We consider a bipartite network of buyers and sellers, where the sellers run locally independent Progressive Second-Price (PSP) auctions, and buyers may participate in multiple auctions, forming a multi-auction market with perfect substitute. The paper develops a projection-based influence framework for decentralized PSP auctions. We formalize primary and expanded influence sets using projections on the active bid index set and show how partial orders on bid prices govern allocation, market shifts, and the emergence of saturated one-hop shells. Our results highlight the robustness of PSP auctions in decentralized environments by introducing saturated components and a structured framework for phase transitions in multi-auction dynamics. This structure ensures deterministic coverage of the strategy space, enabling stable and truthful embedding in the larger game. We further model intra-round dynamics using an index to capture coordinated asynchronous seller updates coupled through buyers' joint constraints. Together, these constructions explain how local interactions propagate across auctions and gives premise for coherent equilibria--without requiring global information or centralized control.

翻译：我们研究一个由买方和卖方构成的二分网络，其中卖方运行局部独立的渐进次价拍卖，而买方可能参与多个拍卖，形成一个具有完美替代性的多拍卖市场。本文为去中心化的渐进次价拍卖开发了一种基于投影的影响力框架。我们通过在活跃投标指数集上的投影形式化定义了主要影响力集与扩展影响力集，并展示了投标价格上的偏序如何支配分配、市场转移以及饱和单跳壳的出现。我们的结果通过引入饱和分量以及一个用于多拍卖动态相变的结构化框架，突显了渐进次价拍卖在去中心化环境中的鲁棒性。该结构确保了策略空间的确定性覆盖，从而能够在更大的博弈中实现稳定且真实的嵌入。我们进一步使用一个指数来建模轮内动态，该指数捕捉通过买方联合约束耦合的协调异步卖方更新。这些构造共同解释了局部互动如何跨拍卖传播，并为形成连贯均衡提供了前提——无需全局信息或集中控制。