For two-player games of perfect information such as Checker, Chess, Go, etc.,
we introduce ``uniqueness'' properties.
For any game position,
we say roughly that
it has a uniqueness property (or, a unique solution property)
if a winning strategy (if it exists) is forced to be unique.
Depending on the way that winning strategy is forced,
a uniqueness property is classified as (bi-)weak, (bi-)strong, and global.
We prove that
any reasonable two-player game G is extended to a game $\Gext$
in such a way that
any original position of G has the bi-strong uniqueness property.
For the global uniqueness property,
we introduce a simple game over Boolean formulas
with the global uniqueness property,
and prove that
any reasonable two-player game with the global uniqueness property
can be converted to a subgame of this game.
Based on these results,
we give a new characterization to some standard complexity classes
such as PSPACE and EXP.