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.)
1) Lottery tickets are bought using income that is after tax, after debt, and after loss of government benefits.
2) Many people buy more than one lottery ticket; they spend hundreds of dollars per year or more.
3) There was a period during which poor folks had reason to legitimately distrust banks and played the illegal numbers game as a sort of stochastic savings mechanism, up to 600-to-1 payouts on 1000-to-1 odds, which meant they did get large units of cash occasionally. Post-FDIC this is no longer a realistic motive and the odds on the government lotteries are worse.
4) Yes, your life can suck, yes, the lottery can seem like the only way out. But this is not a reasoned decision based on having literally no better life-improving use for hundreds of after-tax dollars. It is based on the lure and temptation of easy money to a mind that can't multiply.
5) Those who buy tickets will not win the lottery. If you think the chance is worth talking about, you've fallen prey to the fallacy yourself. In ordinary conversation odds of one in a hundred million of being wrong would correspond to a Godlike level of calibrated confidence. Therefore I say simply, "You WILL NOT win th... (read more)
FWIW, Charles Karelis makes this argument extensively in his book The Persistence of Poverty.
While it's plausible that utility functions are sigmoidal, it's not obviously true, and it's certainly not true of many of the utility functions generally used in the literature.
Moreover, even if experienced-utility (e.g. emotional state) functions are sigmoidal, that doesn't imply that decision-utility functions are, except in the special case that individuals are risk-neutral with respect to experienced utility. More generally than that, a consistent decision-utility function can be any positive monotonic transform of an experienced utility function.
EDIT: I should have added that the implication of that last point is that you can rationalize a lot of behavior just by assuming a particular level of risk preference. You can't rationalize literally anything (consistency is still a constraint), but you can rationalize a lot. All of this makes it especially important to argue explicitly for the particular form of happiness/utility function you're relying on.
(EDITED again to hopefully overcome ambiguities in the way different people are using the terms happiness and utility.)
IAWY right up to the penultimate sentence. Humans continuously modify their utility functions to maintain a steady level of happiness. A change in your utility function's input--like winning the lottery, or suffering a permanent injury--has only a temporary effect. The day you collect your winnings, you're super-happy; a year later, you're no happier than you were when you bought the ticket. If you're considering picking up a crack habit, you had better realize that in a year your baseline happiness will be no higher than it is now, despite all the thi... (read more)
I have a non-rhetorical question for you: do you actually think a significant fraction of people playing the lottery and taking crack cocaine actually maximize utility that way?
If everything comes out exactly right, this can make a case for playing the lottery being better than doing nothing risky but it can't possibly make the case that the lottery isn't massively worse than other forms of gambling. Even if the numbers games are gone going to a casino offers the same opportunity at far better odds and allows you to choose the point on the curve where gambling stops being efficient. I do think however the point that negative-expectation risks can be rational is well taken.
A good reason for not playing the lottery is that you can get better odds by playing roulette, or using other forms of gambling. I am unimpressed by arguing against gambling in general because it's average dollar payoff is negative. That argument is ridiculous.
The discussion about lotteries that I presume lead to this thread was correct, though. It didn't talk about expected winnings, it talked about utility. There are cases where playing the lottery has a high utility - and if the utility is too low, then you shouldn't play.
I'm skeptical that lottery player utility is well modeled as in the convex section of a sigmoid. I'd want to see more analysis to that effect.
My attempt to liven up this post by talking about crack and lotteries has killed many minds here. If you're driven to write a long reply about crack and lotteries, perhaps you can spare one sentence in it to respond to this more general point:
We are inclined to use expected return when we should use expected utility.This quick-and-dirty reasoning works well when we are reasoning, as we often are, about small changes in utility for ourselves or for other people in our same social class; because a line is a good local approximation to a curve. It works les... (read more)
Perhaps this just indicates that I lead too sheltered a life, but I think most people don't have $< utility function is concave just as it is for positive $. So I'm skeptical of the claim that "poor folks" playing the lottery or using crack are generally maximizing their expected utility.
And I can't speak for anyone else, but I don't think I've ever said anything like "those fools don't deserve our help if they're going to make such stupid decisions" about people playing the lottery or taking crack, and if I ever did I'm pretty sure it wouldn't be the desperately poor and/or miserable ones that I had in mind.
I think this post is going to contribute to semantic confusion; when most of us talk about utilons, I think we're talking about the output of a utility function.
I don't think your examples are that plausible in the real world, at least not in terms of the reasoning you give. In your scenarios, it would be much better to hide the money away somewhere and let it accumulate, pretending to the world (and to Uncle Sam) that you spent it on crack or whatever, than to actually spend it on crack.
Having said that, if we determine the rationality of some behavior relative to the actual utility function of the individual, then we can see that for some (possible) utility functions, it would be rational to play the lottery and... (read more)
I like this post. That's a point I think needed to be made.
Before reading this, the way I saw it was that for quite a lot of people, there's something akin to a potential barrier as it exists in chemical reactions, for what they can expect of their life. Unless you can invest enough X (energy, time, money, etc.), then what you're trying to do won't work on average. You could also see it as an escape velocity, or the break-even point in a chain reaction getting critical.
To illustrate in the case of a lottery, many people can't expect to ever be able to get ... (read more)
To see a plot of the utility function, I posted one here:
Doesn't this make some very big assumptions about the fixity of people's circumstances? If my life is so bad that smoking crack begins to seem rational, then surely, taking actual steps to improve my life would be more rational. Similarly, I imagine that the $5 spent on a lottery ticket could be better spent on something that was a positive first step toward improving even the worst of circumstances. Seems the only way this wouldn't be true would be if you simply assert, by fiat, that the person's circumstances are immutable, but I'm not sure whether this accords with reality. (One's politics are clearly implicated here.)
On a whim, I once played the lottery on the theory that the Many Worlds Interpretation is true, and some branch of me would win. I like to think he's out there somewhere.
(Of course, if MWI really is true, then some other me in some other branch would have played the lottery even if I hadn't, so strictly speaking I didn't even need to...)
You haven't explained why relatively happy people play the lottery. The answer is that they can't understand how small the probability of winning is. (Nor can I, by the way; I only understand it mathematically. To make me understand, you could do something like phrase it in terms of coin flips.)
"For someone with $ << 0, the marginal utility of $5 to them is minimal. "
I'm a newbie, which will soon be obvious, but I don't think the utility function is being applied correctly. At each value of U (the worth that a person has at his disposal in goods), we have the utility that can be purchased with U. (So u is negative for U<0 because you get negative things for owing money.)
I understand that if someone is greatly in debt, their utility may not change much if you increase or decrease their debt by some amount. This is why the utilit... (read more)
The real reason not to say "those fools don't deserve our help" is that it doesn't make sense for materialist consequentialists to weight utility based on who deserves what.
Some people who buy lottery tickets argue that a lottery ticket is a small price to pay for the chance of being a millionaire.
While the expected return of the lottery ticket is negative, they place an extra value on the chance of being a millionaire, in addition to the expected return.
For comparison, suppose there is another lottery with the same negative expected return, but the maximum you can win is $5 (corresponding with a much higher probability of winning so that the expected value is the same). Then players will be less interested -- because you've ... (read more)
I think your EDIT is much clearer, and more accurate than your original formulation.
In response to the (IMHO unnecessarily snarky, but perhaps I'm reading in too much) explanation for the edit:
It is possible simultaneously to (a) think that "some [lottery players] may be making much more rational decisions than we think"; (b) think that it's still irrational for them to play the lottery; and (c) not define "rational" as "the unattainable goal of perfect utility maximization."
This just means that you think playing the lottery is really silly.
Dollars in, utilons out. Otherwise what are dollars?
It may be perfectly rational for crabs in a bucket to pull each other down in an attempt to escape individually... from the perspective of a mere individual.
From a perspective of survival of the tribe, it's suicidal, and irrational to boot.
Crabs, of course, do not have tribes.