Self-control often depends on how we compare our options over time.

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Without free will, how might we show contrition?

Why do most people find it difficult to make a personal behavior plan and stick to it? If we’d rather not drink, why don’t we just not drink? If we’d rather not eat junk food, why don’t we just not eat junk food? If we want to get our taxes done this week, why do we keep putting it off until next week.

This essay will shed some light on this problem.

Let’s begin with two parables.

Two ParablesConsider this parable of an ancient astronomer.

Now consider a parable of a weak-willed farmer.

Weakness of Will and Preference ReversalsOur weak-willed farmer assigns less value to drinking when it is a dozen hours away, and assigns more value to drinking as it draws closer in time. This is not, in and of itself, irrational. We should value similar opportunities more when they are near and less when they are far away. 100 dollars today should be worth more to us than 100 dollars a year from now. And a bird in the hand is very often worth two in the bush.

That’s not why we judge him weak-willed.

His desire for drinking grows quite quickly as his normal drinking time approaches. But the rate at which his desire grows is not in and of itself irrational either. There is no single growth rate for valuing future outcomes that’s rational for everyone. What’s rational for a given person depends on many things. It depends on his values, how much he trusts future outcomes to be available, and even on how long he expects to live. Who are we to say how quickly the allure of drinking should grow for the farmer as he gets closer to his drinking hour?

No, what causes us to judge the farmer irrational is that the apparent value of his future opportunities do not all grow at the same rate over time.

The value of drinking grows faster than the value of waking up without a hangover. And that causes him to suffer what is known in the literature as a “preference reversal”.

At 8 A.M. he prefers waking up fresh the next day, and at 8 P.M. he prefers drinking.

How Can we Explain Preference Reversals?If we were completely rational, we would discount all our opportunities at the same rate, and we would discount according to an exponential curve. Here’s what that would look like:

In the above diagram the bottom curve represents our farmer’s preference for drinking (a “smaller, sooner”, or “SS” reward), and the top curve represents his preference for waking up fresh (a “larger, later”, or “LL” reward).

T1 is the point in time in the morning when the farmer first compares the two options. T2 is the time he normally starts drinking. And T3 is the next morning.

At T1, the farmer finds the prospect of drinking that night to be less desirable than the prospect of waking up fresh the next morning. So it does not surprise us that he would decide, in the morning, that he will not drink that night.

As we move from T1 toward T2, the prospect of drinking that night grows sharply in value. But here’s the thing: the prospect of waking up fresh grows in value just as quickly. So the ratio between the two options remains constant.

If the farmer discounted future rewards exponentially like this, he would not suffer a preference reversal. And we would not expect him to drink that night, since he would prefer to wake up fresh the next morning.

But our farmer (along with all of us, and every other kind of animal we’ve ever tested) is vulnerable to preference reversals. And that has led some psychologists and economists (e.g, Herrnstein, Ainslie) to suggest that we don’t discount future rewards according to an exponential function, but, rather, according to a hyperbolic function.

A “hyperbolic” function? What does that look like?

Hyperbolic curves are bowed more severely in the middle than exponential curves. Agents who discount hyperbolically don’t discount at the same rate over time. Instead the value assigned to a future prospect grows quite slowly at first, and then quite rapidly as the prospect draws near.

The present value of the smaller sooner reward will start spiking while the present value of the larger later reward is still just ramping up. And, if the value of the smaller sooner curve rises above the value of the larger later curve, there will be a small window of time in which the agent prefers SS to LL.

Thus, hyperbolic curves bring the potential for preference reversals.

Having a tendency toward preference reversals can be bad for us. It makes us vulnerable to addictions we’d rather not have. It means we can work toward one goal at one point in time, and completely change directions as another opportunity gets closer. And it means that other people can take advantage of us. Imagine, for instance, that the farmer has a friend who buy’s alcohol from the farmer in the morning, when the farmer values it less, and sells it back to him at night for a higher price. This friend could make a profit every day for doing next to nothing. Payday loans work the same way.

Why Hyperbolic Discount Curves?The term ‘hyperbolic discount curve’ is a mouthful. It sounds complicated, and we might wonder why nature would endow us with such a complicated way of adjusting the value of various rewards over time -- especially when it leaves us vulnerable to addictions and predatory lenders.

But hyperbolic curves are not actually complicated at all. In fact, they are among the simplest and most natural curves in the world. The most basic hyperbolic curve is just 1/x. Hyperbolic curves can be generated with simple ratios. The projection of objects on our retinas (the apparent size of objects) as we move closer to them follows a hyperbolic growth curve.

And that’s where the parable of the Sun and the Moon comes in. The apparent sizes of the Sun and the Moon grow according to hyperbolic curves as we move toward them (or shrink according to hyperbolic curves as we move away from them).

When viewed from Earth’s surface, the Sun and the Moon appear about the same size.

If we go away from them (assuming the Moon is situated between the Earth and the Sun, offset just a bit so we can see both clearly), the apparent size of the Moon will shrink faster than that of the Sun, and that will make the Sun appear larger than the Moon.

And if, instead, we travel toward them, as did the ancient astronomer in the parable, the apparent size of the Moon will grow faster than that of the Sun, and that will make the Moon appear larger than the Sun.

To show the full diagram, we would have to continue for several meters to the right, and the Sun’s apparent size would spike about 400 times higher than that of the moon. I’ll leave it to your imagination to complete the diagram.

The analogy can be made even more explicit:

Both formulas give us hyperbolic curves. And both seem like natural way to estimate relative sizes and values.

And this analogy gives us another way to look at our preference reversals. When we approach a smaller sooner reward (such as junk food, video games, or alcohol) in lieu of a larger later reward (like health, productive achievement, and waking up fresh with a healthy liver), we are like the giddy ancient astronomer choosing the Moon for his kingdom, even though, when he has more perspective, he prefers the Sun.

What Now?First, a caveat: hyperbolic curves are not the only way to model preference reversals. They have their merits. They are simple. They seem natural due to the useful analogy between visual perception and perceptions of value. And they allow economists and psychologists to make new use of all the formulas they developed using exponential discount curves. Often they can just plug in the hyperbolic formula for the exponential formula and calculate away.

But all we really need to model preference reversals is a bowed discount distribution. And there are many ways to model that. In fact, I hope to cover another plausible candidate in a future post.

But that’s all beside the point. Regardless of how we model preference reversals, a big question begs to be asked. It’s nice to understand why we’re irrational, but can we actually use this new understanding to stop being so irrational.

I think the answer is “yes”. It might not be as much as we might hope for, but any edge we can get in the battle for self-control is worth investigating.

To see how we might get started, check out this post: Will Power and Game Theory.