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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

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• 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

0.2

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Real networks

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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).

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

1 2 3 4 5 6 7 8 Wilcoxon p values

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