The canonical function game

The canonical function game is a game of length ω ₁ introduced by W. Hugh Woodin which falls inside a class of games known as Neeman games. Using large cardinals, we show that it is possible to force that the game is not determined. We also discuss the relationship between this result and Σ² ₂ absoluteness, cardinality spectra and Π₂ maximality for H(ω ₂) relative to the Continuum Hypothesis.