Evolutionary Games and Population Dynamics
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Transcript of Evolutionary Games and Population Dynamics
Evolutionary Games and Population Dynamics
Oskar Morgenstern (1902-1977)John von Neumann (1903-1957)
John Nash (b. 1930)
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
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
John Maynard Smith (1920-2004)
• Population of players
(not necessarily rational)
• Subgroups meet and interact
• Strategies: Types of behaviour
• Successful strategies spread in population
Evolutionary Game Theory
Population setting
Population Dynamics
Example: Moran Process
Discrete time
Continuous time
Replicator Dynamics
so remainsit ,homogenous is population if
allfor 0)( then0)0( if
imitationor einheritanc throughspread strategies
population of evolution predicts
ttxx ii
Replicator dynamics and Nash equilibria
Replicator equation
))())((()(
invariant equ. leaves of columns toconstants adding
))((
jij
i
j
i
Tiii
AxAxx
x
x
x
A
AxxAxxx
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
Replicator equation for n=2
• Dominance
• Bistability
• stable coexistence
Example dominance
s)cooperator of (freq. 0
05
5-0
lyequivalentor
015
5-10
Dilemma) s(Prisoner' gameDonation
x
Vampire Bat (Desmodus rotundus)
Vampire Bat (Desmodus rotundus)
Vampire Bats
Blood donation as a Prisoner‘s Dilemma?
Wilkinson, Nature 1990
The trait should vanish
Repeated Interactions? (or kin selection?)
Example bistability
AllD and TFT Strategies
game previous of roundssix play
Dilemma sPrisoner' Iterated
Example bistability
Tat)Tit For of (frequ. 1or 0
045-
5-0
lyequivalentor
015
5-60
Dilemmas Prisoner'Iterated
xx
Example coexistence
Example coexistence
Innerspecific conflicts
Ritual fighting
Konrad Lorenz:
…arterhaltende Funktion
Maynard Smith and Price, 1974:
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
If n=3 strategies
• Example: Rock-Paper-Scissors
Rock-Paper-Scissors
/3)(1/3,1/3,1 mequilibriu Nash Unique
011-S
1-01P
11-0R
SPR
matrix Payoff
Rock-Paper-Scissors
dynamics Replicator
011-
1-01
11-0
matrix Payoff
Generalized Rock-Paper-Scissors
z
ba
ab
ba
A
mequilibriu Nash Unique
1321
0
0
0
21
31
32
Generalized Rock-Paper-Scissors
Bacterial Game Dynamics
Escherichia coli
Type A: wild type
Bacterial Game Dynamics
Escherichia coli
Type A: wild typeType B: mutant producing colicin
(toxic) and an immunity protein
Bacterial Game Dynamics
Escherichia coli
Type A: wild typeType B: mutant producing colicin
(toxic) and an immunity proteinType C: produces only the immunity
protein
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)
Mating behavior
• Uta stansburiana (lizards)
• (Sinervo and Lively, Nature, 1998)
Mating behavior
• males: 3 morphs (inheritable)
Rock-Paper-Scissors in Nature
• males: 3 morphs (inheritable)
• A: monogamous, guards female
Rock-Paper-Scissors in Nature
• males: 3 morphs (inheritable)
• A: monogamous, guards female
• B: polygamous, guards harem (less efficiently)
Rock Paper Scissors in human interactions
• Example: three players divide some goods
• Any pair forms a majority
• Shifting coalitions
Phase portraits of Replicator equations:
attractors chaotic
cycleslimit severalor one
1,...,1
)(
Volterra-Lotka with equiv.
classif. no 4for
ni
ybryy
n
jijiii