Dynamic Operational Planning in Warfare: A Stochastic Game Approach to Military Campaigns

TL;DR

Dynamic Operational Planning in Warfare presents a two-player discounted zero-sum stochastic game for campaign-level decision making under uncertainty, formalizing states as $\mathcal{S}=\{1,2\}^{\mathcal{O}}$ and seeking Markov perfect equilibria. It establishes an isotonicity property of the optimal value function $V^*$ on a partially ordered state space and leverages this to reduce both state and action spaces, enabling efficient Shapley-style value iteration with an accelerated variant that exploits pure equilibria. A detailed case study demonstrates rich equilibrium behavior and shows substantial computational gains, providing actionable operational insights for campaign analysts. The framework offers a scalable foundation for dynamic operational planning with clear avenues for future work on multiple adversaries and learning-based approximations.

Abstract

We study a two-player discounted zero-sum stochastic game model for dynamic operational planning in military campaigns. At each stage, the players manage multiple commanders who order military actions on objectives that have an open line of control. When a battle over the control of an objective occurs, its stochastic outcome depends on the actions and the enabling support provided by the control of other objectives. Each player aims to maximize the cumulative number of objectives they control, weighted by their criticality. To solve this large-scale stochastic game, we derive properties of its Markov perfect equilibria by leveraging the logistics and military operational command and control structure. We show the consequential isotonicity of the optimal value function with respect to the partially ordered state space, which in turn leads to a significant reduction of the state and action spaces. We also accelerate Shapley's value iteration algorithm by eliminating dominated actions and investigating pure equilibria of the matrix game solved at each iteration. We demonstrate the computational value of our equilibrium results on a case study that reflects representative operational-level military campaigns with geopolitical implications. Our analysis reveals a complex interplay between the game's parameters and dynamics in equilibrium, resulting in new military insights for campaign analysts.

Figures (9)

• Figure 1: Department of Defense Modeling and Simulation Hierarchy wade20.
• Figure 2: Campaign example with 6 objectives and 2 bases for each player. Player 1 controls objectives 3, 5, and 6. Player 2 controls objectives 1, 2, and 4. Player 1 orders Commander 1 to attack objective 1 and Commander 2 to reinforce objective 6.
• Figure 3: Probabilities of battle outcomes when Player $-i$ attacks objective $o$.
• Figure 4: Counterexample to the isotonicity of the optimal value function without Assumption \ref{['ass:rfc']}.
• Figure 5: Axis classification: In c1, the front (denoted F) for both players is at objective 4. In c2 the front for both players is at objective 5. In pf the front for both players is at objectives 10 and 11. Finally, for sf the front is at objective 14 for Player 1, and at objective 15 for Player 2.

Theorems & Definitions (3)

• Theorem 1
• Proposition 1
• Proposition 2