I just met an interesting question but did not know how to approach it...
Suppose two gunmen (A and B) are moving in a straight line towards each other in a fixed speed. Each of them only have one bullet in the gun. The distance between them is $1$ meter. They have the same probability of hitting each other when shooting. The probability of a shooter knocking the other out, is $1-x$, where $x$ is the distance between them.
The thing is, A has a radar, letting him check whether B has fired. So you can imagine that if he knows B already shot and missed, he will keep walking until he has probability $1$ of shooting his opponent. We assume that once someone decides to shoot and is successful in hitting his opponent (shooting takes no time), the victim will immediately die without any ability to shoot back.
According to the final situation, their utility is defined as follows: if $A$ is alive while $B$ is dead, $A$ gets $U_{1}$, $B$ gets $0$ (similar if $A$ dies and $B$ survives); if both die, they both get $U_{2}$; if they all survive, they both get $U_{3}$. We have, $U_1>U_3>U_2$>0.
What is their best strategy respectively? Assume that they want to maximize their utility first, and if two strategies provides same utility, they prefer the one that gives the opponent less utility.
Thank you for your help! I know the problem is really long.