Applications of Game Theory to Homeland Security

Margaret (Midge) CozzensDIMACS

Rutgers UniversityNovember 12, 2010 at

Embry-Riddle University

To predict what opponents will do to maximize impact or minimize cost;

To determine what dterrence strategies can be employed

N Players (may be coalitions of players) M Strategies for each player, may or may

not be the same for each player Each player has a utility function (what

they value) The utility functions are used to

determine the payoffs for each player for each pair of strategies

The optimal strategy is determined for each player, usually at a Nash Equilibrium

Two players: Comcast and Verizon Each has two strategies: double

television advertising or utilize social networks for marketing

The utility function for each is a single thing – yearly profit

The payoff values are determined by profit = revenue - cost

Comcast can increase profit by $200,000 if it increases its TV ads and Verizon does also. It decreases its profit by $200,000 if Verizon uses viral marketing at the same time.

Comcast can increase profit by $600,000 if it uses viral marketing and Verizon uses more TV ads. It can increase profit by $200,000 if both use viral marketing.

Using P = R – C and survey data: Verizon can increase profit by $500,000

using viral marketing tools and Comcast does not use viral marketing. It can increase profit by $200,000 if Comcast also uses viral marketing.

Verizon can increase profit by $100,00 if it increases TV ads and Comcast does also. It can increase profit by -$300,000 if it increases TV ads, and Comcast uses viral marketing.

Increased profit Increase TV ads Viral marketing

Increase TV ads $200,000,$100,000 -$200,000,$500,000

Viral Marketing $600,000,-$300,000 $200,000,$200,000

C

V

A pure strategy for a player is for that player to use the same strategy all of the time, (ex. every month or every year)

Step 1: for the row player, and each strategy of the row player, find the column player’s best response and indicate in blue the corresponding entry.

Step 2: for the column player, and each strategy of the column player, find the row player’s best response and indicate in red the corresponding entry.

Step 3: Identify the entry(s) of the payoff matrix which both colors. These are Nash equilibrium for the game.

Increased profit Increase TV ads Viral marketing

Increase TV ads $200,000,$100,000 -$200,000,$500,000

Viral Marketing $600,000,-$300,000 $200,000,$200,000

C

V

Not surprisingly, the optimal strategy for both Comcast and Verizon is to use viral marketing

The Nash Equilibrium is (viral mark,viral mark) at (200,000, 200,000) which says that the worst each could do is gain $200,000 by following the viral marketing strategy.

A Nash Equilibrium is a set (pair) of actions a* such that for every player i, ai* is preferred to every other possible action of player i.

If the game is repeated, equivalently decisions are made once a year by Comcast and Verizon, with no change to the payoff matrix, a mixed strategy could be employed.

In this example, there is a pure strategy for each, so working out the mixed strategy equations would yield the exact same answer.

The two players are the US and Al Qaeda. Either can be coalitions, for example US and England.

A utility function is a quantification of a person’s preferences with respect to certain objects or outcomes. Thus, a utility function assigns a number to each possible object or outcome, and ends up with an aggregated utility value. The expected utility of an action is the sum of the probabilities multiplied by the utilities.

In the US-Al Qaeda conflict, the four objects of utility (or motivating factors) are money, lives, reputational capital, and chits in heaven, which vary for each player. For example, while money has relatively high utility for both the US and Al Qaeda, reputational capital (a measure of a country’s reputation, or good standing) has higher utility for the US than for Al Qaeda. Another motivating factor that has higher utility for the US than for Al Qaeda is saving lives. However, chits in heaven (each death of an American in the US-Al Qaeda war increases the likelihood of an Al Qaeda soldier gaining entry to heaven) has very high utility for members of Al Qaeda, but is meaningless to the US.

We used information from the 9/11 Commission Report to come up with equivalency scales for the four motivating factors. These scales allow us to account for the differences in utility of the four motivating factors for the US and Al Qaeda, so that their payoffs in each of our game models would actually be comparable. The four key facts from the 9/11 Commission Report are:

o the total of about 3,000 people died in 9/11; o there were only about 20 plane hijackers, all Arabs

commissioned by Al Qaeda; o it cost Al Qaeda somewhere between $400,000 and

$500,000 to execute 9/11; and o it will cost the US about $700 million to rebuild the

World Trade Center.

USA: -$700,000,000 = -3,000 lives, -3,000 lives = -1 RC, -1 RC= 0 chits in heaven

Utility 4-tuple: (-7, -3, -1, 0)

Note: money in 100 million and lives in thousands lost 

Al Qaeda: -$500,000 = -3,000 lives, -3,000 lives = 1 RC, 1 RC = 2 chits in heaven

Utility 4-tuple: (-.005, -3, 1, 2)

Don’t Attack (Al Qaeda) Attack (Al Qaeda)

Don’t Attack (US) (6, -1) (-11, 3)

Attack (US) (-7, -2.5) (-11.5, 5)

Non zero-sum game

Don’t Attack (Al Qaeda) Attack (Al Qaeda)

Don’t Attack (US) (6, -1) (-11, 3)

Attack (US) (-7, -2.5) (-11.5, 5)

Non zero-sum game

US

AQ

Nash equilibrium with AQ attacking and US not attacking!

Consider a sequential game, where one player starts it off and the other player acts in succession.

Increase the incentives to change the motivation and thus the payoffs.

Use backward induction – work backwards

The only information a player has is what has gone before.

Find a Bayesian Equilibrium – a probabilistic equilibrium determined by values of p, q, and r.

If q = .7 and r - .6If p > .6, then the Bayesian equilibrium is Al Qaeda

attacking, followed by the US attacking vigorously, followed by Al Qaeda attacking vigorously, with a payoff of -27p(1-q) for the US and a payoff of 10p(1-q) for Al Qaeda.

If p < .6, then the Bayesian equilibrium is Al Qaeda attacking vigorously, followed by the US attacking vigorously, followed by Al Qaeda attacking vigorously, with a payoff of -29(1-p)(1-r) for the US and a payoff of 12(1-p)(1-r) for Al Qaeda.

The only alternative seems to be incentives, but how much and of what type?

Suppose the US were to offer Al Qaeda 400 million dollars (4 units in our payoff scale) in exchange for them signing a contract pledging not to attack the US?

The repercussions of breaking this contract would have to be negative enough to deter Al Qaeda from so doing, at least -1 in reputational capital.

Don’t Attack (Al Qaeda) Attack (Al Qaeda)

Don’t Attack (US) (2, 3) (-11, 2)

Attack (US) (-11, 1.5) (-11.5, 5)

New Payoff Matrix

Here the US should not attack and Al Qaeda should not also - this is the tipping point

Suppose that the US does not come up with the money, so there is no loss of money to the US, but NATO, or others contribute the 4 million.

Don’t Attack (Al Qaeda) Attack (Al Qaeda)Don’t Attack (US) (6, 3) (-11, 2)Attack (US) (-7, 1.5) (-11.5, 5)

The Nash equilibrium for this game is neither the USA nor Al Qaeda attacking, which is what we wanted to achieve.

Ricin is a poison found naturally in castor beans. If castor beans are chewed and swallowed, the released ricin can cause injury.  Ricin can be made from the waste material left over from processing castor beans.

It can be in the form of a powder, a mist, or a pellet, or it can be dissolved in water or weak acid.

It is a stable substance under normal conditions, but can be inactivated by heat above 80 degrees centigrade.

The resulting waste mash of processing castor beans contains between 30,000 to 50,000 tons of ricin

It would take a deliberate act to make ricin and use it to poison people. Accidental exposure to ricin is highly unlikely, except through the ingestion of castor beans.

Ricin works by getting inside the cells of a person’s body and preventing the cells from making the proteins they need. Without the proteins, cells die. Eventually this is harmful to the whole body, and death may occur.

Effects of ricin poisoning depend on whether ricin was inhaled, ingested, or injected.

If we suspect that people have inhaled ricin, a potential clue would be that a large number of people who had been close to each other suddenly developed fever, cough, and excess fluid in their lungs. These symptoms could be followed by severe breathing problems and possibly death.

Respiratory symptoms start within 12 hours of inhaling ricin.

Death occurs from 36 to 48 hours after exposure.

Inhalation: The Pepsi Center located in downtown Denver holds 20,000 people per event at maximum capacity. If there were to be full attendance where half of the people were male and half were female, then a terrorist would only need to put about 68 grams or about 0.15 pounds of ricin in the ventilation system in order to kill all 20,000 people in attendance.

Injection – trace amounts added to flu shots. Ingestion – trace amounts in food supply.

Perhaps the most famous incidents of domestic use of ricin took place during October of 2003 and February of 2004. Two letters containing ricin were found in October 2003, both signed “Fallen Angel.” One of these letters was discovered at a Greenville, South Carolina postal facility and the other was found in an off-site processing facility for the White House. Both letters threatened to “turn DC into a ghost town” if changes in how truckers’ hours are regulated were not changed [18]. The FBI is still looking for the person(s) connected to these incidents

In February 2004, ricin was discovered in Senate Majority Leader Bill Frist’s office. This discovery prompted closure of three Senate office buildings. Law enforcement officials are unsure as to how the ricin got into the office.

More recently, a 22 year old man from Ocala, Florida was arrested after authorities found a stash of ricin being stored in a cardboard box in his home. The sheriff’s office was tipped off by an informant who saw the man carrying a vial of ricin in a local nightclub.

Put 1000 μg (per person) of ricin in a food source (Food).

Poison a water supply with 1000 μg (per person) of ricin (Water).

Inject a group of people with ricin from a tainted shot supply of 1000 μg of ricin,.

Distribute ricin into an Air Vent (AV) in a building of an amount two times the lethal dose, i.e. in the Pepsi Center example, 136g would be released.

Drop ricin via a Crop Duster (CD) over a city, 8 metric tons per 100 km2.

No Action

1. Use of nasal swab combined with symptomatic care.2. Some sort of detector to locate ricin in the air/water.3. Intelligence: Knowing a priori where and how an

attack will happen.4. Reverse osmosis filtration in public water systems.5. Tamper proof seals on foods.6. Vaccinate the public.7. Immediate Symptomatic Care.8. Constant use of a gas mask.9. No action for first 24 hours then provide adequate

symptomatic care.10. No action for first 48 hours then provide adequate

symptomatic care.11. No action.