Some Properties of Pomax Games
Rebecca Staffas, Jonas Sj · KTH Publication Database DiVA (KTH Royal Institute of Technology) · 2014
We study pomax games, a family of combinatorial games which are always integer-valued. Specifically, we study games played on Young diagrams and Boolean lattices. We find a linear algorithm for computing the values of pomax games played on Young diagrams with only two rows. Some of the statements involved in this proof hold also for games on general Young diagrams. For pomax games on Boolean lattices, we introduce the concept of upper and lower games and use that as a tool to study the distribution of possible game values. We prove that games that equal any sufficiently small even integer can always be found, and that the density of distinct game values converges when the game size tends to infinity. Based on computational evidence, we conjecture that if the upper and lower games of some game are identical, then the value of that game is the sum of the upper and lower game.