I wondered the other day whether there is an optimum strategy for the famous children's game "it" in a formalised, game-theoretic sense. The game is as follows:
The "play area" is an $m \times n$ units (squared) grid. The "players" are two circles with some initial starting centers $\Omega_0$ & $\Omega_1$ and radii $r_0$ and $r_1$. When a "frame" or unit of time passes, each player is allowed to move their circle's center in any direction by up to 0.1 units of distance. The boundaries for this are that the circles' centers cannot move into such situations where their circumference goes beyond the boundary of the play area.
Player 1 is "it" and so their goal is to have a non-zero area intersection between it and Player 2's circles, while Player 2 wants to avoid such a situation for the longest possible time. The question is, if moves in a frame happen simultaneously (in that, both players choose where they want to move without knowledge of where the other player is going to go and then the frame is carried out and they are moved), is there an optimum strategy for Player 1 in the form of a function which takes the current centers of both players' circles, their radii, the size of the play area and outputs an optimum direction an distance below or equal to 0.1 units to travel in?
Further, if such a thing exists, is it a winning strategy?