Equilibria Extremal Optimization - FSEGArodica.lung/prezentare1.pdf · Equilibria Extremal...
Transcript of 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
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CEBSS Outline Game theory Nash Extremal Optimization Application...
1 Game theory
2 Nash Extremal Optimization
3 Application...
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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
...
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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
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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
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CEBSS Outline Game theory Nash Extremal Optimization Application...
Multi-player games
Nash Extremal Optimization - Cournot
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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.
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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.
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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.
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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%
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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
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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
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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!
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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
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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