Protagonist and immersion. Thoughts?

I’m in the process of creating a game and I’m not sure which route to take. The main character is a detective and I’m not sure if i wan’t the antagonist to call him detective the whole game or I should find a name for my main character. Do you think leaving the name blank would help the immersion of the player since he can think he is the main character or do you feel that a main character with no name feel like the story has not enough depth?
I want to know what you guys think.

Lots of games we know the name of the person we are playing… Ms. Croft, Mr. Nukem, etc. I think having a great narrative will help immerse people into your game…