What is the hat game? How does one win at the hat game? It is almost break, but come to find out in the course of a live demonstration, with audience participation! But once we've solved the hat game, what else would we mathematicians do but generalize it? Multiple colours is too easy; what about infinitely-many players? Do they need the axiom of choice to help them out, and if so, what heavy machinery — such as ultrafilters and Vitali transversals — is essential, and what can be done without? We shall see how an old model, the symmetric Solovay model, which is best-known for providing a mathematical universe where all sets of reals are Lebesgue-measurable, provides the context for answering these questions. But new techniques are needed. See how virtual objects which don't really exist in the "real world'' have profound effects on the real world as we catch a glimpse of geometric set theory, an exciting new body of techniques due to Larson and Zapletal which enable the study of the choice-complexity of a broad range of mathematical objects. Just one payoff is complexity bounds for a vast generalization of the hat game!