There's something a bit unsatisfying about the concept of language games because clearly there are aspects of those games which are not linguistic in a formal sense. It strikes me that it is the 'game' that counts, and a term which sees language games in the same light as other games (for example, ping-pong) is useful.
All utterances in a language game are forms of agency, just as hitting a ping-pong ball is. All are constrained by the rules of the game. So I wonder if 'Agency games' is a better term for what we are talking about.
One of the interesting things that this exposes is the different levels of rules which affect different aspects of rule-based agency. For example, the physical space of a classroom can preclude certain types of agency (embodying rules in the environment), and this can sometimes be in conflict with what students are asked to do. (Don't sit students in rows and then say 'work in groups of 5'). Most interesting is the physical disposition of students and the physical demands of preferred learning styles for students (although I don't like the 'learning styles' discourse!). Doing maths whilst hunched over a desk may be impossible for the student for whom doing maths is only possible walking around and thinking.
By examining 'agency games' in this way, we can start to unpick some of the impossible games that some learners are asked to play (and often don't have a choice in playing!)