game theoretical insights in strategic patrolling: model and analysis nicola gatti –...

Game Theoretical Insights in Strategic Patrolling: Model and AnalysisNicola Gatti – [email protected]

DEI, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy

2

Nicola GattiECAI 2008

Topic, Results, and Outline

• Topic• Study of strategic models for capturing patrolling situations

in presence of opponents• Main results

• Modeling result:• Problems in the current state-of-the-art• Proposal of an alternative model

• Computational result:• Exploitation of game theoretical analysis for reducing the solving

algorithm complexity

• Outline• Strategic patrolling state-of-the-art• Proposal of an alternative model• Towards integration between game theoretical analysis and

algorithmic game theory • Conclusions and future works

3


Game Theory Groundings for Strategic Patrolling

• Definition of game• Protocol: rules of the game (e.g., number of players,

sequential structure, available actions)• Strategic-form games: the players act simultaneously

(e.g., rock-paper-scissors)• Extensive-form games: the players act according to a

given sequential structure (e.g., chess)

• Strategies: players’ behavior in the game

• Solution: a strategy profile σ = (σ1, …, σn) that is somehow in equilibrium• Nash equilibrium: the players act simultaneously

without meeting themselves before playing the game [Nash, 1950]

• Leader-follower equilibrium: a player can commit to a specific strategy and the follower acts on the basis of the commitment [von Stengel and Zamir, 2004]

4


von Neumann’s Hide-and-Seek Game

1 2 3

4 5 6

7 8 9

S

H

S H

H

6


Paruchuri et al.’s Strategic Patrolling (1)

1 2 3

4 5 6

7 8 9

G

R

7



• Assumptions:• Time is discretized in turns• Time needed by the guard to patrol one area is exactly 1

turn• Time needed by the guard to move between two areas is

negligible with respect to time needed to patrol an area• Time needed by the robber to rob an area is d turns• The robber can observe the strategy of the guard

• Game protocol:• Two–player:

• Guard• Robber

• General–sum: each player assigns each area and the robber’s caught a value

• Strategic–form: the players act simultaneously• Actions:

• Guard: a route of d areas, e.g. <1, 2, …, d>• Robber: a single area

8



• Solution concept: leader-follower equilibrium• Strategies: the guard randomizes over a portion of the

actions, while the robber follows a pure strategy• Multiple types: the payoffs of the robber could be known

with uncertainty by the guard• By Harsanyi transformation: the robber can be of

different types (each type has a specific payoff) according to a given probability distribution

• Solving algorithms:• Multi Linear Programming [Conitzer and Sandholm,

2005]• Mixed Integer Linear Programming [Paruchuri et al.,

2008]

9


Problems in Paruchuri et al.’s Strategic Patrolling (1)

robber

gu

ard

1 2 3

<1,2>/<2,1>

1, -1 1, -1 0.66, 1

<1,3>/<3,1>

1 ,-1 0.66, 1 1, -1

<2,3>/<3,2>

0.66, 1 1, -1 1, -1

A simple setting•3 areas

•1 type

•Two turns are needed by the robber to rob an area (d=2)

•Each player has the same evaluations over the areas

10



1 2 3

R

G

Guard’s optimal strategy (.16 <1,2>, .16 <2,1>, .16 <1,3>, .16 <3,1>, .16 <2,3>, .16 <3,2>)

Robber’s optimal strategy (2)

realization <3,1>

R

G

The robber’s expected utility is -.33

realization <1,2>

G

11



1 2 3

R

G

Guard’s optimal strategy (.16 <1,2>, .16 <2,1>, .16 <1,3>, .16 <3,1>, .16 <2,3>, .16 <3,2>)

realization <1,2>

RG

The robber’s expected utility is .33

realization <3,2>

G

12



• The model by Paruchuri et al. does not consider all the possible implications due to the observation of the robber

• According to the assumption of observation, the robber can enter an area when the guard is patrolling and not exclusively when the guard starts to patrol a route

13


An Alternative Strategic Patrolling Model

• The “natural” model is an extensive-form game wherein• Guard: the next area to patrol• Robber: the area to enter or wait

• In this work we search for a strategic-form model alternative to Paruchuri et al.’s model

• The proposed model is a strategic-form model wherein• Guard: the next area to patrol• Robber: the area to enter

and the guard’s strategy will be the same at each turn• In this way the robber cannot improve its expected utility

by waiting• In this model no “consistency“problem there is (the proof

can be found in the paper)

15


Searching for a Nash Equilibrium

• We use the strategic patrolling as case study for the integration of game theoretical analysis and algorithmic game theory

• Idea• Game theoretical analysis allows one to derive some

insights• Singularities: some strategy profiles are never of

equilibrium independently of the values of the parameters (payoffs)

• Regularities: some strategy profiles are of equilibrium with a probability higher than others

• These insights can be exploited to improve searching efficiency and to make hard problems affordable

16


One Robber Type Analysis

• Proposition 1: Independently of the number of the robber’s types, at the equilibrium the guard will randomize over all the possible actions

• On the basis of Proposition 1, except for a null-measure subspace of the parameters, with one type of robber the Nash equilibrium:• Is unique, and• Prescribes that both the guard and the robber will

randomize over all their available actions• In this case the Nash equilibrium can be computed in

closed form as a single problem of linear programming

17


More Robber Types Analysis (1)

• With more types, the equilibrium cannot be computed in closed form

• Anyway, game theoretical insights can be exploited to reduce the complexity of the search

• Searching in the space of the supports• A complete method for searching a Nash equilibrium is

to enumerate all the possible strategy supports and check them one by one

(A strategy support is the set of actions over which agents randomize with a strict positive probability)

• Anyway, such a space rises exponentially in the number of players’ actions and then heuristics are needed

• [Porter et al., 2005] provides some heuristics for ordering the supports and shows that their approach is more efficient than Lemke-Howson algorithm

18


More Robber Types Analysis (2)

• By Proposition 1, the support of the guard will be the whole set of actions

• The supports of all the robber’s types can depict as a matrix

M =

• By game theoretical analysis we can:• Reduce the space of the matrices M• Produce an ordering where the first Ms are the most

probable to lead to an equilibrium

Area 1 … Area n

Type 1 1 … 0

Type 2 0 … 1

… … … …

Type m

1 … 0

19


Experimental Results

• We have studied random settings with 4, 5, 6, 7 areas and different number of robber’s types

• Our approach outperforms Porter et al. approach in term of computational time, dramatically reducing the space of the search

• Our approach outperforms Multi-LP algorithm, although the computation of a Nash equilibrium is harder than the computation of a leader-follower equilibrium

types (with 4 areas)

6 7 8 9 10 11 12

Porter 23.15 67.14 132.31 301.20 621.41 >1000 >1000

Ours 0.190 0.352 0.720 1.015 1.532 1.852 2.231

20


Conclusions and Future Works

• Conclusions• Analysis of state-of-the-art model of strategic patrolling• Proposal of a strategic model in normal-form• Attempt to exploit game theoretical analysis to improve

the algorithm efficiency• Future works

• Patrolling models and solving algorithms• Exploiting game theoretical analysis in algorithmic

game theory

21


Thank you for your attention!

game theoretical insights in strategic patrolling: model and analysis nicola gatti –...

Documents

game solution

game nash

guard game protocol

nicola gattiecai

s strategic patrolling

neumanns hideandseek

game theoretical insights

game theory groundings