Thursday, July 2, 2015

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.