Analysis of the Order Flow Auction under Proposer-Builder Separation

In this paper, we consider the impact of the order flow auction (OFA) in the context of the proposer-builder separation (PBS) mechanism through a game-theoretic perspective. The OFA is designed to improve user welfare by redistributing maximal extractable value (MEV) to the users, in which two auctions take place: the order flow auction and the block-building auction. We formulate the OFA as a multiplayer game, and focus our analyses on the case of two competing players (builders). We prove the existence and uniqueness of a Nash equilibrium for the two-player game, and derive a closed-form solution by solving a quartic equation. Our result shows that the builder with a competitive advantage pays a relatively lower cost, leading to centralization in the builder space. In contrast, the proposer's shares evolve as a martingale process, which implies decentralization in the proposer (or, validator) space. Our analyses rely on various tools from stochastic processes, convex optimization, and polynomial equations. We also conduct numerical studies to corroborate our findings, and explore other features of the OFA under the PBS mechanism.