a market mechanism for prediction markets where the two sides of a bet are geometrically coupled — buying one directly suppresses the other
proposed by Nick Williams and Vitalik Buterin, Ethereum Research, 2020
the core idea
standard prediction markets bound prices to [0,1] because shares settle to fixed payouts of $$C(s_{YES}, s_{NO}) = \lambda \sqrt{s_{YES}^2 + s_{NO}^2}$$. ICBS presents an alternative: settlement rebalances reserves rather than paying fixed amounts. prices are not bounded to [0,1]. instead, the ratio of reserves encodes the market's probability forecast.
in traditional bonding curves, buying token A doesn't affect token B's price. ICBS couples them: buying YES pushes NO's price down, and vice versa. this creates genuine opposition between beliefs rather than independent liquidity pools.
the cost function
$$C(s_{YES}, s_{NO}) = \lambda \sqrt{s_{YES}^2 + s_{NO}^2}$$
where $s_{YES}$ and $s_{NO}$ are token supplies and $\lambda$ is a fixed scaling constant set at deployment.
geometrically: this is the Euclidean distance from the origin in the $(s_{YES}, s_{NO})$ plane. iso-cost curves are circles — every point at distance $r$ from the origin costs $\lambda \cdot r$. trading moves outward from the origin.
$\lambda$ is fixed at deployment by the initial deposit $D$:
$$\lambda = \frac{D}{\sqrt{s_{YES}^2 + s_{NO}^2}}$$
for a 50/50 split at initial price $1, a \$100 deposit creates $s_{YES} = s_{NO} = 50$ tokens, giving $\lambda = 100/\sqrt{50^2 + 50^2} \approx 1.414$. markets of different sizes have identical percentage-based price dynamics — enabling cross-market comparison.
prices and inverse coupling
prices emerge as partial derivatives of the cost function:
$$p_{YES} = \frac{\partial C}{\partial s_{YES}} = \lambda \cdot \frac{s_{YES}}{\sqrt{s_{YES}^2 + s_{NO}^2}}$$
$$p_{NO} = \frac{\partial C}{\partial s_{NO}} = \lambda \cdot \frac{s_{NO}}{\sqrt{s_{YES}^2 + s_{NO}^2}}$$
each token's price increases with its own supply but is suppressed by the opposing side:
$$\frac{\partial p_{YES}}{\partial s_{NO}} = -\lambda \cdot \frac{s_{YES} \cdot s_{NO}}{(s_{YES}^2 + s_{NO}^2)^{3/2}} < 0$$
buying NO directly lowers YES's price. this is the inverse coupling that gives ICBS its name — and that makes it the correct market structure for an epistemic system where TRUE and FALSE are genuine opposites, not independent assets.
the invariant: TVL = cost function
virtual reserves are defined as $r = s \cdot p$:
$$r_{YES} = \lambda \cdot \frac{s_{YES}^2}{\sqrt{s_{YES}^2 + s_{NO}^2}}, \quad r_{NO} = \lambda \cdot \frac{s_{NO}^2}{\sqrt{s_{YES}^2 + s_{NO}^2}}$$
total value locked:
$$TVL = r_{YES} + r_{NO} = \lambda\sqrt{s_{YES}^2 + s_{NO}^2} = C(s_{YES}, s_{NO})$$
TVL always equals the cost function — the on-manifold property. this ensures solvency: total claimable value always matches what the vault holds. reserves can rebalance at settlement without minting or burning tokens.
the market's current probability forecast:
$$q = \frac{r_{YES}}{r_{YES} + r_{NO}}$$
settlement
at resolution, actual outcome $x \in \{0, 1\}$ determines settlement factors:
$$f_{YES} = \frac{x}{q}, \quad f_{NO} = \frac{1-x}{1-q}$$
if the event happens ($x = 1$): YES holders gain ($f_{YES} > 1$), NO holders lose ($f_{NO} < 1$). if it doesn't ($x = 0$): NO holders gain, YES holders lose. reserves rebalance directly:
$$r'_{YES} = r_{YES} \cdot f_{YES}, \quad r'_{NO} = r_{NO} \cdot f_{NO}$$
total vault balance is preserved: $r'_{YES} + r'_{NO} = r_{YES} + r_{NO}$. capital flows from incorrect predictions to correct ones without external capital injection.
settlement uses square-root scaling of the supply parameter $\sigma$ (converting display to virtual supply). scaling $\sigma$ by $\sqrt{f}$ makes virtual supplies scale by $\sqrt{f}$, making reserves (proportional to supply$^2$ via the norm) scale by exactly $f$.
key properties
self-scaling liquidity. buying moves supply further from the origin. TVL $= \lambda\sqrt{s_{YES}^2 + s_{NO}^2}$, so trading volume automatically grows liquidity. no external LPs needed. markets bootstrap from minimal deposits and scale organically. this differs fundamentally from LMSR, where the subsidy parameter $b$ caps liquidity.
early conviction rewards. prices range from 0 to $\lambda$:
$$\lim_{s_{YES} \to \infty,\, s_{NO} \text{ fixed}} p_{YES} = \lambda$$
early traders who buy near zero can see prices approach $\lambda$, yielding arbitrarily large returns. unlike LMSR's fixed [0,1] bounds, ICBS rewards early conviction rather than just tracking consensus. this aligns incentives toward surfacing private knowledge early.
geometric simplicity. only square roots — no fractional powers, no exponentials. the mechanism is computationally tractable and the geometry is intuitive.
ICBS vs LMSR
| ICBS | LMSR | |
|---|---|---|
| price bounds | [0, λ] | [0, 1] |
| liquidity | self-scaling (trading grows TVL) | capped by subsidy parameter b |
| external LPs | none needed | none needed |
| settlement | reserve rebalancing | fixed MATH_PLACEHOLDER_391 payouts |
| early conviction | rewarded (prices can approach λ) | not specially rewarded |
| probability encoding | ratio of reserves | direct price |
| loss bound | none (market maker takes risk) | b·ln(2) per market |
connection to cyber
the inverse coupling property is the market analog of inhibition: buying FALSE directly suppresses the effective weight of YES in the market, exactly as negative weights suppress activations in neural networks. the geometry makes this explicit — the two sides move on a circle, so amplifying one necessarily suppresses the other.
the self-scaling liquidity property solves the bootstrapping problem for the cybergraph: every cyberlink that attracts market activity automatically deepens its own liquidity. the most-contested edges (the epistemically important ones) become the most liquid, yielding the most accurate prices. this is the Lindy effect on the market structure.
the settlement factors $f_{YES} = x/q$ and $f_{NO} = (1-x)/(1-q)$ are inverse probability weights — the same mathematical structure that appears in importance sampling, in the serum scoring formula, and in the KL divergence terms that measure information gain. this is not coincidental: all three are instances of proper scoring rules applied to belief elicitation.
the on-manifold property (TVL = cost function) ensures the market remains solvent as cyberlinks accumulate, without requiring external capital injection. the cybergraph itself is the liquidity — structural knowledge (cyberlinks) bootstraps epistemic knowledge (market prices).
see veritas for how ICBS fits into the full truth-discovery protocol. see inhibition for the connection to inhibitory weights in the tri-kernel. see serum for the scoring layer that sits above the market mechanism. see market for the broader design.