cps 173 security games
DESCRIPTION
CPS 173 Security games. Vincent Conitzer [email protected]. 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 - PowerPoint PPT PresentationTRANSCRIPT
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]