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.