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In the following, a “max” tree consists only of max nodes, whereas an “expectimax” tree consists of a max node at the root with alternating layers of chance and max nodes. At chance nodes, all outcome probabilities are nonzero. The goal is to find the value of the root with a bounded-depth search.

  1. Assuming that leaf values are finite but unbounded, is pruning (as in alpha–beta) ever possible in a max tree? Give an example, or explain why not.

  2. Is pruning ever possible in an expectimax tree under the same conditions? Give an example, or explain why not.

  3. If leaf values are constrained to be in the range $[0,1]$, is pruning ever possible in a max tree? Give an example, or explain why not.

  4. If leaf values are constrained to be in the range $[0,1]$, is pruning ever possible in an expectimax tree? Give an example (qualitatively different from your example in (e), if any), or explain why not.

  5. If leaf values are constrained to be nonnegative, is pruning ever possible in a max tree? Give an example, or explain why not.

  6. If leaf values are constrained to be nonnegative, is pruning ever possible in an expectimax tree? Give an example, or explain why not.

  7. Consider the outcomes of a chance node in an expectimax tree. Which of the following evaluation orders is most likely to yield pruning opportunities: (i) Lowest probability first; (ii) Highest probability first; (iii) Doesn’t make any difference?