“Games” are about rules, not rewards. Rewards are, maybe, incentives we use to get people playing, but its clear they aren’t necessary. Rule-creation and the simple act of rule-following (the act of simple-rule-following) are all that’s required and such systems occur everywhere. See Calvinball and Nomic, a favorite of Hofstadter, or Wittgenstein’s long list of language game manifestations in his Philosophical Investigations:
Giving orders, and obeying them—
Describing the appearance of an object, or giving its measurements
Constructing an object from a description (a drawing)—
Reporting an event—
Speculating about an event—
Forming and testing a hypothesis—
Presenting the results of an experiment in tables and diagrams—
Making up a story; and reading it—
Making a joke; telling it—
Solving a problem in practical arithmetic—
Translating from one language into another—
Asking, thanking, cursing, greeting, praying.
—It is interesting to compare the multiplicity of the tools in language and of the ways they are used, the multiplicity of kinds of word and sentence, with what logicians have said about the structure of language.