Upload
naoki-masuda
View
414
Download
0
Embed Size (px)
DESCRIPTION
Nakamura and Masuda, BMC Evolutionary Biology, 12, 213 (2012).
Citation preview
Groupwise information sharingpromotes ingroup favoritism
in indirect reciprocity
Mitsuhiro Nakamura & Naoki MasudaDepartment of Mathematical Informatics
The University of Tokyo, Japan
1
M. Nakamura & N. Masuda. BMC Evol Biol 2012, 12:213http:/www.biomedcentral.com/1471-2148/12/213
Indirect reciprocity
!! "#
"#
Cost of help Benefit
Later, the cost of help is compensated
by others’ help
Alexander, Hamilton, Nowak & Sigmund
2
▶ A mechanism for sustaining cooperation
What stabilizes cooperationin indirect reciprocity?
1. Apposite reputation assignment rules
2. Apposite sharing of reputation information in the population
3
Reputation assignment rules
4
G B
C G G
D B B
Image scoring (IM)
Donor’s action:cooperation (C) or defection (D)
Recipient’s reputation:
good (G) or bad (B)
▶ C is good and D is bad
▶ Not ESS (Leimar & Hammerstein, Proc R Soc B 2001)
Reputation assignment rules
5
G B
C G G
D B G
G B
C G B
D B G
C toward a B player is B!
Simple standing (ST) Stern judging (JG)
D against a B player is G
D against a B player is G
▶ ESS (e.g., Ohtsuki & Iwasa, JTB 2004)
6
1. Apposite reputation assignment rules
2. Apposite sharing of reputation information in the population
Incomplete information sharing
ignored
Group structure (not well-mixed)
ignored
▶ We assumed groupwise information sharing and (unexpectedly) found the emergence of ingroup favoritism in indirect reciprocity
What stabilizes cooperationin indirect reciprocity?
Ingroup favoritism
▶ Humans help members in the same group (ingroup) more often than those in the other group (outgroup).
▶ Connection between ingroup favoritism and indirect reciprocity has been suggested by social psychologists (Mifune, Hashimoto & Yamagishi, Evol Hum Behav 2010)
7
Tajfel et al., 1971
Explanations for ingroup favoritism
▶ Green-beard effect (e.g., Jansen & van Baalen, Nature 2006)
▶ Tag mutation and limited dispersal (Fu et al., Sci Rep 2012)
▶ Gene-culture co-evolution (Ihara, Proc R Soc B 2007)
▶ Intergroup conflict (e.g., Choi & Bowles, Science 2007)
▶ Disease aversion (Faulkner et al., Group Proc Int Rel 2004)
▶ Direct reciprocity (Cosmides & Toobey, Ethol Sociobiol 1989)
▶ Indirect Reciprocity (Yamagishi et al., Adv Group Proc 1999)
8
Model
▶ Donation game in a group-structured population (ingroup game occurs with prob. θ)
▶ Observers in each group assign reputations to players based on a common assignment rule
▶ Observers assign wrong reputations with prob. µ << 1
9
!! "#
"#
"$
"#
Reputation dynamics
10
dd� �� (�) = −�� (�) + �
� �∈{G�B}M
�θ�� (� �) + (1 − θ)�−� (� �)�M�
� �=1Φ�� � (σ (��� )� ��� � )
▶ where,
�� (�)
�−� (�) ≡ �� �=�
�� (�)/(M − 1)
σ (�)Φ�(�� ��)
Prob. that a player in group k has reputation vector r in the eyes of M observers
Donor’s action: σ (G) = C� σ (B) = DProb. that an observer assigns r when the observer
observes action a toward recipient with reputation r’
r=(G,G,B)
!! "#
"#
"$
"#
Group 1 Group 2
Group 3
scalar
Ingroup reputation dynamics
11
dd� �in(�) = −�in(�) + �
��∈{G�B}
�θ�in(��) + (1 − θ)�out(��)� Φ�(σ (��)� ��)
dd� �� (�) = −�� (�) + �
� �∈{G�B}M
�θ�� (� �) + (1 − θ)�−� (� �)�M�
� �=1Φ�� � (σ (��� )� ��� � )
Outgroup reputation dynamics
12
dd� �out(�) = −�out(�) + �
��∈{G�B}
����∈{G�B}�
θ�in(��)�out(���) + (1 − θ)� 1
M − 1 �out(��)�in(���) +�
1 − 1M − 1
��out(��)�out(���)
��Φ�(σ (��)� ���)
dd� �� (�) = −�� (�) + �
� �∈{G�B}M
�θ�� (� �) + (1 − θ)�−� (� �)�M�
� �=1Φ�� � (σ (��� )� ��� � )
Results: Cooperativeness and ingroup bias
13
Rule
IM
ST
JG
12
12
1 − µ1 − µ 1
2
1 − µθ1 − µ 1 + θ
θ1 + θ
2 − µθ
12
12 − µ
µθ
0�∗in(G) �∗out(G) ψ ρ
Prob. CIngroup
biasFrac. G
(ingroup)Frac. G
(outgroup)
ρ ≡ �∗in(G) − �∗out(G)ψ ≡ θ�∗in(G) + (1 − θ)�∗out(G)
θ
ψ
(a)
0 0.5 1
0
0.5
1
ST, theoryST, M = 2ST, M = 10JG, theoryJG, M = 2JG, M = 10
θ
ρ
(b)
0 0.5 1
0
0.25
0.5
Results: Individual-based simulations
14
−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
IM ST
JG
Play
er
Group
Prob. C
GB
N=300, µ=.01, M=3, θ=.6
Ingroup bias
N=103, µ=.01
Results: Cases with error in actions
15
θ
ψ
(a)
0 0.5 1
0
0.5
1
ST, theoryST, � = 0.01ST, � = 0.1JG, theoryJG, � = 0.01JG, � = 0.1
θ
ρ
(b)
0 0.5 1
0
0.25
0.5
N=103, µ=.01, M=10
▶ Donors fail in cooperation with prob. ε Prob. C
Ingroup bias
Results: Evolutionary stability
16
▶ Conditions under which players using reputations are stable against invasion by unconditional cooperators and defectors:
1 < �� < 1
1 − θ
ST
� (M−1)(1+θ)1+(M−3)θ+Mθ2 < �
� < M−11−Mθ if 0 ≤ θ < 1
M(M−1)(1+θ)
1+(M−3)θ+Mθ2 < �� if 1
M ≤ θ ≤ 1
JG
→ �� > 1
θ (M → ∞)
public reputation: θ = 1private reputation: θ → 1/M, M → ∞
Results: Mixed assignment rules
17
0 0.5 10
0.5
1�a�
Α
Ψ
ST JG
M � 2, � 0.6M ��, � 0.6M � 2, � 0.2M ��, � 0.2
0 0.5 10
0.25
0.5�b�
Α
Ρ
ST JG0 0.5 11
2
3
4
5�c�
M � 2Θ� 0.6
ΑST JG
b�c
0 0.5 11
2
3
4
5�d�
M ��� 0.6
ΑST JG
b�c
0 0.5 11
2
3
4
5�e�
M � 2Θ� 0.2
ΑST JG
b�c
0 0.5 11
2
3
4
5�f�
M ��� 0.2
ΑST JGb�c
▶ Observers use JG with prob. α and ST with prob. 1-α
Results: Heterogeneous assignment rules
18
ψST,ψ
JG,ρ
ST,ρ
JG
(a)
0 2 4 6 8
0
0.5
1
ψSTψJG
ρSTρJG
(b)
0 5 10 15 20
0
0.5
1
ψSTψJG
ρSTρJG
m
πJG−
πST
(c)
0 2 4 6 8
-0.1
0
0.1
0.2b = 2b = 4b = 6
m
(d)
0 5 10 15 20
-0.1
0
0.1
0.2b = 2b = 4b = 6
▶ Different groups use different rules (either ST or JG)
Number of JG groups
M=8 M=20
Conclusions
▶ Indirect reciprocity with group-structured information sharing yields ingroup favoritism.
▶ Ingroup bias is severer than under JG than under ST.
19