Economic Bounds

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Settlement probability inequality

$Q > 1 - 1/M$ is the system-level condition for finite expected escalation cost under proportional per-round costs.

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Derivation

Assume some proportional cost $C > 0$ per dispute round (e.g. a burn fee or capital cost). With multiplier $M$ and per-round settlement probability $Q$, the dispute probability is $P = 1 - Q$. The total expected cost of the oracle game is: For this geometric series to converge, $M \cdot P < 1$: If $Q$ violates this inequality, the total cost of the oracle game diverges. Under any oracle parameterization, the equilibrium no-dispute band widens as needed to ensure the $Q$ inequality holds for the choice of $M$. Since each round settles with probability $Q$, expected rounds to settlement is $1/Q$. The inequality bounds this: Any committed oracle manipulation or delay strategy that loses money internally in each round blows up divergently if this condition is violated, even if the other participants are net profitable. Note that the escalation halt breaks the divergence by capping the oracle game size. However, as long as the escalation halt is high enough relative to external notional, the cost to manipulate through the geometric growth phase is economically similar — the manipulator still faces near-divergent costs before reaching the halt. The per-round cost $C$ is assumed positive in the derivation above, but in practice a given participant’s effective per-round cost can be net negative when external notional dynamics or continuation value subsidize the dispute. In these regimes, the divergence argument does not directly bind that participant, and the settlement probability inequality becomes a necessary but not sufficient condition for finite manipulation cost. See Attack Vectors for the full treatment.

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Distribution-agnostic proof

From Oracle Accuracy & Cost, the disputer’s break-even is derived against the worst-case adversarial counter-dispute. Define: where $Q(A)$ is the double-no-touch probability at barriers $\pm A$ under whatever price process governs the settlement period. A dispute at distance $D$ is profitable when $D > C \cdot M$. When disputers are profitable, this implies $Q(D) > 1 - 1/M$. Since $C$ is the maximum of $(1 - Q(A)) \cdot A$ over all $A$, evaluating at $A = D$ gives: Dividing by $D$: This holds for any distribution — the only requirement is that $C$ is the maximum of $(1 - Q(A)) \cdot A$. The settlement probability inequality is a direct consequence of the definition of the adversarial optimum, not a property of any particular distributional family.

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Minimum cost to delay

Assume a protocol fee $F_p$ that is either burned on each dispute or routed to oracle game dependents. The oracle game liquidity at round $n$ is:

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Total system burn

If $N_d$ disputes occur (from any source), the total protocol fee burned across those rounds is: This closed form is pre-escalation halt; with halt the cost becomes piecewise (geometric growth then flat). The number of rounds required to delay for some target horizon $X$ given settlement time $T$ per round is:

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Attacker-paid versus honest-paid split

Not all $N_d$ rounds are paid for by the attacker. Honest market activity — arbitrageurs correcting price movements — also resets the settlement timer at no cost to the attacker. In the honest-only process, expected rounds to settlement is $1/Q$, capped by $M/(M-1)$. If the attacker only disputes when the game would otherwise settle (which happens with per-round probability $Q$), the expected cost splits cleanly: Applying the settlement probability inequality $Q > 1 - 1/M$: The attacker pays at least $(1 - 1/M)$ of the total burn and the honest network pays at most $1/M$. For small $M$ (e.g. $M = 1.1$), the floor is only ~9.1% — most of the burn comes from honest activity. For $M > 2$, the attacker bears the majority. For delay targets within the honest baseline of $\sim M/(M-1)$ expected rounds, the attacker’s cost can be significantly lower than $C_{\text{total}}$. Any swap fee or protocol fee meaningfully higher than zero widens the no-dispute band (see Oracle Accuracy & Cost), which pushes $Q$ higher. Higher $Q$ means fewer honest rounds per attacker dispute and a larger share of the total burn falling on the attacker. Honest disputes also yield less delay per round than attacker disputes. Honest disputes fire as soon as the price moves past the dispute barrier, so their mean time per round is less than $T$. An attacker maximizing delay would wait until the end of the settlement window, getting close to $T$ of delay per dispute.

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Example

Assume $F_p = 1\%$, $T = 10$ minutes, $M = 2$, and $L_0 =$ $100k. An attacker tries to delay for 2 hours, requiring $N_d = \lceil 120 / 10 \rceil = 12$ rounds. Over $4 million in protocol fees are burned across 12 rounds. With $M = 2$, $Q > 1/2$, so the attacker pays strictly more than half of the total burn in expectation.