Monday, November 10, 2014

A lottery is a tax on... people who are good at reasoning about risk-adjusted returns?

Rescued from the drafts folder because John Oliver has rendered it timely.

People who consider themselves smart sometimes joke that a lottery is a tax on people who are bad at math.

The root of this reasoning is that the expected return on investment for a lottery ticket is negative. It can't help but be otherwise: the lottery turns a profit, hence the payout multiplied by the probability of winning must be less than the price of the ticket. Therefore, the reasoning goes, the people who buy lottery tickets must be incapable of figuring this out. Ha ha, let us laugh at the rubes and congratulate ourselves on our superiority.

One rejoinder is that the true value of a lottery ticket, to the buyer, is the entertainment value of the fantasy of winning. One obtains the fantasy with probability 1 and thus as long as the entertainment value of this fantasy exceeds the price of the ticket, the buyer comes out ahead.

There is something to this. But I think a stronger claim can be made.

A lottery is the mirror image of catastrophe insurance. Note that buying insurance also has a negative expected return. Provided their actuarial tables are accurate, insurance companies turn a profit. Therefore the probability of being compensated for your loss, multiplied by the compensation, must be less than the cost of the insurance premiums. But nobody says that insurance is a tax on people who are bad at math. Quite the opposite: buying insurance is viewed as a sign of prudence.

The issue, of course, is that the naive expected return calculation fails to adequately consider the nature of risk and catastrophe. At certain thresholds, financial consequences as experienced by human beings become strongly nonlinear, probably due to the declining marginal utility of money. Suffering complete ruin due to, say, a car accident entails such severe consequences that we are willing to accept a modestly negative expected return in exchange for the assurance that it will simply never occur.

A lottery is simply the flip side of this coin. The extremely rare event is a hugely positive one, instead of a hugely negative one, huge enough to produce qualitative rather than merely quantitative changes to your lifestyle. And one accepts a modestly negative expected return, not so that one can avoid the risk, but that one can be exposed to the risk.

If you are a member of the educated, affluent middle class, there is an excellent chance that your instinct rebels, and you're already hunting for the flaw in this reasoning. Surely there's something sad, and not prudent, about those largely working-class souls who buy a lottery ticket every week, rather than rolling that $52 per year into a safe index fund at a post-tax rate of return of roughly 2.5% per annum, at which rate their investment, if it somehow survived unperturbed the ups and downs of a life that is considerably more exposed to financial risk than a middle-class person's, might compound to the princely sum of a couple months' rent by the time they die, or alternatively enough to pay for a slightly nicer coffin.

And maybe there is a flaw in this reasoning. I'm not completely convinced myself and I'm not going to start buying lottery tickets. But I'm honestly having trouble finding the flaw. Any indictment of spending (small amounts of) money on lottery tickets must surely also apply to buying insurance. If it is worth overpaying a little bit to eliminate the possibility of a hugely negative outcome, surely it is worth overpaying a little bit to create the possibility of a hugely positive outcome. The situations are symmetric, and I think one can only break the symmetry by admitting the validity of loss-aversion (usually viewed as irrational by economists) or something similar.

Alternatively, if you have access to the actuarial tables of your insurance company, then perhaps you can argue that a lottery is, quantitatively, simply a worse deal than your insurance typically is... but you probably don't have access to those actuarial tables. And I sincerely doubt that the widespread middle-class snobbery towards lottery players is based on quantitative calculations of this sort. (Actually, I strongly suspect that it is based on a fallacious, gut-instinct "folk probability" feeling that gambling on any extremely remote event, like winning the lottery, is somehow inherently foolish.)

So what's the flaw? I ask this question non-rhetorically; that is, I am genuinely curious about the answer.

p.s. None of the above, of course, is to say that I think the taxation effect of lotteries, which is staggeringly regressive, is a good thing. I would strongly support the replacement of lotteries with progressive tax increases plus transfer payments! Gambling addiction is bad too. But these things seem distinct from the notion that playing the lottery in small amounts is irrational in some game-theoretic sense.

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