If you like math notation, here's one way to think of a formal social system:

1. A set of moves $V = \{ m_1, m_2, ... \}$ is available to players. All together, the moves are the design's vocabulary $V$. On Facebook you can post, comment, like. At a tango milonga you can make eye contact across the room, look away, ask someone to dance. At a farmer's market you can buy things, apply for a stall, set up a stall, etc. Generally, all players knows what the vocabulary is.
2. At any time $t$, there's a state of play $s_t$, which gives some common or shared knowledge among all players or an important subset of them. At the milonga this includes who's danced with who; at the farmer's market, who's approved for a stall, who's bought what, etc. (A move $m_k$ by a player $p_j$ will advance the state of play somehow $\Delta(s_i, p_j, m_k) \to s_{i+1}$.)
3. There's a legitimation function $L(s, p_i)$, which says which moves $\{ m_1, m_2, ... \}$ are acceptable from player $p_i$ in a given state of play $s$. At the milonga, you can only accept a dance you've been offered; at the market, you can only set up a stall if you're approved.

Generally, people will only enter a social system if they can see (a) what the moves are, (b) which are legitimate when, and (c) how things are progressing with the state of play. Only then do they see how the system works, and what they could get by participating.

So, the clearer and simpler the system, the more people will play.

A person can operate in multiple systems at once. Say I bring a rose to the woman I'm courting, while she sells bread at the market. I buy a ciabatta while I'm there. The rose advances the state of play of one system; the purchase, another.

(Occasionally, a single act advances more than one system: when I read my poem aloud, I might win your heart and the poetry contest in the same breath.)

Often systems are nested. The legitimate actions in the queue for a dance club are distinct from those inside. You can call this one nested system, or two systems, one gating another.

Vocabulary

Speech and action within a system can be irrelevant or illegitimate. It can be context for a move, or it can be a move itself.

• Irrelevant speech and action is anything outside the vocabulary and which doesn't advance the state of play. If I scratch my beard at the market, nothing happens.
• Illegitimate speech or action is allowed vocabulary that's illegitimate, given my role and the state of play. This would include selling bread without approval, or taking bread without offering money. While sometimes ignored, more often illegitimate speech and action gets the actor removed from the system.

Anything else is a move in the vocabulary of the system, or context for such a move.

Many moves require context—voting requires proposals, for instance. You can share an idea you have, but it becomes a proposal for people to vote on when you say something like "that's my proposal". In this case, you give the context, and then you make your move based on it.

Oftentimes, moves have previous moves as context. Some moves are feedback to other moves, people can like, clap, cheer, review, thank, or comment on a previous move.

Sometimes, the whole of a system operates inside a particular context. All communication may be expected to be around a shared goal, a shared topic, a set of shared values, etc.

State of Play

The state of play is the hardest thing to define, because it can vary so much. In chess, the state of play includes the current positions on the gameboard, but also whatever the players know about one anothers' past strengths and weaknesses.

In general, the state of play includes all information players use to choose their moves, and anything that, when they make the move, changes as a result.

This includes information...

About the players themselves

• Who's there? Do participants have the sense of one another's presence, in the same "place" at the same time? For how long? async fleeting durational Are unknown future listeners / responders also "in the room"? eternal present time
• What are their roles? Social systems often define a vocabulary of roles, like original poster, buyer, seller, influencer, invited expert, etc.
• Status. Are some participants celebrities? Are they ranked somehow? Or is everyone presented as equal?

About how players connect, and who sees each move

• How Connections are Made. Do people friend / follow / join?
• The Shape of Connections. Does the whole group listen to everything said, or do people talk in pairs, or does information flow through a network structure? whole group pairs subgroups network
• Consistency. Is the group together for a duration, or do people come and go? consistent transient
• Relational Depth. Do people know each other well? strangers friends anonymous acquaintances
• Size. Does the speaker know exactly who will see their speech? Is it a large group or small?unclear large small
• Matchmaking. If people make their own connections, what information is used to decide who to connect with? shallow markings deep context human intros algorithms

Legitimacy

Finally, there are always constraints upon who can perform which moves, in which states of play. You can only vote once there's been a proposal. Only members can vote. The guru speaks first. We begin with the national anthem. Etc.

These rules connect the state of play (including roles, status, etc) with the vocabulary.

Relevant considerations include...

• Selection. Is the speaker recognized as invited by a trustworthy person? Are responders? invited random
• Dependencies. Do they need one another, to accomplish something together? independent dependent interdependent
• Pace and Polish. Is the rate of communication fast or slow? Are things slowly crafted? Contemplated? Or quickly tossed back and forth?