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Technologos comments on The Absent-Minded Driver - Less Wrong
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Hm. This would actually appear to be a distinct class of cases in which randomness is "useful", but it's important to note that a simple deterministic algorithm would do better if we were allowed to remember our own past actions - i.e. this is a very special case. I should probably think about it further, but offhand, it looks like the reason for randomized-optimality is that the optimal deterministic algorithm has been prohibited in a way that makes all remaining deterministic algorithms stupid.
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.