Groupwise information sharingpromotes ingroup favoritism

in indirect reciprocity

Mitsuhiro Nakamura & Naoki MasudaDepartment of Mathematical Informatics

The University of Tokyo, Japan

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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

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▶ A mechanism for sustaining cooperation

What stabilizes cooperationin indirect reciprocity?

1. Apposite reputation assignment rules

2. Apposite sharing of reputation information in the population

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Reputation assignment rules

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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

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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)

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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)

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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)

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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

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!! "#

"#

"$

"#

Reputation dynamics

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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)

!! "#

"#

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Group 1 Group 2

Group 3

scalar

Ingroup reputation dynamics

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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

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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

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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

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−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

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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

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▶ 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

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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

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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.

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