Equilibria and Learning in Modular Marketplaces

TL;DR

This work analyzes a modular API marketplace where module owners set prices per API call and a centralized platform selects a feasible subset under a budget to maximize buyer value. It introduces a bang-per-buck greedy selection rule and proves the existence of $oldsymbol{\epsilon}$-equilibria, along with constant-factor approximation guarantees to the optimal value $ ext{OPT}$ under budget and matroid constraints; it also demonstrates that decentralized no-regret learning by module owners converges to these equilibria. The results unify procurement auction literature with decentralized pricing dynamics, showing that simple greedy mechanisms coupled with no-regret updates can yield approximately efficient outcomes in a first-price procurement-like setting. The findings have practical implications for scalable design of modular AI ecosystems (e.g., LLM plugins, specialized models) by providing robust, learnable pricing and allocation rules that tolerate strategic behavior and information asymmetry. Overall, the paper advances understanding of equilibria, efficiency, and learnability in decentralized modular marketplaces with budgets and combinatorial feasibility constraints.

Abstract

We envision a marketplace where diverse entities offer specialized "modules" through APIs, allowing users to compose the outputs of these modules for complex tasks within a given budget. This paper studies the market design problem in such an ecosystem, where module owners strategically set prices for their APIs (to maximize their profit) and a central platform orchestrates the aggregation of module outputs at query-time. One can also think about this as a first-price procurement auction with budgets. The first observation is that if the platform's algorithm is to find the optimal set of modules then this could result in a poor outcome, in the sense that there are price equilibria which provide arbitrarily low value for the user. We show that under a suitable version of the "bang-per-buck" algorithm for the knapsack problem, an $\varepsilon$-approximate equilibrium always exists, for any arbitrary $\varepsilon > 0$. Further, our first main result shows that with this algorithm any such equilibrium provides a constant approximation to the optimal value that the buyer could get under various constraints including (i) a budget constraint and (ii) a budget and a matroid constraint. Finally, we demonstrate that these efficient equilibria can be learned through decentralized price adjustments by module owners using no-regret learning algorithms.

Figures (2)

• Figure 1: Sample run for equilibrium construction. See main text for description.
• Figure 2: This figure gives a pictorial sketch of how we go from an equilibrium where some rejected modules may be bidding above cost to a potentially worse equilibrium where strictly fewer rejected modules are bidding above cost.

Theorems & Definitions (71)

• Definition 2.1
• Definition 2.2: Multiplicative Weight Type Learning Algorithm
• Theorem 4.1
• proof
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Theorem 4.5
• Claim 5.1
• Theorem 5.2