As a thinking tool it means approaching a situation from the opposite end of the natural starting point. Most of us tend to think one way about a problem: forward. Inversion allows us to flip the problem around and think backward.

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Instead of wasting your time trying to disprove complex scenarios, you can make decisions more confidently by basing them on the explanation that has the fewest moving parts. In a complex world, using this model helps us avoid paranoia and ideology.

By not generally assuming that bad results are the fault of a bad actor, we look for options instead of missing opportunities. This model reminds us that people do make mistakes. It demands that we ask if there is another reasonable explanation for the events that have occurred.

The explanation most likely to be right is the one that contains the least amount of intent. The mathematics of permutations and combinations leads us to understand the practical probabilities of the world around us, how things can be ordered, and how we should think about things. The introduction of algebra allowed us to demonstrate mathematically and abstractly that two seemingly different things could be the same. By manipulating symbols, we can demonstrate equivalence or inequivalence, the use of which led humanity to untold engineering and technical abilities.

Knowing at least the basics of algebra can allow us to understand a variety of important results. Though the human brain has trouble comprehending it, much of the world is composed of random, non-sequential, non-ordered events. A stochastic process is a random statistical process and encompasses a wide variety of processes in which the movement of an individual variable can be impossible to predict but can be thought through probabilistically. The wide variety of stochastic methods helps us describe systems of variables through probabilities without necessarily being able to determine the position of any individual variable over time.

Compounding is the process by which we add interest to a fixed sum, which then earns interest on the previous sum and the newly added interest, and then earns interest on that amount, and so on ad infinitum. It is an exponential effect, rather than a linear, or additive, effect. Money is not the only thing that compounds; ideas and relationships do as well. In tangible realms, compounding is always subject to physical limits and diminishing returns; intangibles can compound more freely. Compounding also leads to the time value of money, which underlies all of modern finance.

Multiplying by Zero. Any reasonably educated person knows that any number multiplied by zero, no matter how large the number, is still zero. This is true in human systems as well as mathematical ones. In some systems, a failure in one area can negate great effort in all other areas. Insurance companies and subscription services are well aware of the concept of churn — every year, a certain number of customers are lost and must be replaced.

One of the fundamental underlying assumptions of probability is that as more instances of an event occur, the actual results will converge on the expected ones. The opposite of this model is the law of small numbers, which states that small samples can and should be looked at with great skepticism.

For example, the Richter scale describes the power of earthquakes on a power-law distribution scale: an 8 is 10x more destructive than a 7, and a 9 is 10x more destructive than an 8. This is true of all power-law distributions. Regression to the Mean. In a normally distributed system, long deviations from the average will tend to return to that average with an increasing number of observations: the so-called Law of Large Numbers.

We are often fooled by regression to the mean, as with a sick patient improving spontaneously around the same time they begin taking an herbal remedy, or a poorly performing sports team going on a winning streak. We must be careful not to confuse statistically likely events with causal ones. In many, perhaps most, systems, quantitative description down to a precise figure is either impossible or useless or both. For example, estimating the distance between our galaxy and the next one over is a matter of knowing not the precise number of miles, but how many zeroes are after the 1.

Is the distance about 1 million miles or about 1 billion?

### On the Charts

This thought habit can help us escape useless precision. One of the most important principles of systems is that they are sensitive to scale. Properties or behaviors tend to change when you scale them up or down. In studying complex systems, we must always be roughly quantifying — in orders of magnitude, at least — the scale at which we are observing, analyzing, or predicting the system.

Related to scale, most important real-world results are subject to an eventual decrease of incremental value. A good example would be a poor family: Give them enough money to thrive, and they are no longer poor. But after a certain point, additional money will not improve their lot; there is a clear diminishing return of additional dollars at some roughly quantifiable point. Often, the law of diminishing returns veers into negative territory — i.

The Pareto Principle is an example of a power-law type of statistical distribution — as distinguished from a traditional bell curve — and is demonstrated in various phenomena ranging from wealth to city populations to important human habits. Feedback Loops and Homeostasis. All complex systems are subject to positive and negative feedback loops whereby A causes B, which in turn influences A and C , and so on — with higher-order effects frequently resulting from continual movement of the loop.

In a homeostatic system, a change in A is often brought back into line by an opposite change in B to maintain the balance of the system, as with the temperature of the human body or the behavior of an organizational culture.

## Volume Five banjo tabs from Patton Wages - Bluegrass Today

In a world such as ours, governed by chaos dynamics, small changes perturbations in initial conditions have massive downstream effects as near-infinite feedback loops occur; this phenomenon is also called the butterfly effect. This means that some aspects of physical systems like the weather more than a few days from now as well as social systems the behavior of a group of human beings over a long period are fundamentally unpredictable.

A strong network effect is a good example of preferential attachment; a market with 10x more buyers and sellers than the next largest market will tend to have a preferential attachment dynamic. Higher-level behavior tends to emerge from the interaction of lower-order components. The result is frequently not linear — not a matter of simple addition — but rather non-linear, or exponential.

An important resulting property of emergent behavior is that it cannot be predicted from simply studying the component parts. We find that in most systems there are irreducible quantitative properties, such as complexity, minimums, time, and length.

Below the irreducible level, the desired result simply does not occur. One cannot get several women pregnant to reduce the amount of time needed to have one child, and one cannot reduce a successfully built automobile to a single part. These results are, to a defined point, irreducible. Tragedy of the Commons. A concept introduced by the economist and ecologist Garrett Hardin, the Tragedy of the Commons states that in a system where a common resource is shared, with no individual responsible for the wellbeing of the resource, it will tend to be depleted over time.

The Tragedy is reducible to incentives: Unless people collaborate, each individual derives more personal benefit than the cost that he or she incurs, and therefore depletes the resource for fear of missing out. We see a similar result in human systems, as with bad behavior driving out good behavior in a crumbling moral system, or bad practices driving out good practices in a crumbling economic system. Algorithms are best known for their use in modern computing, but are a feature of biological life as well.

For example, human DNA contains an algorithm for building a human being. Popularized by Nassim Taleb, the sliding scale of fragility, robustness, and antifragility refers to the responsiveness of a system to incremental negative variability. A robust system or object tends to be neutral to the additional negativity variability, and of course, an antifragile system benefits: If there were a cup that got stronger when dropped from 6 feet than when dropped from 1 foot, it would be termed antifragile.

A critical model of the engineering profession is that of backup systems. A good engineer never assumes the perfect reliability of the components of the system. He or she builds in redundancy to protect the integrity of the total system. Without the application of this robustness principle, tangible and intangible systems tend to fail over time. Margin of Safety. Similarly, engineers have also developed the habit of adding a margin for error into all calculations.

In an unknown world, driving a 9,pound bus over a bridge built to hold precisely 9, pounds is rarely seen as intelligent. Thus, on the whole, few modern bridges ever fail. In practical life outside of physical engineering, we can often profitably give ourselves margins as robust as the bridge system. A system becomes critical when it is about to jump discretely from one phase to another. The marginal utility of the last unit before the phase change is wildly higher than any unit before it. A frequently cited example is water turning from a liquid to a vapor when heated to a specific temperature.

A network tends to become more valuable as nodes are added to the network: this is known as the network effect. An easy example is contrasting the development of the electricity system and the telephone system. Only with additional telephones does the phone network gain value.

This network effect is widespread in the modern world and creates immense value for organizations and customers alike. In many systems, improvement is at best, or at times only, a result of removing bad elements rather than of adding good elements. This is a credo built into the modern medical profession: First, do no harm. Similarly, if one has a group of children behaving badly, removal of the instigator is often much more effective than any form of punishment meted out to the whole group. The Lindy Effect refers to the life expectancy of a non-perishable object or idea being related to its current lifespan.

Although a human being who is 90 and lives to 95 does not add 5 years to his or her life expectancy, non-perishables lengthen their life expectancy as they continually survive. The renormalization group technique allows us to think about physical and social systems at different scales. An idea from physics, and a complicated one at that, the application of a renormalization group to social systems allows us to understand why a small number of stubborn individuals can have a disproportionate impact if those around them follow suit on increasingly large scales.

A system is spring-loaded if it is coiled in a certain direction, positive or negative. Positively spring-loading systems and relationships is important in a fundamentally unpredictable world to help protect us against negative events.

## Handbook of Public Economics, Volume 5

The reverse can be very destructive. Complex Adaptive Systems. A complex adaptive system, as distinguished from a complex system in general, is one that can understand itself and change based on that understanding. Complex adaptive systems are social systems.

The difference is best illustrated by thinking about weather prediction contrasted to stock market prediction.