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"blame game" is correct and usable in written English.
You can use it to refer to situations in which one or more parties is trying to assign blame for a problem. For example: "Neither side wanted to take responsibility, so they just kept playing the blame game."
Dictionary
blame game
noun
A situation in which people attempt to blame others rather than trying to resolve a problem.
Exact(60)
But the blame game persists.
The blame game continues.
It's a blame game.
"He never played the blame game".
The "blame game is counterproductive," he said.
Another thing about the media blame game?
The blame game has already begun.
Into a ceaseless, tedious blame game, too.
Everybody plays the blame game sometime.
The blame game is in full swing".
2. Be careful with the blame game.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com