Cities and power laws

Cities are perhaps the greatest human-made constructs. They have their good and bad characteristics. The bigger the cities, the more their good and bad characteristics strengthen their affect people. Cities have their physical lifelines. Physical flows of people, resources, and energy combined compose the metabolism of the cities. Cities look and feel different, but power laws affect them. These scaling relationships have hidden regularities. Ideas, such as patents and businesses in cities follow power laws too. If you are not familiar with power laws, you can learn about them here.

Power laws in cities are similar around the world

Power laws have the same scales around the world. For example, the amount of people living in the second biggest city in a country has about half the amount of people in the biggest city. And the third biggest city has about one-third of the people in the biggest city. Cities have national characteristics. The scale of different metrics depending on the culture, economy, and individuality of each national system. Scaling laws within the countries are similar, but the physical and mental flows differ between countries. If the biggest city in one country has 10 million people and the second biggest city has 5 million, then the biggest city in another country has 4 million people and the second biggest city has 2 million people.

15 percent rule

Cities have different power-law scales, but the most common and important exponent is 0.85 or 1.15 depending on which parameter is scaled. What 0.85 exponent means is that when the size of the city doubles you need 15 percent less of something else per capita to achieve a certain goal. 1.15 means that you will get 15 percent more of something per capita when the city size doubles. These exponents are common in the flow of resources and energy in cities. They are also common in social networks. When the city size quadruples, it needs only about 72 percent of something per capita and gets about 32 percent more of something per capita. Thus, the bigger the city the more efficient it becomes. Getting something more or less per capita isn´t always a bad or a good thing.

Bigger cities need less infrastructure per capita than smaller cities. Physical flows like roads, water and gas lines, and electrical networks all scale to 0.85. A city with 2 million people needs only 185 percent of the infrastructure compared to a city with a million people. The reason for this is that the end-users don´t have to build everything only to themselves. For example, gasoline stations need less space because the economy of scale affects it. Gasoline stations can have bigger gas supplies in a city because they have more potential customers living in the same area. Therefore, you need a smaller amount of them. Most of the physical infrastructure follows this double the size and needs 15 percent fewer resources per capita rule around the world. At least in places from where you can find official statistics about these parameters.

When the city size doubles, it produces 15 percent higher wages, more patents, more crimes, and more sexually transmitted diseases per capita. Most socioeconomic parameters follow this rule, including the speed of walking. These scaling laws make cities more efficient and people more productive in them. They also bring unwanted consequences, but most people in cities have better lives because of them. The best way to create greater nations is to enable the growth of their cities. Of course, nations should also focus on limiting the bad consequences of growing cities before they grow too big. The glorious past of the cities won´t guarantee a glorious future for them. The latter is just more probable consequence.

Some parameters don´t follow the 15 percent rule

Some parameters have a scaling exponent of close to one like the number of houses, and jobs per capita. Doubling the city size doubles the number of businesses. The diversity of the businesses stay pretty much the same. New kinds of businesses increase only by 5 percent when the city size doubles. What happens is that when some businesses become successful in a city, people living in it establishes new businesses that support the success stories. There are many other power-law exponents concerning cities. Let's forget them, at least for now.

There are limits to growth for cities. They don´t grow forever. When the maintenance costs in cities become too large, the growth stops. Maybe the biggest bottlenecks for the growth of the cities come from energy supplies. Growing cities need more energy. If the supply of energy can´t move as fast as they need for energy grows, growth will eventually stop. It also stops when the amount of people wanting to move to a city diminishes.

Understanding power laws is important if you want to understand the world. Power laws are more common than most people think. You can learn more about them from the book Scale, by Geoffrey West. I recommend you to read it.

-TT

Pareto´s law (80/20 rule)

Pareto´s law is a power law. You probably know 80/20 rule, but may have never heard of Pareto´s law. It is the same law. Pareto was an Italian social scientist who first noticed that 20 percent of the Italian population owned 80 percent of the assets in Italy. The inverse was also true, 80 percent of the people owned 20 percent of the assets. In reality, this rule should be restated. The better way to describe it is to say that ”A high proportion of effects come from a low proportion of causes or effort. For example, a small proportion of employees create the most added value to the company and its clients. 80/20 proportion can also be a 1/50, or 30/70, and other proportions. One of the most extreme proportions is that only a few percents of authors that write books sell almost all the books from the total amount of books sold.

Pareto´s law in human endeavors

A small proportion of employees add much more value than others. In simple tasks like machine users or employees in simple office work, the best one percent of the employees are three times more productive than the least productive one percent. In a little bit harder tasks like in retailing or mechanics, the best one percent are twelve times more productive than the last one percent. In the hardest occupations like insurance salesmen, sales negotiators, doctors, and lawyers, the most productive one percent were compared with average employees. The most productive one percent was twice as productive as an average employee. The biggest difference was between the best one percent of programmers and the average programmers and they were twelve times as productive as an average coder. Working with these people is one of the best ways of becoming more productive. If you are an employer, you have to pay these people a lot more than for average employees.

In business, a small number of products and clients can produce the most sales, profits, and losses. There are many examples, like Apple. Iphones bring the most profits and sales for the company. Apple has also many other products and other services. Many businesses have only one big client. When its business goes south, the subcontractor goes bankrupt. If you work for this subcontractor, your job is not as safe as a job in a business that has many clients. If this job is your only source of income, you might be in trouble, unless your expertise is needed elsewhere.

In investing, a small proportion of your investments bring you the most of your investment returns. And a small proportion of your investments bring the most of your losses. It is hard to know which investments bring losses in advance. Most people expect returns in all of their investments. This is a very unlikely turn of event. Only liars can do it. Beware of investing in businesses that have only one big client. When things go south you lose most of your money or all of them. Pareto´s law doesn´t only work with personal investing, it also works in stock indices. Most of the returns from stock indices come from a small number of stocks. And also most of the daily fluctuations have a small effect on indices. This means that few daily fluctuations have the most effect on stock indices.

A minority of your actions take most of your daily time. For example, you work approximately eight hours, sleep approximately from seven to nine hours, and watch television and spend time in social media both take a few hours a day. These actions take most of your time and the majority of actions like eating, exercising, commuting and other small tasks take a minority of your day. Most of your work effort goes to a minority amount of tasks, most of your time watching television goes to few numbers of channels or a few numbers of shows.

Some problems in using Pareto´s principle

Pareto´s principle is not easy to use. Most of the time you don´t know which efforts produce most of the results. Without the right measuring sticks, you can´t know. And it is hard to forecast which actions are crucial to get the best results. You need too many data points to forecast the future when Pareto´s rule is working. For example, according to Nassim Taleb, you need 10^15 data points to have any chance of understanding how these power laws like 20/80 rule work. You have to use a constant trial and error learning for a long time to achieve any meaningful results.

-TT

Risks

Definitions

A risk can be defined as ”A potential of gaining or losing something of value.” or ”An exposure to the chance of injury or loss.”

Personal risks

First I would like to say that risks affect on you in many different levels. For example, they affect on you through corporations, nations and global events like wars. I will keep things in personal level. Humans are very good in understanding risks related to their survival. Especially, when their intuitive decision-making systems are on. You can avoid an imminent threat to your survival without even noticing it. For example, changing your direction from the normal route, because of something is not feeling right. Survival in these cases means not hurting yourself physically in an imminent danger. You are not good in avoiding long-term risks. Modern world has less imminent threats than your brain thinks. This creates many problems that even the risk management experts do not understand.

You have many personal risks other than risks of physiological harm. Financial risk is maybe the most common risk you think about. Losing your job, or inability to pay your debt can cause you harm. You may lose your reputation if you do something really stupid or someone spreads rumours about you. You also have a risk of not being able to adapt to the changing world. Doing what you have always done before is your path of least resistance. Changing things is hard, even when it is necessary. People are also increasing their technology risks, because their dependence on technology is rising. Personal risks can compound in a long-term. Taking small, but unnecessary risks without suffering the consequences fast may lead to severe problems later on. For example, eating crappy food may not cause you any harm for decades, until one day you get a heart attack without any warnings.

Risk Management

Risk management is about probabilities. You need to understand them to understand risk. Repeatable events with fixed probabilities are easier to understand. For example, risks at the roulette table and lottery are fairly easily quantifiable. When you cannot calculate a probability of something happening, you cannot really understand risks. Uncertainty is not well understood. Managing risks without understanding power laws is one of the most common causes of financial destruction for institutions and in a personal level. The other common cause is not understanding complex systems and the second order effects in them.

You have to consider your risk appetite too. Some people cannot sleep well with moderate risks in their lives and some people cannot sleep well without them. Risks are not all bad things. Progress comes from taking risks. Some of them are managed and some of them are not. Humans have developed as a race by avoiding risks concerning on survival. Risk-seeking is about functioning against your basic instincts. It is hard, but many times worth all the effort. When you understand the risks you are taking, it is easier to get better results. You have to ask yourself some questions to understand risks better:

• What are the consequences of not doing it? What will it cost you? What are the unwanted outcomes that can come true? How will you react if they happen to you?
• What are the positive consequences? What will you benefit? How will you react? How do you feel about the outcoming benefits now?

Risks can also be quantified in terms of impact of their consequences. You should use a simple scale in how you want to rate the impact of the risks. Make it reasonable. For example, you can use a three-level scale of impact for risks. Risks with high, medium or low impact. Think about high risks all the time. Risks with medium impact are not that important, but you should check them regularly. Low risks are not that relevant. You should still check them occasionally. You should also remember that low and medium impact risks can become high risks through compounding. In this case, it means taking many small risks many times.

Asymmetric risk

There are two kinds of asymmetric risk. Good ones and the bad ones. An asymmetric risk means that the reward is a lot bigger than the risk taken, or the reward is a lot smaller than the risk taken. In other words, the expected payoff is high or low, depending on the risk taken. Nobody wants to take an asymmetric risk where the expected payoff is crappy. Most often, this happens because the risk taker do not understand what he/she is doing. This happens to the most respected experts too. Especially, financial market participants do not understand risks. One reason is that they haven´t noticed the power laws in the markets or understand them. Many assumptions about the returns in the financial markets are made by thinking through a normal distribution. The problem is that returns in financial markets do not really realize through them. Most of the profits are made with a small number of stocks or during the small number of days, etc. In other words, extreme outcomes in financial markets have higher probabilities than using normal distribution tells you.

It is easy to make a statement that you should avoid all the asymmetric risks, where the downside is enormous compared to the expected payoff. For example, not going to see a doctor, when you have a severe chest pain. Not going to a doctor can cause a death and going to the doctor can save you. You can also take lots of asymmetric risks where the probabilities are against you. When expected payoff is high, you need smaller amounts of successes. This doesn´t mean that all the risks come true and payoff is always high. You should be ready for many losses with these kinds of risks. Overall, the expected payoff will be good in the long run.

Sources:

Smart -> Risk, Andrew Holmes
Risky Strategy, Jamie MacAlister
Fooled by Randomness, Nassim Taleb

-TT

Power laws

Definitions

A power law can be defined as ”A relationship between two quantities such that one is proportional to a fixed power of the other”. It can also be defined as ”A relative change in one quantity results in a proportional relative change in another”.

Basics about the scales of power law

When you find out what the value of the power is in the power law, you can see how things scale. When the power is 1, dependent variable doubles, when the independent variable doubles. When the power is more than 1, the dependent variable more than doubles. For example, when the power is 2, dependent variable quadrubles, when the independent variable doubles. If the power is below 1, then the dependent variable is less than a doubling independent variable.

Power laws in nature

You can see many power laws working in nature. For example, animals in the world are roughly cube shaped. The amount of skin goes up by the factor of 4, when the surface area (m²) is doubled. In the same time the volume (m³) of guts goes up by a factor of 8. Animals lose heat through their skin and generate it through their guts. When the animals grow, their bone structure has to change as the bones´ ability to support the animal grows slower than the weight of the animal. Compare square with a cube. What this basically means that animals with lots of weight need to have strong bones to support it. For example, an elephant has thick legs, because they carry a lot of weight.

The metabolic rates of animals scale to their mass to the ¾ power. It tells you that the amount of the energy you need to survive. It basically means that the larger you are, the less energy you need compared to your per unit of mass. These kind of scaling laws apply to the oxygen intake and heart rate. Breathing and heart rates scale with mass to the – ¼ power. These power laws are approximations. They do not give exact truths. They are estimates based on existing data. These power laws tell about physical limits of each living thing. You cannot find a giant with thin legs or live forever.

Power laws in human constructions like cities

These laws apply to human behavior like wars and human constructions like languages and cities. For example, Lewis Fry Richardson found out a power law about wars. The higher the number of people killed in the war, the more time it takes to a war as destructive as the former one. You can also find same kind of power law working in languages. How often you use a word is described by an exponent of -1. The word that is the second most common in language is found half the times compared to the most common word. This power law applies also the population of cities in different countries and the sizes of companies. The largest city in the country has approximately twice the population of the second largest city. There are many power laws found in city sizes. For example, the bigger the population of the city, the less each person uses gasoline and the less road they need. All these power laws can be used in planning future cities.

Pareto´s law

Pareto´s law means that the minority of causes, inputs or efforts usually lead to a majority of the results, outputs or rewards. This rule states that there is an in-built imbalance between causes and results, inputs and outputs. Sometimes, this rule is defined as a 20/80-rule. For example, 20% of the customers bring 80% of the profits for the company. I wouldn´t really describe Pareto´s law with 20/80 rule, because these scales have so much variation. For example, only a small percentage of the authors sell almost 100% of the books. You have to invert, when you are using Pareto´s law too. For example, you need to understand that most of the bad things that happen have only few reasons. Most of your losses come from few sources. These rules apply to your relationships too. Most people are insignificant for your well-being, but few people cause most of the bad things in your life. Pareto´s law also means that small changes or lucky events may give big results. For example, when Oprah has recommended some products, their sales have gone through the roof.

Universe is full of power laws. You should understand these non-linearities as well as linear cause-and-effect relationships. These laws are much more usual than everybody thinks. Most of the people never notice them. Using power laws to your advantage is hard, but the results are worth the effort. Most of the misunderstood risks come from power laws. Do not ignore them.

Sources:

A Crude Look at the Whole, John Miller

Antifragile, Nassim Taleb

The 8020 Principle, Richard Koch

These power laws are related to misunderstood risks. They will be the topic for the next week.

-TT