Cooperation in Social Dilemmas through Position Uncertainty

Transcription

1 Cooperation in Social Dilemmas through Position Uncertainty Andrea Gallice and Ignacio Monzón Università di Torino and Collegio Carlo Alberto North American Summer Meetings of the Econometric Society St. Louis - June 16th, 2017 Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 1 / 27

2 Introduction and Motivation Social Dilemmas Conflict between individual incentives and collective interests Examples: Voluntary provision of a public good Prisoners dilemma Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 2 / 27

3 Introduction and Motivation Public Good Provision Charity seeks voluntary donations for a large project Several wealthy donors: They obtain utility from project But would rather have somebody else paying for it How should the charity contact them? Simultaneously or sequentially? If sequentially, Letting them know their order? Informing them of the actions of others before? Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 3 / 27

4 Introduction and Motivation Our Contribution Full contribution not expected outcome: Finite number of self-interested agents One-shot interaction Our novel mechanism achieves full contribution: Donors are contacted sequentially They do not know their order in the sequence (position uncertainty) They observe the actions of some of their predecessors Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 4 / 27

5 Introduction and Motivation Intuition: An example Example: Each agent observes her immediate predecessor. Assume simple profile of play: Agents defect if they observe defection They contribute otherwise First agent happy to contribute, so everybody after her does so Last agent defects Contribution unravels! What if agents do not know their position? If return from contributions is high enough, contribute Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 5 / 27

6 Introduction and Motivation Related Literature 1 Infinitely repeated games: Friedman [REStud, 1971], Dal Bó and Fréchette [WP, 2016] 2 Finitely repeated interactions: Kreps, Milgrom, Roberts and Wilson [JET, 1982], Andreoni and Miller [EJ, 1993], Fehr and Gächter [AER, 2000] 3 Sequential moves in PGG: Varian [JPubE, 1994], Andreoni, Brown and Vesterlund [GEB, 2002], Andreoni [JPE, 1998], Romano and Yildirim [JPubE, 2001], Vesterlund [JPubE, 2003] 4 Position Uncertainty: Nishihara [ET, 1997], Gershkov and Szentes [JET, 2009], Monzón and Rapp [JET, 2014], Doval and Ely [WP, 2016], Salcedo [WP, 2017] Details Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 6 / 27

7 The Model Position Uncertainty n potential contributors, indexed by i {1,...,n} n positions in a sequence, indexed by t {1,...,n} Agents do not know their position in the sequence (all permutations are equally likely) Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 7 / 27

8 The Model Actions and Payoffs Agent i chooses action a i {C, D}, either to contribute one unit to the common pool or to contribute zero (defect) G i j =i 1 { a j = C } : number of opponents who contribute Payoffs: u i (C, G i ) = r n (G i + 1) 1 and u i (D, G i ) = r n G i Total amount invested multiplied by return from contributions r > 1, and then split among all agents For any G i, defection dominates contribution (r < n). Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 8 / 27

9 The Model Sampling Agents observe the actions of their m immediate predecessors The sample ξ = (ξ,ξ ) Ξ specifies: The number of people the agent observes: ξ {0,...,m} How many of those contributed: ξ {0,...,ξ } Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 9 / 27

10 The Model Strategies and Equilibrium Strategy σ i (C ξ) : Ξ [0,1], probability of playing contribution Conditional on sample (observed) Not on position (unobserved) Agents form beliefs µ about their position and the history of past play Equilibrium concept: sequential equilibrium (σ,µ ) Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 10 / 27

11 The Model Beliefs: Example. All nodes with 3 agents Three possible samples with m = 1:. 1 C D C D 3 3 C D 3 C D C D C D C D Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 11 / 27

12 The Model Beliefs: Example. All nodes with 3 agents Three possible samples with m = 1:. ξ = (0,0) 1 C D C D 3 3 C D 3 C D C D C D C D Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 11 / 27

13 The Model Beliefs: Example. All nodes with 3 agents Three possible samples with m = 1:. ξ = (0,0) 1 ξ = (1,0) 2 C D 2 3 C D 3 3 C D 3 C D C D C D C D Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 11 / 27

14 The Model Beliefs: Example. All nodes with 3 agents Three possible samples with m = 1:. ξ = (0,0) 1 ξ = (1,0) 2 C D 2 ξ = (1,1) 3 C D 3 3 C D 3 C D C D C D C D Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 11 / 27

15 Results Our Main Result: Full Contribution Proposition 1: Full Contribution with Position Uncertainty ( ) a If r n m+1 m 1, then there exists an equilibrium in which all agents contribute. b If instead r < 2 ( ) 1 + n m+1 m 1, then no agent contributes in equilibrium. Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 12 / 27

16 Results Comments on Proposition 1.a. 1 Simple strategy in equilibrium: contribute unless you observe defection 2 Equal probability on all deviations 3 We deal with cases m 2 and m = 1 separately Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 13 / 27

17 Results Sample size m 2 Samples sizes m 2. Equilibrium Let Ξ C denote all samples without defection: ξ = (ξ,ξ ) Ξ C ξ = ξ. Lemma 1: Full Contribution with Sample Size m 2 Consider the following profile of play: { σi 1 if ξ Ξ C (C ξ) = 0 if ξ Ξ C for all i I Then ((σ,µ ) is a ) sequential equilibrium of the game whenever r n m+1 m 1. Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 14 / 27

18 Results Sample size m 2 Lemma 1. No defectors Agent observes ξ Ξ C on the equilibrium path! If she contributes everybody does so. Payoffs rn n 1 = r 1 If she defects all her successors defect. Payoffs n r n+m 1 2 ( ) agent contributes whenever r n m+1 m 1 (Same condition guarantees contribution if ξ < m) Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 15 / 27

19 Results Sample size m 2 Lemma 1. Defectors Agent observes defection off the equilibrium path! Equilibrium strategy calls for defection Key intuition: no overturning agents who observe defection cannot prevent successors from defecting Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 16 / 27

20 Results Sample size m 2 No overturning. Why? Agent who observes defection in position t If successor receives sample ξ t+1 Ξ C defection never stops ξ t+1 has defection for sure when Sample ξ t Ξ C has less than m actions Sample ξ t Ξ C has more than one person defecting Hope only for ξ t = (m,m 1): m observed actions only one defection Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 17 / 27

21 Results Sample size m 2 No overturning. Why? (cont) Let m = 2 and ξ t = (2,1). Then, either h t = (...,C, D) or h t = (..., D,C) If a t = C, h t+1 = (..., D,C) or h t+1 = (...,C,C) Overturning only possible if sole defector is in position t m But histories with sole defector in position t m have zero probability h t = (...,C, D) can have only one mistake h t = (..., D,C) has at least two mistakes Agent who receives ξ t = (m,m 1) cannot stop defection. Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 18 / 27

22 Results Sample size m = 1 Sample size m = 1. Overturning When m = 1, overturning is possible! An agent can prevent further defection by contributing Then, our simple pure equilibrium cannot be sustained with high r: Agents would contribute when they observe defection A mixed equilibrium sustains full contribution: Agents forgive defection with probability γ [0,1) Forgiveness makes defection more attractive Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 19 / 27

23 Results Sample size m = 1 Sample size m = 1. Mixed Equilibrium Lemma 2: Full Contribution with Sample Size m = 1 Consider the following profile of play: { σi 1 if ξ {(0,0),(1,1)} (C ξ) = γ if ξ = (1,0) for all i I For any r 2 there exists γ [0,1) such that (σ,µ ) is a sequential equilibrium of the game. Figure Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 20 / 27

24 Results No contribution for low r Proposition 1.b. No contribution for low r Lemma 3: No Contribution for Low r ( ) Whenever r < n m+1 m 1, no agent contributes in equilibrium. Why contribute? To get successors to contribute themselves. When is contribution most tempting? All successors contribute when agent contributes, but defect otherwise Benefit = r/n(# successors who contribute +1) Cost = 1 # successors who contribute n/r 1. However, on average the number of future contributors is n/2 Then, when r is low, contribution cannot emerge. Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 21 / 27

25 Results Robustness Noisy Information on Positions Agents may have some information about their positions. Agent in position t receives noisy signal S t : { t with probability λ ( 1 S t = n,1) τ with probability (1 λ)/(n 1) for all τ = t The further in the sequence, the more attractive defection becomes Worst case scenario: Agent observes sample of contribution and signal s = n. Does she still contribute? Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 22 / 27

26 Results Robustness Noisy Information on Positions (cont) Lemma 4: Noisy Information on Positions Let m 2 and consider the profile σ i (C ξ) = { 1 if ξ Ξ C 0 if ξ Ξ C for all i I Then, (σ,µ ) is a sequential equilibrium of the game if and only if [ r 2n 2 + ] (n 1 m) (n m) 1 n 1/(1 λ) m When λ = 1/n, condition as in Proposition 1 The larger the signal quality λ, the larger the return to contributions r needed to sustain full contribution. Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 23 / 27

27 Results Robustness Noisy Information on Positions (cont) λ 1 Equilibrium with full contribution 1 n ( ) n m+1 m 1 n r Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 24 / 27

28 Results Robustness Prisoners Dilemma Agents choose sequentially whether to cooperate or defect Agent i obtains the sum of the payoffs from n pairwise interactions Usual (per pair) payoffs: C D C 1 l D 1 + g 0 Agent i receives: u i (C, G i ) = G i (n 1 G i ) l or u i (D, G i ) = (1 + g)g i. Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 25 / 27

29 Results Robustness Prisoners Dilemma Cooperation in the Prisoners dilemma Full cooperation is an equilibrium whenever g 1 2m n + m 1 No restriction on l since on the equilbrium path nobody defects Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 26 / 27

30 Conclusion Conclusion Simple mechanism to achieve full contribution in a public good game Even with finite number of rational agents and one-shot interactions Our mechanism features sequential decisions, position uncertainty, and partial observation of past actions Same mechanism attains cooperation in other social dilemmas (PD) Results robust to some information on positions Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 27 / 27

31 Related Literature I: How to Achieve Cooperation? 1 Infinitely repeated interactions: Rational agents cooperate to induce others to do the same Friedman [REStud, 1971], Dal Bó and Fréchette [WP, 2016] 2 Finitely repeated interactions: Rational agents should not cooperate in the PD. (backwards induction) Small likelihood of behavioral agents: Kreps, Milgrom, Roberts, and Wilson [JET, 1982] Warm glow: Andreoni and Miller [EJ, 1993]. Punishment: Fehr and Gächter [AER, 2000] Our paper: finite, one-shot games with fully rational agents Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 1 / 4

32 Related Literature II: Sequential Contribution in PGG Sequential moves lower contributions: Varian [JPubE, 1994] Experimental evidence shows the opposite: Andreoni, Brown and Vesterlund [GEB, 2002] Sequentiality allows for the release of valuable info. This is useful: to coordinate when there are multiple equilibria: Andreoni [JPE, 1998] when there are warm-glow effects: Romano and Yildirim [JPubE, 2001] when the quality of the PG is uncertain: Vesterlund [JPubE, 2003] Our paper: sequential structure, partial observations of predecessors moves, position uncertainty. Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 2 / 4

33 Related Literature III: Position Uncertainty Monzón and Rapp [JET, 2014]: Observational Learning with PU Standard approach in information design: principal can choose flexibly which information about past actions to communicate Nishihara [ET, 1997]: Cooperation in a PD can arise if all agents are informed immediately when defection occurs Gershkov and Szentes [JET, 2009]: Agents can be convinced to acquire and report costly information in committees Doval and Ely [WP, 2016], Salcedo [WP, 2017]: Recommendations a principal should send to make agents choose the socially optimal action Our paper: Agents directly observe the actions of some of their predecessors. Natural environment. Harder off-the-equilibrium beliefs Return Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 3 / 4

34 Probability of Forgiveness as a Function of r γ 1 pure equilibrium Example with n = 10 and m = 1 10 r Return Gallice & Monzón (U. Torino, Collegio Carlo Alberto) Cooperation through Position Uncertainty NASMES St Louis 4 / 4