Error and Complexity

The learning curve model that I developed based on observations of how fast tasks are performed has yielded another valuable insight. The difference between expected performance of a task and actual performance over time, will tend to spike and then fall off gradually, which is likely to confound typically linear (and often overly-optimistic) approaches to scheduling its completion. Furthermore, the timing and amount of the spike will vary with the complexity of the task, which may not even be explicitly factored into expectations. The math shows that the timing and amount of this spike in what could be considered "error" is theoretically predictable.

In "Units of Completion," I suggested that the complexity of a task could be assessed in terms of a number of units that are simultaneously performed during the task, and which define how closely we can measure its completion. I've taken this another step, by using the concept of a unit to identify the highest meaningful efficiency that could be used to establish ideal expectations.

For example, if my task is to edit a page with 500 words, then the highest completion I could reliably measure is 499 words (500 minus one), which is 499/500 or 99.8% of the total. That fraction is also the highest meaningful efficiency, which translates into an expectation of editing 499 words in the best-case time. With average editing ability, I would have an efficiency of 50% instead of 99.8%; so during the best-case time, I will have only edited 50% of the total, or 250 words. If I'm responsible for meeting a schedule based on 99.8% efficiency, at the end of the best-case time I will be behind by 249 words (499 minus 250), or 49.8%, which is my error at that time, and it will take nine times that long to reach zero error. A manager tracking my progress up to the best-case time would see an even worse picture, because my error would reach a peak of 67% when I was at just 40% of the best-case time. Ideally, of course, the manager should plan for the actual time to achieve zero error, and not care what happens until then.

Editing ten pages instead of one could be treated as a single task, and the same fractions would simply apply to the larger number of words, with the minimum allowable error at the end now being ten words instead of one. If, however, this error was still kept at one word, then the highest completion (and the highest meaningful efficiency) would increase to 99.98% (or 4999/5000 words) and have significant side-effects which might together be considered a major degradation in performance. For one, the manager would now need to allow more than 12 times the new best-case time (which accounts for all ten pages) for me to reach zero error. My maximum error would increase to nearly 74%, occurring at 32% of the best-case time; and at the best-case time, my error would be slightly higher, at 50.0%.

My actual efficiency in editing pages is higher than the average, more like 70% than 50%, which would decrease the time and value of the maximum error, as well as the amount of time to reach zero error in each case. There would still appear to be a degrading effect on performance as the complexity increased (for example, the maximum error would have increased from 54% to 62%), and that could still raise an unnecessary "red flag" by a manager who was looking too closely and didn't expect it.

The real world is certainly messier than this theoretical discussion might imply. As I described in "Units of Completion," a lot depends on whether the task you think you're evaluating is one of these idealized pure tasks, a parallel combination of pure tasks, or a sequence of pure tasks. Since my analysis is based on actual observations (as are the other models I've developed), the behaviors I've identified are potentially observable in actual situations, and are therefore subject to test. They suggest a reasonable set of explanations for what may be unresolved or even unrecognized issues in actual applications, which is why I've brought them up.

One such issue, which I alluded to and can foresee, is an increase in waste: wasted time, wasted effort, and wasted physical resources. For example, a coordinated "task" such as a major industrial or government project might be terminated because of commitment to unrealistic planning goals that could not be met, and the waste of discontinuing it would be added to the loss of opportunity for meeting the needs it was intended to address. Spikes in what I've called "error" might result in the waste of resources to correct problems that don't exist, which rings true as a consequence of too much complexity. If more realistic schedules are impractical, either because they demand resources that aren't available, or because of competition with others who do not acknowledge their necessity, then the gains of previous effort should be preserved as much as possible until a new and more effective task -- or set of tasks -- can be devised. If preservation cannot be done, waste seems inevitable, and the ultimate objectives of the task are too important to abandon, then cooperation (rather than competition) may be needed between multiple entities who can together address the impediments to success.

Tree Line

In the late 1990s, I was on a group hike that nearly ended in disaster. About a dozen of us were above tree line on a mountain just as a thunderstorm threatened to move in. Our goal was to reach a small lake near the top, but the thunderstorm made it too dangerous to continue. A handful of people insisted on going anyway. Since it was an organized hike, the entire group needed to return to the trailhead together, so the rest of us waited at an abandoned mine so the others could find us when they returned from the lake.

We hunkered down in what little cover we could find just as the thunderstorm moved over us and began dumping torrents of rain. In the distance, we saw a couple of people become trapped on a rock face, and we were soon joined by a larger group of hikers who were less prepared than we were. We assisted the newcomers and debated just how safe we really were. The storm was bigger than we hoped, and it became clear to most of us that the risk of staying was too great. During a brief lull in the rain, we and the newcomers made a dash for the trees. Luckily, the rest of our group had made the same decision, abandoning their trip to the lake, and joined us at tree line. After hiking down the mountain as fast as possible, we encountered emergency vehicles waiting for the hikers we had seen on rock face.

I was reminded of this story recently as more bad news came in about humanity's sabotage of natural systems. Honeybees, critical to the survival of plants, are losing habitat because of climate change. Meanwhile, scientists have documented a mass die-off of seabirds that suggests serious problems with ocean ecosystems. Catastrophic seal level rise may now be inevitable, again due to climate change; and a new study indicates that we humans are critically reducing the collection and availability of energy necessary for ecosystems – and us – to function.

Like hikers determined to get as far above tree line as possible, we have defined "progress" as distance from Nature. We have done the equivalent of cutting down trees to fuel our ascent, altering the weather in the process and spawning the thunderstorm that threatens to maroon us, and then kill us. Heading back down the mountain is perceived as an act of cowardice, giving up on our dreams; so some people go ahead, while others compromise by waiting in the brush just above tree line until they come to their senses. Meanwhile, the risk is growing that lightning will cause the remaining trees to burn, cutting off escape, and that the thunderstorm will grow and last a very long time.

We can do the equivalent of retreating below tree line, and try to grow as many trees back as possible to reduce the risk; we could hunker down and hope the storm passes; or we could follow our original plan and keep going up. The option you choose depends on what you value; and if you value people's lives above arbitrary personal attainment, then the choice is obvious.

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.