Strategy-Stealing is Non-Constructive

TL;DR

This paper investigates the computational complexity of finding winning moves in combinatorial games where strategy-stealing proves a first player's win, establishing PSPACE-hardness for certain game classes.

Contribution

It demonstrates that locating winning strategies in these games is computationally hard, even when the existence of a winning move is guaranteed by strategy-stealing.

Findings

Proves PSPACE-hardness for Minimum Poset Games

Proves PSPACE-hardness for Symmetric Maker-Maker Games

Shows complexity of constructive strategies in strategy-stealing arguments

Abstract

In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing. This work is about the complexity behind these proofs: how hard is it to actually find a winning move in a game, when you know by strategy-stealing that one exists? We prove that this problem is PSPACE-hard already for Minimum Poset Games and Symmetric Maker-Maker Games, which are simple classes of games that capture two of the main types of strategy-stealing arguments in the current literature.