We introduce the formalism of infinite-state reactive modules to reason about the strategic behaviour of autonomous agents in a setting where data are explicitly exhibited in the systems description and in the specification language. Technically, we endow reactive modules with an infinite domain of interpretation for individual variables, and introduce FO-ATL, a first-order version of alternating time temporal logic, for the specification of properties of interest. We show that their verification is decidable for classes of data types of interest. This result is proved by defining a first-order version of alternating bisimulations and finite bisimilar abstractions. We illustrate the formal machinery by applying it to English and sealed bid auctions. In particular, we show that strategic properties of agents in auctions, including manipulability and collusion, can be expressed and verified in this framework.