Picker–Chooser fixed graph games

Given a fixed graph H and a positive integer n, a Picker–Chooser H  -game is a biased game played on the edge set of KnKn in which Picker is trying to force many copies of H and Chooser is trying to prevent him from doing so. In this paper we conjecture that the value of the game is roughly the same as the expected number of copies of H   in the random graph G(n,p)G(n,p) and prove our conjecture for special classes of graphs H such as complete graphs and trees.