By Jeffrey Breinholt*

This last week while I was on vacation in California, two articles caught my attention. On the Counterterrorism Blog, Roderick Jones described how virtual worlds are helping Western governments combat terrorism, by offering realistic computer-based simulations to government personnel involved in operational decisionmaking, and as communication platforms. Meanwhile, an article in LifeScience, by Heather Whipps describes the utility of academic mathematics to fighting al Qaida.

These articles pick up on something I have been writing about - the need for more empiricism in counterterrorism. They came at a time when I was diving into a number of new books on applied math.

Whipps’ article focused on mathematical efforts to model the ideal terrorist cell, which uses graph theory to understand how networks operate and evolve. If we can identify terrorist networks and how they work, we might be better at identifying the best target nodes to eliminate in order to maximize disruption. Whipps describes the work of Jonathan Farley from Cal Tech, and quotes him as saying, “I am open to someone telling me this is nonsense,” while cautioning, “I do not get the impression that the U.S. government cares about mathematics, even though you can prove you can get better results with less money.”

While I do not think this claim is entirely accurate - the FBI employs trained empiricists in its information office and in the Terrorist Financing Operations Section, and they are doing some amazing things with Bank Secrecy Act data - I understand Farley’s sentiments. Official counterterrorism efforts have not yet been integrated with university research centers, in part because of their distinct cultures and different methods, and, within the counterterrorism ranks, the press of other matters.

This realization hit me several months ago, after I published an article which applied a branch of math known as game theory to the problem of effective counterterrorism cooperation between two allied countries. Game theory explains why two countries will not be able to cooperate fully, no matter how motivated they are at the individual level. It also offers a way out of the morass. Unfortunately, I am not sure whether the article was read by anyone. It may have been too academic for my colleagues in the Department of Justice, while too elementary to trained mathematicians and economists. (For those interested in reading it, it is here).

The science Farley is advocating is the math of networks, and it seems to be the one area of math that has garnered attention within national security circles, so much so that one of its tools - link analysis - is now a standard part of the FBI agent’s laptop. However, Farley is absolutely correct that we have barely scratched the surface on applying this science in a strategic way through the help of experts in academia. Tools like link analysis are generally used for descriptive rather than prescriptive purposes; we have not seen much application of network math to terrorist forecasting and operational planning. Hopefully, that is changing.

What about the part of mathematics that I tried to apply in my article? I had no idea how valuable game theory might be until a read a recent book by Dallas Morning News science reporter Tom Siegfied, entitled A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature (Joseph Henry Press 2006).

Science is almost always based on analogy. When Adam Smith posited that an invisible hand assures the efficient distribution of goods and services in a free market, he was not referring literally to the hand of God. Rather, he was describing a process that appeared to follow a rational course, as if i guided by a higher power. Charles Darwin, in describing evolution, was not referring to a fitness contest that organisms voluntarily enter in hopes of winning the prize of survival. Instead, he was using an analogy to describe why the offspring of more adaptable members of particular species tend to pass their lineage to successive generations.

Game theory is like a scientific analogy on steroids. Its practitioners set up artificial situations, and analyze how hypothetical participants should act if they want to maximize the resulting rewards under the rules of the game. From there, scientists try to apply the lessons from these analogies to the real world.

The most famous practitioner of game theory was John Nash, the Princeton professor featured in Ron Howard’s 2001 bio-pic “A Beautiful Mind.” Nash won the 1994 Nobel Prize in Economics for his work in describing an equilibrium that game theory players will reach in their individual decisions over repeated games.

I had wanted to know more about Nash’s theory, since I liked the movie so much that I used it as a teaching tool. My favorite scene involves Nash’s discussion with his wife, who claims that he cannot reason his way out of the paranoid schizophrenia that plagues him. The Nash character, played by Russell Crowe, defiantly asks “Why not?” A man of science, he refuses to believe that there is not a rational way of the dilemma that is his mental illness. He ultimately proves his point. He can will his mental illness away to the point of functionality, by simply ignoring the fictitous characters that he sees as he walks around the Princeton campus, even if it occasionally involves the awkward moment of asking a bystander whether something he sees is actually there. The refusal to believe that science and reason cannot be used against what plagues us, no matter how bad it seems to get, is an attitude I hope we adopt, without apologies.

It appears that game theory has much to offer in conjuring appropriate responses to al Qaida, a point that Siegfried makes in passing in Beautiful Math. Surprisingly, this conclusion comes not from the power of game theory in universalizing the findings arrived in artificial settings. Quite the opposite. The amazing thing about game theory is that it is not universal. Its tools can be used to highlight cultural differences in player strategies, and providing the scientific means of identifying them. Once identified, these differences can be exploited by official counterterrorism actions.

The classic tool of game theory is called the Prisoner’s Dilemma, which originated in a short story by Edgar Allen Poe. The authorities arrest two people they believe are part of a criminal conspiracy, and separate them into two rooms. The authorities have enough admissible evidence to convict one of them, though for a crime that is less than what is believed to be their full criminal culpability. The goal is to get one of them to talk.

Each prisoner is given a binary choice: either cooperate with the cops, or stay silent. If both stay silent, they will each receive a one-year sentence. If one confesses while the other stays silent, the confessor will walk away without punishment while the non-confessor gets five years. If both confess, they each get three years. If you are one of the prisoners, what do you do?

The inability of the prisoners to communicate with each other is what makes this situation so pernicious. If you are in that situation, your best choice depends on a prediction of what your fellow prisoner will do. If you assume that she is more self-serving than cooperative, you will likely decide to confess. On the other hand, if you believe that she has the will of G. Gordon Liddy, you might decide to follow her likely lead and stay mum.

The Nash Equilibrium posits the existence of a point at which each player’s mixed strategy in a repeating game maximizes his payoff if the strategies of the others are held fixed. In other words, if you are playing the game, there is at least one combination of strategies that, if you change yours and the other players do not, you’ll do worse. When all of the players reach this point, the game is at equilibrium.

(Note that in a single play of the Prisoner’s Dilemma, the existence of the Nash Equilibrium does not provide an answer to the immediate question of what you should do as a prisoner. The only fool-proof method involves an analogy to quantum physics, which shows atomic particles can simultaneously be in two different states at once. It strikes me that, by analogy, the quantum solution to the prisoner’s dilemma would be a choice to confess if that’s what your opponent does, while staying silent if she opts that way - in other words, two different (conditional) choices simultaneously. This is not something permitted by the rules of the Prisoner’s Dilemma. For a primer on the prospects of a quantum computing, see Ian Ayres’ Supercrunchers: Why Thinking-By-Numbers is the New Way to be Smart (Bantam 2007))

Is there evidence that the Nash Equilibrium works in real life? It has been established as a human strategy in a variety of contexts where the players are college students in post-industrial societies. Amazingly, it has also been established in the animal world. In Cambridge, England, there was a duck pond. Researchers went to different parts of the pond, and started throwing in pieces of bread of the same size. The only difference was the rate at which the bread was thrown. On one side, bread was thrown every five seconds. At the other, it was every ten seconds. Nash’s theory would suggest that the gaggle of ducks would divide into two groups whose size depended on each duck maximizing the amount of bread it receives - two thirds of the duck at the five-second locations, and one-third at the ten-second location.

That is precisely what happened. The ducks reached the Nash Equilibrium. They did so is less than a minute.

The amazed researchers than added another variable, throwing bread pieces of different sizes, Though it took a little longer, the ducks eventually divided themselves into groups that were predicted by the Nash Equilibrium in this new situation. The ducks operated according to game theory. It seems that to direct your actions to maximize your individual payout, you need not even be human.

Here’s the kicker, and why game theory is potentially so powerful in counterterrorism: it seems that culture impacts the extent to which your choices follow game theory orthodoxy. Researchers went to eastern Peru, and ran some Machiguenga farmers through some classic game theory devices. It turns out they consistently play in a different way than the American college students who were tested. This had a number of explanations. Perhaps the inapplicability of the Nash Equilibrium reflects the importance of sharing and maximizing utility across the Machiguenga community, as opposed to the intrinsic value of the payoff to individual players who were socialized to think of the community first. Maybe a player choice that violates a cultural norm creates dissonance which offsets the relative benefit of maximizing utility for oneself personally, thereby throwing off the equilibrium point.

So game theory might not be as powerful a predictor as originally thought, because its rules are not universal across cultures. Bad news? Perhaps for its committed proponents, unless one also considers advances in physiological testing described by Siegfried.

It turns out we can measure brain activity while people are involved in various game theory tests. These tests suggest that the observable transfer of dopamine - the chemical that makes you feel good - can be used a fair proxy for rewards during the play. How powerful is this method? Game theory tools and brain activity testing can determine how risk-averse and risk-loving players are, an attribute that impacts the decisions they make in times of uncertainty. Observable brain activity can accurately predict how different players will react to a series of tests that increase in risk-awareness, even when the measurements occur before their decisions have deviated. In other words, brain scans can identify the risk-takers and play-it-safers just after the beginning of the game, even while their behaviors are identical. For some people, for example, the rewards of dopamine follow more significantly when they get satisfaction from punishing their opponent, even when the alternative would result in pecuniary benefits. Does that attribute remind you of anyone currently living in a cave?

The combination of game theory and physiolology has spawned a series of new scientific fields, like neuroeconomics and sociophysics. They offer some stunning counterterrorism prospects that have not yet been fully considered. Consider what has been described as the WOG Factor. WOG was a 19th century British pejorative to describe Imperial subjects, and is an acronym for “worthy Oriental gentlemen.” The WOG Factor - not something you hear much about now, since the terms “WOG” or “Oriental” are considered offensive - referred to the fact that people from nether regions do not follow the same sort of patterns at westerners. This makes it hard for us to predict what they will do when faced with a choice we think is obvious. This is one of the problems we have in conjuring appropriate responses to Bin Laden and Al-Zawahiri, or to committed insurgents in Iraq. We tend to view them and their motivations through a western lens.

Just imagine what game theory and brain activity measurements could offer to our ability to understand the actions and motivations of our current enemies, many of whom are now in U.S. custody.

Is this ethical? I cannot imagine that anyone would argue that the type of testing done on U.S. college students amounts to torture. On the issue of consent, I can envision real games, involving real prison-like rewards, that would result in willing participants among the inmate population at Guantanamo. In scientific parlance, this would make them consenting subjects. The benefits? Imagine if we can the ability to discover the hidden code for the jihadist mindset, so that we could more effectively choose strategies for altering their calculus over the long term? This would be entirely different - and perhaps more significant - than the process of acquiring information within their personal knowledge. My bet is that some of their choices are as cultural and subliminal as that of the Peruvian farmers, as just as foreign to our way of thinking.

There are bound to be people who argue that it is unfair for advanced societies to use technology to understand their enemies in the Third World, where they do not have the same technology to use against us. To that, my response is simple: we invented it, and we did not start this war.

Some of the promising mathematical applications to counterterrorism are more purely numeric. In addition to Beautiful Math, there is another book I would recommend to anyone interested in how mathematics could be harnessed in fight terrorism: Keith Devlin and Gary Lorden’s Solving Crime with Mathematics: The Numbers Behind Numb3rs (Plumb Books, 2007). Written as a companion guide to the CBS television series, it is a highly accessible description of how math can work for law enforcers. It contains chapters on data mining, image enhancement, DNA and network analysis, along with game theory, all of which have been used by the characters in the show. My favorite revelation of the book, however, involved a mathematical formula for determining whether we are seeing the beginnings of an epidemic, or a pattern that will become manifest over time.

In math, this is referred to a “changepoint detection.” It works like this: you have a list of observable phenomena that are considered to be precursors to something bad that may happen in the future. A few occasional occurrences of this precursor activity are not big problem, and are expected. However, when these occurrences get too frequent, you face the question of whether it is the start of an epidemic. For example, in medical epidemics, the observable phenomenon might be the rate at which people appear at hospitals complaining about certain symptoms. Declaring an epidemic would trigger the full of responses. If it is done too early and an epidemic is not actually in progress, resources are wasted and people will becomes less trusting when the government announces the next one. This makes pinpointing the start of an actual epidemic of tricky proposition. Exactly when do the precursors reach the point of increasing frequency that we can be confident that we are seeing the beginning stages of an epidemic?

Statistics have developed a number of good numerical tests for changepoint detection. The most effective one was apparently developed by a British mathematician named E.S. Page.

Let’s assume that something happens on average of once a month, but that it will be considered an epidemic if it accelerates to a weekly occurrence. Page’s model starts with a numerical index S, which will change everyday depending on whether the event occurs. When S reaches a certain number (typically 50), one can conclude that an epidemic is in progress. If S ever drop below 1 upon its daily recalculation, it is automatically reset at 1, which is the daily minimum. From there, it is just a matter of daily recalibration to see if S has reached 50.

The recalibtration is easy enough for a pocket calculator. Every day, S is updated, by multiplying it by the probability of whatever happened that day assuming that the epidemic is occurring, and dividing it by the probability of whatever happened, assuming that the epidemic is not occurring.

For example, if you are worried about a non-epidemic monthly occurrence becoming a weekly occurrence, you want to know when enough occurrences are happening that the rate is now once every seven days, as opposed to once every 30 days. The probability-if-we-assume-epidemic is 1/7, while the probability-if-we-assume-normalcy is 1/30. .

So every day, we have either a “yes” if the event occurs, or a “no” if the event does not occur. If “yes,” we multiply the previous day’s P by 1/7 and divide the result by 1/30. If “no,” we multiply the previous day’s P by 6/7, and divide the result by 29/30. If P stays below 50 despite a number of consecutive “yes” days, the epidemic has not started despite the trends in that direction. Once P reaches 50, however, we are seeing a significant enough shift from the status quo to the frequency rate that we fear to conclude we should declare epidemic and respond appropriately. The beauty of Page’s model, as noted in Solving Crime With Numbers, is that we can respond without waiting for the actual observed frequency to reach once every seven days, by which time effective responses may be too late.

This statistical tool of changepoint detection has immense significance for counterterrorism operational decisions, where our chosen responses carry many of the same risks and costs of announcing a health epidemic. Like the science described in Beautiful Math, It also has great applicability to a number of other issues that are of interest to Counterterrorism Blog readers and contributors.

(As always, the views expressed in this post are the author’s own, and do not reflect those of the Department of Justice.)

*Counterterrorism Blog
December 4, 2007
http://counterterrorismblog.org/2007/
12/the_promise_of_mathematics_to.php
Cross-posted with permission

---

---

Related: War Against Islamo-fascism, Terrorist Groups, Technology

---