In any given game, it seems there are two ways to play a character whose decisions drive the story. You can make the decisions you would make if you were the character, or you can make the decisions you think the character would make.
The former approach is common to most modern games, particularly ones that feature customizable characters, silent protagonists, or open worlds. These games offer a blank slate for the player to fill as they see fit. Since the player is indistinct from the character they play, they can allow their own background to influence their decisions despite the fictional context. Galatea, for example, is a game that features two characters: the titular Galatea and you, an art critic. That’s all you know about yourself to begin with. The entire game is based upon your interactions with the one other character in the room, but it supplies you with several versatile commands and many topics of interest for you to construct your own personality with. What topics you fixate on, where and when you touch her, whether you view her as an artwork or a human—it all depends on you. The decisions you make reflect upon you, because what is a person but the series of decisions they’ve made throughout their life?
The latter approach is less common, but it has deeper roots. The most famous example of a game that makes use of this is Dungeons and Dragons. Though you can still customize your character, this is a roleplaying game and so you are generally discouraged from making a character that resembles you. Instead you are meant to spend the campaign portraying a character whose words or actions would not line up with your own if you were in that same situation. Every decision you make must be “in character,” which at once frees you from moral responsibility and burdens you with narrative plausibility. Photopia is a prime example of this. It leaps between the perspectives of several different characters and provides limited dialogue choices that progress the story in one direction regardless of what is chosen. As such, every decision you make is automatically in character. The responsibility of finding the best possible ending has been lifted from your hands, but you are left with a growing feeling of helplessness as the story veers towards tragedy. The game will end the same way every time because the characters don’t know what kind of story they’re living in. Only the player does.