Game characterizations for the number of quantifiers

A game that characterizes equivalence of structures with respect to all first-order sentences containing a given number of quantifiers was introduced by Immerman in 1981. We define three other games and prove that they are all equivalent to the Immerman game, and hence also give a characterization for the number of quantifiers needed for separating structures. In the Immerman game, Duplicator has a canonical optimal strategy, and hence Duplicator can be completely removed from the game by replacing her moves with default moves given by this optimal strategy. On the other hand, in the last two of our games there is no such optimal strategy for Duplicator. Thus, the Immerman game can be regarded as a one-player game, but two of our games are genuine two-player games.