Nakamura and Masuda, BMC Evolutionary Biology, 12, 213 (2012).

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