  Search has played a fundamental role in computer game research since the very
beginning. And while online search has been commonly used in perfect
information games such as Chess and Go, online search methods for imperfect
information games have only been introduced relatively recently. This paper
addresses the question of what is a sound online algorithm in an imperfect
information setting of two-player zero-sum games. We argue that
the~fixed-strategy~definitions of exploitability and $\epsilon$-Nash equilibria
are ill-suited to measure an online algorithm's worst-case performance. We thus
formalize $\epsilon$-soundness, a concept that connects the worst-case
performance of an online algorithm to the performance of an $\epsilon$-Nash
equilibrium. As $\epsilon$-soundness can be difficult to compute in general, we
introduce a consistency framework -- a hierarchy that connects an online
algorithm's behavior to a Nash equilibrium. These multiple levels of
consistency describe in what sense an online algorithm plays "just like a fixed
Nash equilibrium". These notions further illustrate the difference between
perfect and imperfect information settings, as the same consistency guarantees
have different worst-case online performance in perfect and imperfect
information games. The definitions of soundness and the consistency hierarchy
finally provide appropriate tools to analyze online algorithms in repeated
imperfect information games. We thus inspect some of the previous online
algorithms in a new light, bringing new insights into their worst-case
performance guarantees.

Computer Science and Game Theory