Equilibria Extremal Optimization - FSEGA rodica.lung/prezentare1.pdf · PDF file Equilibria Extremal Optimization Rodica Ioana Lung Facultatea de S˘tiint˘e Economice ˘si Gestiunea

CEBSS Outline Game theory Nash Extremal Optimization Application...

Equilibria Extremal Optimization

Rodica Ioana Lung

Facultatea de Ştiinţe Economice şi 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, conflicting

objectives;

Solution concepts:

I Nash equilibrium (NE): no player can improve its payoff by unilateral deviation

I Berge equilibrium (BE): players maximize each others payoffs

I Pareto optimal solution/equilibrium: no objective can be increased 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, conflicting

objectives;

Solution concepts:

I Nash equilibrium (NE): no player can improve its payoff by unilateral deviation

I Berge equilibrium (BE): players maximize each others payoffs

I Pareto optimal solution/equilibrium: no objective can be increased without decreasing another objective

...

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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 ascendancy relation: @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 q p-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 the inequality

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},

b�(s, q) = card{i ∈ N, ui (s) < ui (si , q−i ) + �, 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) = 2s

2 1 + 3s1 − s22 − 3s2,

si ∈ [−2, 1], i = 1, 2.

Berge-Zhukovskii equilibrium: (1, 1); Payoffs (−1, 1). → � ∈ {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 the aggregate 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 Nash equilibrium

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: repeat 3: 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) then 7: set sbest := s; 8: end if 9: 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

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 the

game; 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 the community;

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, [

1 10 · 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 if if 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 if end for Output: individual from A having the best modularity;

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CEBSS Outline Game theory Nash Extremal Optimization Application...

Results - MNEO

books dolphins football karate 0

0.2

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Real networks

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mixing parameter

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LFR small

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Infomap Louvain modOpt OSLOM MNEO

Figure: Comparisons with other methods - mean NMI (with error bars) values obtained in 30 runs (30 instances for each GN and LFR networks and 30 independent runs for the real-world networks).

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CEBSS Outline Game theory Nash Extremal Optimization Application...

pMNEO

z o

ut =

8

z ou

t= 7

z o

ut =

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 8 Wilcoxon p values

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

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p=25% p=50% p=75% p=100% Infomap Louvain ModOpt Oslom

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

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p=25% p=50% p=75% p=100