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    can you do this in a shader, to make a game look this way?

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      Check out “Return of the Obra Dinn” by Lucas Pope, I have not played it but I hear good things. This is an article about how it achieves it’s dithered look

      To answer your question, Atkinson dithering was designed to be super efficient on low-end processors but does not easily translate to a shader because of the dependencies between pixels. At best, you would have to do multiple passes.

      Quite apart from this, Atkinson dithering only really works for still images. For video, the error manifests as strange temporal artifacts that are very distracting. The aforementioned Return of the Obra Dinn uses a combination of techniques and a lot of effort to enumerate the look of Atkinson dithering while animating.