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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
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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
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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
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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)
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1 informed individuals in group of 100.
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10 informed individuals in group of 100.
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Animal groups may be led by asmall number of individuals
Difference in preference
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Competing preferencesCompeting preferences
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Leadership
•• Influence of leadershipInfluence of leadership•• Emergence of leadershipEmergence of leadership
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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