Equilibria Extremal Optimization - FSEGArodica.lung/prezentare1.pdf · Equilibria Extremal Optimization Rodica Ioana Lung Facultatea de S˘tiint˘e Economice ˘si Gestiunea Afacerilor

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


1 Game theory

2 Nash Extremal Optimization

3 Application...

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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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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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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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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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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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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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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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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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Multi-player games - probabilistic Nash ascendancy

Differential evolution

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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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Multi-player games

Nash Extremal Optimization - Cournot

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Multi-player games

Nash Extremal Optimization - Cournot

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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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Community structure - definitions

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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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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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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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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


1 2 3 4 5 6 7 8

12345678

0

0.2

0.4

0.6

0.8

1

0

0.5

1


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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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



1 2 3 4 5 6 7 8

12345678 0

0.2

0.4

0.6

0.8

1

0

0.5

1


1 2 3 4 5 6 7 8

12345678 0

0.2

0.4

0.6

0.8

1

0

0.5

1


1 2 3 4 5 6 7 8

12345678 0

0.2

0.4

0.6

0.8

1

0

0.5

1


1 2 3 4 5 6 7 8

12345678 0

0.2

0.4

0.6

0.8

1

0

0.5

1


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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pMNEO

k

arat

e

foo

tbal

l

dol

phin

s

b

ooks

Real networks

0

0.5

1



1 2 3 4 5 6 7 8

12345678 0

0.2

0.4

0.6

0.8

1

0

0.5

1


1 2 3 4 5 6 7 8

12345678 0

0.2

0.4

0.6

0.8

1

0

0.5

1


1 2 3 4 5 6 7 8

12345678 0

0.2

0.4

0.6

0.8

1

0

0.5

1


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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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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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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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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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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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


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

Documents

Equilibria Extremal Optimization - FSEGArodica.lung/prezentare1.pdf · Equilibria Extremal Optimization Rodica Ioana Lung Facultatea de S˘tiint˘e Economice ˘si Gestiunea Afacerilor