Exercise 5.18

This question considers pruning in games with chance nodes. Figure trivial-chance-game-figure shows the complete game tree for a trivial game. Assume that the leaf nodes are to be evaluated in left-to-right order, and that before a leaf node is evaluated, we know nothing about its value—the range of possible values is $-\infty$ to $\infty$.

1. Copy the figure, mark the value of all the internal nodes, and indicate the best move at the root with an arrow.

2. Given the values of the first six leaves, do we need to evaluate the seventh and eighth leaves? Given the values of the first seven leaves, do we need to evaluate the eighth leaf? Explain your answers.

3. Suppose the leaf node values are known to lie between –2 and 2 inclusive. After the first two leaves are evaluated, what is the value range for the left-hand chance node?

4. Circle all the leaves that need not be evaluated under the assumption in (c).