Distribution Markets: Why Prediction Markets Should Think in Curves, Not Buckets
tl;dr: We propose a new prediction market primitive where traders submit full probability distributions over continuous outcomes (e.g., "ETH will be $3,200 ± $200"), rather than buying discrete Yes/No shares in fixed buckets. This eliminates liquidity fragmentation, captures trader confidence, and produces a richer, more composable information signal.
The Problem: Discrete Markets Waste Information
If you've used Polymarket, you've seen markets like this:
"Solana price on April 23?"
- 11 discrete buckets: <$40, $40-50, $50-60, ..., $120-130, >$130
- $11,234 in volume split across 11 separate liquidity pools
- Only one bucket resolves YES
This design is simple, but it discards almost everything interesting about what traders actually believe.
| What discrete markets destroy | What we lose |
|---|---|
| Precision | A trader who thinks ETH = $3,247.63 must round to the nearest bucket |
| Confidence | Someone "80% sure" and someone "51% sure" make the exact same trade |
| Liquidity | Each bucket is its own tiny AMM; spreads are wide everywhere |
| Composability | You cannot derive "P(ETH > $3,500)" without summing bucket prices manually |
In short: discrete buckets force traders to approximate their beliefs, fragment liquidity, and throw away the rich information contained in a full distribution.
The Idea: Trade Distributions, Not Binary Shares
In a Distribution Market, traders don't buy Yes/No shares. They submit a full probability distribution over a continuous outcome — for example, a Normal distribution N(μ, σ) predicting the price of ETH on May 1.
- μ (mean): Where they think the price will land
- σ (standard deviation): How confident they are (narrow = confident, wide = uncertain)
The market maintains a capital-weighted consensus distribution — a mixture of every trader's view, weighted by how much capital they put behind it.
At resolution, if ETH lands at x*, your payout depends on one simple ratio:
If you assigned more probability to the realized outcome than the market consensus did, you earn more than your collateral. If you assigned less, you earn less. This is a proper scoring rule: it is mathematically optimal for you to report your true beliefs.
Most importantly, the sum of all payouts always equals the total collateral. This is not an approximation — it is an exact algebraic identity. The contract can never become insolvent.
Three Immediate Advantages
1. Infinite Granularity
There are no bucket boundaries. A trader submits μ = 3247.63, and the market price at every real number is simultaneously discoverable.
2. Confidence Is Tradeable
σ is not a side effect — it's a first-class tradeable parameter. A trader who is highly confident submits a narrow σ and pays more for that precision. A trader who is uncertain submits a wide σ and pays less. The market explicitly prices uncertainty.
3. No Liquidity Fragmentation
All trading concentrates in a single pool. With the same total volume as a discrete market, a Distribution Market has deeper liquidity at every price point, tighter spreads, and better price discovery.
The Multimodal Mixture (And Why It's a Feature)
A natural question: if traders disagree — say, bears cluster at $2,900 and bulls at $3,800 — the consensus distribution becomes multimodal with a low-density "gap" between them.
If ETH resolves in that gap, everyone gets near-zero payout. This looks like a bug, but it's not. Genuine disagreement is real information. The market should display the consensus honestly — two peaks, a flag for "low-confidence zone between $X and $Y" — rather than forcing a false unimodal consensus.
We are also exploring a hybrid scoring fallback: when the market density falls below a threshold in gap zones, transition to a distance-based scoring against a reference Normal. This prevents the "total loss in the gap" failure mode without breaking the proper scoring rule under normal conditions.
Scaling: Merkle Proofs for Gas Efficiency
The resolution step requires computing each trader's probability density at the realized outcome — an O(N) loop. This is a one-time cost paid by the resolver; trade gas stays cheap.
For very large markets, we can move computation off-chain and verify it on-chain with a Merkle proof: the resolver submits a Merkle root of all (trader, density) pairs, and each trader claims by providing a Merkle path proving their individual value. Resolution gas becomes constant regardless of participant count.
Where This Fits
Our work is heavily inspired by Paradigm's Distribution Markets paper (December 2024). We adopt their core insights — function-space markets, L2 norm cost functions, and proper scoring rules — but make two practical simplifications for an on-chain MVP:
| Paradigm's Vision | Our MVP |
|---|---|
| Non-parametric functions (any PDF) | Parametric Normal distributions (O(1) storage) |
| Continuous function-space AMM | Capital-weighted average at resolution + Merkle proofs |
| Full L2 ball invariant | L2 norm used for fee scaling + minimum collateral |
We view this as a practical first step toward the full vision to move closer to non-parametric, continuous-function markets.
What's Next
We are currently building in three parallel tracks:
- Agent-based simulations — to validate the scoring rule, stress-test fee calibration, and generate concrete worked examples
- Solidity implementation — Foundry + PRB-Math, with both O(N) resolution and Merkle-proof variants
- Testnet deployment — Gnosis Chiado pilot with a single ETH spot-price market
If you're a prediction market researcher, mechanism designer, or smart contract engineer, we'd love your feedback.
→ Read the full technical proposal: https://github.com/gabrielfior/distribution-markets/blob/main/docs/distribution-markets-proposal.md
Built with Scaffold-ETH 2. Targeting Gnosis Chain for testnet launch.