Execution Price Impact as the Token Universe Grows

Multiswap places a universe of $n$ Reserve Assets in one pool. This article compares its marginal execution-price impact with an equivalent Uniswap v2 star containing the same assets, initial relative prices, and aggregate initial reserves token for token.

The universe contains $n-1$ assets $A_i$ and one generic stablecoin called $\mathrm{USD}$. The first comparison studies a trade $A\to\mathrm{USD}$. The second studies an $A\to B$ trade, which Multiswap executes as one interaction while a Uniswap v2 star routes through USD.

For equal starting universes, the Multiswap execution-depth advantage is

$$\boxed{ \frac{n-1}{n(1-e_s)}. }$$

The advantage increases with the number of assets $n$ and the scale elasticity $e_s$. Multiswap reaches parity with Uniswap v2 at

$$\boxed{ e_s=\frac1n. }$$

For the $A\to B$ comparison, the Multiswap advantage in the receive-side $B/\mathrm{USD}$ execution-price response is

$$\boxed{ \frac{1}{1-e_s}. }$$

Scope. Both comparisons use initial execution-price slopes and equal starting universes with the same assets, token quantities, and relative prices. The $A\to B$ comparison follows one Multiswap interaction and the corresponding two-pool Uniswap v2 route.

1. Prices and execution prices

Every Multiswap Reserve Asset $i$ has reserve

$$a_i>0,$$

scale

$$s_i>0,$$

and price in the internal Scale numeraire

$$\boxed{ P_i=\frac{s_i}{a_i}. }$$

The price of $A$ in terms of USD is

$$\boxed{ P_{A,\mathrm{USD}} = \frac{P_A}{P_{\mathrm{USD}}} = \frac{s_A/a_A}{s_{\mathrm{USD}}/a_{\mathrm{USD}}}. }$$

For any Reserve Asset,

$$\boxed{ P_{A,A}=1. }$$

The execution price of an $A\to\mathrm{USD}$ swap is the quantity of USD received per unit of $A$ paid:

$$\boxed{ P_{A,\mathrm{USD}}^{\mathrm{exec}} = -\frac{da_{\mathrm{USD}}}{da_A}. }$$

At the initial state,

$$P_{A,\mathrm{USD}}^{\mathrm{exec}} = P_{A,\mathrm{USD}}.$$

The comparison uses the initial slope

$$\boxed{ \left. \frac{dP_{A,\mathrm{USD}}^{\mathrm{exec}}}{dr_A} \right|_{r_A=0}, }$$

where

$$r_A=\frac{da_A}{a_A}.$$

2. Equal starting universes

The Uniswap v2 star contains $n-1$ pools:

$$A_i\longleftrightarrow\mathrm{USD}.$$

Each spoke begins with equal USD value on its two sides. Equal spokes contain the same initial quantity of USD and the same initial USD value of their other asset.

The equivalent Multiswap pool contains the same $n$ assets, the same aggregate quantity of every asset, and the same initial relative prices.

For each non-USD Reserve Asset,

$$a_iP_{i,\mathrm{USD}} = a_{\mathrm{USD},i}.$$

Since

$$P_{i,\mathrm{USD}} = \frac{P_i}{P_{\mathrm{USD}}},$$

we have

$$s_i = a_iP_i = a_{\mathrm{USD},i}P_{\mathrm{USD}}.$$

The aggregate USD reserve is

$$a_{\mathrm{USD}} = \sum_{i=1}^{n-1}a_{\mathrm{USD},i}.$$

Its Multiswap scale is

$$\begin{aligned} s_{\mathrm{USD}} &= a_{\mathrm{USD}}P_{\mathrm{USD}} \\[4pt] &= \sum_{i=1}^{n-1} a_{\mathrm{USD},i}P_{\mathrm{USD}} \\[4pt] &= \sum_{i=1}^{n-1}s_i. \end{aligned}$$

For equal spokes, $s_i=s_A$ for every non-USD Reserve Asset. Therefore

$$\boxed{ s_{\mathrm{USD}}=(n-1)s_A. }$$

The initial reserve of $A$ is equal in both systems. The same

$$r_A=\frac{da_A}{a_A}$$

therefore represents the same quantity $da_A$.

3. Multiswap execution-price impact

Post-Trade Elasticity uses scale elasticity $e_s$ and price elasticity $e_P$:

$$\boxed{ e_P=1-e_s, \qquad 0\le e_s<1. }$$

For each participating Reserve Asset,

$$\boxed{ \frac{P_i'}{P_i} = (1+r_i)^{-e_P}. }$$

Multiswap settles the swap at the post-trade relative price:

$$\boxed{ P_{A,\mathrm{USD}}^{\mathrm{exec}} = P_{A,\mathrm{USD}}'. }$$

For $A\to\mathrm{USD}$,

$$\begin{aligned} \frac{P_{A,\mathrm{USD}}'}{P_{A,\mathrm{USD}}} &= \frac{P_A'/P_A} {P_{\mathrm{USD}}'/P_{\mathrm{USD}}} \\[4pt] &= \left( \frac{1+r_{\mathrm{USD}}} {1+r_A} \right)^{e_P}. \end{aligned}$$

The two reserve changes satisfy

$$\boxed{ s_A\frac{r_A}{(1+r_A)^{e_P}} + s_{\mathrm{USD}} \frac{r_{\mathrm{USD}}} {(1+r_{\mathrm{USD}})^{e_P}} = 0. }$$

Differentiating at the initial state gives

$$\boxed{ \left. \frac{dr_{\mathrm{USD}}}{dr_A} \right|_{r_A=0} = -\frac{s_A}{s_{\mathrm{USD}}}. }$$

Therefore,

$$\boxed{ \left. \frac{dP_{A,\mathrm{USD}}^{\mathrm{exec}}}{dr_A} \right|_{r_A=0,\mathrm{MS}} = -e_PP_{A,\mathrm{USD}} \left( 1+\frac{s_A}{s_{\mathrm{USD}}} \right). }$$

For the equal starting universe,

$$s_{\mathrm{USD}}=(n-1)s_A,$$

so

$$\boxed{ \left. \frac{dP_{A,\mathrm{USD}}^{\mathrm{exec}}}{dr_A} \right|_{r_A=0,\mathrm{MS}} = -\frac{n(1-e_s)}{n-1} P_{A,\mathrm{USD}}. }$$

4. Uniswap v2 execution-price impact

Consider the $A/\mathrm{USD}$ spoke in the equivalent Uniswap v2 star.

Its initial relative marginal price follows from

$$\begin{aligned} P_{A,\mathrm{USD}} &= \frac{P_A}{P_{\mathrm{USD}}} \\[4pt] &= \frac{s_A/a_A} {s_{\mathrm{USD}}/a_{\mathrm{USD}}}. \end{aligned}$$

The two sides of the pool have equal scale at this price, giving

$$\boxed{ P_{A,\mathrm{USD}} = \frac{a_{\mathrm{USD}}}{a_A}. }$$

After the user pays

$$da_A=a_Ar_A,$$

the constant-product relation gives the USD amount received:

$$-da_{\mathrm{USD}} = a_{\mathrm{USD}} \frac{r_A}{1+r_A}.$$

The execution price is therefore

$$\begin{aligned} P_{A,\mathrm{USD}}^{\mathrm{exec}} &= -\frac{da_{\mathrm{USD}}}{da_A} \\[4pt] &= \frac{ a_{\mathrm{USD}}r_A/(1+r_A) }{ a_Ar_A } \\[4pt] &= \frac{P_{A,\mathrm{USD}}}{1+r_A}. \end{aligned}$$

Differentiating at the initial state gives

$$\boxed{ \left. \frac{dP_{A,\mathrm{USD}}^{\mathrm{exec}}}{dr_A} \right|_{r_A=0,\mathrm{V2}} = -P_{A,\mathrm{USD}}. }$$

5. Multiswap advantage

Place the Uniswap v2 execution-price slope in the numerator so that a larger ratio represents a larger Multiswap advantage:

$$\boxed{ \frac{ \left| dP_{A,\mathrm{USD}}^{\mathrm{exec}}/dr_A \right|_{\mathrm{V2}} }{ \left| dP_{A,\mathrm{USD}}^{\mathrm{exec}}/dr_A \right|_{\mathrm{MS}} } = \frac{1} {e_P\left(1+s_A/s_{\mathrm{USD}}\right)}. }$$

For equal starting universes,

$$s_{\mathrm{USD}}=(n-1)s_A,$$

and therefore

$$\boxed{ \text{Multiswap advantage} = \frac{n-1}{ne_P} = \frac{n-1}{n(1-e_s)}. }$$

Parity occurs when this ratio equals one:

$$\boxed{ e_s=\frac1n. }$$

For

$$e_s>\frac1n,$$

Multiswap has less marginal execution-price impact than the equivalent Uniswap v2 star.

6. Advantage matrix

Each entry is the Uniswap v2 execution-price-impact slope divided by the Multiswap execution-price-impact slope.

Assets $n$	$e_s=0$	$e_s=1/512$	$e_s=1/128$	$e_s=1/32$	$e_s=1/8$	$e_s=1/2$
$2$	$0.5\times$	$0.501\times$	$0.504\times$	$0.516\times$	$0.571\times$	$1\times$
$8$	$0.875\times$	$0.877\times$	$0.882\times$	$0.903\times$	$1\times$	$1.75\times$
$32$	$0.969\times$	$0.971\times$	$0.976\times$	$1\times$	$1.107\times$	$1.938\times$
$128$	$0.992\times$	$0.994\times$	$1\times$	$1.024\times$	$1.134\times$	$1.984\times$
$512$	$0.998\times$	$1\times$	$1.006\times$	$1.030\times$	$1.141\times$	$1.996\times$
$n\to\infty$	$1\times$	$1.002\times$	$1.008\times$	$1.032\times$	$1.143\times$	$2\times$

Assets $n$	$e_s=0.6$	$e_s=0.7$	$e_s=0.8$	$e_s=0.9$	$e_s=0.95$	$e_s=0.975$
$2$	$1.25\times$	$1.667\times$	$2.5\times$	$5\times$	$10\times$	$20\times$
$8$	$2.188\times$	$2.917\times$	$4.375\times$	$8.75\times$	$17.5\times$	$35\times$
$32$	$2.422\times$	$3.229\times$	$4.844\times$	$9.688\times$	$19.375\times$	$38.75\times$
$128$	$2.480\times$	$3.307\times$	$4.961\times$	$9.922\times$	$19.844\times$	$39.688\times$
$512$	$2.495\times$	$3.327\times$	$4.990\times$	$9.980\times$	$19.961\times$	$39.922\times$
$n\to\infty$	$2.5\times$	$3.333\times$	$5\times$	$10\times$	$20\times$	$40\times$

An entry above $1\times$ means Multiswap has less marginal execution-price impact. An entry of $1\times$ means parity. An entry below $1\times$ means Uniswap v2 has less.

7. Direct Multiswap interaction and routed Uniswap v2 trade

Now consider a trade between two non-USD assets:

$$A\longrightarrow B.$$

Multiswap executes this as one interaction. The participating Reserve Assets satisfy

$$r_A>0, \qquad r_B<0, \qquad r_{\mathrm{USD}}=0.$$

Since the USD reserve remains fixed,

$$P_{\mathrm{USD}}'=P_{\mathrm{USD}}.$$

The value-flow equation is

$$s_A\frac{r_A}{(1+r_A)^{e_P}} + s_B\frac{r_B}{(1+r_B)^{e_P}} =0.$$

Differentiating at the initial state gives

$$\boxed{ \left. \frac{dr_B}{dr_A} \right|_{r_A=0} = -\frac{s_A}{s_B}. }$$

Multiswap settles at the post-trade relative price. Its receive-side $B/\mathrm{USD}$ execution price therefore satisfies

$$\frac{P_{B,\mathrm{USD}}^{\mathrm{exec}}}{P_{B,\mathrm{USD}}} = \frac{P_{B,\mathrm{USD}}'}{P_{B,\mathrm{USD}}} = (1+r_B)^{-e_P}.$$

Therefore,

$$\boxed{ \left. \frac{dP_{B,\mathrm{USD}}^{\mathrm{exec}}}{dr_A} \right|_{r_A=0,\mathrm{MS}} = e_P\frac{s_A}{s_B}P_{B,\mathrm{USD}}. }$$

For the equal starting universe, $s_A=s_B$, giving

$$\boxed{ \left. \frac{dP_{B,\mathrm{USD}}^{\mathrm{exec}}}{dr_A} \right|_{r_A=0,\mathrm{MS}} = (1-e_s)P_{B,\mathrm{USD}}. }$$

The Uniswap v2 star executes the same economic trade as two interactions:

$$A\longrightarrow\mathrm{USD}, \qquad \mathrm{USD}\longrightarrow B.$$

The USD received from the $A/\mathrm{USD}$ spoke is

$$-da_{\mathrm{USD},A} = a_{\mathrm{USD},A} \frac{r_A}{1+r_A}.$$

That quantity becomes the USD paid into the $B/\mathrm{USD}$ spoke:

$$r_{\mathrm{USD},B} = \frac{-da_{\mathrm{USD},A}}{a_{\mathrm{USD},B}}.$$

Equal spokes satisfy

$$a_{\mathrm{USD},A}=a_{\mathrm{USD},B},$$

so

$$r_{\mathrm{USD},B} = \frac{r_A}{1+r_A}.$$

Therefore,

$$\left. \frac{dr_{\mathrm{USD},B}}{dr_A} \right|_{r_A=0} =1.$$

In the constant-product $B/\mathrm{USD}$ spoke, the amount of $B$ received is

$$-da_B = a_B \frac{r_{\mathrm{USD},B}}{1+r_{\mathrm{USD},B}}.$$

The execution price is the USD paid per unit of $B$ received:

$$\begin{aligned} P_{B,\mathrm{USD}}^{\mathrm{exec}} &= \frac{da_{\mathrm{USD},B}}{-da_B} \\[4pt] &= P_{B,\mathrm{USD}} (1+r_{\mathrm{USD},B}). \end{aligned}$$

Differentiating at the initial state gives

$$\boxed{ \left. \frac{dP_{B,\mathrm{USD}}^{\mathrm{exec}}}{dr_A} \right|_{r_A=0,\mathrm{V2}} = P_{B,\mathrm{USD}}. }$$

Place the Uniswap v2 slope in the numerator:

$$\boxed{ \frac{ \left.dP_{B,\mathrm{USD}}^{\mathrm{exec}}/dr_A\right|_{\mathrm{V2}} }{ \left.dP_{B,\mathrm{USD}}^{\mathrm{exec}}/dr_A\right|_{\mathrm{MS}} } = \frac{1}{e_P} = \frac{1}{1-e_s}. }$$

This advantage depends on $e_s$ and is independent of $n$.

Trade	$e_s=0$	$e_s=1/512$	$e_s=1/128$	$e_s=1/32$	$e_s=1/8$	$e_s=1/2$
$A\to B$	$1\times$	$1.002\times$	$1.008\times$	$1.032\times$	$1.143\times$	$2\times$

Trade	$e_s=0.6$	$e_s=0.7$	$e_s=0.8$	$e_s=0.9$	$e_s=0.95$	$e_s=0.975$
$A\to B$	$2.5\times$	$3.333\times$	$5\times$	$10\times$	$20\times$	$40\times$

8. The result

The Multiswap execution-depth advantage is

$$\boxed{ \frac{n-1}{n(1-e_s)}. }$$

For fixed $e_s$, this advantage increases with $n$ and approaches

$$\boxed{ \frac{1}{1-e_s}. }$$

For fixed $n$, the advantage increases with $e_s$.

At $n=2$, parity occurs at

$$\boxed{ e_s=0.5. }$$

For the 512-asset universe and

$$e_s=0.95,$$

the advantage is approximately

$$\boxed{ 19.961\times. }$$

Multiswap combines Post-Trade Elasticity with the unified USD state of the complete $n$-asset pool. The resulting execution-depth advantage increases with both the size of the universe and scale elasticity.

For a trade $A\to B$, Multiswap uses one interaction between the participating Reserve Assets. The equivalent Uniswap v2 star route composes two pool interactions through USD. The resulting advantage in the receive-side $B/\mathrm{USD}$ execution-price response is

$$\boxed{ \frac{1}{1-e_s}, }$$

which increases with scale elasticity and applies across universe sizes.