Cooperation, Reputation & Gossiping
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Transcript of Cooperation, Reputation & Gossiping
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Introduction Indirect Reciprocity Gossip Results
Cooperation, Reputation & Gossiping
V.A. Traag1, P. Van Dooren1, Y.E. Nesterov2
1ICTEAMUniversite Catholique de Louvain
2COREUniversite Catholique de Louvain
15 April 2011
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Introduction Indirect Reciprocity Gossip Results
Outline
1 Introduction
2 Indirect Reciprocity
3 Gossip
4 Results
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Introduction Indirect Reciprocity Gossip Results
Cooperation
Cooperation (and defection)
• Organizations (also Wikipedia, open source software, . . . )◮ Why do people contribute?
• Worker ants in colonies◮ Why do workers help without individual benefit?
• Prudent parasites in hosts◮ Why do parasites not replicate faster?
• Human body◮ Why do cells not replicate faster?
Central question
If defecting (not cooperating) is a real option, why (and how) hascooperation evolved?
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Introduction Indirect Reciprocity Gossip Results
Formal cooperation (and defection)
Prisoner’s Dilemma
• The game knows two options, donating or not donating.
• Donate at a cost c > 0 to benefit someone else with benefitb > c .
• Agents are paired, and play a round of donating or not.
• Cooperators C donate, defectors D do not donate.
This can be summarized in the payoff matrix
A =
(C D
C b − c −c
D b 0
)
Defectors dominate
Whatever strategy you encounter (C or D), always better to defect.
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Introduction Indirect Reciprocity Gossip Results
Dynamical View
• Natural to model game dynamics in an evolutionary context.
• Survival of the fittest (fitness = payoff).
Definition (Replicator equation)
Population with i = 1, . . . , n different mixed strategies pi
xi Relative abundance (frequency)
p =∑
i pixi Average strategy
fi = p⊺
i Ap Expected payoff
f = p⊺Ap Average payoff
Evolution of the population given by
xi = xi (fi − f ) = xi ((pi − p)⊺Ap).
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Introduction Indirect Reciprocity Gossip Results
Overview
What are possibly mechanisms to get cooperation?Payoff matrix
A =
(C D
C b − c −c
D b 0
)
Mechanisms
• Kin selection (r > cb)
Cooperate because offspring benefits of your cooperation. Basisof ‘selfish gene’, or ‘inclusive fitness’.
• Direct reciprocity (w > cb)
Cooperate because of possible future payoffs.
• Indirect reciprocity (q > cb)
Cooperate because someone else may cooperate with you in thefuture.
Nowak. Science (2006) vol. 314 (5805) pp. 1560-1563
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Introduction Indirect Reciprocity Gossip Results
Some strategies
Example (Always)
Defect/cooperate on all rounds
Other CDDDDCC
AllD DDDDDDD
AllC CCCCCCCC
Example (Tit-for-tat)
Start cooperating, then repeatopponent.
Other CDDDDCC
TFT CCDDDDC
Example (Win-Stay, Lose-Shift)
Change strategy if losing, keep itotherwise.
Other CDDDDCC
WSLS CCDCDCC
Example (Generous Tit-for-tat)
As TFT, but cooperates afterdefection with probability p.
Other CDDDDCC
GTFT CCDDCDC
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Introduction Indirect Reciprocity Gossip Results
Insufficient explanation
Why is kin selection and reciprocity not sufficient?
Insufficient explanation
• Humans cooperate also with non-kin.
• Humans cooperate in non-iterative situations.
Indirect reciprocity
• Cooperate if cooperated with others in the past.
• Brings reputation into play.
• How to respond to reputation?
• How to determine new reputation?
Nowak and Sigmund. Nature (2005) vol. 437 (7063) pp. 1291-1298
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Introduction Indirect Reciprocity Gossip Results
Image score
Definition (Image score, reputation)
• Integer status −5 ≤ Si ≤ 5 known to all.
• If cooperate increase (with 1).
• If defect decrease (with 1).
Definition (Discriminator Strategy)
• Cooperative threshold −5 ≤ kj ≤ 6.
• If status Si ≥ kj cooperate, otherwise defect.
• Strategy kj = −5 corresponds to AllC.
• Strategy kj = 6 corresponds to AllD.
Nowak and Sigmund. Nature (1998) vol. 393 (6685) pp. 573-577.
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Introduction Indirect Reciprocity Gossip Results
Other reputation dynamics
Morals
• Defecting a defector, good or bad?
• What action should be regarded as good?
• When to cooperate, when to defect?
GG GB BG BB
C ∗ ∗ ∗ ∗
D ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
Reputation of donor and recipientAction of donor
New reputation can beeither Good or Bad
Action can be eitherCooperate or Defect
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Introduction Indirect Reciprocity Gossip Results
Some reputation dynamics
GG GB BG BB
C G G G G
D B B B BImage scoring
C G G G G
D B G B BStanding
C G B G B
D B G B BJudging
C G B G B
D B B B BShunning
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Introduction Indirect Reciprocity Gossip Results
Leading eight
Best strategies
• In total 2, 048 different possible strategies.
• There are 8 strategies (leading eight) that perform best (highestpayoff, and ESS).
GG GB BG BB
C G ∗ G ∗
D B G B ∗
C D C ×
Maintainance of cooperation
Mark defectors
Punish defectors
Forgive defectors
Apologize
Ohtsuki and Iwasa. Journal of Theoretical Biology (2004) vol. 231 (1) pp. 107-120
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Introduction Indirect Reciprocity Gossip Results
Subjective reputation
Subjective reputation
• Unrealistic that everybody knows the reputation of everybody.
• Introduce a subjective (private) reputation.
• ‘Observe’ only a few interactions.
Observing
• Probability q of observing an interaction.
• Cooperation declines with lower q.
• Diverging reputations cause further errors.
• Good may defect bad, but not all agree on who’s bad.
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Introduction Indirect Reciprocity Gossip Results
Synchronize reputations
Synchronizing reputations
• Spread local information to synchronize reputations.
• Players ‘gossip’ about each other to share information.
• Start gossip, spread gossip and how to interpret gossip?
Lying, cheating and defecting
• Possibly ‘false’ gossips spread.
• Spread rumours unconditionally allows liars to invade.
• Liars cannot invade conditional rumour spreaders.
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Introduction Indirect Reciprocity Gossip Results
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 (∆Iij (m)),◮ Gossip (∆Sij (k ,m)).
Decision to cooperate
The decision to cooperate αij(m) =
{0 if Rij(m) < 01 if Rij(m) ≥ 0
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Introduction Indirect Reciprocity Gossip Results
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?
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Introduction Indirect Reciprocity Gossip Results
Social strategy
G B
C G B
D B G
Reputation of k , or αik(m).
Action of j , or αjk(m)
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)
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Introduction Indirect Reciprocity Gossip Results
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)
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Introduction Indirect Reciprocity Gossip Results
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)
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Introduction Indirect Reciprocity Gossip Results
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.
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Introduction Indirect Reciprocity Gossip Results
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
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Introduction Indirect Reciprocity Gossip Results
Phase portraits Individual
C D
Gossipers
Individual PrejudicedC D
Gossipers
Individual Trusting
In ‘friendly’ environment, being individually prejudiced pays off.
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Introduction Indirect Reciprocity Gossip Results
Phase portraits Social
C D
Gossipers
Social PrejudicedC D
Gossipers
Social Trusting
In ‘hostile’ environment, being socially trusting pays off.
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Introduction Indirect Reciprocity Gossip Results
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)
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Introduction Indirect Reciprocity Gossip Results
Thank you for your attention.
Questions?