More generally, game theory often produces situations where randomization is clearly the rational strategy when you do not know with certainty the other player's action. The example given in this post is straightforwardly convertible into a game involving Nature, as many games do, where Nature plays first and puts you into one of two states; if you are in X and play Continue, Nature plays again and the game loops. The solution to that game is the same as that derived above, I believe.

The point is, for a broad class of games/decision problems where Nature has a substantial role, we might run into similar issues that can be resolved a similar way.