When Equivalent Representations Produce Different Responses: Prompt-Sensitive Stackelberg Games with LLM Followers

Classical Stackelberg models assume that a follower implements a stable best-response function. We test this assumption when the follower is a large language model (LLM). Using an exactly solvable quadratic Stackelberg game, we hold the underlying utilities fixed while presenting the follower’s objective through four representations constructed to encode the same objective: a direct equation, an expanded quadratic, plain language, and an applied robot-control framing. Across 2,640 primary model calls, these representations induce substantially different actions, follower regret, and response distributions. The maximum pairwise Wasserstein distance between prompt-conditioned responses reaches $0.339$. Although the baseline leader commitment changes little, additional parameter regimes exhibit substantially different empirical and prompt-robust commitments. Targeted bootstrap follow-ups confirm positive worst-prompt utility gains in two such regimes. These results suggest that prompt representation can act as a structured source of follower uncertainty in LLM-based strategic systems.