Gun Control, Part 3: Back to the Present

This is the third installment in a series on the gun control issue. In our first episode we examined data that demonstrate a correlation between fewer guns and an increased variance in the homicide rate. In the second we explored game theory as a hypothesis to explain this phenomenon. This time around we’ll look into another hypothesis, one that can explain why past gun control laws have failed to keep guns out of the hands of criminals and what this means for future legislative efforts.

To begin, we have to go back to school. Way back. You’re a four-year old living in the Bay Area in 1972. You are friends with a girl whose dad is a professor at Stanford. His name is Walter Mischel, and you are about to help him understand something that will inform our conversation on this issue.

One day your parents drop you off at Dr. Mischel’s lab for play time. After some opening activities, you find yourself alone with a researcher, sitting at a kid-sized table in an otherwise empty room, staring down a giant marshmallow. You tell the researcher that you like marshmallows very much. She smiles and says that the marshmallow is yours, and you can eat if you want to.

But wait, there’s more: She says she has to leave the room for about 15 minutes, and if you haven’t eaten your marshmallow when she returns, she will give you another one, too! The researcher exits the room and you are left alone with your marshmallow, weighing your options.

You’re a precocious little preschooler, future Stanford material, so you calculate your rate of return on this trade:

  • You have one marshmallow
  • If you invest that marshmallow for 15 minutes in this grad student’s scheme, you will earn a second marshmallow
  • Your return on investment, therefore, is 1 marshmallow/1 marshmallow = 100%
  • Your rate of return is 100%/15 minutes, or a whopping (24 – 1) x 100 = 1500% per hour! At that rate you’d score more marshmallows than there are atoms in the observable universe in a little more than three days, assuming you could keep the grad student on the hook that long.

Fifteen minutes later the researcher returns and sees you rubbing your little hands together with glee, marshmallow still on the table. “How about double-or-nothing?” you eagerly ask. You’ve decided that you don’t need Stanford after all. You’re on your way to starting your own hedge fund!

Your less-strategic friends didn’t fare so well in the same study. A few of them ate the marshmallow right away. Others made an effort to delay eating the marshmallow but just couldn’t pull it off: Fewer than 1/3 of the study participants were able to wait the full 15 minutes and earn the second marshmallow.

You can read more about the marshmallow experiment here.

Let’s jump ahead a few decades and play a grown-up version of this game, called Zero-Coupon Bond. It works like this:

I need to raise some money, and you happen to have some money. I promise that I can pay you $100 one year from today, but I can’t pay anything until then. How much money will you lend me today in exchange for $100 one year out?

Let’s leave creditworthiness out of the question, I have plenty of assets for collateral. The real issue for you is the opportunity cost of your money, i.e. the return you could make on the money you lend me if you were to invest it somewhere else.

Let’s say you offer to lend me $90 today in exchange for $100 a year from now. I calculate that the cost of that capital is ($100 – $90) / $90 = $10 / $90 = 11.1% per year.

“Dude,” I object. “That’s, like, 1100 bips higher than the 1-year T-Bill rate. What kind of friend are you?”

You admit that the rate is a bit loan-sharky. You blame your experience at Stanford 41 years ago for your unrealistic expected rates of return. We agree to split the difference: You loan me $95 today, I will pay you $100 a year from now. That’s an interest rate of $5 / $95, or 5.26% per year. Not great, but at least I didn’t have to pay an entire universe of marshmallows.

We call this a zero-coupon bond because it is a bond (a promise to pay) that features no coupon payments, regular payments of interest (like you pay on your mortgage) before the bond matures. The principal ($95) is paid back with interest ($5) in a single payment at maturity. And we like zero-coupon bonds because they are really simple instruments that let us experiment on the relationship between present value and future value. They’re the financial equivalents of lab mice.

So here’s a thought experiment: What if you had priced the bond at only $94? Maybe I wouldn’t have sold you my promise to pay. What if I had asked $96? Maybe you wouldn’t have bought it. It turns out that $95 was a very special price for us: You and I were both at least indifferent to $95 today or $100 a year from now, and so the deal was done.

But it’s not $5 that are important, rather the interest rate those dollars represent. This rate, 5.26% per year in our example, helps us establish a relationship between present value and future value. In finance we refer to this as a discount rate, the rate at which we discount future value to bring it back to the present so we can compare what we spend today with what we will earn in the future, or vice-versa. The higher the discount rate, the lower the present value of future cash flows.

And this discount rate has a special place in decision science. In a business the discount rate for future cash flows is the firm’s cost of capital. A business that raises capital at a cost of 12% will not undertake projects that yield future returns of less than 12% per year if its managers are rational. The business is indifferent to projects that yield exactly the discount rate and it will generally consider investing in projects whose yield exceeds the discount rate. Managers sometimes refer to the discount rate as the “hurdle rate”, because it is the rate of return that an investment needs to “clear” in order to be considered.

It turns out that each of us humans is running around with our own individual hurdle rate in our head. We are faced with a constant stream of decisions that trade off future vs. present rewards:

  • Do I sleep in, or do I wake up and go to work? Sleeping in pays off now, work pays off later.
  • Do I have dessert? That would definitely pay off now, but skipping it generally pays off later.
  • Do I smoke the next cigarette? Smoking might pay off now, not smoking pays off later.
  • Do I lie, or do I tell the truth? Lying pays off now, telling the truth pays off later.
  • Do I contribute to my 401(k) with each paycheck, or do I spend that money on entertainment? You get the idea.

For those decisions that clear your individual hurdle rate, you tend to choose the future reward. For those that don’t clear the hurdle rate, you tend to choose the present reward.

So what is the range of hurdle rates that we encounter in society?

It’s huge. Frighteningly so.

Recall the marshmallow experiment above: A return of the entire universe in 3 days failed to clear the hurdle rate of more than 2/3 of preschoolers. Children in general have incomprehensibly high hurdle rates for future rewards as most parents will attest.

Drug addicts also have high discount rates when compared with the general population. Many acknowledge that they take enormous risks in order to get their next fix, then do it anyway, over and over again.

Persons who are mentally ill exhibit impaired decision-making similar to that of substance abusers.

And these three classes of people are disproportionately represented in the US prison population:

It seems safe to assume that that the discount rate among those who end up incarcerated is quite high, meaning that criminals likely to choose present rewards over future rewards, every time.

Here’s an illustrative example:

Going back to our robbery vignette from episode 2, let’s consider that you have $100 in your pocket and I have nothing in mine. I can choose to rob you and get $100 right now. I know that I have a pretty good chance of getting your money even if you put up a fight or try to flee.

I also know that if I commit a robbery, I run the risk of getting caught and going to jail for, say, a year. For the sake of easy math, I’m going to value my freedom at $100,000 per year, or the opportunity cost of all the fun I could have during that time as a free man.

So assuming I pull off the robbery and get caught, my payout would look like this:

Gun Control 3-1

I get $100 today and lose $100,000 over the next year. Pretty lousy deal at face value, isn’t it? Things don’t even look very good in present-value terms for a rational person whose discount rate is, say, 10%:

Gun Control 3-2

The opportunity cost of the first year of future incarceration is discounted by 10% (divided by 1 + 10% = 1.1)  to arrive at a present value of -$90,909. The “net present value” or NPV is the sum of what I get immediately ($100) and the discounted future value of what that decision costs me over time ($90,909). So the NPV of a decision to rob you is $100 + -$90,909 = -$90,809. If I think like most adults, there’s no way I’m going to risk the robbery. Getting caught would be just too expensive.

But let’s use this model to ask a rather interesting question: How high would my discount rate need to be to justify the robbery? In other words, how much would I have to discount the future punishment in order to make robbing you today seem like a good idea?

It turns out that if my discount rate was 99,900% per year, I’d be indifferent to robbing you given the payouts above. That rate seems astronomically high, doesn’t it? If a person with that kind of discount rate was your lender in the Zero Coupon Bond game above, they’d insist on lending you only $0.10 today in exchange for a $100 payment one year from now. If a business used that discount rate to allocate capital, it would invest only if it could double its money in about a month.

But a discount rate of 99,900% per year equates to about 2% per day, or 0.08% per hour, about 1.8 million times less than the discount rate of most preschoolers as measured in the Stanford marshmallow experiment. The disturbing conclusion is that this simple crime would absolutely pay off in NPV terms for someone with the discount rate of a child, or a drug addict, or one who is mentally ill. In other words, for precisely the classes of people who tend to end up in prison in the United States.

We who consider ourselves to be rational adults struggle to understand this fact, but that doesn’t stop it from scaring us. And we respond like we’re dealing with rational adults: If 1 year in prison for robbery doesn’t deter the crime, let’s make it 2 years.

But how does a criminal view another year in prison? We’ll keep the other terms (amount of money stolen, opportunity cost of a year in jail, discount rate) of the deal the same:

Gun Control 3-3

Wait a minute, what happened here? The NPV is now $100 + -$100 + $0 = $0, and the perpetrator is still indifferent to the crime!

Perhaps we “rational” adults forgot that discount rates, like other interest rates, compound exponentially over time. We discount the Year 1 punishment back to present value by dividing by 1 + the discount rate, or 1 + 99,900% = 1000. We calculate the PV of the second year of prison by dividing its opportunity cost by 1 + the discount rate squared, or 1,000,000. And for the third year we will divide by 1 + the discount rate cubed, or 1,000,000,000, etc.

So if the perp’s discount rate wasn’t big enough to discourage the crime given a 1-year sentence, adding more years isn’t going to make a bit of difference. You could threaten him with an automatic life sentence, and the crime would still pay off in NPV terms.

The threat of additional incarceration does nothing to dissuade those who are already predisposed to view crime as a good investment. This idea is counter-intuitive for those of us who have reasonable discount rates and are therefore motivated to stay out of jail, we naturally view obeying the law as a better investment. Incarceration thus becomes a means of removing the high-discount-rate persons from society. And there sure seem to be a lot of them lately:



What’s causing this? And what can be done to change the status quo?

First, we need to recognize that even those with the highest discount rates for future consequences cannot discount immediate ones. Consider the internet café armed robbery video we discussed in the last episode: The bad guys viewed robbing the café and its patrons as a good investment in spite of the high likelihood that they would be incarcerated. However, they quickly (and dramatically) changed their minds when faced with the threat of immediate death. Their discount rate didn’t change, but the consequences were moved out of the future and into the present where the discount rate is irrelevant. Criminals respond better to immediate consequences than to future ones.

Second, we need to recognize and begin to address the futility of the growing mountain of new laws that are aimed exclusively at people who have no intention of obeying existing ones. These legislative actions seem to be popular and therefore help our officials justify their bids for re-election, but do they do any real good? And how do we calculate the cost of already-law-abiding people having to navigate an endless web of rules?

Finally, we can go back to the Stanford marshmallow experiment and internalize a few of its findings from follow-up studies of the same participants:

  • Having a low discount rate in preschool (i.e. tending to choose future rather than present rewards) was correlated with being described by parents and peers as more competent ten years later.
  • Choosing future rewards in preschool was correlated with achieving a higher SAT score in high school.
  • A 2011 study of the same participants indicates that the ability to choose future rewards over present ones remains with a person for life. Brain imaging showed key differences between the marshmallow-eaters and non-eaters in areas linked to decision-making and addictions.

Most importantly, a similar study in 2012 found an underlying factor that significantly affected the subjects’ ability to choose the future reward: The participants were divided into two groups, one that was primed with a broken promise before being presented with the first marshmallow (the unreliable tester group), and one that was primed with a fulfilled promise (the reliable tester group). Subjects in the reliable tester group went four times longer before eating the marshmallow as compared with subjects in the unreliable tester group.

Whoa there, pardner.

Are you saying that the way a preschooler makes decisions is correlated with the way he will make decisions as an adult?


And are you saying that these behaviors in preschool ultimately predict the way his adult brain will work at the cellular level?


And you’re saying that, while he’s still in preschool and this behavior is not yet hard-wired into his neurons, this tendency to choose future rewards — that will make him more competent, more successful, and even keep him out of jail — can be increased by something as simple as keeping your promises to him?

That’s exactly right.

But none of these facts will help anyone get re-elected. And so this time around we’ll get a new set of gun control laws, which will be obeyed by the law-abiding citizens and completely disregarded by the criminals. All perfectly rational, of course.

3 thoughts on “Gun Control, Part 3: Back to the Present

  1. Zach, a wonderful discussion on the futility of rational politicians holding office. So yes we will have more laws and we will still wonder why they don’t work. Thanks for sharing.

  2. Mathew23:4 about summed up a politicians actions !
    and to confirm this
    when i hear politicians say or brag abut the laws they passed , i think f how law laden we have become !

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