An Introduction To Hyperbolic Discounting

Part Of: [Breakdown of Will] sequence

Table Of Contents

  • What Is Akrasia?
  • Utility Curves, In 200 Words Or Less!
  • Choosing Marshmallows
  • Devil In The (Hyperbolic) Details
  • The Self As A Population
  • Takeaways

What Is Akrasia?

Do you agree or disagree with the following?

In a prosperous society, most misery is self-inflicted. We smoke, eat and drink to excess, and become addicted to drugs, gambling, credit card abuse, destructive emotional relationships, and simple procrastination, usually while attempting not to do so.

It would seem that behavior contradicting one’s own desires is, at least, a frustratingly common human experience. Aristotle called this kind of experience akrasia. Here’s the apostle Paul’s description:

I do not understand what I do. For what I want to do I do not do, but what I hate I do. (Romans 7:15)

The phenomenon of akrasia, and the entire subject of willpower generally, is controversial (a biasing attractor). Nevertheless, both its description and underlying mechanisms are empirically tractable. Let us now proceed to help Paul understand, from a cognitive perspective, the contradictions emerging from his brain.

We begin our journey with the economic concept of utility.

Utility Curves, In 200 Words Or Less!

Let utility here represent the strength with which a person desires a thing. This value may change over time. A utility curve, then, simply charts the relationship between utility and time. For example:

Hyperbolic- Utility Curve Outline

Let’s zoom in on this toy example, and name three temporal locations:

  • Let tbeginning represent the time I inform you about a future reward.
  • Let treward represent the time you receive the reward.
  • Let tmiddle represent some intermediate time, between the above.

Consider the case when NOW = tbeginning. At that time, we see that the choice is valued at 5 “utils”.

Hyperbolic- Utility Curve T_beginning

Consider what happens as the knife edge of the present (the red line) advances.  At NOW = tmiddle, the utility of the choice (the strength of our preference for it) doubles:

Hyperbolic- Utility Curve T_middle (2)

Increasing utility curves also go by the name discounted utility, which stems from a different view of the x-axis (at the decision point looking towards the past, or setting x to be in units of time delay). Discounted utility reflect something of human psychology: given a fixed reward, other things equal, receiving it more quickly is more valuable.

This concludes our extremely complicated foray into economic theory. 😛 As you’ll see, utility curves present a nice canvas on which we can paint human decision-making.

Choosing Marshmallows

Everyday instances of akrasia tend to be rather involved. Consider the decision to maintain destructive emotional relationships: the underlying causal graph is rather difficult to parse.

Let’s simplify. Ever heard of the Stanford Marshmallow Experiment?

In these studies, a child was offered a choice between one small reward (sometimes a marshmallow) provided immediately or two small rewards if he or she waited until the tester returned (after an absence of approximately 15 minutes). In follow-up studies, the researchers found that children who were able to wait longer for the preferred rewards tended to have better life outcomes, as measured by SAT scores, educational attainment, body mass index (BMI) and other life measures.

Naming the alternatives:

  • SS reward: Call the immediate, one-marshmallow option the SS (smaller-sooner) reward.
  • LL reward: Call the delayed, two-marshmallow option the LL (larger-later) reward.

Marshmallows are simply a playful vehicle to transport concepts. Why are we tempted to reach for SS despite knowing our long-term interests lie with LL?

Here’s one representation of the above experiment (LL is the orange curve, SS is green):

Hyperbolic- Utility Curve Two Option Choice

Our definition of utility was very simple: a measure of preference strength. This article’s model of choice will be equally straightforward: humans always select the choice with higher utility.

The option will people select? Always the orange curve. No matter how far the knife edge of the present advances, the utility of LL always exceeds that of SS:

Hyperbolic- Utility Curve Exponential Self (1)

Shockingly, economists like to model utility curves like these with mathematical formulas, rather than Google Drawings. These utility relationships can be produced with exponential functions; let us call them exponential discount curves.

Devil In The (Hyperbolic) Details

But the above utility curves are not the only one that could be implemented in the brain. Even if we held Utility(tbeginning) and Utility(treward) constant, the rate at which Utility(NOW) increases may vary. Consider what happens when most of the utility obtains close to reward-time (when the utility curves form a “hockey stick”):

Hyperbolic- Utility Curve Hyperbolic Choice (1)

Let us quickly ground this alternative in a mathematical formalism. A function that fits our “hockey stick” criteria is the hyperbolic function; so we will name the above a hyperbolic discount curve.

Notice that the above “overlap” is highly significant – it indicates different choices at different times:

Hyperbolic- Utility Curve Hyperbolic Selves (1)

This is the birthplace of akrasia – the cradle of “sin nature” – where SS (smaller-sooner) rewards temporarily outweigh LL (larger-later) rewards.

The Self As A Population

Consider the story of Odysseus and the sirens:

Odysseus was curious as to what the Sirens sang to him, and so, on the advice of Circe, he had all of his sailors plug their ears with beeswax and tie him to the mast. He ordered his men to leave him tied tightly to the mast, no matter how much he would beg. When he heard their beautiful song, he ordered the sailors to untie him but they bound him tighter.

With this powerful illustration of akrasia, we are tempted to view Odysseus as two separate people. Pre-siren Odysseus is intent on sailing past the sirens, but post-siren Odysseus is desperate to approach them. We even see pre-siren Odysseus restricting the freedoms of post-siren Odysseus…

How can identity be divided against itself? This becomes possible if we are, in part, the sum of our preferences. I am me because my utility for composing this article exceeds my utility attached to watching a football game.

Hyperbolic discounting provides a tool to quantify this concept of competing selvesConsider again the above image. The person you are between t1 and t2 makes choices differently than the You of all other times.

Another example, using this language of warfare between successive selves:

Looking at a day a month from now, I’d sooner feel awake and alive in the morning than stay up all night reading Wikipedia. But when that evening comes, it’s likely my preferences will reverse; the distance to the morning will be relatively greater, and so my happiness then will be discounted more strongly compared to my present enjoyment, and another groggy morning will await me. To my horror, my future self has different interests to my present self. Consider, too, the alcoholic who moves to a town in which alcohol is not sold, anticipating a change in desires and deliberately constraining their own future self.


  • Behavior contradicting your desires (akrasia) can be explained by appealing to the rate at which preferences diminish over time (utility discount curve).
  • A useful way of reasoning about hyperbolic discount curves is warfare between successive “yous”.

Next Up: [Willpower As Preference Bundling]


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