Prices & AMM

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Implied probability

Market prices represent implied probability. A YES token trading at 45¢ implies roughly a 45% chance the outcome occurs. YES + NO prices should sum to approximately $1.00 (plus the overround fee).

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The AMM venue

Most markets today trade through an automated market maker (AMM) — a pool of YES and NO tokens that sets prices based on supply and demand. When you place a prediction:

1. USDC is minted into YES + NO tokens (fees skimmed).
2. You swap away the side you do not want.
3. The AMM price updates for the next trader.

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LS-LMSR in depth

ezpz.fi prices AMM trades with LS-LMSR — liquidity-sensitive Logarithmic Market Scoring Rule (Othman, Sandholm, Pennock, Reeves, EC 2010). It is a cost-function market maker: the pool tracks how many outcome tokens it has sold and charges USDC equal to the change in a convex cost function $C(q)$. Compared to a static LMSR (fixed liquidity parameter $b$), LS-LMSR lets depth grow with open interest. More tokens outstanding → larger effective liquidity → smaller price impact per dollar traded. The trade-off is that vig (house edge) is embedded in the pricing rule rather than added as a separate fee on top.

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State

Each pool tracks binary quantities (in micro-token units): $$q = (q_{\text{yes}},\ q_{\text{no}})$$ These are the net YES and NO tokens sold by the pool since inception, including the maker’s seed quantities. There is no reserve-ratio formula like $x \cdot y = k$.

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Liquidity parameter

Liquidity scales with total outstanding quantity $T = q_{\text{yes}} + q_{\text{no}}$: $$b(q) = \alpha \,(q_{\text{yes}} + q_{\text{no}}) = \alpha T$$ where $\alpha > 0$ is fixed at pool creation. When $\alpha$ is larger, the pool is deeper (more vig, less slippage). At uniform prices the total overround is approximately: $$p_{\text{yes}} + p_{\text{no}} = 1 + 2\alpha \ln 2$$ So a target vig of $v$ cents implies: $$\alpha \approx \frac{v}{2 \ln 2}$$ (e.g. $v = 0.05 \Rightarrow \alpha \approx 0.036$ for a ~5¢ overround at even odds).

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Cost function

The pool’s cost function is the LS-LMSR log-sum-exp form: $$C(q) = b(q)\,\ln\!\left(e^{q_{\text{yes}}/b(q)} + e^{q_{\text{no}}/b(q)}\right)$$ Numerically stable equivalent (used on-chain): let $m = \max(q_{\text{yes}}, q_{\text{no}})$, $d = m - \min(q_{\text{yes}}, q_{\text{no}})$, and $b = \alpha T$. Then: $$C(q) = m + b\,\ln\!\left(1 + e^{-d/b}\right)$$ Only the difference $d/b$ is exponentiated, which keeps fixed-point math safe at large $T$.

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Trade pricing algorithm

Trades are not priced at a displayed marginal quote. The economic rule is always the cost-function difference. Buy $\Delta$ tokens of side $i$ (e.g. YES): $$\text{cost} = C(q_i + \Delta,\ q_{\neg i}) - C(q_{\text{yes}}, q_{\text{no}})$$ Sell $\Delta$ tokens of side $i$: $$\text{payout} = C(q_{\text{yes}}, q_{\text{no}}) - C(q_i - \Delta,\ q_{\neg i})$$ On-chain, USDC amounts are rounded in the pool’s favor (buys rounded up, sells rounded down). Because $C$ is convex, $\Delta C$ per token rises as you trade more in one direction — that is AMM slippage.

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Marginal prices (display only)

The marginal price of side $i$ is the partial derivative: $$p_i = \frac{\partial C}{\partial q_i}$$ For binary LS-LMSR, the gradient includes an extra $\alpha$ term because $b$ depends on $q$. In scale-free coordinates $u = q_{\max}/T$ and $w = e^{-d/b}$: $$p_{\max} = u + g + (1-u)\frac{1-w}{1+w}, \qquad p_{\min} = u + g - u\frac{1-w}{1+w}$$ where $g = \alpha \ln(1+w)$. Properties: $$p_{\text{yes}} + p_{\text{no}} = 1 + v(q) \geq 1$$ The excess $v(q) \geq 0$ is embedded vig — it accrues to pool NAV, not a separate skim. At heavy skew the favorite’s raw marginal price can exceed $1; charts clamp display to $(0,1)$ but trades still use $\Delta C$.

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Implied fair probability

When a maker seeds at “60¢ YES”, that means the vig-free softmax probability, not the vig-inclusive marginal: $$\sigma_{\text{yes}} = \frac{e^{q_{\text{yes}}/b}}{e^{q_{\text{yes}}/b} + e^{q_{\text{no}}/b}} = \frac{1}{1 + e^{-(q_{\text{yes}} - q_{\text{no}})/b}}$$ This $\sigma$ is what “60% odds” means in the authoring UI. Raw marginals $p_i$ sit at or above $\sigma_i$ — the spread is the rule’s built-in margin.

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Seeding at chosen odds

At pool creation the maker supplies seed USDC $S$ and declared probability $p^*$ (in basis points). The program derives initial $(q_{\text{yes}}, q_{\text{no}})$ so that:

1. $\sigma_{\text{yes}} \approx p^*$ (within one price tick), and
2. Worst-case loss is bounded by the seed.

Let $P = \max(p^*, 1-p^*)$, logit $L = \ln\!\big(P/(1-P)\big)$, and $\rho = (1-P)/P$. Define: $$K = L + \ln(1+\rho), \qquad T = \frac{S}{\alpha K}$$ Then assign total quantity $T$ across sides with skew $\alpha T L = q_{\max} - q_{\min}$, putting $q_{\max}$ on YES when $p^* \geq \tfrac{1}{2}$. The worst-case loss at the seed state is: $$\text{WCL}(q_0) = C(q_0) - \min(q_{\text{yes},0}, q_{\text{no},0}) = \alpha T \ln\!\frac{1}{1-P} \leq S$$ That is why seed USDC is the maker’s collateral against subsidy — the pool cannot lose more than the seed in the adversarial orthant bound.

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Why depth grows with volume

Static LMSR keeps $b$ fixed, so a popular market eventually becomes too shallow — each marginal trade moves prices sharply. LS-LMSR ties $b = \alpha T$ to open interest. As players buy YES or NO: $$T = q_{\text{yes}} + q_{\text{no}} \uparrow \quad \Rightarrow \quad b \uparrow$$ A larger $b$ flattens the cost curve locally, so the same $ stake moves prices less after the market has absorbed more flow. Depth is endogenous to activity, not a fixed constant set only at launch.

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What you see in the UI

See Architecture for on-chain program layout and Venues for AMM vs CLOB.

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Liquidity

Makers seed initial liquidity when publishing a market. The pool holds balanced YES/NO reserves. LP providers earn swap fees from every trade regardless of who wins the market.

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Price display

On market and event pages you see:

• Current price — best available AMM price for each side
• Trade preview — stake, fees, shares received, and potential payout before you confirm
• Charts — price history where available (sports and crypto events)

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CLOB pricing (future)

Order-book markets will show bid/ask spreads and depth instead of AMM curves. CLOB trading is not available to players yet. See Venues.