Infinite games with uncertain moves

TL;DR

This paper investigates how uncertainty about available moves in infinite two-player games affects the complexity of winning sets within the Borel hierarchy, revealing a pattern of complexity jumps at certain levels.

Contribution

It introduces a novel analysis of infinite games with uncertain moves, demonstrating how Borel hierarchy levels change under such conditions.

Findings

Sets at every alternate Borel level jump to the next higher level

Uncertainty about moves causes complexity increases in the hierarchy

Results have implications for game theory and descriptive set theory

Abstract

We study infinite two-player games where one of the players is unsure about the set of moves available to the other player. In particular, the set of moves of the other player is a strict superset of what she assumes it to be. We explore what happens to sets in various levels of the Borel hierarchy under such a situation. We show that the sets at every alternate level of the hierarchy jump to the next higher level.