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

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  • 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

    ...

    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 ascendancy relation: @q ∈ S such that q Nash ascends s.

    I Nash non-dominated = NE.

    Rodica Ioana Lung

  • 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

    Rodica Ioana Lung

  • 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

    Rodica Ioana Lung

  • 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

    Rodica Ioana Lung

  • 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}.

    Rodica Ioana Lung

  • 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}.

    Rodica Ioana Lung

  • CEBSS Outline Game theory Nash Extremal Optimization Application...

    Multi-player games - probabilistic Nash ascendancy

    Differential evolution

    Rodica Ioana Lung

  • 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 .

    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...

    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

    Rodica Ioana Lung

  • CEBSS Outline Game theory Nash Extremal Optimization Application...

    Community structure - definitions

    Rodica Ioana Lung

  • 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.

    Rodica Ioana Lung

  • 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;

    Rodica Ioana Lung

  • CEBSS Outline Game theory Nash Extremal Optimization Application...

    Results - MNEO

    books dolphins football karate 0

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    N M

    I

    Real networks

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    z out

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    0.1 0.2 0.3 0.4 0.5 0.6 0

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    mixing parameter

    N M

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    LFR small

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    mixing parameter

    N M

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    LFR big

    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).

    Rodica Ioana Lung

  • CEBSS Outline Game theory Nash Extremal Optimization Application...

    pMNEO

    z o

    ut =

    8

    z ou

    t= 7

    z o

    ut =

    6

    GN benchmark

    0

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    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

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