1.
Tag-based indirect reciprocity by
incomplete social information
Naoki Masuda1 and Hisashi Ohtsuki2
The University of Tokyo, Japan
2
Harvard University
http://www.stat.t.u-tokyo.ac.jp/~masuda
1
Ref: Masuda & Ohtsuki, Proc. R. Soc. B, 274,
689-695 (2007).

2.
Prisoner’s Dilemma
Opponent
Cooperate
Defect
Cooperate
(3, 3)
(0, 5)
Defect
(5, 0)
(1, 1)
Self
unique Nash equilibrium

3.
A Prisoner’s Dilemma
• A donor may donate
cost c to benefit the
recipient by b (>c).
• If each player serves
as donor and
recipient in different
(random) pairings, the
game is symmetric
PD.
recipient
donor
C
(-c, b)
D
(0, 0)
(b > c)
C
D
C (b-c, b-c)
(-c, b)
D
(0, 0)
(b, -c)

4.
Origins of altruism
• Kin selection
• Direct reciprocity
– Iterated Prisoner’s dilemma
•
•
•
•
•
Spatial reciprocity
Indirect reciprocity
Network reciprocity
Group selection
Others
• Is ‘helping similar others’ a viable (stable)
strategy?

5.
An affirmative answer by
Riolo, Cohen & Axelrod, Nature 2001
• b=1.0, c=0.1
• Player i has
– Tag wi ∈ [ 0,1]
– Tolerance µi ∈ [ 0,1]
• i cooperates with j if w j − wi ≤ µi
• Players copy tag and tolerance of successful others.
• mutation:
– Random allocation of tag
– Neutral drift of tolerance
• Results of their numerical simulations of evol dynamics:
– Donation rate is maintained high (~ 75%).
– The mean tolerance level is small (0.01-0.03).
– With some sudden changes though.

6.
However, rebuttal
by Roberts & Sherratt (Nature 2002)
Criticism 1
i was assumed to coopreate if
w j − wi ≤ µi &
µi ∈ [ 0,1]
Criticism 2
Neutral drift & µi ∈ [ 0,1]
A player cooperate with birds
with exactly the same feather
Random walk with
reflecting boundary
Cooperation is lost if µi ∈ [ 0,1] is
replaced by µi ∈ [ − 10 −6 ,1]
Positive bias. Why
mutation increases
generosity?

7.
We establish a viable model of tagbased reciprocity.
[
]
µi ∈ − 10 −6 ,1
• Use a kind of
• q: prob that μj is public to others
• If player i gets to know μj <|wj-wi|, i does
not donate even if μi ≥|wj-wi|
• q=0 → eventually ALLD (μi <0)
• q=1 → eventually ALLC (μi takes max)
• No mutation of tags

8.
2-tag model
• Same or different only.
tag
tolerance
wi ∈ [ 0,1]
[
]
µi ∈ − 10 ,1
−6
{
→ wi ∈ w , w
→
a
b
}
µi ∈ { − 1,0,1}
μ
phenotype
-1 no donate (D)
0
tag user
1
donate (C)

9.
Payoffs of 6 subpopulations
tag = a
tag = b
h: assortativity

10.
Replicator dynamics
• Symmetric case
– Full theoretical analysis
(global analysis)
• Asymmetric case
6 vars, 4 dim
note: no tag evolution
– Best-response theory
(local analysis only)
– Numerical simulations

11.
Symmetric case
Small q
μ
phenotype
-1 no donate (D)
0
tag user
1
donate (C)
Intermediate q
Large q
c (1 − q )
A≡
< 1 ⇒ bq > c
( b − c) q
is the condition for tag users
to emerge.

12.
With assortativity h
b = 1, c = 0.3
q = 0.5, h = 0
q = 0.5, h = 0.8

13.
Asymmetric case (best response)
A=
p >A
b
1
p1b ≤ A
A − (1 − ( t + h − ht ) ) p1b
X=
.
t + h − ht
μ
Among 9 pure strategies, only (μa,μb)=(-1,-1), (1,0), (0,-1), (0,0), and (1,1) are viable.
c (1 − q )
,
( b − c) q
phenotype
-1 no donate (D)
0
tag user
1
donate (C)

14.
Basin areas (numerical)
(1, 1)
(-1, -1)
μ
phenotype
-1 no donate (D)
0
(0, -1)
(-1, 0)
(0, 0)
q
tag user
1
donate (C)

15.
Best response (continuous tag)
• Any μi = μ is ESS if
bq>c
• If μi is uniformly
distributed,
bq − c
µ opt =
,
( b − c) q
µ opt = 0,
( bq > c )
( bq ≤ c )
optimal μ
b/c=4
2
1.2
q

16.
Numerical simulations
noiseless case
noisy case
μ
n = 800
b=1
c = 0.3
q

17.
Conclusions
• Tag-based indirect reciprocity is viable
when publicity of tolerance is intermediate.
– Large publicity → cooperation prevails
– Small publicity → defection prevails
• Future work: network version?