This game should give the students an appropriate mental model of objects. Objects are active. Objects provide services. Objects request services. Objects do only what they are designed to do.
Computation occurs through interaction.
They can do this as a joint project, with the person playing each class contributing the interactions of objects of that class. You can discuss object oriented design prior to this exercise or not. You can use it as an introduction to design or not. If you use it as an introduction you can then show the students your design using UML or other notation. You can actually use such an exercise on the first day of class, prior to any discussion of objects.