In cross-silo federated learning (FL), companies collaboratively train a shared global model without sharing heterogeneous data. Prior related work focused on algorithm development to tackle data heterogeneity. However, the dual problem of coopetition, i.e., FL collaboration and market competition, remains under-explored. This paper studies the FL coopetition using a dynamic two-period game model. In period 1, an incumbent company trains a local model and provides model-based services at a chosen price to users. In period 2, an entrant company enters, and both companies decide whether to engage in FL collaboration and then compete in selling model-based services at different prices to users. Analyzing the two-period game is challenging due to data heterogeneity, and that the incumbent's period one pricing has a temporal impact on coopetition in period 2, resulting in a non-concave problem. To address this issue, we decompose the problem into several concave sub-problems and develop an algorithm that achieves a global optimum. Numerical results on three public datasets show two interesting insights. First, FL training brings model performance gain as well as competition loss, and collaboration occurs only when the performance gain outweighs the loss. Second, data heterogeneity can incentivize the incumbent to limit market penetration in period 1 and promote price competition in period 2.

翻译：在跨机构联邦学习（FL）中，各公司协作训练共享的全局模型而无需共享异构数据。先前相关工作主要关注应对数据异构性的算法开发。然而，竞合的双重问题——即FL协作与市场竞争——仍未得到充分探索。本文采用动态两阶段博弈模型研究FL中的竞合关系。在第一阶段，一家在位公司训练本地模型，并以选定价格向用户提供基于模型的服务。在第二阶段，一家新进入公司加入市场，两家公司决定是否参与FL协作，随后以不同价格向用户销售基于模型的服务并展开竞争。由于数据异构性以及第一阶段在位公司的定价对第二阶段竞合关系产生跨期影响，导致问题非凹，分析该两阶段博弈具有挑战性。为解决此问题，我们将原问题分解为若干凹子问题，并开发了一种能达到全局最优的算法。在三个公开数据集上的数值实验结果揭示了两个重要发现：首先，FL训练在带来模型性能提升的同时也会引发竞争损失，仅当性能增益超过损失时协作才会发生；其次，数据异构性可能激励在位公司在第一阶段限制市场渗透，并在第二阶段加剧价格竞争。