Flipping the Winner of a Poset Game

TL;DR

This paper introduces an efficient method to invert the winning strategy in poset games, establishing their computational complexity and linking them to Boolean formula problems.

Contribution

It provides a novel inductive procedure to flip the winner in poset games and demonstrates an efficient construction to relate poset games to Boolean formulas.

Findings

Efficient method to change the winning player in poset games.

Reduction of Boolean formulas to poset games to establish complexity bounds.

Solves the problem of constructing games with inverted winning strategies.

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 not 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.