts leaves less diffusion variance, reducing barrier crossings. This is not a theorem covering all possible distributions, and we did not test every family. Behavior can differ under real-world price action, but empirically the result appears reasonably robust. If we define C=max⁡A (1−Q(A))⋅AC = \max_A\, (1 - Q(A)) \cdot A, drift can produce C>0.65C > 0.65: 
Drift (σ units)     C      Maximizing A    Q(A*)
    0.00          0.654       0.842        0.223
    0.10          0.655       0.843        0.223
    0.20          0.657       0.849        0.226
    0.30          0.661       0.858        0.229
    0.50          0.675       0.889        0.240
    0.75          0.706       0.961        0.265
    1.00          0.757       1.085        0.302
    1.50          0.945       1.437        0.343
    2.00          1.219       1.765        0.310
    3.00          1.886       2.439        0.227
Drift is measured in units of μT/σT\mu T / \sigma\sqrt{T} — i.e. drift per settlement window scaled by volatility. It has to be extreme to meaningfully move CC. Up to ∼0.3σ\sim 0.3\sigma of drift per settlement period, CC barely budges from 0.65. Even at 0.5σ0.5\sigma it’s only 0.675. It takes 1σ1\sigma+ of drift to start mattering — for coherent asset choices and timeframes, it seems typical drift does not impact things much. For extreme scenarios (LUNA collapse, extreme funding stress, etc), drift can start to matter. Finally, when we include swap fees and protocol fees, the disputer’s EV becomes:
EV=(D−Fs−Fp)−(1−Q(A))⋅(A−Fs)⋅MEV = (D - F_s - F_p) - (1 - Q(A)) \cdot (A - F_s) \cdot M
Where FsF_s is the swap fee and FpF_p is the protocol fee. As a conservative threshold, fees are purely additive to required edge, so dispute distance shifts from D>0.65⋅M⋅σ(T)D > 0.65 \cdot M \cdot \sigma(T) to roughly D>0.65⋅M⋅σ(T)+Fs+FpD > 0.65 \cdot M \cdot \sigma(T) + F_s + F_p.

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Self-consistent honest barrier

The 0.65⋅M0.65 \cdot M threshold is a conservative upper bound. It maximizes (1−Q(A))⋅A(1 - Q(A)) \cdot A over all possible adversarial distances AA, peaking at A∗≈0.84σA^* \approx 0.84\sigma. But when 0.65⋅M<A∗0.65 \cdot M < A^* (i.e. M≲1.3M \lesssim 1.3 in a normal), the honest dispute barrier is below A∗A^*, so the price never actually reaches A∗A^*. Someone disputes at bb first, preempting the adversary. This means the relevant (1−Q(A))⋅A(1 - Q(A)) \cdot A is always evaluated below the global maximum, and the honest barrier tightens. In this context, every honest disputer uses the same barrier bb. Since the next honest disputer also disputes at bb, the marginal disputer’s EV at the self-consistent barrier is:
EV=b−(1−Q(b))⋅b⋅M=0EV = b - (1 - Q(b)) \cdot b \cdot M = 0
which simplifies to:
Q(b)=1−1MQ(b) = 1 - \frac{1}{M}
This is similar to the inequality from economic bounds, but here it appears as the marginal honest disputer indifference condition. Next, we find the bb that satisfies Q(b)=1−1/MQ(b) = 1 - 1/M and denote this b∗b^*. The self-consistent barrier b∗b^* is tighter than 0.65⋅M⋅σ(T)0.65 \cdot M \cdot \sigma(T) when 0.65⋅M<A∗0.65 \cdot M < A^* (i.e. M≲1.3M \lesssim 1.3 in a normal): 
M        0.65·M·σ     b*       Q(b*)
1.001      0.65σ     0.42σ     ~1e-3
1.01       0.66σ     0.50σ     0.010
1.05       0.69σ     0.61σ     0.048
1.10       0.72σ     0.68σ     0.091
1.15       0.75σ     0.74σ     0.130
1.20       0.78σ     0.78σ     0.167
For M≳1.3M \gtrsim 1.3, honest barriers exceed A∗A^* and the preemption no longer applies, since a counter-disputer can act at A∗A^* before any honest participant would, so the vanilla 0.65⋅M0.65 \cdot M bound holds.

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Accuracy under manipulation

Under selective delay, a manipulator can choose to start another round if they don’t like where the price is about to settle. This turns the “safe” interior of the barriers into an additional source of loss for the honest disputer from the swap portion of the counter-dispute. As a conservative threshold, we can assume the honest disputer does not try to calculate the attacker’s optimal policy, and just assumes they always get counter-disputed at the end of the period. The honest disputer’s EV now picks up an extra term:
EV=D−(1−Q(A))⋅A⋅M−Q(A)⋅Rˉ(A)⋅MEV = D - (1 - Q(A)) \cdot A \cdot M - Q(A) \cdot \bar{R}(A) \cdot M
Where Rˉ(A)\bar{R}(A) is the conditional surviving mean. This is the expected absolute return at the end of the settlement window, given that the price never left ±A\pm A. Simplifying:
EV=D−[A−Q(A)⋅(A−Rˉ(A))]⋅MEV = D - \left[A - Q(A) \cdot (A - \bar{R}(A))\right] \cdot M
Applying the same self-consistent logic as the honest barrier above, we set EV=0EV = 0 at the marginal barrier bb where the next counter-dispute also occurs at bb:
Q(b)⋅(b−Rˉ(b))=b⋅M−1MQ(b) \cdot (b - \bar{R}(b)) = b \cdot \frac{M - 1}{M}
Under Brownian motion, Rˉ(b)/b≈0.363\bar{R}(b) / b \approx 0.363 for moderate barrier levels. In this regime, the manipulation equation simplifies: dividing both sides by bb,
Q(b)≈1−1M0.637Q(b) \approx \frac{1 - \frac{1}{M}}{0.637}
which is the same clean form as the no-manipulation case, just with a rescaled threshold. The full manipulation damage function f(A)=(1−Q(A))⋅A+Q(A)⋅Rˉ(A)f(A) = (1 - Q(A)) \cdot A + Q(A) \cdot \bar{R}(A) is monotonically increasing, asymptoting to E[∣W1∣]=2/π≈0.798σ\mathbb{E}[|W_1|] = \sqrt{2/\pi} \approx 0.798\sigma. Unlike the vanilla (1−Q(A))⋅A(1 - Q(A)) \cdot A which peaks at A∗≈0.84σA^* \approx 0.84\sigma, there is no finite argmax. The damage to the honest disputer increases with wider barriers. This means honest preemption always tightens the barrier under manipulation, for all MM. For small MM, the Rˉ(b)/b≈0.363\bar{R}(b)/b \approx 0.363 approximation holds and the self-consistent barriers are: 
M       b* (no manip)     b* (under manip)     Q(b*)
1.01         0.50σ              0.53σ            0.016
1.05         0.61σ              0.66σ            0.075
1.066        0.64σ              0.69σ            0.097
1.10         0.68σ              0.75σ            0.143
1.15         0.74σ              0.82σ            0.205
For larger MM, the 0.3630.363 ratio decays and b∗b^* converges to 2/π⋅M⋅σ(T)≈0.798⋅M⋅σ(T)\sqrt{2/\pi} \cdot M \cdot \sigma(T) \approx 0.798 \cdot M \cdot \sigma(T): 
M       b* (under manip)     0.798·M·σ
1.5           1.18σ            1.20σ
2.0           1.59σ            1.60σ
3.0           2.39σ            2.39σ
5.0           3.99σ            3.99σ
For illustration, assuming honest dispute barriers are calibrated for a manipulator but no manipulator is present, the expected rounds and effective rounds (accounting for early termination when the price hits b∗b^* before the end of the settlement window) are: 
M       b*       Q(b*)     E[rounds]     E[τ|hit]     eff rounds
1.01    0.53σ    0.016       63.46         0.27T          17.60
1.05    0.66σ    0.075       13.34         0.36T           5.46
1.066   0.69σ    0.095       10.48         0.38T           4.60
1.10    0.75σ    0.142        7.04         0.41T           3.50
1.15    0.82σ    0.203        4.92         0.45T           2.76
1.5     1.18σ    0.525        1.91         0.58T           1.52
2.0     1.59σ    0.776        1.29         0.68T           1.19
3.0     2.39σ    0.966        1.03         0.80T           1.03
5.0     3.99σ    ~1.0         1.00           —             1.00
Expected rounds is 1/Q(b∗)1 / Q(b^*). Effective rounds accounts for the fact that disputed rounds end early. The total expected time is the number of disputed rounds times E[τ∣τ<T]\mathbb{E}[\tau \mid \tau < T], plus one full settling round.

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Cost

The initial reporter bears the loss term of being disputed at adversarial distance AA, losing (1−Q(A))⋅A(1 - Q(A)) \cdot A in expectation on their position of size LL, assuming Fs=0F_s = 0 in the oracle game. The multiplier does not apply — the initial reporter’s position is at size LL, not L⋅ML \cdot M. The initial reporter must be compensated at least this amount, which in the simplified example above is ~0.65⋅σ(T)⋅L0.65 \cdot \sigma(T) \cdot L. The cost to use the oracle is therefore the initial reporter reward, which scales with both the initial liquidity and the volatility over the settlement time. Larger oracle games (higher LL) and longer settlement times (higher σ(T)\sigma(T)) increase the cost. Adding a swap fee FsF_s can dramatically reduce the initial reporter’s cost. The initial reporter can only be disputed profitably when the price moves past FsF_s, so their expected loss drops to (1−Q(A))⋅(A−Fs)(1 - Q(A)) \cdot (A - F_s). For a swap fee on the order of σ(T)\sigma(T) or larger, the initial reporter’s risk approaches zero. The tradeoff is worse accuracy — the no-dispute band widens by FsF_s, so the settled oracle price can be farther from the true price. The initial reporter bounty contract can be used to find the market-clearing payment in any environment. The bounty starts small and escalates exponentially each round (e.g. 1.5x per round). The first reporter to submit claims whatever the bounty has grown to at that time. This means the market itself determines the fair compensation — if the bounty is too low for current conditions (high volatility, high gas), no one reports until it grows enough. The realized payout reveals the true cost of initial reporting under those conditions.