Risk, Information and Insurance (Roger Bucks)

It is the year 2099. The economy is heavily dependent on a rare mineral, unobtainium, which can only be found among the Asteroids. The market for unobtainium is one of perfect competition, since a very large number of miners are searching among the Asteroid belt in small spacecraft and the market absorbs all their output at the going price of §19 (Nineteen hyperbucks) per shipload. It costs each miner §1 (one hyperbuck) to visit a candidate Asteroid, and 15% of the trips succeed in finding the mineral, with a net gain of §18; the rest return empty-handed, with a net loss of §1. The Asteroid belt is huge, and unobtainium will surely become technologically obsolete before the total supply is noticeably depleted.

Since one hyperbuck is a lot of money, our hero, Roger Bucks, is risk averse. He would be just willing to exchange 3 hyperbucks cash for a lottery in which he had a 50% chance of winning 18 hyperbucks and a 50% chance of losing 2 hyperbucks. The exponential model of Roger's utility function is U(X) = 1.09574(1-e-.1218756X), but we will use a simpler model: U(X) = , CME(u) = 20u2 - 2.

Roger can reduce his uncertainty about the next Asteroid on his list to visit by sending a robot probe to it, at a cost of §0.30. If the asteroid has unobtainium there is a 75% chance the probe will correctly report "positive," while if there is no unobtainium, the probability that the probe will erroneously report "positive" is 30%.

Whether he sends a probe or not, he also has the option of buying an insurance policy from Lloyd's of Luna. The premium is §9.10; if he returns empty-handed, the insurance company refunds his premium and covers his §1 expenses; otherwise they keep the §9.10, reducing his net to §8.90.