Optimal Protocol Fee

Abstract

We analyze the fee-taking strategy for a protocol that captures a share of liquidity provider (LP) fees in an automated market maker. Under constant taker flow and a simple equilibrium condition for liquidity supply, we derive the optimal protocol fee share that maximizes protocol profit while accounting for TVL dynamics. We show that the protocol maximizes profit by capturing one-third of the total fee pool, leaving two-thirds to LPs.

1. Model Setup

Let:

v = volume (exogenous, constant)
F = total LP fee rate (exogenous, fixed)
f = LP fee share (fraction of F retained by LPs)
g = protocol fee share (fraction of F retained by the protocol)
r = short-term market rate (exogenous, includes risk premium)
L = total value locked (endogenous)

We assume f + g = 1 (protocol and LPs partition the fee).

2. Market Equilibrium

In equilibrium, LP earnings equal the opportunity cost:

v f = r L

Thus:

L = \frac{v f}{r}

The TVL adjusts until LPs are indifferent between deploying capital here versus the outside market rate r.

3. Protocol Profit Maximization

Protocol profit is:

p = f g L = f g \cdot \frac{v f}{r} = \frac{v}{r} f^2 g

Substituting f = 1 - g:

p(g) = \frac{v}{r} (1 - g)^2 g \quad \text{for } g \in [0,1]

Let h(g) = g(1 - g)^2. Taking the derivative:

h'(g) = (1 - g)^2 - 2 g (1 - g) = (1 - g)(1 - 3 g)

4. Solution

Setting h'(g) = 0 yields critical points at g = 1 and g = \frac{1}{3}.

Evaluating:

h(0) = 0
h(1) = 0
h\!\left(\frac{1}{3}\right) = \frac{1}{3} \cdot \left(\frac{2}{3}\right)^2 = \frac{4}{27}

The maximum occurs at $g^ = \frac{1}{3}$*, giving:

Quantity	Value
Protocol fee share	`g^* = \frac{1}{3}`
LP fee share	`f^* = \frac{2}{3}`
Equilibrium TVL	`L^* = \frac{2 v}{3 r}`
Max protocol profit	`p^* = \frac{4 v}{27 r}`

Conclusion

Under the assumptions:

Constant exogenous volume (v),
Fixed total fee rate (F),
LPs supply capital until fee income equals opportunity cost (rL),
Protocol profit is proportional to both the fee split and TVL,

the protocol’s optimal fee policy is:

Protocol share of the fee pool:
\phi_P^* = \frac{1}{3}
LP share of the fee pool:
\phi_L^* = \frac{2}{3}

Equivalently, in absolute terms:

Optimal protocol fee rate: F/3,
Optimal LP fee rate: 2F/3.

This fee split balances extracting revenue from trades against maintaining sufficiently attractive LP returns to support a large equilibrium TVL.

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