Twenty (simple) questions

In the game of “20 questions”, Alice picks an object, person, or place, and Bob attempts to reveal it using as few Yes/No questions as possible (20 is the gold standard). To spice up the game, suppose that Alice picks her object according to a distribution known to Bob, and Bob is interested in minimizing the expected number of questions that he asks. Huffman came up (unknowingly) with an optimal strategy for this game. There is one problem with his strategy - it could potentially use arbitrary questions, which might be awkward.

What if we restrict Bob to a bounded inventory of questions, which he needs to choose ahead of time, before Alice decides on her distribution?

Joint work with Yuval Dagan, Ariel Gabizon, and Shay Moran; and with Idan Mehalel.