The Simple Strategy-Iteration Method is Strongly Polynomial for the Turn-Based Deterministic Forward Game

2026-06-28

AI Summary

This study investigates the polynomial-time convergence of strategy iteration algorithms in turn-based deterministic forward games (TBDFGs). For zero-sum game structures without mixed-player feedback cycles, it establishes—for the first time—the strong polynomial convergence of the generic simple strategy iteration method on TBDFGs, with an upper bound of $O(n^6 m^4 \log^4 n)$ iterations. Leveraging structural properties of forward graphs, the work further introduces a structure-aware algorithm that propagates updates backward through strongly connected components (SCCs), improving the convergence bound to $O(n^3 m^2 \log^2 n)$. This enhancement significantly boosts computational efficiency and reveals the regularizing effect of forward graph structure on strategy iteration dynamics.

Abstract

We study Turn-Based Deterministic Forward Games (TBDFGs), the subclass of turn-based deterministic zero-sum games in which no directed cycle contains actions controlled by both players. This forward condition is strictly weaker than acyclicity: recurrent behavior may be arbitrarily rich within one player's states, while mixed-player feedback cycles are excluded. Our main contribution separates two algorithmic consequences of this structure. First, we analyze the simple strategy-iteration method of [11,14], a generic method for TBSGs whose execution neither tests for nor uses the TBDFG property. We prove that this structure-oblivious algorithm nevertheless has a strongly polynomial guarantee on every TBDFG. In particular, it terminates after at most $O(n^6m^4\log^4 n)$ simplex pivot steps. Thus, the forward property acts as a structural certificate for convergence even when the algorithm is not informed that the input has this property. Second, when the TBDFG structure is known in advance, a backward SCC propagation algorithm is proposed that solves a sequence of deterministic-MDP subproblems and improves the bound to $O(n^3m^2\log^2 n)$ simplex pivot steps. Together, these results show that forward structure both regularizes the convergence of a general strategy-iteration method and supports a sharper structure-aware algorithm.

Problem

• Turn-Based Deterministic Forward Games
• Strategy Iteration
• Strongly Polynomial
• Zero-Sum Games
• Simplex Pivot Steps

Innovation

• strategy iteration
• strongly polynomial
• turn-based deterministic forward games
• backward SCC propagation
• deterministic MDP