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

ASSORTATIVITY EVOLVING FROM SOCIAL DILEMMAS

Heinrich H. Nax1 Alexandros Rigos2

1ETH Zürich 2University of Leicester

Controversies in Game Theory II 2015-05-27

NAX & RIGOS ASSORTATIVITY & SOCIAL DILEMMAS TITLE

THE TRAGEDY OF THE COMMONS

TRAGEDY OF THE COMMONS

A situation where individuals acting independently and rationally according to each’s self-interest behave contrary to the best interests

• f the whole group by depleting some common resource.

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PD WITH RANDOM INTERACTIONS LEADS TO TRAGEDY

OF THE COMMONS

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.

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ASSORTATIVITY AND THE TRAGEDY OF THE COMMONS

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.

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LOCAL INTERACTIONS

Interact mostly with your neighbours. (Boyd and Richerson, 2002; Grund, Waloszek, and Helbing, 2013)⇒ Spatial assortativity.

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KIN SELECTION

Interact mostly with your relatives. (Hamilton, 1964)

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GREEN BEARD EFFECT

Recognise others’ types. (Dawkins, 1976)

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HOMOPHILY

You mostly interact with people that are like you in some characteristic. Alger and Weibull (2012) (social preference assortativity)

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MERITOCRACY

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.

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MAIN IDEA OF THE TALK

We know that assortativity (of any kind) may help to resolve the Tragedy of the Commons.

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MAIN 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?

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MAIN 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?

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SPATIAL ASSORTATIVITY À LA GRUND, WALOSZEK & HELBING (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 ρi that indicates how much they “care” about their Moore neighbours. They all start

• ff with ρi = 0.

They decide using the best-response rule (best-responding to the actions of their neighbours in the previous period.

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SPATIAL ASSORTATIVITY À LA GRUND, WALOSZEK & HELBING (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 ρi that indicates how much they “care” about their Moore neighbours. They all start

• ff with ρi = 0.

They decide using the best-response rule (best-responding to the actions of their neighbours in the previous period.

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SUMMARY OF THE MODEL (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, ρi])) with probability 0.8 U([ρi, 1]) with probability 0.2

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MAIN RESULTS

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).

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POSSIBLE EXTENSIONS: ENDOGENISATION

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.

1

In the model all a, b and c coincide.

2

Extension 1: Let range a be subject to an evolutionary process (endogenising the extent of social preferences).

3

Extension 2: Let range b be subject to an evolutionary process (endogenising migration).

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We saw that: Spatial Assortativity ⇒ Social Preferences ⇒ Cooperation We will now turn to Action Assortativity.

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MERITOCRACY (ACTION ASSORTATIVITY)

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.

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MAIN IDEA: ENDOGENISE 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.

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QUESTIONS 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?

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SOCIAL DILEMMAS

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

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SOCIAL DILEMMA CLASSIFICATION

C D C r, r a, 1 D 1, a 0,0

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a r a = 2r − 1 a = 0 a = r PD MHD VD UD

• 1. Prisoners’ Dilemma (PD): 2r > 1 + a and a < 0.
• 2. Missing Hero Dilemma (MHD): 2r < 1 + a and a < 0.
• 3. Volunteer’s Dilemma (VD): 2r < 1 + a and a > 0.
• 4. Underprovision Dilemma (UD): 2r > 1 + a and a > 0.

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SOCIAL DILEMMA CLASSIFICATION

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a r a = 2r − 1 a = 0 a = r PD MHD VD UD

Efficient Outcome C-D (a > 2r − 1) C-C (a < 2r − 1) Best Reply C (a > 0) VD UD vs D D (a < 0) MHD PD

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THE MODEL

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)(πC − πD) We call a Social Dilemma G = (r, a) and a level of meritocracy α an environment E = (G, α)

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ENVIRONMENT EQUILIBRIA

Given an environment E, we can calculate the Evolutionarily Stable States (ESSs). We call them environment equilibria (EE). These are states that are “attracting” states that are arbitrarily close to them. In Social Dilemmas, EE generically exist.

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VOTING FOR MERITOCRACY

Now the population will get to select how the meritocracy itself evolves. Each player can vote either for an increase or a decrease of meritocracy. Players who get good outcomes when matched in homogeneous (heterogeneous) groups are more (less) likely to vote for more meritocracy. They use a utility voting rule. The more extreme their payoffs are the more opinionated they are i.e. they vote for one direction or another with higher probability. Assortativity follows a replicator-like dynamic based on the number of votes received: ˙ α = α(1 − α)(v+ − v−)

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CO-EVOLUTION

So, the co-evolution of meritocracy and cooperation is described by the system of equations ˙ x = x(1 − x)(πC − πD) (1) ˙ α = α(1 − α)(v+ − v−) (2)

DEFINITION (FULL EQUILIBRIUM)

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.

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FULL EQUILIBRIA

PROPOSITION

For any Social Dilemma G, all full equilibria have either α∗ = 0 or α∗ = 1. In particular MHDs have only one full equilibrium with α∗ = 0 PDs and UDs have two full equilibria: one with α∗ = 0 and one with α∗ = 1. VDs have all the range of full equilibria, depending on the parameter values.

For small r → 1 FE with α∗ = 0. For intermediate r → 2 FE: one with α∗ = 0 and one with α∗ = 1. For high r → 1 FE with α∗ = 1.

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DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

MHD (r=0.20, a=-0.40)

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DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

PD (r=0.30, a=-0.60)

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DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

UD (r=0.70, a=0.10)

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DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

VD (r=0.40, a=0.10)

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DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

VD (r=0.55, a=0.40)

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DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

VD (r=0.70, a=0.60)

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MERITOCRACY ROBUSTNESS – STABILITY

We want to quantify how stable the possible outcomes (full equilibria) are. Since we get either full or null meritocracy, we can measure how robust (full) meritocracy is. We do that by an “invasion barrier” approach (how “hard” is it to get from α = 0 to α = 1?).

DEFINITION (MERITOCRACY ROBUSTNESS)

Full meritocracy is ̺-robust for some ̺ ∈ [0, 1] if, for every ̺′ ∈ (̺, 1], all convex combinations ̺′(x∗

1, 1) + (1 − ̺′)(x∗ 0, 0) lie in the basin of

attraction of (x∗

1, 1) and, for every ̺′′ ∈ [0, ̺), all convex combinations

̺′′(x∗

1, 1) + (1 − ̺′′)(x∗ 0, 0) lie in the basin of attraction of (x∗ 0, 0).

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MERITOCRACY ROBUSTNESS – EXAMPLE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

PD (r=0.30, a=-0.60)

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MERITOCRACY ROBUSTNESS – EXAMPLE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

PD (r=0.30, a=-0.60)

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MERITOCRACY ROBUSTNESS – EXAMPLE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

PD (r=0.30, a=-0.60)

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MERITOCRACY ROBUSTNESS – EXAMPLE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

PD (r=0.30, a=-0.60)

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MERITOCRACY ROBUSTNESS – EXAMPLE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α

PD (r=0.30, a=-0.60)

̺

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MERITOCRACY ROBUSTNESS – RESULTS

• 1
• 0.5

0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a r 0.2 0.4 0.6 0.8 1 PD MHD VD UD

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MERITOCRACY ROBUSTNESS AS A MEASURE OF THE EFFICIENCY OF THE VOTING MECHANISM

The efficiency level that can be reached by any level of meritocracy is maximised when α = 1 (the assortative optimum). So, meritocracy robustness ̺ can be seen as a measure of the expected full equilibrium efficiency relative to full assortativity (the assortative optimum).

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CONCLUSION

We endogenised the level of meritocracy for a matching process in democratic regimes. In such regimes and for our class of social dilemmas only full or null meritocracy can be stable. The endogenisation of meritocracy can help mitigate tragedies of the commons. The situations where it helps the most are UD and some VD. Partial help for PD. No help for MHD. Future work: Test the results of our research by running experiments.

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