Are Bundles Good Deals for FOML?

Bundled products are often offered as good deals to customers. When we bundle quantifiers and modalities together (as in $\mathrm{\exists }x\mathrm{\square }$, $\Diamond \forall x$ etc.) in first-order modal logic (FOML), we get new logical operators whose combinations produce interesting fragments of FOML without any restriction on the arity of predicates, the number of variables, or the modal scope. It is well-known that finding decidable fragments of FOML is hard, so we may ask: do bundled fragments that exploit the distinct expressivity of FOML constitute good deals in balancing the expressivity and complexity? There are a few positive earlier results on some particular fragments. In this paper, we try to fully map the terrain of bundled fragments of FOML in (un)decidability, and in the cases without a definite answer yet, we show that they lack the finite model property. Moreover, whether the logics are interpreted over constant domains (across states/worlds) or increasing domains presents another layer of complexity. We also present the $\mathit{\text{loosely}}\mathit{\text{bundled}}\mathit{\text{fragment}}$, which generalizes the bundles and yet retain decidability (over increasing domain models).