Parabola Markets

The Problem

Binary markets discard your knowledge.

Ask "Will the Lakers–Celtics game go over 220?" and you're forced into yes or no. But you know far more than that.

Standard sportsbook

Forced binary, buried vig

You think the total will land around 224. The line is fixed at 220.5 with a ~5% house edge baked in. You can't say how confident you are or where you think it truly peaks.

Lakers vs Celtics — Total
Over 220.5 −115 (house: 53.5%)
Under 220.5 −105 (house: 51.2%)

Parabola

Express your full belief

Submit the distribution you actually believe. The closer your distribution to the realized outcome, the higher your payout. Your confidence (σ) matters just as much as your direction (μ).

Lakers vs Celtics — Total
Your estimate μ = 224 σ = 5.8
Crowd μ = 219 σ = 9.4

Your Edge

More ways to be right.

In a binary market, you either picked the right side or you didn't. In an estimation market, being a little right still pays — and being very precise pays a lot.

NBA example · same game, same information

What you can express	Binary	Parabola
You believe: total lands around 224, probably 218–230
Direction (over/under)	✓ yes/no	✓
How sure you are	✗ ignored	✓
"Likely between 218–230"	✗ no market	✓
"Crowd is too uncertain"	✗ no market	✓
Your edge captured	Partial	Full

Can a confidence interval beat an over/under?

Yes. If you submit N(224, σ=5) and the crowd is at N(219, σ=9), you're implicitly saying the total lands between 215–230 with 85% probability. The crowd says only 56%. That gap — on a range the sportsbook doesn't even offer — is pure captured value. You earn for being right about the range, not just the side.

Markets that can't exist in binary

Volatility bet

You think the outcome will be extreme — but you don't know which direction. Bet a wide distribution. Binary markets have no way to express this.

Certainty bet

You agree with consensus on direction but think the crowd is overconfident. Bet a tighter σ — lower collateral, higher payout per dollar if you're right.

Interval bet

"It'll definitely land somewhere in this range." Express a narrow distribution centered on that range. There's no binary equivalent.

Tail bet

You think the consensus is sleeping on a tail outcome. Submit a distribution with heavier tails than the crowd to capture the underpriced extreme scenarios.

Live comparison — Lakers vs Celtics total

What the math actually says

Under the log scoring rule, your score after outcome x is log Q(x). Taking expectations under the true distribution P*:

            𝔼[score | report Q] = −H(P*) − KL(P* ‖ Q)

            P* = true dist · Q = reported · H = entropy of P*

            Since KL ≥ 0, score is maximised when Q = P*.

            Caveat: Q must assign positive probability wherever P* does, or KL blows up.

This shows log scoring is strictly proper: the unique best strategy is to report your true belief. Ordinary fixed-payoff binary bets do not automatically have this property — they reward being directionally right, not reporting a calibrated probability. A binary contract can be made proper (e.g. via log-score or Brier-score payoffs), but a standard over/under payout is not one of those.

NBA example: KL(N(224,5) ‖ N(219,9)) = log(9/5) + (25+25)/162 − ½ ≈ 0.40 — a single number capturing both the mean shift and the tighter certainty.

How It Works

Four steps. Infinite nuance.

01

Pick a market

Sports totals, macro data releases, crypto prices, election margins — any outcome with a numeric realization.

02

Submit a distribution

Choose μ (where you expect the outcome) and σ (how certain you are). More conviction → tighter σ → less collateral.

03

Sign on Solana

Your distribution is an on-chain transaction. The AMM quotes you immediately against the current crowd distribution.

04

Settle and collect

A proper scoring rule pays proportional to how close your distribution was to the realized outcome. Precision compounds.

Perpetual Estimation Markets

Trade the distribution continuously.

Distribution perps let you hold a long or short position on how the crowd's collective estimate evolves over time — a perpetual on crowd belief itself.

AMM distribution vs Pyth oracle anchor

Long +0.014%/hr

Short −0.014%/hr

The AMM maintains a market-implied distribution relative to an oracle reference. Traders express views over the whole distribution — not a single binary event. Funding is based on a signed, statewise discrepancy (log-density ratios), so you are rewarded when the market distribution moves toward the one you supplied.

Long mean: you think the market's implied mean is too low. You profit as the distribution's centre shifts up toward the oracle. A binary contract cannot express this cleanly across the full distribution.
Short uncertainty: you think the market is overestimating variance. Submit a tighter distribution. You profit as the crowd's σ compresses — a dimension binary markets don't price at all.
Signed funding: unlike raw KL (which is unsigned), the funding rule uses statewise log-density ratios so it's always clear who pays whom and why.
No expiry: hold as long as you want. Exit any time by taking the opposing side.

Weather example

A binary market asks only: "Will Chicago exceed 75°F on Thursday?" That collapses your entire forecast into a single threshold probability. Suppose you believe the temperature will be around 72°F with low variance, while the market is wider and centred elsewhere. A distribution market lets you express both: the expected level and the confidence around it. You can be rewarded for being right about the shape of the distribution — not just which side of an arbitrary line the outcome lands on.

Early Access

Ready to bet the full distribution?

Parabola is live on Solana devnet. Join the waitlist to get early access.

Join the waitlist or reach out on X →

Markets for NBA totals. Brent crude. election margins. Solana prices. Bitcoin's year-end. Fed rate decisions.