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.

翻译：我们设想一个市场，其中多样化的实体通过API提供专门的“模块”，允许用户在给定预算内组合这些模块的输出以完成复杂任务。本文研究此类生态系统中的市场设计问题：模块所有者战略性地为其API定价（以最大化利润），而中央平台在查询时协调模块输出的聚合。这也可视为带预算约束的一价采购拍卖。首先观察到，若平台算法旨在寻找最优模块集合，则可能导致不良结果——存在某些价格均衡会为用户提供任意低的价值。我们证明，在背包问题“性价比”算法的适当变体下，对于任意$\varepsilon > 0$，总存在$\varepsilon$-近似均衡。进一步，我们的首要结果表明：采用该算法时，任何此类均衡均能为买家在多重约束下（包括（i）预算约束及（ii）预算与拟阵约束）获得的最优价值提供常数近似。最后，我们论证模块所有者可通过无遗憾学习算法进行分散式价格调整，从而习得这些高效均衡。