Ra#onal'Dynamics:'Coopera#on'in'Selfish'

Environments,'Segrega#on

Ozalp'Babaoglu'Dipar#mento'di'Informa#ca'

Università'di'Bologna'www.cs.unibo.it/babaoglu/

©'Babaoglu'2014

Structure'vs'dynamics

■ Structure'—'shape'of'the'network'■ degree'distribu#on'■ clustering'■ diameter'

■ Dynamics'—'what'is'happening'in'the'network'■ naviga#on'■ gossiping'■ topology'building'(newscast,'cyclone,'TQMan)'■ aggrega#on'■ synchroniza#on'

■ Structure'and'dynamics'are'oTen'interrelated'■ effects'of'topology'on'aggrega#on

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Ra#onal'dynamics

■ So'far,'the'dynamics'have'been'“blind”'—'nodes'have'no'“free'will”'or'“purpose”'but'pass'around'informa#on'blindly'

■ When'in'fact,'nodes'are'oTen'individuals'or'other'ac#ve'en##es'with'intent,'goals'and'selfQinterests'

■ This'results'in'ra.onal1dynamics1■ Game1theory1is'a'tool'for'studying'ra#onal'dynamics'

■ Strategies'—'model'intent'■ U.lity'—'measure'achievement'of'goals

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Ra#onal'dynamics'Elements'of'game'theory

■ Set'of'par#cipants'called'players'■ Each'player'has'a'set'of'op#ons'for'behavior'called'

strategies'■ For'each'choice'of'strategies,'a'player'receives'a'payoff'that'

may'depend'on'the'strategies'selected'by'other'players'■ Summarized'in'the'form'of'a'payoff1matrix

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Ra#onal'dynamics'Elements'of'game'theory

■ Each'players'knows'everything'about'the'structure'of'the'game:'■ who'the'other'players'are'■ the'set'of'all'possible'strategies'■ the'payoff'matrix'■ but'does'not'know'the'strategies'chosen'by'the'other'players'

■ Players'are'ra.onal'—'each'tries'to'maximize'her'own'payoff,'given'her'beliefs'about'the'strategies'used'by'other'players

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Ra#onal'dynamics'Iterated'network'games

■ Large'number'of'players'(nodes'in'the'network)'■ A'network'mediates'the'interac#ons'between'players'■ Payoffs'depend'only'on'local'interac#ons'(between'nodes'

that'are'neighbors'in'the'network)'■ Payoff'matrix'specifies'value'for'each'configura#on'of'local'

neighborhood'(without'exhaus#ve'enumera#on)'■ Interested'not'in'“oneQshot”'outcomes'but'in'the'dynamics'

of'iterated'plays'■ The'node’s'overall'u.lity'is'the'running'average'of'its'

payoffs'from'repeated'interac#ons

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Coopera#on'in'selfish'environments'PeerQtoQpeer'applica#ons

■ PeerQtoQpeer'applica#ons'such'as'file'sharing'are'totally'decentralized'and'“open”'—'anyone'can'join'them'

■ They'are'subject'to'“freeQriding”'—'selfish'users'that'enjoy'the'benefits'without'contribu#ng'their'share'■ they'download'but'do'not'allow'uploads'■ they'store'their'files'but'do'not'contribute'disk'space'for'others'

■ High'levels'of'freeQriding'can'render'these'systems'useless'■ How'to'reduce'the'level'of'selfishness'(and'increase'the'

level'of'coopera#on)?'■ “CopyQandQwire”'algorithm

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Coopera#on'in'selfish'environments'“CopyQandQrewire”

■ Two'logically'dis#nct'networks:'■ Random'overlay'network'to'maintain'connec#vity'■ Applica#onQdependent'interac#on'network'

■ Periodically,'node'P'compares'its'u#lity'with'that'of'a'peer'Q'selected'at'random'(from'the'connec#vity'network)'

■ If'Q'has'been'achieving'higher'u#lity'■ P'copies'Q ’s'strategy'■ P'rewires'its'links'to'the'neighbors'of'Q'

■ With'(very)'small'probability,'node'P'■ “Mutates”'its'strategy'(picks'an'alterna#ve'strategy'at'random)'■ Drops'all'of'its'current'links'■ Links'to'a'random'node

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Coopera#on'in'selfish'environments'Gossip'framework'instan#a#on

■ Style'of'interac#on:''pull'■ Local'state'S:''Current'u#lity,'strategy'and'neighborhood'

within'an'interac#on'network'■ Method'SelectPeer():''Single'random'sample'■ Method'Update():''Copy'strategy'and'neighborhood'if'

the'peer'is'achieving'beher'u#lity

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Coopera#on'in'selfish'environments'“CopyQandQrewire”'algorithm

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B

C A

DE F

H

J K

G

Connec#vity'network

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Compare'u#li#es

Coopera#on'in'selfish'environments'“CopyQandQrewire”'algorithm

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Copy

B

C A

DE F

H

J K

G

Applica#on'dependent'interac#on'network

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Coopera#on'in'selfish'environments'“CopyQandQrewire”'algorithm

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B

C A

DE F

H

J K

G

Copy

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Coopera#on'in'selfish'environments'“CopyQandQrewire”'algorithm

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B

C A

DE F

H

J K

G

RewireMutate

Copy

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Coopera#on'in'selfish'environments'“CopyQandQrewire”'algorithm

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B

C A

DE F

H

J K

G

RewireMutate

Copy

Drop'all'linksLink'to'random'node

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Coopera#on'in'selfish'environments'Prisoner’s'Dilemma

■ Prisoner’s'Dilemma'(PD)'is'an'abstract'game'that'captures'the'conflict'between'“individual'ra#onality”'and'“common'good”'

■ Two'guilty'individuals'have'been'captured'and'while'being'interrogated'in'separate'rooms,'are'offered'a'deal'by'the'police'

■ Each'prisoner'can'choose'between'“Confess”'(C)'and'“Deny”'(D)'■ Each'prisoner'must'act'unilaterally'—'no'collusion,'conversa#on'

■ Confession'leads'to'higher'individual'u#lity'—'selfishness'■ Denial'leads'to'higher'global'u#lity'—'coopera#on

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D CD −2,'−2 −10,'−1C −1,'−10 −8,'−8

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Coopera#on'in'selfish'environments'Prisoner’s'Dilemma

■ Note'that'(C,'C)'represents'an'equilibrium'state'—'neither'prisoner'can'improve'her'payoff'by'changing'her'strategy'

■ No'other'pair'of'strategies'is'an'equilibrium'—'some'prisoner'is'always'beher'off'by'changing'her'strategy'

■ “Dilemma”'because'both'prisoners'would'have'been'much'beher'off'if'both'had'chosen'“Deny”'

■ But'(D,'D)'is'not'an'equilibrium'state'■ In'general,'just'because'the'players'are'at'equilibrium'in'a'

game'does'not'mean'that'they'are'happy

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Coopera#on'in'selfish'environments'Prisoner’s'Dilemma

■ Test'the'“copyQandQrewire”'algorithm'with'repeated'itera#ons'of'Prisoner’s'Dilemma'played'on'the'interac#on'network'■ Only'pure'strategies'are'played'(always'C'or'always'D)'■ In'each'round,'a'node'plays'with'one'random'neighbor'selected'from'the'interac#on'network'

■ Muta#on:'flip'current'strategy'■ U#lity:'average'payoff'achieved'so'far

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Coopera#on'in'selfish'environments'Simula#on'results

■ 500'nodes'■ Ini#al'state:'

■ All'nodes'are'selfish'■ Random'interac#on'network

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Coopera#on'in'selfish'environments'Prisoner’s'Dilemma'@'cycle'180

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Small'clusters'of'selfish'nodes'start'to'form

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Coopera#on'in'selfish'environments'Prisoner’s'Dilemma'@'cycle'220

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Giant'coopera#ve'cluster'emerges

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Coopera#on'in'selfish'environments'Prisoner’s'Dilemma'@'cycle'230

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Coopera#ve'cluster'starts'to'break'apart

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Coopera#on'in'selfish'environments'Prisoner’s'Dilemma'@'cycle'300

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Small'coopera#ve'clusters'formed,'selfish'nodes'become'isolated

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Coopera#on'in'selfish'environments'Phase'transi#on

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%'of'c

oope

ra#v

e'no

des

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Coopera#on'in'selfish'environments'Resul#ng'dynamics

■ “CopyQandQrewire”'causes'the'interac#on'network'to'evolve,'resul#ng'in'the'nodes'to'“move”'in'search'of'beher'neighborhoods'

■ Equilibrium'states'achieve'very'high'levels'of'coopera#on'■ GroupQlike'selec#on'between'clusters'

■ Clusters'of'coopera#ve'nodes'grow'and'persist'■ Selfish'nodes'tend'to'become'isolated'

■ Can'be'seen'as'a'“strategic”'(as'opposed'to'“stochas#c”)'network'forma#on'process

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Homophily

■ Homophily'—'“Birds'of'a'feather'flock'together”'■ Individuals'seek'similar'individuals'■ Homophily'is'observed'across'race,'gender,'age,'religion,'

income'in'a'wide'variety'of'networks'—'neighborhood,'friendship,'marriage,'loans,'etc.'

■ Some'reasons'for'homophily:'■ Opportunity'■ Social'pressure'■ Cost/benefit'■ Social'compe##on'

■ Peer'effects'—'related'but'different'property'where'individuals'adopt'the'behavior'of'their'peers

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Homophily'Segrega#on

■ Popula#on'of'individuals'■ Each'individual'has'a'“type”'■ Individuals'achieve'a'“u#lity”'based'on'the'types'of'other'

individuals'in'their'neighborhood'in'comparison'to'their'own'type'

■ Individuals'care'about'where'they'live'■ Individuals'can'move'if'they'are'not'happy'about'their'

current'neighborhood

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Segrega#on'in'the'wild'(race)

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New'York

CaucasianAfrican'AmericanLa#noAsian

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Segrega#on'in'the'wild'(race)

28LA

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Segrega#on'in'the'wild'(race)

29Houston

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Segrega#on'in'the'wild'(race)

30Washington'DC

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Segrega#on'in'the'wild'(income)

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>'$200K'

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Schelling’s'threshold'model'Neighborhood

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Neighborhood'of'loca#on'X

“stay”

t ='3/7'='42.8%

1 2 3

4 X 5

6 7 8

1 2 3

4 X 5

6 7 8

“go”

1 2 3

4 X 5

6 7 8

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■ Run'NetLogo'“Segrega#on”

Schelling’s'threshold'model'Cascading'moves

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1 2 3

4 X 5

6 7 8

1 2 3

4 X 5

6 7 8

Exodus

1 2 3

4 X 5

6 7 8

Genesis

1 2 3

4 X 5

6 7 8

1 2 3

4 5

6 7 8

1 2 3

4 5

6 7 8

t ='2/7'='28.5%

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Schelling’s'threshold'model'As'an'iterated'network'game

■ Played'on'a'k×k'grid,'with'large'number'of'players'■ Strategy'of'a'player'is'where'it'decides'to'live'—'there'are''

k2 possible'loca#ons'■ Fixed'grid'network'mediates'interac#ons'by'limi#ng'any'

player’s'payoffs'to'depend'only'on'the'neighboring'cells'

■ For'the'occupant'of'loca#on'X,'the'payoff'matrix'would's#ll'need'to'have'29'entries

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1 2 3

4 X 5

6 7 8

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Schelling’s'threshold'model'As'an'iterated'network'game

■ Let'h'denote'the'percentage'of'neighboring'cells'that'have'the'same'type'as'the'occupant'of'loca#on'X'

■ For'Schelling’s'Model'with'threshold't,'the'payoff'matrix'for'occupant'of'loca#on'X'can'be'condensed'to'just'2'entries:'■ if'h≥t'then'payoff'is'1'■ if'h

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Schelling’s'threshold'model'Remarks

■ Difficult'to'infer'collec.ve1behavior'from'individual1preferences'■ Tolerance'of'51%'(almost'total'desegrega.on)'led'to'97%'of'the'individuals'having'similar'type'neighbors'(almost'total'segrega#on)'

■ True'also'in'other'decentralized'systems'■ Possible'counter'examples'in'centralized'systems'(people'not'

allowed'to'move'unilaterally'but'are'told'where'to'live'by'a'central'authority)'

■ Similar'to'solving'a'system'of'equa#ons'(individual'preferences'or'PageRanks)'in'a'constrained'system

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