Consider the following decision tree, where the probabilities for each event node are shown in parentheses.

The dollar amount given next to each branch is the cash flow generated along that branch, where these intermediate cash flows add up to the total net cash flow shown to the right of each terminal branch. (The unknown amount for the top branch is represented by the variable x. ) The decision maker has a utility function U (y) = y1/3 where y is the total net cash flow after a terminal branch. The resulting utilities for the various terminal branches are shown to the right of the decision tree.
Use these utilities to analyze the decision tree. Then determine the value of x for which the decision maker is indifferent between decision alternatives A1 and A2.

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Payoff Utility $x 1/3 $x +$7,600 $6,859 19 A1 (0.60) -$600 $590 -$1,331 -11 (0.40) -$141 A2 $0 (0.50) $16 -$125 -5 +$141 (0.50)