Why do people both buy lottery tickets and insurance against losses?

Hat tip to Tyler Cowen. He says:

For a long time I didn't think much of Milton's seminal 1948 article on expected utility theory, but it turns out the joke is on me...

The Friedman-Savage piece starts with an obvious puzzle: why do people both buy lottery tickets and insurance against losses? That would seem to make them both risk-loving and risk-averse at the same time. The proffered answer is simple: part of the utility function is concave, and part is convex. Across the lower range we wish to play it safe, but above a certain margina we are willing to take gambles (by the way, here is some evidence, and why it might follow from market constraints).

I admit that I have bought about 100 lottery tickets in my lifetime, whenever the (overstated) pot is over $20 million, which usually yields the (false) expected value > 1.0. The reasons why the pot and expected value are false:

1) The pot figure is a total of future payments, usually over 20 years. The lump sum payout option is about half of the advertised pot, and that is before taxes.

2) There is a material risk of splitting the pot with identical tickets.

But even knowing this, I still have spent the $100 over the last 20 years. Silly me.

Since we are on the topic of utility, awhile back, I posted a reference to Pascal's Wager. Jason Ruspini recently linked to Alex Tabarrok's paper on the rationality of taking the wager, which I had read a long time ago. If you ever find yourself estimating your personal utilities beyond your estimated life expectancy, I suggest you read the whole paper.