The lottery came up in a recent comment, with the claim that the expected return is negative - and the implicit conclusion that it's irrational to play the lottery. So I will explain why this is not the case.
It's convenient to reason using units of equivalent value. Dollars, for instance. A utility function u(U) maps some bag of goods U (which might be dollars) into a value or ranking. In general, u(kn) / u(n) < k. This is because a utility function is (typically) defined in terms of marginal utility. The marginal utility to you of your first dollar is much greater than the marginal utility to you of your 1,000,000th dollar. It increases the possible actions available to you much more than your 1,000,000th dollar does.
Utility functions are sigmoidal. A serviceable utility function over one dimension might be u(U) = k * ([1 / (1 + e-U)] - .5). It's steep around U=0, and shallow for U >> 0 and U << 0.
Sounds like I'm making a dry, academic mathematical point, doesn't it? But it's not academic. It's crucial. Because neglecting this point leads us to make elementary errors such as asserting that it isn't rational to play the lottery or become addicted to crack cocaine.
For someone with $ << 0, the marginal utility of $5 to them is minimal. They're probably never going to get out of debt; someone has a lien on their income and it's going to be taken from them anyway; and if they're $5 richer it might mean they'll lose $4 in government benefits. It can be perfectly reasonable, in terms of expected utility, for them to play the lottery.
Not in terms of expected dollars. Dollars are the input to the utility function.
Rationally, you might expect that u(U) = 0 for all U < 0. Because you can always kill yourself. Once your life is so bad that you'd like to kill yourself, it could make perfect sense to play the lottery, if you thought that winning it would help. Or to take crack cocaine, if it gives you a few short intervals over the next year that are worth living.
Why is this important?
Because we look at poor folks playing the lottery, and taking crack cocaine, and we laugh at them and say, Those fools don't deserve our help if they're going to make such stupid decisions.
When in reality, some of them may be making <EDITED> much more rational decisions than we think. </EDITED>
If that doesn't give you a chill, you don't understand.
(I changed the penultimate line in response to numerous comments indicating that the commenters reserve the word "rational" for the unobtainable goal of perfect utility maximization. I note that such a definition defines itself into being irrational, since it is almost certainly not the best possible definition.)
Please amplify on "#1 is wrong".
This is a very common conversation in science. Some of it is conducted improperly, which is annoying, but I would hardly categorize the whole thing as unhelpful. In particular, the "improper" critiques usually consist of hypothesizing more and more elaborate hidden mechanisms with no evidence to support them as alternatives.
But we know hyperbolic discounting exists. We know that people are insensitive to the smallness of small probabilities.
When the other mechanism is nailed down by other evidence (hyperbolic discounting (for crack), or neglect of the tinyness of tiny odds (for lottery tickets)) and the new mechanism is not known, then A->B, C->B, C steals the evidence that C provides for A. You need to provide new D with A->D, C!->B. Where the implication from C to B is imperfect then B goes on providing some trickle of evidence to A but if the implications are equally strong then the trickle does not distinguish between A and C as opposed to other hypotheses and the prior odds win out.
In particular, the notion that ticket buyers really are making an expected utility calculation says that decreasing the odds of a lottery win by a factor of 10 (while perhaps multiplying the number of tickets sold by 10 and keeping the price constant, so that the number of lottery winners reported in the media is constant), will decrease the price they are willing to pay for a given lottery ticket by a factor of 10. Are you willing to make that prediction? I'd expect ticket sales to remain pretty much the same.
That's an interesting point.
If, as I said