Noisy Duels
Martin Fox, George Kimeldorf · SIAM Journal on Applied Mathematics · 1969
Let $G_{mn} (P_1 ,P_2 )$ be the noisy duel in which the first player has m bullets with accuracy function $P_1 $ and the second player has n bullets with accuracy function $P_2 $, where m, n, $P_1 $ and $P_2 $ are known to both players. Results are well known for the duels in which $P_1 = P_2 $ or when $m = n = 1$. The following theorem is proved: If $P_1 $ and $P_2 $ are nondecreasing and continuous on $[0,1]$ with $P_1 (0) = P_2 (0) = 0$ and $P_1 (1) = P_2 (1) = 1$, then the game $G_{mn} (P_1 ,P_2 )$ has a value. We discuss the structure of $\varepsilon $-good strategies and introduce the concept of a good first-shot time. It is shown that although good strategies may not exist, at least one of the players always has a good first-shot time.