Noisy fighter-bomber duel

The classical duel is a zero-sum game of two players of the following type. The players have certain resources and use them during a given time interval with the goal of achieving success. Use of the resource γ at the moment t leads to success with the probability depending on the amount of resource γ and the time t only (it is usually assumed that the probability of success increases with time). As soon as one player reaches the goal he receives his profit, which is equal to his opponent’s loss, and the game ends. Various assumptions about the ways the players use their resources and about the players receiving information about the opponent’s behavior during the game define various kinds of duels [1, 2]. Models were considered where the players’ resources were discrete (discrete firing duels), infinitely divisible (continuous firing duels), continuous for one of the players and discrete for the other one (mixed duels, or fighter-bomber duels) [3, 4]. Researchers studied noisy duels [3, 5], where every player at a given moment of time had complete information about his opponent’s behavior up to that moment, and silent duels, where no such information was available. At the present time, duels are considered as classical models of competition [1, 6]. The author is grateful to Leonid Positselski for his help in editing this paper and translating it into English.