Cooperation in n - player Prisoner's Dilemma threshold game

Cooperation in n - player
Prisoner’s Dilemma threshold game
Boza, G.
(1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3)
1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University
2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University
3-Collegium Budapest, Institute for Advanced Study
Budapest, Hungary
4-IIASA, International Institute for Applied Systems Analysis
Laxenburg, Austria

Reciprocal altruism
Individual 1.
Studying cooperation
cost
helping act
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
>
if cost benefit
<
cost benefit
benefit
Individual 2.
X

Game theory: a tool for studying cooperation
The Prisoner’s Dilemma Game
• conflict of interest: the dilemma
• two players
Player 1
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
Player 2

n-players game: Public Goods Game
conflict of interest
n-players (n > 2)
public good: non-excludable, non-rival resource
benefit function (fitness function)
score
T (cheat)
R
P
0
number of cooperators
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
Defectors Free-riders
Cooperators

n-player games in natural systems
…on land…
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
…and in water

n-player Threshold Game
score
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
T
R
0
number of cooperators

n-player Threshold Game
score
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
T
R
0
number of cooperators

n-player Threshold Game
score
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
T
R
0
number of cooperators

n-player Threshold Game
score
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
T
R
0
number of cooperators

n-player Prisoner’s Dilemma Threshold Game
n-players from the population of N randomly chosen: Well – mixed population
group size (n)
threshold value (TV)
cost of cooperation (c)
benefit of cooperation (b)
focal
C
D
CC
b-c
willingness to cooperate (x), evolving trait
b
partners
CD
b-c
0
DD
x = 1 always cooperates
x = 0 always defects
Bach, L. A., Helvik, T., Christiansen, F. B., (2006). The evolution of n-player cooperation – threshold games
and ESS bifurcations, Journal of Theoretical Biology, 238: 426-434.
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
-c
0
3
2
1

x
1
0.5
0
Results with well-mixed population structure
0 0.5
1
TV = 3 / 3
c
x
1
0.5
0
0 0.5
1
TV = 2 / 3
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
c
Changing parameter:
c – cost of cooperation
TV – threshold value
x
1
0.5
0
0 0.5
1
TV = 1 / 3
c

Game with spatial population structure
cellular automaton
von Neumann – neighborhood:
focal individual +
4 closest individual on the grid
Moore – neighborhood:
focal individual +
8 closest individual on the grid
asynchronous update
proportional update rule
p
d
i
i = n
∑
j = 1
TWO GROUP FORMING SCENARIOS
d
j
= defectors
= cooperators
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy

Results with spatial population structure
Scenario 1: fixed group composition (1 area – 1 group)
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy

Results with spatial population structure
Scenario 1: fixed group composition (1 area – 1 group)
Scenario 2: group composition corresponding to focal individual
(focal individual – 1 focal’s group + 8 neighbors’ group)
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy

Results with spatial population structure
Scenario 1: fixed group composition (1 area – 1 group)
Scenario 2: group composition corresponding to focal individual
(focal individual – 1 focal’s group + 8 neighbors’ group)
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy

Results with spatial population structure
Scenario 1: fixed group composition (1 area – 1 group)
Defectors overtake
Scenario 2: group composition corresponding to focal individual
(focal individual – 1 focal’s group + 8 neighbors’ group)
Cooperators prevail more
than in well-mixed population
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy

TV=4
Results with spatial population structure II.
1
0.8
0.6
0.4
0.2
Population structure
Well –mixed Spatial
1
x x
0
0 0.2 0.4 0.6 0.8 1
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
1.
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
c 1
c
2. 0.8
0.6
0.4
0.2
x
0
0 0.2 0.4 0.6 0.8 1
c

Results with well-mixed population structure II.
n
Maximal cost what
cooperators still can bear
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy
1
0.8
0.6
0.4
0.2
0
1
0.5
Changing parameter:
n – group size
0 5 10 15 20 25 30 35 40 45 50
0 10 20 30 40 50
n

Conclusions
High levels of cooperation can evolve in n-player Prisoner’s Dilemma
Threshold Game
What matters: …..
• the size of the group
• the threshold value
• interpretation of (localized) interaction group
• Update rule (competition rule)
Thank you for your attention !
Boza et al. - Cooperation in n-player Prisoner’s Dilemma threshold game - ECEM’07, Trieste, Italy