Lotteries: A sucker’s game or a rational choice?

2009 May 31
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By David Annis
Article ID: 1323

I enjoy playing the lottery. But I often hear it described as a tax on those who can’t calculate the odds.  I think that view is wrong for four reasons.

The first reason that the lottery is worth paying money for is because of the highly positive-skewed outcome.

The premise of the argument that lotteries are a tax on the stupid, that paying for something with a negative expected financial return is one that its proponents apply only to the lottery.  Many things we buy are bad deals: movie theater popcorn is far more expensive than anywhere else, yet we don’t think of movie theater popcorn as a tax on those not smart enough to feed themselves at home.

This could be you.

One could argue that lottery tickets – unlike popcorn – return only monetary rewards. However, lottery tickets are not the only thing that we buy which pay back money and have negative expected return.   Every year I pay for homeowner’s insurance, knowing that the odds are stacked against me. In fact, homeowners would be better off investing their insurance payments and taking their chances.  Since most homeowners realize that part of every dollar sent to the insurance company goes to administration and corporate profits, why do they continue to pay for it?  It’s because humans have a large aversion for highly negatively skewed outcomes (like the small probability that your house burns to the ground) and a strong desire for highly positively skewed outcomes (like winning the lottery jackpot).

Those who rail against the lottery seldom rail against homeowners insurance, despite the fact that it too has a negative expected return.  If I can pay a premium for a set of outcomes that eliminated a negative skew, why can’t I pay a premium for a highly positively-skewed outcome?

The second reason that the lottery is worth paying for is entertainment value.

Playing the lottery can be as entertaining as a movie, at a fraction of the price.  Thinking about the gifts I’ll get my wife, the college libraries I’ll endow to ensure my kids get into a good school, and the new car I’ll buy gives me great pleasure.

The third reasons are civic duty and charitable giving.

Lottery revenues are often used to fund education, a worthy goal.  I donate regularly to the schools through various fundraisers and through the Okemos Education Foundation.  The value of the tax deduction I get is less than the typical 50% of revenues that are used for prizes in a lottery.  Why shouldn’t I give through the lottery as well?

Finally, the fourth reason is that the lottery is sometimes a good deal.

Let’s take as an example Michigan’s Lotto 47. In this game six numbers between 1 and 47 are randomly selected. The goal is to match three or more of these six numbers.  Tickets cost $1 USD and prizes and odds are detailed below.

Numbers Matched Odds Prize Value
6 1 in 10,737,573 Jackpot
5 1 in 43,649 $2,500 $0.057
4 1 in 873 $100 $0.115
3 1 in 50 $5 $0.10

Ignoring the jackpot for now, we get roughly 26% of every dollar back in prizes.  So, when the jackpot reaches a level that returns more than 74 cents per dollar we would be wise to invest.  That occurs when the present value of the jackpot reaches $7,945,804.02.  We need to adjust the advertised jackpot for two things; to adjust the prize to the present value of the annuity payments and the chance that we split the prize with another lucky winner.

As a rule of thumb, the present value of a lottery annuity is roughly half the total value of the payments, so the break-even prize if you were guaranteed not to split the prize would be $16,000,000.

Calculating the odds of splitting the prize is difficult, because the number of bets varies over time and is related to the size of the prize.  We can, however, approximate the odds of splitting the prize.  In 2008, total sales of Lotto 47 tickets were  64,129,000 or  616,625 per drawing.  If bets were evenly distributed across drawings, we’d have a 616,625/10,737,573 or 5.74% chance of splitting the prize. As the prize grows, more people bet. So to be safe, let’s add 10% to the break-even prize.  Using these rough odds, it pays to buy a ticket when the prize exceeds $17.6 million.

So, the next time someone claims that the lotto is a tax on ignorance, ask them if – taking into account charitable contributions, entertainment value, and the value of positive skew, how big would a jackpot need to grow before they would consider playing the lottery. That cost is as rational as any other.


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12 Comments
2009 May 31
Sal Governale permalink

This was retarded. What’s next? A skeptical look at whether or not a tramp stamp tattoo is a wise choice?

2009 June 1
Navin Kumar permalink

Chill Sal,

It’s articles like this that point out that one shouldn’t be only skeptical of Canonized Debate Topics. It’s worth remembering that there’s hidden value everywhere.

Fun article.

2009 June 1
Andy Kaiser permalink

In counterpoint to Sal, I liked this one a lot. I’d always been exactly like David described at the beginning – thinking that those who play lotteries are paying a “stupid tax”.

This article changed my mind.

It was, frankly, enlightening to consider all points of playing a lotto, including odds calculation for when it’s mathematically a good deal to play.

Andy

2009 June 1
David Annis permalink

Sal,

I’m sorry that you didn’t find the article amusing. I wrote it because I have been told on more than one occasion that I am stupid to play the lottery and because I believe that it is important to challenge conventional wisdom about things other than religion and hostility to science.

Not that I think that the article is perfect; I really simplified the calculation of the odds of splitting a prize – because there are a huge number of variables that go into accurately calculating the real odds. To do it right, you need to account for increasing sales as jackpot sizes increase, the size of prizes in alternate games, the fact that humans don’t really pick numbers randomly, and many other variables.

David

2009 June 1
Rob Steenwyk permalink

Sal, I hardly find this article retarded, I find it very interesting. In my opinion, this site should not just be about showing the implausibility of Big Foot, but also about the every day things we don’t even think to question. There are thousands of articles out there on why the evidence supporting Big Foot (or insert big topic here) is not valid, but taking a look at the lotto? Not so much.

Excellent work David, keep it up.

2009 June 2
dave permalink

“Using these rough odds, it pays to buy a ticket when the prize exceeds $17.6 million.”

Did you adjust for taxation?

2009 June 2
David Annis permalink

In theory, it doesn’t matter – gambling losses are deductible, so if we played all of the numbers at a break even jackpot, the losses and the gain would cancel out – net tax liability zero. In practice, I’d like to have the problem of only having bought one ticket and having the tax liability for the jackpot.

David

2009 June 3
David Annis permalink

Dave,

My answer above was a bit flippant. It assumed that you bought one of each ticket in a break even drawing and deducted the losses on the losing tickets. The truth is that the real world is more complex, so you have a valid point. I’ve tried to do a quick set of rough calculations below that account for income taxes.

In the real world, I am unlikely to play all numbers in a drawing. Gambling losses are only deductible only to the extent that they are offset by winnings and losses can not be carried forward to a future year. So, in any given year I’ll not be able to deduct most of my losses. The truth is that I don’t deduct any of my losses – perhaps I should.

Winnings under $600 aren’t reported to the IRS and nobody claims them, even though we are supposed to. So, we can assume for a typical ticket 21.5% is returned tax free. That leaves 78.5% that will be taxed. If we assume that we pay roughly 33% in income taxes that means we lose 25.9% of wagers to taxes – making the break even jackpot rise to about $23.8 million.

David

2009 June 5
Humanity Blues permalink

Some well put points. That being said, and I don’t want to come off as judgmental here, but I notice a lot of poor people at gas stations waste $20 dollars at a time buying lotto tickets and I’m pretty sure they’re wasting their money. Regardless, I guess it’s their choice. I buy a lottery ticket about once a month. It’s funny. I always feel deep down I’m gonna win even though there’s such a horrible probability in my favor.

2009 June 5
David Annis permalink

Humanity,

I understand the concern that the lottery is essentially a regressive tax.

I would like to see better investment vehicles made available and advertised to the poor so they don’t use the lottery as an alternative to traditional investment vehicles.

However, I don’t want to patronize the poor by shielding them from bad choices. Certainly they buy all sorts of things that I would not approve of: toy guns, cigarettes, whiskey, real guns, Twinkies, and Frosted Flakes, but we can’t all spend ideally. We all make many bad choices in life. In addition to my lotto habit I have a weakness for Little Debbie Nutty bars and I spend money on toys my kids don’t really need. If I were poor I’d want to be able to continue to indulge my weaknesses.

David

2009 June 7
noisician permalink

my 2 cents:
—
sure, i grant #4 – there are sometimes unusual situations where the odds turn in the gambler’s favor. fine, “professional” gamblers who work out the odds, game the system as much as possible, and buy tickets only when the odds are in their favor (and possibly buy large blocks of tickets to cover every #) don’t fall into the “tax on the stupid” category. but this person is not who we’re talking about. he is not the typical lottery player or a person who buys tickets on a regular basis.
—
the rest of the arguments are even less convincing.
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#1 – even if we grant your assertion that insurance is a bad investment, this doesn’t somehow make gambling on lottery tickets a good investment! you MIGHT have an argument that insurance is as stupid as the lottery, but you’d need to present more info.
—
#2 – please. this is why lottery is called tax on the stupid. people start thinking about spending all that prize money and they get stupid! … if you’re into wishful thinking, i’ve got another great way for you to spend your money: go to church every Sunday and donate whatever you didn’t gamble away, and then think about all the good blessings the magic sky fairy will be sending your way! entertaining!
—
#3 – this is a rationalization to make you feel better about your poor decision to blow a lot of cash on gambling, NOT a motivation to gamble.
—
(sorry if i came off harsh, i’m just cranky tonight.)
—

2009 August 28
David Annis permalink

Tonight’s Mega Millions is a case where it seems the odds are in the player’s favor.
“The odds of winning the prize, the second-biggest in North American lottery history, are about 1 in 176 million. It’s worth about $210.4 million if the winner chooses to be paid in one lump sum instead of annual installments.”
(from http://www.bloomberg.com/apps/news?pid=20601103&sid=aitwmwHXxGpU )
David

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