Towards Performatively Stable Equilibria in Decision-Dependent Games for Arbitrary Data Distribution Maps

TMLR Paper5764 Authors

Abstract: In decision-dependent games, multiple players optimize their decisions under a data distribution that shifts with their joint actions, creating complex dynamics in applications like market pricing. A practical consequence of these dynamics is the \textit{performatively stable equilibrium}, where each player's strategy is a best response under the induced distribution. Prior work relies on $\beta$-smoothness, assuming Lipschitz continuity of loss function gradients with respect to the data distribution, which is impractical as the data distribution maps, i.e., the relationship between joint decision and the resulting distribution shifts, are typically unknown, rendering $\beta$ unobtainable. To overcome this limitation, we propose a gradient-based sensitivity measure that directly quantifies the impact of decision-induced distribution shifts. Leveraging this measure, we derive convergence guarantees for performatively stable equilibria under a practically feasible assumption of strong monotonicity. Accordingly, we develop a sensitivity-informed repeated retraining algorithm that adjusts players' loss functions based on the sensitivity measure, guaranteeing convergence to performatively stable equilibria for arbitrary data distribution maps. Experiments on prediction error minimization game, Cournot competition, and revenue maximization game show that our approach outperforms state-of-the-art baselines, achieving lower losses and faster convergence.

Submission Length: Regular submission (no more than 12 pages of main content)

Changes Since Last Submission:

We have:

- enhanced related work discussion by adding proper citations in Section 1 (third paragraph), Section 3.2 (second paragraph), and clarifying the connections and distinctions in Section 3.1 (first and second paragraphs, Corollary 1), Section 3.2 (last paragraph), and Section 5.2 (first paragraph).

- improved clarity of notations and problem definition in Section 1.1 (second paragraph), Section 2 (second paragraph).

- strengthened theoretical results by providing additional lemmas and theorems for finite sample analysis in Appendices E, F, and G.2, updating the probabilistic guarantee, motivation, and presentation in Section 3.3 (Theorem 3), Section 4 (last paragraph and Theorem 4), and Appendices C and G.

- improved methodological detail in Section 3.3 (last paragraph), Section 4 (second and third paragraphs, Algorithm 1).

- enhanced experimental presentation in Section 5.1.1 (first paragraph), Section 5.2.1 (first paragraph), Section 5.3.1 (first paragraph), and Appendix J.2.

Assigned Action Editor: ~Marc_Lanctot1

Submission Number: 5764