Noisy fighter-bomber duel

We discuss a duel-type game in which Player I uses his resource continuously and Player II distributes it by discrete portions. Each player knows how much resources he and his opponent have at every moment of time. The solution of the game is given in an explicit form. Keywords: noisy duel, payoff, strategy, the value of a game, consumption of resource.

💡 Research Summary

The paper introduces and solves a novel variant of the classic noisy duel, in which the two opponents consume their limited resources in fundamentally different ways. Player I (the “fighter”) expends his resource continuously over time, while Player II (the “bomber”) allocates his resource in discrete packets or “portions.” Both players have perfect, instantaneous knowledge of the remaining quantities of their own and their opponent’s resources at every moment. The authors formulate the interaction as a zero‑sum stochastic game with a payoff that depends on the probability of each player’s elimination, which in turn is a function of the instantaneous consumption rates.

The model is built on two monotone hazard‑rate functions: λ_I(u) maps the continuous consumption rate u chosen by Player I to an instantaneous death probability, and λ_II(k) maps the size k of a discrete packet chosen by Player II to a similar probability. By assuming that these functions are increasing (more resource spent yields a higher chance of eliminating the opponent), the authors capture the essential trade‑off between resource intensity and effectiveness. The state of the game is described by the pair (x, y), where x and y denote the remaining resources of Players I and II, respectively. The value function V(x, y) represents the maximal probability that Player II wins when both players act optimally from that state.

Because the control actions are of two different natures—continuous control for Player I and impulse (discrete) control for Player II—the optimality conditions combine a Hamilton‑Jacobi‑Bellman (HJB) partial differential equation with an impulse‑control jump condition. The HJB part reads

 max_{u≥0}{‑u V_x + λ_I(u)