Games on the Sperner Triangle

TL;DR

This paper introduces a new two-player game based on Sperner's lemma, demonstrating its properties, computational complexity, and providing an online playable version, contributing to combinatorial game theory and computational complexity.

Contribution

It presents a novel Sperner-based game with proven properties, complexity results, and an accessible online implementation, expanding the application of fixed-point theorems in game design.

Findings

The game always has a winner.

Deciding the winning strategy is PSPACE-complete.

The game is balanced with no decisive advantage for either player.

Abstract

We create a new two-player game on the Sperner Triangle based on Sperner's lemma. Our game has simple rules and several desirable properties. First, the game is always certain to have a winner. Second, like many other interesting games such as Hex and Geography, we prove that deciding whether one can win our game is a PSPACE-complete problem. Third, there is an elegant balance in the game such that neither the first nor the second player always has a decisive advantage. We provide a web-based version of the game, playable at: http://cs-people.bu.edu/paithan/spernerGame/ . In addition we propose other games, also based on fixed-point theorems.