Presentation at Alife 2011, Paris, April 15, 2011.

1. Cooperation, Reputation & Gossiping
V.A. Traag1, P. Van Dooren1, Y.E. Nesterov2
1ICTEAM
Universit´e Catholique de Louvain
2CORE
Universit´e Catholique de Louvain
15 April 2011

2. 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?

3. Proposed model
Basics
1 Each agent has a reputation of the other: Rij (m)
2 Everybody plays and cooperates/defects based on local
reputation
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) < 0
1 if Rij (m) ≥ 0

4. Gossiping
Consider all neighbours k when updating the reputation Rij
i j
k
The link to
be updated.
Does i ‘like’ k?
Will k gossip to i?
What action
has j taken
to k?

5. 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 as
either 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)

6. Individual strategy
C D
C + −
D − +
Action of j, or αji (m).
Action of j, or αij (m)
Action is considered as
either 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)

7. 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) +
λ
1
n − 2
k=i,j
αki (m)(2αik(m) − 1)(2αjk(m) − 1)
Social strategy
Reputation dynamics
Rij (m + 1) = Rij (m) + ∆Rij (m)

8. 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.

9. 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 cooperation
vs. 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

10. Phase portraits Individual
C D
Gossipers
Individual Prejudiced
C D
Gossipers
Individual Trusting
In ‘friendly’ environment, being individually prejudiced pays off.

11. Phase portraits Social
C D
Gossipers
Social Prejudiced
C D
Gossipers
Social Trusting
In ‘hostile’ environment, being socially trusting pays off.

12. 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 from
individual 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)

13. Thank you for your attention.
Questions?