If PH = Σ^P_k for some fixed k, then determining whether Verifier has a winning strategy for n-round verification games would be no harder than for k-round games for all n ≥ k

Consider, for instance the following variation on the standard rules of Go: (i) the game is played on an \(n \times n\) board; (ii) the winner of the game is the player with the most stones at the end of \(n^2\) rounds. e. the player who moves first)? [30] What these games have in common is that the definition of a winning strategy for the player who moves first involves the alternation of existential and universal quantifiers in a manner which mimics the definition of the classes \(\Sigma^P_n\)