Continuous Versus Discrete Time in Dynamic Common Pool Resource Game Experiments

Abstract

We study the impact of discrete versus continuous time on the behavior of agents in the context of a dynamic common pool resource game. To this purpose, we consider a linear quadratic model and conduct a lab experiment in which agents exploit a renewable resource with an infinite horizon. We use a differential game for continuous time and derive its discrete time approximation. In the single agent setting, we fail to detect, on a battery of indicators, any difference between agents’ behavior in discrete and continuous time. Conversely, in the two-player setting, significantly more agents can be classified as myopic and end up with a low resource level in discrete time. Continuous time seems to allow for better cooperation and thus greater sustainability of the resource than does discrete time.

Appendices

Appendices

Appendix A: The Discretization of the Continuous Time Model

This section presents the procedure adopted to discretize the continuous time model. Let’s consider the following continuous time model:

$$\begin{aligned}&\max \limits _{w(t)} \int _{0}^{\infty }e^{-rt}f(w(t),H(t))dt \nonumber \\&s.t\; \left\{ \begin{array}{ll} \dot{H}(t) = R-\alpha w(t) \\ H(0) = H_{0}\ge 0, H_{0} given \\ H(t)\ge 0\\ w(t)\ge 0 \end{array} \right. \end{aligned}$$

(9)

For the discretization of the model above, let’s consider $\tau$ as the discretization step and n as a period. Time is discretized into intervals of length $\tau$, such that the differential equation and the payoff are approximated in each interval $n\tau$, $(n+1)\tau$. Thus, the discretization of the objective function gives:

$$\begin{aligned} \int _{n\tau }^{(n+1)\tau }e^{-rt}f(w(t),H(t))dt& = \left[ -\frac{e^{-rt}}{r}f(w(n),H(n))\right] ^{(n+1)\tau }_{n\tau }\\& = -\frac{e^{-r(n+1)\tau }}{r}f(w(n),H(n))-\left( -\frac{e^{-rn\tau }}{r}\right) f(w(n),H(n))\\& = \frac{e^{-rn\tau }}{r}\left( -e^{-r\tau }f(w(n),H(n))\right) +\frac{e^{-rn\tau }}{r}f(w(n),H(n))\\& = f(w(n),H(n))\frac{e^{-rn\tau }}{r}\left( -e^{-r\tau }+1\right) \\ \int _{n\tau }^{(n+1)\tau }e^{-rt}f(w(t),H(t))dt& = f(w(n),H(n))e^{-rn\tau }\left( \frac{1-e^{-r\tau }}{r}\right) \end{aligned}$$

Using Taylor’s first order limited development of $e^{-r\tau }$ gives :

$$\begin{aligned} e^{-r\tau }&\simeq 1-r\tau \end{aligned}$$

Thus, the objective function becomes:

$$\begin{aligned} \int _{n\tau }^{(n+1)\tau }e^{-rt}f(w(t),H(t))dt&\simeq f(w(n),H(n))(1-r\tau )^n\left( \dfrac{1-(1-r\tau )}{r}\right) \\& = f(w(n),H(n))(1-r\tau )^n\left( \dfrac{1-1+r\tau }{r}\right) \\ \int _{n\tau }^{(n+1)\tau }e^{-rt}f(w(t),H(t))dt& = f(w(n),H(n))(1-r\tau )^n\tau \end{aligned}$$

The discretization of the dynamics gives:

$$\begin{aligned} H(n+1) = H(n)+\left( R-\alpha w(n) \right) \tau \end{aligned}$$

The discrete time problem can be defined as:

$$\begin{aligned}&\max \limits _{w(n)} \sum \limits _{n = 0}^{\infty } (1-r\tau )^{n}\left[ aw(n)-\frac{b}{2}w(n)^{2}-max\left( 0, c_{0}-c_{1}H(n)\right) w(n)\right] \tau \nonumber \\&s.t\; \left\{ \begin{array}{ll} H(n+1) = H(n)+\tau \left( R-\alpha w(n)\right) \\ H(0) = H_{0}\ge 0, H_{0} given \\ H(n)\ge 0\\ w(n)\ge 0 \end{array} \right. \end{aligned}$$

(10)

The discrete time model therefore converges towards the continuous time model when the discretization step $\tau$ tends toward zero.

In order to see the degree of the approximations used in the experience, with the parameters chosen in the model, Fig. 10 shows the feedback trajectory in continuous time and in the discretizations ($\tau = 0.1$ and $\tau = 1$).

Fig. 10

Feedback equilibrum in continuous and discret time

Appendix B: Figures from Experimental Instructions

See Figs. 11, 12, 13.

Fig. 11

Total revenue from extraction

Fig. 12

Unitary cost of extraction

Fig. 13

Decision-making screen shot. We follow a hypothetical subject who chooses his extraction rate at random