DekantPM vs Paradigm distribution markets

Paradigm Research published Distribution Markets (Dave White, 2024) describing a continuous prediction-market design over an infinite-dimensional payout-function space. DekantPM is a finite-dimensional discretization of the same idea, with one critical difference around solvency.

What Paradigm proposed

Each trader holds a payout function f: ℝ → ℝ₊, where f(v) is the dollars paid if outcome v resolves. The AMM enforces:

‖f‖₂ = √(∫ f(v)² dv) = k

An L₂-norm constraint over a continuous function space. Rational traders post payout functions proportional to their belief density, and the AMM aggregates these into an implied distribution.

This is mathematically clean and infinite-dimensional, which means the system theoretically captures any real-valued belief distribution, not just Normals.

The backing problem

Paradigm's design has a subtle solvency issue:

In infinite dimensions, ‖f‖₂ = k does not bound max(f).

A function with very narrow spike at one point can have small L₂ norm while having arbitrarily large value at the spike. Concretely: a spike of height h and width w has ‖·‖₂² = h²w, so h = √(k²/w) - for any norm k, you can make h as large as you like by choosing w small enough.

This means a trader could in principle accumulate a payout function whose value at some v exceeds the AMM's collateral. Paradigm addresses this with an explicit cap:

‖f‖₂ = k    AND    max(f) ≤ b

The second constraint must be tracked separately (it's not implied by the first). This complicates the AMM: any operation that would push max(f) above b must be rejected, and tracking max(f) for arbitrary functions is non-trivial.

How DekantPM sidesteps it

DekantPM discretizes outcomes into N bins. The position becomes a vector x ∈ ℝᴺ with:

‖x‖₂ = √(Σ xᵢ²) = k

In finite dimensions, this constraint automatically implies:

xᵢ² ≤ k²    →    |xᵢ| ≤ k    (for every i)

The vault holds collateral ≥ k. The maximum claim against any single outcome is bounded by xᵢ, which is bounded by k, which is what the vault holds. Solvency is automatic, no extra constraint needed.

This is the key tradeoff:

	Paradigm (infinite-dim)	DekantPM (discretized)
Belief expressivity	Any function `f: ℝ → ℝ₊`	Only piecewise-constant on `N` bins
Solvency	Requires `max(f) ≤ b` as a separate constraint	Automatic from L₂ norm in finite dims
Tractability on-chain	Hard (must track `max(f)`)	Easy (linear pass over `N` outcomes)
Convergence	Already in the limit	As `N → ∞`, converges to Paradigm's design

Practical implications

For a trader, the discretization shows up as:

You can only express Normal-shaped beliefs (or compositions thereof). The infinite-dimensional version would let you post any payout function - including bimodal, skewed, or fat-tailed ones - directly.
Resolution is bin-quantized. The continuous version pays continuously based on f(v_resolved); DekantPM pays only on the bin containing v_resolved.

For a developer or LP:

The protocol is provably solvent at every state. No max(f) watchdog needed.
Per-instruction compute is O(N). With N ≤ 256, every operation fits in a single Solana transaction.
The protocol converges toward Paradigm's design as N increases. A larger N gives finer resolution and a closer approximation to the continuous integral, at the cost of more compute units per trade.

Why this is the right tradeoff for an on-chain market

Solana's per-transaction compute budget is finite. Tracking an additional max-payout invariant for arbitrary functions is expensive - every trade would need to recompute the bound. Tracking a finite-dimensional L₂ norm is cheap and the bound is implicit in the math.

You give up the elegance of unrestricted function space. You get a protocol that runs at full speed inside a single Solana instruction.

When you'd want Paradigm's version instead

Off-chain or in a settlement layer with much larger compute budgets, infinite-dimensional distribution markets make sense - especially for markets where the belief shape isn't well-modeled by a Normal (e.g., recovery rates, lottery payouts, fat-tailed outcomes).

DekantPM is positioned for the on-chain, retail-trader case where Normal-shaped curves cover most legitimate beliefs and where the on-chain solvency story has to be airtight.