We begin by examining a strange game described by Leonard Mlodinow in his book The Drunkard's Walk:
"Suppose the state of California made its citizens the following offer: Of all those who pay the dollar or two to enter, most will receive nothing, one person will receive a fortune, and one person will be put to death in a violent manner."
Couched in such language, the game sounds like it comes from a post-apocalyptic American dystopia. Or a reality show on Fox. But actually, Mlodinow is just talking about the California state lottery. Most people simply cough up a buck; one (usually) person eventually hits the jackpot, and the increased traffic---on average, after factoring in some reasonable assumptions and stats from the NHTSA---causes about one extra motor vehicle fatality per game (p.78). This doesn't exactly sound like a game you'd want to play. And, as a math teacher, I'm nigh on required to agree with that sentiment. But I don't.
Even though it's always smoldering in the background, the recently ginormous Mega Millions jackpot has fanned the flames of lottery hatred, particularly among the mathosphere. But I'm here to present the minority opinion. The lottery isn't such a bad bet for the average citizen. At least it's not as bad a bet as it's often made out to be.
Let's look at some of the common anti-lottery arguments, and why they're not particularly strong. Oh, and before my credibility dips all the way down to zero, I should say that I am in no way employed by any lottery organization, and actually I've never even purchased a ticket. Feel better? Moving on.
The lottery is a tax on ignorance.
This is trotted out pretty frequently, I suspect, because (a) it's pithy and quotable, and (b) it pretty much ends any meaningful discussion on the matter. It's like calling religion "the opiate of the masses." It seeks to make the opposing position seem automatically ridiculous. How can you have a reasoned debate after that? You can't, really, which is one good reason to dismiss this kind of argumentation out-of-hand. But besides belonging to a class of bad arguments, this particular one is awfully thin. Maybe it wasn't always, but it is now.
Calling the lottery a "tax on ignorance" is like putting warning labels on cigarettes. There was a time when the public was legitimately unaware of the dangers of smoking. Hell, doctors even sponsored cigarette brands. And when the damaging effects of tobacco came to light, people needed to be warned that it was, in fact, not such a wonderful idea to use it. Hence, warning labels. But in 2012, they're redundant. I just can't believe that there is one person in this country who has been to even a single day of public school who thinks that cigarettes are safe. Ever surprise a smoker by telling him that cigarettes will kill him? Didn't think so.
Ever surprise a lotto player by telling him he'll never win? Didn't think so. No one is ignorant of the astronomical odds against hitting the jackpot. Maybe there was a time when people were being duped, but that time has long passed. We can certainly talk about whether it's okay for the government to make money off of people's hopes and dreams of a fantasy life, but let's not pretend that people are unaware of the fantasy.
You can take the money you would have spent on lottery tickets and invest it instead.
Let's ignore, for a moment, that a savings account is currently losing you about 2% per year (in real dollars), and that an index fund over the past five years has lost you something like 8% per year. I mean, those quote investments are certainly still better than the lottery, which loses you just shy of 100% per year. But why are we mentioning the lottery in the same breath as investments, anyway?
Let's reformulate the above heading: "You can take the money you would have spent on X and invest it instead." And that sentence is true for anything you happen to spend your money on. Why are lotto tickets so special? In what sense is buying a lottery ticket more a waste of money than buying a King Size Snickers? Neither one of those things gives you a monetary return on your investment. But of course that's not what we expect out of a Snickers. And, I submit, it's not really what we expect out of a lottery ticket, either. What we expect out of both of those purchases is utility. And obviously there is some utility to be had in both cases (about a dollar's worth), since people are willing to pay it. Why pay a dollar to make you fatter and increase your risk of diabetes? I don't know. Why pay a dollar to spend a few days wistfully imagining a life that includes indoor hot tubs? I don't know. They're equally silly, and offer roughly equivalent utility to a great many people.
Purchasing a lottery ticket has a negative expectation.
This is my favorite mathematical argument, because it's terrible. I will grant you that this is almost always (but not quite) true. A $1 lottery purchase normally has an expected value of very nearly -$1. When the jackpot gets very large, the expectation becomes slightly less negative, and when the jackpot gets hugely large, the expectation might even creep into the black. But all of that is really beside the point.
First of all, the negative expected value is very tiny. For most people who buy lottery tickets, that expenditure is trivial. I certainly waste way, way more money per week in buying "sure things" than most of the lotto faithful do in gambles. But that's not really the point, either.
The point is this. Saying you shouldn't make any gambles with negative expected value is to tacitly imply that you should also be in favor of gambles with positive expectation and be indifferent to gambles with zero expectation. This would make you a completely rational actor. It would also make you a complete idiot.
Are you indifferent to betting $100,000 with me on the flip of a fair coin? I'm sure as hell not! I'll do you one better: I'll pay you $1.3 million dollars if it comes up heads, and you pay me $1 million even if it comes up tails. Now that bet has a positive expected value for you...wanna gamble? Of course you don't. Monetary expectation has almost nothing to do with your willingness to engage in risk. It's your expected utility that you're worried about, and utility is not linear with money. Over small intervals of the domain, it might be approximately linear, and so it's tempting to equate the two, but they're very different globally, as our coin-flipping gambles show. A dollar's worth of utility lost is absolutely trivial (to me), but the potential utility that comes with hundred of millions of dollars, even with very small probability, more than counters that loss. I'm basically free-rolling: paying nothing for the chance at something. In other words it's possible (even normal) for me to have a negative expectation in money, but a positive expectation in utility. And that's the only expectation that really matters.
Of course for some people the lottery is terrible. People have gambling problems. People spend way too much money on all kinds of things they probably shouldn't. But that doesn't mean that everyone---or even most people---that play are suckers. Eating the occasional King Size Snickers probably won't get your foot chopped off; smoking the occasional cigarette probably won't kill you (sorry, kids), and buying the occasional lottery ticket will likely have about zero net impact on your finances. Besides, isn't it worth it to dream, for even a day, of having indoor hot tubs? They're so bubbly.