This assumes that the question is asked only once, but then, to which of the 20 copies will it be asked?
Every copy that is in a green room is asked the question (so either 2 or 18 copies total are asked). If all answer Play, we play. If all answer Don’t Play, we don’t. In any other case we fine all 20 copies some huge amount; this is intended to make them agree beforehand on what answer to give. (This is reworded from the OP.)
For your other thought experiment—if there aren’t actual N copies being asked the question, then there’s no dilemma; you (the only copy) simply update on the evidence available (that the room is green). So yes, the original problem requires copies being asked in parallel to introduce the possibility that you’re hurting other copies of yourself by giving a self-serving answer. Whereas if you’re the only copy, you always give a self-serving answer, i.e. play only if the room is green.
Every copy that is in a green room is asked the question (so either 2 or 18 copies total are asked). If all answer Play, we play. If all answer Don’t Play, we don’t. In any other case we fine all 20 copies some huge amount; this is intended to make them agree beforehand on what answer to give. (This is reworded from the OP.)
For your other thought experiment—if there aren’t actual N copies being asked the question, then there’s no dilemma; you (the only copy) simply update on the evidence available (that the room is green). So yes, the original problem requires copies being asked in parallel to introduce the possibility that you’re hurting other copies of yourself by giving a self-serving answer. Whereas if you’re the only copy, you always give a self-serving answer, i.e. play only if the room is green.