1. Solving Large-Scale Low-Rank Zero-Sum Security Games
of Incomplete Information
Anmol Monga
New York University
am6704@nyu.edu
December 2, 2016
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2. Table of content
1 Introduction
Robust Linear Programming
Game Theory
2 Problem Formulation
Zero-Sum Game
Uncertainty in game matrix
Non-robust method
Robust Method
3 Application
4 Conclusion
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3. Introduction
What is uncertainty?
incomplete information
Information is effected by noise
Past methods:
John C Harsanyi introduced a novel method to solve games with
incomplete information ”Games with incomplete information played
by bayesian players” .
Bayesian games is the standard method to solve games with
incomplete information.
Require past statistics to model the uncertainty.
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4. Introduction
We propose:
We define a low ranked constraint and polyhedral constraint over the
incomplete matrix.
We solve for the worst case strategy in the game under the
constraints.
It makes game strategy robust to uncertainty.
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5. Robust Linear Programming
Linear programming problem:
min
x∈RN
cT
x , s.t. aT
i x ≤ bi , i = 1,··· ,N. (1)
c ∈ RN,ai ∈ RN and bi ∈ R are external data and are prone to
uncertainty.
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6. Robust Linear Programming
Figure: Uncertain polyhedral constraint
Robust counterpart:
min
x∈RN
cT
x , s.t. aT
i x ≤ bi , ai ∈ Ui , i = 1,··· ,N
min
x∈RN
cT
x , s.t. max
ai ∈Ui
aT
i x ≤ bi , i = 1,··· ,N
Ui determines the set over which the uncertainty of ai exist.
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7. Game Theory
What?
Mathematics of human interaction.
Model cooperation and conflict between rational and irrational
individual.
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8. Key Elements of a Game
Players : Who is interacting?
Strategies: What are their options?
Payoffs : What are their incentives?
Information : What do they know?
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10. Zero-Sum Game
What?
Two person non-cooperative zero sum finite game is a confrontational
game.
Player P1 minimises his payoff and player P2 maximises his payoffs.
In zero sum game there is always a winner and a looser.
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12. Pure-Strategy
The upper value is the minimum of the maximum value of the rows
and the corresponding row index is the strategy of P1.
upper value = max
j∈{1,···,m}
Ai∗j ≤ max
j∈{1,···,m}
Aij , ∀i ∈ {1,··· ,n}
The lower value is the maximum of the minimum value of the
columns and the corresponding column index is the strategy of P2.
lower value = min
i∈{1,···,n}
Aij∗ ≥ min
i∈{1,···,n}
Aij , ∀j ∈ {1,··· ,m}
i and j represents the row and column index in the game matrix A. i∗
and j∗ are the pure strategy equilibrium.
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13. Mixed Strategy Equilibrium
The pure strategy may not have saddle point equilibrium.
We can repeat the game over and over again and calculate the
probability with which a strategy can be played. This is called mixed
strategy equilibrium.
mixed strategy equilibrium :
min
y∈Y
max
z∈Z
y Az (2)
where,
Y = {y ∈ Rn
: y ≥ 0 and
n
∑
i=1
yi = 1}
Z = {z ∈ Rm
: z ≥ 0 and
m
∑
i=1
zi = 1}.
(3)
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14. Mixed Strategy Equilibrium
The mixed strategy equilibrium of player 1 and player 2 can be
solved using two linear programming problems.
V = max
˜y∈Rn
1T
n ˜y s.t. A ˜y ≤ 1m , ˜y ≥ 0.
V = min
˜z∈Rm
1T
m ˜z s.t. A˜z ≥ 1n , ˜z ≥ 0.
1m = [1,··· ,1]T
1×m , 1n = [1,··· ,1]T
1×n
The strategies of player P1 and player P2 are y = ˜y
V and z = ˜z
V
respectively.
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15. Uncertainty in game matrix
A =
P1/P2 d1 d2 d3 d4 d5 ··· ··· ··· dm
b1 5±0.2 4 × 2 1 ··· ···
b2 3±0.2 1 2 4 1±0.2
b3 0 × 1±0.2 0 1 ×
b4 5 1 2 1 1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. ×
.
.
.
bn ×
A few coefficients in the game matrix A are unknown.
The coefficients in the game matrix are prone to error.
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16. Modelling Uncertainty of Game Matrix
Uncertainty can be modelled using bounded, closed and convex sets.
The low rank constraint on the game matrix:
Ub ∈ {A ∈ Rn×m
: rank(A) ≤ r}. (4)
A set comprising of all the noisy game matrix:
Uc = {A ∈ Rn×m
:| Aij − ¯Aij |≤ Mij where, (i,j) ∈ Ω}. (5)
Ω refers to a set of known coefficients in the game matrix A. ¯A is
error free game matrix.
Solve the zero-sum game over all game matrix in set Uc ∩Ub
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17. Modelling Uncertainty in Game matrix
Relax low rank set Ub
Ub ∈ {A ∈ Rn×m
: A ∗ ≤ σ}. (6)
Relax the polyhedral set Uc
Uc = {A ∈ Rn×m
:| P( ¯Aij )−P(Aij ) |≤ P(Mij )}. (7)
where,
P(A) =
Aij s.t. (i,j) ∈ Ω
0 otherwise
. (8)
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19. Modelling Uncertainty in Game Matrix
There are two ways to solve the action strategy of game matrix A defined
in the set Uc ∩Ub
Non-robust method: Randomly choosing a matrix A in set Uc ∩Ub
and then calculating the strategy for the game
Robust method: Calculating a strategy which is feasible for all the
matrices in set Uc ∩Ub
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20. Non-robust method
Choose the lowest rank matrix in set Uc ∩Ub.
Asol = argmin
A∈Uc ,A≥0
A ∗. (9)
Vnr =max
˜y∈Rn
1T
n ˜y s.t. Asol ˜y ≤ 1m , ˜y ≥ 0
Vnr = min
˜z∈Rm
1T
m ˜z s.t. Asol ˜z ≥ 1n , ˜z ≥ 0.
(10)
The strategies of player P1 and P2 are y = ˜y
Vnr
and z = ˜z
Vnr
respectively.
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21. Robust Method
We can calculate the robust strategy for player P1 and player P2 which is
feasible over all matrices in Uc ∩Ub by :
¯V =max
˜y∈Rn
1T
n ˜y
s.t.A ˜y ≤ 1m , ˜y ≥ 0, A ∈ Uc ∩Ub , A ≥ 0.
(11)
V = min
˜z∈Rm
1T
m ˜z
s.t.A˜z ≥ 1n , ˜z ≥ 0, A ∈ Uc ∩Ub , A ≥ 0.
(12)
The strategies of player P1 and P2 are y = ˜y
¯V
and z = ˜z
V respectively.
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22. Robust Method
¯V =max
˜y∈Rn
1T
n ˜y
s.t. max
A∈Ub∩Uc ,A≥0
aj ˜y ≤ 1m ∀j, ˜y ≥ 0
(13)
Where ,aj is the jth column of matrix A.
V = min
˜z∈Rm
1T
m ˜z
s.t. min
A∈Ub∩Uc ,A≥0
bi ˜z ≥ 1n ∀i, ˜z ≥ 0.
(14)
Where, bi is the ith row of matrix A.
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23. Robust Method
The optimisation problem to obtain a robust strategy for player P1 and
player P2 are
¯V = min
˜y∈Rn,Yl ∈Rn×m,Kl ∈Rn×m
+
−1T
n ˜y
s.t.σ ˜yeT
l +P(Yl )+Kl 2 + P(M),| P(Yl ) | − P( ¯A),P(Yl ) ≤ 1,
˜y ≥ 0, l = 1,....,m.
(15)
V = min
˜z∈Rm,Gl ∈Rn×m,Fl ∈Rn×m
+
1T
m ˜z
s.t.1+σ el ˜zT
−P(Gl )−Fl 2 + P(M),|P(Gl )| − P(Gl ),P( ¯A) ≤ 0,
˜z ≥ 0, l = 1,......,n.
(16)
The strategies of player P1 and P2 are y = ˜y
¯V
and z = ˜z
V respectively.
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24. Example
A centralized sensor network comprising of 4 sensors T,P,H, W and a
server S:
Figure: Sensor Network
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25. Example
(a) (b) (c)
Figure: Attack-defence configuration
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27. Example
Correlated Strategies:
Its very likely that two random strategies involves attacking the same
sensor.
Hence, the attack strategies are correlated.
Similarly, the defence strategies are correlated.
Enables modelling incomplete information as low rank constraint.
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29. Result
index
0 5 10 15
probabilty
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
probability distribution of attacker strategy
nonrobust attacker strategy
orignal attacker strategy
robust attacker strategy
index
0 5 10 15
probabilty
0
0.1
0.2
0.3
0.4
0.5
0.6
probability distribution of defender strategy
nonrobust defender strategy
orignal defender strategy
robust defender strategy
Figure: We compare the action strategy of game matrix A obtained using robust
method, non-robust method and the complete matrix.
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30. Result
percentage of unknowns
0 5 10 15 20 25 30 35 40 45
deviationfromorignalstrategy
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
The change in defender strategy due to change in no. of unknowns
deviation in defender strategy
variance in defender strategy
percentage of unknowns
0 5 10 15 20 25 30 35 40 45
deviationfromorignalstrategy
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
The change in attacker strategy due to change in no. of unknowns
deviation in defender strategy
variance in attacker strategy
Figure: Effectiveness of the robust method in approximating the original complete
game as we increase the number of unknowns in the game matrix .
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31. Result
From the result shown in the previous two slides, we can draw the
conclusion that for a game with incomplete information:
1 The robust method does a better job at predicting the action
strategy of the corresponding complete game matrix when compared
to non-robust method.
2 As we increase the number of unknowns, robust method is less able
to approximate the strategy of complete game from an incomplete
game matrix.
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32. Conclusion
In this paper we have provided two methods to solve a zero-sum
game with incomplete information: Robust method and Non-robust
method.
We have shown that the robust method approximates the player
strategy much better than the non-robust method.
We have also defined a deterministic model for the uncertainty in the
game matrix.
In future work we plan to use the ADMM algorithm to run the robust
method for a very large zero-sum game with incomplete information .
We can extend the robust method to non-cooperative non-zero sum
game.
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