Games of Strategy are Games of Reading and Writing
1.
Schelling defines games of strategy as any situation in which each player's best choice of action depends on the actions (he expects) the other player will take (and vice-versa, reflexively). This is in contrast to games of skill and games of chance. "Strategy," then, is the study of conflicting parties' behaviors as they are premised on "the interdependence of the adversaries' decisions and on their expectations about each others' behaviors."
We will call "script" a set of moves, from start to end, across a game. Games of strategy are definitionally marked by their lack of a player-independent & globally optimal ("PIG-optimal") script—that is, a script which will produce the best possible outcome in every game against every possible player.
(Or at the very least, these games of strategy have not had their PIG-optimal strategy discovered yet, so a successful move, or script, is functionally indexical—"indexical" as in, "player and context-dependent.")
When moves are indexical, it means that their efficacy depends on the opponent's future move(s). In "Rock, Paper, Scissors," no single move is PIG-optimal. The efficacy of playing "rock" fully depends on whether the opponent will play rock (resulting in stalemate), paper (defeat), or scissors (victory).
2.
Strategy games span a continuum between cooperative and adversarial.
In games tending toward the cooperative side, it behooves players to be legible, and thus predictable. This is because, if any individual player's best move depends on other players' next move(s), then Pareto optimal aggregate play among all players requires players perfectly anticipating each other (so they are each, individually, able to make the best move dependent on each other). In a cooperative game of Rock, Paper, Scissors where winning consisted of matching moves (e.g. two rocks, two papers, etc), it behooves players to get into a rhythm, to repeat themselves, etc.
This goes some way, I think, toward explaining social customs and conventions, the inculcation of habit. It is a pro-social gesture to be legible, insofar as it allows other players in a social space to optimize around you. This reciprocal self-legibilizing allows for spontaneous organization, where sophisticated behavioral patterns emerge in e.g. conversational turn-taking, team sports, drivers who efficiently and safely manage an intersection with a broken traffic light, etc. Meanwhile, those who are unpredictable or illegible are "creepy," "unsettling," etc...
(Convention—settling on a shared script—is also computationally tractable, whereas assessing other players' next move in order to make your next move, when their move depends on their assessment of your next move, which depends on... is an NP-hard recursive problem. More on this soon.)
In games tending toward the adversarial side, it behooves players to be illegible, and thus unpredictable. This is because, if any individual player's move efficacy depends on his opponent's next move(s), then that player is disadvantaged if his opponent can anticipate, and thus thwart, his choice. For a competent opponent to know that he is about to play "rock" is equivalent to losing. So long as he can be anticipated, he cannot win, because in a game like Rock Paper Scissors, moves are fully indexical in their "quality" or efficacy. There is no "better" or "worse" move in a vacuum; only better or worse moves given an opponent's move.
Another high-level strategy, for adversarial games, is what I'll call "pseudo-legibility." This approach involves setting up an opponent expectation, by intentionally leaking information, or displaying a pattern, during low-stakes rounds of the game. Then, when a high-stakes round approaches, the player "cashes out" by breaking his pattern. Here he has flipped the script: by making himself appear predictable, he has in actuality made his opponent predictable.
For instance, a poker player may feign a tell, and then display that tell several times during small pots. Later in the game, when a large pots (and a good hand) comes around, he can once again display the tell, and lead his opponent to believe he is bluffing. At this point, he can just "take his opponent's money": he can "play" him by ratcheting up the bet and clearing house.
3.
Note that there is in some sense a player-independent optimal strategy for Rock, Paper, Scissors, which is pure randomness. The problem is that pure randomness will secure a lower win percentage against sub-optimal players than a player-dependent strategy would have. Randomness is only "optimal" in that, if one has no idea who his opponent is, it is the optimal strategy. We will call this a player-blind optimal strategy, or PB-optimal. In a one-round game of Rock, Paper, Scissors against an anonymous opponent—or in a multi-round, blindfolded game, where neither player learns their opponent's moves (and by extension, the outcome of the round) until all rounds are over—pure randomness is the most effective strategy.
(This has something to do with "exploitability"—@natural-hazard has written that "Traditional game theory sacrifices any plausible yet uncertain edge you might have for the guarantee that you will never be taken advantage of." But I have yet to integrate this concept of exploitability into my broader understanding.)
One interesting property of PB-optimal strategies is that they are also the optimal strategies when playing against other PB-optimal players. That is, the PB-optimal strategy becomes more and more effective the better the players one is competing against. In a tournament of professional Rock, Paper, Scissors players, winners would be those best at cognitively simulating randomness. In a tournament of amateur Rock, Paper, Scissors players, winners would be those best at reading their opponents, at "playing the player."
(Also, in some meaningful sense, winners would be playing the room: If, thus far in the tournament, you've noticed that players are rock-heavy, you might be inclined to play paper OR, alternatively, hypothesize that, since other players have thus far had outsized success—premium returns—on playing paper, then a preference for scissors may be a successful tactic.)
4.
Playing the player involves what we can call generalized reading. There are conceptual reasons to call it this, but the idea is bone-achingly simple: to play a player—to recognize his patterns and regularities, in order to exploit them—one must interpret the player. Look for clues, watch his face for tells, pay attention to how he responds to certain moves, or what biases or preferences exist in his play. And inevitably, if one plays with patterns, then these patterns are on display—all moves are publicly visible in their entirety by both players. One cannot play "rock" while hiding that one is playing rock. In this sense, information inevitably "leaks" in the process of play.
PB-optimal play, in other words, becomes optimal when player moves do not leak information. Only in the "blinded" version of Rock, Paper, Scissors can one "play rock" while hiding "playing rock."
In a game like Rock, Paper, Scissors, virtually all players are equally competent at selecting and deploying moves, conditioned on knowing what their opponent will play. (Each player knows rock beats scissors, scissors paper, paper rock. And each player is equally physically competent at using a given move; there is no level of skill that bars them from countering rock with paper.) Therefore, success in the game is one-to-one with success at anticipating one's opponent. This makes it an extreme case of strategic indexicality, the property of strategies depending on opponent strategy. But there is always, in games of strategies, an element of indexicality to a given move's efficacy—otherwise it would not be a game of strategy, it would have a PIG-optimal solution.
Randomness is PB-optimal insofar as it prevents the opponent predicting, at a rate higher than 1/3, what move one will play next. Illegibility is chosen deliberately as a descriptive term here: one cannot be "read" by one's opponent, when one plays randomly; there is no pattern to pull out of gameplay. It is ironic, and perhaps deeply important, that the strategy which is optimal when blindfolded is also the strategy which blindfolds (one's opponent.)
Pseudo-legibility, then, is a strategy of generalized writing. Insofar as your opponent is basing his moves on what he predicts you will do, and he is making predictions based on his interpretations of your previous play (including e.g. body language), then it is advantageous to present him with a false reading, a false interpretation. By planting such a false reading in his mind, one can thereby begin predicting his behavior while simultaneously remaining unpredictable oneself.
This occurs in situations of mixed conflict and coordination. When one feels like they have been "led on," or that they operated under a false impression, they feel they have been "played." They believed the Other was presenting—pro-socially and honestly—a certain pattern, and thus presented their own pattern pro-socially and honestly. This makes them manipulable to the Other, while the Other remains un-manipulable to them. (Because the Other's real patterns are unknown.)
5.
Strategy games get weirdly recursive, in their induction patterns, because one is not modeling what "is" but what their opponent believes. Simultaneously, the opponent is modeling what his opponent believes. Thus, one must model what one's opponent believes one believes one's opponent believes... etc. The "level" at which this mutual modeling bottoms out matters quite a bit, insofar as a player who is thinking too many "levels" deep will mis-model his opponent.
In this sense, strategy games share many structural similarities with Keynesian Beauty Contests, the world of fashion, stock bubbles, sexual fitness etc. A female of a species inclined to find those males handsome which other females find handsome is, in some sense, looking for genetics that will produce a sexually desirable offspring, to the extent that the sexual taste of females in her offspring's generation can be modeled via the sexual taste of females in her generation.
Hyperstition is at its most effective in such systems. The difficult part is breaking inertia, giving the meme a kick-start. But once it picks up momentum, this momentum builds via the positive feedback loops of a Matthew Effect. The meme goes "viral." This is known as runaway selection, and it can come at the cost of an organism's non-sexual fitness. That is, in optimizing for some intra-species game, it becomes less optimized for the inter-species game. Which, of course, is the opposite of how these things are "supposed" to work.
I'll let @mclegibilist take it away, about how the socially valuable byproducts of strategy games are what we really care about, when we organize, host, regulate, and manage such games.