Performative prediction models account for feedback loops in decision-making processes where predictions influence future data distributions. While existing work largely assumes insensitivity of data distributions to small strategy changes, this assumption usually fails in real-world competitive (i.e. multi-agent) settings. For example, in Bertrand-type competitions, a small reduction in one firm's price can lead that firm to capture the entire demand, while all others sharply lose all of their customers. We study a representative setting of multi-agent performative prediction in which insensitivity assumptions do not hold, and investigate the convergence of natural dynamics. To do so, we focus on a specific game that we call the ''Bank Game'', where two lenders compete over interest rates and credit score thresholds. Consumers act similarly as to in a Bertrand Competition, with each consumer selecting the firm with the lowest interest rate that they are eligible for based on the firms' credit thresholds. Our analysis characterizes the equilibria of this game and demonstrates that when both firms use a common and natural no-regret learning dynamic -- exponential weights -- with proper initialization, the dynamics always converge to stable outcomes despite the general-sum structure. Notably, our setting admits multiple stable equilibria, with convergence dependent on initial conditions. We also provide theoretical convergence results in the stochastic case when the utility matrix is not fully known, but each learner can observe sufficiently many samples of consumers at each time step to estimate it, showing robustness to slight mis-specifications. Finally, we provide experimental results that validate our theoretical findings.

翻译：表演性预测模型考虑了决策过程中的反馈循环，其中预测会影响未来的数据分布。现有研究大多假设数据分布对策略的微小变化具有不敏感性，然而这一假设在现实世界的竞争性（即多智能体）场景中通常不成立。例如，在伯特兰德型竞争中，一家公司价格的微小下调可能导致其占据全部市场需求，而其他所有公司则急剧失去全部客户。我们研究了一个不满足不敏感性假设的多智能体表演性预测代表性场景，并探究了自然动态的收敛性。为此，我们聚焦于一个称为“银行博弈”的特定博弈，其中两家贷款机构在利率和信用评分阈值上展开竞争。消费者的行为类似于伯特兰德竞争，每位消费者选择其符合资格（基于机构的信用阈值）且利率最低的机构。我们的分析刻画了该博弈的均衡，并证明当两家机构采用一种常见且自然的无遗憾学习动态——指数权重法——并进行适当初始化时，尽管博弈具有一般和结构，该动态始终收敛至稳定结果。值得注意的是，我们的设定允许多个稳定均衡存在，收敛性取决于初始条件。我们还提供了在效用矩阵不完全已知的随机情况下的理论收敛结果，此时每位学习者在每个时间步可以观测足够多的消费者样本来估计该矩阵，这表明了模型对轻微设定错误的鲁棒性。最后，我们通过实验结果验证了理论发现。