S PATIAL A SSORTATIVITY LA G RUND , W ALOSZEK & H ELBING (2013) - PowerPoint PPT Presentation

A SSORTATIVITY E VOLVING FROM S OCIAL D ILEMMAS Heinrich H. Nax 1 Alexandros Rigos 2 1 ETH Zürich 2 University of Leicester Controversies in Game Theory II 2015-05-27 N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS T ITLE

T HE T RAGEDY OF THE C OMMONS T RAGEDY OF THE C OMMONS A situation where individuals acting independently and rationally according to each’s self-interest behave contrary to the best interests of the whole group by depleting some common resource. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 1 / 31

PD WITH R ANDOM I NTERACTIONS L EADS TO T RAGEDY OF THE C OMMONS Consider a big population playing a PD. C D C 11,11 -2,8 D 8,-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). ⇒ Tragedy of the Commons. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 2 / 31

A SSORTATIVITY AND THE T RAGEDY OF THE C OMMONS In the same PD as before. . . C D C 11,11 -2,8 D 8,-2 0,0 . . . individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs. Extreme case: Cs always meet Cs! Then the tragedy of the commons can be overcome. ⇒ Assortativity can help with the tragedy of the commons. It can be the outcome of different processes. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 3 / 31

L OCAL I NTERACTIONS Interact mostly with your neighbours. (Boyd and Richerson, 2002; Grund, Waloszek, and Helbing, 2013) ⇒ Spatial assortativity. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 4 / 31

K IN S ELECTION Interact mostly with your relatives. (Hamilton, 1964) N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 5 / 31

G REEN BEARD EFFECT Recognise others’ types. (Dawkins, 1976) N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 6 / 31

H OMOPHILY You mostly interact with people that are like you in some characteristic. Alger and Weibull (2012) (social preference assortativity) N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 7 / 31

M ERITOCRACY Mechanism that assorts individuals according to their level of cooperation (Nax, Murphy, and Helbing, 2014). (action assortativity) Individuals that contribute more to a public good have a higher chance of getting into groups with other high contributors. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 8 / 31

M AIN IDEA OF THE TALK We know that assortativity (of any kind) may help to resolve the Tragedy of the Commons. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 9 / 31

M AIN IDEA OF THE TALK We know that assortativity (of any kind) may help to resolve the Tragedy of the Commons. What levels of assortativity would be the outcomes of natural processes? N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 9 / 31

M AIN IDEA OF THE TALK We know that assortativity (of any kind) may help to resolve the Tragedy of the Commons. What levels of assortativity would be the outcomes of natural processes? What are the outcomes of such more general processes? N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 9 / 31

S PATIAL A SSORTATIVITY À LA G RUND , W ALOSZEK & H ELBING (2013) There is an L × L lattice with a torus topology partly occupied (60%) by individuals. Indiviudals can either Cooperate (C) or Defect (D). All individuals initially defect. Each gets payoff depending on the outcomes of interactions with their Moore neighbours according to a Prisoners’ Dilemma: C D C R,R S,T D S,T P ,P Individuals also have a “friendliness” parameter ρ i that indicates how much they “care” about their Moore neighbours. They all start off with ρ i = 0. They decide using the best-response rule (best-responding to the actions of their neighbours in the previous period. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 10 / 31

S PATIAL A SSORTATIVITY À LA G RUND , W ALOSZEK & H ELBING (2013) There is an L × L lattice with a torus topology partly occupied (60%) by individuals. Indiviudals can either Cooperate (C) or Defect (D). All individuals initially defect. Each gets payoff depending on the outcomes of interactions with their Moore neighbours according to a Prisoners’ Dilemma: C D C R,R S,T D S,T P ,P Individuals also have a “friendliness” parameter ρ i that indicates how much they “care” about their Moore neighbours. They all start off with ρ i = 0. They decide using the best-response rule (best-responding to the actions of their neighbours in the previous period. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 10 / 31

S UMMARY OF THE M ODEL ( CONT ’ D ) Individuals “die” with a probability β . Individuals also reproduce (asexually) with a probability proportional to their payoff. Offspring take the place of dead individuals. Reproduction is local with a probability v and random with probability 1 − v . Children inherit their parent’s friendliness parameter with probability 1 − µ and mutate with probability µ . Mutants’ friendliness parameter is determined by a random draw from the distribution: U ([ 0 , ρ i ])) with probability 0.8 U ([ ρ i , 1 ]) with probability 0.2 N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 11 / 31

M AIN R ESULTS Cooperation sets off if we get an unconditional cooperator and spreads if around him/her there are individuals with a high enough friendliness trait. More local reproduction leads to higher levels of cooperation and friendliness . Explains the co-existence of different social preferences (cooperators/defectors/conditional cooperators). N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 12 / 31

P OSSIBLE EXTENSIONS : E NDOGENISATION There are 3 “ranges:” a The range containing individuals about whom I care. b The range within which my children will be born. c The range of individuals whose payoffs my choices affect. In the model all a , b and c coincide. 1 Extension 1: Let range a be subject to an evolutionary process 2 (endogenising the extent of social preferences). Extension 2: Let range b be subject to an evolutionary process 3 (endogenising migration). N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 13 / 31

We saw that: Spatial Assortativity ⇒ Social Preferences ⇒ Cooperation We will now turn to Action Assortativity. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 14 / 31

M ERITOCRACY (A CTION A SSORTATIVITY ) Players take actions and get matched accordingly. More cooperative players have a higher probability of getting matched to cooperative players. Heinrich’s talk yesterday. High enough levels of meritocracy can lead to more efficient and more equitable outcomes. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 15 / 31

M AIN IDEA : E NDOGENISE THE ASSORTATIVE PROCESS Study Social Dilemmas that differ in their strategic structure efficiency structure Given a certain level of assortativity (meritocracy), we know what the (stable) outcome(s) will be (Jensen and Rigos, 2014). Let the population decide upon their own level of meritocracy based on a “voting” rule. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 16 / 31

Q UESTIONS WE AIM TO ANSWER What are the strategic outcomes of such processes? What happens with respect to efficiency? To what extent can populations endogenise the solution to a tragedy-of-the-commons problem? N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 17 / 31

S OCIAL D ILEMMAS C D C r , r a , 1 D 1 , a 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 N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 18 / 31

S OCIAL D ILEMMA C LASSIFICATION 1 a = 2 r − 1 0 . 8 a = 0 a = r 0 . 6 0 . 4 UD C D 0 . 2 VD 0 a C r , r a , 1 − 0 . 2 D 1 , a 0,0 − 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 1. Prisoners’ Dilemma (PD): 2 r > 1 + a and a < 0. 2. Missing Hero Dilemma (MHD): 2 r < 1 + a and a < 0. 3. Volunteer’s Dilemma (VD): 2 r < 1 + a and a > 0. 4. Underprovision Dilemma (UD): 2 r > 1 + a and a > 0. N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 19 / 31

S OCIAL D ILEMMA C LASSIFICATION 1 a = 2 r − 1 0 . 8 a = 0 a = r 0 . 6 0 . 4 UD 0 . 2 VD a 0 − 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) VD UD vs D D ( a < 0) MHD PD N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 20 / 31

T HE M ODEL continuum of players in continuous time x is the proportion of cooperators (C-players) at each date t they get matched in pairs the matching takes place according to a meritocratic rule α = 0 → random matching α = 1 → perfectly meritocratic matching players imitate more successful individuals → replicator dynamics ˙ x = x ( 1 − x )( π C − π D ) We call a Social Dilemma G = ( r , a ) and a level of meritocracy α an environment E = ( G , α ) N AX & R IGOS A SSORTATIVITY & S OCIAL D ILEMMAS 21 / 31