Introduction Model Results Discussion

Observations on Cooperation Yuval Heller (Bar Ilan) and Erik Mohlin (Lund)

Erice 2017

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Introduction Model Results Discussion

Motivating Example Alice interacts with a remote trader, Bob. Both agents have opportunities to shirk/cheat. Alice obtains anecdotal evidence about Bob’s actions in a couple of past interactions Alice considers this information when deciding how to act. Alice is unlikely to interact with Bob again. Future partners may ask Bob about Alice’s behavior. Research question Can cooperation be sustained in such environments? Heller & Mohlin

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Introduction Model Results Discussion

Underlying Game: The Prisoner’s Dilemma (PD)

c c

1

d

1+g

d 1

−l

−l

0

g>0 - gain of a greedy player.

1+g 0

l>0 - loss if the partner defects. g
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Introduction Model Results Discussion

Brief Summary of Results 1

Novel behavior supports stable cooperation. (uniqueness in the restricted set of stationary strategies).

2

Stable cooperation requires observation of 2+ of interactions

3

Observation of partner’s past actions: g > l: Only defection is stable. g < l: Cooperation is stable (and robust to any noise). l +1

4

Observation of action profiles: Cooperation is stable iff g <

5

Optimal feedback: Observing partner’s actions against cooperation. Heller & Mohlin

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.

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Introduction Model Results Discussion

Observation Structure and Environment Strategies and Steady States Solution Concept

Observation Structure and Environment Basic stationary model (focus of the presentation): Each player privately observes a sample of k actions played by his partner (against other opponents). Agents are restricted to stationary strategies. IID sampling from the partner’s stationary behavior. Set of possible signals - m ∈ {0, 1, 2, ..., k} (interpreted as the number of observed defections).

Alternative model: unrestricted set of strategies, observing the last k actions. All the results hold except uniqueness. Heller & Mohlin

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Stationary Strategies Definition (Strategy - s : {0, ..., k} → ∆ ({c, d})) Mapping assigning a mixed action for each possible observation. Interpretation: The agent’s behavior conditional on the observed signal.

Strategy distribution – Distribution σ over the set of strategies (with a finite support). Interpretation: Heterogeneous population. Example of a Strategy Distribution

supp (σ ) = {su , s1 , s2 } su ≡ 50%

s1 (m) =

   c   d

σ (su ) = ε, σ (s1 ) = m=0 s2 (m) = m≥1

1−ε 6 ,

σ (s2 ) =

   c

m≤1

  d

m≥2

5·(1−ε ) 6

Consistent Signal Profile Definition (Signal profile - θ : supp (σ ) −→ ∆ (M)) θ (s) is interpreted as the distribution of signals observed by agents who are matched with a partner who plays strategy s.

Definition (Consistent signal profile ) Signal profile θ and strategy distribution σ jointly induce a behavior profile: a distribution of actions for each strategy. The behavior profile induce a signal profile of observed actions.

Consistency: The induced signal profile is θ .

A strategy distribution may admit multiple signal profiles. Ignored in this presentation.

Introduction Model Results Discussion

Observation Structure and Environment Strategies and Steady States Solution Concept

Commitment Strategies (“Crazy” Agents)

We refine our solution concept by requiring robustness to the presence of few “crazy” agents (`a la Kreps et al., 1982). Definition (Distribution of Commitments – (Sc , λ )) Sc is a finite set of commitment strategies, and λ ∈ ∆ (Sc ) is a distribution over these strategies. We assume that at least one of the commitment strategies is totally mixed.

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Introduction Model Results Discussion

Observation Structure and Environment Strategies and Steady States Solution Concept

Nash in Perturbed Environment Definition (Perturbed Environment ) A fraction ε of committed agents play a strategy according to λ ∈ ∆ (SC ). Definition (Nash equilibrium in a perturbed environment.) π (σ ∗ ) ≥ πs (σ ∗ ) for every strategy s, where π (σ ∗ ) denote the mean payoff of the 1 − ε “normal” agents, and πs (σ ∗ ) denotes the payoff to strategy s.

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Introduction Model Results Discussion

Observation Structure and Environment Strategies and Steady States Solution Concept

Perfect Equilibrium

Definition (Perfect equilibrium σ ∗ ) The limit of Nash equilibria in some converging sequence of perturbed environments.

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Introduction Model Results Discussion

Observation Structure and Environment Strategies and Steady States Solution Concept

Perfect Equilibrium

Definition (Perfect equilibrium σ ∗ ) The limit of Nash equilibria in some converging sequence of perturbed environments. Definition (Strictly perfect action a∗ ) The limit behavior of Nash equilibria in any converging sequence of perturbed environments.

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Introduction Model Results Discussion

Taxonomy of PDs Observation of Actions Other Observation Structures

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Results

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Prisoner’s Dilemma - Taxonomy Offensive (submodular, Takahashi, 2010) PD l < g : stronger incentive to defect against cooperative partner than defective partner. Defensive (supermodular) PD l > g . Acute PD g >

l+1 2 :

defection against cooperator gives more than half of

what opponent looses. Mild (mildly tempting) PD g <

l+1 2 .

Introduction Model Results Discussion

Taxonomy of PDs Observation of Actions Other Observation Structures

Stable Defection in any PD

Claim Defection is strictly perfect equilibrium action in any PD.

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Defection is the Unique Outcome in Offensive PDs Proposition Assume an offensive PD (l < g ) with observation of any number of actions. If σ ∗ is a perfect equilibrium then everyone defects.

Defection is the Unique Outcome in Offensive PDs Proposition Assume an offensive PD (l < g ) with observation of any number of actions. If σ ∗ is a perfect equilibrium then everyone defects.

Intuition: Assume to the contrary that σ ∗ 6≡ d Direct gain from defecting decreases in the partner’s prob. of defection. The indirect loss is independent of the current partner’s behavior. ⇒ Incumbents are less likely to defect when observing more defections. ⇒ If Alice always defects, she outperforms the incumbents.

Defection is the Unique Outcome in Offensive PDs Proposition Assume an offensive PD (l < g ) with observation of any number of actions. If σ ∗ is a perfect equilibrium then everyone defects.

Intuition: Assume to the contrary that σ ∗ 6≡ d Direct gain from defecting decreases in the partner’s prob. of defection. The indirect loss is independent of the current partner’s behavior. ⇒ Incumbents are less likely to defect when observing more defections. ⇒ If Alice always defects, she outperforms the incumbents.

Remark (Alternative model with unrestricted set of strategies) A weaker result: Full cooperation isn’t a perfect equilibrium.

Stable Cooperation in Defensive PD Proposition Assume g ≤ l and observing k ≥ 2 actions. Cooperation is strictly perfect. Moreover there is essentially a unique strategy distribution that supports cooperation (uniqueness relies on the restriction to stationary strategies).

Essentially Unique Stable State Everyone cooperates when observing no defections. Everyone defects when observing ≥ 2 defections. 0
1 k

of the incumbents defect when observing 1 defection

(i.e., q of the agents follow s 1 , and the remaining 1−q follow s 2 ). The value of q depends on the commitment strategies.

Introduction Model Results Discussion

Taxonomy of PDs Observation of Actions Other Observation Structures

Other Observation Structures

What happens if the signal about the partner depends also on the behavior of other opponents against her? We study three observation structures: 1

The entire action profile.

2

Mutual cooperation or not (=conflict). Signals:{CC , not − CC }.

3

Observing actions against cooperation.

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Signals:{CC , DC , CD, DD}.

Signals:{CC , DC , ?D}.

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Stable Cooperation when Observing Mutual Cooperation Proposition If players observe conflicts (i.e., CC or not) in at least two interactions, then cooperation is a perfect equilibrium iff the PD is mild (g <

l +1 2

).

Intuition Mild PDs: The perfect state is similar to the previous results. Players condition their play on the number of observed conflicts. Acute PDs: involvement in a conflict has to be punished with probability of at least 1/2. Because both players have to be punished, each conflict induces at least one additional conflict ⇒ Conflicts are “contagious”.

Stable Cooperation when Observing Action Profiles Proposition If players observe action profiles in at least two interactions, then cooperation is a perfect equilibrium iff the PD is mild (g <

l +1 2

).

Intuition The high frequency of punishments required in acute PDs implies that the partner is more likely to defect when observing mutual defection (relative to observing the partner to be the sole defector). ⇒ Agents “punish” partners who defect against another defector ⇒ destabilizes cooperation. In mild PDs, cooperation is stable (with essentially the same unique supporting behavior).

Introduction Model Results Discussion

Taxonomy of PDs Observation of Actions Other Observation Structures

Observing Actions Against Cooperation Proposition If players observe actions against cooperation (i.e., {CC , DC , ?D}) in at least two interactions, then cooperation is perfect in any Prisoner’s Dilemma.

Intuition No indirect loss of defecting against a defector, since it is not observed. Makes it easier to incentivize agents to deter defection. Providing more information to agents may harm cooperation. Heller & Mohlin

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Introduction Model Results Discussion

Related Literature and Contribution Conclusion

Related literature (Partial List): Community Enforcement 1

Contagious equilibria (e.g., Kandori 1992; Ellison, 1994).

2

Applications of belief-free equilibria (Takahashi, 10; Deb, 12).

3

Image scoring (e.g., Nowak & Sigmund, 98).

4

Exogenous reputation mechanisms (e.g., Sugden, 86; Kandori, 92).

5

Structured populations (Cooper & Wallace, 04; Alger & Weibull, 13).

6

Observation of preferences (e.g., Dekel et al., 07; Herold, 12).

Our Main Methodological Contributions Robustness to few crazy agents. Heller & Mohlin

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Companion Projects and Directions for Future Research Companion working papers: 1

“When is Social Learning Path-Dependent?”: When does a distribution of stationary strategies uniquely determines the consistent behavior?

2

“Coevolution of deception and preferences”: Players can deceive others about their preferences and intentions.

Directions for future research: Experiment to test the theoretical predictions. Realistic, yet tractable, model of online feedback. Studying non-negligible noise levels.

Introduction Model Results Discussion

Related Literature and Contribution Conclusion

Conclusion Introducing robustness against few crazy agents into the setup of community enforcement (Prisoner’s Dilemma with random matching). 1

Unique novel behavior supports stable cooperation.

2

Stable cooperation requires observation of 2+ interactions

3

Observation of partner’s past actions: g > l: Only defection is stable. g < l: Cooperation is stable.

4

Observation of action profiles: Cooperation is stable iff g <

5

Optimal feedback: Observing actions against cooperation. Heller & Mohlin

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Summary of Results - When is Cooperation Stable?

Category of PD

Mild (g <

l+1 2 )

Acute (g >

l+1 2 )

Actions

Defen.

Y

Offen.

N

Defen.

Y

Offen.

N

Conflicts

Action profiles

Y

Y

N

N

Action against Coop.

Y

Stable cooperation requires observation of 2+ interactions. Observing a single interaction: Cooperation is not stable if g > 1.

Introduction Model Results Discussion

Related Literature and Contribution Conclusion

Backup Slides

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Influence of Cheap Talk Introducing cheap-talk with unrestricted language destabilize the perfect equilibrium in which everyone defects. Experimenting agents use a secret handshake to cooperate among themselves (Robson, 1990). Implications (observation of actions + cheap-talk): Defensive PD - Only the cooperative equilibrium is stable. Offensive PD - No stable equilibrium. The population state cycles between the defective and the cooperative equilibrium (as in the one-shot PD, see Wiseman & Yilankaya, 2001).

Introduction Model Results Discussion

Related Literature and Contribution Conclusion

Steady States and Payoffs - Details

Back

Fact (standard fixed point argument) Each strategy distribution admits a consistent behavior (not necessarily unique).

Example (k = 3; Each agent plays the mode (frequently observed action).) 3 consistent behaviors: full cooperation, no cooperation, uniform mixing.

The Payoff of each incumbent strategy (s ∈ supp (σ )) and the average payoff in the population are defined in a standard way. πs (σ , η) = ∑s 0 σ (s 0 ) · π (ηs (s 0 ) , ηs 0 (s)), π (σ , η) = ∑s∈supp(σ ) σ (s) · πs (σ , η). Heller & Mohlin

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Introduction Model Results Discussion

Related Literature and Contribution Conclusion

Illustration of Stable Cooperation in Defensive PD

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Introduction Model Results Discussion

Related Literature and Contribution Conclusion

Illustration of Unstable Cooperation in Offensive PD

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