The Mathematics of Social Life

Introduction
We can think of a person's personality as a vector in Fⁿ. The dimensions of F represent the dimensions of the goals and interests that one might entertain—what one values, broadly speaking. Some people are more interested in pursuing money than others; some people have more an inclination towards spirituality than others; some people give more importance to their family than others. Thus, we can say that money, spirituality, and family represent some of the nearly infinite dimensions of this space (but we think of n as finite for simplicity). The zero vector would signify balance within all dimensions.

Like R, F is continuous, infinite, and contains postive and negative values; just as value traits come in shades of grey, have no known limits, and always work in pairs of opposites. Although personality doesn't just include values, they are its most important component. Values are what drive action and shape our life's trajectory, which is what we're interested in modeling.

These personality vectors represent a static snapshot of a person's personality. However, people's goals and interests are, of course, constantly changing: at any given moment, a person has a certain place she's at in life, and a certain place she'd like to go. This defines what we'll call a change vector in Fⁿ. The vector points towards the new personality vector one wants to have, and the magnitude represents the rate at which the change is happening. Continuing the previous example, we can say that a person with a penitant heart who wants to focus more attention on his family rather than on money would have a change vector that points 1) less on the money dimension 2) more on the spirtuality dimension and 3) more on the family dimension.

Over the course of our lives, the personality vector traces a trajectory through Fⁿ that describes how our goals and interests have changed. These trajectories are useful in describing, characterizing, and categorizing people's lives. When we say, for example, that someone has meandered through life, we may mean that the distance traveled along the trajectory is much larger than the distance traveled in Fⁿ.

The change vectors are first derivatives of our life's trajectory. At each moment in time, the vectors are tangent to the trajectory, and mark where we're immediately heading. Because our personalities have inertia, our life's trajectory follows the vector unless the vector changes: until we receive some new information, or have some new experience, our thought pattern will not change.

Society
A society is a well-defined set of people. Societies can be both big and small, ranging from small towns, to nations, to the entire world. Societies can also be defined in terms other than geography, such as national origin, occupation, or marital status.

Societies do not have a uniform distribution of personalities. Instead, they tend to cluster in certain regions of the space (this is where stereotypes come from). We can describe the distribution of personalities of a society with a density map on Fⁿ.

Not all people within a society are equally important to us; we tend to think that some have more of a claim on our consideration than others. A social network is thus defined as a graph of all social claims on oneself that one considers meaningful. The graph is organized as a series of concentric rings, with us at the center and each progressively farther rings representing more distant/less valued relationships.

Hypothesis on Nature vs. Nurture
When we are born, we have an initial personality vector in Fⁿ. This is the nature component, since it limits the possibility of our life's trajectory to within the neighborhood of that point. However, once that point is established, we can move in any direction. This is the nurture component.

An analogy is a cow tied by a rope to a tree: the cow can only move as far as the rope will let her, but within the radius of the rope she can move in any direction.

It's hard to say whether babies are born with change vectors as well, i.e. whether they have a conception of what they want to do with life. But even if they did, it would be almost inseparable from the immediate influence that the family exerts on the newborn.

Interaction
Interaction is comprised of two components: liking and influence. They are separate matters, since we are often influenced by people that we do not necessarily like.

We often gauge how much we like something or someone by how much it aligns with our goals and interests. That is, liking is a measure of the similarity between two vectors, i.e. the angle between them. An angle of 0 measures complete liking and an angle of pi measures complete dislike. If a vector is perpendicular to another, then it is neither heading in a similar direction on any dimension, nor is it heading away; perpendicular vectors thus signal indifference.

Note that we are talking about personalities, not genes. People with very similar personalities may look completely different, talk different, and manifest their goals and interests in different ways—but their goals and interests are similar.

Liking, like influence, happens not just through face-to-face encounters with other people, but also by learning about the work they produce. Our work crystallizes our personality in time, making it possible for distant, unknown people to like and be influenced by us, even without directly meeting. Reading the works of an author reveals the idea vector of the author at that time; learning about the social works of a great leader is a way of meeting their vector.

Influence is a question of what influences the change vectors. Since influence can occur to varying degrees, we can think of it as lying on a spectrum. One one extreme, we have complete influence, where both parties fully and equally influence each other. In complete influence, two people leave the interaction with the same new "goal" personality vector, one that is the sum of their previous individual personality vectors: components of the vector moving in opposite directions would cancel out, and components moving in the same direction would build on each other. However, because personality change is continuous, and occurs over time, the change in personality vector is not instantaneous. Instead, after the interaction, both parties leave with change vectors that now point towards their new goal personality vector. Over time, if these change vectors are unaltered, then their personalities will become the same.

On the other end of the spectrum, we have a complete lack of influence, where the interaction does not affect the change vectors at all. Most interaction is either of this latter type (since we can't afford to constantly reevaluate ourselves every time we interact), or somewhere in between.

Because individuals have free will, personality change is a stochastic event, and hard to pin down. However, we do know that a person is more likely to be influenced one direction or another based on the distribution of her social network within Fⁿ.

We can simplify the process of influence by thinking of it as a random event where a person chooses to be influenced from someone in her social network. That is, every t periods a person decides to be influenced. She samples someone at random from her social network and is influenced, let's say completely influenced. Because her social network is unevenly distributed, she is more likely to sample certain people over others, causing her to be more influenced in a certain direction. The family, as a child's first social network, has the first say in influencing the child's change vector. However, as the child begins her process of socialization within larger society, the familial claims weaken and other people gain stronger influence.

Social Life
When we find ourselves in good company, it is because the personalities of the people involved complement each other. Similarly, in tight knit circles of friends, the personality vectors are linearly dependent on each other.

Social cliques are subspaces of Fⁿ. These cliques are, by definition, closed and bounded, and represent clusters of a socially defined personality type.

Large social gatherings, such as parties, are also home to the formation of subspaces. Like-minded people tend to gravitate towards each other and form social circles, and at times these circles don't even mix. A good party, though, is one where the maximal subspace of the party includes every member, meaning that everyone has interest in each other.

History
We can think of all the people in the world as also representing a bounded region in Fⁿ. The edges of the space represent the boundaries of the present human experience. As people have interacted more and more throughout history, they have provided more and more opportunities for ideas to change, pushing personality vectors in new directions and expanding the human experience.

The region of high density of a density map on all of humanity is the zeitgeist of the era. It represents the certain general paradigm of thinking, acting, and being for people at that time.

Photoshop Lab, Volume 3

I found a website with a lot of cool Photoshop tutorials (see here), which has inpired me to create the following new set of projects. Enjoy!

I used this tutorial for the elephant one and this tutorial (plus this one, for the background) for the boxer one.

Typography

Much to my surprise, I'm finding that learning Photoshop hasn't just been about picking up some cool technical skills, but also about discovering this whole new world of graphic design, a world that produces all the visual media I consume all day everyday but which I've taken entirely for granted.

Today's latest discovery: typography.

typography |tīˈpägrəfē|
noun
the art or process of setting and arranging types and printing from them.

Typography has two parts to it. On one hand, it's about the design and creation of new fonts. This is, perhaps, the world's most underappreciated art form. I never thought of type design myself until I got into Photoshop and looked up the blog I Love Typography, where you see how the process of type creation involves all the trappings of traditional artwork, including sketches, concept drawings, and numerous edits. Typography is also perhaps the world's most useful art, since it's what drives all the blogs, advertisements, flyers, and other visual media that we use. It's nice then that typography has gotten a recent boost from the MacArthur Foundation, which recently awarded a grant to type designer Matthew Carter, known for fonts like Georgia (this one) and Verdana (this one).

The second side of typography is making pictures out of words. I've really taken a liking to it. Here are some of my favorite examples, collected from around the Internet:

You can find a lot more examples of stunning typographic artwork here. Here's my own attempt at the art form:

I based the design on what I learned in this tutorial.

Making Sense of Political Campaigns

The mid-term elections are coming up in only a few weeks, which means our nation's ritual democratic exercise in absurdity—that is, political campaigning—is fully underway.

In trying to figure out why these campaigns always become so ridiculous (e.g. by running ads that compare the opponent to the Taliban, or start off with "I am not a witch"), it's easy to blame voter stupidity. It's tempting to believe that if only everyone were smart and educated, politicians wouldn't be able to confuse or fear-monger voters so easily, and, hopefully, we could have a less chaotic democracy.

However, William Riker, a political philosopher at the University of Rochester, argues convincingly that democracy can be inherently unstable. His analysis in Liberalism Against Populism, which I present below, applies Social Choice theory to democracy in a fascinating way. It's one of the most powerful theories I've heard.

Preliminaries
Riker employs a lot of technical vocabulary and notation, which I'll define here first.

X=(x, y, z, ...) represents a set of alternatives. Alternatives can be anything—bundles of goods, motions on the House floor, values, or candidates running for election.

Preference is a relation between two alternatives, with three possibilities: either xPy (x is preferred to y), xIy (x is just as good as y), or xRy (x is at least as good as y; this condition is also known as weak preference, with xPy representing strong preference).

Rationality requires that preferences be complete, reflexive, and transitive. Completeness means that a person should be able to come up with a preference relation for any two options. Reflexive means that an option x is at least as good as itself (this is to ensure people aren't schizophrenic, basically). Transitive means that if xPy and yPz, then xPz.¹

A Social Welfare Function (SWF) is a function that takes a set of individual preferences and gives an overall group preference. If, for example, I'm in a club with three members that has to decide on a name, and there are three names avaiable, then the SWF takes our individual rankings and comes up with a ranking for the group. For this ranking to make any sense, then we require it to follow the same rationality conditions that we require the individual members themselves—completness, reflexivity, and transitivity.

Condorcet Voting is a democratic procedure for choosing among three or more alternatives that involves taking simple majority votes on all pairwise alternatives. To illustrate, we have three people—A, B, and C—who are choosing among alternatives x, y, and z. They have the following preference orderings:

A: x y z
B: y x z
C: z x y

When we take pairwise votes we ask how many people prefer x to y, y to z, and z to x. In this example it turns out that on the vote x or y, x wins; on the vote y or z, y wins; and on the vote z or x, x wins. This gives us social preferences of xPy, yPz, and xPz, which together gives us a social ordering of x, y, z; x is the winner.

The Condorcet Voting Paradox is when individual preferences are such that the Condorcet Voting procedure doesn't yield a transitive social ordering, but rather cycles. If, for example, our three people (A, B, C) have the following preferences,

A: x y z
B: y z x
C: z x y

then we have xPy, yPz, zPx, or in other words, x beats y, y beats z, but z beats x! The social ordering is cyclical, x, y, z, x, which makes each option equally valid as the social choice.

Arrow's Impossibility Theorem
The formal problem is that we want a fair, reliable, logical, and democratic way of choosing from three or more alternatives. In other words, we want a democratic way of aggregating individual preferences that really reflects "the popular will."

The starting point for the analysis is Arrow's Impossibility Theorem, which shows that there is no SWF for three or more alternatives that also satisfies "some reasonable conditions of fairness on the method and a condition of logicality of the result" (Riker 116).

This may sound rather abstract, but Riker takes this formal problem and applies it quickly to democracy by modeling democracy as a SWF: voters have certain individual ranking of candidates, and democracy amalgamates those individual preferences into a social choice.

The way the proof works is that first Arrow gives six basic, logical conditions that we can generally agree a SWF should have, and then shows how they contain a contradiction. This method is called axiomatic social choice theory, because the theories are built up from axioms, or conditions. The axioms are (following Riker 116-119):

1. Universal admissibility of individual orderings: People should be able to rank alternatives in any way they want, as long as it's rational.
2. Monotonicity: If a person raises the value of a winning alternative, it cannot become a loser; or, if a person lowers the value of a losing alternative, it cannot become a winnner.
3. Citizens' sovereignty or Nonimposition: An alternative x is imposed if it is the winner no matter what set of individual preference orderings are given.
4. Unanimity: If everyone preferes x to y, then the SWF should not choose y. Riker notes there are only two ways of violating unanimity: one, if the SWF is not monotonic, and two, if the method of amalgamation of preferences imposes x regardless of the set of preferences.
5. Independence of Irrelevant Alternatives: Adding alternatives to an original set of profiles shouldn't affect the SWF's relative ranking of the original alternatives. For example, if in the original profile (x, y, z) everyone prefers x to y, adding w to the list shouldn't make y somehow place higher than x. Similarly, subtracting alternatives shouldn't affect the relative ordering of the remaining alternatives. If everyone prefers x to y, subtracting z shouldn't cause y to beat x.
6. Nondictatorship: There should be no person whose preference xPy sets the social choice as x, regardless of what other people think.

The proof of Arrow's Theorem involves exploiting the Universal Domain axiom to allow for Condorcet Voting Paradox preferences, showing that these preferences contain a dictator for one alternative, and then showing that if we have a dictator over one alternative then he's a dictator over all alternatives.

When faced with this result, one's natural reacction is to question the necessity of some of the axioms. Independence of Irrelevant Alternatives seems particularly suspect, since, for example, it exludes all plurality voting systems. Nevertheless, these axioms are important. They ensure that the result of the aggregation process is not manipulable, and to remove any one of these is to open up the system to agenda setting, strategic voting, and other forms of manipulation.²

Double Peaked Preferences and Strategic Voting
Arrow's theorem only applies when we have cycles. When do cycles occur in politics?

In politics, our alternatives are almost always positioned along a spectrum, most commonly the liberal/conservative one. To model what cyclical preferences would look like in actual politics, we take our alternatives x, y, z to represent candidates from various parts of the liberal/conservative spectrum. The Condorcet Voting Paradox over these candidates would look like:

Both A and B's preferences make sense: the farther they move away from their preferred candidate, the less they like the alternatives. C, on the other hand, has strange preferences. As he moves away from his most preferred candidate, z, he likes the alternatives less and less. But after a certain point, as he moves even farther away towards x, he starts liking the candidate more!

What makes A and B's preferences sound is that they are single-peaked. When you graph the preferences on a political axis, they only have one peak, and as you move further from the peak preference only decreases. By the opposite token, what makes C's preferences illogical is that they are double-peaked. Double-peaked preferences come in many shapes, but they all have the property of increasing preference at some point as one moves away from the preferred alternative.

Why would double-peaked preferences occur? Riker argues they result from disagreement about the meaning of the political spectrum. Person C is probably not only be looking at the candidates in terms of the liberal/conservative axis, but also the environmental axis, the trustworthy axis, or even the good-looking axis. This transforms a 1-dimensional problem into a multi-dimensional problem. As long as these various axes overlap, then there will be no disagreement about the axis; but the more orthogonal these axes are, the more likely we are to see double-peaked preferences.

Critics of Riker have argued that studies show that Americans overwhelmingly agree on the liberal/conservative axis. That seems to save democracy from Arrow problems, right? Riker counters that even if society doesn't actually have cyclical preferences, individuals can vote strategically to create them. When we have cycles, we have no true social choice, which means the result depends entirely on how one sets the agenda and the voting system. Strategic voting thus allows for manipulation of the social choice.

Riker's argument is especially strong because he not only shows that it is theoretically possible to strategically vote, but also provides examples from U.S. history where he argues that it's happened, such as the repeated defeat of the Powell amendment to the federal education bill in the House of Representatives.

Chaos in Politics
Thus, when Riker claimed that democracy was inherently choatic, he was arguing that democracy is susceptible to cycles, and cycles allow for manipulation of the social choice.

I see Riker's argument applying to political campaigning in that campaign managers often design campaigns to shift the terms of the debate. Ads will come out that arguing "this election isn't about small government, it's about jobs"; or, as is more often the case, "this election isn't about policy, it's about who looks more presidential/American/empathetic/working-class." In Rikerian terms, campaign managers who find their candidates losing on the traditional left/right policy axis try to find other orthogonal axes to divide voters, which creates cycles and allows for maniuplation. Right now these other axes are stupid because we have stupid voters. But even if all voters were smart, there always exist new axes (e.g. freedom) that bisect the traditional left/right spectrum and divide even intelligent voters.

Endnotes:
1. We have good reasons to believe that these conditions make sense; if you need convincing, see Chapter 1 of On Philosophy, Politics, and Economicsby Gerry Gaus.

2. In fact, one very infamous and controversial paper, "Agenda Influence and Its Implications," exploits the fact that all voting systems have a weak spot. Levine, president of the local flying club, has to decide how to structure the voting for the club's new round of purchases. His problem is that his preferences go against what everyone else in the group wants. Levine turns to Plott, an economist, and the two of them devise a strategic voting system that sets up the votes in such a way that Levine gets what he wants—and he does! Not surprisingly, after the paper is publihed, Plott and Levine get a lot of flak for their "unethical" study.