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badger comments on Single player extensive-form games as a model of UDT - Less Wrong
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The first option is standard. When the second interpretation comes up, those strategies are referred to as behavior strategies.
If every information set is visited at most once in the course of play, then the game satisfies no-absent-mindedness and every behavior strategy can be represented as a standard mixed strategy (but some mixed strategies don't have equivalent behavior strategies).
Kuhn's theorem says the game has perfect recall (roughly players never forget anything and there is a clear progression of time) if and only if mixed and behavior strategies are equivalent.
Thank you, I did not know the terminology.
The types of games we care about that inspire us to have to use UDT do not have perfect recall, so whether or not behavior strategies are possible is an important question. It also feels like an empirical question.