evolutionary games and population dynamics · population dynamics four dynamical scenarios...

Evolutionary Games and Population Dynamics

Maintenance of Cooperation in Public Goods Games

Christoph HauertProgram for Evolutionary Dynamics, Harvard University

Group defense and group foraging

Predator inspection and alarm calls

Major transitions in the evolution of life, e.g. the formation of multicellular organisms.

Social welfare: health care, pension plans, unemployment compensation…

(Global) sustainability: greenhouse gases, drinking water, fisheries…

Conflict of interest between individual and community performance.

The problem of cooperationExamples

Definition Cooperators sustain common good at some cost while

defectors attempt to exploit the resource by avoiding the costly contributions.

Groups of cooperators do better than groups of defectors.

Defectors outperform cooperators in each group.

Hence the dilemma.

Prominent examples: Prisoner’s dilemma Snowdrift game Public goods game.

Social dilemmas

Public goods game

Group of four players

Endowment of one dollar

Investment in common pool

Experimenter doubles amount in pool and divides it equally among all players

Full cooperation yields two dollars.

Each invested dollar returns only 50 cents to the investor.

Rational players invest nothing.

Sample game

Payoffs in groups of size N with k cooperators:

Average payoffs in large populations with x cooperators and interactions in random groups:

PD(k) =r

Nk

PC(k) = PD(k)! 1

Public goods gameEvolutionary dynamics

fD = xr

N(N ! 1)

fC = (x(N ! 1) + 1)r

N! 1

Evolutionary fate of cooperators:

Classical case:Defection is dominant, cooperators go extinct.

High returns, small groups:Cooperation is dominant, defectors go extinct.

However, in each group defectors still outperform cooperators - Simpson’s paradox.

Evolutionary dynamicsReplicator equation

x = x(1! x) (fC ! fD)r < N

r > N

Population dynamics

Introduce third type indicating vacant space z:

b: baseline birthrated: death rate

Variations in populations densities can lead to variations in the interaction group size.

Feedback can maintain cooperation and lead to persistent populations ( b < d ).

Variable population densities

x = x(z(b + fC)! d)y = y(z(b + fD)! d)z = (1! z)(!b + d)! z(xfC + yfD)

Average payoffs in groups of size S:

Average payoffs in large populations:

Population dynamicsVariable population densities

PD(S) =r

S

S!1!

m=0

"x

1! z

#m "y

1! z

#S!1!m "S ! 1

m

#m

= rx

1! z

"1! 1

S

#

PC(S) = PD +r

S! 1.

fD = rx

1! z

!1! 1! zN

N(1! z)

"

fC = fD ! F (z)F (z) = 1 + (r ! 1)zN!1 ! r

N

1! zN

1! zwhere

Population dynamics

In absence of cooperators, defectors disappear.

Cooperators can persist.

Sufficiently high initial densities.

Death rates below threshold.

Homogeneous populations

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

fraction of cooperators x

de

ath

ra

te d

Population dynamics

Fraction of cooperators in population :

N: maximum group sizer: multiplication factor of public good

d: death rate

Interior fixed point

Transformation of variables

u = !zu(1! u)F (z)

z = !(1! z)(uz(r ! 1)!1! zN!1

"! d)

u =x

x + y

Q = (u, z)F (z) = 0

u =d

z(r ! 1) (1! zN!1)

Population dynamics

Four dynamical scenarios

Co-existence in stable interior fixed point.

Oscillations with decreasing amplitude.

Population cannot recover if densities too low or exploitation too high.

Heterogeneous populations

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

rela

tive fra

ction o

f coopera

tors

f=

x

x+

yextinction

cooperation

extinction

oscillations,

co-existence

population density x+y

ba

population density x+yc d0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

extinction

rela

tive fra

ction o

f coopera

tors

f=

x

x+

y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

extinction

oscillations,

extinction

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

rela

tive fra

ction o

f coopera

tors

f=

x

x+

y

extinction

cooperation

extinction

oscillations,

co-existence

population density x+y

ba

population density x+yc d0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

extinction

rela

tive fra

ction o

f coopera

tors

f=

x

x+

y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

extinction

oscillations,

extinction

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

rela

tive

fra

ctio

n o

f co

op

era

tors

f=

x

x+

y

extinction

cooperation

extinction

oscillations,

co-existence

population density x+y

ba

population density x+yc d0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

extinctionre

lative

fra

ctio

n o

f co

op

era

tors

f=

x

x+

y

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

extinction

oscillations,

extinction

Population dynamics

All scenarios:a. co-existenceb. cooperationc. oscillations, extinctiond. extinction

Increasing death rate d: dynamics changes from left to right.

Decreasing returns of public

good r: dynamics changes from top to bottom.

Heterogeneous populations

Extend analysis to arbitrary baseline birth rates

Potential for Hopf-bifurcations.

Analysis of more general social dilemmas

Snowdrift game

Spatial structure

Lattice games versus games in continuous space

OutlookWork in progress

OutlookSpatial structure

Eco-evolutionary feedback can stabilize cooperation at intermediate frequencies.

Cooperation, population increase, large groups ⇔ Defection, population decline, small groups.

Fails for pairwise Prisoner’s Dilemma interactions.

Effective group size cannot vary.

Spatial structure

Stabilizes cooperation.

Mechanism closely related to voluntary interactions in public goods games.

The abundance of the risk averse loner strategy controls the effective groups size of the public goods interactions.

Conclusions

Tutorials:http://www.univie.ac.at/virtuallabs

References:Hauert, Holmes & Doebeli (2006) Proc. R. Soc. Lond B.Hauert, DeMonte, Hofbauer & Sigmund (2002) Science.

Michael Doebeli, UBC, Vancouver BC.

Miranda Holmes, Courant Institute, NY University.

Joe Yuichiro Wakano, University of Tokyo.

Martin Nowak, Program for Evolutionary Dynamics, Harvard University.

Acknowledgments