I NTRODUCTION T HE PD AND THE T RAGEDY OF THE C OMMONS ( CONT ’ D ) Prisoner B C D C -1,-1 -5,0 Prisoner A D 0,-5 -4,-4 What is the problem? In the outcome (Nash Equilibrium), they get 8 years in prison together. By playing C, they could have been getting 2 years in prison together. The incentives are not there for the “efficient” outcome to occur. ⇒ Tragedy of the Commons. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 5 / 30
I NTRODUCTION PD WITH R ANDOM I NTERACTIONS L EADS TO T RAGEDY OF THE C OMMONS Consider a large population playing a Prisoners’ Dilemma. C D C 9,9 -2,11 D 11,-2 0,0 HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 6 / 30
I NTRODUCTION PD WITH R ANDOM I NTERACTIONS L EADS TO T RAGEDY OF THE C OMMONS Consider a large population playing a Prisoners’ Dilemma. C D C 9,9 -2,11 D 11,-2 0,0 They follow different strategies and meet at random. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 6 / 30
I NTRODUCTION PD WITH R ANDOM I NTERACTIONS L EADS TO T RAGEDY OF THE C OMMONS Consider a large population playing a Prisoners’ Dilemma. C D C 9,9 -2,11 D 11,-2 0,0 They follow different strategies and meet at random. More successful strategies evolve more quickly (because of replication or imitation). HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 6 / 30
I NTRODUCTION PD WITH R ANDOM I NTERACTIONS L EADS TO T RAGEDY OF THE C OMMONS Consider a large population playing a Prisoners’ Dilemma. C D C 9,9 -2,11 D 11,-2 0,0 They follow different strategies and meet at random. More successful strategies evolve more quickly (because of replication or imitation). The unique Nash equilibrium is D-D and all individuals following D is the unique Evolutionarily Stable Strategy (ESS). HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 6 / 30
I NTRODUCTION PD WITH R ANDOM I NTERACTIONS L EADS TO T RAGEDY OF THE C OMMONS Consider a large population playing a Prisoners’ Dilemma. C D C 9,9 -2,11 D 11,-2 0,0 They follow different strategies and meet at random. More successful strategies evolve more quickly (because of replication or imitation). The unique Nash equilibrium is D-D and all individuals following D is the unique Evolutionarily Stable Strategy (ESS). A population full of D-individuals cannot be invaded by C-individuals. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 6 / 30
I NTRODUCTION PD WITH R ANDOM I NTERACTIONS L EADS TO T RAGEDY OF THE C OMMONS Consider a large population playing a Prisoners’ Dilemma. C D C 9,9 -2,11 D 11,-2 0,0 They follow different strategies and meet at random. More successful strategies evolve more quickly (because of replication or imitation). The unique Nash equilibrium is D-D and all individuals following D is the unique Evolutionarily Stable Strategy (ESS). A population full of D-individuals cannot be invaded by C-individuals. ⇒ Tragedy of the Commons. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 6 / 30
I NTRODUCTION A SSORTATIVITY AND THE T RAGEDY OF THE C OMMONS In the same PD as before.. . C D C 9,9 -2,11 D 11,-2 0,0 HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 7 / 30
I NTRODUCTION A SSORTATIVITY AND THE T RAGEDY OF THE C OMMONS In the same PD as before.. . C D C 9,9 -2,11 D 11,-2 0,0 . . . let individuals meet assortatively. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 7 / 30
I NTRODUCTION A SSORTATIVITY AND THE T RAGEDY OF THE C OMMONS In the same PD as before.. . C D C 9,9 -2,11 D 11,-2 0,0 . . . let individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 7 / 30
I NTRODUCTION A SSORTATIVITY AND THE T RAGEDY OF THE C OMMONS In the same PD as before.. . C D C 9,9 -2,11 D 11,-2 0,0 . . . let individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs. Extreme case: Cs always meet Cs! HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 7 / 30
I NTRODUCTION A SSORTATIVITY AND THE T RAGEDY OF THE C OMMONS In the same PD as before.. . C D C 9,9 -2,11 D 11,-2 0,0 . . . let individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs. Extreme case: Cs always meet Cs! Then efficient outcomes can be reached. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 7 / 30
I NTRODUCTION A SSORTATIVITY AND THE T RAGEDY OF THE C OMMONS In the same PD as before.. . C D C 9,9 -2,11 D 11,-2 0,0 . . . let individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs. Extreme case: Cs always meet Cs! Then efficient outcomes can be reached. ⇒ Assortativity can help overcome the tragedy of the commons. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 7 / 30
I NTRODUCTION A SSORTATIVITY AND THE T RAGEDY OF THE C OMMONS In the same PD as before.. . C D C 9,9 -2,11 D 11,-2 0,0 . . . let individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs. Extreme case: Cs always meet Cs! Then efficient outcomes can be reached. ⇒ Assortativity can help overcome the tragedy of the commons. Assortativity can be the outcome of different processes. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 7 / 30
I NTRODUCTION L OCAL I NTERACTIONS Interact mostly with your neighbours. (Boyd and Richerson, 2002; Grund, Waloszek, and Helbing, 2013) ⇒ Spatial assortativity. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 8 / 30
I NTRODUCTION K IN S ELECTION Interact mostly with your relatives. (Hamilton, 1964) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 9 / 30
I NTRODUCTION G REENBEARD EFFECT Recognise others’ types. (Dawkins, 1976) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 10 / 30
I NTRODUCTION H OMOPHILY You mostly interact with people that are like you in some characteristic. Alger and Weibull (2012) (social preference assortativity) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 11 / 30
Q UESTIONS We know that assortativity (of any kind) can theoretically help resolve the Tragedy of the Commons. (Jensen and Rigos, 2014, among others) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 12 / 30
Q UESTIONS We know that assortativity (of any kind) can theoretically help resolve the Tragedy of the Commons. (Jensen and Rigos, 2014, among others) The assortativity level is usually taken as given. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 12 / 30
Q UESTIONS We know that assortativity (of any kind) can theoretically help resolve the Tragedy of the Commons. (Jensen and Rigos, 2014, among others) The assortativity level is usually taken as given. What levels of assortativity would be the outcomes of natural processes? HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 12 / 30
Q UESTIONS We know that assortativity (of any kind) can theoretically help resolve the Tragedy of the Commons. (Jensen and Rigos, 2014, among others) The assortativity level is usually taken as given. What levels of assortativity would be the outcomes of natural processes? To what extent can populations endogenise the solution to a tragedy-of-the-commons problem? HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 12 / 30
Q UESTIONS We know that assortativity (of any kind) can theoretically help resolve the Tragedy of the Commons. (Jensen and Rigos, 2014, among others) The assortativity level is usually taken as given. What levels of assortativity would be the outcomes of natural processes? To what extent can populations endogenise the solution to a tragedy-of-the-commons problem? ⇒ Endogenise assortativity HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 12 / 30
M AIN I DEA Study Social Dilemmas that differ in their HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 13 / 30
M AIN I DEA Study Social Dilemmas that differ in their strategic structure HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 13 / 30
M AIN I DEA Study Social Dilemmas that differ in their strategic structure efficiency structure HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 13 / 30
M AIN I DEA Study Social Dilemmas that differ in their strategic structure efficiency structure Let the population decide upon their own level of assortativity based on a “voting” rule. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 13 / 30
M AIN I DEA Study Social Dilemmas that differ in their strategic structure efficiency structure Let the population decide upon their own level of assortativity based on a “voting” rule. Think about it as a system with agents who interact, get results and give feedback to a central planner. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 13 / 30
S OCIAL D ILEMMAS C D r , r a , 1 C 1 , a D 0,0 We keep r ∈ ( 0 , 1 ) and a ∈ ( − 1 , r ) so that C always increases the payoff of the other player (“public goods” character) D is always a best response to C HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 14 / 30
S OCIAL D ILEMMAS S OCIAL D ILEMMA C LASSIFICATION 1 a = 2 r − 1 0 . 8 a = 0 C D a = r 0 . 6 r , r a , 1 C 0 . 4 UD 1 , a D 0,0 0 . 2 SD 0 a − 0 . 2 C D − 0 . 4 MHD 0 . 8 , 0 . 8 − 0 . 2 , 1 C − 0 . 6 PD − 0 . 8 1 , − 0 . 2 D 0,0 − 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 r 1. Prisoners’ Dilemma (PD): 2 r > 1 + a and a < 0. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 15 / 30
S OCIAL D ILEMMAS S OCIAL D ILEMMA C LASSIFICATION 1 a = 2 r − 1 0 . 8 a = 0 C D a = r 0 . 6 r , r a , 1 C 0 . 4 UD 1 , a D 0,0 0 . 2 SD 0 a − 0 . 2 C D − 0 . 4 MHD 0 . 3 , 0 . 3 − 0 . 2 , 1 C − 0 . 6 PD − 0 . 8 1 , − 0 . 2 D 0,0 − 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 r 1. Prisoners’ Dilemma (PD): 2 r > 1 + a and a < 0. 2. Missing Hero Dilemma (MHD): 2 r < 1 + a and a < 0. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 15 / 30
S OCIAL D ILEMMAS S OCIAL D ILEMMA C LASSIFICATION 1 a = 2 r − 1 0 . 8 a = 0 C D a = r 0 . 6 r , r a , 1 C 0 . 4 UD 1 , a D 0,0 0 . 2 SD 0 a − 0 . 2 C D − 0 . 4 MHD 0 . 3 , 0 . 3 0 . 2 , 1 C − 0 . 6 PD − 0 . 8 1 , 0 . 2 D 0,0 − 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 r 1. Prisoners’ Dilemma (PD): 2 r > 1 + a and a < 0. 2. Missing Hero Dilemma (MHD): 2 r < 1 + a and a < 0. 3. SnowDrift Game (SD): 2 r < 1 + a and a > 0. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 15 / 30
S OCIAL D ILEMMAS S OCIAL D ILEMMA C LASSIFICATION 1 a = 2 r − 1 0 . 8 a = 0 C D a = r 0 . 6 r , r a , 1 C 0 . 4 UD 1 , a D 0,0 0 . 2 SD 0 a − 0 . 2 C D − 0 . 4 MHD 0 . 8 , 0 . 8 0 . 2 , 1 C − 0 . 6 PD − 0 . 8 1 , 0 . 2 D 0,0 − 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 r 1. Prisoners’ Dilemma (PD): 2 r > 1 + a and a < 0. 2. Missing Hero Dilemma (MHD): 2 r < 1 + a and a < 0. 3. SnowDrift Game (SD): 2 r < 1 + a and a > 0. 4. Underprovision Dilemma (UD): 2 r > 1 + a and a > 0. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 15 / 30
S OCIAL D ILEMMAS S OCIAL D ILEMMA C LASSIFICATION 1 a = 2 r − 1 0 . 8 a = 0 a = r 0 . 6 0 . 4 UD 0 . 2 SD 0 a − 0 . 2 − 0 . 4 MHD − 0 . 6 PD − 0 . 8 − 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 r Efficient Outcome C-D ( a > 2 r − 1) C-C ( a < 2 r − 1) Best Reply C ( a > 0) SD UD vs D D ( a < 0) MHD PD HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 16 / 30
M ODEL T HE M ODEL Continuum of individuals in continuous time HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 17 / 30
M ODEL T HE M ODEL Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 17 / 30
M ODEL T HE M ODEL Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) At each time t they get matched in pairs (how? → next slide) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 17 / 30
M ODEL T HE M ODEL Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) At each time t they get matched in pairs (how? → next slide) More successful behaviour is imitated by more individuals ⇒ Replicator Dynamics x = x ( 1 − x )( π C − π D ) ˙ HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 17 / 30
M ODEL T HE M ODEL Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) At each time t they get matched in pairs (how? → next slide) More successful behaviour is imitated by more individuals ⇒ Replicator Dynamics x = x ( 1 − x )( π C − π D ) ˙ π C ( π D ) is the average payoff of Cooperators (Defectors) in the population. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 17 / 30
M ODEL T HE M ODEL Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) At each time t they get matched in pairs (how? → next slide) More successful behaviour is imitated by more individuals ⇒ Replicator Dynamics x = x ( 1 − x )( π C − π D ) ˙ π C ( π D ) is the average payoff of Cooperators (Defectors) in the population. We are looking for asymptotically stable states of the above dynamics. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 17 / 30
M ODEL T HE M ODEL Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) At each time t they get matched in pairs (how? → next slide) More successful behaviour is imitated by more individuals ⇒ Replicator Dynamics x = x ( 1 − x )( π C − π D ) ˙ π C ( π D ) is the average payoff of Cooperators (Defectors) in the population. We are looking for asymptotically stable states of the above dynamics. These states generically exist. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 17 / 30
M ODEL M ATCHING The matching process follows a rule with a constant index of assortativity α ( ∈ [ 0 , 1 ] ). HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 18 / 30
M ODEL M ATCHING The matching process follows a rule with a constant index of assortativity α ( ∈ [ 0 , 1 ] ). α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 18 / 30
M ODEL M ATCHING The matching process follows a rule with a constant index of assortativity α ( ∈ [ 0 , 1 ] ). α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) α = 1 → full assortativity (C’s meet C’s and D’s meet D’s for sure ) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 18 / 30
M ODEL M ATCHING The matching process follows a rule with a constant index of assortativity α ( ∈ [ 0 , 1 ] ). α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) α = 1 → full assortativity (C’s meet C’s and D’s meet D’s for sure ) More generally: α = Pr ( C meets C ) − Pr ( D meets C ) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 18 / 30
M ODEL M ATCHING The matching process follows a rule with a constant index of assortativity α ( ∈ [ 0 , 1 ] ). α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) α = 1 → full assortativity (C’s meet C’s and D’s meet D’s for sure ) More generally: α = Pr ( C meets C ) − Pr ( D meets C ) Proportions of pair types under assortative matching with assortativity α : C-C: f 1 ( x , α ) = α x + ( 1 − α ) x 2 C-D: f 2 ( x , α ) = 2 x ( 1 − x )( 1 − α ) D-D: f 3 ( x , α ) = α ( 1 − x ) + ( 1 − α )( 1 − x ) 2 HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 18 / 30
M ODEL M ATCHING The matching process follows a rule with a constant index of assortativity α ( ∈ [ 0 , 1 ] ). α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) α = 1 → full assortativity (C’s meet C’s and D’s meet D’s for sure ) More generally: α = Pr ( C meets C ) − Pr ( D meets C ) Proportions of pair types under assortative matching with assortativity α : C-C: f 1 ( x , α ) = α x + ( 1 − α ) x 2 C-D: f 2 ( x , α ) = 2 x ( 1 − x )( 1 − α ) D-D: f 3 ( x , α ) = α ( 1 − x ) + ( 1 − α )( 1 − x ) 2 π C ( x , α ) = ( r · f 1 ( x , α ) + a · f 2 ( x , α ) / 2 ) / x π D ( x , α ) = ( 0 · f 3 ( x , α ) + 1 · f 2 ( x , α ) / 2 ) / ( 1 − x ) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 18 / 30
M ODEL M ATCHING The matching process follows a rule with a constant index of assortativity α ( ∈ [ 0 , 1 ] ). α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) α = 1 → full assortativity (C’s meet C’s and D’s meet D’s for sure ) More generally: α = Pr ( C meets C ) − Pr ( D meets C ) Proportions of pair types under assortative matching with assortativity α : C-C: f 1 ( x , α ) = α x + ( 1 − α ) x 2 C-D: f 2 ( x , α ) = 2 x ( 1 − x )( 1 − α ) D-D: f 3 ( x , α ) = α ( 1 − x ) + ( 1 − α )( 1 − x ) 2 π C ( x , α ) = ( r · f 1 ( x , α ) + a · f 2 ( x , α ) / 2 ) / x π D ( x , α ) = ( 0 · f 3 ( x , α ) + 1 · f 2 ( x , α ) / 2 ) / ( 1 − x ) In asymptotically stable states (ESS) of the dynamics: � More Cooperators Higher assortativity ⇒ Higher efficiency (avg. payoff) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 18 / 30
M ODEL V OTING FOR A SSORTATIVITY Now the population will get to (dynamically) select how assortativity itself evolves. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 19 / 30
M ODEL V OTING FOR A SSORTATIVITY Now the population will get to (dynamically) select how assortativity itself evolves. Each individual gets one vote and can vote either for an increase or a decrease of assortativity. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 19 / 30
M ODEL V OTING FOR A SSORTATIVITY Now the population will get to (dynamically) select how assortativity itself evolves. Each individual gets one vote and can vote either for an increase or a decrease of assortativity. They compare their payoff to the highest and the lowest payoff they can receive in the Social Dilemma. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 19 / 30
M ODEL V OTING FOR A SSORTATIVITY Now the population will get to (dynamically) select how assortativity itself evolves. Each individual gets one vote and can vote either for an increase or a decrease of assortativity. They compare their payoff to the highest and the lowest payoff they can receive in the Social Dilemma. They vote using a logit choice rule based on the payoffs they receive. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 19 / 30
M ODEL V OTING ( FEEDBACK ) Agents matched in Homogeneous pairs (either C-C or D-D): probability to vote for an increase in assortativity increasing in payoff. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 20 / 30
M ODEL V OTING ( FEEDBACK ) Agents matched in Homogeneous pairs (either C-C or D-D): probability to vote for an increase in assortativity increasing in payoff. Heterogeneous pairs (C-D): probability to vote for an increase in assortativity decreasing in payoff. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 20 / 30
M ODEL V OTING ( FEEDBACK ) Agents matched in Homogeneous pairs (either C-C or D-D): probability to vote for an increase in assortativity increasing in payoff. Heterogeneous pairs (C-D): probability to vote for an increase in assortativity decreasing in payoff. Example: A Defector gets matched to a Cooperator and receives the highest payoff in the game. Then, he/she votes for a decrease in assortativity for sure . HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 20 / 30
M ODEL C OUNTING THE VOTES v + : Number of votes for more assortativity v − : Number of votes for less assortativity HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 21 / 30
M ODEL C OUNTING THE VOTES v + : Number of votes for more assortativity v − : Number of votes for less assortativity Assortativity follows a replicator-like dynamic based on the number of votes received: α = α ( 1 − α )( v + − v − ) ˙ HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 21 / 30
M ODEL C O -E VOLUTION x = x ( 1 − x )( π C − π D ) ˙ ( 1 ) α = α ( 1 − α )( v + − v − ) ˙ ( 2 ) HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 21 / 30
R ESULTS F ULL E QUILIBRIA D EFINITION (F ULL E QUILIBRIUM ) For a Social Dilemma G , a full equilibrium is a pair ( x ∗ , α ∗ ) that is an asymptotically stable solution of the system (1) – (2). In Social Dilemmas full equilibria always exist. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 22 / 30
R ESULTS F ULL E QUILIBRIA D EFINITION (F ULL E QUILIBRIUM ) For a Social Dilemma G , a full equilibrium is a pair ( x ∗ , α ∗ ) that is an asymptotically stable solution of the system (1) – (2). In Social Dilemmas full equilibria always exist. O BSERVATION For any Social Dilemma G , all full equilibria have either α ∗ = 0 ( x 0 ) or α ∗ = 1 ( x 1 ). HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 22 / 30
R ESULTS D YNAMICS PD ( r =0.30, a =-0.60) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 23 / 30
R ESULTS D YNAMICS MHD ( r =0.20, a =-0.40) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 23 / 30
R ESULTS D YNAMICS SD ( r =0.40, a =0.10) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 23 / 30
R ESULTS D YNAMICS SD ( r =0.55, a =0.40) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 23 / 30
R ESULTS D YNAMICS SD ( r =0.80, a =0.70) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 23 / 30
R ESULTS D YNAMICS UD ( r =0.80, a =0.40) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 23 / 30
R ESULTS F ULL A SSORTATIVITY R OBUSTNESS We want to measure how stable are the possible outcomes (full equilibria). HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 24 / 30
R ESULTS F ULL A SSORTATIVITY R OBUSTNESS We want to measure how stable are the possible outcomes (full equilibria). Efficiency is increasing in equilibrium assortativity ⇒ Robustness of the equilibrium with α = 1 is a measure of efficiency. HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 24 / 30
R ESULTS F ULL A SSORTATIVITY R OBUSTNESS We want to measure how stable are the possible outcomes (full equilibria). Efficiency is increasing in equilibrium assortativity ⇒ Robustness of the equilibrium with α = 1 is a measure of efficiency. We use an invasion barrier approach. F ULL A SSORTATIVITY R OBUSTNESS “What is the largest invasion a population at x 1 can sustain?” HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 24 / 30
R ESULTS F ULL A SSORTATIVITY R OBUSTNESS – E XAMPLE PD ( r =0.30, a =-0.60) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 25 / 30
R ESULTS F ULL A SSORTATIVITY R OBUSTNESS – E XAMPLE PD ( r =0.30, a =-0.60) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 25 / 30
R ESULTS F ULL A SSORTATIVITY R OBUSTNESS – E XAMPLE PD ( r =0.30, a =-0.60) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 25 / 30
R ESULTS F ULL A SSORTATIVITY R OBUSTNESS – E XAMPLE PD ( r =0.30, a =-0.60) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 25 / 30
R ESULTS F ULL A SSORTATIVITY R OBUSTNESS – E XAMPLE PD ( r =0.30, a =-0.60) 1 0 . 9 0 . 8 0 . 7 0 . 6 0 . 5 x 0 . 4 0 . 3 0 . 2 0 . 1 0 ̺ G 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 α HH N AX AND A R IGOS A SSORTATIVITY IN S OCIAL D ILEMMAS 25 / 30
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