Archive for September 16th, 2013

Determinacy of Borel games V

September 16, 2013

This post is a kind of postscript to the previous four. It consists of miscellaneous observations that shed some light on the proof of Borel determinacy. I’m writing it mainly for my own benefit, but there seems to be no harm in posting it, just in case anyone else finds it interesting.

An attempt to lift that sort of works and sort of doesn’t

One way to get some further insight into the proof of Borel determinacy is to look at some auxiliary games that don’t work as lifts. A particularly natural candidate is this. Let T be an infinite (pruned) tree, let A\subset[T] and let G be the game G(T,A). Now define a game G' as follows. Player I plays a move of the form (x_1,\sigma), where \sigma is a strategy for G with first move x_1. Player II plays a move of the form (x_2,\tau), where \tau is a strategy for G and x_2=\tau(x_1). Thereafter, the two players must play the strategies \sigma and \tau.

Clearly the outcome of this game is decided after the first two moves: if \sigma beats \tau then Player I wins and otherwise Player II wins.