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It is said there are four kinds of work the paragraph must do: establish setting, further plot (sometimes split into creating and resolving tension), deepen characters, and reinforce theme. While the exact number, names, and even delivery unit may be argued, it's a good guideline. Words have tasks to accomplish, and the good writer cuts what doesn't pull its weight.
When we move writing into the interactive realm, we can look at interactivity through this lens. Does interactivity require more of the paragraph? If so, it seems that any new tasks would relate in some way to changes wrought by gameplay: creating it, reporting it, reacting to it. And just so we're on the same page, by "writing" we are considering either text that explicitly appears in the game, or spoken dialogue. We are not discussing scriptwriting -- most plot events in videogames are acted out visually, not transmitted to the player through language, so do not apply here.
I propose four more tasks for the paragraph.
Teaching how to play the game, that is, how to effect meaningful change within the game, is the first task for writing. Out-of-world methods like explicit tutorials are artless methods akin to a speculative fiction novel infodumping the workings of its magic or technology. An improvement is typographic conventions on text such as color or boldface to non-invasively emphasize important locations, items, characters, or inputs. But ideally, the player learns what's needed by learning from others in the world: characters explain how to use a new item or ability, request work of the player, or demonstrate tactics.
When writing is used in that way, the writing itself is static. How to play the game doesn't itself change. The current state of the game does however, requiring the paragraph to comprise many strung-together snippets of text or pre-recorded dialogue, re-constructed during play. While graphics may be the best solution for monitoring widget factories, games which deal less in producing physical resources than in producing, say, social change, could do worse than weaving that into mutable dialogue. Frequently the most skipped and least interactive part of a videogame, dialogue then becomes as vital as a heads-up display, and therefore at least as interesting. And that isn't the only narrative solution for conveying gamestate. Literature frequently uses symbolism to remark on a character's -- or the world's -- development or degradation.
It isn't enough to give players the inputs and outputs to a game. They must see that their actions tie to a distinct change, and sometimes effects aren't immediate. This happens frequently when the player interacts with characters, as characters will hide their true feelings, their true state. Proving agency is the third task for the paragraph, and like plot it has a two-part throw-and-catch structure. Dialogue, facial expressions, body language, even scene-ending descriptions of setting can foreshadow an effect caused by player agency. Later, when the effect becomes apparent, proving agency's involvement can be done through dialogue, parallel language, or flashbacks.
Every now and then, a player may feel the need to explore whether and why the game's cats are invincible, offer commentary on the game itself, or just poke at odd corners in general. It's nice if the game responds with appreciation, humor, or an army of vindictive chickens. Games and stories both exist for the purposes of amusement, but as authored interactive media, can interact with the player as a player. Indulging the player deepens the connection even across replays, through optional content such as fourth-wall breaks, responses to bizarre behavior, or performance of thoughtful but non-essential gestures, such as trekking back to the beginning areas of the game, to a party member's house, merely to break the news of his death to his mother.
Interactive paragraphs may require some amount of prose generation. It may be initially as simple as requiring one noun phrase in place of another, or for worlds that accumulate a long history of player agency, larger swatches of generated prose. There is of course some amount of overhead in stitching together so many little snippets of text, not to mention that convincingly blending snippets of recorded dialogue is still in its infancy. But stories are made of paragraphs, games are made of tasks, and paragraphs have tasks to accomplish.