This was cross-posted on The Asexual Agenda.
In Psychology & Sexuality‘s special issue on asexuality, there was an article called A Mystery Wrapped in an Enigma, which was basically an interview with several of the article authors. A few people
have complained that this article revealed that “many of the
researchers seem wildly out of touch with actual ace communities and
discourses.”
Therefore, it might be an interesting exercise to see how people
actively involved in ace communities would respond to the same
questions. I’m going to answer the questions, and I invite you to
answer one or more of them in the comments or on your blogs.
What motivated your initial engagement with asexuality research?
I initially started reading (a)sexuality research during the time
that I was questioning the extent of my asexuality. What I really
wanted to know: how does science break down attraction into components?
AVEN had one way of breaking it down (and since this was four years
ago, it was not the quite same way it’s broken down now), but I wanted a
second opinion. I didn’t get much of an answer from academic research,
and perhaps it was foolish to expect one. But I continue to be
interested in academic research as a second opinion on asexual issues.
Are there issues particular to asexuality research which differentiates it from other forms of sexualities research?
I think the main difference is the community structure. Because the
community is very centralized, a lot of asexuals will have more or less
the same conceptual understandings. Because it is primarily
internet-based, we transcend national boundaries (but language
boundaries not so much). Because asexual awareness is very low, there
is a huge population of “potential asexuals”, people who do not identify
as asexual, but who might identify under the right circumstances. All
of this has a big influence on what kind of research is feasible.
Unlike Bishop, I don’t really have a problem with classifying
asexuality as sexualities research. I wouldn’t usually use “PC” in a
pejorative sense, but I think this is a case of being excessively PC.
How can researchers disentangle asexuality from diagnostic entities such as ‘hypoactive sexual desire disorder?’
There are probably some clear-cut cases, such as the happy asexual,
or the person who suddenly loses their libido and is very distressed
about it. But there are always going to be some ambiguous cases. How
do we tell between lifelong HSDD, and a potential asexual who is
distressed about not fitting into the norm? I’m not convinced that
there is in fact a difference, and I suspect it’s really a moral
question about the best way to respond to such people.
In ambiguous cases, I would like clinicians offering the possibility
of asexuality to their patients, and seeing how they react. And I would
like to see research evaluating the results of such an approach.
In a recent review article, C.J. Chasin raises several provocative questions. First, do you think asexuality should be categorised as a sexual orientation or as a meta-category akin to sexual? Second, how might asexual men and women rationalize or justify engaging in sexual activity and what are the potential consequences of doing so?
I’m with Hinderliter, in that I can’t think of any empirical
difference between asexuality being a sexual orientation, and being a
meta-category. I don’t think it’s a scientific question at all. It’s a
political question. Is it more politically beneficial to consider
asexuality an orientation or as a meta-category? I think it is better
to think of it as an orientation, because: a) the parallel to LGB is
educationally useful, and b) conceptualizing gray-A as half of a sexual
orientation is needlessly confusing.
I think the second question is loaded. Asking how asexual people
“rationalize” engaging in sexual activity suggests that asexuals who
have sex are necessarily disempowered, and they only have sex for the
“wrong” reasons. According to Aicken et al. (2013),
most potential asexuals who are having sex enjoy it, suggesting that it
isn’t always a kind of disempowerment. The researchers had several
good answers for why asexuals might have sex (satisfying partners,
expressing intimacy, desire to reproduce), but I would add that some
asexuals may also have sex for pleasure. It would be cool to see
research on the relative prevalance of these different motivations.
How is the negativity directed toward asexual individuals
similar to/different from the negativity directed at other sexual
minorities (e.g., gay men, lesbian women, etc.)?
There are a lot of different ways in which individuals and society
express homophobia, biphobia, transphobia, and ace-hate. It’s hard to
compare the different categories when each category is so varied!
There are some notable differences in the canonical forms of
negativity directed at different groups. Asexuals experience erasure
and denial, whereas gay men experience bullying, and trans women
experience serious violence. These canonical forms of negativity partly
reflect reality, but they also reflect back-and-forth political
framing. Bisexual erasure doesn’t get much attention, nor does the fact
that bisexuals have higher rates of suicide than gay and lesbian folk.
There’s also a tendency to see these as being problems for white men,
when usually minorities and women are disproportionately affected (eg). Research is what helps us tangle out the details.
What directions do you expect asexuality research to take over the next decade?
I expect a lot of it to rehash what most people within the asexuality
community already know. Maybe years down the line they’ll “discover”
gray-As and demis. The queer and feminist theorists will “discover”
that asexuals have the potential to break down romantic/non-romantic
boundaries. Etc. etc. There will probably also be a lot of research on
HSDD. And that’s all important research.
Although, what I’d like to see is more information that
would be news to me. Academic research can give us knowledge that we in
the community would otherwise not have access to. In particular, we
don’t know much about potential asexuals, since by definition we don’t
hang out much. And we know very little about the relative prevalence of
different experiences.
Sunday, March 31, 2013
Thursday, March 28, 2013
The fallacious slippery slope
I find that the best way to talk about fallacies and biases is to wait for a good example to come around. Recent discussion of same-sex marriage offers some excellent examples of slippery slope arguments. Chris Hallquist recently highlighted an example:
I remember learning about logical fallacies in grade school English, and they would always include the "slippery slope fallacy" among others. I think this is wrong. A slippery slope argument is not necessarily fallacious. It depends on how it's used. Therefore, I call this a fallacious slippery slope argument to distinguish from those slippery slope arguments which are not fallacious.
First, I wish to divide slippery slope arguments into two categories (which are my own creation):
1. The slippery slope of reasoning
2. The slippery slope of consequences
Example of a slippery slope of reasoning: Suppose I claimed that all the best things are green. You could counter, "But if you follow that slippery slope, you must also believe that cats (which are not green) are not among the best things! Clearly this is absurd."
Another example: "If you believe that same-sex marriage should be legal, then you should also believe that man-dog marriage should be legal. This is clearly absurd."
This kind of slippery slope argument is no different from an argument ad absurdum. In a mathematical argument, we'd call it proof by contradiction. It's a logically valid argument, it's just that it's often unsound (ie the conclusions follow from the premises, it's just that the premises are wrong). In the green example, the person forgot to show that it is absurd to believe that cats are not the best things. In the example of same-sex marriage, the person forgot to show that if same-sex marriage is legal, then man-dog marriage should be legal. But if we granted the premises, the conclusions would follow.
William Lane Craig uses a slippery slope of consequences. He does not say that if we support same-sex marriage, we must logically also support man-dog marriage. Rather, he says that if same-sex marriage is legal, this would lead people to also legalize man-dog marriage in the future. The argument is not about what people should conclude from prior beliefs, it's about what people will do, how people will behave.
It's basically a moral argument. Lane Craig doesn't want people to marry dogs, so from his perspective it's worth taking actions to avoid this.
But there's something rather strange about this argument. We are free agents. We can either choose to allow man-dog marriage or not. We will choose according to our preferences. If we prefer to allow man-dog marriage, why should we prevent getting what we prefer? If we prefer not to allow man-dog marriage, then why would we choose to allow man-dog marriage?
Mind you, the slippery slope of consequences still isn't necessarily fallacious. There are some situations where we might worry that our future selves will be irrational. Or that other people will be irrational. Or that we'll have different preferences in the future. And there are game-theoretic situations where it's better to have fewer options (I've recently been reading about decision theory).
But none of those situations are relevant here. Making laws is a very deliberative process, not something that's decided on the spot on an irrational whim. And it's hard to imagine a non-trivial game-theoretic situation. Usually there are people who want a law, and people who don't, and that's that.
So despite the slippery slope argument being an acceptable argument in general, William Lane Craig manages to use a form that is completely fallacious.
--------------------------
On a related note, my boyfriend pointed out a slippery slope argument made by Supreme Court Justice Scalia, when he argued in favor of anti-sodomy laws in 2003:
By turning marriage into a socially constructed reality that doesn’t have a nature, marriage can then be whatever you want it to be. Not just the union of a man and another man, but also even two men and a woman–three partners in marriage. Or it could be a man and a child. Or maybe even a man and his dog, if he feels close enough to his pet to want to marry it.(In context, William Lane Craig is arguing that long-term relationships are so uncommon among gay men, that they couldn't really be fighting for marriage for its own sake. Instead, their real goal must be to "deconstruct marriage". But I will ignore this context to focus on the slippery slope only.)
--William Lane Craig
I remember learning about logical fallacies in grade school English, and they would always include the "slippery slope fallacy" among others. I think this is wrong. A slippery slope argument is not necessarily fallacious. It depends on how it's used. Therefore, I call this a fallacious slippery slope argument to distinguish from those slippery slope arguments which are not fallacious.
First, I wish to divide slippery slope arguments into two categories (which are my own creation):
1. The slippery slope of reasoning
2. The slippery slope of consequences
Example of a slippery slope of reasoning: Suppose I claimed that all the best things are green. You could counter, "But if you follow that slippery slope, you must also believe that cats (which are not green) are not among the best things! Clearly this is absurd."
Another example: "If you believe that same-sex marriage should be legal, then you should also believe that man-dog marriage should be legal. This is clearly absurd."
This kind of slippery slope argument is no different from an argument ad absurdum. In a mathematical argument, we'd call it proof by contradiction. It's a logically valid argument, it's just that it's often unsound (ie the conclusions follow from the premises, it's just that the premises are wrong). In the green example, the person forgot to show that it is absurd to believe that cats are not the best things. In the example of same-sex marriage, the person forgot to show that if same-sex marriage is legal, then man-dog marriage should be legal. But if we granted the premises, the conclusions would follow.
William Lane Craig uses a slippery slope of consequences. He does not say that if we support same-sex marriage, we must logically also support man-dog marriage. Rather, he says that if same-sex marriage is legal, this would lead people to also legalize man-dog marriage in the future. The argument is not about what people should conclude from prior beliefs, it's about what people will do, how people will behave.
It's basically a moral argument. Lane Craig doesn't want people to marry dogs, so from his perspective it's worth taking actions to avoid this.
But there's something rather strange about this argument. We are free agents. We can either choose to allow man-dog marriage or not. We will choose according to our preferences. If we prefer to allow man-dog marriage, why should we prevent getting what we prefer? If we prefer not to allow man-dog marriage, then why would we choose to allow man-dog marriage?
Mind you, the slippery slope of consequences still isn't necessarily fallacious. There are some situations where we might worry that our future selves will be irrational. Or that other people will be irrational. Or that we'll have different preferences in the future. And there are game-theoretic situations where it's better to have fewer options (I've recently been reading about decision theory).
But none of those situations are relevant here. Making laws is a very deliberative process, not something that's decided on the spot on an irrational whim. And it's hard to imagine a non-trivial game-theoretic situation. Usually there are people who want a law, and people who don't, and that's that.
So despite the slippery slope argument being an acceptable argument in general, William Lane Craig manages to use a form that is completely fallacious.
--------------------------
On a related note, my boyfriend pointed out a slippery slope argument made by Supreme Court Justice Scalia, when he argued in favor of anti-sodomy laws in 2003:
"Today’s opinion dismantles the structure of constitutional law that has permitted a distinction to be made between heterosexual and homosexual unions, insofar as formal recognition in marriage is concerned. If moral disapprobation of homosexual conduct is 'no legitimate state interest' for purposes of proscribing that conduct; and if, as the Court coos (casting aside all pretense of neutrality), '[w]hen sexuality finds overt expression in intimate conduct with another person, the conduct can be but one element in a personal bond that is more enduring,' what justification could there possibly be for denying the benefits of marriage to homosexual couples exercising '[t]he liberty protected by the Constitution'?"I'm sure by now Scalia has thought of a few justifications for denying same-sex marriage that do not rely on anti-sodomy laws.
-Lawrence v. Texas, 539 U.S. 558, 605-06 (2003) (Scalia, J., dissenting) (internal citations omitted).
Monday, March 25, 2013
Improving on the queue
Speaking of social science technologies...
I grew up in Los Angeles, so I went to Disneyland a lot. By now I've seen everything a hundred times, so the experience is not so enjoyable. But when I was a kid, the problem was that I couldn't get enough of the rides, because the lines were so long.
When I was 11, Disneyland introduced the Fastpass system. At the front of each major ride there were these magical boxes. You'd put your park ticket in, and you'd get out a Fastpass which had a time printed on it. The time is calculated based on how many other people have gotten Fastpasses so far that day. At the printed time, you could get in a special line at the ride which moved much faster than the normal line. The drawback was that you could only have a single fast pass per park ticket.
The fastpass sure was gratifying, because we'd get to zoom past lines that would normally take over an hour. But the rides can only serve a certain number of people per unit time, so it couldn't possibly benefit everyone, could it? Was the benefit of the Fastpass all an illusion, or was there more to it?
My hypothesis is that there is a real benefit, and it has to do with game theory. Normal queues are a prisoner's dilemma, and the Fastpass system mitigates the problem.
The swimming pool analogy
Imagine, for instance, that we're in a swimming pool, and the diving board is so popular that there's always a line for it. We all prefer staying in the swimming pool, but we're also willing to stand out of the water for a while in order to try the diving board. The shorter the line is, the more people willing to get into it, which makes the line longer again. So the length of the line reaches a sort of Nash equilibrium.
However, the equilibrium line length is not the best line length. The best line length would be zero. Instead of everyone lining up, people could stay in the pool. Then people would voluntarily go up to the diving board at a steady pace, always maintaining a line length of zero. In this scenario, just as many people get to use the diving board, and no one has to stand for long outside of the pool. To get in line is to take the defecting strategy in a prisoner's dilemma.
And unfortunately, this is a prisoner's dilemma that involves many people (as many as there are in the swimming pool). So it's virtual certainty that everyone will defect. Tragedy of the commons.
You could imagine queueing systems that attempt to defeat the prisoner's dilemma. For example, you could have people write their names on a list (on a waterproof board?), while one person calls people to the diving board at the appropriate times. You'd have to restrict the number of times people can sign their name to prevent people from just signing over and over with virtually no cost to themselves. This new system would allow everyone to stay in the swimming pool. Unfortunately, since the cost of diving is lower, the "line" would be much longer, consisting of many people who only marginally enjoy diving.
The Fastpass
The Disney Fastpass system is similar to the queuing system I proposed for the swimming pool. Rather than having everyone wait in line, they get Fastpasses, which is just like signing your name on a board. The time on the Fastpass is the time when the ride calls you up. The Fastpass system doesn't allow more people to enjoy the ride, but it does allow people to spend less time in line overall.
What do people do with that extra time? If you really like this one ride, you might just get in the normal line over and over again. But if that's your behavior, then it's unlikely that the Fastpass system benefits you. The Fastpass line makes the normal line slower.* So even though you're getting the benefits of the Fastpass line every few hours, on average you're doing no better, or possibly worse.
*There's also an equilibriating response--fewer people are willing to line up in a slow line. The wait is the product of the speed of the line and the number of people in it. Despite the equilibriating response, the total wait will be longer.
A more beneficial way to spend the time is to watch shows or go on other less popular rides. These attractions would normally operate at less than full capacity. But because of the Fastpass system, more people may have time to take these rides and watch these shows.
Another alternative (one which I suspect is Disneyland's favorite) is to use the extra time to patronize shops and restaurants.
That's my hypothesis. But what does Disneyland think are the advantages of Fastpass? Take a look at the Fastpass patent:
I grew up in Los Angeles, so I went to Disneyland a lot. By now I've seen everything a hundred times, so the experience is not so enjoyable. But when I was a kid, the problem was that I couldn't get enough of the rides, because the lines were so long.
When I was 11, Disneyland introduced the Fastpass system. At the front of each major ride there were these magical boxes. You'd put your park ticket in, and you'd get out a Fastpass which had a time printed on it. The time is calculated based on how many other people have gotten Fastpasses so far that day. At the printed time, you could get in a special line at the ride which moved much faster than the normal line. The drawback was that you could only have a single fast pass per park ticket.
The fastpass sure was gratifying, because we'd get to zoom past lines that would normally take over an hour. But the rides can only serve a certain number of people per unit time, so it couldn't possibly benefit everyone, could it? Was the benefit of the Fastpass all an illusion, or was there more to it?
My hypothesis is that there is a real benefit, and it has to do with game theory. Normal queues are a prisoner's dilemma, and the Fastpass system mitigates the problem.
The swimming pool analogy
Imagine, for instance, that we're in a swimming pool, and the diving board is so popular that there's always a line for it. We all prefer staying in the swimming pool, but we're also willing to stand out of the water for a while in order to try the diving board. The shorter the line is, the more people willing to get into it, which makes the line longer again. So the length of the line reaches a sort of Nash equilibrium.
However, the equilibrium line length is not the best line length. The best line length would be zero. Instead of everyone lining up, people could stay in the pool. Then people would voluntarily go up to the diving board at a steady pace, always maintaining a line length of zero. In this scenario, just as many people get to use the diving board, and no one has to stand for long outside of the pool. To get in line is to take the defecting strategy in a prisoner's dilemma.
And unfortunately, this is a prisoner's dilemma that involves many people (as many as there are in the swimming pool). So it's virtual certainty that everyone will defect. Tragedy of the commons.
You could imagine queueing systems that attempt to defeat the prisoner's dilemma. For example, you could have people write their names on a list (on a waterproof board?), while one person calls people to the diving board at the appropriate times. You'd have to restrict the number of times people can sign their name to prevent people from just signing over and over with virtually no cost to themselves. This new system would allow everyone to stay in the swimming pool. Unfortunately, since the cost of diving is lower, the "line" would be much longer, consisting of many people who only marginally enjoy diving.
The Fastpass
The Disney Fastpass system is similar to the queuing system I proposed for the swimming pool. Rather than having everyone wait in line, they get Fastpasses, which is just like signing your name on a board. The time on the Fastpass is the time when the ride calls you up. The Fastpass system doesn't allow more people to enjoy the ride, but it does allow people to spend less time in line overall.
What do people do with that extra time? If you really like this one ride, you might just get in the normal line over and over again. But if that's your behavior, then it's unlikely that the Fastpass system benefits you. The Fastpass line makes the normal line slower.* So even though you're getting the benefits of the Fastpass line every few hours, on average you're doing no better, or possibly worse.
*There's also an equilibriating response--fewer people are willing to line up in a slow line. The wait is the product of the speed of the line and the number of people in it. Despite the equilibriating response, the total wait will be longer.
A more beneficial way to spend the time is to watch shows or go on other less popular rides. These attractions would normally operate at less than full capacity. But because of the Fastpass system, more people may have time to take these rides and watch these shows.
Another alternative (one which I suspect is Disneyland's favorite) is to use the extra time to patronize shops and restaurants.
That's my hypothesis. But what does Disneyland think are the advantages of Fastpass? Take a look at the Fastpass patent:
Not only is the customer frustrated at not being able to access more attractions, but the amusement park itself suffers from having underutilized attractions because the customers are waiting in line for other attractions. Instead of waiting in line for a single attraction, a customer could be riding other attractions, eating food, shopping at stores, playing games, or other activities.Yep. I got it right. It's a brilliant idea IMO.
Saturday, March 16, 2013
Wednesday, March 13, 2013
Social science technologies
Here's an interesting paper I found via Physics Today:
Probably a few of these technologies get a really bad rap, especially marketing and management. The paper says:
A recent poll showed that most people think of science as technology and engineering—life-saving drugs, computers, space exploration, and so on. This was, in fact, the promise of the founders of modern science in the 17th century. It is less commonly understood that social and behavioral sciences have also produced technologies and engineering that dominate our everyday lives. These include polling, marketing, management, insurance, and public health programs.The paper details each of those technologies and gives some specific improvements brought by research in those areas. For example, it was social science research that developed the surgical safety checklist, which measurably reduces the risk of complications in surgery. One of the "management" examples was the realization that workers at a Fiat auto plant had to bend and stretch to work, and that new machinery could eliminate this problem and increase productivity per worker.
Probably a few of these technologies get a really bad rap, especially marketing and management. The paper says:
It is commonly recognized that the same knowledge about atomic structures that brings us nuclear medicine can also bring us nuclear winter. The same knowledge about operantSo it's not the technology itself that's bad, it's how you use it. Do you buy this?
conditioning can bring relief from terrifying phobias or, as with advertisements for tobacco, it can kill.
Monday, March 11, 2013
The Gini coefficient of log normal wealth
I recently learned how to use $\LaTeX$ and I found out how to display it in Blogger. I use MathJax. Let me know if it doesn't display properly. I use a script-blocker, and I had to allow MathJax in order for it to work. (ETA: Also, it apparently doesn't work in my rss feed.)
I might as well take the excuse to write a post with lots of math in it. I'm going to talk about wealth distribution, because I saw a video about it recently.
(Via Pharyngula) This is not new information to me, because I remember when the study by Norton and Ariely made news in 2011. But very nice presentation!
The goal of the calculation
Back in 2011, I observed that the wealth distribution (desired, imagined, and actual) appeared to follow a log normal distribution, which is about the simplest distribution of wealth I can think of. The log normal distribution is $$D(y) = \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{y^2}{2\sigma^2}}$$ where y is the log of the wealth, D(y) is the density of people with respect to y, and $\sigma$ is the standard deviation.* Of course, the "density with respect to the log of wealth" is intuitively meaningless, so I'll convert to the density with respect to wealth. $\sigma$ is also intuitively meaningless, so I'll let $\sigma = \log{N}$. Here, N is the ratio of wealth between two people who are a standard deviation apart. For example, if I'm at the mean, and you're one standard deviation above the mean, then you own N times as much wealth.
*Note that I'm setting the mean of y to zero, which is the same as setting the median wealth to 1. This can always be done by appropriate choice of units.
With these changes, we can rewrite the log normal distribution as $$D(x) = \frac{1}{x\log{N} \sqrt{2\pi}}e^{-\frac{(\log{x})^2}{2(\log{N})^2}}$$ where x is the wealth, and D(x) is the density of people who have wealth x.
This model has a single parameter, N. However, this is not the standard way of measuring wealth inequality. The standard way is using the Gini coefficient. The Gini coefficient is a number between 0 and 1. 0 represents a situation where everyone has exactly equal amount of wealth. 1 represents a situation where one person has all the wealth. The Gini coefficient is represented graphically here:
The Lorenz curve is the plot of cumulative wealth vs cumulative population when the population is arranged from poorest to wealthiest. To illustrate, say we're given a percentage, 40%. So we look at the poorest 40% of the population, and determine what fraction of the total wealth they own. Say that they own 1%. The Lorenz curve will contain the point (0.4,0.01).
The Gini coefficient ("G") is defined as twice the area of A.
So the question I'm going to answer is, how does G relate to N in a log normal wealth distribution?
The calculation
$\DeclareMathOperator{\erf}{erf}$
Starting with a log normal distribution, I'm going to calculate the Gini coefficient. First thing we need to do is calculate the number of people who own wealth x or less. Let's call this function P(x) (P stands for population). We can calculate it from $$P(x) = \int_0^x D(x') \mathrm{d}x'$$ It's simpler to evaluate this integral if we integrate with respect to $y = \log{x}$ rather than x. So we get $$P(x) = \int_{-\infty}^{\log{x}} D(y) \mathrm{d}y$$ Substituting in D(y), $$P(x) = \int_{-\infty}^{\log{x}} \frac{1}{\log{N} \sqrt{2\pi}}e^{-\frac{y^2}{2(\log{N})^2}} \mathrm{d}y$$ $$P(x) = \frac{1}{2}(\erf{(\frac{\log{x}}{\log{N}\sqrt{2}})} + 1)$$ erf is the error function, which is basically defined as the integral of a normal distribution.
The next thing we need to do is calculate W(x), which is the total amount of wealth owned by people who own wealth x or less. It can be calculated similarly to P(x). $$W(x) = \int_0^x x' D(x') \mathrm{d}x'$$ $$W(x) = \int_{-\infty}^{\log{x}} e^y D(y) \mathrm{d}y$$ $$W(x) = \int_{-\infty}^{\log{x}} \frac{1}{\log{N} \sqrt{2\pi}}e^{-\frac{y^2}{2(\log{N})^2} + y} \mathrm{d}y$$ $$W(x) = \frac{1}{\log{N} \sqrt{2\pi}} e^{(log{N})^2/2} \int_{-\infty}^{\log{x}} e^{-\frac{(y-(\log{N})^2)^2}{2(\log{N})^2}} \mathrm{d}y$$ $$W(x) = \frac{1}{2} e^{(log{N})^2/2} (\erf{( \frac{\log{x}}{\log{N}\sqrt{2}} - \frac{\log{N}}{\sqrt{2}})} + 1 )$$ For this to really be meaningful, instead of the total wealth, I want to talk about the fraction of the total wealth owned by people who own x or less. Let's call this fraction F(x). $$F(x) = \frac{W(x)}{W(\infty)} = \frac{1}{2} (\erf{( \frac{\log{x}}{\log{N}\sqrt{2}} - \frac{\log{N}}{\sqrt{2}})} + 1 )$$ I'm a very visual person, so I'm going to show plots of F(x) and P(x).
I might as well take the excuse to write a post with lots of math in it. I'm going to talk about wealth distribution, because I saw a video about it recently.
(Via Pharyngula) This is not new information to me, because I remember when the study by Norton and Ariely made news in 2011. But very nice presentation!
The goal of the calculation
Back in 2011, I observed that the wealth distribution (desired, imagined, and actual) appeared to follow a log normal distribution, which is about the simplest distribution of wealth I can think of. The log normal distribution is $$D(y) = \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{y^2}{2\sigma^2}}$$ where y is the log of the wealth, D(y) is the density of people with respect to y, and $\sigma$ is the standard deviation.* Of course, the "density with respect to the log of wealth" is intuitively meaningless, so I'll convert to the density with respect to wealth. $\sigma$ is also intuitively meaningless, so I'll let $\sigma = \log{N}$. Here, N is the ratio of wealth between two people who are a standard deviation apart. For example, if I'm at the mean, and you're one standard deviation above the mean, then you own N times as much wealth.
*Note that I'm setting the mean of y to zero, which is the same as setting the median wealth to 1. This can always be done by appropriate choice of units.
With these changes, we can rewrite the log normal distribution as $$D(x) = \frac{1}{x\log{N} \sqrt{2\pi}}e^{-\frac{(\log{x})^2}{2(\log{N})^2}}$$ where x is the wealth, and D(x) is the density of people who have wealth x.
This model has a single parameter, N. However, this is not the standard way of measuring wealth inequality. The standard way is using the Gini coefficient. The Gini coefficient is a number between 0 and 1. 0 represents a situation where everyone has exactly equal amount of wealth. 1 represents a situation where one person has all the wealth. The Gini coefficient is represented graphically here:
The Lorenz curve is the plot of cumulative wealth vs cumulative population when the population is arranged from poorest to wealthiest. To illustrate, say we're given a percentage, 40%. So we look at the poorest 40% of the population, and determine what fraction of the total wealth they own. Say that they own 1%. The Lorenz curve will contain the point (0.4,0.01).
The Gini coefficient ("G") is defined as twice the area of A.
So the question I'm going to answer is, how does G relate to N in a log normal wealth distribution?
The calculation
$\DeclareMathOperator{\erf}{erf}$
Starting with a log normal distribution, I'm going to calculate the Gini coefficient. First thing we need to do is calculate the number of people who own wealth x or less. Let's call this function P(x) (P stands for population). We can calculate it from $$P(x) = \int_0^x D(x') \mathrm{d}x'$$ It's simpler to evaluate this integral if we integrate with respect to $y = \log{x}$ rather than x. So we get $$P(x) = \int_{-\infty}^{\log{x}} D(y) \mathrm{d}y$$ Substituting in D(y), $$P(x) = \int_{-\infty}^{\log{x}} \frac{1}{\log{N} \sqrt{2\pi}}e^{-\frac{y^2}{2(\log{N})^2}} \mathrm{d}y$$ $$P(x) = \frac{1}{2}(\erf{(\frac{\log{x}}{\log{N}\sqrt{2}})} + 1)$$ erf is the error function, which is basically defined as the integral of a normal distribution.
The next thing we need to do is calculate W(x), which is the total amount of wealth owned by people who own wealth x or less. It can be calculated similarly to P(x). $$W(x) = \int_0^x x' D(x') \mathrm{d}x'$$ $$W(x) = \int_{-\infty}^{\log{x}} e^y D(y) \mathrm{d}y$$ $$W(x) = \int_{-\infty}^{\log{x}} \frac{1}{\log{N} \sqrt{2\pi}}e^{-\frac{y^2}{2(\log{N})^2} + y} \mathrm{d}y$$ $$W(x) = \frac{1}{\log{N} \sqrt{2\pi}} e^{(log{N})^2/2} \int_{-\infty}^{\log{x}} e^{-\frac{(y-(\log{N})^2)^2}{2(\log{N})^2}} \mathrm{d}y$$ $$W(x) = \frac{1}{2} e^{(log{N})^2/2} (\erf{( \frac{\log{x}}{\log{N}\sqrt{2}} - \frac{\log{N}}{\sqrt{2}})} + 1 )$$ For this to really be meaningful, instead of the total wealth, I want to talk about the fraction of the total wealth owned by people who own x or less. Let's call this fraction F(x). $$F(x) = \frac{W(x)}{W(\infty)} = \frac{1}{2} (\erf{( \frac{\log{x}}{\log{N}\sqrt{2}} - \frac{\log{N}}{\sqrt{2}})} + 1 )$$ I'm a very visual person, so I'm going to show plots of F(x) and P(x).
In this image, x=1 is the median wealth, and N=2.
Now, what we really want is the Lorenz curve. As I explained earlier, the Lorenz curve is the cumulative wealth vs the cumulative population when people are sorted from poorest to wealthiest. By definition, every point (P(x),F(x)) is on the Lorenz curve. But I'd like an explicit formula, which I'll call L(p). $$L(p) = F( P^{-1}(p) )$$ At this point, it's just elementary plugging in and simplification. Skipping to the result, $$L(p) = \frac{1}{2} ( \erf{(\erf{^{-1}(2p-1)} - \frac{\log{N}}{\sqrt{2}} )} + 1)$$ Here is a plot of L(p) for N=2:
G is defined as twice the area of A as shown above. In mathematical terms, $$G = 1 - 2 \int_0^1 L(p) \mathrm{d}p$$ This simplifies to $$G = -\int_0^1 \erf{(\erf{^{-1}(2p-1)} - \frac{\log{N}}{\sqrt{2}})} \mathrm{d}p$$ And that's where we stop, because this function is not integrable. Instead I'll use Mathematica to numerically evaluate and plot G as a function of N.
Note that the plot only shows $N \geq 1$ because values of N below 1 are meaningless.
Concluding remarks
Isn't $\LaTeX$ great? Now I can scare off my readers with math equations that are better formatted than ever!
It's somewhat difficult to find Gini coefficients for the US. As far as income goes, it's somewhere between .378 and .486 depending on the study. But the above youtube video is about wealth inequality, which is much greater. It appears that in 1984, the Gini coefficient was 0.84 in 1989, and 0.801 in 2000 (it's unclear whether this is a change over time, or if it's just from differences between studies). In any case, it's pretty high.
Previously, I determined that N is about 6.5 in the US, because this led to a distribution that looked rather like the one reported by Norton and Ariely. The corresponding Gini coefficient is 0.814. That's quite close! When people tried guessing the amount of inequality, they came up with a distribution with N about 2.7, which corresponds to G = 0.518. When people were asked about the ideal amount of inequality, they gave a distribution with N about 1.5, which corresponds to G = 0.226.
I don't really understand the economic significance of wealth inequality (or income inequality for that matter). But the high degree of inequality in the US is clearly an unhappy situation.
Wednesday, March 6, 2013
Attacks on my boyfriend
This was cross-posted on The Asexual Agenda.
I'm openly asexual, and I also pay attention to media mentions of asexuality. So I've heard all the standard attacks and denials. I can handle them.
But here's what I think is particularly nasty: attacks on my boyfriend.
This happened a few times in my boyfriend's circle of friends. It's a fairly typical circle of friends in that it consists of people who are mostly the same age, race, and social class. This particular circle consists mostly of gay white educated young men. They're my friends too, of course, and I have nothing against them.
A year ago, one of these men, named J, found out that my boyfriend and I were going to an asexual meetup that weekend. Meetups are something we do on occasion. We go to a cafe and have casual conversations about whatever people like. More often than not, what we discuss has nothing to do with asexuality. J seemed to have a different image in mind though. J accused my boyfriend, over instant messages, of getting into a sexless relationship for me. He said I was trying to convert him to asexuality.
On a more recent occasion, my boyfriend went to a movie night. This is something we do every few weeks, but this particular week I was out of town so he was there without me. The host of the movie night says to my boyfriend, "So, I heard you were asexual. What's that about?" My boyfriend had to explain that I was asexual, but he wasn't. This led to a situation where all his friends were quizzing him on asexuality.
My boyfriend felt very uncomfortable, because he felt put on the spot to defend the legitimacy of our relationship to his friends. He felt like he was in a double-bind. First they assumed that we're in a sexless relationship, and then they questioned the legitimacy of sexless relationships. My boyfriend wanted to inform them that our relationship is sexually active, but also didn't want to imply that sexless relationships were somehow less legitimate.
That's what bothered my boyfriend the most, but I was more bothered by the larger pattern of behavior. They pounced on him when I wasn't there. It felt like they were using underhanded tactics to hit me at my weak spot. And they've never mentioned any of it to me, even though my boyfriend said they should redirect questions to me.
It's true that I'm not as close of a friend to them, and that may explain their behavior. But if they were really interested in learning about asexuality, they should have asked the more knowledgeable person. My boyfriend is not asexual. He gets all his information about asexuality secondhand through me. He doesn't necessarily know how to respond to all the standard attacks. And why should he have to? Asexuality isn't his own lived experience.
On another occasion, a different friend said to me in front of my boyfriend, "You're the most sexually active asexual I know." That was awkward, and assumed knowledge about our sex lives. But you know, that wasn't as bad, because at least he said it to my face.
I'm openly asexual, and I also pay attention to media mentions of asexuality. So I've heard all the standard attacks and denials. I can handle them.
But here's what I think is particularly nasty: attacks on my boyfriend.
This happened a few times in my boyfriend's circle of friends. It's a fairly typical circle of friends in that it consists of people who are mostly the same age, race, and social class. This particular circle consists mostly of gay white educated young men. They're my friends too, of course, and I have nothing against them.
A year ago, one of these men, named J, found out that my boyfriend and I were going to an asexual meetup that weekend. Meetups are something we do on occasion. We go to a cafe and have casual conversations about whatever people like. More often than not, what we discuss has nothing to do with asexuality. J seemed to have a different image in mind though. J accused my boyfriend, over instant messages, of getting into a sexless relationship for me. He said I was trying to convert him to asexuality.
On a more recent occasion, my boyfriend went to a movie night. This is something we do every few weeks, but this particular week I was out of town so he was there without me. The host of the movie night says to my boyfriend, "So, I heard you were asexual. What's that about?" My boyfriend had to explain that I was asexual, but he wasn't. This led to a situation where all his friends were quizzing him on asexuality.
My boyfriend felt very uncomfortable, because he felt put on the spot to defend the legitimacy of our relationship to his friends. He felt like he was in a double-bind. First they assumed that we're in a sexless relationship, and then they questioned the legitimacy of sexless relationships. My boyfriend wanted to inform them that our relationship is sexually active, but also didn't want to imply that sexless relationships were somehow less legitimate.
That's what bothered my boyfriend the most, but I was more bothered by the larger pattern of behavior. They pounced on him when I wasn't there. It felt like they were using underhanded tactics to hit me at my weak spot. And they've never mentioned any of it to me, even though my boyfriend said they should redirect questions to me.
It's true that I'm not as close of a friend to them, and that may explain their behavior. But if they were really interested in learning about asexuality, they should have asked the more knowledgeable person. My boyfriend is not asexual. He gets all his information about asexuality secondhand through me. He doesn't necessarily know how to respond to all the standard attacks. And why should he have to? Asexuality isn't his own lived experience.
On another occasion, a different friend said to me in front of my boyfriend, "You're the most sexually active asexual I know." That was awkward, and assumed knowledge about our sex lives. But you know, that wasn't as bad, because at least he said it to my face.
Categories:
about me,
asexuality,
lgbta
Monday, March 4, 2013
The burden of proof and God
One of the more tedious arguments concerning gods is the argument
over who has the burden of proof. Whereas many atheists argue that the
theist must first make the argument for the existence of gods, their
opponents argue that this is a cop out. For example, on NY Times:
I'm somewhat influenced by the role of burden of proof in law, though law does not necessarily provide a paragon of debate format. In a court of law, attorneys set out to prove certain facts. But producing the arguments and evidence costs money, so the law must specify whose role it is to prove the facts, and to what extent they must be proven. In most criminal cases, the plaintiff must prove "beyond reasonable doubt", and in most civil cases, the burden of proof is a "preponderance of the evidence". If you're wondering how to interpret that, it depends whether you ask the defendant's or plaintiff's attorney.
(Please correct me if I made any error with regard to law.)
In the case of proving God, it is only sensible that the theist has the burden of proof. The atheist doesn't know beforehand what particular god or gods the theist believes in, and doesn't know what arguments to use. Theists keep on telling us that not all religious people believe the same things, and I wholeheartedly agree. That's why it's the theist's role to explain what they believe and why.
But this does not confer an advantage to atheists, any more than it confers an advantage to the defendant in a court of law. It just means that the theist makes the first argument. If the argument, taken at face value, meets the burden of proof, then it is up to the atheist to counter it. In other words, the burden of proof shifts to the atheists. The burden of proof shifts back and forth indefinitely, and does not give an advantage to either side.
By saying theists have the burden of proof, I only mean that they have the initial burden of proof. It is not meant as a "get out of an argument free" card.
Contemporary atheists often assert that there is no need for them to provide arguments showing that religious claims are false. Rather, they say, the very lack of good arguments for religious claims provides a solid basis for rejecting them. The case against God is, as they frequently put it, the same as the case against Santa Claus, the Easter Bunny or the Tooth Fairy. This is what we might call the “no-arguments” argument for atheism.I take the side of atheists; I think theists have the burden of proof. This is not about giving atheists an unfair advantage in the debate, nor is it about making a "no-arguments" argument. In fact, I do not believe it is an advantage, fair or otherwise, at all. It's simply about who takes which role.
I'm somewhat influenced by the role of burden of proof in law, though law does not necessarily provide a paragon of debate format. In a court of law, attorneys set out to prove certain facts. But producing the arguments and evidence costs money, so the law must specify whose role it is to prove the facts, and to what extent they must be proven. In most criminal cases, the plaintiff must prove "beyond reasonable doubt", and in most civil cases, the burden of proof is a "preponderance of the evidence". If you're wondering how to interpret that, it depends whether you ask the defendant's or plaintiff's attorney.
(Please correct me if I made any error with regard to law.)
In the case of proving God, it is only sensible that the theist has the burden of proof. The atheist doesn't know beforehand what particular god or gods the theist believes in, and doesn't know what arguments to use. Theists keep on telling us that not all religious people believe the same things, and I wholeheartedly agree. That's why it's the theist's role to explain what they believe and why.
But this does not confer an advantage to atheists, any more than it confers an advantage to the defendant in a court of law. It just means that the theist makes the first argument. If the argument, taken at face value, meets the burden of proof, then it is up to the atheist to counter it. In other words, the burden of proof shifts to the atheists. The burden of proof shifts back and forth indefinitely, and does not give an advantage to either side.
By saying theists have the burden of proof, I only mean that they have the initial burden of proof. It is not meant as a "get out of an argument free" card.
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