1. The first shipper should declare a value of 640 gold pieces. If the inspectors purchase at that price, then the next ship pays no toll whatsoever. So the net receipts to the two ships are 1,640 gold pieces.
Conversely, if the inspectors accept the toll based on 640 gold pieces, then the first shipper will pay 160 gold pieces in toll and receive a net sale price of (1000 – 160) = 840 gold pieces. The second ship will then (based on the solution to the warm-up problem) declare its value at 800 gold pieces and will receive a net value of 800 gold pieces whether the shipment is purchased or not. In the two alternatives, the shippers receive a total of 1,640 gold pieces.
2. First the shippers should all value their cargo at the same amount V (less than or equal to 1,000). Suppose the inspectors purchase two shipments and accept the tolls on the other three. Then the five ship owners receive (2*V) for the purchased shipments and (3*(1000 - 0.25V)). The total is 2V + 3000 - 0.75V = 1.25V + 3000. But then it is also possible that the inspectors would decide to accept the toll, in which case the shippers receive (5 * (1000 - 0.25V)) = 5000 - 1.25V.
Setting these two answers as equal (so that the ship owners receive the optimum return either way), we get:
At this rate, the shippers receive 4,000 gold pieces.
3. Intuitively, sending in the ships one at a time should be a better strategy, because once the warehouses are filled, the shippers can put the value at 0. In practice (this solution demands a careful case analysis), altogether they get only about 60 more gold pieces.