Social norms Social norms and Global and Local Interaction in a and Global and Local Interaction in a

Common Pool ResourceCommon Pool ResourceJoelle Noailly, Cees Withagen, Jeroen van den Bergh,

Faculty of Economics and Business Administration

Free University, Amsterdam

Tilburg University

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OutlineOutline

1. Introduction

2. The Common Pool Resource game

3. Interaction on the circle; static resource

4. Interaction on the circle; resource dynamics

5. Interaction on the 2D torus

6. Conclusions

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1. Introduction 1. Introduction

• Social dilemmas and CPR

• Cooperative behaviour: observations:Ostrom (1990), Acheson

(1988),McKean (1992), Sethi and Somanathan (1996)

experiments: Ostrom (1990), Fehr and Gächter (2002) analytical models: Fehr and Schmidt (1999), Sethi and Somanathan (1996)

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ExplanationsExplanations

Dasgupta (1993):

• small communities act as states (but spontaneous and destructive actions)

• Folk theorem (but multiple equilibria, possible changing over time)

• internalization of social norms

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Testing robustnessTesting robustness

• Sethi and Somathan analyse an evolutionary game without spatial features.

Does spatial disaggregation matter?Cooperation?

• Analytical and numerical approach.

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2. CPR game; 2. CPR game; Key notionsKey notions• n players. • xi: individual effort, total effort X=xi

• w: wage rate • F: total extraction as function of total effort

F is increasing, strictly concave, F(0)=0,

F’(0)>w, F’()<w • Payoff player i: πi(xi,X)=xiZ(X) with

Z(X)=F(X)/X-w (average profit)

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Standard equilibrium conceptsStandard equilibrium concepts

Social optimum: F’(XP)=w

restrained level of resource exploitationNash equilibrium:

(n-1)F(XC)/XC+F’(XC)=nw

suboptimalFree entry: F(XO)=w

erosion of profits

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Evolutionary modellingEvolutionary modelling

• n players• 3 strategies

co-operation: nC, xl

defection: nD, xh, δ (sanction)

enforcement: nE, xl, γ (enforcement cost)

• properties:

XP nxl<nxh<XO

• replicator dynamics

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Global interaction (S&S)Global interaction (S&S)

• All defectors are punished by all enforcers:

C=xlZ(X)

E=xlZ(X)-nD

D=xhZ(X)-nE

• Replicator dynamics:

dnk/dt=nk[ k- ] with average payoff

• 2 types of equilibria:only defectors (‘all D’)

mix of cooperators and enforcers (CE)

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Spatial local interactionSpatial local interaction

Motivation:

Resources are spatially distributed and cause spatial externalities: pollution in adjacent areas, fisheries, water

Bounded rationality (spatial myopia)

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Local interaction Local interaction

• Profits

C=xlZ(X)

mE=xlZ(X)-m, m=0,1,2,3,4,...

kD=xhZ(X)-k, k=0,1,2,3,4,...• Interaction on

circle: two direct neighbours

torus: four direct neighbours• Players observe direct neighbours

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Imitation and replicationImitation and replication• Simple rule: imitate best strategy in

neighbourhood

• Sophisticated rule: imitate (on average) best strategy in neighbourhood

• Profit ranking ambiguous, unlike Eshel et al.

(1998)• Assumption: Z(X)>0 for all X

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3. Sophisticated interaction on the 3. Sophisticated interaction on the circle (static resource)circle (static resource)

Definition in terms of profits (regardless of X) • Sanction rate is relatively low if

0D> C= 0E> 1D> 1E> 2D> 2E

• Sanction rate is relatively very low if it is low and ½[0E+ 1E]<1D

• Sanction rate is relatively moderately low if it is low and ½[0E+ 1E]>1D

• Sanction rate is relatively high if 0D> C= 0E> 1E> 1D >2E>2D

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Lemma on classification Lemma on classification of of sanctionsanctionss

• Sanction rate is relatively low if

<, (xh-xl)Z(nxl)<2-• Sanction rate is relatively very low if

<, (xh-xl)Z(nxl)<2- and -½< (xh-xl)Z(nxh)

• Sanction rate is relatively moderately low if

<, (xh-xl)Z(nxl)<2- and -½> (xh-xl)Z(nxh)

• Sanction rate is relatively high if

(xh-xl)Z(nxl)<- and -2< (xh-xl)Z(nxh)

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Limit statesLimit states

• Equilibrium: no agent changes strategy

• Blinkers: states rotate

• Cycling: reproduction in two periods (occurs)

Neglect ‘allC’,’allD’,’allE’, CE

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Lemma on equilibriumLemma on equilibrium

i) CED:never; ii) CD: never;

iii) DED: never; iv) EDE: never.Proof i) and ii) evident. iii) DED never with low sanction. Suppose high

sanction. Surrounding D’s not punished twice (EDE is ruled out in high sanction case). Hence enforcer switches to defection

iv) idem

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Low sanction rate Low sanction rate

• Relatively very low sanction:Neither DE nor CDE equilibriumNeither DE nor CDE blinker

• Relatively moderately low sanction:DE requires n>4. If n=5 then EEEDD. Minimal cluster

of E’s is 3CDE requires n>8. If n=9 then CEEEDDEEENeither DE nor CDE blinker

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High sanction rateHigh sanction rate

• DE requires n>4. If n=5 then EEDDD. Defectors in minimal cluster of 3

• CDE requires n>7. If n=8 then CEEDDDEE

• No DE blinkers

• CDE blinker requires n>3. If n=4 then CDDE

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New insights New insights

• CDE equilibrium occurs, contrary to S&S

• No CD equilibrium

• In DE only few enforcers required, contrary to S&S

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Stochastic stability: Stochastic stability: TTheoryheory

State is ordered vector of CDE’s

CCDDE ”=“ CDDEE ”=“ EDDCC

Transition matrix based on replicator dynamicsTransition matrix based on mutationSolve from T=, 0, =1

Stochastic stability of CDE is problematic

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““Stochastic stability”: SimulationsStochastic stability”: Simulations

• F(X)=X

• Fixed: n=100, xl=100, xh=120, =1000, =0.5, =300, w=5

• Varying: (initial value 280)

• nxl=XP

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““Stochastic stability”: Time scaleStochastic stability”: Time scale

• nE=50, nC=nD=25• Initial ordering CEDE• Constitutes CDE equilibrium• Mutation with probability 5/1000• 100 simulation runs for different fixed horizons• After 10000 rounds 24% in CDE• After 30000 rounds 22% in CDE

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““Stochastic stability”: SStochastic stability”: Shareshares and and spatial distributionspatial distribution

• All nC, nD, and nE take values 0,5,10,15,…with sum equal to 100: (no allC, no allD, no allCE: 190 possibilities).

• For each z(0) 100 spatial distributions.

• For each z(0) and each spatial distribution 100 runs.

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““Stochastic stability”: ResultsStochastic stability”: Results

• D: 32%; CE:4%; DE:33%

CDE equilibrium: 29%

CDE cycling: 2%

• High CDE likelihood also found for other sanction levels

• Additional results on shares and spatial distribution (in section 5)

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4. 4. Sophisticated interaction on the Sophisticated interaction on the circle (dynamic resource)circle (dynamic resource)

Regeneration according to logistic growth:

G(N)=rN(1-N/K)

Resource stock is depleted and regenerated after each round:

N(t+1)=N(t)+G(N(t))-F(X(t),N(t))

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BenchmarkBenchmark

Max subject to Steady state . Determine such that if

N̂

laLnaX NN ˆ

dtwXNXpFe rt ]),([0

XNGN )(

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AnalysisAnalysis

• xht(Nt)=ahNt

• xlt(Nt)=alNt

• Consider kD. If nD increases, aggregate profits decrease-for given stock-, but also stock decreases.

• Simulations show that likelihood of CDE increases

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5. Simple interaction on torus5. Simple interaction on torus

• Profits

C=xlZ(X)

mE=xlZ(X)-m, m=0,1,2,3,4

kD=xhZ(X)-k, k=0,1,2,3,4

• Neighborhood

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SimulationsSimulations

• Mainly simulations using CORMAS

• F(X)=X

• Fixed n=100 (10x10 grid), xl=2, xh=4, =100, =0.2, w=0.2, =0.1, =0.4

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Example 1Example 1z(0)=(5%, 2%, 93%)z(0)=(5%, 2%, 93%)

=3 =2 =1

=4 =5 >5

Defector

Cooperator

Enforcer

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Example 2Example 2z(0)=(30%;40%,30%)z(0)=(30%;40%,30%)

=3 =2 =1 =4

=5 =6

Defector

Cooperator

Enforcer

=7 >7

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ObservationsObservations

• CDE equilibrium exists.• CDE equilibrium exhibits clusters: groups of 5

enforcers and/or co-operators offer ‘protection’ to central player.

• Defecting cluster survives• If central player is E, he will ‘protect’ enforcers in

the neighbourhood• Clusters can grow

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Observations (continued)Observations (continued)

• Defectors subject to severe punishment imitate enforcers or cooperators (C and E eliminate D)

• Punishing enforcers revert to co-operation when there are co-operators in the neighbourhood (C eliminate E)

• Hence initially rise in co-operation.• If C eliminates E quickly then D equilibrium

emerges.• If E eliminates D quickly then CE equilibrium

emerges.

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Simulation: spatial distribution Simulation: spatial distribution z(0)z(0)=(30%;40%;30%).=(30%;40%;30%).

Random spatial distribution

• No CD, C or E equilibria.

• Strategies CDE can coexist in the long-run.

C

D E

z0

z50

C

D E

z0

z50

C

D E

z0 z50

C

D E

z0

z50

D

Z(50)=(1;0;0)

DE

Z(50)=(0.91;0.09;0)

CDE

Z(50)=(0.37;0.29;0.34)

CE

Z(50)=(0;0.26;0.74)

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Simulation: Simulation: sharesshares and spatial and spatial distributiondistribution

• nC, nD and nE take values 0,5,10,15,…with sum equal to 100: 231 possibilities.

• For each z(0) 100 spatial distributions.• For each z(0) and each spatial distribution 100

runs• Interpretation of dots.

Consider picture D. Take some orange dot. Of all spatial distributions with the given z(0) approximately 70% converge to D-equilibrium

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allD equilibriaallD equilibria

• D equilibria

D E

C

10.8 - 0.990.6 - 0.790.4 - 0.590.2 - 0.390.01 - 0.19

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DE equilibriaDE equilibria

• DE equilibria

D E

C

10.8 - 0.990.6 - 0.790.4 - 0.590.2 - 0.390.01 - 0.19

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CE equilibriaCE equilibria

• CE equilibria

10.8 - 0.990.6 - 0.790.4 - 0.590.2 - 0.390.01 - 0.19

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CDE equilibriaCDE equilibria

• CDE equilibria

D E

C

0.8 - 1

0.6 - 0.79

0.4 - 0.59

0.2 - 0.39

0.01 - 0.19

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SummarySummary

• C: 0.4% (0.4%)*

• D: 41% (79%)

• E 3% (3%)

• CE: 18% (23%)

• DE: 18% (0%)

• CDE: 20% (0%)

*between brackets sophisticated rule

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Variation in price and sanctionVariation in price and sanction

• D equilibria best attained forlow sanctionshigh harvest pricesmall population: total effort decreases, profits increase. In contrast with S&S (there higher n makes detection more difficult)

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DynamicsDynamics

• xht(Nt)=λxhNtθ

• xlt(Nt)=λxlNtθ

• r=0.5; NK=1000;N(0)=500; λ=0.05; θ=0.5

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CDE equilibriumCDE equilibrium exists exists

10 20 30 40 50 60rounds

20

40

60

80

100strategies

10 20 30 40 50 60rounds

420

440

460

480

500

stock level

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D equilibriumD equilibrium

10 20 30 40 50 60rounds

20

40

60

80

100strategies

10 20 30 40 50 60rounds

420

440

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480

500

stock level

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CDE equilibriumCDE equilibrium

10 20 30 40 50 60rounds

20

40

60

80

100strategies

10 20 30 40 50 60rounds

420

440

460

480

500

stock level

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D equilibriumD equilibrium

10 20 30 40 50 60rounds

20

40

60

80

100strategies

10 20 30 40 50 60rounds

420

440

460

480

500

stock level

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6. Conclusions6. Conclusions

• Results S&S not robust: more equilibria possible with spatial interaction

• Co-operators and enforcers can survive in large group of defectors

• Interactions lead to more co-operative outcomes• Diversity of equilibria is maintained with resource

dynamics

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Future researchFuture research

• More on resource dynamics

• Alternative replicator dynamics

Relevant average payoffs

Local Nash equilibria

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Future researchFuture research

• More analysis with current resource dynamics;Does fall/rise in stock accelerate/delay convergence to particular strategy equilibrium?

• Resourcespecification of alternative temporal dynamics,adding spatial heterogeneity,adding spatial connectivity and dynamics.

• Coevolutionary dynamics:resource size and composition - fish, pests.