Mathematical Thinking

Asymmetrical Thinking: Expose and exploit imbalances

The idea that we need to “think outside of the box” to solve problems is frustratingly vague. Fortunately, there are several mental strategies that can promote creativity, including thinking nonlinearly and counterfactually. But one of the most underutilized processes is asymmetrical thinking, which involves flipping around and reshuffling ideas to unearth hidden imbalances.

This mindset is useful across disciplines, including physics, economics, biology, business, and investing. Its origins, however, are from geometry.

The dance of symmetry and asymmetry

The basis of geometry is symmetry, which describes the property of an object being unaffected by undergoing some transformation. Consider an equilateral triangle. If we move the whole shape an inch to the right, the angles and dimensions of the triangle are unchanged; that is, they are symmetrical with regards to this simple translation.

On the flip side, asymmetry is simply the absence of symmetry, indicating that an object is affected by undergoing some transformation. If we take our original triangle and pinch or stretch it, the lengths and angles of the triangle will change—they are asymmetrical with regards to pinching or stretching.

Ancient scholars, particularly Euclid, harnessed the concepts of symmetry and asymmetry in shapes to discover revolutionary geometrical and mathematical principles, setting the foundation for modern physics, engineering, and more.¹

Outside of geometry, asymmetry offers an invaluable mental model: it can be extremely insightful to flip things—whether shapes, ideas, relationships, or strategies—backwards and forwards and see whether what is true in one direction is also true in the opposite direction—in other words, whether there is symmetry. In our lives, finding asymmetries presents opportunities for unearthing unique insights, stimulating creative ideas, and creating leverage to drive outsized impact.

Asymmetries all over

Physics

In physics, practically all laws of nature originate in symmetries and asymmetries. Emmy Noether’s groundbreaking 1915 theorem connected symmetries in nature with the universal laws of conservation. Her work showed that whenever we see some sort of symmetry in nature, there must also exist a corresponding “conserved” element which preserves the symmetry.²

For example, it’s well-established that the laws of physics are uniform throughout the universe. Regardless of where we are, the laws of nature remain unchanged (symmetrical), just as the angles of our triangle did not change when we moved it. This symmetry gives birth to principles such as the the “law of conservation of momentum,” which holds that unless an outside force (such as friction or air resistance) intervenes, the momentum of a system will remain unchanged.³

Biology

Nature abundantly showcases symmetry.

Most vertebrates, from humans to elephants, have two “halves” that are roughly equal. This property, called “bilateral symmetry,” is believed to be advantageous for efficient movement and centralized control of sensory organs in the “head” of the organism. Greater symmetry also correlates with higher rates of reproduction, since many organisms with greater symmetry tend to be preferred as mates (example: facial symmetry in humans).⁴

However, it’s important to note that asymmetry is also an important and widespread trait, even in humans. Consider the phenomenon of “handedness,” the left lung being smaller than the right, or the left and right brain controlling different cognitive functions.

Statistics

Picture the symmetrical “bell curve” of the normal distribution. For any given random observation, there’s an equal probability that it will fall above or below the central average of the data—such as with human height and weight.

However, many real-world events generate results that are in fact asymmetrical (or “skewed”) rather than normal. Some produce “power-law” distributions in which a few extreme values dominate over the modest majority, such as with the frequency of words in most languages, the magnitude of earthquakes, and city populations.

Business and Investing

In the world of venture capital (“VC”), investors bet on risky, early-stage ventures, which carry both high potential growth and high risk of failure.

Unsurprisingly, the financial returns of early-stage startups are power-law distributed. Most investments generate low or negative returns, but a few bets generate enormous returns. VCs are essentially in pursuit of asymmetrical returns, aiming to own a piece of occasional breakout successes such as Google or OpenAI.

They gladly accept that many (even most) of their investments will generate little or no return, as long as one or two investments become wildly successful. The most they can lose is 1x their money, but the extreme winners could generate 10x, 100x, or more.

Using asymmetry as a strategy

In the competitive realm, one indirect strategy involves using your relative advantage to impose asymmetric costs on the opposition. In military conflicts, for example, weaker insurgent factions may compensate by using asymmetric warfare tactics, such as by attacking the opponent’s electrical grids, roads, or water supply.

In business, the basic concept of competitive advantage is rooted in differences—in the asymmetries among competitors. The task falls to the leaders to identify the asymmetries that can be turned into advantage and exploited, such as a valuable patent, strong network effects, or significant economies of scale.⁵

A great example is Netflix’s critical insight around 2011 to commit substantial resources towards producing original content. Historically, Netflix and its competitors licensed a portfolio of content produced by other companies. This meant that every new subscriber increased content costs. With Netflix’s landmark pivot to producing its own original content—particularly early shows such as Lilyhammer and House of Cards—Netflix turned content into a fixed cost.⁶ With original content, more subscribers don’t directly increase content costs.

This strategy created a cost asymmetry between Netflix and its competitors: Netflix’s content costs would become cheaper over time on a per-subscriber basis, as it spread its fixed costs over a larger number of subscribers! Netflix would lead a global revolution in the production of original streaming content.

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Asymmetry is much more than an academic concept—it offers a powerful new lens through which we can view the world. By identifying and leveraging the hidden asymmetries in our lives, we can open the door to inventive solutions and strategies. As we navigate challenges in our own lives and careers, we should consider how asymmetrical thinking might offer unexpected answers.

Local vs. Global Peaks: Balancing exploration and exploitation to reach our pinnacle

A local optimum is a solution that is optimal within a neighboring set of candidate solutions—a point from which no small change can generate improvement. However, this local peak may still be far from the global optimum—the optimal solution among all possible solutions, not just among nearby alternatives.

This valuable model can teach us about the inherent tradeoff between capitalizing on our current opportunities and pursuing new ones—whether in biological ecosystems, businesses, or machine learning. We can use it to better understand the complex environments we operate in, and to design more effective strategies to achieve our goals.

Getting stuck

Picture a rugged plane comprised of many peaks and valleys of various elevations, with numerous individuals or groups competing to reach the highest peaks. Nearby points tend to have similar levels of “fitness.” The landscape itself may shift dynamically, altering the peaks and valleys and transforming the available paths to reach them. This model is known as a “fitness landscape,” an extremely useful metaphor for thinking about optimization amidst local and global peaks in a variety of applications—including systems, biology, computer science, and business.¹

In complex systems (such as an industry or an ecosystem), it is easy to get stuck on local peaks as the ground shifts beneath our feet (undermining our position), especially if we fail to survey new territory. We won’t know precisely how the landscape will shift, so the only way to sustain progress in the long-term is, simply, to explore.

Sometimes, we may even have to go down (temporarily worsen our situation) in order to ascend a higher peak. And this requires a lot of courage. For example, Netflix’s stock fell by almost 80% from its peak after CEO Reed Hastings announced they were getting out of the DVD business in 2011. Ten years later, Netflix had pioneered the video streaming industry, and its stock price had grown by nearly 1,300%!

Evolutionary searches can never relax. We must constantly experiment with new ideas and strategies to find better solutions and adapt as the landscape shifts.

Faster than the speed of evolution

In biological evolution, we can visualize the competition for genetic dominance as a rugged fitness landscape in which the peaks and valleys represent the highs and lows of evolutionary fitness across an ecosystem. Higher peaks represent species or organisms that are better adapted to their environment—that is, ones that are more successful than their nearby competitors at causing their own replication.

Evolution is capable of creating remarkably complex and useful features, such as the human body’s ability to repair itself or the peacock’s brilliant tail. However, because it optimizes only for the ability of genes to spread through the population, evolution will inevitably reach only local peaks of fitness within a given environment.² It can favor genes that are useless (the human appendix), suboptimal (women’s narrow birth canals), or even destructive to the species. For instance, the peacock’s large, colorful tail that helps it find mates also makes it more vulnerable to predators.³

When the landscape shifts, even a highly adapted species will be unable to evolve toward a worse (less well-fitted) state than its current one in order to begin ascending a new, higher evolutionary peak. If the environment shifts faster than the species can adapt to it, mass extinctions can occur.⁴

Fortunately, we humans don’t need to be bound by evolutionary timescales. Often, we can find better hills to climb.

Let’s look to computer science and business to see why.

Getting un-stuck

Algorithms provide useful insights into optimization and into overcoming local peaks.

The simplest optimization algorithm is known as “gradient ascent,” in which the program just keeps going “up.” For instance, a video site such as YouTube might be programmed to continue recommending videos that resemble your past content consumption. But “dumb” algorithms like this one maximize only short-term advantage, leading us to local peaks but not to global ones. What if the user’s content preferences change? What if the viewer gets bored by stale recommendations? What if repetitive videos trap the user in a filter bubble?

Randomness and experimentation can help us “pogo-jump” to higher peaks that simple gradient ascent would not reach. For example, a “jitter” involves making a few random small changes (even if they seem counterproductive) when it looks like you are stuck on a local peak, then resuming hill-climbing. A “random-restart” involves completely scrambling our solution when we reach a local peak—which is particularly useful when there are lots of local peaks.⁵

Perhaps our video site should recommend random pieces of viral content even if the viewer hasn’t watched similar clips previously. Or show clips that contrast sharply with past viewing habits (for nuance or contrarian content). Only experimentation can reveal whether we are climbing the best hill.

The explore/exploit tradeoff

In business, it is useful to picture the strategic environment as a rugged landscape, with each “local peak” representing a coherent bundle of mutually reinforcing choices.

Every organization needs to balance experiments in exploitation of its current businesses with experiments in exploration for future innovations. In the short-term, simple “gradient ascent” strategies (keep going up) help ensure the company is exploiting its current strengths and opportunities. Over the long-term, however, companies must make occasional medium- or long-distance “pogo jumps” to prevent getting stuck on local peaks and, sometimes, to make drastic improvements. The key problem with many organizations is that when the environment seems stable, they stop experimenting because it seems costly and inefficient, and because it sometimes creates internal competition.⁶

This was Reed Hastings’s revelation about Netflix in 2011: its wildly successful DVD-by-mail business was merely a local peak. The landscape had shifted. The new global peak, he believed (correctly), was streaming.

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The overall lesson is that because the environment is uncertain and always changing, good strategy requires individuals and organizations to carefully cultivate and protect a portfolio of strategic experiments, creating valuable options for the future.

Even when it seems we are at a “peak,” there may be even higher peaks that we cannot yet see, and the peaks themselves are constantly shifting! In such an environment, complacency is a death sentence.

Compounding: Why we should actively trade ideas, but not stocks

“Spend each day trying to be a little wiser than you were when you woke up. Discharge your duties faithfully and well. Step by step you get ahead, but not necessarily in fast spurts… Slug it out one inch at a time, day by day. At the end of the day—if you live long enough—most people get what they deserve.”
Charlie Munger, Poor Charlie’s Almanack (2005, pg. 138)

There is perhaps no more fundamental idea that reminds us of the value of the twin virtues of patience and discipline than the phenomenon of compound interest.

Compounding describes the process by which a fixed quantity (such as a savings account) grows by accruing “interest” at a certain rate, then earning interest on the original quantity plus interest on the newly added interest, and so on.

Not so simple

Compounding is a powerful example of a reinforcing (positive) feedback loop, which produces an exponential growth effect in which the absolute growth in the quantity increases over time. In contrast, “simple interest” produces linear growth in which the balance increases by a constant absolute amount each period.

For example, a savings account with an initial balance of $10,000 that earns 5% simple interest will grow by exactly $500 each year (see simplified chart below). If that same $10,000 were to instead earn 5% compound interest, the balance would grow by $500 in Year 1, $525 in Year 2, $551 in Year 3, and (skipping ahead) $776 in Year 10, etc.—generating a nonlinear increase in value over time.

simple interest grows linearly, compound interest grows exponentially

The mathematical phenomenon of compounding is one of the most powerful concepts to understand, with applications for our personal financial management, our habits and productivity, and indeed our pursuit of wisdom generally.

The best for last

Compounding teaches us to be patient, because most of the benefits of compounding come at the end! Whether we’re starting to build up our retirement savings, creating new relationships, or establishing better habits, we may not see huge benefits up front. But if we combine the discipline and patience to keep making incremental improvements, over time we can generate enormous results.

“Habits are the compound interest of self-improvement. They don’t seem like much on any given day, but over the months and years their effects can accumulate to an incredible degree.”
James Clear (2018, on Twitter), author of “Atomic Habits”

In financial decisions, it is critical to value cash flows not based on their absolute value today, but on their opportunity cost—the potential value of that cash flow if we had instead invested it and allowed it to compound over time. Any use of money must justify the opportunity cost of foregone compound interest on those funds—and that amount could be huge.

Disciplined investors are “cursed” with viewing investments and expenses through this lens. Warren Buffett famously quipped that his worst investment ever was actually his purchase of Berkshire Hathaway. He estimated that if he had simply taken the amount he paid in 1962 and invested it at the rate of return he would go on to earn over his career, he would have accrued $200bn more wealth.¹

Stop trading so much

If I could summarize the one lesson about money that I’ve learned in my own career in finance and strategy, it would be that people (especially men) generally overestimate their own financial acumen.

Overconfidence in finance leads us to transact much more often than we should—and transactions are costly, because they counteract the power of compounding.² Every time we sell a stock or asset, we pay some transaction fees, and we owe taxes on any gains we accrue. As compounding teaches us, the value of these costs rises exponentially over time, since we could have simply let those funds compound freely.

“Beware of little expenses: a small leak will sink a great ship.”
Benjamin Franklin

The more actively that individual investors trade, the more money they typically lose. A fascinating study observed that if we break out the returns of individual investors into tiers based on how frequently they trade, the net returns of every group except for the least frequent traders are lower than the net return from simply investing in an S&P 500 index fund. And the group with the heaviest traders generated the lowest returns, by far.³

My advice: unless we are market geniuses (most of us aren’t), we probably shouldn’t be trading frequently. Most likely, we would be better off in the long-term by investing the majority of our portfolios in low-cost, passive “index funds” (which simply mirror the returns of a market index)—only infrequently checking our balance or executing transactions.

Learning begets learning

Perhaps the most powerful case of compounding is knowledge growth itself.

Wisdom is not a matter of collecting facts and clever examples. Being a polymath is the “simple interest” version of learning. Rather, the way we compound our knowledge is by incrementally building up a self-supporting, interconnected foundation of ideas and explanatory frameworks—a “latticework,” to borrow Charlie Munger’s terminology.

The human brain learns by association, the cognitive processes in which we attempt to draw meaningful connections between a new piece of information and our prior knowledge. We compare, combine, and create variations between different ideas.

Developing a strong foundation of models and ideas in our heads enables a positive feedback loop: more information supplies more possible connections, which helps us improve our knowledge, which makes it easier to add more information and make even more meaningful connections, and so on.⁴ In other words, learning compounds to facilitate more learning.

This happens through independent reading, research, and elaboration—by following our curiosity and thinking critically about different (often incompatible) ideas and how we might combine them with other knowledge. The more diverse these ideas are across disciplines (physics, systems, economics, etc.), the stronger the foundation will be.

By committing to a lifelong process of learning in this associative, multidisciplinary way, our knowledge can truly compound to empower us to solve an incredible range of problems.

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The key lesson? Diligently pursue incremental improvements, then simply be patient. Let compounding work its magic! With time, the odds tip heavily in your favor.

Power Laws: The hidden forces behind all sorts of inequalities

A power law describes the relationship between two variables in which one variable varies as an exponent of the other. For example, if the side of a square is doubled, the area is multiplied by a factor of four.

Power laws are nonlinear relationships, where the collective output changes by more than the proportional change in input. These relationships contrast sharply with linear functions (which are simpler and more intuitive), where the changes in input and output are proportional.

The dynamics of power laws help describe many extreme phenomena we observe in the world, from natural disasters to the success of startups. Anyone who aspires to maximize their impact can benefit from learning about the nature of power laws—and how to exploit them.

Far from normal

Power laws produce distributions very different from the nice, symmetrical, bell-shaped “normal distributions.” In normally distributed systems, our observations will have a meaningful central tendency (the average) and increasingly rare deviations from that average—such as with human height and weight. Most people aren’t that far from the average height, and the shortest 1% of people and tallest 1% differ in height by only around 14 inches.¹

In contrast, the distributions produced by power laws don’t peak around a typical value; rather, the range of values is much wider—with a majority of observations of modest values on one end, and a “fat tail” of rare but extreme outcomes on the other end. While the tallest people in the world aren’t 10x taller than the shortest people, with power laws, the most extreme events can be orders of magnitude greater than the least extreme.

The distributions of a many physical, biological, and man-made phenomena approximately follow a power-law distribution. For example:

The frequencies of words in most languages — If we want to learn a new language, we would do well to start with the small fraction of words (as few as 135 of them) that make up the majority of usage (“Zipf’s law”).
The populations of cities — As of 2019, the U.S. had more than 19,000 cities, though just 37 cities housed the majority of the population. The largest one, New York City, housed over 8 million residents alone, more than double the next-closest city.²
The size of lunar craters — The Moon has countless small craters from millions of years of minor collisions with other interstellar material, and a few enormous craters from exponentially larger collisions.
The frequencies of family names — There are millions of rare or obscure last names, but approximately 1 in 68 people on the entire Earth has the last name “Wang.”³
The magnitude of earthquakes — The Richter scale is logarithmic: each increase of 1 on the scale equates to a 10x increase in strength. For instance, a magnitude 5.0 earthquake is ten-times more destructive than a magnitude 4.0 earthquake, but is one-tenth as common.⁴

Other examples include the size of computer files, the number of views on web pages, the sales of most branded products (e.g., books, music), and individual incomes and wealth.⁵ In each case, there is a minority that supplies a majority of the outputs.

Sandpiles and avalanches

We should always be wary of the potential for power-law effects whenever we are dealing with systems comprised of many interacting parts—in other words, with complex systems (such as economies, ecosystems, or the climate).⁶ The various components and the feedback loops that govern them tend to cause such systems to evolve into a very delicate state of balance, a dynamic equilibrium. When forces push the system outside its equilibrium bounds, the system may shift into a new, discrete phase.

As we shall see, when a system “tips” out of equilibrium and into a phase change, the results commonly follow a power-law distribution.

Consider a pile of sand on a countertop. You drop additional grains, one-by-one, onto the pile, steadily increasing the pile’s slope (its key control parameter). At first, each new grain does little; the pile remains roughly in equilibrium. Eventually, however, the pile’s slope will increase to an unstable “critical” threshold, beyond which the next grain may cause an avalanche, a type of phase transition.

At this critical stage, we can’t say for certain whether the next grain will cause an avalanche, or how big that avalanche will be. We do know, however, that the probability of an avalanche is much higher near the tipping point, and that avalanches of any size are possible, but smaller avalanches will happen far more often. A power law!

We see these same power-law dynamics in all manner of complex systems, which essentially “adapt” themselves through a series of “avalanches” to maintain overall stability.⁷ Examples include the extinction of species in nature, price bubbles and bursts in financial markets, traffic jams, or earthquakes relieving pressure from grinding tectonic plates.

The engine of venture capital

The dynamics of power laws are central to our economy and to innovation in general. Progress occurs through the trial-and-error process of startups trying new and different ways of creating value. These experiments require financial capital from investors who can accept substantial risk, since most startups fail.

Venture capital investors seek to earn outsized investment returns not by having a large proportion of their investments do well, but having one or two “grand slams” that generate massive returns (think Facebook or Tesla). It would not be surprising for a venture fund’s one or two big winners to return more than all their other investments combined. The most that VCs can lose is 1x their investment, but there is (theoretically) no cap to how much they can gain if an investment is successful.

Union Square Ventures, for example, invested in Coinbase in 2013 at a share price of about $0.20, and achieved a massive return when Coinbase opened its initial public offering at $381 in 2021—a valuation of around $100bn and an increase of over 4,000x.⁸

Feed the winners, starve the losers

A common corollary to power laws is the “80/20 rule” (or, the Pareto principle), which states that for many events 80% of effects (output) come from 20% of the causes (inputs). Mathematically, the rule approximates a power-law distribution. Phenomena roughly following this rule have been observed in income distribution, software coding, business results, quality control, infection transmission, and elsewhere.

While often the exact number varies, the principle reveals that most of the work we’re doing only generates a small amount of our overall results. The key lies in identifying “the 20%”—of activities, individuals, projects, products, businesses, grievances, etc.—that are driving a disproportionate share of the outcomes, and concentrating efforts towards them.

Corporate “turnaround” master Don Bibeault famously relies on the 80/20 rule to drive transformation in struggling businesses, recognizing that most businesses spend too much time satisfying customers, selling products, and preserving marginal employees that make little or no contribution to the bottom line. The key to implementing an 80/20 policy is redeploying resources away from the “marginal many” to the “critical few” that account for current results and future opportunity—a tactic Bibeault calls “feed the winners and starve the losers.”⁹

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Power laws teach us that some inputs are much more important than others, and they can explain many of the extreme results we observe in the world. Concentrating our efforts towards unlocking (or avoiding) the outliers of power laws can provide substantial leverage towards helping us achieve our goals.

When seeking an effective strategy or solution, we should ask ourselves where the most “power” in the situation might be hidden!

Exponential Growth and Decay: Grow fast or die trying

Exponential growth occurs whenever a stock of some material or quantity increases or replicates itself in constant proportion to how much there already is. This is a multiplicative effect, in which each step is more extreme than the preceding one. As an example, consider a stock of 1,000 hogs that, given its rates of fertility and mortality, grows exponentially at 10% per year. In the first year, there’s an increase of 100 hogs, then 110 in the second year, then 121 in the third year, and so on—illustrating the snowballing effect.

This type of growth contrasts sharply with linear growth, in which the stock changes by a constant quantity each period (an additive effect). If the hog population grew by a fixed 100 hogs annually, the implied percentage growth rate would diminish over time—from 10% in the first year to 9.1% in the second, and so on.

Exponential patterns are ubiquitous, from biological phenomena such as population growth and disease spread, to economic phenomena such as GDP growth and compound interest, to technological trends such as network effects in communications networks and improvements in the processing power of computers (“Moore’s law”).

Despite the prevalence of exponential progressions, our human intuition often fails to appreciate their speed and chaotic potential. Instead, we gravitate towards linear thinking because it serves us well in most practical circumstances. We can commit fewer errors and better explain and predict the world once we understand the power of exponential growth, and its equally powerful ability to unravel.

Feedback loops: Accelerators and regulators

The underlying driver of exponential growth lies in reinforcing (positive) feedback loops, which exist whenever a system (such as a virus or a savings account) can self-multiply or grow as a constant fraction of itself. These amplifying forces generate exponential growth, producing either virtuous or vicious cycles.

In contrast, balancing (negative) feedback loops are stabilizing, goal-seeking functions that aim to maintain a system in a given range of acceptable parameters—in a “dynamic equilibrium.” Consider how a thermostat regulates the temperature of a home, or how our bodies induce perspiration and shivering to stabilize our body temperatures.¹

In physical systems that are growing exponentially, there must be at least one positive feedback loop propelling the growth, but there must also be at least one negative feedback loop constraining the growth, because no physical system can grow forever in a finite environment.²

The inevitable decay

Consider a virus such as COVID-19, which initially spreads exponentially through the population as each infected person infects multiple others. It might seem like an uncontrollable plague.

But again, nothing grows forever. The flip side of exponential growth is exponential decay, when a quantity decreases at a rate proportional to its current value. If we withdraw 10% of the funds in our savings account every period, the accounts’ value will decay exponentially in a downward reinforcing feedback loop.

Let’s return to our example of a virus such as COVID-19 or smallpox. Over time, balancing feedback loops will kick in to combat the spread. In the worst case, the virus could simply start running out of people to infect because so many get sick. But eventually, even highly contagious viruses run out of steam. Our bodies will start to develop antibodies to increase immunity. Widespread vaccinations can achieve the same effect. Governments, organizations, and individuals may adapt their behavior to mitigate the risk and impact of the virus (wearing masks, providing aid, social distancing, etc.).

Once the average number of people that one infected person infects (the so-called “R-knot”) falls below the critical level of 1, the exponential growth turns to exponential decay, and the virus begins to die out. This is known as “herd immunity,” when there are not enough new hosts to whom the virus can continue to spread. The key insight for epidemic control is that “perfection” is not necessary; we don’t have to stop all transmission, just enough transmission to achieve herd immunity.³

We can find exponential decay progressions in a variety of real-world applications, including the biological half-life of chemicals or drugs, the rate of radioactive decay by which nuclear material disintegrates, the decrease in atmospheric pressure at increasing heights above sea level, and the effectiveness of advertising messages over time.

Many phenomena exhibit both exponential growth and exponential decay, at different phases in their progressions.

The ascent and fall of Clubhouse

In 2020-21, Clubhouse, a live audio chatroom app, experienced a meteoric rise, growing to tens of millions of downloads within months. Its exponential growth was fueled by network effects, a type of positive feedback loop in which each new user makes the network more valuable to all other users. Notable celebrities and influencers joined in, creating more buzz and attracting more users around the app’s aura of novelty and exclusivity.

However, after its initial surge, Clubhouse ran into numerous challenges. Its pandemic-driven novelty diminished. Many celebrities and business leaders moved on. It was also difficult for users to engage consistently with live audio due to their busy schedules and fickle attention spans; podcasts and audiobooks remained much more convenient.

Perhaps above all, existing social media giants such as Facebook and Twitter appreciated the potential of the live audio format and moved rather quickly to copy Clubhouse’s functionality. It turned out that live audio was more promising as a feature of the existing platforms—which already had hundreds of millions of users—rather than as a standalone service that would need to bootstrap network effects from the ground up. Users could simply engage with live audio content within their existing social media routines.

The app’s new downloads declined by over 90% between February and April 2021.⁴ As signups slowed and users either churned or migrated to a competing service, the app became less valuable to both new and existing users—exacerbating its decline. Clubhouse experienced first-hand the capricious potential for exponential growth to unravel.

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Whether with a virus, a population, or a company, we must remember that infinite exponential growth is mostly a theoretical construct. In the physical and practical world, there are always limits. Exponential growth may only occur across a particular scale of observation—such as during the initial contagion of a virus (when few people have been exposed), or during the early period of a new product’s life (when novelty is high and competition is low). Balancing feedback loops ultimately tame exponential progressions.

We must not underestimate the speed with which exponential forces can generate explosive growth—or equally rapid decline!