Captain Galaxy and Commander Glarcon are locked in mortal combat. Each mans a battle tank armed with N photonic missiles which move at the speed of light. They move toward each other at constant velocity=v on a 1-dimensional track, unable to stop or reverse direction. Assume v << c. The probability of scoring a kill with a missile is described by a function f(d) which monotonically increases from 0 to 1 as the distance between the tanks decreases from infinity to 0. If the distance closes to 0 and no missiles are fired, both tanks are destroyed in the collision. Assume each combatant attempts to maximize their own probability of survival.

Note that this is not strictly a zero-sum game, since it is possible for neither player to survive. But it is impossible for both to survive.

The state of the game is thus described by three variables:

d=distance between the players

N1= number of own missiles remaining

N2= number of opponent’s missiles remaining

A strategy S(d,N1,N2) would describe a combatant’ actions (shoot or don’t shoot) for all possible states.

If each player has exactly one missile what is the optimal strategy? Clearly, if the first player shoots and misses, the 2nd will win by waiting for d to approach 0 and then make a last minute shot. What if each player has exactly two missiles? What if each player has N missiles?

It may simplify the problem to assume f(d) is proportionate to 1/d or 1/d^2 and then solve the general case.