The Happiness Approximation

One of the cornerstones of my understanding of global population and consumption is the mathematical relationship between happiness and ecological footprint (our individual consumption of ecological resources). While trying to derive real significance from the convenient abstraction, I gained insights into how much of the natural world we can safely use, the purpose and mechanics of the global economy, and how people's values influence the survival of our species. Yet still I didn't fully appreciate why it had the form it did.

In the mean time, I grappled with the dynamics of human behavior and how to derive a means of successfully communicating the lessons I was learning so they could be used to hopefully extend the lifetime of our species, which my research showed is uncomfortably limited. One approach was to focus on complexity and how it influences both comprehension and the completion of basic tasks. I studied the implications of a general relationship I had discovered in my own experience between efficiency and the timing of progress in various activities, including the propagation of error in product development and messages as represented by my writing and that of others.

I suspected that task completion might be related to the pace and nature of global consumption and population growth. While I couldn't find a close correlation, I did discover that our species is attempting to use all of the Earth's resources that don't threaten the planet's ability to meet our basic needs, and we have historically been doing so in at least four different ways at a speed that varies with how much we consume.

A few weeks ago I had the latest of a string of epiphanies that has marked my research since the beginning. I accepted the possibility that the timing of consumption might be independent of the timing of task completion, and sought to generalize how they might fit together to describe how much resources are consumed during a task. In the process, I derived a relationship of the form found between happiness and ecological footprint. I was then able to apply the generalization, which I came to call the "happiness approximation" to how happiness has changed over human history and how it could conceivably change over a person's lifetime.

As I studied the consequences of the happiness approximation, I homed in on the special cases of completion and consumption that involve full use of resources in the completion of a task, which would be optimum where resources are limited. A simple statistical simulation showed that optimum completion averaged out to 76%, and that general completion had an average of 87%. Averaging those two values produced the 82% that the happiness relationship had identified as maximum average happiness for a population (and also happens to be close to the 80% used as a rule of thumb for realistic completion by project managers.

The timing of task completion also rang a bell of familiarity. The minimum amount of time it takes to complete a task (in the simulation) is 19, which is close to the 20 years that is the lowest life expectancy in my historical data (at the earliest year: 10,000 BC). If this correlation is correct, then it is the earliest point in a person's life that they can achieve 100% happiness, which according to the simulation only happens in 14% of cases.

I also modeled a better match to traditional expectations of task completion time, including something close to achievement of historical values of happiness and life expectancy in 2015. It never achieved total or optimum completion, and still took three units of time to reach maximum consumption, which is what I typically use for planning purposes (where one unit of time represents best possible achievement). Only 2% of a general population achieves that or better, which speaks to how selective an organization must be to realistically promise such performance; much less than 1% could be expected to achieve the perfect performance that an organization might optimistically promise in one unit of time.

Units of Completion

Further investigation of the application of task completion time to the global variables defining humanity's past and future has yielded another surprising insight. People over time have apparently collaborated over time in a series of tasks focused on increasing life expectancy to successively higher values.

With an average efficiency of 50%, and knowing the maximum we could achieve, we would expect to halve the difference between our current value and that maximum during each of several attempts (assuming the success we achieved during each attempt was preserved). Each attempt would take the same amount of time as the best-case (going from zero to the maximum with 100% efficiency).

This isn't what happened. It took millions of years to complete the first pass, which we seem to have treated as a single task on its own, achieving a life expectancy of 35 years in 1900. The second pass also proceeded as a separate task aimed at 53 years, which we completed in just 61 years. There were six more such tasks after that, each taking significantly less time than the task before it, and we reached what my Half-Earth Hypothesis projects as the maximum (71 years) in 2011.

For at least two years after that, life expectancy decreased. Though the projection indicates that the decrease is temporary, it also shows that it is an artifact of our being at the peak, and a more pronounced and sustained decrease is imminent if we continue to increase our consumption of ecological resources.

My analysis also yielded another interesting insight, which relates to earlier study of complexity. Although I stand by my observation that performance of a single, continuous task is unlikely to reliably and measurably exceed 95% completion because of the influence of unknown and uncontrollable variables (otherwise known as "luck"), there is a granularity of real tasks that can be used to define a target value that incorporates what I think of as "resolution," or "acceptable error."

If we think of a task as the effectively-simultaneous manipulation or creation of a number of observable units, equivalent to what I've described as "interactions" in the description of an event, our uncertainty in assessing completion of the task will have a maximum value equal to one unit, which, as a fraction of the total, is the reciprocal of the number of units. The amount of completion we can verify, therefore, is one minus this fraction. For example, if a task consists of writing a 500-word page, and the result is defined by the number of words, then the maximum meaningful completion is 0.998, or 99.8%, which we can then use to calculate the expected time in terms of the best-case completion time. Luckily for those of us who like doing calculations in our heads, the completion time for the average person (at 50% efficiency) equals the number of times we must multiply 2 by itself to get the number of units; so for the example of 500 words, we know that the completion time is about 9. For 95% completion, the number of units is 20, which corresponds to a completion time close to 4.

What really got me excited about this was the discovery of new significance for two of the critical numbers coming out of the Half-Earth Hypothesis. Recall that the maximum amount of ecological resources we can consume is limited by the need to conserve the living and non-living providers of the basic resources we need to survive, which means we must leave alone the equivalent of half the renewable resources provided by Earth's biosphere (thus the source of the name "Half-Earth"). The sum of what we would be consuming at that point, plus the species providing our basic sustenance, can be no more than 82% of the total, and those species need an additional 15% of the total to meet their needs for survival. It turns out that, regardless of efficiency, 82% completion requires half as much time as 97% (82% plus 15%); and 97% as a maximum value corresponds to 31 units, which is easily remembered as the largest number of days in a month as well as around the smallest valid sample size for statistical analysis. Of course, 97% is also close to the 95% that I've observed as maximum reliable completion.

I am keenly aware, and must remind readers, that these discoveries and the reasoning behind them are best considered as hypotheses that remain to be extensively tested. They represent informed opinions and interesting patterns in data that may be either misleading or groundbreaking. At the very least, I find them extremely interesting, and worthy of further study and discussion to help bring into new context the facts we know and might find later.

Efficiency Revisited

In "Efficiency and Completion Time," I discussed how the amount of time it takes to complete a generic task typically takes longer than the minimum possible time. I've since explored the math behind that observation, and how it might vary across the world's population using insights described in "Changing People."

It turns out that what I called the "efficiency" of performing a task is equivalent to what we commonly know as a compounding interest rate on a credit card we're not paying off, where we are exponentially depleting the remaining amount of the task. If efficiency varies randomly from zero to one across the population, then the 68% of people in the central bulge of the bell curve will have efficiencies between about 0.4 and 0.6. That is, a typical person will accomplish between 40% and 60% of the remaining part of a task during a period of time that is the best case for accomplishing the entire task, which includes preparation and perfect luck. Very skilled people working on state-of-the-art projects, in my experience, are at the high end of "typical" efficiency, which is at the low end of the top 16% of the population that includes only one person who can achieve the task in the best-case time.

Realistically there is a limit to luck, which based on my testing experience is around 95%. This means that multiple attempts at a task are unlikely to all achieve more than 95% of it. Typical people take about 3 to 6 times as long as the best case to complete 95% of a task. Only 47 people on our whole planet could accomplish 95% of a task in the best-case time or less (notably, at the beginning of civilization this number was ten – at least two breeding couples – with this efficiency).

Of course, tasks are extremely variable, which would seem to make these kinds of comparisons meaningless. That variability is, however, covered by the definition of best-case time, which includes the influence of such things as technological aids, the education available to the population, and inherent complexity. Efficiency is explicitly tied to just what humans can do, which is subject to finitely limited physical and mental capabilities that influence the performance of all tasks, so the time to complete a task must be measured as the time that humans are involved in activities that support it, which includes education, skill development, and acquisition of resources.

It is also important to note that each person may not have the same efficiency for all tasks: for example, someone with 20% efficiency for one task may have 80% efficiency for another task; the bell curve simply represents the number of people who have given efficiencies. Tasks may also have common aspects that allow people to have about the same efficiency for multiple tasks, especially those that enable them to meet their basic needs in a natural environment.

In this context, our social infrastructure, which includes education, can be seen as a means of accelerating the effective completion level of common tasks to a point where further action on specific tasks takes a reasonable amount of time, especially those that affect the goals of the society – the most critical being its survival. This also involves preparing people to cooperate (another common task) so they can collectively benefit from each other's strengths (high efficiencies), and offset each other's weaknesses (low efficiencies) under changing conditions that force adaptation through performance of modified or additional tasks. With a random distribution of efficiencies across the world's population, it would take seven times the best time to prepare 99% of the population to do a common task in the best time; that factor, and the limit of time available (perhaps 13 hours per day) would determine how many common tasks could be prepared for.

One of the most common tasks in human history is our joint effort to dominate the natural world, which materially translates into maximizing our consumption of ecological resources (our ecological footprint). Our achievement of the task on a per-person basis has accelerated, with the best-case time since the beginning of civilization shrinking through radical changes from more than 220,000 years to around one year, and the appearance that we may have already achieved the task around 2011 and are now going backwards.