One of the major outstanding foundational problems about Boolean functions is the sensitivity conjecture, which asserts that the degree of a Boolean function is bounded above by some fixed power of its sensitivity. We propose an attack on the sensitivity conjecture in terms of a novel two-player communication game. A lower bound of the form $n^{\Omega(1)}$ on the cost of this game would imply the sensitivity conjecture.

To investigate the problem of bounding the cost of the game, three natural (stronger) variants of the question are considered. For two of these variants, protocols are presented that show that the hoped-for lower bound does not hold. These protocols satisfy a certain monotonicity property, and we show that the cost of any monotone protocol satisfies a strong lower bound in the original variant.

There is an easy upper bound of $\sqrt{n}$ on the cost of the game. We also improve slightly on this upper bound. This game and its connection to the sensitivity conjecture was independently discovered by Andy Drucker (arXiv:1706.07890).

A preliminary version of this paper appeared in the Proceedings of the 6th Innovations in Theoretical Computer Science conference, 2015.