A single aggregate demand curve can represent AMM order flow only when every trader shares the same homothetic preference. With any other preference structure, the distribution of holdings remains a state variable. No single utility function then reproduces aggregate best responses across pool states.
The aggregation condition is the result to remember. The paper presents Theorem 1's price equality and Proposition 6's Pareto result as a mechanism-specific analogue of the First Welfare Theorem.
Inside the closed pool
The model has one constant-function pool, with trading function phi and inventory r0. Two traders hold two assets in portfolios r1 and r2. Fees, an external trading venue and liquidity provision by the traders are absent. A market state is the triple (r0, r1, r2). Valid trades preserve the pool's invariant and leave all parties with nonnegative holdings. At a unilateral no-trade equilibrium, neither trader can improve through a single valid swap against the pool while the other remains still.
There is no data: no sample, no period, no universe.
The paper proves theorems and illustrates them with hand-built constant-product calibrations. Example 2's limiting state is reported rounded to three decimal places; Table 4 gives reserves to four figures. Its economics follows from replacing the fixed, linear Walrasian budget line with the pool's liquidity curve, which shifts whenever a trader uses it. Traders therefore impose price impact by construction. The pool participates by carrying inventory rather than quoting a price vector.
Theorem 1 supplies the characterization. Under Assumptions (I) to (IV) on phi, u1 and u2, an interior state is a unilateral equilibrium exactly when p(phi, r0) = p(u1, r1) = p(u2, r2). The pool's marginal price and both traders' marginal rates of substitution must equal one common number. The authors describe this as tangency to the pool's feasible frontier, rather than optimization on a shared linear budget set with clearing. Their modelling consequence matters: aggregate trader holdings are not conserved because the pool absorbs one asset while supplying the other. Total system inventory alone is conserved.
One feasible set can carry two prices
Equilibrium can support multiple prices. Example 3 contains two unilateral equilibria in the same feasible set, with identical totals (4, 9/4) and invariant level 1, at prices 1/4 and 1/2. Trader utilities are u1 = x^(1/8) y^(1/2) and u2 = x^(1/4) y^(1/4), while phi = (xy)^(1/4). Separate homotheticity for each trader achieves nothing. Proposition 5 identifies the stronger requirement: a common ratio-price map across traders determines the price, although the holdings split remains free.
Efficiency, bounded by the invariant
Proposition 6 gives the corresponding welfare statement: every individually rational unilateral equilibrium is Pareto optimal. The abstract includes the qualification, "relative to the fixed CFMM invariant", and the authors later explain that the result makes no efficiency claim relative to the larger pure-exchange feasible set. Pool inventory belongs to the allocation under comparison. The proposition describes the pool's own stopping point and supplies no evidence that routing through a CFMM is competitive.
Example 8 shows the gap most clearly at reserve scale lambda = 1. The pool begins at (1, 2), priced at 2.000. Traders start from (9,1) and (1,9), each with logarithmic utility 2.197. By the eightieth trade, both have reached 3.234. Under the Walrasian allocation, each receives (5,5), worth log 25, about 3.219. Both traders exceed the competitive benchmark.
The authors address the likely misreading twice. The pool creates no resources, since the mechanisms use different feasible sets, and logarithmic utility values are not representation-invariant welfare magnitudes. They retain only the ordering as economically relevant. I agree with that treatment. The 0.015 has no cardinal content.
The aggregation knife edge
Proposition 7 constructs a weak representative agent for n traders at any fixed equilibrium. It uses weighted sup-convolution with weights w_i = 1 / (d u_i / dy at r_i). These weights normalize each weighted gradient to (p0, 1). The construction works for precisely that reason, yet offers little away from the chosen state because its weights come from the equilibrium already solved.
Theorem 3 gives the sharp result. A state-independent representative agent exists if and only if u1, u2 and v are equivalent to one common homothetic preference. The proof uses the tangent-intercept transform A_f(q) = inf_x {qx + f(x)}. This converts the infimal convolution of frontiers into pointwise addition, and any one-parameter family closed under that addition must form a dilation family.
Remark 4 places the same obstruction on the supply side. Combining heterogeneous liquidity curves into one CFMM with a state-independent liquidity family demands the same collapse condition. Heterogeneous curves may still be combined locally or routed across. The resulting object need not equal a single CFMM drawn from a state-independent liquidity family.
Who gets the first trade?
Theorem 5 proves that alternating utility-maximizing trades converge. In Example 2, the pool starts with reserves (100,100), while the traders begin at (9,1) and (1,9). After four trades their positions are (4.928, 4.927) and (5.072, 5.073). The limit is (4.928, 4.928) and (5.072, 5.072), where both indifference prices and the pool price equal 1. The Walrasian allocation (5,5),(5,5) has the same price, lies off that path and requires a three-trade cooperative sequence.
Corollary 3 establishes the first-round order effect. When both indifference prices lie on the same side of the pool price, and neither opening trade crosses the other trader's price, the first mover gains more. When the pool price lies strictly between them, moving first leaves the trader worse off because the trade shifts the price toward the other trader. Conjecture 1 says this ordering may survive to the limit. It remains unproven, supported by the two constant-product calibrations in Examples 6 and 7.
Table 4 examines a different effect and does not test the conjecture. Trader 1 moves first in every row, so the table compares traders rather than trading orders. Holding the initial pool price at 2, trader 1's late-stage utility increases with reserve scale: 3.207 at lambda = 0.5, 3.234 at lambda = 1, and 3.297 at lambda = 2. Trader 2's utility declines from 3.247 to 3.234 and then 3.192. The authors see the first-round effect against agent 1 at lambda = 0.5, where trader 1 finishes below trader 2. At the reported late-stage iterate, the pool price is 1.048 and 1.091 at t = 80, then 1.167 at t = 32 for lambda = 2.
Reachability and rational choice
Valid reachability extends strictly beyond the individually rational set. Example 5(i) moves from ((1,1),(9,1),(1,9)) to ((1,1),(1.345,1.345),(8.655,8.655)) in three trades. The mechanism permits that state, although rational traders would never reach it. Mechanical permission and rational arrival are separate issues.
Theorem 4 answers the reachability question affirmatively for any interior feasible state. A finite valid trading sequence can reach it while keeping all intermediate states interior. The proof starts from Lemma 6's local controllability, using a two-trade loop for horizontal displacement and a four-trade loop for vertical displacement.
The assumptions also exclude shifted or truncated constant-product pools, at least when read literally, including those used to represent concentrated liquidity through virtual reserves (Uniswap V3). Assumptions (I) and (II) impose the axes as asymptotes and allow prices across all of (0, infinity). A shifted constant-product pool confines attainable prices to [beta^2/e^c, e^c/alpha^2], with x in [0, e^c/beta - alpha]. Concentrated-liquidity behaviour occupies that bounded interval. The authors discuss the truncated case informally through one-sided conditions and say they do not develop a full boundary theory.
We could not test any of this. The model requires pool reserves at every swap, the invariant, the ordered trade sequence and each trader's utility. Our crypto data consist of price bars. They contain none of the pool state and cannot recover preferences even in principle. This remains a modelling note.
The caveat that decides its reach
The closed market is the authors' own caveat. They describe the closed-market assumption as a modeling restriction intended to isolate the internal equilibrium force created by the liquidity curve, and they disclaim any suggestion that decentralized-asset markets are typically isolated. Fees, outside arbitrage and strategic liquidity provision are additional wedges deliberately omitted.
A version with fees and an outside arbitrageur would change my view if the common-homothetic collapse condition still held. Then the aggregation result would become a constraint on pooled demand models rather than a benchmark curiosity.