Chapter six

Asymptotes, or, how to stop chasing happiness so it stops running.

There is a particular sentence I have heard, in some version, from every well-meaning adult in my life, and which I would like to take apart in this chapter, with surgical care, because it has done more damage than almost any other piece of advice in the human repertoire.

The sentence is, just focus on being happy.

I am sure you have heard it too. I am sure it has been delivered to you in good faith, by people who love you, at moments when you were in trouble, and I am sure it has rolled off you like rain off a windscreen, because at no point in your life, ever, has there been a moment when you could have, by simply focusing on it, become happy. The advice contains an invisible mathematical error that is, on inspection, the same mathematical error contained in just relax, just stop worrying about it, just be confident, and approximately ninety percent of the entire genre of advice your relatives give you at weddings.

The error is this. Happiness is not the kind of object that yields to direct optimization. It is, in the technical language of philosophy and economics, a byproduct. Byproducts have a particular and slightly cruel property, which is that the harder you aim at them, the further they retreat. There are several names for this in different disciplines. Philosophers call it the paradox of hedonism. Economists call it second-order goals. Mathematicians, when they describe systems that exhibit this property, draw a curve called an asymptote, which is a line that another line approaches forever, getting arbitrarily close, without ever touching.

This chapter is about that curve.

Let me first describe what an asymptote actually is, because the word gets thrown around loosely and the strict version is, in this case, exactly what we need.

Consider the function y = 1 / x. When x is large, y is small. When x is enormous, y is tiny. As x heads off to infinity, y heads down toward zero. It does not, however, ever reach zero. For any value of x you choose, no matter how large, the value of y is still positive, still a little above zero. The horizontal line at y = 0 is the asymptote of the curve. The curve is in love with the line. The curve is also, mathematically, never going to get there.

If you replaced x with effort spent chasing happiness, and the line at y = 0 with the feeling you are chasing, you would have a perfectly serviceable model of an extremely common kind of human suffering. The effort goes up. The distance to the goal shrinks. The distance to the goal never reaches zero. The chaser experiences this, in real time, as I am so close, I just need a little more, and the chaser is, in a strict sense, correct about being close, and the chaser is also, in a strict sense, never going to arrive. The geometry is the trap. The trap is the geometry.

I want to make a distinction here, because it matters. The trouble is not that happiness is unreachable. The trouble is that chasing happiness, as a direct goal-seeking activity, is a strategy that has, built into its very geometry, the property that it never converges. The strategy is the asymptote. Happiness is not. The happiness, if you set the strategy down, is sitting right there.

Let us try the other shape of the trap, because the same phenomenon shows up in a second form that is, if anything, even more useful for the anxious mind to understand.

Imagine that happiness is not a destination you approach. Imagine, instead, that happiness is a function of how much of it you have, sampled in the moment you check. Call this function H(e), where e is the effort you are currently putting into being happy. We will sketch what H looks like.

At zero effort, H is, for most people, modest. You are not actively trying. You take what you get. The baseline is roughly the mood you happen to be in.

As e increases, H increases. Pursuing things you find meaningful, exercising, sleeping well, reading, talking to people you love, all of these are positive-effort activities that, in moderation, raise the mood. The curve, in this region, climbs.

At some intermediate value of e, the curve peaks. The peak is the sweet spot. It is the level of intentional living at which you are doing enough to feel alive but not so much that you are auditing yourself for happiness in real time.

Past the peak, the curve declines. The more effort you put into being happy, the less happy you become. The decline is not a small effect. It is, in many adult lives, the single dominant feature of the curve. The harder you try to be happy, the more you are watching yourself for happiness, and watching yourself for happiness is, by a curious property of the human mind, the most reliable known method of not feeling any.

The shape of the curve is sometimes called, in optimization theory, a concave function with an interior maximum. It has, importantly, a single highest point, and that point is not at either extreme. It is in the middle. The two failure modes, mathematically, are symmetric. You can fail by not trying. You can also, equally, fail by trying too hard. The anxious brain, having been told all its life to just focus on being happy, sits to the right of the peak and applies more and more effort, watching its happiness slowly drain, and concludes, on the basis of the slope it is observing, that the answer is yet more effort.

This is the mathematical signature of a strategy that has the sign of its derivative wrong. The brain is reading I am moving away from the goal, therefore I should accelerate. The correct reading is I am moving away from the goal, therefore I should slow down. The brain has, in the most precise sense, mistaken the direction of its own gradient.

Let me describe, in code, what the chase looks like as a process, because there is a piece of comedy in the algorithm that I would like to deliver in its native form.

def chase_happiness(target, position, learning_rate, iterations):
    """
    Try to reach a target by moving toward it. Specifically,
    move a fixed proportion of the remaining distance, every
    iteration. Watch what happens.

    Inputs:
        target:        the value we are chasing (call it 1.0)
        position:      our current happiness (call it 0.0)
        learning_rate: how aggressively we close the gap each step
        iterations:    how many attempts we make
    """
    history = [position]

    for _ in range(iterations):
        gap = target - position
        position = position + learning_rate * gap
        history.append(position)

    return history


# A modest chaser, who closes 50% of the remaining gap each step:
modest = chase_happiness(1.0, 0.0, 0.5, 20)
# -> 0.0, 0.5, 0.75, 0.875, 0.9375, 0.96875, ... 0.99999...
# The chaser gets arbitrarily close. The chaser never arrives.

# An aggressive chaser, who tries to close 95% of the gap each step:
aggressive = chase_happiness(1.0, 0.0, 0.95, 20)
# -> 0.0, 0.95, 0.9975, 0.999875, ... 0.99999999...
# The aggressive chaser gets there faster. The aggressive chaser
# also never arrives.

# An extremely aggressive chaser, who tries to overshoot, closing
# 150% of the gap each step:
overshooter = chase_happiness(1.0, 0.0, 1.5, 20)
# -> 0.0, 1.5, 0.75, 1.125, 0.9375, 1.03125, ...
# The overshooter oscillates around the target, never settling.
# This is, in case it is not obvious, every anxious overachiever
# you have ever met, including, on several occasions, me.

The simulation makes the geometry visible. A reasonable chaser closes half the gap and gets nearer and nearer without arriving. An aggressive chaser closes most of the gap each step and still does not arrive. An overshooter, the chaser who tries so hard that they blow past the target and have to come back, ends up oscillating forever around the very thing they were trying to reach, getting close to it on every odd step, overshooting on every even step, and never, in any iteration, sitting still inside it.

This last case, the overshooter, is worth dwelling on for a moment. It is the chaser who, having heard that effort produces happiness, applies so much effort that he passes through happiness without recognizing it. He arrives at the dinner he had been looking forward to, sits down, finds himself wondering whether he is happy enough at the dinner, decides he is not, applies more effort to enjoy the dinner, leaves the dinner having mostly thought about whether he was enjoying it, and concludes from the experience that dinners are insufficient. This is, in case anyone is wondering, a portrait of the man writing this paragraph, taken roughly between the ages of twenty-five and thirty-five.

The cure, mathematically, is not what the well-meaning relatives think it is. The cure is not to try less. The cure is to stop optimizing for the wrong variable.

Here is the trick, and it is a piece of optimization theory so old and so well established that I cannot quite believe nobody told you about it in school.

Happiness is not the kind of object you can put on the right-hand side of an optimization. It is the kind of object that emerges as a byproduct of optimizing for other things. You optimize for a piece of work you find meaningful, and happiness leaks out of the work. You optimize for a relationship in which you are attentive to another person, and happiness leaks out of the attention. You optimize for the long slow project of becoming someone you respect, and happiness leaks out of the becoming. In every case, happiness is on the left-hand side, downstream of the activity. You cannot grab it directly. You can only set the activity in motion and notice, afterward, with some surprise, that the bowl that was supposed to be empty has been quietly filling.

The mathematical name for this property is that happiness is a fixed point of a particular function, not the output of a particular optimization. The Epilogue of this book will return to fixed points and explain them in full. For now, I will say only this. The way to reach a fixed point is not to chase it. The way to reach a fixed point is to do the activity the fixed point is the natural rest position of, and to let the system settle there on its own. The system settles. You do not push the system into settling. The settling is what systems do when you stop perturbing them.

If you do this, with patience, the curve in Figure 6.2 starts to feel different. You are no longer trying to push your effort further to the right, past the peak. You are not, in fact, watching the curve at all. You are doing the work the curve is a byproduct of, and the curve is, on its own, doing the thing curves do, which is to find its shape. The peak is not something you arrive at. The peak is something you discover, in retrospect, you have been sitting near for several months without noticing.

I want to be precise about one more thing, because it is, I think, where this chapter could be misread, and the misreading would cost us the rest of the book.

I am not saying you should not pursue things. I am not saying ambition is bad. I am not saying that the way to be happy is to have no goals and lie quietly on the floor. The floor is, in fact, a local minimum in the sense of Chapter 5, and lying on it is one of the failure modes the previous chapter was specifically trying to lift people out of. The two chapters do not contradict each other. They describe two different traps. The Chapter 5 trap is being stuck and refusing to move. The Chapter 6 trap is moving too hard at the wrong target.

The combined diagnosis is something like this. Move. But do not move toward happiness directly. Move toward the things that produce it as a byproduct, at the level of effort that puts you near the peak of the curve, with the understanding that the peak is not exactly knowable from the inside, and that you will, on most days, be slightly off it in one direction or another, and that this is fine.

This is, I will grant, less catchy than just focus on being happy. It is also true, which the catchy version is not. The reader who internalizes this distinction, even loosely, even partially, will spend less of the rest of their life in the specific failure mode of having sprinted past the dinner of their own life while monitoring it for sufficient pleasantness.

Chapter 7 takes a different turn, into linear algebra, and into the question of who you are when life multiplies you by a matrix and forgets to ask. The chapter looks, mathematically, at the directions in which your personality does not change under transformation, and it will, in passing, finally close the loop on the not-father algorithm I described in Chapter 2.

For now, the page closes here. The asymptote is the strategy, not the goal. The goal is sitting on the other side of the strategy you have been running, and the goal will be there when you set the strategy down, and the goal will not, if you are patient, run.