The Computational Complexity of the Ungar Games on Distributive Lattices

The Ungar game, introduced by Defant et al., is a combinatorial game played on a finite lattice, which originates from a geometric transformation introduced by Ungar to resolve Scott’s slope problem. When played on a distributive lattice $L$, the Ungar game reduces to repeatedly choosing and removing a non-empty set of maximal elements from a finite poset $P$, where $L$ is represented as the lattice $J(P)$ of the order ideals of $P$ by Birkhoff’s representation theorem. Defant et al. raised the open question of whether determining the outcome of the Ungar game on $J(P)$ is PSPACE-complete with respect to $\mathrm{\mid }P\mathrm{\mid }$. In this paper, we settle this open question in the affirmative. More precisely, we prove that the problem is PSPACE-complete even when restricted to posets $P$ with maximum chain length $l(P)\le 3$ such that the underlying graph of the Hasse diagram of $P$ is bipartite, where the length of a chain is defined as the number of elements in the chain minus one. Furthermore, we show that the problem is NP-complete for $l(P)\le 2$, whereas the case $l(P)\le 1$ is known to belong to LOGSPACE.