Model Description
This is a simple evolutionary game model (based on replicator equations) that couples evolution of users' social norms and renewable resource dynamics. The reference article, Sethi and Somanathan (1996), provides the following overview of the model. "The problem of extracting commonly owned renewable resources is examined within an evolutionary-game-theoretic framework. It is shown that cooperative behavior guided by norms of restraint and punishment may be stable in a well-defined sense against invasion by narrowly self-interested behavior. The resource-stock dynamics are integrated with the evolutionary-game dynamics. Effects of changes in prices, technology, and social cohesion on extraction behavior and the long-run stock are analyzed. When threshold values of the parameters are crossed, social norms can break down leading generally to the lowering of the long-run stock, and possibly to its extinction".
Reference
Sethi, R., & Somanathan, E. (1996). The Evolution of Social Norms in Common Property Resource Use. The American Economic Review, 86(4), 766-788.
Scenarios
Sethi and Somanathan (1996) does not provide explicit parameter values for readers to replicate their finding. Nevertheless, two regimes of system behavior can still be tested by traversing the parameter space. In the first regime, the combined population of cooperators (s1) and enforcers (s3) is not stable and their social norms evolve in such a way that all of the population eventually turn into defectors (s2). Set the y-axis to the auxilary variable 's13' (combined population of cooperators and enforcers). Set the x-axis to the resource level (k). Set the parameters as the following: p=20, q=0.001, b=0.001, w=0.1, n=8, r=0.2, delta=0.5, gamma=0.001, and k_max=100. Under this setting, the fraction of cooperators and enforcers eventuallybecomes zero (extinction) and the resource level declines significantly.
In the second regime, the population of cooperators and enforcers stays stable at some fixed point and defectors cannot invade. This happens when the penalty/punishment (delta) for defectors becomes sufficiently large. To see this, gradually vary the punishment/penalty parameter (delta) from 0.5 to 7 (rest of the parameters stay same as above). You will see that at some point, cooperators and enforcers replace defectors and the resource level decreases only slightly from the initial value. You can also get the same transition by changing other paramters, e.g., cost of monitoring defectors (gamma).
$\Large \frac{dk}{dt}=rk(1-\frac{k}{k_{max}})-qkX$ |
| Rate of change in resource stock |
$\Large X=((s_{1}+s_{3})x_{po}+s_{2}x_{nash})N$ |
| Total harvesting effort |
$\Large s_{i}=s_{i}(\pi_{i}-\pi_{avg}) \ \ for \ \ i=1,2,3$ |
| Rates of change in different user strategies: cooperator, defector, and punisher (i.e., replicator dynamics) |
$\Large \pi_{1}=\frac{x_{po}}{X}(pqkX-bX^2)$ |
| Payoff of cooperator strategy |
$\Large \pi_{2}=\frac{x_{nash}}{X}(pqkX-bX^2)-\delta(s_{3}N)$ |
| Payoff of defector strategy |
$\Large \pi_{3}=\pi_{1}-\gamma(s_{2}N)$ |
| Payoff of punisher strategy |
$\Large \pi_{avg}=s_{1}\pi_{1}+s_{2}\pi_{2}+s_{3}\pi_{3}$ |
| Average payoff of all strategies |
$\Large x_{nash}=\max_{x_{i}} \frac{x_{i}}{X}(pq\sum x_{i}-b\sum x_{i}^2)$ |
| A harvest level that maximizes individual benefit (Nash equilibrium) |
$\Large x_{po}=\frac{\max_{\sum x_{i}} pq\sum x_{i}-b\sum x_{i}^2}{N}$ |
| A harvest level that maximizes group benefit (socially optimal) |
#=====define parameters par p=20, q=0.001, b=0.001, w=0.1, n=8, r=0.7 par delta=0.5, gamma=0.001, k_max=100 #=====define some hidden variables===== #============economic functions #==================demographicfunctions xn=(p*q*k-w)/(b*(n+1)) xe=(p*q*k-w)/(2*b*n) s3=1-s1-s2 E=((s1+s3)*n*xe+s2*n*xn) A=(p*q*k*E-b*E^2)/E pi_1=xe*(A-w) pi_2=xn*(A-w)-delta*s3*n pi_3=pi_1-gamma*s2*n pi_avg=s1*pi_1+s2*pi_2+s3*pi_3 #=====auxiliary quantities================= aux coop=s1+s3 aux xnash=xn aux xpo=xe #aux crit=1-(xn-xe)*(A-w)/(delta*n) #======right hand sides dk/dt=r*k*(1-k/k_max)-q*k*E ds1/dt=s1*(pi_1-pi_avg) ds2/dt=s2*(pi_2-pi_avg) #=============initial data init k=50, s1=0.5, s2=0.25 @ meth=qualrk @ bound=100 @ total=150 @ dt=0.1 @ xp=k,yp=s2 @ xlo=-5,xhi=119,ylo=-0.1,yhi=1.2 done
Yu JHD, Arizona State University.
Bozicevic M, Arizona State University.
