Game Theoretical Insights in Strategic
Patrolling: Model and Analysis
Nicola Gatti – [email protected]
DEI, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy
Topic, Results, and Outline
2
• 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
ECAI 2008
Nicola Gatti
Game Theory Groundings for Strategic Patrolling
3
• 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., rockpaper-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]
ECAI 2008
Nicola Gatti
von Neumann’s Hide-and-Seek Game
4
S
1
H
2
3
4S
5
H
6
7
8
9
H
ECAI 2008
Nicola Gatti
Paruchuri et al.’s Strategic Patrolling (1)
6
G
1
2
3
4
5
6
7
8
9
R
ECAI 2008
Nicola Gatti
Paruchuri et al.’s Strategic Patrolling (2)
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
ECAI 2008
Nicola Gatti
Paruchuri et al.’s Strategic Patrolling (3)
• 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]
ECAI 2008
Nicola Gatti
8
Problems in Paruchuri et al.’s Strategic Patrolling (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
robber
guard
ECAI 2008
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
Nicola Gatti
9
Problems in Paruchuri et al.’s Strategic Patrolling (2)
Guard’s optimal strategy (.16 <1,2>, .16 <2,1>, .16 <1,3>, .16 <3,1>, .16 <2,3>, .16 <3,2>)
G
realization <3,1>
<1,2>
G
G
1
2
R
3
The robber’s expected utility is -.33
R
Robber’s optimal strategy (2)
ECAI 2008
Nicola Gatti
10
Problems in Paruchuri et al.’s Strategic Patrolling (2)
Guard’s optimal strategy (.16 <1,2>, .16 <2,1>, .16 <1,3>, .16 <3,1>, .16 <2,3>, .16 <3,2>)
G
realization <1,2>
<3,2>
G
G
R1
2
3
The robber’s expected utility is .33
R
ECAI 2008
Nicola Gatti
11
Problems in Paruchuri et al.’s Strategic Patrolling (3)
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
ECAI 2008
Nicola Gatti
An Alternative Strategic Patrolling Model
13
• 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)
ECAI 2008
Nicola Gatti
Searching for a Nash Equilibrium
15
• 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
ECAI 2008
Nicola Gatti
One Robber Type Analysis
16
• 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
ECAI 2008
Nicola Gatti
More Robber Types Analysis (1)
17
• 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
ECAI 2008
Nicola Gatti
More Robber Types Analysis (2)
18
• 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=
Area 1
…
Area n
Type 1
1
…
0
Type 2
0
…
1
…
…
…
…
Type m
1
…
0
• 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
ECAI 2008
Nicola Gatti
Experimental Results
19
• 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
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
• Our approach outperforms Multi-LP algorithm, although the
computation of a Nash equilibrium is harder than the computation
of a leader-follower equilibrium
ECAI 2008
Nicola Gatti
Conclusions and Future Works
20
• 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
ECAI 2008
Nicola Gatti
21
Thank you for your attention!
ECAI 2008
Nicola Gatti