The Denumerable Setting: Full Visibility

Abstract

With two colors, it is easy to produce a predictor for a denumerably infinite set of agents ensuring infinitely many will guess correctly. For example, even if agents only see higher-numbered agents, one can have an agent guess red if he sees infinitely many red hats, and guess green otherwise. If there are infinitely many red hats, everyone will guess red and the agents with red hats will be correct; if there are finitely many red hats, everyone will guess green, and the cofinitely many agents with green hats will be correct. But something much more striking is true: there is a predictor ensuring that all but finitely many—not just infinitely many—are correct, and this is what Gabay and O’Connor obtained using the axiom of choice.

This chapter presents that result, and uses it to derive a theorem of Hendrik Lenstra that exhibits a predictor ensuring that the guesses are either all correct or all incorrect. We also consider the kind of sequential guessing that arose in Chapter 2, and we prove an equivalence between this and Lenstra’s context. We also examine the necessity of the axiom of choice in these results, both via the property of Baire and via square bracket partition relations.