Tag-based indirect reciprocity by incomplete social information

Naoki Masuda1 and Hisashi Ohtsuki2

1The University of Tokyo, Japan2Harvard University

http://www.stat.t.u-tokyo.ac.jp/~masuda

Ref: Masuda & Ohtsuki, Proc. R. Soc. B, 274, 689-695 (2007).

Prisoner’s Dilemma

Cooperate Defect

Cooperate (3, 3) (0, 5)

Defect (5, 0) (1, 1)

SelfSelfOpponenOpponentt

unique Nash equilibrium

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

C (-c, b)

D (0, 0)

C D

C (b-c, b-c) (-c, b)

D (b, -c) (0, 0)

(b > c)

donor

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?

• b=1.0, c=0.1• Player i has

– Tag– Tolerance

• i cooperates with j if

• 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.

An affirmative answer byRiolo, Cohen & Axelrod, Nature 2001

1,0iw 1,0i

iij ww

Cooperation is lost if is replaced by

However, rebuttal by Roberts & Sherratt (Nature 2002)

iij ww

Criticism 1 Criticism 2

1,0ii was assumed to coopreate if

&

A player cooperate with birds with exactly the same feather

1,0i 1,10 6i

1,0iNeutral drift &

Random walk with reflecting boundary

Positive bias. Why mutation increases generosity?

• 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

We establish a viable model of tag-based reciprocity.

1,10 6i

• Same or different only.

2-tag model

1,0,11,10

,1,06

ii

baii wwww

μ phenotype

-1 no donate (D)

0 tag user

1 donate (C)

tag

tolerance

Payoffs of 6 subpopulationstag = a

tag = b

h: assortativity

Replicator dynamics

• Symmetric case

– Full theoretical analysis (global analysis)

• Asymmetric case– Best-response theory

(local analysis only)– Numerical simulations

note: no tag evolution

6 vars, 4 dim

Symmetric case

μ phenotype

-1 no donate (D)

0 tag user

1 donate (C)

Small q Intermediate q Large q

cbq

qcb

qcA

11

is the condition for tag users to emerge.

With assortativity h

q = 0.5, h = 0 q = 0.5, h = 0.8

b = 1, c = 0.3

Asymmetric case (best response)

μ phenotype

-1 no donate (D)

0 tag user

1 donate (C)

Apb 1

.

1

,1

1

htht

phthtAX

qcb

qcA

b

Among 9 pure strategies, only (μa,μb)=(-1,-1), (-1,0), (0,-1), (0,0), and (1,1) are viable.

Apb 1

Basin areas (numerical)

q

(-1, 0)(0, -1)

(1, 1)(-1, -1)

(0, 0)

μ phenotype

-1 no donate (D)

0 tag user

1 donate (C)

Best response (continuous tag)

• Any μi = μ is ESS if bq>c

• If μi is uniformly distributed,

optimal μ

q

1.2

b/c=4

2 cbq

cbqqcb

cbq

opt

opt

,0

,

Numerical simulations

q

μ

noiseless case

noisy case

n = 800

b = 1

c = 0.3

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?