Differential Games

Origin. Rufus Isaacs (RAND) developed differential game theory (1965); Soviet contributions by Pontryagin, Krasovskii, and Subbotin extended the theory, particularly for pursuit-evasion games.

Mechanism. A differential game is a dynamic system controlled by multiple players with conflicting objectives. Each player's optimal strategy depends on the other's strategy and on the state trajectory. The solution concept is typically a saddle point: neither player can improve by unilaterally changing strategy. Soviet work characterized the "value" of the game — the outcome under optimal play — and the optimal feedback strategies.

Procedure. Model the dynamics (how the state evolves given both players' controls), the objectives (what each player maximizes or minimizes), and the constraints (admissible controls). Formulate the Hamilton-Jacobi-Isaacs equation for the value function. Solve (analytically or numerically) to find the value at each state. Extract the optimal strategies as functions of state. The strategies are feedback: the optimal action depends on the current state, not on the full history.

Applies to. Military engagement, pursuit-evasion (missile guidance, intercept), competitive economics, security games, any dynamic conflict where both parties optimize simultaneously.

Limitations. The Hamilton-Jacobi-Isaacs equation is a partial differential equation that is difficult to solve except in special cases. Numerical methods scale poorly with dimension. The saddle point may not exist in pure strategies, requiring mixed strategies. Also: the theory assumes both players are fully rational and have common knowledge of the game structure.