There is already much discussion regarding how games should provide players with meaningful choices and impacting outcomes. In addition to these ideas, immersion can also be granted by the game's ability to reflect these meaningful choices/outcomes on the physical identity of the avatar, as well as their resulting position within the game society. By changing the physical appearance or social reception of a character according to the decisions made in the game, the player is immersed in a realistic sense of social justice. For instance, a character that's been blowing up cars and killing police throughout the entire game shouldn't be able to walk into an orphanage and be kindly greeted by a smiling nun. By the same token, a character that's donated half of their wealth to the needy and has purged the city of a deadly virus shouldn't be greeted by a scowling officer that tells them, "Stay out of trouble!" The best example I can give of this principle working effectively is within the first Fable. By doing evil deeds like murdering citizens or stealing, the character grows horns, starts balding, attracts flies due to foul odor, as well as many other small, odious traits. And in a response to these characteristics, society treats him (I say "him" because the avatar could only be male) like a monster - running, screaming, and making nasty remarks as they are interacted with. In this way, the morality of the choices that players make become meaningful to the way the avatar is perceived by both the player, as well as the in-game society. This sense of realism (not the "growing horns" part but more of the avatar's reflection and treatment within society) and social justice is a great way to promote meaningful consequences and increase player immersion by holding player morality accountable.

Below is an image from Fable that shows the physical difference between making morally good (top) versus evil (bottom) choices.
Fable alignment
Example #2: Be God (Global Game Jam 2015 - Recife) The avatar (some sort of god) changes according to the player's answers. It follows image bellow:
Choices2 0

bad god example

Choices3 0

Good god example

Example #3: Mass Effect's Shepard.

Spanning 3 full games and a multitude of choices, many players craft their main protagonist of the mass effect series to have a personality. While many players simply ask at every turn, "what would I do," or just pick the correctly colored choice for the outcome they want, the game lets those players who decide to engage it play any sort of character they want. If you play in a consistent way, by the end of the game your appearance will reflect the personality you've decided to follow.

Pure paragon vs pure renegade by shotgunchief

While it's not as distinct as with Fable, there is a very stark difference between a character who has made, "good," choices (your scars will heal) and one who has made, "bad," choices (the scars get worse and take on a synthetic glow).

Disappointing ending aside, this system of choices got players to become very invested in a character that has almost no original personality. My Shepard may look and act completely differently than yours because of the things we've chosen to do as we play.

Example #4 Dragon Age

In Dragon Age depending on what you choose to do people will react differently to you. your standing in the world is affected the very moment you chose a certain race to start with. Over time you can see how your actions affects the world around you.