Cooperation, Reputation & Gossiping

V.A. Traag1, P. Van Dooren1, Y.E. Nesterov2

1ICTEAM

Universite Catholique de Louvain

2CORE

Universite Catholique de Louvain

15 April 2011

Motivation

Indirect reciprocity

• Indirect reciprocity ‘good’ explanation for human cooperation.

• Usual approach: reputation dynamics.

• Cavity: spreading of reputation tacitly assumed.

Gossiping, our approach

• Make spreading of reputation explicit: gossip.

• What is result of gossiping?

• Who will cooperate with whom?

Proposed model

Basics

1 Each agent has a reputation of the other: Rij(m)

2 Everybody plays and cooperates/defects based on localreputation

3 Everybody gossips the result of the interaction

4 New reputation Rij(m + 1) based on:◮ Own observation,◮ Gossip.

Decision to cooperate

The decision to cooperate αij(m) =

{0 if Rij(m) < 01 if Rij(m) ≥ 0

Gossiping

Consider all neighbours k when updating the reputation Rij

i j

k

The link tobe updated.

Does i ‘like’ k?

Will k gossip to i?

What actionhas j takento k?

Social strategy

G B

C G B

D B G

Reputation of k , or αik(t).

Action of j , or αjk(t)

Action is considered aseither Good or Bad

Social strategy

• Cooperation vs. good agent and defection vs. bad agent is good

• Change in reputation due to gossiping with neighbour k

∆Sij(k , m) = αki (m)(2αik(m) − 1)(2αjk(m) − 1)

Individual strategy

C D

C + −

D − +

Action of j , or αji (m).

Action of j , or αij(m)

Action is considered aseither Good or Bad

Individual strategy

• +1 for ‘good’ actions, −1 for ‘bad’ actions to reputation

• We currently study WSLS-like: Consider CC and DD as good.

∆Iij(m) = (2αij(m) − 1)(2αji (m) − 1)

Reputation dynamics

Combine individual & social strategies

Combine with social influence parameter 0 ≤ λ ≤ 1

∆Rij(m) = (1 − λ)

Individual strategy︷ ︸︸ ︷

(2αij(m) − 1)(2αji (m) − 1) +

λ1

n − 2

∑

k 6=i ,j

αki (m)(2αik(m) − 1)(2αjk(m) − 1)

︸ ︷︷ ︸

Social strategy

Reputation dynamics

Rij(m + 1) = Rij(m) + ∆Rij(m)

Cooperative fixed points

Fixed point

• For which networks do we have αij(m + 1) = αij(m)?

• Good reputation remains good, bad reputation remains bad

Undirected case

• If αij(m) = αji (m), fixed points are groups

• Cooperate within groups, defect between groups

• Implies it is (weakly) social balanced

• Can have q groups if

λ >q

q + 1

More social influence may lead to more fragmented cooperation.

Evolutionary dynamics

Four different regimes (Cooperate with prob p on first round)

p < 1/2 p > 1/2

λ < 1/2 Individualistic prejudiced

• Defect vs. cooperators

• Cycles of cooperation vs.defectors

Individualistic trusting

• Cooperate vs.cooperators

• Cycles of cooperationvs. defectors

λ > 1/2 Social prejudiced

• Cooperate vs. cooperators(except second round)

• Defect vs. defectors(except second round)

Social trusting

• Cooperate vs.cooperators

• Defect vs. defectors

Phase portraits Individual

C D

Gossipers

Individual PrejudicedC D

Gossipers

Individual Trusting

In ‘friendly’ environment, being individually prejudiced pays off.

Phase portraits Social

C D

Gossipers

Social PrejudicedC D

Gossipers

Social Trusting

In ‘hostile’ environment, being socially trusting pays off.

Conclusions

Proposed model

• Proposed model for gossiping and reputation dynamics

• Interesting possible cooperative network structure

• Evolutionary stable for some parameter range

• More socially oriented strategy could have developed fromindividual strategy

Shortcomings

• Actual convergence to fixed point not investigated

• Characterize directed fixed points

• Evolutionary dynamics investigated in limit of large n

• Interact all-to-all unrealistic, e.g. restrict to graph

• Gossip perhaps passed on further (cascades of gossip)

Thank you for your attention.

Questions?