The risk of flooding in land adjacent to the River Nudd has recently increased. This is because of a combination of high spring tides and the development by farmers of more efficient drainage systems in the nearby hills, which means that, after heavy rainfall, water enters the river more quickly. A tidal barrier is being constructed at the mouth of the river, but the Hartland River Authority has to decide how to provide flood protection in the two years before the barrier is completed. Flooding is only likely to occur during the spring high-tide period and the height of the river at this time cannot be predicted with any certainty. In the event of flooding occurring in any one year the Authority will have to pay out compensation of about $2 million. Currently, the Authority is considering three options. First, it could do nothing and hope that flooding will not occur in either of the next two years. The river’s natural banks will stop flooding as long as the height of the water is less than 9.5 feet. It is estimated that there is a probability of 0.37 that the height of the river will exceed this figure in any one year. Alternatively, the Authority could erect a cheap temporary barrier to a height of 11 feet. This barrier would cost $0.9 million to erect and it is thought that there is a probability of only 0.09 that the height of the river would exceed this barrier. However, if the water did rise above the barrier in the first year, it is thought that there is a 30% chance that the barrier would be damaged, rendering it totally ineffective for the second year. The Authority would then have to decide whether to effect repairs to the barrier at a cost of $0.7 million or whether to leave the river unprotected for the second year. The third option would involve erecting a more expensive barrier. The fixed cost of erecting this type of barrier would be $0.4 million and there would be an additional cost of $0.1 million for each foot in the barrier’s height. For technical reasons, the height of this barrier would be either 11 or 13 feet, and it is thought that there would be no chance of the barrier being damaged if flooding did occur. The probability of the river’s height exceeding the 13-foot barrier in any one year is estimated to be only 0.004.

(a) Draw a decision tree to represent the River Authority’s problem.

(b) Determine the optimum policy for the Authority, assuming that their objective is to minimize expected costs. (For simplicity, you should ignore time preferences for money.)