Resolving Zeckhauser’s paradox

Zeckhauser’s paradox has puzzled and entertained many rationality enthusiasts for almost half a century. You are forced to play a Russian Roulette with a 6-chamber revolver containing either (A) two bullets, or (B) four bullets. Would you pay more to remove the two bullets in (A) than you would to remove one in (B)? Most would say yes, but rational considerations based on the classical utility theory suggest you should not. We discuss a possible solution within the classical framework, by explicitly stating and accounting for more detailed preferences in terms of fewer bullets and smaller debt. To a large extent, the paradox arises due to a surreptitious trespassing of Savage’s Small-World utilities implied by a limited set of preferences to govern a larger world containing potentially conflicting preferences. To avoid logical issues associated with death in the roulette, we also describe a non-fatal game-show version, where you choose one box out of six that could be either empty or contain prize money. Here, the paradox arises when you pay from the prize money, but not when you pay from your own money. In summary, the paradox provides a useful lesson about the normative role of the utility function as a rational guide for our decisions and preferences.