We propose a dialogue game protocol for purchase negotiation dialogues which identifies appropriate speech acts, defines constraints on their utterances, and specifies the different sub-tasks agents need to perform in order to engage in dialogues according to this protocol. Our formalism combines a dialogue game similar to those in the philosophy of argumentation with a model of rational consumer purchase decision behaviour adopted from marketing theory. In addition to the dialogue game protocol, we present a portfolio of decision mechanisms for the participating agents engaged in the dialogue and use these to provide our formalism with an operational semantics. We show that these decision mechanisms are sufficient to generate automated purchase decision dialogues between autonomous software agents interacting according to our proposed dialogue game protocol.