The idea-space in human brains is gameable
I’m not an expert on Topology, but one of the ways I think about the games I like is that they make use of the idea-space inside the human brain as a gameable space. Now, by that I don’t mean that you can imagine places that aren’t real and think up activities people might engage in in places like that. What I mean is that the way we think actually provides “dimensions” along which you can design meaningful interactions in a game. From my reading of what contemporary psychology and cognitive science tell us, we’re capable of perceiving the appropriateness or congruence of matches between ideas. You know the confident “that feels right!” feeling you get when you figure out what the answer to a riddle must be, or when you come up with the perfectly apt humorous remark? Or the “that’s not right” feeling you get when Hollywood miscasts a part in a movie adaptation of a story you know? That’s what it subjectively feels like to have different levels of connection between ideas, which is apparently how the intuitive side of our cognition works. Even though we don’t have scientific units for it, we can get a feel for how “librarian-y” someone is by intuitively comparing them against the idea of “librarian” we have in our brains. We can even get a feel for how weaselly someone is.
This is the entire basis of the game Apples to Apples. In it, one player puts an “adjective” card on the table (for example, “Hot and Sticky”). Then all the other players consult their hand of “noun” cards and put forward the one they think the initial player will select as the “best match” (for example, “The Equator”, “Cinnamon Buns”, “The Sports Illustrated Swimsuit Issue”). Some people have a hard time grasping that there is no single way that people are required to make that “best match” comparison. It’s not always “the most similar” or “the most opposite”. Sometimes it’s the ill-defined “funniest”, but even inveterate jokesters will sometimes feel compelled to pick a straightforward match if it’s dead-on. The way it works is that the player compares the “adjective” they put forward to the various options and picks the one that feels like the “best match”. We don’t need to put a name to a comparison to feel how strong it is. Strictly speaking Apples to Apples tends to be about emphasizing the minor variations between people rather than the commonality because it asks the player to pick a “best” match on each round (thus the way to win, if you care about that, is to “play the player” and put forward cards with matches that are likely to resonate especially strongly), but it illustrates the point that there are dimensions of play that games can lean beyond simple factors like tallest/short, fast/slow, near/far, big/small, etc. Personally I’m not a huge fan of the gameplay in Apples to Apples (my sense of humor tends to run a little more cerebral and surrealistic than average so my joke answers nearly always lose out to the more obvious jokes) but since it uses this abstraction as the central element of play it’s a useful example.
While they don’t always foreground it the way Apples to Apples does, Roleplaying games make heavy use of this concept to inform and constrain play. The old-school “puzzle solving realism” style of play, for example, leans heavily on the ability of humans to mutually imagine “that’s what would happen!” to explore the consequences of poking things with ten-foot poles or pouring acid on them. The Burning Wheel family of games orients players to judge characters by looking through the lens of written character Beliefs, rewarding players for acting along (or dramatically against) the line of those Beliefs. Games with oracle mechanics like Ganakagok use abstract concepts to guide play (“figure out the most ‘Woman of Storms’ way to conclude this scene”). Even something as fuzzy as “what’s the most dramatically appropriate (or dramatically ironic) thing?” or the dreaded “what’s best for The Story?” can be used in a game context. Stories and storytelling have a huge role in human culture and the way that human minds work, so it shouldn’t be surprising that we have a lot of intuitions related to stories and imagination. These intuitions can be built into the “space” of play in these games in the same way that features of human locomotion are as important dimension of play in sports as ball-physics and field geometry.