How Powerful is Your Vote?
Power is the ability to act according to your will by exerting control over your environment. The more power you have, the more needs and desires you can fulfill.
But things get complicated as soon as your environment becomes polluted with other people. Resources are scarce, and this leads to conflicts. Power over nature can extend to power over people.
Your power is diminished to the extent that others can exert their power over you and your environment, and vice versa. In the extreme case, a totalitarian dictator can exert absolute power over everyone else. Democracy promises to mitigate this risk by distributing power evenly among the population.
“Empowerment” is often regarded as synonymous with voting. The vote is the means by which marginalized groups could break their shackles, have their say, and win back the freedoms that they have lost to the power of others.
But how much power does a vote really bestow?
Empowerment by Voting?
Under conditions of scarcity, the exertion of power takes the form of a choice. A choice is an “all or nothing,” “A or B,” binary event. You can gain one thing only by giving up another.
A single vote is not a choice; the whole election process results in only one choice. Regardless of how many votes are cast, the result is a single “A or B” outcome.
As an isolated individual, you make 100% of all choices. You are not all-powerful, because your available choices are still limited by natural and technological constraints, but you are in full control.
Let’s introduce one other person, and empower each of you with an equal vote on all choices. You now only have 50% of the power you previously held. So does Person #2.
Now Person #3 arrives, and you and Person #2 vote unanimously to empower Person #3 with an equal vote. This noble choice has knocked all three of you down to 33%.
There is a simple mathematical relationship at work here. The power of any individual’s vote can be calculated as:
Figure 1 shows the hyperbolic curve produced by this function (orange curve), up to 350 million people. The US population is around 330 million.
Well, that doesn’t look like much. It seems to immediately drop from 100% to zero.
Changing the horizontal axis to a logarithmic scale reveals more detail at the lower end:
We see the effects of 2 and 3 voters, as discussed above. By moving to the right along the curve, we can consider increasing population sizes.
At 10 voters, your voting power is 10%. This could be a small club, sports team, or business. They say that if 10% of a population is passionate about a cause, the others will often follow. So an individual may have some sway at this scale.
At 100 voters, your power drops to 1%. This could be a community group, village council, class president, or medium-sized business. 1% already sounds low, but you could potentially form a voting bloc on an issue by canvassing people one at a time. Empowerment at this scale takes some work.
Let’s jump to 10,000 voters. This is somewhere around the median population for a typical U.S. city. You are empowered with 0.01% of the choice for mayor.
0.01 percent. You call that “empowerment?”
I call that “zero.” And that’s at the scale of small town local politics, where “your vote really matters.”
Your coveted vote for President of these United States gives you a whopping 0.000,000,8% of the choice. That’s less than one millionth of one percent of a choice, or 8 nano-choices.
Less, if there’s a strong voter turnout.
The measurement error when counting votes is probably a few orders of magnitude higher than this, so that your vote is truly worth zero.
In an election, you have no choice. You are 100% powerless.
Empowerment by Isolation?
There are stereotypes of the right wing sovereign man or left wing hippie who disconnects from society to live off the land in survivalist fashion. This isolation is an attempt at re-empowerment by moving to the left along the curve, so that your choices are no longer subject to the will of anyone else.
Is this what it takes to empower yourself?
The isolationist has overlooked a key assumption in the chart above. He may throw off the burden of society and move towards 100%. But what does he end up with? 100% of what?
We can more formally define an individual’s power to make his own choices:
The first term represents the total number of choices that are available to an individual at any given time. In the previous section, we assumed a fixed number of 100% for this term.
Under this isolationist scenario, “Individual Production Power” represents the number of choices available as a result of the individual’s skills, knowledge, and real capital goods (tools, machinery, house, local resources, etc.)
“Control over Choices” is the “Individual Voting Power” as shown in Figure 2 above, so that we can rewrite Individual Power as:
The isolationist has restored the second term to 100% by reducing “All Voters” to “1 Voter.” However, his power is limited to whatever he can produce with his own two hands. He can increase his number of available choices by learning new skills or building capital goods, but he does so without any trade or outside assistance. 1
Isolationism does not lead to empowerment.
Empowerment by Voting Bloc?
Political analysts pore over demographic breakdowns of election exit polls to build narratives that a particular group, or bloc, of voters swayed the result. This may provide some insight for the marketing team of a future candidate, but is of little use to an individual voter.
If 100% of voters agree with you on everything, then the result is effectively the same as the isolated individual case. You have 100% control, since nobody will oppose any of your choices. Here, democracy is superfluous.
This is why the issue of slavery does not appear on any modern ballot; it is no longer a contentious issue. Issues are only discussed if they are sufficiently controversial for each side to energize a different voting bloc. This makes democracy inherently polarizing.
So in reality, some people will agree with you and some won’t.
To calculate the voting power of a bloc, the “Bloc Size,” or number of people in a sympathetic bloc, replaces “1 Voter” in the numerator:
Figure 3 shows “Bloc Voting Power” curves for various Bloc Sizes:
If everyone else disagrees with you on every issue, then your Bloc Size=1 (orange curve), which is identical to the original curve shown in Figure 2. You are powerless.
And… you’re probably a libertarian.
Since there are often many like-minded people on a given issue, and a group of like-minded people may be able to convert others, it is possible to achieve one of the curves above, or something in between.
This possibility is the primary reason for belief in “Empowerment by Democracy.”
This idea is typically applied to a group of people assumed to share common interests, such as socioeconomic class, geographic location, culture, gender, or ethnicity. Under the somewhat dubious assumption of “identity politics” where votes are cast based on group identity rather than consideration of individual issues, these people are counted in a bloc.
Since these are often numerical minority groups, they need to sway other voters to support their cause in order to win a majority vote.
Majoritarian democracy inherently and inevitably disempowers minorities; yet proponents of democracy encourage “empowerment” of these groups through voting. These advocates are either earnest and innumerate, or are cynically courting votes for a particular party.2
If you want to disempower minorities, tell them that voting can empower them. If you want to empower minorities, keep reading.3
Empowerment by Activism?
“Disempowerment by Democracy” is manifest in every protest, petition, and political campaign. Each of these strategies is an attempt by a small, powerless group of people to influence a larger group and form a majority voting bloc in hopes of amplifying their power.
A nationwide petition with 100,000 signatures is represented by the blue curve in Figure 3. This vital issue is worth 0.1% of the President’s attention.
The green curve in Figure 3 shows the power of a “million man march.” This is a massive effort of activism. 1 million people taking a day off from work costs roughly $60 million in lost income (or lost leisure time). This expense, around 10% of a presidential campaign budget, buys a measly 1% of the result for a national vote.
If the publicity goes viral, so that each protester wins over 10 converts, crank that up to… 11%.
It’s something, but is it enough? Could you do something more beneficial with that time and effort?
The amount of “Individual Bloc Voting Power” that you can exert over a voting bloc is determined by:
“Converted Voters” is the number of people whom you can personally convince to join your bloc, despite their previous disagreement with you. Your friends who already agree with you don’t count. How many of your friends have ever changed their mind about politics because of something you said? Ten? One?
If you miraculously convert 100 friends from “voting against Politician A,” to “voting against Politician B,” your Individual Bloc Voting Power for President jumps from 0.000,000,8% (orange curve) to 0.000,08% (red curve).
In a relative sense, that’s certainly something. In an absolute sense, it’s still absolutely nothing.
Add to this the uncertainty that your own efforts may still fail to build a large enough bloc against your opposition, who may have formed their own bloc.
This isn’t empowering anyone, except for a few people who think shouting things that rhyme with “hey hey, ho ho” is a pleasant way to spend an afternoon.
Adjustment: 50% Plus One
The above calculations implicitly assume that an almost unanimous vote is required to win an election.
However, election success typically requires only a simple majority, meaning a bloc size of 50% of all voters, plus one extra vote.
This can be applied by reducing the “Number of Voters” in the denominator by 50%, plus one. This effectively doubles the power of any size bloc.
The Myth of the Marginal Voter
An effect of the 50%+1 simple majority rule is the potential for a single “marginal” vote to sway the result. This is the critical “plus one” vote in the event of a tiebreaker.
The power of a marginal vote depends on the “Margin Size,” or difference between the majority and minority. We will call this m for short, and we’ll introduce the variable n as shorthand for “All Voters.”
Since many elections are close, this can dramatically increase the voting power of the marginal voter. If there is a perfect 50/50 split, the single “plus one” marginal voter has 100% power and decides the result!
How likely is this to occur? The probability of determination by m marginal votes is:4
Where:
n = Total Number of Voters (All Voters)
m = Margin Size (number of marginal votes)
! : factorial operator, where n! = n(n-1)(n-2)…3x2x1
For a single marginal voter, m=1.
The probability that an election will be decided by a single vote decreases similarly to Individual Voting Power (Figure 2), but at a slower rate, as population increases.5
The conditional probability that a single vote will sway the election and that your vote will be this marginal vote is:
For the US Presidential election, this is 0.01% * 0.000,000,8%= 0.000,000,000,008%, or 1 in 1.3 trillion.
However, this last calculation is meaningless!
The marginal voter is always part of the majority. In fact, any vote in the majority could be considered the marginal vote. It is impossible to identify any particular voter (or bloc) as the marginal vote.
There is no “marginal voter.”
But It Was So Close!
The illusion of the marginal vote is buoyed by the propensity of elections to be decided by narrow margins.
A US Presidential election is considered a “landslide” if it is won by only a few percentage points. As it turns out, this is a correct interpretation!
In any election, m=1 is actually the most likely margin size, since it is the mean (peak) of the binomial bell curve as shown in Figure 5.
However, as shown in Figures 4 and 5, this is still a small probability, especially at larger total numbers of voters.
The probability that an election will have a margin smaller than m is given by summing the probability of a particular margin size over the possible margin sizes from 1 to m.
Figures 6 and 7 show this cumulative probability distribution curve for various total numbers of voters, with the margin size represented as number of voters m (Figure 6) and percentage of all voters m% (Figure 7), where:
This shows that as the number of voters (n) increases, the expected margin size (m) (x-axis) in percentage terms is reduced for a given probability (y-axis).6 Note that the curves appear to reverse order when switching between number of voters (m) and percent of all voters (m%) on the x axis.
With 10,000 voters, 99.9% of elections should be decided by a margin of less than 300 voters (3%)
99.9% of US Presidential elections should be decided by a margin of less than 40,000 voters (0.3%) (popular vote, ignoring the electoral college).
This sounds encouraging for “Empowerment by Democracy.” Did your 0.000,000,8% just become 1/40,000 = 0.002,5%?
Unfortunately, no. You still only have one vote, which puts you at the very left side of these curves. Unless you can convince 40,000 people to change their vote, the result in the previous section stands.
Empowerment by Popular Mandate
Actual Presidential elections often have much wider margins than just 0.3%. Why is this?
The above margin calculations treat each vote as a 50%/50% random event. A similar outcome would be expected if every voter simply flipped a coin.
The expected margins for 99.9% of elections could arbitrarily be considered a “noise floor,” meaning that for smaller margin sizes, the result is indistinguishable from random statistical noise.
If the actual margin is significantly larger than this expected margin, then a clear signal of preference has been received. This signal is the vaunted “Popular Mandate.”
We can quantify this signal in terms of “Shannon Information.” This is an alternative way to compare very low probabilities. Shannon information is measured in bits, just like the digital bits in a computer.
The less probable an event is, the more information its occurrence conveys. Small margins are more expected, so they convey very little new information. As mentioned above, they are just noise.
Large margins convey much more information, which scales parabolically (on the order of m2).7
Elections with more voters convey more information for a given margin size when expressed in percent of all voters (m%).
We can rank US presidents by the amount of information in their popular mandate. As we have seen above, smaller margins are expected for elections with more total voters. So if the total number of voters varies, it is not an “apples for apples” comparison to simply rank based on the percent margin. This approach corrects for differences in probability of a margin size due to the total number of voters.
And the top prize goes to the people’s hero… Richard Nixon!
Elections which were decided by the electoral college, contrary to the popular majority, are at the bottom, and are shown as having negative information. This is a convention chosen here to show the “Unpopular Mandate” despite the technical win. The total information for these five elections is shown in maroon.
Trump’s 2016 campaign pulls up the rear, with negative 46,000 bits. Better luck next time, sport!
In comparison to a simple percent margin ranking, elections with lower total votes tend to rank lower, and those with more total votes tend to rank higher. This reflects the lower probability of larger margins for larger populations.
Due to population growth since 1788, earlier elections tend to rank relatively lower. Despite Washington’s unopposed victories, he has much less of a popular mandate than Nixon because a unanimous decision in a population of 4 million is around 10900,000 times more likely than a 23% margin in a population of 200 million. That’s a one with 900 thousand zeros behind it.
Interestingly, most elections have a clear signal. In physics, a “5 sigma” signal, corresponding to 0.000,000,3% probability, or 22 bits, is sufficient to declare a newly discovered particle. Only three elections are noisier than this threshold.
1960 (JFK v Nixon), a famously tight race with 0.17% margin, has 141 bits. This result had a 1 in 1045 chance of randomly occurring.
There are around 1080 atoms in the known universe. 266 bits is like randomly selecting any single atom, and it happens to be the one you were looking for. That’s a pretty strong signal. Every election above 1884 (Cleveland v Harrison) exceeds this, most of them by orders of magnitude.
But what does this all mean? Has this bit of nerdmongering provided us with any real information?
It demonstrates that the implied assumption of “every voter flips a coin” in the previous section does not typically hold in real elections. There tend to be surprisingly strong margins which indicate a clear preference among the electorate.
However, strong margins don’t help your individual voting power. As discussed above, you are at peak power when you are the only marginal vote. A higher probability of a large margin is a lower probability of your vote mattering.
A large margin does empower the victorious politician to claim that his choices represent “the will of the people.” That’s a bold claim to make based on a few kilobytes of information.
The opposition voters might disagree, along with the 40% of the population who abstain. Every US President has been elected by no more than 23% of the total population.
23 percent. You call that a “Popular Mandate?”
I call that a scam. Voting is for suckers.
I propose that politicians should have an information budget which limits the amount of legislation they can enact to the number of bits given by their popular margin. This ~5,000 word article (plain text, zipped) is about 12.5kB, so a 50kB (400,000 bit) margin earns you about 20,000 words. More than 60,000 pages of new regulations are currently written each year, so this will be a tight diet.
Who’s with me? Let’s form a voting bloc!
Empowerment by Abstention
In every election, some people abstain from voting. Even where voting is compulsory, invalid “Donkey Votes” are cast or fines are paid in lieu of voting. Case in point:
THIS IS 2020, PEOPLE, WHY AM I THE ONLY ONE VOTING AGAINST SLAVERY?!!!!!11
Voters should welcome this. Every abstainer increases the voting power of all active voters.
When I abstain, my 0.000,000,8% voting power is distributed among the 130 million (minus one) people who do vote for the US President.
Each voter is now empowered with an additional 0.000,000,000,000,006% of a choice. That’s 6 femtochoices.
I expect a proportionate amount of gratitude (6 femtothanks) from everyone. You’re femtowelcome.
While this is an absurdly small number, “Get Out the Vote” campaigns are counterproductive in principle, unless done selectively to form a bloc.
In our general formulation, abstainers are considered by using only the number of participating voters, rather than the entire population.8
You may be concerned that by abstaining, you will become a victim of your ideological adversaries’ idiotic and tyrannical policies, rather than having the chance to force your own idiotic and tyrannical wise and benevolent policies upon them.9 You are correct; you are already a victim of many idiotic and tyrannical policies. Your vote won’t change this fact.
Maybe everybody would be better off if fewer of their choices were subject to the influence of their political opponents.
The Scope of Democracy
To avoid complications and present the best case scenario for “Empowerment by Democracy,” our analysis here assumes “one man, one vote” direct democracy. We also assume that there is no coercion, corruption, intimidation, bribery, gerrymandering, voter fraud, voter suppression, miscounting, or any of the other shenanigans that plague real world elections.
This is democracy at its theoretical best.
The above charts show how your life would be affected by a “Totalitarian Democracy,” in which every choice you make is ultimately decided by a referendum.
The “Isolated Individual” and “Totalitarian Democracy” scenarios considered above represent extremes of a spectrum, with reality somewhere between these two cases.
Let’s assume that only 30% of your choices are affected by democratically decided laws. If this sounds high, compare your total taxes paid (federal, state, local, social security including the employer payroll tax, sales tax, property tax, etc.) to your gross income. Then estimate the percentage of your daily activities that are restricted, mandated, or otherwise influenced by various laws, regulations and surveillance. 30% is probably low-balling it in the land of the free.
This means that Person #2 would reduce your power by 15% (½ of 30%) to 85%. Person #3 would reduce it to 80%, and so on.
In a large population, your “Amount of Control Over Choices” would asymptotically approach 70%, instead of zero. This function is:
To generalize, where d is defined as the percentage of choices determined by democratic voting:
Which simplifies to:
Figure 10 shows various values of d as separate curves:
Proponents of democracy want to increase d so that more decisions are made by voting. This moves the curve down, towards totalitarian democracy (yellow curve).
Advocates of limited government want to increase Individual Power by reducing the value of d, thus causing the curve to move up towards the anarchic case of 100% individual choice (orange curve).
A lower value of d also decreases the amount of power bestowed by a vote. This is a trade-off between control over your own choices and control over the choices of others. d=30% means that your vote only empowers you to control 30% of other people’s choices, compared to 100% under “totalitarian democracy.”
So here, too, while your Individual Power may be stronger, the case for “Empowerment by Democracy” grows weaker.
Empowerment by Markets
Voting is a zero sum game; to empower one person, you necessarily disempower everyone else. The result is an entire population of disempowered people fighting each other rather than challenging the politicians whom their votes have empowered.
Democracy is the passive-aggressive realization of the Hobbesian “war of all against all.”
Markets, however, are positive sum. You choose to give up something of less value to gain something of greater value. Two people will choose to trade only if they both expect to benefit. This simple fact is the seed of real empowerment.
When two people trade goods and services, the additional gains realized by each party, whether in money, real goods, or psychological satisfaction, are profits. These profits incentivize both people to produce more than they need, in order to reap the benefits of trade.
The more value that people can produce, the more power there is to go around. Scarcity causes conflicts. Abundance eliminates them.
In a market, you control 100% of each choice. The choices available to you are limited only by your budget, which is determined by the value that you can provide to others.
Most of these decisions seem less glamorous than weighing in for 0.000,000,8% of the choice to unleash some new national healthcare scheme that ostensibly promises everyone access to more choices.
But you make hundreds of 100% anarchic choices each week, each of which has a direct, immediate, positive impact on your life: choosing leisure activities, enjoying a meal, creating something, or buying a new tube of toothpaste.
Compare this “Empowerment by Toothpaste” to your uncertain 0.000,000,8% of a choice every 4 years. Which of these choices are really empowering you?
In contrast to the “disempowerment curves” above, we can construct a similar “empowerment curve” for the market. Again, we will set the arbitrary benchmark for an isolated individual at 100%.
A second person introduces a significant leap in possibility and efficiency. Two people can perform tasks that are impossible for one person, such as carrying heavy objects or making babies. They can also specialize according to their individual strengths.
As more people enter the market (perhaps due to all of the baby-making), so do more diverse goods and services. Larger and more complex capital investments become feasible, further increasing the abundance of goods through greater productivity. This, in turn, increases the number of choices available to each individual.
The representative curve of this real profit, or “Market Power,” increases along with population. This is not a specific measurable metric like GDP, but can be assumed to scale with population in a similar fashion.
In his brilliant book Scale, Geoffrey West shows that these returns to scale can be generally estimated using a “superlinear power law.” For every doubling of population, metrics such as incomes and GDP per capita increase by 15%.10
In base 10, this is an exponent of 0.2016. That should be easy for politics junkies to remember.
This relation is called, appropriately, a “Power Law.”
For a city of 100,000 people, this is a 10X (1,000%) increase. For a city of 20 million, it’s 30X.
Extrapolating beyond West’s data, for the US population of 330 million, this could be up to a 50X increase! For a globally connected market of 7.8 billion people this could be up to a 98x increase.11
That’s what I call “empowerment.”
Markets overwhelmingly increase power for each individual as population increases. In the developing world, impoverished people migrate to slums near big cities to gain access to these benefits of markets.12 Even a destitute homeless person will benefit from a larger population, since a wealthier society can afford to better support him. A few coins from a stranger will benefit him much more than his vote ever will.
Under democracy, society is a counterproductive burden. But in a free market, it is the engine that drives empowerment.
The General Theory of Empowerment
All of the above can be combined into a single formula to show the interplay between “Empowerment by Markets” and “Disempowerment by Democracy.”
Let’s call this “The General Theory of Empowerment.”
Starting from our original definition of Individual Power, we can plug in the various components defined above. We will ignore the “probability of a single vote margin” element to avoid complications with various margins and bloc sizes:
Market Power includes your Individual Production Power. This means that:
Which finally yields:
For simplicity, we will assume the following:
- The Required Majority to win is 50%+1
- 100% participation rate
This will give an idea of how having more or less democracy affects Individual Power within a market economy.
Zooming in on the y-axis, with a dotted line showing the “break even” point:
Curves with more choices determined by voting (high d) ultimately reduce individual power, since the democracy component is dominant. Under “totalitarian democracy” (yellow curve), any contributions from the market are crushed.
For any value of d<100%, there is a “break even” point where the power curve can cross over 100%, escaping the drag of democracy. The market component counterbalances, then overwhelms, the democracy component.
As d is reduced, this occurs at lower population sizes, and the benefits of a large market economy can be realized. However, it never achieves the empowerment of a purely free market (orange line)
A sympathetic voting bloc improves the situation (for those in the bloc) by preserving the d=0 condition up until the point where that bloc is no longer dominant. Once the bloc size is less than 50%+1, “Disempowerment by Democracy” kicks in.
Figures 14 and 15 show these curves for a Bloc Size of 10,000:
Zooming in the y-axis:
In US Presidential elections, typically around 40% of the Total Population votes. This has the same effect on your voting power as if you had increased your bloc size by 1/40%=2.5x. It doesn’t mean you have a bigger bloc size, but each vote in your bloc has 2.5 times more power.
In this case, the curves look as follows:
Empower Yourself, and Everyone Else
If you were unable to vote, what would you do? How would you seek to empower yourself?
How powerless would your vote have to be before it is no different than being unable to vote? Less than 0.01%? Less than 0.000,000,8%?
Politics has it backwards. Elections play an increasingly significant role at larger scales of government, yet just a handful of voters renders your vote powerless.
Democracy does not empower anyone. It disempowers everyone, except for the politicians whose own choices are granted legitimacy by the resulting “popular mandate.” This is one-way empowerment, from the masses to a minority of elites. Voting only encourages them.
Democracy gives only the illusion of empowerment. Thinking that you have some measure of control, even a vanishingly small one, you relinquish your choices to the electorate.
You choose to disempower yourself.
But you are not powerless. Free markets promote two-way empowerment. Through cooperation, competition, investment, and trade, you can empower others.
If you can form a voting bloc, you can form a buyers group and negotiate discounts or demand new products and services from companies who will fall all over themselves to secure that many dedicated customers. They would do the same for your political rivals, and you would be OK with that because it doesn’t harm your interests. You all benefit from the economy of scale.
Markets promote cooperation among diverse, even antagonistic, groups of people. Democracy promotes antagonism among otherwise friendly groups of people.
By accepting your powerlessness under democracy, you can focus on real empowerment. You can stop wasting time, energy, and friendships arguing over politics. You can stop expecting everyone else to make choices for you.
Don’t protest; produce. The choice to abstain is a first step towards empowering yourself, and everyone else.
Don’t vote.
I’m starting to feel like my vote doesn’t matter…
Share this:
- In reality, people who live like this are typically not 100% isolated, so they can benefit from access to markets. I am describing the extreme case as a benchmark.
- I’m not being partisan here. While the left is more likely to use the term “empowerment,” the right also engages in this practice, e.g. with religious, pro-gun rights, or anti-immigration groups.
- It should be clear that I am not being discriminatory here; I don’t want anyone to vote. I want everyone to actually, not just symbolically, empower themselves. Voting does not achieve this.
- To explain this formula, we count the number of unique combinations of votes yielding 50%*n+m/2. This majority size is given by (n+m)/2 voters. The number of combinations is given by the binomial coefficient formula, with k=(n+m)/2. This gives the number of combinations of this size majority voters for candidate A.
The same number of combinations are possible for candidate B, so we can multiply this by 2 since we are agnostic as to which outcome prevails.
The total number of vote combinations is 2n. So we divide the number of majority combinations by 2n to get the probability.
This can also be calculated (e.g. in a spreadsheet) using a binomial distribution function with 50% trial probability, n trials, and (n+m)/2 successes. This is the method I used to produce these charts.
Bayesian inference could also be used to incorporate data from historical margins. This would likely result in a larger expected margin size.
- Approximately on the order of O(n-0.485)
- While this seems counterintuitive, it is due to the nature of a discrete probability distribution. The sum of all possibilities for the binomial “bell curve” must add up to 100%. As n increases, the bell curve flattens as probability is distributed from the middle to the tails, which now have more elements. Figure 5 shows these curves for various values of n.
- In fact, in order to avoid long tail probability precision errors in Excel (which is limited to around 10-300), some higher values in this chart are estimated using an extrapolated quadratic curve fit. I have checked this against actual values calculated using Wolfram Alpha and it is surprisingly accurate. I leave the proof as an exercise for the reader, because I haven’t done a proof.
- This participation rate is based on the total population, not just eligible voters. So it will be lower than the rate that is typically reported.
- At the risk of sounding partisan, I assume that the reader is on team “wise and benevolent.”
- West, G., 2017. Scale. New York: Penguin Press.
- West’s work measures these metrics for individual cities ranging from 30,000 to 25 million people, not entire countries. The benefits to scaling for cities derive from efficiencies of physical infrastructure shared by large populations. So the relation may differ when considering the population of a small village or entire large country.
There are also complications in positive and negative directions due to, e.g. infrastructure congestion and benefits of interconnected global trade, which are beyond the scope of this article.
- For a brief synopsis of the relative benefits of slums over rural villages, see https://foreignpolicy.com/2012/08/13/in-praise-of-slums/.
For a more thorough discussion, see Glaeser, E, 2017. Triumph of the City. New York : Penguin Press.
We have also discussed migration to slums in Anarchitecture episode 10.
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