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