The Ungar Games on Graded Posets

For a poset $P$, an Ungar move sends $P$ to $P\setminus T$, where $T$ is some subset of maximal elements of $P$. With these Ungar moves, Defant, Kravitz, and Williams define the Ungar games, where two players alternate making nontrivial Ungar moves until one player cannot make a move and loses. We characterize the second-player wins on graded posets. We first prove recursive characterizations of second-player wins before using these results to give classifications of the second-player wins in terms of boolean circuits. We also generalize Defant, Kravitz, and Williams' work on Young's Lattice $J(\mathbb{N}^2)$ to the higher-dimensional $J(\mathbb{N}^d)$.