# Why splitting a trade across pools costs less: the convexity behind route splitting

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> Updated: 2026-07-29

Price impact is convex in size, so one large fill is never cheaper than the same size split across independent pools — that is Jensen's inequality, not a heuristic. The optimal split equalises marginal price across venues, which means shares proportional to real depth, never an even division.

Send an order to one pool and you pay for every unit of depth you consume, at a price that gets worse as you go. Send the same order across several independent pools and each one is asked for a smaller bite, at a price that stays closer to the top of its book. The difference is not a routing trick or a marketing claim: it is a mathematical property of the cost curve, and it can be stated precisely enough to compute.

## Impact is convex, and convexity is the whole argument

For a constant-product pool the output for an input x against a reserve R is a strictly concave function of x — each additional unit buys less than the one before it. The cost you pay, price impact, is therefore CONVEX in size: it does not grow linearly, it accelerates. Doubling the order more than doubles the impact.

Once you know the cost function is convex, the result follows from a theorem rather than from intuition. Jensen's inequality states that for a convex cost, the cost of the average is no greater than the average of the costs — equivalently, splitting a quantity into parts and paying each part's cost is never worse than paying the cost of the whole at once, and is strictly better whenever the curve is genuinely curved. Splitting cannot lose. That is a proof, not a preference.

> equation: C(Σxᵢ) ≥ Σ C(xᵢ) for C convex, xᵢ ≥ 0 — The convexity argument: paying the cost of one large fill is never cheaper than paying the sum of the costs of its parts across independent pools — Jensen's inequality, applied to price impact.

## The optimum: equalise the marginal price, not the share

Knowing that splitting helps does not say how to split. The naive answer — divide equally across the candidates — is usually wrong and can be catastrophic: an even split between a deep pool and a near-empty one sends half the order into a book that cannot fill it, destroying more value than routing to the deep pool alone would have.

The correct condition comes from constrained optimisation. Maximise total output subject to the shares summing to the order size, and the first-order condition says every pool that receives a non-zero share must have the SAME marginal output at its allocation. If one venue is still paying a better rate at the margin than another, you are allocating wrongly: move a unit from the worse margin to the better one and the total rises. The optimum is reached exactly when no such move helps — when the marginal prices are equal.

The practical consequence is that shares must be proportional to real depth, not to venue count. A pool holding ten times the capital absorbs roughly ten times the size before its marginal price degrades to the same level. Allocating by depth is the discrete approximation of equalising the margin; allocating evenly is an approximation of nothing.

> equation: ∂fᵢ/∂xᵢ = λ for every i with xᵢ > 0, Σ xᵢ = X — The Lagrangian condition for an output-maximising split: at the optimum every used venue pays the same marginal rate λ. Unequal margins mean value is still on the table.

## Why the gains die off — the case for a bounded leg budget

If splitting is free improvement, why not split across everything? Because the benefit is concave in the number of venues while the cost is linear in it. The first split removes the most expensive part of the curve — the tail of the single fill where impact accelerates hardest. The second removes less. By the fourth or fifth the marginal saving is small, while every additional leg adds a fixed execution cost and one more pool whose state can move against you between quote and inclusion.

So the leg budget is an optimisation, not a limitation: extend it until the expected marginal saving from one more venue falls below that venue's marginal cost, then stop. A small bounded budget captures nearly all of the available amortisation; an unbounded one pays real costs for diminishing arithmetic.

## Independence is the load-bearing assumption

The argument above quietly assumes the pools are INDEPENDENT — that filling one does not move the price of another. When that holds, the splits genuinely see fresh books. When it does not, the assumption fails and the maths goes with it: two venues that are really the same liquidity wearing different clothes, or two pools an arbitrage bot keeps welded together within the same block, do not offer two independent bites of depth. A splitter that counts them twice believes it has diversified when it has concentrated.

This is why depth must be measured as capital that actually exists rather than as advertised numbers, and why the split should be computed against what each venue can really absorb. The convexity result is only as good as the independence of the curves you are summing over — a statistical assumption, and one worth testing rather than assuming.

## Slippage, separately

Splitting reduces IMPACT — the cost you cause yourself against finite depth, which is knowable before you sign. It does not, by itself, reduce SLIPPAGE — the market moving between quote and inclusion, which is a random variable nobody can predict. But it changes that variable's shape: the realised outcome becomes an average across several venues rather than a single draw, so extreme deviations partially cancel and the variance of the fill narrows even when its mean is unchanged. That is diversification, in the ordinary statistical sense, applied to execution.

Reduced variance is not protection, though — the tail still exists. The bound on the worst case has to come from somewhere else: an enforced minimum-output floor, re-derived from the real trade at execution, that reverts rather than settles below it. Splitting lowers the expected cost and tightens the distribution; the floor is what caps the tail.

**Verify it yourself:** Quote a size against a single pool, then quote the same size split across several independent pools of comparable total depth, and compare outputs — the split is never worse and improves with size, exactly as convexity predicts; then split evenly instead of by depth and watch the advantage collapse

Related: https://blazephoenix.xyz/learn/capital-anchored-split · https://blazephoenix.xyz/learn/price-impact-explained · https://blazephoenix.xyz/learn/iron-floor · https://blazephoenix.xyz/learn/phantom-liquidity