Games, Groups, Norms andSocieties

Simon Levin, UCI 2008 http://www.n-line.co.uk/2006/04/18/china_traffic/

• We are here to honor a classic text

Games and DecisionsIntroduction and Critical Survey

Luce and Raiffa

“Raiffa opted to cover 2-persongames and statistical decision

theory, and I focused on n-persongames…and information theory”

Duncan Luce, 1988

Unique challenges

•• Collective dynamicsCollective dynamics–– Relation between individuals and groupsRelation between individuals and groups

•• Multiple scalesMultiple scales–– Dynamic of norms and societiesDynamic of norms and societies

•• Behavior/ecology and evolutionBehavior/ecology and evolution–– What should individuals do?What should individuals do?

•• Proximate Proximate vsvs. ultimate explanations. ultimate explanations

A fundamental insight ofA fundamental insight ofevolutionary theory is thatevolutionary theory is that

ultimate and proximateultimate and proximateexplanations need not coincideexplanations need not coincide

Evolutionary biology

• Proximate vs. ultimate cause

www.waynesthisandthat.com

• Initial reasons for pattern may simplyprovide template for evolution of adaptation

Such observations hold forgroups

• Initial reasons for aggregation may simply providetemplate for evolution of adaptive behavior

www.wildcrest.com/Frantz/

www.birminghamzoo.com

Pattern formation

•• Symmetry breakingSymmetry breaking•• Reinforcement and stabilizationReinforcement and stabilization

www.nature.ca/notebooks

Formation of societies

•• Random associationsRandom associations•• Active aggregationActive aggregation•• Stabilization of group boundariesStabilization of group boundaries•• Customs, norms, lawsCustoms, norms, laws•• Institutions, religions, societiesInstitutions, religions, societies

Even phytoplanktonare patchily distributed

spiff.ucsd.edu

Random inhomogeneities lead to reinforcement:Attraction and repulsion in gregarious animals

Tony Sinclair

AnimalAnimal groupsgroups like this bird flock emergelike this bird flock emerge from from individuals following local rulesindividuals following local rules

What is the relationship betweenan individual agent

...and how it responds to itsneighbors and local environment

......and the macroscopic properties of ensembles of such agents?and the macroscopic properties of ensembles of such agents?

How do individuals learn therules, the social norms?

•• Non-human animal groupsNon-human animal groups•• Beijing trafficBeijing traffic•• SocietiesSocieties

Games and collective search

Grunbaum

There is a long and rich history ofthe application of mathematics to

ecology

Vito Volterra 1860-1940

Fluctuations of the Adriatic Fisheries

VariantsVariants on on VolterraVolterra’’s s original equationsoriginal equationsexhibit robust limit-cycle behaviorexhibit robust limit-cycle behavior

!

dx /dt = a(x,y)x(t)

dy /dt = b(x,y)y(t)

www.vanderbilt.edu/AnS

Evolutionary theory also has a rich mathematical historyEvolutionary theory also has a rich mathematical history

R.A.FisherR.A.Fisher J.B.S.J.B.S.HaldaneHaldaneSewall Sewall WrightWright

The challenge remains to meldthese two scales

Place ecological interactions withinPlace ecological interactions withinan evolutionary frameworkan evolutionary framework

!

dx /dt = f (x;",E)

d" /dt = #g(x;",E)

To do so, must embed this systemTo do so, must embed this system in an even higher-order system in an even higher-order system

Ecological

Evolutionary

Fast scale:

Slow scale:

Approaches to marrying ecologyand evolution

•• OptimizationOptimization•• Game TheoryGame Theory•• CoevolutionCoevolution

–– TightTight–– DiffuseDiffuse

www.dkimages.com

Evolution and the Theory ofEvolution and the Theory ofGamesGames

““Evolution is an existentialist gameEvolution is an existentialist game””

LBSlobodkin

Darwin saw natural selection as aprocess of gradualgradual adaptation in a

changing environment

www.biology-online.org

Too easily, however, thistransmogrified into

Evolution as optimizationEvolution as optimization

www.thehitsdoctor.com

Why Optimization?

FisherFisher’’s fundamental theorems fundamental theoremof natural selection:of natural selection:

The mean fitness will increaseThe mean fitness will increasetowards a maximum.towards a maximum.

Selection as hill-climbing findsmaxima

Hence, an optimization principle emergesHence, an optimization principle emerges

!

dw /dt = s(pq /w )(dw /dp)2

!

w

•• Genetic constraints (epistasis, linkage)Genetic constraints (epistasis, linkage)•• Temporal change in the landscapeTemporal change in the landscape•• Frequency dependenceFrequency dependence•• CoevolutionCoevolution

But there are problems with thisseductive picture

Indeed,Indeed,

The deepest problems involvefrequency-dependencefrequency-dependence and

coevolutioncoevolution

encyclopedia.laborlawtalk.com

Because of coevolution andfrequency-dependence

•• Optimization must give way to game theoryOptimization must give way to game theory

To deal with this,To deal with this,Maynard Smith introduced the game-theoreticMaynard Smith introduced the game-theoretic

notion of the evolutionarily stable strategy (ESS): notion of the evolutionarily stable strategy (ESS):

www.pbs.org

A strategy that, once established,A strategy that, once established, cannot be invadedcannot be invaded

Things become more complicated ifwe study the dynamics of such games

and how strategies changehttp://www-eco.enst-bretagne.fr/~phan

Modified Hawks vs. Doves

Maynard SmithMaynard Smith

Case 3:

[ ] -0.6 0.9 -0.9 0.7

Hawks and Doves

Durrett and Levin,1994/Buttel/Case 3

Spatially restricted competitionSpatially restricted competition

•• Hawks Hawks outcompeteoutcompetedoves locallydoves locally

•• Then hawks go extinctThen hawks go extinctlocallylocally

•• Doves Doves recolonize recolonize emptyemptyareasareas

In this example, viscosity iscrucial

But anomalies also can arise without itBut anomalies also can arise without it

Evolutionary dynamics ofphenotypes

•• r(v,u) is the fitness of a rare phenotype v invading ar(v,u) is the fitness of a rare phenotype v invading apopulation in which u is establishedpopulation in which u is established

•• r(v,u) typically is the linearized growth rate of the v-r(v,u) typically is the linearized growth rate of the v-phenotype population near (0, u*)phenotype population near (0, u*)

•• More generally, dominant eigenvalue or Floquet exponentMore generally, dominant eigenvalue or Floquet exponent

Henceforth, assume scalar phenotypesHenceforth, assume scalar phenotypes

The fitness surface is now dynamic

!

w

Focus just on invasion dynamicsat critical points

Piotr Zacny

Resident u

Invader vConvergence-stable

u=vr=0

!

"r /"u+ "r /"v = 0

So critical points with respect to u and v coincide on diagonal

ESS (evolutionarily stable strategy)

r r ((vv, , uu) is maximized as a function of ) is maximized as a function of vv at at v v == u u

!

"r

"v=0,

" 2r

"v 2# 0

uu

But the notion of ESS turns outto be just a beginning

•• There may be several ESSesThere may be several ESSes•• ESS may not be reachableESS may not be reachable

Need complementary notions

•• Neighborhood invader strategyNeighborhood invader strategy•• Convergence stable strategyConvergence stable strategy

Resident u

Invader vConvergence-stable

u=vr=0

!

"r /"v > 0

A resident to the left can be invaded from the right

Resident u

Invader vConvergence-stable

r=0

!

"r /"v # 0

!

"r /"v # 0

A resident to the right can be invaded from the left

Resident u

Invader vConvergence-stable

r=0

!

"r /"v # 0

!

"r /"v # 0

!

" 2r /"u"v +" 2r /"v 2 # 0

!

"r /"v = 0

Or, equivalentlyOr, equivalently

Convergence-stable strategy

!2r

!v2"!

2r

!u2

(attracting in space of phenotypes)(attracting in space of phenotypes)

Hence, an ESS may not be attracting

And an attracting strategy may not be an ESS

This leads to a powerful way tounderstand observed strategies

•• Begin with a basic dynamical modelBegin with a basic dynamical model•• Allow (heritable) variation in the traits of interactingAllow (heritable) variation in the traits of interacting

individualsindividuals•• Explore the adaptive dynamics of such systems, includingExplore the adaptive dynamics of such systems, including

–– continuously stable strategies (convergence-stable continuously stable strategies (convergence-stable ESSesESSes))–– evolutionary branching and possibleevolutionary branching and possible–– coexistence of typescoexistence of types

The evolution of altruism andcooperation

•• AltruismAltruism was a puzzle for Darwin

www.csiro.au

Even bacteria cooperate

www.cs.montana.edu/~ross

Link between group living and communication

Quorum Sensing Slime Biofilms

Low cell density High cell density

Pseudomonas aeruginosa Slime OFF Slime ON

Vibrio cholerae Slime ON Slime OFF

Extracellular Polymers (Slime)

Key

Cell that makespolymer

Cell that cannotmake polymer

Extracellularpolymer

Nutrient Diffusion

Nadell, Xavier, Levin, Foster

Biofilm formation and quorum sensing

Constitutive Slime-producer

Slime

QS Strain (below quorum)

QS Strain (above quorum)

Nadell, Xavier, Levin, Foster

WhatWhat’’s happening?s happening?

Similar ideas may be applied toother animals

• Slime molds• Insects• Krill• Birds• Fish•• UngulatesUngulates

Couzin

Fundamental questions

•• How are individual decisions affected byHow are individual decisions affected bythe social context?the social context?

•• How does the social context emerge andHow does the social context emerge andevolve?evolve?

Issues

•• ExploExploration ration vsvs. Exploitation. Exploitation•• DiscountingDiscounting•• Costs/benefits of leadershipCosts/benefits of leadership•• Group sizeGroup size

Group membership providesbenefits, to some extent in

competition with other groups

www.sit.edu

Group membership providesadvantages over being solitaryBut those benefits may decrease as group size increases

http://humwww.ucsc.edu/gruesz/war/scene.jpg

In many animal species, individualsassemble themselves into

aggregations

www.public.iastate.edu/~jhale

Macroscopic patterns emergewhen individuals follow one

another…among humans

web-japan.org

…as well as other animals

www.nomadafricantravel.co.uk

…leading to fascinatinggeometries

www.travellersworldwide.com

www.pigeon.psy.tufts.edu

Individuals imitate othersIndividuals imitate others’’ behavior behavior

And fads and customs proliferate

www.tattoobyshad.com

…uniformity prevails

Formation of societies

•• Random associationsRandom associations•• Active aggregationActive aggregation•• Stabilization of group boundariesStabilization of group boundaries•• Customs, norms, lawsCustoms, norms, laws•• Institutions, religions, societiesInstitutions, religions, societies

• Simple memes: Threshold voter modelSimple memes: Threshold voter model (the traditional,oversimplified fare)

Problems of scale

• Simple memes (the traditional,oversimplified fare)

• Clusters of memesClusters of memes (traits or behaviors are not independent)

Problems of scale

24857

13657

Focalindividual

Neighbor

Labels

Social norms, multiple traits/opinions Durrett and Levin, JEBO

*Religion*Religion*Ethnicity*Ethnicity*Political party*Political party

Related to a model of Axelrod

24857

13657

Focalindividual

Neighbor

Labels Attitudes

Social norms, multiple traits/opinions

*Abortion rights*Abortion rights*Stem-cell research*Stem-cell research*Gay marriage*Gay marriage

Homophilous Homophilous ImitationImitation

Analogies to Schelling’s model

Formation of cooperative groups

•• Imitation alone can lead to formation ofImitation alone can lead to formation ofstable groupsstable groups

Formation of cooperative groups

• Imitation alone can lead to formation ofstable groups–– Opinions and attitudes on diverse issues mayOpinions and attitudes on diverse issues may

get bundled as get bundled as ““frozen accidentsfrozen accidents””

Formation of cooperative groups

• Imitation alone can lead to formation ofstable groups

•• Existence of groups can produce collectiveExistence of groups can produce collectivebenefitsbenefits

Formation of societies

•• Random associationsRandom associations•• Active aggregationActive aggregation•• Stabilization of group boundariesStabilization of group boundaries•• Customs, norms, lawsCustoms, norms, laws•• Institutions, religions, societiesInstitutions, religions, societies

Formation of cooperative groups

• Imitation alone can lead to formation ofstable groups

• Existence of groups can produce collectivebenefits

•• Collective benefits can lead to selection forCollective benefits can lead to selection forimitation, higher thresholdsimitation, higher thresholds

Extensions

• More complex webs of interaction (smallworlds)

• Asymmetric imitation•• Power structurePower structure

Extensions

• More complex webs of interaction (smallworlds)

• Asymmetric imitation• Power structure•• Payoffs (Fitness differences)Payoffs (Fitness differences)

Role of leadershipCouzinCouzin,, Franks, Krause, LevinFranks, Krause, Levin

g1

Col

lect

ive

deci

sion

-mak

ing

Trend setter

Copier

So the direction chosen by informed individuals mustreconcile these tendencies.

si(t)

di(t+Δt) = si(t) + ω gi(t)si(t) + ω gi(t)

Col

lect

ive

deci

sion

-mak

ing

gi(t)

1 informed individuals in group of 100.

Col

lect

ive

deci

sion

-mak

ing

10 informed individuals in group of 100.

Col

lect

ive

deci

sion

-mak

ing

Animal groups may be led by asmall number of individuals

Difference in preference

Col

lect

ive

deci

sion

-mak

ing

Competing preferencesCompeting preferences

0 20 40 60 80 100 120 140 160 180

0

30

60

90

120

150

180

210

240

270

300

330-120

-150

180

150

120

90

60

30

0

-30

-60

-90

0 20 40 60 80 100 120 140 160 180

0

30

60

90

120

150

180

210

240

270

300

330-120

-150

180

150

120

90

60

30

0

-30

-60

-90

-120

-150

180

150

120

90

60

30

0

-30

-60

-90

Difference in preference

Col

lect

ive

deci

sion

-mak

ing

Leadership

•• Influence of leadershipInfluence of leadership•• Emergence of leadershipEmergence of leadership

g1

Col

lect

ive

deci

sion

-mak

ing

Why do individuals use particular strategies?Why do individuals use particular strategies?

How does selection shape the trade-off between trackingHow does selection shape the trade-off between tracking resources and tracking other individuals?resources and tracking other individuals?

What is the value of information?What is the value of information?

Can this be extended to dynamicsin abstract opinion spaces?

What determines who the leaders are?What determines who the leaders are?

• Simple memesSimple memes

(the traditional,oversimplified fare)(the traditional,oversimplified fare)

•• Clusters of memes Clusters of memes

(traits or behaviors are not independent)(traits or behaviors are not independent)

•• Systems of justice, morality Systems of justice, morality

(collective dynamics of whole systems exhibit unique emergent(collective dynamics of whole systems exhibit unique emergentproperties)properties)

Problems of scale

Many social norms can only beunderstood in broader contexts than those

in which they are observed

•• Charitable givingCharitable giving•• Ultimatum gameUltimatum game•• Fehr Fehr experimentsexperiments

Broader questions

•• How do groups become stabilized?How do groups become stabilized?•• Political parties (Political parties (DuvergerDuverger’’s s law)law)•• ReligionsReligions•• SocietiesSocieties•• LawsLaws•• Problems of the Global CommonsProblems of the Global Commons

Need expanded game-theoreticframework

•• Rewards for adherence to group normsRewards for adherence to group norms•• Historical effectsHistorical effects•• Meta-game contextMeta-game context•• HeuristicsHeuristics•• Multiple scales, in which group dynamicsMultiple scales, in which group dynamics

consideredconsidered

www.dentsply.ca

I hope to have this worked out forthe 60th anniversary

Thank you Thank you