When going first is worth something
Three authors work out when a constant-function market maker is at rest, and when the order of two traders decides who gets the better fill.
2 minAlgorithmic & AI Trading
Muqiao Huang, Ruodu Wang and Yiyun Wang posted Equilibrium in closed constant-function market maker economies on 24 August. The setting is deliberately small: two assets, two traders, one CFMM, and no outside market to arbitrage against. Closed, in other words, because the only prices are the ones the participants make.
The condition for rest
The central result is a clean characterisation. An interior state is a unilateral no-trade equilibrium exactly when the CFMM's marginal price equals both traders' marginal rates of substitution. Not approximately, and not for one of them: the pool price has to sit at the point where neither trader would improve by moving.
From there, individually rational equilibria are Pareto optimal, a representative agent can be constructed by weighted sup-convolution, and every feasible state turns out to be reachable through more than one valid sequence of trades.
The part with a desk application
The authors also give conditions determining whether trading first helps or hurts. That is the question a market maker actually faces, and it usually gets answered by folklore: go first, get the better price. The paper says it depends, and specifies on what.
The caveat is the model. Two traders and no external market is a long way from a live pool with arbitrage bots watching every block. What a closed model buys is a statement that is exactly true about a simple world, which is a firmer foundation than a statement that is approximately true about a complicated one.
Retold from arXiv. This is a summary in our own words; follow the link for the original reporting.