Mathematical Aspects of Terrorism Hazard

The following article, written for Catastrophe Risk Management by Dr. Gordon Woo, catastrophe research consultant for RMS, was first published in April 2002.

From ancient Babylonian times, when money lent for trading was forfeited if the trader was mugged on his travels, the pursuit of commerce has benefited from insurance against criminal activity. Financial protection has long been provided against a battery of malevolent human actions: from burglary and arson to hijacking, piracy, kidnapping and mugging. Whatever the vagaries of human nature, criminal intent is one of the more predictable traits, and many types of crime are sufficiently frequent to generate substantial volumes of statistical data. Where past loss data are available, an actuarial approach to fitting a loss curve may be feasible. Natural catastrophes are notoriously low-frequency high-severity events, the risk of which cannot be evaluated using experience data alone. As such, mathematical models of the underlying stochastic processes have been developed in order to quantify their risk.

On a single day last September, al-Qaeda elevated terrorism to the category of catastrophe risk. Just as the loss experience from Hurricane Andrew and the Northridge earthquake could not have been extrapolated from scanning a few decades of historical claims data, so the multi-billion dollar loss from the World Trade Center disaster was of a different order of magnitude from previous terrorism losses, including the earlier truck bomb attack on the World Trade Center in 1993. Unfortunately, whereas there exist physical laws which allow the scaling of natural hazard events from small to large, such laws do not exist for terrorism events. In the context of flood hazard in the Netherlands, a leading Dutch mathematician, Laurence de Haan, entitled a seminal paper, ‘Fighting the Arch-enemy with Mathematics’. This begs the question: what mathematics can be used to fight terrorism? Even if acts of terrorism are not governed by physical laws, they are governed by strategies. The great Chinese master of the art of warfare, Sun Tze, wrote in the first millennium B.C. that ‘what is of supreme importance in war is to attack the enemy’s strategy’. An understanding of terrorist strategy should be of value to counter-terrorist forces and insurers alike.

The analysis of human conflict has a mathematical guise called ‘game theory’. Much of the early work on game theory was done during World War II, and intensive research was further undertaken during the subsequent Cold War era. Popularization of the concepts of game theory may be ascribed to Sylvia Nasar, whose book on the life of the economics Nobel laureate John Nash was turned into an Oscar-winning movie: ‘A Beautiful Mind’. During the early 1950’s, John Nash consulted on strategy issues for the RAND corporation, which remains to this day at the forefront of quantitative conflict research.

Although the study of recreational board games gave rise to its name, game theory is the general study of mathematical models of conflict between intelligent rational decision-makers. Seasoned chess players may lose games on occasion to dumb irrational players, but the most formidable opponents are both intelligent and rational. As a London schoolboy, Omar Saeed Sheikh, the self-confessed kidnapper of the Wall Street Journalist Daniel Pearl, was a champion at chess. He later enrolled to study mathematics at the London School of Economics. The threat posed by terrorists would be misjudged if their intelligence and rationality were underestimated or insulted. The national intelligence services should have learned this lesson. The first female head of MI5, Stella Rimington, ignominiously recalls in her autobiography how she heard about one of the most devastating IRA bomb attacks on London, through watching CNN in New Zealand.

Terrorists may be intelligent, but rational also? According to a dictionary definition, behaviour is rational if it is endowed with reason. What this reason is, in the case of Islamic Jihad, is explained lucidly in the book, ‘Milestones’ by Sayyid Qutb. The Milestones to which the title refers are steps along the way to the establishment of Islamic states. This is a Utopian political vision of a god-fearing government with no barriers of race and class, and where women are not exploited for their physical attractions to work as air stewardesses. (This is Sayyid Qutb’s own prophetic example). Such a vision was anathema to the secular Egyptian government of the 1950’s. The book was banned, and its Egyptian author was executed. In writing a book which espoused the virtues of martyrdom, Sayyid Qutb must have reasoned that martyrdom was his own inevitable destiny. Through the fluency of his pen and the style of his death, Sayyid Qutb has inspired and motivated a generation of Islamic radicals, including Osama Bin Laden himself.

For one who resolutely believes Sayyid Qutb in the promise of paradise open to martyrs, the commitment of a suicidal act of violence to further the cause of Islamic statehood is rational. But the preparedness, even willingness, of Islamic militants to become martyrs itself opens up new strategies in the analysis of the conflict between terrorist and counter-terrorist forces. These new strategies offer fresh game theory insights into terrorism hazard. Consider the grenade game, which is one of the standard illustrations of game theory. This game involves two players A and B. First, player A chooses between giving player B $1000 or nothing. Secondly, player B observes player A’s move, and then chooses whether or not to explode a grenade that will kill both players. Suppose that player B threatens to explode the grenade unless player A pays the $1000. In the conventional analysis of this game, if player A believed the threat, his best response is to pay the $1000. However, since this threat involves an act of suicide, the threat by player B is regarded as not credible.

Another game theory paradigm is the timing of firing in a duel. If two protagonists, each armed with a pistol with a single bullet, walk towards each other, at what point should one fire? The later one fires, the more sure of hitting the target, but the greater the chance of being hit oneself. In the conventional game theory analysis of this contest, the payoff, if a duelist succeeds in hitting his opponent, is + 1, and the payoff, if he is hit by his opponent, is –1. However, if paradise is the payoff for martyrdom, then an Islamic militant would wish to be maximally sure of hitting the target, and would tend to fire later. Taking sufficient time to achieve mission success is a trait of al-Qaeda. The patience and diligence with which al-Qaeda operations are planned reflect underlying fundamentalist belief in the high payoff of a suicide mission.

In 1994, Algerian terrorists planned to fly a jet into the Eiffel tower in Paris. Unbeknown to both MI5 and CIA, as early as 1995, dissident Afghan waiters in London were soliciting American signatories on applications for flight training in USA. The planning for September 11 had begun at least seven years earlier. Faced with the contrasting prospects of paradise, if they succeeded, or prison, if they failed, the leaders of the suicide hijack mission were rational in taking meticulous care over every detail of their planning.

Not just the preparation time, but also the swarm attack is a feature of al-Qaeda strategy which is comprehensible in game theory terms. In an al-Qaeda training manual, found in an apartment in Manchester, England, missions are listed as including the destruction of embassies, urban bridges, and centres of vital economic interest. If one specific class of target is selected for attack, (e.g. embassies, bridges, ports, etc.), defences would inevitably be strengthened after a strike, making a second attempt more difficult. Already this has happened with US airport security. Hence an opportunist terrorist strategy would be to launch a simultaneous attack on many individual targets within the same class, so stretching homeland defence. Al-Qaeda have managed to synchronize surprise attacks on US embassies and landmark buildings. In conventional military strategy, the casualty rate resulting from such simultaneous attacks might be prohibitive. The strategist, Sun Tze, argued against using troops like a swarm of ants; a strategy bound to lead to high casualties.

The social insect metaphor is intriguing for a terrorist network such as al-Qaeda, prepared to launch martyrdom missions. Astonishing levels of spatial swarm intelligence are achievable by colonies of ants, which can fulfill their programmed functions without the need for central instruction. If terrorists depended heavily on communication with a command hub, swarm attacks might be quite susceptible to counter-intelligence. However, participants may operate essentially individually, and may not be stationed together in any one locality. Instead, they may form emergent virtual cells, the members of which would be dispersed over the world, communicating via the internet to plan an attack.

The lessons for insurers from this strategic analysis reinforce those which might be drawn from the historical record of terrorism. Aggregate urban exposures need to be tightly controlled, as do exposures to specific classes of property or infrastructure, which might be specially targeted. Catastrophe loss correlated over a wide geographical area is not just a signature of a natural disaster; it can result from synchronized terrorist attacks. This would be alarming enough if they were attacks using explosives or missiles. If they were to be multiple attacks using weapons of mass destruction, such as nuclear devices, salvation might depend on the presence of a genius, such as imagined in the movie, ‘A Beautiful Mind’, with powers of discerning obscure messages of prior intelligence.