A rich man is forced to play a version of Russian Roulette. The cylinder of a six-shooter revolver containing two bullets is spun and the barrel is then pointed at the head. He is now offered the opportunity of paying money to have the two bullets removed before the trigger is fired. It turns out that the payment can be made as high as $10,000,000 before he becomes indifferent between paying and taking the risk of being shot.

(a) Why would the rich man also be indifferent between having the trigger pulled when the revolver contains four bullets and paying $10,000,000 to have one of the bullets removed before the trigger is pulled? (Assume that he is rational in the sense of VNM expected utility theory).

(b) Why would the rich man not be willing to pay as much as $10,000,000 to have one bullet removed from a revolver containing only one bul- let?

(c) Would you be willing to pay more to have one bullet removed from a revolver containing four bullets or to have one bullet removed from a revolver containing one bullet? If the answer is yes, then you are not rational in the sense of VNM EU theory. But perhaps you prefer it that way!

Hints: Assume that the player doesn't care if whether paying $1m and being shot or not paying and being shot (death gives the same utility regardless). There three different outcomes being described here: being shot, not being shot, and not being shot and paying $1m. Its safe to assume that U (death) < U (live and pay $1m) < U (live and keep money)

Without loss of generality, we can normalize U (death) = 0 and U (live and keep money) = 1. This is because the VNM expected utility function preserves preference ordering from any affine transformation. An affine transformation is just a linear transformation, such as f (x) = Ax + B.