October 31, 2018

Investment Theory

## Reflections on the Mega Millions Lottery: Can a Lottery Ticket with Great Odds Still be a Bad Bet?

*By Victor Haghani and James White* ^{1}

Last week, two of our favorite financial journalists – the WSJ’s Jason Zweig and Bloomberg’s Matt Levine – covered the Mega Millions lottery featuring an eye-catching jackpot of $1.54 billion.^{2} Jason’s What Investors Can Learn from Gamblers suggested that we use this lottery as a case study to help us make better investment decisions more generally, and that’s our intent as well. Matt’s humorous take in It’s OK to Get Distracted by Mega Millions is to “live a little!” and “If you are a hardworking hedge-fund analyst desperate to quit your daily grind, it’s okay to spend this morning buying lottery tickets and the rest of today and tomorrow fantasizing about what you’ll say to your boss when you win.” He posted a spreadsheet that calculated the Expected Value of a ticket, and took a flier himself (boringly, we didn’t). Taken together, these articles suggest that if the Expected Value of the ticket is less than what you must pay for it, then you should pass…unless you get a net hedonic kick from the experience of buying it.

But what’s left unexplored is this: what if the Expected Value is actually greater than the cost? This isn’t an entirely hypothetical question, as it’s sometimes the case that the combination of rapid growth of a rolling jackpot and the tendency of many players to crowd around certain numbers and patterns results in the opportunity to play the lottery with a positive edge.^{3} What should you do then?

Well, it depends. Specifically, it depends on how much the odds are in your favor, how much the ticket costs relative to your wealth, and the extent to which increases in your wealth have diminishing marginal value to you. This “diminishing marginal value of wealth,” encoded in what financial economists call your Utility function, is at the crux of the decision, and it’s what makes Expected Utility a better decision-metric than Expected Value. Using Expected Value implicitly assumes that making $10 feels just as good as losing $10 feels bad, which just isn’t how it works for most people.^{4} Using utility preferences instead of monetary values takes account of the asymmetry between winning and losing (imperfectly in many cases) but it usually gets us in the ballpark of the right decision.^{5}

Let’s illustrate with an example: say you have $100,000 of savings, and that you’re very tolerant of financial risk.^{6} To make our point really clear, let’s also say that a $10 ticket gives you a probability of winning the $1.5 billion jackpot 100 times more favorable than the fair odds. In other words, the Expected Value of the ticket is $1,000 but you get to buy it for just $10.

You know you’ve got a great deal based on Expected Value, but what about Expected Utility? Well, despite the Expected Value of the ticket being 100x its face amount, the Expected Utility of buying the ticket is negative – in fact, *really* negative.^{7} You’d need to be able to buy the ticket for less than $0.65 to have a good investment from an Expected Utility perspective, ignoring its fantasy-fun-value. Even if you had $1 million of savings instead of $100,000, you’d still find buying the ticket for $10 a negative Expected Utility proposition.

Is this something we should pay attention to, or is there a problem with Utility theory here? We’re paying attention: Expected Utility puts weights on the almost-certain loss versus the almost-impossible huge gain in a way that is both intuitively appealing (to us, at least) and theoretically sound. Sure, if you could play this lottery at these super-favorable odds ten million times, you’d have a 99.9% chance of winning at least once…but that’s not what you’re faced with here, not to mention a few practical problems like coming up with $100 million of cash to buy the tickets. Also, the utility-based decision rule isn’t saying that you shouldn’t play the lottery when the odds are in your favor, it’s just saying that $10 is too big of a bet relative to $100,000 or even $1,000,000 of wealth. Your optimal bet size, with those great odds and $100,000 of wealth, would be to spend about a dime on a ticket.

We agree with Matt that we shouldn’t overthink what we do with $10 for a lottery ticket.^{8} After all, as he put it, “Ten bucks won’t even get you into a movie, and isn’t this more fun to think about?” But there are investments out there that can look like a lot like lottery tickets, with positive Expected Value arising from a small probability of a big payout and a high probability of a relatively small loss. Viewed through the lens of Expected Value, we might be tempted to commit some serious dough to these lottery-ticket style investments. That’s when we need to coolly pull Expected Utility out of our toolkit to get a clearer picture of an investment’s attractiveness and how much of it to take on board.

- Victor is the Founder and CIO of Elm Partners, and James is Elm’s CEO.
**Past returns are not indicative of future performance.**This not is not an offer or solicitation to invest. - Or a lump sum cash value of $878 million. $1.54 billion was the estimated annuity value. Both are pre-tax figures.
- For a more detailed description of these positive odds opportunities, or in their words, “fantastic expected rates of return offered by certain lottery drawings”, and also a more detailed discussion of the ideas in this note, see “Finding Good Bets in the Lottery and Why You Shouldn’t Take Them,” by Aaron Abrams and Skip Garibaldi, Mathematical Association of America (2010), which won the Lester R. Ford Award for outstanding papers in mathematics in 2011.
- We are skeptical of assertions that some people have utility curves that display an increasing marginal utility of wealth arising from small increases in wealth being transformational in terms of a person’s quality of life, at least in the case of the financial circumstances of the typical reader of our research notes.
- This is all in the literature. For a more in depth discussion, see our note: A Sharper Lens for Sizing Up Nickels and Steamrollers.
- We’ll assume your utility, or loosely speaking your happiness, goes up with the natural log of your wealth, which is to say you exhibit log-utility,
*U(W) = ln(W)*. This is a very risk tolerant utility versus wealth schedule, at least according to a survey we recently conducted and wrote about. In our survey, we didn’t find any of our respondents nearly that risk-tolerant. Log utility would suggest that you’d take a 50/50 gamble, where you double your wealth on heads versus losing 49% of it on tails, a gamble which most people wouldn’t think twice to turn down – especially for people who are older and have more financial savings versus their human capital. - Here’s the calculation: the utility of his current $100,000 of wealth is
*ln(100,000) = 11.5129*. If he wins the lottery, his utility goes up to*ln(1,500,099,990)*and if he loses the lottery his utility goes to*ln(99,990)*. The expected value of a 1:1,500,000 chance of winning the lottery and 1,499,999:1,500,000 of not winning is 11.5128, which is lower than 11.5129, so buying that ticket is a negative Expected Utility decision. In fact, we can figure out just how bad it is, and the answer is that it’s like having $9 less wealth to start with. - It’s been observed that people do weird things with small sums. So, someone with $100,000 of wealth might turn down a fair coin flip that’ll win $110 on heads versus lose $100 on tails, even though all that would be needed, even for a pretty risk-averse investor (e.g. the average from our survey), is better than win $100.31 versus lose $100. It’s fascinating that people tend to play the lottery when they shouldn’t, but then refuse symmetric but small favorable bets when they should take them.