The "core" conjecture for approval-based elections

I will discuss a combinatorial conjecture which states that a "core" (or "stable") solution exists for any instance of an election where each voter presents a set of approved candidates (without ranking). Some of the concepts surrounding this conjecture go back to the work of Thiele on elections in Denmark in 1895. The conjecture is still open but approximate variants of it have been proved. I will describe our latest result in this direction, a 4-approximately stable solution for ABC elections (with Drew Gao and Kimi Sun), and some technical tools which might be of independent interest: market equilibria and strongly Rayleigh distributions.