An arXiv preprint examines a closed, fee-free economy with one constant-function market maker (CFMM), two assets and two traders. A CFMM is a market-making mechanism whose price depends on the inventory it holds. Under the paper’s assumptions, an interior unilateral equilibrium—where neither trader has a profitable one-sided move—occurs exactly when the CFMM price matches both traders’ marginal rates of substitution, or their marginal willingness to exchange one asset for another. The result comes from formal mathematical analysis, not an empirical sample: proofs establish the propositions, while numerical calculations illustrate selected trading paths.
The efficiency claim is deliberately narrow
That equality does not turn the model into a general efficiency claim. For an interior initial state, every unilateral equilibrium that is individually rational relative to that start is Pareto optimal relative to the fixed total inventory and CFMM invariant. In ordinary language, within those constraints, there is no permitted alternative that makes one trader better off without making the other worse off. The comparison does not extend to the larger feasible set of a Walrasian pure-exchange economy.
One representative agent, or many?
The paper also asks whether two traders can be represented as one. At a chosen equilibrium, a weak representative utility can be constructed through weighted sup-convolution, a mathematical way to combine their utility frontiers. But the weights are local planner weights determined by equilibrium marginal utilities, so the construction is generally not state-independent.
A stronger, state-independent representative agent exists exactly when the aggregate frontiers collapse to one family of utility level curves—equivalently, when the traders share a common homothetic preference. Under that condition, any two interior unilateral equilibria in the same feasible set have the same equilibrium price, although their allocations can still differ.
Which states can the market reach?
Reachability is broader than welfare improvement. Starting from an interior state, every interior feasible state can be reached through a finite sequence of valid trades, with all intermediate states kept interior. But “valid” here describes mechanical possibility: the result does not say that every trade improves utility or is utility-maximizing.
When traders instead alternate utility-maximizing trades, the model gives a convergence result. From an interior start, the sequence converges to an interior unilateral equilibrium. When it begins from the paper’s specified initial state, the limit is individually rational and Pareto optimal within the model’s feasible set.
Examples put the theory on a path
A worked numerical path makes the convergence result concrete. Its computed limit was approximately a CFMM position of (100, 100), holdings of (4.928, 4.928) for the first trader and (5.072, 5.072) for the second, with all three prices tending to 1. The Walrasian benchmark gave each trader (5, 5). These are rounded numerical approximations from a worked calculation, not observations from a market.
A separate numerical comparison examined reserve scale. At the intermediate scale λ = 1, both reported utility values exceeded log 25, approximately 3.219. The example says both traders can be better off than the corresponding Walrasian allocation, but not for every reserve scale or initial state. The paper presents this as a numerical comparison rather than a general comparative-static theorem, and the utility figures use a logarithmic normalization.
Who moves first matters
The sharpest path-dependence result concerns the first round of trading. The first mover is favored when both traders initially want the same direction of trade and the first trade does not cross the other trader’s indifference price—the price at which that trader is just indifferent. But the first mover is disadvantaged when the initial CFMM price lies between the two traders’ indifference prices. The effect therefore depends on the starting configuration rather than following a single rule.
The authors go further only as a conjecture: they propose that the first-round welfare ordering persists at limiting equilibria. Numerical examples support that idea, but the general statement remains unproved.
A benchmark with tight boundaries
A separate result shows that, with initial price alignment, a finite sequence of valid trades can implement the Walrasian allocation. This is not a convergence result: the trades need not improve utility and need not be best responses.
The model is deliberately narrow. It has one CFMM, two assets and two traders, with no external trading venue and no liquidity provision by traders. It excludes fees, outside venues, strategic liquidity provision, stochastic arrivals and more than two assets, leaving those mechanisms as future-work directions. The analysis also does not prove that every interior individually rational state is reachable through improving trades. The supplied document is an arXiv v1 preprint dated 24 Aug 2026.
Paper data and sources
Original title: Equilibrium in closed constant-function market maker economies
Authors: Muqiao Huang, Ruodu Wang, Yiyun Wang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-24
DOI: Not available
Original paper · Full text