Evolutionary Games and Population Dynamics
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Transcript of Evolutionary Games and Population Dynamics
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Evolutionary Games and Population Dynamics
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Oskar Morgenstern (1902-1977)John von Neumann (1903-1957)
John Nash (b. 1930)
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Nash-Equilibrium
• Arbitrarily many players
• each has arbitrarily many strategies
• there always exists an equilibrium solution
• no player can improve payoff by deviating
• each strategy best reply to the others
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Nash equilibria can be ‚inefficient‘
game Dilemma Prisoners'
mequilibriuNash only theis ),(
015D
510C
DC
D and C strategies
not or euro) 5cost own (at player -coon
euro 15confer can player each :gameDonation
DD
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John Maynard Smith (1920-2004)
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• Population of players
(not necessarily rational)
• Subgroups meet and interact
• Strategies: Types of behaviour
• Successful strategies spread in population
Evolutionary Game Theory
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Population setting
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Population Dynamics
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Example: Moran Process
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Discrete time
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Continuous time
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Replicator Dynamics
so remainsit ,homogenous is population if
allfor 0)( then0)0( if
imitationor einheritanc throughspread strategies
population of evolution predicts
ttxx ii
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Replicator dynamics and Nash equilibria
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Replicator equation
))())((()(
invariant equ. leaves of columns toconstants adding
))((
jij
i
j
i
Tiii
AxAxx
x
x
x
A
AxxAxxx
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Replicator equation for n=2
)0 i.e. 1, and 0 between (provided
and 1,0for points fixed
])()[1(
0
0ly equivalentor
1 ,
2221
1211
21
abxba
axxx
xbaaxxx
b
a
aa
aa
xxxx
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Replicator equation for n=2
• Dominance
• Bistability
• stable coexistence
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Example dominance
s)cooperator of (freq. 0
05
5-0
lyequivalentor
015
5-10
Dilemma) s(Prisoner' gameDonation
x
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Vampire Bat (Desmodus rotundus)
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Vampire Bat (Desmodus rotundus)
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Vampire Bats
Blood donation as a Prisoner‘s Dilemma?
Wilkinson, Nature 1990
The trait should vanish
Repeated Interactions? (or kin selection?)
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Example bistability
AllD and TFT Strategies
game previous of roundssix play
Dilemma sPrisoner' Iterated
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Example bistability
Tat)Tit For of (frequ. 1or 0
045-
5-0
lyequivalentor
015
5-60
Dilemmas Prisoner'Iterated
xx
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Example coexistence
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Example coexistence
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Innerspecific conflicts
Ritual fighting
Konrad Lorenz:
…arterhaltende Funktion
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Maynard Smith and Price, 1974:
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Example neutrality
(drift) points fixed are points all
6060
6060
round each in cooperate all
ALLC of that 1 TFT, offrequency
Dilemmas Prisoner'iterated
xx
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If n=3 strategies
• Example: Rock-Paper-Scissors
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Rock-Paper-Scissors
/3)(1/3,1/3,1 mequilibriu Nash Unique
011-S
1-01P
11-0R
SPR
matrix Payoff
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Rock-Paper-Scissors
dynamics Replicator
011-
1-01
11-0
matrix Payoff
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Generalized Rock-Paper-Scissors
z
ba
ab
ba
A
mequilibriu Nash Unique
1321
0
0
0
21
31
32
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Generalized Rock-Paper-Scissors
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Bacterial Game Dynamics
Escherichia coli
Type A: wild type
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Bacterial Game Dynamics
Escherichia coli
Type A: wild typeType B: mutant producing colicin
(toxic) and an immunity protein
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Bacterial Game Dynamics
Escherichia coli
Type A: wild typeType B: mutant producing colicin
(toxic) and an immunity proteinType C: produces only the immunity
protein
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Bacterial Game Dynamics
Escherichia coli
Rock-Paper-Scissors cycleNot permanent!Serial transfer (from flask to flask):only one type can survive!(Kerr et al, Nature 2002)
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Mating behavior
• Uta stansburiana (lizards)
• (Sinervo and Lively, Nature, 1998)
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Mating behavior
• males: 3 morphs (inheritable)
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Rock-Paper-Scissors in Nature
• males: 3 morphs (inheritable)
• A: monogamous, guards female
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Rock-Paper-Scissors in Nature
• males: 3 morphs (inheritable)
• A: monogamous, guards female
• B: polygamous, guards harem (less efficiently)
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Rock Paper Scissors in human interactions
• Example: three players divide some goods
• Any pair forms a majority
• Shifting coalitions
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Phase portraits of Replicator equations:
attractors chaotic
cycleslimit severalor one
1,...,1
)(
Volterra-Lotka with equiv.
classif. no 4for
ni
ybryy
n
jijiii
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