This observation provides part of the reason why it is currently expected that the answer to Open Question 3a is positive. For as we can now see, the assertion \(\textbf{PH} = \Sigma^P_k\) for some \(k\) is equivalent to the assertion that the problem of determining whether Verifier has a winning strategy for \(n\)-round verification games is no harder than that of deciding this question for \(k\)-round games for all \(n \geq k\). A related observation bears on the status of Open Question 3b. For note that if \(\textbf{PH} = \textbf{PSPACE}\), then \(\sc{TWO}\ \sc{PLAYER}\ \sc{SAT}\) would be complete for \(\textbf{PH}\) (as it is for \(\textbf{PSPACE}\)). Since \(\textbf{PH}\) is defined as \(\bigcup_{k \in \mathbb{N}} \Sigma^P_k\), it would then follow that \(\sc{TWO}\ \sc{PLAYER}\ \sc{SAT} \in \Sigma^P_k\) for some \(k\). But in this case, we would again have that the problem of determining whether Verifier has a winning strategy for \(n\)-round verification games would be no harder than that of deciding this question for \(k\)-round games for all \(n \geq k\). As this again runs contrary to expectation, it is also widely believed not only that \(\textbf{PH} \subsetneq \textbf{PSPACE}\) but also that the former class differs from the latter in failing to have complete problems.