Evolutionary Games and Population Dynamics Maintenance of Cooperation in Public Goods Games Christoph Hauert Program for Evolutionary Dynamics, Harvard University
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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
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