#### Document Type

Article

#### Publication Date

2011

#### Keywords

poset games, computational complexity, combinatorial game theory, game theory

#### Disciplines

Mathematics | Other Statistics and Probability | Statistical Models | Statistical Theory | Theory and Algorithms

#### Abstract

Partially-ordered set games, also called poset games, are a class of two-player combinatorial games. The playing field consists of a set of elements, some of which are greater than other elements. Two players take turns removing an element and all elements greater than it, and whoever takes the last element wins. Examples of poset games include Nim and Chomp. We investigate the complexity of computing which player of a poset game has a winning strategy. We give an inductive procedure that modifies poset games to change the nim-value which informally captures the winning strategies in the game. For a generic poset game *G*, we describe an efficient method for constructing a game ¬* G* such that the first player has a winning strategy if and only if the second player has a winning strategy on *G*. This solves the long-standing problem of whether this construction can be done efficiently. This construction also allows us to reduce the class of Boolean formulas to poset games, establishing a lower bound on the complexity of poset games.

#### Recommended Citation

Kalinich, A. O. (2012). Flipping the winner of a poset game. *Information Processing Letters*, *112*, 86–89. doi:10.1016/j.ipl.2011.09.016

#### Included in

Mathematics Commons, Other Statistics and Probability Commons, Statistical Models Commons, Statistical Theory Commons, Theory and Algorithms Commons