cps 173 security games
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CPS 173
Security games
Vincent Conitzer
conitzer@cs.duke.edu
Recent deployments in security
• Tambe’s TEAMCORE group at USC
• Airport security
• Where should checkpoints, canine units, etc. be deployed?
• Deployed at LAX and another US airport, being evaluated for
deployment at all US airports
• Federal Air Marshals
• Coast Guard
• …
Security example
action
action
Terminal A Terminal B
Security game
0, 0 -1, 2
-1, 1 0, 0
A
B
A B
Some of the questions raised• Equilibrium selection?
• How should we model temporal / information
structure?
• What structure should utility functions have?
• Do our algorithms scale?
0, 0 -1, 1
1, -1 -5, -5
D
S
D S
2, 2 -1, 0
-7, -8 0, 0
Observing the defender’s
distribution in securityTerminal A
Terminal B
Mo Tu We Th Fr Sa
observe
This model is not uncontroversial… [Pita, Jain, Tambe, Ordóñez, Kraus AIJ’10; Korzhyk, Yin, Kiekintveld, Conitzer, Tambe JAIR’11; Korzhyk, Conitzer, Parr AAMAS’11]
Other nice properties of
commitment to mixed strategies
• Agrees w. Nash in zero-sum games
• No equilibrium selection problem
• Leader’s payoff at least as good as
any Nash eq. or even correlated eq.
(von Stengel & Zamir [GEB ‘10]; see also
Conitzer & Korzhyk [AAAI ‘11], Letchford,
Korzhyk, Conitzer [draft])
≥
0, 0 -1, 1
-1, 1 0, 0
0, 0 -1, 1
1, -1 -5, -5
Discussion about appropriateness of
leadership model in security
applications• Mixed strategy not actually communicated
• Observability of mixed strategies?
– Imperfect observation?
• Does it matter much (close to zero-sum anyway)?
• Modeling follower payoffs?
– Sensitivity to modeling mistakes
• Human players… [Pita et al. 2009]
2, 1 4, 0
1, 0 3, 1
Example security game• 3 airport terminals to defend (A, B, C)
• Defender can place checkpoints at 2 of them
• Attacker can attack any 1 terminal
0, -1 0, -1 -2, 3
0, -1 -1, 1 0, 0
-1, 1 0, -1 0, 0
{A, B}
{A, C}
{B, C}
A B C
• Set of targets T
• Set of security resources W available to the defender (leader)
• Set of schedules
• Resource w can be assigned to one of the schedules in
• Attacker (follower) chooses one target to attack
• Utilities: if the attacked target is defended,
otherwise
•
Security resource allocation games[Kiekintveld, Jain, Tsai, Pita, Ordóñez, Tambe AAMAS’09]
w1
w2
s1
s2
s3
t5
t1
t2t3
t4
Game-theoretic properties of security resource
allocation games [Korzhyk, Yin, Kiekintveld, Conitzer, Tambe
JAIR’11]
• For the defender:
Stackelberg strategies are
also Nash strategies
– minor assumption needed
– not true with multiple attacks
• Interchangeability property for
Nash equilibria (“solvable”)
• no equilibrium selection problem
• still true with multiple attacks [Korzhyk, Conitzer, Parr IJCAI’11]
1, 2 1, 0 2, 2
1, 1 1, 0 2, 1
0, 1 0, 0 0, 1
Compact LP• Cf. ERASER-C algorithm by Kiekintveld et al. [2009]
• Separate LP for every possible t* attacked:
Defender utility
Distributional constraints
Attacker optimality
Marginal probability of t* being defended (?)
Slide 11
Counter-example to the compact LP
• LP suggests that we can cover every
target with probability 1…
• … but in fact we can cover at most 3
targets at a time
w1
w2
.5
.5
.5 .5
Slide 12
tt
t t
Will the compact LP work for
homogeneous resources?• Suppose that every resource can be
assigned to any schedule.
• We can still find a counter-example for
this case: t
t t
.5 .5
.5
t
t t
.5 .5
.5
r rr
3 homogeneous resources
Birkhoff-von Neumann theorem• Every doubly stochastic n x n matrix can be
represented as a convex combination of n x n
permutation matrices
• Decomposition can be found in polynomial time O(n4.5),
and the size is O(n2) [Dulmage and Halperin, 1955]
• Can be extended to rectangular doubly substochastic
matrices
.1 .4 .5
.3 .5 .2
.6 .1 .3
1 0 0
0 0 1
0 1 0
= .10 1 0
0 0 1
1 0 0
+.10 0 1
0 1 0
1 0 0
+.50 1 0
1 0 0
0 0 1
+.3
Slide 14
Schedules of size 1 using BvN
w1
w2
t1
t2
t3
.7
.1
.7
.3
.2 t1 t2 t3
w1 .7 .2 .1
w2 0 .3 .7
0 0 1
0 1 0
0 1 0
0 0 11 0 0
0 1 0
1 0 0
0 0 1
.1 .2.2 .5
Algorithms & complexity[Korzhyk, Conitzer, Parr AAAI’10]
HomogeneousResources
Heterogeneousresources
Schedules
Size 1 PP
(BvN theorem)
Size ≤2, bipartite
Size ≤2
Size ≥3
P(BvN theorem)
P(constraint generation)
NP-hard(SAT)
NP-hard
NP-hardNP-hard(3-COVER)
Slide 16
Placing checkpoints in a city [Tsai, Yin, Kwak, Kempe, Kiekintveld, Tambe AAAI’10; Jain, Korzhyk,
Vaněk, Conitzer, Pěchouček, Tambe AAMAS’11]
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