Collaboration in social networks: Incentives and topology Luca Dall’Asta, Politecnico Torino Matteo Marsili Abdus Salam ICTP, Trieste and Paolo Pin Dept. Economics, Universita’ di Siena Luca Dall’Asta, Matteo Marsili, and Paolo Pin Collaboration in social networks PNAS 2012 Sunday, May 27, 12
The puzzle of cooperation Why do we see so much cooperation around? Failed states, why do societies collapse? Will Euro collapse if Greece drops out? Sunday, May 27, 12
Much has been written on the emergence of cooperation on networks Repeated games, reputation and trust (Myerson 1991) Endogenous network games (Vega-Redondo 2007, Jackson 2008, Goyal 2009) Repeated games on evolving networks (Ellison 1994, Haag Lagunoff 2006, Vega-Redondo 2006). Cooperation in evolutionary games without mutation (Boyd 1999, Hofbauer Sigmund 2003, Poncela et al 2010) Repeated games and punishment on specific structures (Eshel et al 1998, Haag Lagunoff 2007, Fainmesser 2009, Karlan et al 2009) Focus here: social network = pattern of repeated interactions repeated interaction = forward looking behavior collaboration = incentives + credibility of threats How difficult is this in large games on complex structures? Sunday, May 27, 12
Outline The prisoners dilemma Collaboration in repeated interaction: 2 players Collaboration is supported by credible threats of punishment Collaboration in N players games on a network: Local contribution game Conditional collaboration has to be reciprocal and limited to a subset of neighbors How does collaboration depend on incentives and topology? Collaborative equilibria are subgraphs of the social network The complexity of collaboration: Counting collaborative equilibria with message passing Conclusions Sunday, May 27, 12
Defection is the only possible outcome in one shot prisoner’s dilemma C (s=1) D (s=0) C (s=1) 1-x, 1-x -x, 1 1, -x 0, 0 D (s=0) N players on graph G=(N,L) Each player either cooperates (C) or defects (D) with all neighbors X u i ( s i , s − i ) = − X i s i + s j Payoff: 1 for each neighbor that collaborates minus X i (=cost of collaboration) j ∈ ∂ i All D (s i =0) is the only Nash equilibrium s i = 0 , 1 Sunday, May 27, 12
N=2: When the game is played many times cooperation is possible, among other things Strategies become plans of actions, decided at time 0, to optimize future payoffs ∞ � ⇥ s ( t ) i , s ( t ) ⇤ d t u i U i = (1 − d ) d ∈ [0 , 1] , − i t =0 u 2 Cooperation under trigger strategies T: T= {start with C; C as long as opponent plays C, D forever, if opponent plays D} (C,C) 1-x If d is large enough, (T, T) is a Nash equilibrium Folk’s theorem: many other outcomes can be supported as a Nash equilibrium ! d=1 in what follows 1-x u 1 (D,D) Sunday, May 27, 12
But threats should be credible N=3 T T Is it credible that 1 and 2 1 2 punish 3? Not if u 1 (C,C,D) > u 1 (D,D,D) ! ? Players need to condition C only 3 to a subset of their neighbors If i conditions on j, j should condition on i Emergent heterogeneity Sunday, May 27, 12
? 11 18 16 15 29 21 19 10 37 33 8 39 32 26 13 24 7 2 27 35 1 14 - Trees 12 22 9 36 38 4 3 34 - Graphs 31 5 28 30 40 17 6 25 20 23 Sunday, May 27, 12
On trees, Nash equilibria are subtrees Given an undirected tree G=(N,L) k i = | ∂ i | = degree of node i m i = smallest integer larger than X i c i = number of collaborators in ∂ i u i ( s i , s − i ) = c i − X i s i Any collection of disjoint undirected 32 subgraphs Γ =(V, Λ ) of G is a collaborative 2 27 11 19 equilibrium where all i ∈ V cooperate 35 24 25 1 conditionally to neighbors in Γ and | ∂ i ∩ Λ |=m i 23 40 10 36 34 9 16 31 12 Incentives: i ∈ V c i - X i ≥ c i -m i ⇒ m i ≥ X i 5 6 37 26 29 j 13 17 20 18 3 38 21 Reciprocity: i,j ∈ V, if j does not punish i 39 28 33 7 ⇒ i should not punish j when j defects 22 4 i 8 15 30 k Credibility: 14 i,k ∈ V, (i,k) ∈ Λ if k defects c i - 1 - X i < c i - m i ⇒ m i < X i +1 Sunday, May 27, 12
On generic graphs cascades of defection make things more complex Indirect defections: As a result of the defection of j ∈ ∂ i other neighbors k ∈ ∂ i may also defect because of loops A collection of disjoint undirected subgraphs Γ =(V, Λ ) of G is a collaborative equilibrium where all i ∈ V cooperate conditionally to neighbors in Γ and | ∂ i ∩ Λ |=m i provided i) the indirect effects caused by the defection of all j ∈ ∂ i ∩ Λ have the same consequence of the defection of i itself. i) holds provided removing i from V does not disconnect Γ ⎡ X 3 ⎤ = 2 ⎡ X 2 ⎤ = 2 Works on trees, for dimers and loops, for the complete graph Likely works on random graphs and on ⎡ X 1 ⎤ = 2 dense graphs ⎡ X 4 ⎤ = 1 ⎡ X 5 ⎤ = 1 counter example Sunday, May 27, 12
Nash equilibria on random graphs Sunday, May 27, 12
Regular random graphs: k i =k, X i =X for all i k=4 5 28 8 18 16 q=1 40 29 6 7 6 12 40 34 28 X ≤ 1 dimers 20 11 35 38 8 9 27 14 2 21 17 15 4 39 10 29 22 19 1 < X ≤ 2 circuits 12 24 1 36 24 9 19 23 3 37 3 1 22 30 4 17 20 14 10 2 ... 11 37 26 26 33 13 k=4 13 23 30 25 21 25 32 39 15 34 16 31 18 36 33 q=2 27 7 35 32 38 5 31 k=4 11 q-1< X ≤ q q-regular subgraphs 18 16 q=3 15 29 21 19 10 ... 37 33 8 39 32 26 13 24 7 2 27 35 1 14 12 22 9 36 38 4 3 k-1< X ≤ k back to dimers 34 31 5 28 30 40 17 6 25 20 23 Do NE exist? How many? How hard is it to find them? Circuits: Marinari, Monasson, Semerjian 2006 q-regular subgraphs: Pretti, Weigt 2006 Sunday, May 27, 12
Counting NE by message passing x i → j = 1 if i conditions C on j, x i → j = 0 otherwise - there are m i -1 k ∈ ∂ i /j with x k → i =1 ⇒ x i → j =1 - m i k ∈ ∂ i /j with x k → i =1 ⇒ x i → j =0 - no k ∈ ∂ i /j with x k → i =1 ⇒ x i → j =0 k Marginals: µ i → j = P { i ∈ V, i punishes j } i j Circuits: Marinari, Monasson, Semerjian 2006 q-regular subgraphs: Pretti, Weigt 2006 Sunday, May 27, 12
Message passing equations: e − ✏ Z m i − 1 N i \ j → i µ i → j = N i \ j → i + e − ✏ Z m i − 1 N i \ j → i + e − ✏ Z m i Z 0 k N i \ j → i Z q X Y Y V → i = (1 − µ k → i ) I | U | = q µ j → i i U ⊆ V j ∈ U k ∈ V/U e − ✏ Z � i N i → i P { i ∈ C } = Z 0 N i → i + e − ✏ Z � i j N i → i µ i → j µ j → i P { i ∈ Γ j } = µ i → j µ j → i + (1 − µ i → j )(1 − µ j → i ) . Fixed point ⇒ number of subgraphs (entropy) Circuits: Marinari, Monasson, Semerjian 2006 q-regular subgraphs: Pretti, Weigt 2006 Sunday, May 27, 12
Regular random graphs: dimers (m i =1) Exponentially many NE’s 1 1.4 ρ typ (K) s typ (K) 0.8 1.2 0.6 s( ρ ) 1 0.4 K = 4, m = 1 0.8 entropy s( ρ ) 0.2 λ 1 ( ρ ) 0.6 λ 2 ( ρ ) 0 0 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 18 20 ρ K ρ = fraction of cooperators NE ∃ ∀ ρ ∈ [0,1] s( ρ ) = log(number of NE| ρ )/N Sunday, May 27, 12
Regular random graphs: circuits (m i =2) Exponentially many NE’s 1 2 entropy s( ρ ) λ 1 ( ρ ) 0.8 λ 2 ( ρ ) 1.5 0.6 K = 3, m = 2 s( ρ ) 1 0.4 0.5 0.2 ρ typ (K) s typ (K) 0 0 0 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 16 18 20 ρ K ρ = fraction of cooperators NE ∃ ∀ ρ ∈ [0,1] s( ρ ) = log(number of NE| ρ )/N (Marinari, Monasson, Semerjian 2006) Sunday, May 27, 12
Regular random graphs: m i =3 1 1 entropy s( ρ ) ρ typ (K) λ 1 ( ρ ) 0.8 0.8 λ 2 ( ρ ) 0.6 K = 4, m = 3 0.6 s( ρ ) 0 5 10 15 20 3 0.4 2.5 2 s typ (K) 1.5 0.2 1 0.5 0 0 0.6 0.8 1 5 0 10 15 20 ρ K Exponentially many NE’s NE ∄ ∀ ρ < ρ c NE are non-local and fragile (Pretti, Weigt 2006) Sunday, May 27, 12
Heterogeneous random graphs 0.7 Erdös-Rényi: E[k]=4 0.6 0.5 x=0.1 0.4 X i = xk i ! ( " ) x=0.5 x=0.3 0.3 0.2 x=0.7 x=0.72 0.1 1 x = 0. 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0,8 " 0,6 Scale free: hubs collaborate more 0,4 P(k) likely than spokes P C (k) 0,2 0 0 10 20 30 40 50 60 70 k Sunday, May 27, 12
Assortative networks are more conducive to collaboration 0,8 uncorr. disass. assort. 0,6 s( ρ ) 0,4 0,2 Scale free network P(k)~k -2.5 0 0 0,2 0,4 0,6 0,8 1 X i = xk i , x=0.1 ρ Sunday, May 27, 12
Conclusions: Theory Collaboration in repeated prisoners dilemma as a graph theoretical problem: 1- make sure enough neighbors collaborate 2- not credible to monitor more neighbors 3- checks should be reciprocal If incentives to defect (x) is small then cooperation is easy is large i) collaboration requires critical mass ii) Nash equilibria are fragile iii) effect of defection are non-local Topology: Collaboration is easier on i) trees ii) densely connected graphs Collaboration is harder on networks which can be disconnected (e.g. quasi 1d graphs) Sunday, May 27, 12
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