Cooperation in Anonymous Dynamic Social Networks

Brendan LucierUniversity of

Toronto

Brian RogersNorthwestern

University

Nicole ImmorlicaNorthwestern

University

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iLikeDogs

XcutieXSephiroth97

Nice_guy_21

jdhxy

John_Doe

CatLover

trustw0rthy

Jake93726

iLikeDogs

XcutieXSephiroth97

Nice_guy_21

jdhxy

John_Doe

CatLover

trustw0rthy

Jake93726

Explain cooperative behaviour

Goal:

Prisoner’s Dilemma on an Anonymous Network

Prior Work

• [Datta’93], [Ellison’94], [Ghosh-Ray’96], [Kranton’96], [Friedman-Resnick’01], ...

• Common idea: Agents apply strategies that indirectly punish defecting behaviour.

– E.g. contagion of defection, trust-building phase for new links.

• Generally supports only fully cooperative behaviour at equilibrium.

This Work

• We consider a network formation process that runs simultaneously with the prisoner’s dilemma game.

• There are equilibria in which cooperation arises naturally from the interplay between the dilemma game and the network formation game.– Strategies at equilibrium are extremely simple.– Supports stable co-existence of cooperation and defection.

Main idea: defection stunts neighbourhood growth, while cooperation leads to more opportunities to interact with other cooperators.

1. Link Proposal: Nodes sponsoring fewer than K links can propose new links to partners chosen randomly from the population.

2. Link Acceptance: Nodes can accept or reject each proposed link.3. Stage Game: Each node chooses a single action: cooperate or

defect. Prisoner’s Dilemma game played with each neighbour and payoffs are realized.

4. Link Severance: Nodes choose whether or not to sever any incident links.

5. Player Exit: Each node exits the system with probability 1-δ, to be replaced by a new player.

The Model• Directed graph. Each node sponsors up to K links.• Game proceeds in discrete rounds. On each round:

1, 1 -b, 1+a

1+a, -b 0, 0

The Model

• Agents optimize for present value of expected lifetime payoffs. Time discounting is due to exit probability.

• Anonymity: agents are unaware of their partner’s identities, and in particular whether a new partner has been encountered previously.

• Quantity of interest: fraction q ϵ [0,1] of agents who choose cooperation in each round.

Properties of Strategies

A strategy that is...

1. Unforgiving: immediately breaks the link to any partner that defects.

2. Trusting: always accepts proposed links, and keeps the link with any partner that cooperates.

3. Consistent: either cooperates every turn (cooperator), or defects every turn (defector).

Main Results:

1. There are (sequential) equilibria in which all agents apply UTC strategies, and which support either full or partial cooperation.

2. We determine the fraction of agents that will choose cooperation at equilibrium, as a function of the model parameters.

Characterizing UTC Equilibria

• In a UTC strategy, the only choice is between cooperation and defection, made at birth.

• What is the expected utility of each choice?• Idea: Assume q is fixed for all time.– Implies a steady-state of system properties, such

as the number of links proposed each round.• Compute lifetime utilities in this steady-state

as a function of q: uC(q) and uD(q).

Characterizing UTC Equilibria• Utility of choosing cooperation:

q

uD(q)

q

uC(q)

• Utility of choosing defection:

Analysis

• All-Defect (q=0) is an equilibrium in each case.• All-Cooperate (q=1) is an equilibrium in case 2.• Where the curves cross, there is an equilibrium in

which defectors and cooperators coexist.• An equilibrium at q+ is stable; an equilibrium at q- is not.

q q q

Case 1 Case 2 Case 3

q+q- q-

Main Results:

1. There are (sequential) equilibria in which all agents apply UTC strategies, and which support either full or partial cooperation.

2. We determine the fraction of agents that will choose cooperation at equilibrium, as a function of the model parameters.

UTC Strategies RevisitedWhy is it an equilibrium for agents to play unforgiving, trusting, and consistent strategies?

Defectors will always defect, so it is optimal

to break links.

Cooperators will always cooperate, so it is optimal

to keep links. ?Accept proposed links if

expected utility is positive.

Claim: always if q > 2/3

Potential deviation: an agent cooperates to build up social capital, then “cashes in” by defecting and starting over.

Consistency Revisited

Intuition: a cooperator generates utility by building a network of partners (at a cost), then gaining benefit from those links.If an agent could improve utility by switching to defection, it must have been even better to defect every turn.

Lemma: If other agents play UTC strategies, and (1+b) / (1+a) ≥ 1 – (1-q)δ2,

then a consistent strategy maximizes expected utility.

Survival rateCoop rateGame utilities

Note: sufficient to have b ≥ a.

Conclusions• We proposed a model for repeated Prisoner’s Dilemma in an

anonymous social network with sustained interactions. • Our model supports (full and partial) cooperation at equilibrium

with simple strategies.• Key insight: interaction between network formation and

behaviour choices.

Open Questions:• What if agents are heterogeneous?• Different network formation models (preferential attachment,

local search)?• Other applications of the interaction between network games and

network formation.

Thank You

1. Link Proposal: Nodes sponsoring fewer than K links can propose new links to partners chosen randomly from the population.

2. Link Acceptance: Nodes can accept or reject each proposed link.3. Action: Each node chooses an action: cooperate or defect.4. Payoffs: Prisoner’s Dilemma game played with

each neighbour. Payoffs awarded.5. Link Severance: Nodes choose whether or not

to sever any incident links.6. Player Exit: Each node exits the system with

probability 1-δ, to be replaced by a new player.

The Model• Directed graph. Each node sponsors up to K links.• Game proceeds in discrete rounds. On each round:

1, 1 -b, 1+a

1+a, -b 0, 0