The Ungar Games
Colin Defant, Noah Kravitz, Nathan Williams

TL;DR
This paper introduces the Ungar game played on finite lattices, analyzes winning strategies for specific lattice classes, and provides enumerative results and complexity discussions for these combinatorial structures.
Contribution
It defines the Ungar game on finite lattices, characterizes winning positions in various lattice classes, and derives enumerative formulas for Eeta wins in these structures.
Findings
Number of Eeta wins in weak order on $S_n$ is $O(0.95586^nn!)$
Characterization of Eeta wins in Young's lattice intervals including rectangles and root posets
Enumeration of Eeta wins in Tamari lattices
Abstract
Let be a finite lattice. An Ungar move sends an element to the meet of , where is a subset of the set of elements covered by . We introduce the following Ungar game. Starting at the top element of , two players -- Atniss and Eeta -- take turns making nontrivial Ungar moves; the first player who cannot do so loses the game. Atniss plays first. We say is an Atniss win (respectively, Eeta win) if Atniss (respectively, Eeta) has a winning strategy in the Ungar game on . We first prove that the number of principal order ideals in the weak order on that are Eeta wins is . We then consider a broad class of intervals in Young's lattice that includes all principal order ideals, and we characterize the Eeta wins in this class; we deduce precise enumerative results concerning order ideals in rectangles and type- root posets. We…
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Taxonomy
TopicsArtificial Intelligence in Games
