1) The study examines how groupwise sharing of reputation information promotes ingroup favoritism in indirect reciprocity. 2) It finds that when reputation information is shared within groups, rather than across the whole population, individuals come to favor others in their own group when deciding whether to cooperate. 3) The degree of ingroup bias depends on the reputation assignment rule used, with the Stern Judging rule producing stronger ingroup favoritism than the Simple Standing rule.

1. Groupwise information sharing
promotes ingroup favoritism
in indirect reciprocity
Mitsuhiro Nakamura & Naoki Masuda
Department of Mathematical Informatics
The University of Tokyo, Japan
M. Nakamura & N. Masuda. BMC Evol Biol 2012, 12:213
http:/www.biomedcentral.com/1471-2148/12/213 1

2. Indirect reciprocity
Alexander, Hamilton, Nowak & Sigmund
▶ A mechanism for sustaining cooperation
Cost of help Benefit
!! "#
"#
Later, the cost of
help is compensated
by others’ help
2

3. What stabilizes cooperation
in indirect reciprocity?
1. Apposite reputation assignment rules
2. Apposite sharing of reputation information
in the population
3

4. Reputation assignment rules
Image scoring (IM)
Donor’s action: Recipient’s
cooperation (C) G B reputation:
or defection (D) good (G) or bad (B)
C G G
D B B
▶ C is good and D is bad
▶ Not ESS (Leimar & Hammerstein, Proc R Soc B 2001)
4

5. Reputation assignment rules
Simple standing (ST) Stern judging (JG)
G B G B C toward a
B player is B!
C G G C G B
D B G D B G
D against a B D against a B
player is G player is G
▶ ESS (e.g., Ohtsuki & Iwasa, JTB 2004) 5

6. What stabilizes cooperation
in indirect reciprocity?
1. Apposite reputation assignment rules
2. Apposite sharing of reputation information in
the population
Incomplete Group structure
information sharing (not well-mixed)
ignored ignored
▶ We assumed groupwise information sharing and
(unexpectedly) found the emergence of ingroup
favoritism in indirect reciprocity 6

7. Ingroup favoritism
Tajfel et al., 1971
▶ 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

8. 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

9. 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

10. Reputation dynamics
d
M
() = − () + θ ( ) + (1 − θ)− ( )
Φ (σ ( ) )
d
∈{GB}M =1
▶ where, r=(G,G,B)
() Prob. that a player in group k has reputation vector r in
the eyes of M observers
Group 3
− () ≡ ()/(M − 1)
$
Group 1 Group 2
= !! #
σ () Donor’s action: σ (G) = C σ (B) = D # #
Φ ( ) Prob. that an observer assigns r when the observer
scalar observes action a toward recipient with reputation r’ 10

11. Ingroup reputation dynamics
d
M
() = − () + θ ( ) + (1 − θ)− ( )
Φ (σ ( ) )
d
∈{GB}M =1
d
in () = −in () + θin ( ) + (1 − θ)out ( ) Φ (σ ( ) )
d ∈{GB}
11

12. Outgroup reputation dynamics
d
M
() = − () + θ ( ) + (1 − θ)− ( )
Φ (σ ( ) )
d
∈{GB}M =1
d
out () = −out () +
d
∈{GB} ∈{GB}
1 1
θin ( )out ( ) + (1 − θ) out ( )in ( ) + 1 − out ( )out ( ) Φ (σ ( ) )
M −1 M −1
12

13. Results: Cooperativeness and ingroup bias
Frac. G Frac. G Ingroup
(ingroup) (outgroup) Prob. C bias
Rule ∗ (G)
in ∗ (G)
out ψ ρ
1 1 1
IM 2 2 2
0
1+θ µ µ
ST 1−µ 1−µ 1−
θ θ θ
1 1+θ 1
JG 1−µ − µθ −µ
2 2 2
ψ ≡ θ∗ (G) + (1 − θ)∗ (G)
in out ρ ≡ ∗ (G) − ∗ (G)
in out
13

14. Results: Individual-based simulations
(a)
1
IM ST ψ
ST, theory
1.0
1.0
0.5 ST, M = 2
ST, M = 10
0.8
0.8
Player
Prob. C JG, theory
0.6
0.6 JG, M = 2
0.4
0.4
0 JG, M = 10
0.2
0.2
0 0.5 1
JG
0.0
0.0
Group
−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
θ
(b)
1.0
0.5
0.8
G
0.6
B ρ
0.4
0.25
Ingroup bias
0.2
0.0
−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
N=300, µ=.01, N=103, µ=.01 0
M=3, θ=.6 0 0.5 1
14
θ

15. Results: Cases with error in actions
(a)
1
ψ
▶ Donors fail in cooperation 0.5
ST, theory
ST, = 0.01
with prob. ε Prob. C
ST, = 0.1
JG, theory
JG, = 0.01
0 JG, = 0.1
0 0.5 1
θ
(b)
0.5
ρ
0.25
Ingroup bias
N=103, µ=.01, M=10 0
0 0.5 1
15
θ

16. Results: Evolutionary stability
▶ Conditions under which players using reputations
are stable against invasion by unconditional
cooperators and defectors:
ST
1 public reputation: θ = 1
1
1−θ private reputation: θ → 1/M, M → ∞
JG
(M−1)(1+θ) M−1 1
1+(M−3)θ+Mθ 2
1−Mθ if 0 ≤ θ M
(M−1)(1+θ)
1+(M−3)θ+Mθ 2
if 1
M ≤θ≤1
1
→ (M → ∞)
θ 16

17. Results: Mixed assignment rules
▶ Observers use JG with prob. α and ST with prob. 1-α
a b c
1 0.5 5
M2
Θ 0.6
Ψ Ρ 4
bc
0.5 0.25 3
M 2, Θ 0.6
M , Θ 0.6
M 2, Θ 0.2 2
M , Θ 0.2
0 0 1
0 0.5 1 0 0.5 1 0 0.5 1
ST Α JG ST Α JG ST Α JG
d e f
5 5 5
M M2 M
Θ 0.6 Θ 0.2 Θ 0.2
4 4 4
bc
bc
bc
3 3 3
2 2 2
1 1 1
0 0.5 1 0 0.5 1 0 0.5 1
ST Α JG ST Α JG ST Α JG 17

18. Results: Heterogeneous assignment rules
▶ Different groups use different rules (either ST or JG)
(a) (b)
1 1
ψST , ψJG , ρST , ρJG
ψST ρST ψST ρST
ψJG ρJG ψJG ρJG
0.5 0.5
0 0
0 2 4 6 8 0 5 10 15 20
(c) (d)
0.2 M=8 0.2 M=20
b=2 b=2
b=4 b=4
πJG − πST
0.1 b=6 0.1 b=6
0 0
Number of -0.1 -0.1
JG groups 0 2 4 6 8 0 5 10 15 20
18
m m

19. Conclusions
▶ Indirect reciprocity with group-structured
information sharing yields ingroup favoritism.
▶ Ingroup bias is severer than under JG than under ST.
19