On the constructability of solution to a pair of two person search games

Let B denote the collection of all subsets of [0, 1] having exactlyk points, and let R denote the collection of all (Lebesgue measureable) subsets of [0, 1] with total length exactly a < 1. Consider the following two games in which the opponents BLUE and RED chooseB from B andR from R respectively: In the Set Hider Game (SHG) the payoff to RED is 1 ifB ∩R is empty and 0 otherwise. In the Point Hider Game (PHG) the payoff to BLUE is 1 ifB ∩R is empty and 0 otherwise. In this paper we solve the SHG and show that the PHG has no reasonable solution fork>1.