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]

Movement Forecast: Effective Availabilism

Table Of Contents

  • The Availability Cascade
  • Attentional Budget Ethics
  • Effective Availabilism
  • Why Quantification Matters
  • Cascade Reform Technologies
  • Takeaways

The Availability Cascade

The following questions pop up in my Facebook feed all the time.

Why is mental illness, addiction, and suicide only talked about when somebody famous succumbs to their demons?

Why do we only talk about gun control when there is a school shooting?

What is the shape of your answer? Mine begins with a hard look at the nature of attention.

Attention is a lens by which our selves perceive the world. The experience of attention is conscious. However, the control of attention – where it lands, how long it persists – is preconscious. People rarely think to themselves: “now seems an optimal time to think about gun control”. No, the topic of gun control simply appears.

When we pay attention to attention, its flaws become visible. Let me sketch two.

  1. The preconscious control of attention is famously vulnerable to a wide suite of dysrationalia. Like transposons parasitizing your DNA, beliefs parasitize your semantic memory by latching onto your preconscious attention-control software. This is why Evans-Pritchard was so astonished in his anthropological survey of Zande mysticism. This is why your typical cult follower is pathologically unable to pay attention to a certain set of considerations. The first flaw of the attentional lens is that it is a biasing attractor.
  2. Your unconscious mind is subject to the following computational principle: what you see is all there is. This brings us the availability heuristic, the cognitive shortcut your brain uses to travel from “this was brought to mind easily” to “this must be important”. The attentional lens is that the medium distorts its contents. This is nicely summed up in the proverb, “nothing in life is as important as you think it is, while you are thinking about it.” The second flaw of the attentional lens is that bound in a positive feedback loop to memory (“that which I can recall easily, must be important, leads me to discuss more, is something I recall even more easily”).

My treatment of this positive feedback loop was at the level of individual. But that same mechanism must also promote failures at the level of social network. The second flaw writ large – the rippling eddies of attentional currents (as captured by services like Google News) – are known as availability cascades. And thus we have provided a cognitive reason why our social atmosphere disproportionately discusses gun control when school shootings appear in the news.

In electrical engineering, positive feedback typically produces runaway effects: a circuit “hits the rails” (draws maximum current from its power source). What prevents human cognition from doing likewise, from becoming so fixated on one particular memory-attention loop that it cannot escape? Why don’t we spend our days and our nights dreaming of soft drinks, fast food, pharmaceuticals? I would appeal to human boredom as a natural barrier to such a runaway effect.

Attentional Budget Ethics

We have managed to rise above the minutia, and construct a model of political discourse. Turn now to ethics. How should attention be distributed? When is the right time to discuss gun control, to study health care reform, to get clear on border control priorities?

The response profile of such a question is too diverse to treat here, but I would venture most approaches share two principles of attentional budgets:

  1. The Systemic Failure Principle. If a system performance fails to meet some arbitrary criteria of success, that would be an argument for increasing its attentional budget. For example, perhaps the skyrocketing costs of health care would seem to call for more attention than other, relatively more healthy, sectors of public life.
  2. The Low Hanging Fruit Principle. If attention is likely to produce meaningful results, that would be an argument for increasing its attentional budget. For example, perhaps not much benefit would come from a national conversation about improving our cryptographic approaches to e-commerce.

Despite how shockingly agreeable these principles are, I have a feeling that different political parties may yet disagree. In a two party system, for example, you can imagine competing attentional budgets as follows:

Attentional Budgets

Interpret “attentional resources” in a straightforward (measurement-affine) way: let it represent the number of hours devoted to public discussion.

This model of attentional budgets requires a bit more TLC. Research-guiding questions might include:

  • How ought we model overlapping topics?
  • Should budget space be afforded for future topics, topics not yet conceived?
  • Could there be circumstances to justify zero attention allocation?
  • Is it advisable to leave “attentional budget creation” topics out of the budget?
  • How might this model be extended to accomodate time-dependent, diachronic budgeting?

Effective Availibilism

Let us now pull together a vision of how to transcend the attentional cascade.

In our present condition, even very intelligent commentators must resort to the following excuse of a thought: “I have a vague sense that our society is spending too much time on X. Perhaps we shouldn’t talk about it anymore”.

In our envisioned condition, our best political minds would be able to construct the following chain of reasoning: “This year, our society has spent three times more time discussing gun control than discussing energy independence. My attentional budget prescribes this ratio to be closer to 1:1. Let us think of ways to constrain these incessant gun-control availability cascades.”

In other words, I am prophesying the emergence of an effective availabilism movement, in ways analogous to effective altruism. Effective availabilist groups would, I presume, primarily draw from neuropolitical movements more generally.

Notice how effective availabilism relies on, and comes after, of publically-available psychometric data. And this is typical: normative movements often follow innovations in descriptive technology.

Why Quantification Matters

Policy discussions influence votes which affect lives. Despite the obvious need for constructive discourse, a frustrating amount of political exchanges are content-starved. I perceive two potential solutions for this failure of our democracy:

  1. Politics is a mind-killer. By dint of our evolutionary origins, our brains do not natively excel at political reasoning. Group boundaries matter more than analyses, arguments are soldiers. But these are epistemic failure modes. Policy debates should not appear one-sided. Movements to establish the cognitive redemption of politics are already underway. See, for example, Jonathon Haidt’s (“educating the public on evidence-based methods for improving inter-group civility”)
  2. Greasing policy discussions with data would facilitate progress. One of my favorite illustrations of this effect is biofeedback: if you give a human being a graphical representation of her pulse, the additional data augments the brains ability to reason – biofeedback patients are even able to catch their breath faster. In the same way, improving our data streams gives hope of transcending formerly-intractable social debates.

The effective availabilism movement could, in my view, accelerate this second pathway.

Cascade Reform Technologies

It seems clear that availability cascades are susceptible to abuse. Many advertisers and political campaigns don’t execute an aggregated optimization across our national attentional profile. Instead, they simply run a maximization algorithm on their topic of interest (“think about my opponent’s scandal!”).

With modern-day technology (polls, trending Twitter tags, motive abduction, self-monitoring), noticing attentional budget failures can be tricky. With the above technology in place, even subtle attentional budget failures will be easily detectable. We have increased our supply of failures, but how might effective availabilists increase demand (open vectors of reform towards availability cascade failure modes)?

The first, obvious, pathway is to use the same tool – attentional cascades – to counterbalance. If gun control is getting too much attention, effective availabilists will strive to push social media towards a discussion of e.g., campaign finance reform. They could, further, use psychometric data to evaluate whether they have overshot (SuperPACs are now too interesting), and to adjust as necessary.

Other pathways towards reform might be empirically-precise amplification of boredom circuits. Recruit the influential to promote the message that “this topic has been talked to death” could work; as could the targeted use of satire.


  • Pay more attention to the quiet whispers of your mind. “Haven’t I heard about this enough” represents an undiscovered political movement.
  • Social discourse is laced with the rippling tides of availability cascades, and are at present left to their mercy.
  • As hard psychometric data makes its way towards public accessibility, a market of normative attentional budgets will arise.
  • The business of pushing current attentional profiles towards normative budgets will become the impetus of effective availabilism movements.
  • A cottage industry of cognitive technologies to achieve these ends will thereafter crystallize and mature.

Attentional Budgets Usage (1)