Simple Strategy-Iteration with Dantzig's Rule is not Strongly Polynomial on Turn-Based Deterministic Games

We study turn-based deterministic zero-sum games with discounted and long-run average costs. For discounted costs, we establish exponential and superpolynomial iteration lower bounds for the simple strategy-iteration method with Dantzig’s rule under unrestricted costs and unit weights, respectively. To the best of our knowledge, this shows for the first time that the method is not strongly polynomial even when all action costs belong to $\{-1,0,1\}$. For long-run average costs, we establish an exponential iteration lower bound for termination of a Dantzig counterpart with unrestricted costs and prove that it admits a strongly polynomial implementation with unit weights.