Stackelberg Security Games for Security

Fernando Ordóñez

Universidad de Chile

Stackelberg Games for Security

Fernando Ordóñez

Universidad de Chile

Stackelberg Games for Security

Fernando Ordóñez

Milind Tambe, P. Paruchuri, C. Kiekintveld, B. An, J. Pita, M. Jain, J. Tsai, R. Yang, A. Jiang,

M. Brown, E. Shieh… and others

Stackelberg Security Game

4

5

Stackelberg Security Game

Stackelberg Security Game

6

Monday Tuesday

Stackelberg Security Game

7

Wednesday

Outline

• Stackelberg Games

• Deployed Applications

• Challenges in Stackelberg Security Games

– Problem Size

– Uncertainty/rationality

– Model Inputs (data, game definition)

• Ongoing work

Game Theory: Stackelberg Game

• Stackelberg: defender goes first, attacker second

• Non zero sum utilities

• A mixed strategy is optimal for the leader

Police

Adversary

Target #1 Target #2

Patrol #1 7, -4 -2, 3

Patrol #2 -7, 7 4, -3

Game Theory: Stackelberg Game

ARMOR: LAX (2007)

Deployed Security Game Applications

IRIS: FAMS (2009)

GUARDS: TSA (2010) PROTECT: USCG (2011)

Optimization Model (Rational Adversary)

)7( ),(maxarg

)6( 0,1

)5( 1

)4( assignment feasible ,10

)3( 1

)2( =

)1( urcesTotal_Reso .

),( max,

ik

a

ikq

k

k k

jj

jj

jjj

Ti

i

ik

d

ikax

qxUq

q

q

Aa

a

Aax

xts

qxU

A

AConstraint on x to enforce a feasible marginal coverage on targets

USCG Patrols

Port of Boston (Not actual areas)

Challenges in SSG

• Problem Size

• Uncertainty/rationality

• Model Inputs (data, game definition)

• Evaluation

Federal Air Marshals (FAMS)

Strategy 1 Strategy 2 Strategy 3

Strategy 1

Strategy 2

Strategy 3

Strategy 4

Strategy 5

Strategy 6

Strategy 1 Strategy 2 Strategy 3

Strategy 1

Strategy 2

Strategy 3

Strategy 4

Strategy 5

Strategy 6

Multiple Defense Resources

4 Flights 2 Air Marshals

100 Flights 10 Air Marshals

6 Schedules

17,000,000,000,000 Schedules

Pure strategies are joint schedules: Each air marshal assigned to a tour

Payoff duplicates: Depends on target covered

Speedup: Compact Representation

ARMOR Actions

Tour combos

Prob

1 1,2,3 x1

2 1,2,4 x2

3 1,2,5 x3

… … …

120 8,9,10 x120

CompactAction

Tour Prob

1 1 y1

2 2 y2

3 3 y3

… … …

10 10 y10

Attack 1 Attack 2 Attack …

Attack 6

1,2,3 5,-10 4,-8 … -20,9

1,2,4 5,-10 4,-8 … -20,9

1,3,5 5,-10 4,-8 … -20,9

… … … … …

ARMOR: 10 tours, 3 defenders

MILP similar to ARMOR 10 instead of 120 variables y1+y2+y3…+y10 = 3

Algorithm Development

• Tight formulations

• Decomposition Methods

– Column generation

– Constraint generation

• Heuristic Methods

Uncertainty/Rationality

Uncertainty/Rationality

Optimization Model (Partially Rational Adversary)

Fractional and Non-Convex

)4( assignment feasible ,10

)3( 1

)2( =

)1( urcesTotal_Reso .

)( )( max)(

)(

,

jj

jj

jjj

Ti

i

i

d

ie

e

ax

Aa

a

Aax

xts

xUxF

A

A

k

xak

U

xaiU

Playing against Human Adversaries

Experimental Results

PT = Prospect theory QRE = Quantal Response Equilibrium

Model Inputs

Steps to build SSG

1. Gather representative data

2. Define targets to protect

3. Define time periods to protect

4. Types of Attackers

5. Defender and Attacker utilities

1: Relevant Data

• 2 year crime event data

• Horizon: annual averages of crime

– No daily variation

– No seasons

• Baseline patrol strategy

2: Targets

Clustering,

nodes with > 10 events in 20 meters

3/4: Periods/Attacker types 8 attacker types (clustering crime data)

7 Periods (cross police shifts with crime types)

Prob. de un tipo de atacante en un periodo Cluster S1 S2 S3 S4 S5 S6 S7 Total

0 0,234 0,516 0,624 0 0,603 0,562 0,395 1815

1 0,078 0,057 0,048 0,142 0,049 0,079 0,097 679

2 0 0 0 0,47 0 0 0 545

3 0,032 0,018 0,018 0 0,012 0,027 0,05 369

4 0 0 0 0,26 0 0 0 405

5 0,253 0,091 0,063 0,079 0,066 0,093 0,15 808

6 0,023 0,027 0,022 0,048 0,033 0,016 0,024 419

7 0 0 0 0 0 0,223 0,285 575

8 0,381 0,291 0,225 0 0,238 0 0 1110

Total 727 457 1892 1217 939 881 612

5: Utilities

Crime events have a value information

Cluster Avalúo ($)

0 $ 91.175

1 $ 104.448

2 $ 67.976

3 $ 225.985

4 $ 87.650

5 $ 108.717

6 $ 69.481

7 $ 69.246

8 $ 109.174

Cluster Promedio de Utilidad Días Reclusión Tasa Descuento Costo ($)

0 91175 61 40% 319113

1 104448 1752 40% 365568

2 67976 63 40% 237916

3 225610 1746 40% 789636

4 87650 1747 40% 306776

5 108717 1686 40% 380511

6 69481 74 40% 243184

7 69246 1757 40% 242362

8 109174 1739 40% 382109

Results

A frequency with which each node should be protected to maximize utilities

Evaluation

• Computer

• Anectdote

• Tests on field

-1,6

-1,4

-1,2

-1

-0,8

-0,6

-0,4

-0,2

00

0,5 1

1,5 2

2,5 3

3,5 4

4,5 5

5,5 6

Def

en

de

r's

Exp

ect

ed

Uti

lity

Attacker λ value

PASAQ(λ=1.5)

DOBSS(λ=∞)

PASAQ(noise high)

DOBSS(noise high)

Robustness Results: Observation Noise

Patrol Schedules – before/after PROTECT

0

5

10

15

20

25

30

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7C

ou

nt

0

20

40

60

80

100

120

140

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7

Co

un

t

Base Patrol Area

Pre-PROTECT

Post-PROTECT

From the Port of Boston

Conduct pre- and post-PROTECT assessment Effectiveness (tactical deterrence) increased from pre- to post- PROTECT observations

Adversarial Perspective Team (APT)

On going work: protecting the border

Sampled patrols from optimal solution

Research Questions • Efficient algorithms to solve real instances

(patrolling on a network)

• Automatically determine payoff values

• Multiple types of security resources

• Validation