Games with a uniqueness property

Abstract

For two-player games of perfect information such as Chess, we introduce “uniqueness” properties to describe game positions in which a winning strategy (should one exist) is forced to be unique. Depending on how uniqueness is forced, and whether it applies to both players, the uniqueness property is classified as (bi-) weak, (bi-) strong, or global. We prove that any reasonable two-player game G is extendable to a game G∗ with the bi-strong uniqueness property, so that e.g., QBF remains PSPACE-complete under this restriction. For global uniqueness, we introduce a simple game GUPQBF over Boolean formulas with this property, and prove that any reasonable two-player game with global uniqueness is reducible to this game. On the other hand, we also show that GUPQBF resides in “small” counting classes believed properly contained in PSPACE. Our results give a new characterization to some complexity classes such as PSPACE and EXPTIME.