Download PowerPoint Presentation 2

Detection of Nuclear Threats:
Defending Multiple Ports
Jeffrey Victor Truman
17 July 2009
Outline of Presentation
• The Real-World Problem
• Detection
• Multiple-Port Detection
– As a Game
– As a Linear Programming Problem
– Implementing Variable Costs
• Acknowledgements
Inspection and Threat Detection
Suppose we have some inspection policy at a port that can
determine whether a container is good or bad at some
performance level, and that detection at this port depends
linearly on the budget we apply.
We then call the coefficient that measures performance level
the testing power index.
We will then apply this inspection policy to the containers.
Conditional Probability
• For a given test, the probability of “flagging”
an item (as a threat) depends on the truth
about that item: Pr{flag|truth}
• Truth = threat; not threat
• Pr{flag|threat}=d; Pr{flag|not threat}=f
• But some tests can be set to various levels of
“suspiciousness”. We can set the threshold
lower to be more cautious, which increases d
but also increases f.
The ROC Curve (d vs f)
We can plot detection rate d as a function of the
false alarm rate f for a given sensor and
threshold. Using the sensor and threshold
gives us a point (d0,f0). We can also clearly
obtain the points (0,0) and (1,1) by manually
inspecting nothing and everything,
respectively. By using mixed strategies with
suitable probabilities, it is also possible to
operate at any point along a line between any
of the points we already have.
An Example ROC Curve
The C-d Curve (d vs C)
In principle, one might want to minimize Cost (where
Cm is the cost of a miss and Cf the cost of a false
alarm):
Ctotal = Cm(1-d) + Cff = Cm + Cff – Cmd
But for some of the threats we consider, Cm is
enormous, which suggests that we need to maximize
d. However, we cannot afford the inspection policy
of manually checking everything which that would
suggest.
Thus, we plot the cost of inspection and harm to
commerce against d, plotting the optimum point for
a given budget b. [hard problem, not done here]
One Example C-d Curve
Here, we will get our scalar
testing power index from the
initial linear part of the C-d
curve, such as the one
shown to the right, which I
obtained by combining 50 of
the sensor I used to get the
previous ROC curve.
The Multi-Port Problem
• Now, we will consider multiple ports.
• We wish to allocate funding across different
ports to get the best overall detection rate.
Suppose also that our efforts at each port
have a testing power index tp. The number of
undetected threats U   a p (1  bpt p ) ,
p
with constraints  a p c p  C and  bp  B .
p
p
Possible Solutions
• We have tried to model this as a finite game,
which we can then convert into a linear
programming problem.
• One way is to consider our options as some
mixed strategy of purely defending each port
with some probability.
Specifics
• Suppose, for example, that t1 = .4, t2 = .6, and
t3 = .8. Also suppose for now that our
adversary can afford to send one bomb, and
the cost to the adversary of sending it does
not depend on the port targeted.
A Game Matrix and LP
• Then, we have a game with the following
matrix: .6 1 1


 1 .4 1 


 1 1 .2 
• A game specified in this way can be solved by
converting it to a linear programming
problem.
The Solution
• The LP problem is to maximize/minimize (for
the attacker/defender) the expected value of
the game, subject to the probabilities of the
alternatives being nonnegative numbers that
sum to one.
• We can solve this using MATLAB.
The Solution from MATLAB
• The solution is symmetric; we find that
a1=b1=0.4615, a2=b2=0.3077, a3=b3=0.2308,
and the value of the game is 0.8154.
• This means that the ideal strategy for the
attacker and defender is to attack/defend
each port with those probabilities, and that on
average we then expect about an 81.5%
chance of them getting a bomb through.
Accounting for Cost Differences
• However, in general it will not cost our
adversary the same amount to attack each port.
We take this into account by multiplying each
row by the total number of bombs they could
send to each port by devoting their entire
budget to that port.
• Suppose the adversary could send only 1/3 of a
bomb to port 1, 1/2 to port 2, and 1 to port 3.
Another Matrix and a Solution
• This gives another game matrix:
.2 1/ 3 1/ 3
.5 .2

.5


 1 1
.2 
Where ai and bi are the attacker and
defender’s probabilities, the solutions are:
a1  0
b1  0
a2  0.7273 b2  0.2727
a3  0.2727 b3  0.7273
v  0.418182
• One thing to note here is that it is now too
expensive for the adversary to attack port 1.
• We now have only a 42% chance of a bomb
getting through.
Summary
We consider multi-port defense as a finite
game, where the alternatives are focusing
completely on each port, and find a strategy
to optimize resource distribution between the
multiple ports.
Limitations of This Approach
• The main limitation that we expect from this
approach is that it assumes that the C-d curve
is linear. Since it has an initial linear segment,
we expect this approach to work there; this
region is where we do not have enough
money to completely defend any one port.
• This assumption, and thus our approach, fails
as the total budget moves beyond the linear
portion of the curve.
Acknowledgements
• The DIMACS Program 
• Dr. Paul Kantor, my mentor
– Thanks to ONR and DNDO of DHS for support
• Dr. Vladimir Menkov, designer of the C-d curve software
• Bapi Chatterjee, for a helpful zero-sum game solver for
MATLAB
• J.C.C. McKinsey, for his text Introduction to the Theory of
Games (1952).
• Luce and Raiffa, for their text Games and Decisions:
Introduction and Critical Survey (1957).