That is highly relevant, and is basically where I was planning to go with my next post. In particular, see the dice problem in this comment—sometimes throwing away information requires randomizing our probability distribution. I suspect that this idea can be used to rederive Nash equilibria in a somewhat-more-embedded-looking scenario, e.g. our opponent making its decision by running a copy of us to see what we do.
Thanks for pointing out the relevance of your work, I’ll probably use some of it.