Saturday, April 26, 2008

Game Theory and Human Behavior, Part II

Remember the post Game Theory and Human Behavior, Part I? What, you don't? Okay, so I posted it back in February. Obviously it was meant as the first of a series of posts. You might want to read that one again before reading this post, because this one uses some of the concepts I explored in the first post.

The practice of torture is another signaling game. This one is a bit more complicated than the good employee signaling game or signaling in contract bridge, but the premise is the same. Let's call the person doing the torturing the investigator, and the person being tortured the detainee. (I refer to them both as male, since they always are on television and in the movies.) The idea is that the investigator seeks truthful information from the detainee. The investigator doesn't have the same knowledge space as the detainee, although the two may overlap. (If the investigator knew what the detainee knew, he wouldn't have to talk to the detainee at all!)

The detainee, on the other hand, seeks to provide the investigator with only enough information to satisfy him and stop the torture. So the detainee wants to give the investigator the very minimum amount of information that would satisfy his curiosity.

Economist Roger Koppl performed a thorough evaluation of the efficacy of torture in a 2005 paper entitled "Epistemic Systems." In this paper, he determines that torture works only if the following two conditions are met:
  1. The investigator must be able to recognize the truth when he hears it.
  2. The investigator must be able to credibly commit to stopping the torture upon hearing the truth.
If the investigator cannot recognize the truth when he hears it, then he won't find the truth regardless of the method of questioning. If the detainee doesn't believe that the torture will stop upon telling the truth, then he has no incentive to tell the truth.

Here's the problem: These two conditions are both necessary for torture to work, but they almost never occur together in real-life situations.

Consider the first condition. The more the investigator knows about the knowledge space of the detainee, the better he is able to recognize the truth. But how can he recognize the truth when he hears it, if he isn't privy to everything within the knowledge space of the detainee? This is how the detainee can get away with providing the least useful information out of the set of relevant information that he knows, or even worse, information that sounds truthful to the investigator but turns out to be false upon verification.

The second condition is similarly difficult to satisfy. Consider the case of a Chilean rebel, who broke down and confessed the names of the nuns and priests who had sheltered her. Her devout interrogators didn't believe that these holy men and women were involved, and continued to torture her.

Even if torture were effective on actual "bad guys," the consequences of using it on people who are actually innocent bystanders are the things nightmares are made of. In Discover Your Inner Economist, Tyler Cowen discusses the nightmare situation of being mistaken for a spy when you are nothing but a clueless American on vacation. How can you signal to your captors that you are not a spy? Anything that you do, from protesting your innocence, to providing them with information that sounds promising but turns out to be false, is indistinguishable from what a real spy would do in that situation.

There's always the situation in spy movies and political thriller TV shows such as 24, in which the entire population of New York City is in imminent danger from a ticking time bomb, which was set by a fanatical terrorist of some sort. The hero has to beat the information about the location of the bomb out of the bad guy. Usually it's a renegade cop played by somebody like Bruce Willis -- a guy who goes against the rules and saves the day, much to the chagrin and grudging gratitude of his superiors.

While these sorts of movies and shows are entertaining, they are not reality-based. Dr. Jean Maria Arrigo, a social psychologist with a mathematics background, refutes the ticking time bomb scenario in her papers "A Utilitarian Argument Against Torture Interrogation of Terrorists" (Science and Engineering Ethics (2004) 10, pp. 543-572) and "Torture, Terrorism and the State: a Refutation of the Ticking-Bomb Argument" (Journal of Applied Philosophy (2006) 23(3), pp. 355-373), co-authored with Vittorio Bufacchi. I focus upon this second paper.

Bufacchi and Arrigo first formalize the scenario. The dominant elements in the story are that first, a great number of lives are at stake; second, there is a time at which the catastrophic event will occur if it is not prevented; and third, a person privy to knowledge that will allow authorities to intervene and prevent the catastrophic event has been captured. Under these circumstances, some philosophers and legal experts believe that torture may be justified. A naive utilitarian argument might go something like, "we're causing one person some pain so to prevent that person from causing millions of others even more pain."

But this is an argument based on fiction. First, it assumes that torture is necessary for the captive to give up his information, when the opposite is more likely to be true. Consider the case described in Bufacchi and Arrigo's paper (p. 359):

Five foreign terrorists were captured by the local [counterterrorist police team]. All were found under arms with explosives and maps of targets....The question of how many [terrorist] cells were to be sent to the country to other targets was of interest. The first three terrorists were not even questioned, only shot. The next two were asked the question separately. One shot was heard. The officer said to the last terrorist, ‘Do you also want to remain silent?’ The guy began to lay out the entire operation, the training the cells had received, where they were to meet, where the weapons depots were located, and the route that the terrorists were to take to exfiltrate the country....The other cells were picked up along with in-country support personnel.
How was the fifth man chosen, you may wonder?
The group was searched and then fed and given tea as per the Shariat law of the Koran. Those who refused to eat or drink and made intense hostile eye contact were selected as the first three. The body posture decided who went fourth. The youngest, who ate the bread, drank tea, and thanked his captors was determined to be the least experienced. His AK rifle was not even clean...and he did not appear committed to the jihad.
So, they simply profiled the men they captured, picked the one who was most likely to confess first, and just asked him. No torture was necessary to obtain the information they needed. Once he was free from the influence of the more senior terrorists, and felt well-respected by his captors' adherence to local custom, he readily gave the police all the information they needed. Like the majority of "ticking-bomb success stories, the efficacy of torture interrogation is demonstrated only if the case is framed on that premise."

There are several premises upon which the argument for forward-looking interrogational torture is based. I will use Bufacchi and Arrigo's outline of the argument to explain this (Notation: P = premise, C = conclusion).
  • P1: The terrorist has been captured.
  • P2: Torturing the terrorist will make him reveal crucial information about the catastrophic event.
    • C1: The terrorist must be tortured.
    • C2: The crucial information will be revealed.
    • C3: The catastrophe will be averted, saving the lives of many innocent people.
Unfortunately, the conclusions C1, C2, and C3 do not follow from premises P1 and P2. There are some hidden premises that must hold true before C1-3 can follow, and none of these premises are legitimate "from an empirical point of view" (p. 360).

P1 has an invisible, companion premise:
  • P1A: This terrorist possesses crucial information about the catastrophic event.
Remember the overlapping knowledge spaces of the torturer and detainee? Well, they don't completely overlap; if they did, we wouldn't need to extract information from the detainee. So this premise is an assumption, and we could end up torturing some innocent bystander who was indistinguishable (based on our knowledge space) from a real terrorist.

P2 is problematic, too, as I have discussed above. But even if we assume that Koppl's conditions are both met, there are other problems. Many American POW's in Vietnam (e.g., John McCain) withstood a great deal of torture without confessing anything. Similarly, most victims of the Inquisition confessed to no crimes. Countless torture victims in the Algerian Civil War went to their graves with their secrets undiscovered. Terrorists are more than likely trained in the art of withstanding torture, making it even harder to draw any information out of them with this means. Finally, terrorists could be provided with false information to "confess" in order to deliberately throw their captors off the trail -- this tactic was recommended by Sun Tzu more than 2500 years ago!

And even if torture did yield information from the terrorist, what is to say that it will happen in a timely fashion? After all, if the time bomb is going to go off in 24 hours, anyone sufficiently devoted to the cause would use every ounce of their fortitude to resist for that length of time. Furthermore, torture interrogation is not a "quick coercion" -- it is a lengthy "degradation of the subject's resistance" over the course of months, not minutes (p. 361)!

According to Koppl, "It is a fair, but approximate, summary to say that, as a means of extracting information, torture works best when it is needed the least."

I couldn't have said it better myself.

Friday, April 25, 2008

My Ever-Evolving Self-Image: A Mathematical Perspective

The topic of this month's scientiae carnival is "our changing views of ourselves and our careers as we pass through time."

Since I am more mathematician than wordsmith, I thought I might express these changes in graphical form. The x-axis represents time, and the colored, dashed lines represent major life events that have exerted influence upon my life.

First up, the amount of change for the better I must bring to this world (or else face certain doom):
A few data points from the previous graph: {(4, President of the United States + Nobel Peace Prize), (8, Nobel Peace Prize), (12, Nobel Prize in Chemistry/Physics/Medicine), (18, Professor at top-ranked Research I University), (32, Best second-rate mathematician in the world)}.

Next, my level of self-confidence:
A few data points from this graph: {(4, I can never do anything right), (12, Junior High -- need I say more?), (23, Grad School is Hard!), (27, My family -- putting the fun in dysfunctional), (29, I made it out of grad school! I made it!), (32, I got a real job!)}.

Third, how convinced I am that I'm doing the right thing with my life:
A few data points for this graph: {(4, I'll never be able to stop being a total screw-up), (12, I have to be a violinist but I'll never make it), (16, I want to be a scientist even though my mother thinks my talents are best spent elsewhere), (21, Physics is hard!), (27, Why did I ever take my mother's opinion of what I should do in life as my own? WTF was I thinking?), (32, I am definitely in the right place.)}.

Wednesday, April 23, 2008

Power of Two Cake

I know you've been long awaiting a post on my powers-of-two themed cake.  Now that I'm less swamped by work, I can oblige.

But first, a little math.  (I know you all come here for the math!)  Astute commenter Rico mentioned his fascination with the sum of the series 1/2n (where n = 0, 1, 2, ...).  If you add 1 + 1/2 + 1/4 + 1/8 +..., the sum approaches but never reaches 2.  For a given n, the sum of the series equals 2-1/2n.  So for n=4, the sum equals 2 - 1/16 = 1+15/16.

I decided to represent this sum in cake.  In order to do so, I needed to bake two square cakes.  I had one 8" square cake pan, but I needed another, so I did as The Joy of Cooking suggested and squared off a rectangular cake with aluminum foil, putting dried beans on the other side:

The cake was my favorite cake recipe: cocoa devil's food cake from The Joy of Cooking.  You've seen me make this cake twice already, so I won't duplicate it here.

Here's how it turned out: the foil edge is a little uneven, but not bad.  I decided to use it as the base of the cake, to hide the unevenness.

My dad and Marvis were visiting, and Dad helped cut the cake into the proper pieces.  You can see him measuring the fractions with a measuring tape.
Here are the pieces after he was done.  So, if the other square represents one, then we have a 1/2, a 1/4, 1/8, and two 1/16s (one of which will not be used), concretely (or maybe, more aptly, chocolately) demonstrating the sum I showed above.
The frosting is a simple buttercream: butter + milk + powdered sugar + vanilla.  Here's the 1 + 1/2 + 1/4.
And here is the completed cake.
Next I decorated it with chocolate chips, to illustrate positive powers of two.  I put four chocolate chips on the smallest piece, four more on the part of the next sized layer that's showing, eight on the part of the next sized layer that's showing, etc.
So that ideally, if you look straight down on the cake, it just looks like it has 64 evenly-spaced chocolate chips on it.
Here's the cake from its most photogenic angle, complete with tissue paper flowers in the background:
I ate the leftover 1/16th piece while I was decorating the cake.  Then, when I served the cake, I had the 1/16th piece, Jeff had the 1/8th piece, and Dad and Marvis split the 1/4 piece.  We all had equal portions of cake and equal numbers of chocolate chips.  How cool is that?!?

I took the leftover cake to work with me the following Monday, so that Jeff and I wouldn't eat it.  I told everyone it was made in celebration of powers of two.  One of my colleagues emailed me with a message of "Great cake -- terrible reason!"  I had to laugh.

I hope you all enjoyed this mathematically-inspired cake as much as I did!

Friday, April 18, 2008

Busy Busy Busy

I have been so busy for the past two weeks that it's just not funny. I have been burning the candle from both ends, and probably the middle too.

I'm the PI (principal investigator) for a proposal that's due the Monday after this coming Monday (or, as the British call it, Monday-week). This means I am in charge of writing the proposal, figuring out the budget, and filling out all the dumb paperwork. And because we are soliciting money from the gubmint, then there are all kinds of crazy hoops to jump through too. Adding to the confusion is the fact that this is the first time I have ever had these responsibilities, so I don't know what I'm doing at all.

I worked on the proposal all day on Sunday, and it looks like I'm going to need to devote some more time to it this weekend. I hate doing work on the weekend but it doesn't seem like I really have a choice in this case.

Last week, I taught a one-day course on supercomputing to students and professors from colleges and universities in the region who were attending a conference put on by my division. I had to add some new content to the course, unfortunately, so I couldn't just dust of the old slides and reuse them. It was kind of a last-minute thing and I had to make all the slides the day before. But it was a success and I got a lot of positive ratings. My colleague, who organized the conference, declared that I was amazing for daring to do a live demo of writing a parallel program on the fly. He was especially impressed because it compiled and ran correctly on the first try. I attribute that to the excellent audience members, who found my syntax errors so that the compiler didn't have to.

This week, we had our users' meeting. I gave a presentation on how to use our supercomputer. My boss, his boss, and a very important person from the gubmint thought that my presentation was really good. Personally, I thought it was so-so. But hey, if they were pleased, I was pleased!

I'm really glad that the worst of the busy times are over. I still have to finish up the proposal and jump through all the proper hoops, of course, but it's manageable. Now I can get back to my other projects (work-related and other).

Wednesday, April 16, 2008

Power Winners!

Sorry, my vast blogging audience, for my involuntary hiatus. I had intended to announce the winners of the contest on the 8th (=23) but then my life and my job got in the way. So I decided that the 16th, which is two squared, squared, was a good alternative date. I'm future-posting this at 2:32 am, just to get a few extra powers of two in there. I may be awake at that time, but doing all the work I'm supposed to be doing rather than writing this post.

Anyhow, there were some terrific entries in the contest, and it was hard to decide who should win. In keeping with the theme, I will announce them in terms of powers of two.

22 place: A tie between Lost Clown and Rico, who describe their geektastic obsessions with cyclic groups and sums of 1/2N, respectively. You are my kind of people! When I get around to it, I will post pictures of my powers of two cake, which illustrates concretely (actually, chocolately) the fact that the sum of 1/2N is finite.

2-1 place: Rachel, with the bimetalic Toonie, her favorite coin. If you bring those coins south of the border, they'll be worth more than $2. Heck, at the rate the dollar's plummeting, they might be worth $22 American sometime soon!

1st place: Laura, who shared a favorite song featuring the powers of two. See, math and music share a special relationship!


20th place: Pete, whose father, mother, and he are each twins. 'Cause that's just cool.

All those of you who are not my family members or colleagues have won a very exciting prize: your choice of a Tennessee ball cap or t-shirt.

Why? Because a) I live in Tennessee, and you don't, and b) Tennessee is a power-of-two themed state name. It has four (=22) unique letters, each of which appears in it a power-of-two times (T -- once; e -- four times; n and s -- twice each). And, depending on where in the state you live, it is pronounced with either two or four syllables (TEN-see or TEN-uh-say-ee).

If you want to collect your prize, simply leave a comment with your choice, the size (if applicable), and a mailing address to which I can send your prize. The comments are moderated, and I promise not to publish your contact information. If you don't feel comfortable claiming your prize, don't worry about it; that's less work for me.

Friday, April 04, 2008

The Power of Twos

Today is a great day to celebrate the number two! Let me tell you why:
  • The date: it's 4/4/08, or, as powers of two, it's 22/22/23.
  • The timestamp of this post: 4:00 (=22) p.m. EDT, or in the 24-hour clock, 1600 (=24) hours. (Attention my workplace: I created this post in the morning before leaving for work, and used the cool new scheduled posts feature in Blogger in draft.)
  • Ages: Vinny is one (=20), my nephew Byron is 4 (=22), I am 32 (=25), and my dad is 64 (=26). Also, as of tomorrow (i.e., in =20 days) Vinny will be one and a half, which is 20+2-1.
  • Events: I've had my new job for half a year now (=2-1). We've lived here for two (=21) years. My dad and Marvis have been married for four (=22) years (although next weekend it will be five). I took my prelim exam almost four years ago. (Maybe those two almost average out.)
  • Other interesting facts: Four (=22) of the digits 0-9 are powers of two (1, 2, 4, and 8). Also, if we wrote out all these interesting numbers in base two, they would all consist of a single 1 followed by a number of zeros equal to the number of the exponent (e.g., 25 = 1000002).
In honor of this glorious day, I will (of course) be baking a powers-of-two-themed cake. (Stay tuned for pictures.) But also, I wanted to have a little contest for my vast readership.

Within the next 64 hours (that's 2 days and 16 hours, for more powers of two!), leave a comment with your favorite/most interesting connection to powers of two. I will select the "best" 2N of them, where N could be 0 but might be more, and the winner(s) will not only gain fame but also an appropriately themed prize!

Deadline: 8:00 a.m. EDT, Monday, April 7, 2008
Your favorite/most interesting connection to powers of two.

Thursday, April 03, 2008

A Toddler Taxonomy

Kingdom Animalia
     Phylum Mnao: cats, squirrels
     Phylum Ooh ooh aah aah: monkeys, apes, gorillas, Cookie Monster
     Phylum Oof: any other animal (dogs, geese, etc.)

Kingdom Plantae
     Phylum Owish: fruits of the citrus variety
     Phylum Nganga: banana
     Phylum Ppp: grapes
     Phylum Apple: any radially symmetric fruit not of Phylum Owish or Ppp (e.g., apples, pears, kiwi)
     Phylum Twee: trees, bushes