Social norms and Global and Local Interaction in a Common Pool Resource

1. Social norms and Global and Local Interaction in a Common Pool Resource Joelle Noailly, Cees Withagen, Jeroen van den Bergh, Faculty of Economics and Business Administration Free University, Amsterdam Tilburg University

2. Outline 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

3. 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)

4. Explanations 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

5. Testing robustness • Sethi and Somathan analyse an evolutionary game without spatial features. Does spatial disaggregation matter? Cooperation? • Analytical and numerical approach.

6. 2. CPR game; Key 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)

7. Standard equilibrium concepts • Social optimum: F’(XP)=w restrained level of resource exploitation • Nash equilibrium: (n-1)F(XC)/XC+F’(XC)=nw suboptimal • Free entry: F(XO)=w erosion of profits

8. Evolutionary 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

9. 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)

10. Spatial local interaction Motivation: • Resources are spatially distributedand cause spatial externalities: pollution in adjacent areas, fisheries, water • Bounded rationality (spatial myopia)

11. 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

12. Imitation 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

13. 3. Sophisticated interaction on the 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

14. Lemma on classification of sanctions • 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)

15. Limit states • Equilibrium: no agent changes strategy • Blinkers: states rotate • Cycling: reproduction in two periods (occurs) Neglect ‘allC’,’allD’,’allE’, CE

16. Lemma 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

17. Low sanction rate • Relatively very low sanction: Neither DE nor CDE equilibrium Neither DE nor CDE blinker • Relatively moderately low sanction: DE requires n>4. If n=5 then EEEDD. Minimal cluster of E’s is 3 CDE requires n>8. If n=9 then CEEEDDEEE Neither DE nor CDE blinker

18. High 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

19. New insights • CDE equilibrium occurs, contrary to S&S • No CD equilibrium • In DE only few enforcers required, contrary to S&S

20. Stochastic stability: Theory State is ordered vector of CDE’s CCDDE ”=“ CDDEE ”=“ EDDCC Transition matrix based on replicator dynamics Transition matrix based on mutation Solve  from T=, 0, =1 Stochastic stability of CDE is problematic

21. “Stochastic stability”: Simulations • F(X)=X • Fixed: n=100, xl=100, xh=120, =1000, =0.5, =300, w=5 • Varying:  (initial value 280) • nxl=XP

22. “Stochastic 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

23. “Stochastic stability”: Shares and spatial 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.

24. “Stochastic 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)

25. 4. Sophisticated interaction on the 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))

26. Benchmark

27. Analysis • 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

28. 5. 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

29. Simulations • 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

30. Defector Cooperator Enforcer Example 1z(0)=(5%, 2%, 93%) t =1 t =2 t =3 t =4 t >5 t =5

31. Defector Cooperator Enforcer Example 2z(0)=(30%;40%,30%) t =1 t =3 t =4 t =2 t =5 t =6 t =7 t >7

32. Observations • 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

33. 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.

34. C C z50 C C z0 z50 z50 z0 D E D z0 E z0 D E D z50 E Simulation: spatial distribution z(0)=(30%;40%;30%). Random spatial distribution • No CD, C or E equilibria. • Strategies CDE can coexist in the long-run. D Z(50)=(1;0;0) DE Z(50)=(0.91;0.09;0) CE Z(50)=(0;0.26;0.74) CDE Z(50)=(0.37;0.29;0.34)

35. Simulation: shares and spatial distribution • 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

36. C D E allD equilibria • D equilibria

37. C D E DE equilibria • DE equilibria

38. CE equilibria • CE equilibria

39. C D E CDE equilibria • CDE equilibria

40. Summary • C: 0.4% (0.4%)* • D: 41% (79%) • E 3% (3%) • CE: 18% (23%) • DE: 18% (0%) • CDE: 20% (0%) *between brackets sophisticated rule

41. Variation in price and sanction • D equilibria best attained for • low sanctions • high harvest price • small population: total effort decreases, profits increase. In contrast with S&S (there higher n makes detection more difficult)

42. Dynamics • xht(Nt)=λxhNtθ • xlt(Nt)=λxlNtθ • r=0.5; NK=1000;N(0)=500; λ=0.05; θ=0.5

43. CDE equilibrium exists

44. D equilibrium

45. CDE equilibrium

46. D equilibrium

47. 6. 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

48. Future research • More on resource dynamics • Alternative replicator dynamics Relevant average payoffs Local Nash equilibria

49. Future research • More analysis with current resource dynamics; • Does fall/rise in stock accelerate/delay convergence to particular strategy equilibrium? • Resource • specification of alternative temporal dynamics, • adding spatial heterogeneity, • adding spatial connectivity and dynamics. • Coevolutionary dynamics: • resource size and composition - fish, pests.