Anavilhanas Natural Reserve (about 4000 Km 2 )

Transcription

1 Anavilhanas Natural Reserve (about 4000 Km 2 )

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4 A control room receives this alarm signal: what to do?

5 adversarial patrolling with spatially uncertain alarm signals Nicola Basilico, Giuseppe De Nittis, Nicola Gatti

6 Game theory for security of physical environments Objective: provide protection to physical environments against threats. The effort that can be spent in a time unity is limited: finite resources Our constraints: resources can provide security only locally, are finite, and not enough to protect everything at the same time.

7 Game theory for security of physical environments Assume that the threat is the output of a rational decision maker (Attacker) The resource allocation policy should consider the presence of an adversary that, for example, can observe and plan, has preferences. Security Game Defender: controls resources to protect the environment VS Attacker: tries to compromise some area without being detected The proper solution concept prescribes the policy (strategy) to be adopted Seminal works [Flood, 1972; Washburn 1995;] Recent works [Paruchuri et al. 2008; Pita et al. 2008; Kiekintveld et al. 2009; Yang et al. 2011;]

8 The Basic Model Idea: a game theoretical setting where the Defender is supported by an alarm system installed in the environment Environment: undirected graph Target t: v(t) value d(t) penetration time: time units needed to complete an attack during which capture can happen At any stage of the game: The Defender decides where to go next The Attacker decides whether to attack a target or to wait

9 The Basic Model In several applications the Attacker can obtain a correct belief about the Defender s strategy. Worst case assumption: the Defender s strategy is common knowledge of the game. Stackelberg paradigm: a Leader commits to a strategy, a follower best responds to it (In our setting the Defender will be the leader and the Attacker will be the follower.)

10 Leader-Follower (Example) F L A B C D (5,1) (1,0) (6,2) (-1,5) Let s suppose that, before the game begins, L makes the following announcement: L

11 Leader-Follower (Example) F L A B C D (5,1) (1,0) (6,2) (-1,5) Let s suppose that, before the game begins, L makes the following announcement: L F

12 Leader-Follower (Example) F L A B C D (5,1) (1,0) (6,2) (-1,5) Let s suppose that, before the game begins, L makes the following announcement: L I will play C F F

13 Leader-Follower (Example) F C D L A B (5,1) (1,0) (6,2) (-1,5) L

14 Leader-Follower (Example) F C D L A B (5,1) (1,0) (6,2) (-1,5) L Leader follower equilibrium (LFE) L

15 Leader-Follower (Example) F C D L A B (5,1) (1,0) (6,2) (-1,5) L Leader follower equilibrium (LFE) L Two important properties: 1. The follower does not randomize: it chooses the action that maximizes its expected utility. If indifferent between one or more actions, it will break ties in favor of the leader (compliant follower). 2. LFE is not worse than any NE (the leader can always announce a NE)

16 Computing a LFE (general sum) Idea: 1. For each action b of the Follower: Find the best commitment C(b) to announce, given that b will be the action played by F 2. Select the best C(b) Step 1

17 Computing a LFE (general sum) Idea: 1. For each action b of the Follower: Find the best commitment C(b) to announce, given that b will be the action played by F 2. Select the best C(b) Step 1

18 Computing a LFE (general sum) Step 2: We need to solve a LP n times, where n is the number of actions for the Follower For zero-sum games: maxmin strategy (For multiple followers and uncertain types of followers the problem becomes harder.)

19 The Alarm System Each attack at a target t probabilistically generates a signal that is sent to the Defender Example (deterministic): If an attack is present on tagets {8,4,5} generate B If an attack is present on tagets {6,7} generate A Signal A Signal B

20 The Alarm System Each attack at a target t probabilistically generates a signal that is sent to the Defender Signal C Example (non-deterministic): If 8 is attacked signal B is generated with some probability p, signal C with 1-p. Signal A Signal B

21 The Alarm System Each attack at a target t probabilistically generates a signal that is sent to the Defender Signal C Example (non-deterministic): If 8 is attacked signal B is generated with some probability p, signal C with 1-p. Signal A Signal B It is convenient to distinguish between two operation modalities in this setting: - Normal patrolling: if the Defender does not receive any signal it normally patrols the environment; - Signal response: if the Defender receives a signal it must handle it.

22 The signal response phase B Signal A The Defender patrols the environment and now is in v 1 The Attacker attacks v 4 The Alarm system generates with prob. 1 signal B Signal B

23 The signal response phase Upon receiving the signal, the Defender knows that the Attacker might be in v 8, v 4, or v 5 B In principle, it should check each target no later than d(t) 1 8 d=3 4 d=1 5 d=2 1 4 d=1 5 d=2 8 d=3 1 4 d=1 8 d=3 5 d=2 Covering routes (maximal)

24 Game Tree Let s take a look to the game tree, we have three players: - Attacker (A): it can wait or attack a single target; - Alarm system (Nature): given an attack t, generates a signal s with p(s t) - Defender (D): patrols the environment visiting vertices of the graph;

25 The Defender is in some vertex v; Let s start from the Attacker decision node. Game Tree

26 Game Tree A can decide to wait (not to attack) for the current stage;

27 Game Tree A can decide to wait (not to attack) for the current stage (action Δ); The alarm system does not generate any signal

28 Game Tree A can decide to wait (not to attack) for the current stage (action Δ); The alarm system does not generate any signal The Defender selects the next vertex to patrol (normal patrolling mode)

29 A may also decide to attack some target t I ; Game Tree

30 Game Tree A may also decide to attack some target t I ; Now N generates a signal according to some discrete probability distribution;

31 Game Tree A may also decide to attack some target t I ; Now N generates a signal according to some discrete probability distribution; The same signal can be associated to different attacks.

32 Game Tree Once observed a signal, D responds by following some covering route, the game ends

33 Game Tree This subgame encodes the signal response phase: Signal Response Game (SRG-v)

34 Game Tree This subgame encodes the normal patrolling phase: Patrolling Game Infinite horizon (the Attacker can wait forever)

35 Game Tree PG SRG-v Approach: Solve SRG-v for each v (find the optimal response strategy from each vertex v) Solve PG under the assumption that the Defender plays the optimal SRG-v strategy if a signal is received when the current vertex is v

36 The Signal Response Game We can formulate the game in normal form, for vertex 1 Attack 1 Attack n Signal A Route X Route Z 1 Signal B Route W Route Y

37 The Signal Response Game Solving the SRG, Minmax (NE): T is the set of targets, S is the set of signals, R is the set of routes, p(s t) is the probability that signal s is issued when target t is attacked Repeat this for each starting vertex v

38 Building the Game The number of covering routes is, in the worst case, prohibitive: (all the permutations for all the subsets of targets) Some routes will never be played: Dominates Dominates Even if we remove dominated covering routes, their number is still very large

39 Building the Game Idea: can we consider covering sets instead? From to Covering sets are in the worst case: (still exponential but much better than before) Problem: we still need routes operatively! Solution: we find covering sets and then we try to reconstruct routes

40 Building the Game Idea: simultaneously build covering sets and the shortest associated covering route Dynamic programming inspired algorithm: we can compute all the covering routes in Is this the best we can do? If we find a better algorithm we could build an algorithm for Hamiltionan Path which would outperform the best algorithm known in literature (for general graphs).

41 Algorithm Idea: simultaneously build covering sets and the shortest associated covering route Covering set: C Covering route: r Terminal vertex: t Covering set with k target whose shortest covering route ends in t Cost of the associated shortest covering route Shortest path between t and f

42 Algorithm Example 3 D 1 B 3 C A

43 Algorithm Example 3 D k=1 1 3 <{A}->A, 0> B C A

44 Algorithm Example 1 B 3 D 3 C k=1 <{A}->A, 0> k=2 <{A,B}->B, 1> A

45 Algorithm Example 1 B 3 D 3 C k=1 <{A}->A, 0> k=2 <{A,B}->B, 1> <{A,C}->C, 2> A

46 Algorithm Example 1 B 3 D 3 C k=1 <{A}->A, 0> dominated k=2 <{A,B}->B, 1> <{A,C}->C, 2> A

47 Algorithm Example 1 B 3 D 3 C k=1 <{A}->A, 0> dominated k=2 <{A,B}->B, 1> <{A,C}->C, 2> k=3 <{A,B,C}->C, 2> A

48 Algorithm Example 1 B 3 D 3 C k=1 <{A}->A, 0> dominated k=2 <{A,B}->B, 1> <{A,C}->C, 2> k=3 unfeasible <{A,B,C}->B, 3> <{A,B,C}->C, 2> A

49 Algorithm Example 1 B 3 D 3 C k=1 <{A}->A, 0> dominated k=2 <{A,B}->B, 1> <{A,C}->C, 2> k=3 unfeasible <{A,B,C}->B, 3> <{A,B,C}->C, 2> <{A,C,D}->D, 3> A

50 Algorithm Example 1 B 3 D 3 C k=1 <{A}->A, 0> dominated k=2 <{A,B}->B, 1> <{A,C}->C, 2> k=3 unfeasible <{A,B,C}->B, 3> <{A,B,C}->C, 2> dominated <{A,C,D}->D, 3> A

51 Algorithm Example 1 B 3 D 3 C k=1 <{A}->A, 0> dominated k=2 <{A,B}->B, 1> <{A,C}->C, 2> k=3 unfeasible <{A,B,C}->B, 3> <{A,B,C}->C, 2> dominated <{A,C,D}->D, 3> A k=4? All unfeasible

52 Building the Game (some numbers) The edge density is a critical parameter. The more dense the graph, the more difficult to build the game.

53 Building the Game (some numbers) Comparison with an heuristic sub-optimal algorithm. Good news: the heuristic method seems to perform better where the exact algorithm requires the highest computational effort

54 The Patrolling Game Solving the signal response game gives the Defender s strategy to react upon the reception of a signal Patrolling game: what to do when no signal is received? Signal B Signal A It s a Leader-Follower scenario: the Attacker can observe the position of the Defender before playing (we can solve it easily) What is the equilibrium patrolling strategy in the presence of an alarm system?

55 The Patrolling Game Intuitive result: if the alarm system covers all the targets; if no false positive are present; if the false negative rate is below a certain threshold: The equilibrium patrolling strategy is not to patrol: the Defender places at the most convenient vertex of the graph and waits for something to happen. Current work: to allow false positives and non-bounded false negatives.

56 Thank you.