Lotteries, insurance, and star-shaped utility functions

Most agents insure against risks which entail large losses, but few insure against small losses. The fact that organized lotteries charge low ticket prices examplifies this spirit. This weakly risk averse behavior pattern seems to be well structured but is at variance with concavity. We derive axiomatically a class of utility functions, star-shaped, which accommodate these phenomena. Concavity yields risk aversion everywhere and decreasing marginal utility. Star-shaped utility functions exhibit risk aversion at some wealth positions, and average utility from any of these is a decreasing function of wealth.