equilibria extremal optimization - fsegarodica.lung/prezentare1.pdf · equilibria extremal...
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Equilibria Extremal Optimization
Rodica Ioana Lung
Facultatea de Stiinte Economice si Gestiunea Afacerilor
28.05.2015
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
1 Game theory
2 Nash Extremal Optimization
3 Application...
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
Solution concepts
Problem description:
I Game → players, strategies, payoffs;
I Multiobjective optimization → optimize multiple, conflictingobjectives;
Solution concepts:
I Nash equilibrium (NE): no player can improve its payoff byunilateral deviation
I Berge equilibrium (BE): players maximize each others payoffs
I Pareto optimal solution/equilibrium: no objective can beincreased without decreasing another objective
...
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
Solution concepts
Problem description:
I Game → players, strategies, payoffs;
I Multiobjective optimization → optimize multiple, conflictingobjectives;
Solution concepts:
I Nash equilibrium (NE): no player can improve its payoff byunilateral deviation
I Berge equilibrium (BE): players maximize each others payoffs
I Pareto optimal solution/equilibrium: no objective can beincreased without decreasing another objective
...
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
Nash ascendancy relation
Compare two strategy profiles:
I compute k(s, q) = card{i ∈ N|ui (qi , s−i ) > ui (s), qi 6= si}I s Nash ascends q if k(s, q) < k(q, s)
I s non-dominated with respect to the Nash ascendancyrelation: @q ∈ S such that q Nash ascends s.
I Nash non-dominated = NE.
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Probabilistic Nash ascendancy relation
Reduce the number of payoff function evaluations:
I only a percent p of players are tested:
I Ip = {i1, i2, ..., inp} ⊂ N
I then kp(s, q, Ip) = card{i ∈ Ip|ui (s) < ui (qi , s−i ), si 6= qi}.I s ∈ S p−Nash ascends q ∈ S with respect to Ip if
kp(s, q, Ip) < kp(q, s, Ip).
I s ∈ S p-non-dominated @q ∈ S , @Ip ⊂ N such that qp-ascends s with respect to Ip.
I p-non-dominated = Nash equilibria
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Berge generative relation
Definition (Berge-Zhukovskii)
A strategy profile s∗ ∈ S is a Berge-Zhukovskii equilibrium if theinequality
ui (s∗) ≥ ui (s
∗i , s−i )
holds for each player i = 1, ..., n, and all s−i ∈ S−i .
Quality measures:
b(s, q) = card{i ∈ N, ui (s) < ui (si , q−i ), s−i 6= q−i},
bε(s, q) = card{i ∈ N, ui (s) < ui (si , q−i ) + ε, s−i 6= q−i},
The same mechanism → generative relation
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Evolutionary detection - Nash equilibria
Game:
I Payoffs:
u1(y1, y2) = y1
u2(y1, y2) = (0.5− y1)y2
I y1 ∈ [0, 0.5], y2 ∈ [0, 1].
I NE: (0.5, λ), λ ∈ [0, 1].
NSGA-II results
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Evolutionary detection - Berge Zhukovskii equilibria
Example
u1(s1, s2) = −s21 − s1 + s2,
u2(s1, s2) = 2s21 + 3s1 − s2
2 − 3s2,
si ∈ [−2, 1], i = 1, 2.
Berge-Zhukovskii equilibrium: (1, 1);Payoffs (−1, 1).→ ε ∈ {0, 0.1, 0.2, 0.5, 0.9}.
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Cournot oligopoly
I qi , i = 1, ...,N - quantities/ N companies
I market clearing price P(Q) = a− Q, where Q is theaggregate quantity on the market.
I total cost for the company i of producing quantity qi :C (qi ) = cqi .
I payoff for the company i is its profit:
ui (q1, q2, ..., qN) = qiP(Q)− C (qi )
I if Q =∑N
i=1 qi , the Cournot oligopoly has one Nashequilibrium
qi =a− c
N + 1, ∀i ∈ {1, ...,N}.
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Multi-player games - probabilistic Nash ascendancy
Differential evolution
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Nash Extremal Optimization
Algorithm 1 Nash Extremal Optimization procedure
1: Initialize configuration s = (s1, ..., sn) at will; set sbest := s;2: repeat3: For the ’current’ configuration s evaluate ui for each player i ;4: find j satisfying uj ≤ ui for all i , i 6= j , i.e., j has the ”worst
payoff”;5: change sj randomly;6: if (s Nash ascends sbest) then7: set sbest := s;8: end if9: until a termination condition is fulfilled;
10: Return sbest .
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Multi-player games
Nash Extremal Optimization - Cournot
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
Multi-player games
Nash Extremal Optimization - Cournot
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
Community structure detection in social networks
M. E. J. Newman, Communities, modules and large-scale structure in networks, Nature Physics 8, 2531 (2012)doi:10.1038/nphys2162
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Community structure - definitions
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CEBSS Outline Game theory Nash Extremal Optimization Application...
The game
Game Γ = (N, S ,U):
I N = {1, ..., n}, the set of players: network nodes;
I S = S1 × S2 × . . .× Sn, the set of strategy profiles of thegame; s ∈ S , s = (c1, . . . , cn)
I U = {ui}i=1,n, the payoff functions; ui : S → R represents the
payoff of player i , i = 1, n.
ui (s) = f (Ci )− f (Ci\{i}),
f (C ) =kin(C )
(kin(C ) + kout(C ))α,
I kin(C ) - double of the number of internal links in thecommunity;
I kout(C ) - number of external links of the community;I α - a parameter controlling the size of the community.
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Mixed Network Extremal optimization
Randomly initialize all individuals in P and A;Evaluate all individuals in P and A;for NrGen = 0 to MaxGen do
Set mngen = max{
1,[
110 · N · (N − 2)−
NrGenMaxGen
]}Run a NEO iteration for each pair (Pi ,Ai );if Λ iterations were performed on the original network then
Mix network with probability ρ;Randomly re-initialize population A;Evaluate all individuals in P and A;
end ifif Network is mixed and λ iterations were performed then
Restore network to original structure;Randomly re-initialize population A;Evaluate all individuals in P and A;
end ifend forOutput: individual from A having the best modularity;
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Results - MNEO
books dolphins football karate0
0.2
0.4
0.6
0.8
1
NM
I
Real networks
1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
zout
NM
I
GN
0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
mixing parameter
NM
I
LFR small
0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
mixing parameter
NM
I
LFR big
InfomapLouvainmodOptOSLOMMNEO
Figure: Comparisons with other methods - mean NMI (with error bars)values obtained in 30 runs (30 instances for each GN and LFR networksand 30 independent runs for the real-world networks).
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CEBSS Outline Game theory Nash Extremal Optimization Application...
pMNEO
z ou
t=8
z
out=
7
z out=
6
GN benchmark
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8Wilcoxon p values
1 2 3 4 5 6 7 8
12345678
0
0.2
0.4
0.6
0.8
1
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8
12345678
0
0.2
0.4
0.6
0.8
1
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8
12345678
0
0.2
0.4
0.6
0.8
1
Figure: GN6-8; boxplots of NMI values for all methods considered (left). On the right, the color matrixrepresents Wilcoxon p values for each pair of methods considered, numbered in the order they appear in theboxplot. A white square indicates a statistical difference between results.
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
pMNEO
µ=
0.5
µ=0.
4
µ=
0.3
µ=
0.2
µ=0.
1
LFR benchmark, 1000 nodes, small
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8Wilcoxon p values
1 2 3 4 5 6 7 8
12345678 0
0.2
0.4
0.6
0.8
1
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8
12345678 0
0.2
0.4
0.6
0.8
1
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8
12345678 0
0.2
0.4
0.6
0.8
1
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8
12345678 0
0.2
0.4
0.6
0.8
1
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8
12345678 0
0.2
0.4
0.6
0.8
1
Figure: LFR 1000 nodes, S; boxplots of NMI values for all methods considered (left). On the right, the colormatrix represents Wilcoxon p values for each pair of methods considered, numbered in the order they appear in theboxplot. A white square indicates a statistical difference between results.
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
pMNEO
k
arat
e
foo
tbal
l
dol
phin
s
b
ooks
Real networks
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8Wilcoxon p values
1 2 3 4 5 6 7 8
12345678 0
0.2
0.4
0.6
0.8
1
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8
12345678 0
0.2
0.4
0.6
0.8
1
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8
12345678 0
0.2
0.4
0.6
0.8
1
0
0.5
1
p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom
1 2 3 4 5 6 7 8
12345678 0
0.2
0.4
0.6
0.8
1
Figure: real networks; boxplots of NMI values for all methods considered (left). On the right, the color matrixrepresents Wilcoxon p values for each pair of methods considered, numbered in the order they appear in theboxplot. A white square indicates a statistical difference between results.
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
pMNEO
GN LFR 128 LFR 1000 small books dolphins football karate0
10
20
30
40
50
60
70
80
90
100%
p=100%p=75%p=50%p=25%
Figure: pMNEO duration in percents relative to p = 100%
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
ε-Berge equilibria & the community detection problem
0 100 200 300 400 500 600 7000
0.2
0.4
0.6
0.8
1
GN, zout
=7
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
GN, zout
=8
0 100 200 300 400 500 600 7000
0.2
0.4
0.6
0.8
1
LFR S, µ=0.4
generation0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
LFR S, µ=0.5
generation
ε = 0
ε = 10−1
ε = 10−2
ε = 10−3
ε = 10−4
ε = 10−5
Figure: Evolution of NMI for different values of ε
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
ε-Berge equilibria & the community detection problem
1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
NM
I
zout
GN networks
0.2 0.4 0.6µ
LFR S networks
0.2 0.4 0.6µ
LFR B networks
books dolphins football karate
Real networks
network
BEO, ε = 10−3
InfomapOSLOMModularity Optimization
Figure: Berge equilibria - comparisons with other methods
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
Conclusions:
I (some) equilibria can be characterized by the use of”generative relations”;
I generative relations can be embedded in evolutionaryalgorithms to compute corresponding equilibria;
I Extremal optimization is efficient for multi-player games;
I The community detection problem can be approached as agame;
I EO results are promising with both NE and BZ!
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
The work presented here was the result of the collaboration with:
I Noemi Gasko
I Mihai Alexandru Suciu
I Tudor Dan Mihoc
I prof. D. Dumitrescu
Thank you for your attention!
Questions?
Rodica Ioana Lung
CEBSS Outline Game theory Nash Extremal Optimization Application...
The work presented here was the result of the collaboration with:
I Noemi Gasko
I Mihai Alexandru Suciu
I Tudor Dan Mihoc
I prof. D. Dumitrescu
Thank you for your attention!
Questions?
Rodica Ioana Lung