Animals often communicate using signals which seem to be completely arbitrary. These postures and ritualised acts give the impression that they have no other effect than to simply appear as they do to the receiver. Such signals, whose meanings are associated to their form through arbitrary convention, are called conventional signals. Theoreticians have directed much less attention to the topic of conventional signalling than to alternative signal types, such as handicapping signals. This lack of attention has lead to a poor understanding of threat displays and other communication contexts in which signals do not appear costly. We present what we believe to be the simplest possible model of conventional signalling between individuals with conflicting interests. This model requires a more complicated, and realistic, time structure than the action-response games widely used to model handicapped signalling. We demonstrate that this need for extended time structure is due to the exchange of information that conventional signalling requires. Signallers must be in a state of ignorance when choosing a signal, they must later receive information before choosing a subsequent action. The order in which these events happen is critical to conventional signalling. These results demonstrate the necessity of investigating communication with more complicated games than action-response games.