The skeleton of the argument is:

1. Present a particular thought experiment, intended to provoke anthropic reasoning. There are two moderately plausible answers, "50%" and "a billion to one against".
2. Assume for the sake of argument, the answer to the thought experiment is 50%. Note that the "50%" answer corresponds to ignoring the color of the room - "not updating on it" in the Bayesian jargon.
3. The thought experiment is analogous to the Bolzmann-brain hypothesis. In particular, the color of the room corresponds to the ordered-ness of our experiences.
4. With the exception of the ordered-ness of our experiences, a stochastic-all-experience-generator would be consistent with all observations.
5. Occam's Razor: Use the simplest possible hypothesis consistent with observations.
6. A stochastic-all-experience-generator would be a simple hypothesis.
7. From 3, 4, 5, and 6, predict that the universe is a stochastic all-experience generator.
8. From 7, some very unpleasant consequences.
9. From 8, reject the assumption.

I think the argument can be improved.

According to the minimum description length notion of science, we have a model and a sequence of observations. A "better" model is one that is short and compresses the observations well. The stochastic-all-experience-generator is a short model, but it doesn't compress our observations. I think this is basically saying that according to the MDL version of Occam's Razor, 6 is false.

The article claims that the stochastic-all-experience-generator is a "simple" model of the world and would defeat more common-sense models of the world in an Occam's Razor-off in the absence of some sort of anthropic defense. That claim (6) might be true, but it needs more support.