Margaret (Midge) Cozzens
DIMACS
Rutgers University
November 12, 2010 at
Embry-Riddle University
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To predict what opponents will do to maximize
impact or minimize cost;
To determine what dterrence strategies can be
employed
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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
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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
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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.
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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.
V
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
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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.
V
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
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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.
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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.
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The two players are the US and Al Qaeda.
Either can be coalitions, for example US and
England.
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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)
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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)
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Non zero-sum game
Don’t Attack (Al Qaeda)
Don’t Attack (US)
Attack (US)
Attack (Al Qaeda)
(6, -1)
(-11, 3)
(-7, -2.5)
(-11.5, 5)
Non zero-sum game
AQ
Don’t Attack (Al Qaeda)
Don’t Attack (US)
Attack (Al Qaeda)
(6, -1)
(-11, 3)
(-7, -2.5)
(-11.5, 5)
US
Attack (US)
Nash equilibrium with AQ attacking and US not attacking!
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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.
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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 - .6
If 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.
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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.
New Payoff Matrix
Don’t Attack (Al Qaeda)
Don’t Attack (US)
Attack (US)
Attack (Al Qaeda)
(2, 3)
(-11, 2)
(-11, 1.5)
(-11.5, 5)
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.
The Nash equilibrium for this game is neither the USA nor Al
Qaeda attacking, which is what we wanted to achieve.
Don’t Attack (Al
Qaeda)
Attack (Al Qaeda)
Don’t Attack
(US)
(6, 3)
(-11, 2)
Attack (US)
(-7, 1.5)
(-11.5, 5)
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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
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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.
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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.
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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
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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.
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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.