Michael Rabin in his very influential 1969 paper Decidability of Second-Order Theories and Automata on Infinite Trees studied, among other things, the fragment $L_I$ of monadic second-order logic that allows quantification over ideals of Boolean algebras, and he showed that countable Boolean algebras have a decidable theory in this language.
I am interested in comparing this language $L_I$ with other seemingly weaker languages such as $L_{\kappa\lambda}$ or a language with game quantifiers. For instance, what kind of sentence $\phi \in L_I$ has no/some equivalent sentence in the latter group of languages? Are there anything that can easily be seen about this problem?
(I looked for such literature by looking at the papers that cites the paper of Rabin's - of which there are many! - on MathSciNet, but with no avail.)