On Limits to the Use of Linear Markov Strategies in Common Property Natural Resource Games

Abstract

We derive conditions that must be satisfied by the primitives of the problem in order for an equilibrium in linear Markov strategies to exist in some common property natural resource differential games. These conditions impose restrictions on the admissible form of the natural growth function, given a benefit function, or on the admissible form of the benefit function, given a natural growth function.

Appendix

In this appendix, we verify the conditions for existence of an equilibrium in affine strategies, given the utility function (17).

Assume the strategies φ _j (x(t)) of the n − 1 players j ≠ i to be given by φ _j (x(t)) = γ _j  + δ _j x(t). Then the equation of motion facing player i in choosing his own decision rule c _i  = φ _i (x(t)) becomes:

$$\dot{x} = g(x)-c_{i}-x\sum_{j\neq i}\delta _{\!j} - \sum_{j\neq i}\gamma_j,$$

(43)

and the current value Hamiltonian associated to his problem becomes:

$$H(x,c_{i},\lambda_{i}) = u(c_{i})+\lambda_{i}\left[ g(x)-c_{i}-x\sum_{j\neq i}\delta _{\!j} + \sum_{j\neq i}\gamma_j\right].$$

(44)

Assume φ _i (x(t)) = γ _i  + δ _i x(t) to be a solution. Then:

$$\frac{\dot{c}_i}{c_i}=\frac{\delta_i\dot{x}}{\delta_i x + \gamma_i},$$

or, substituting from Eq. 43:

$$\frac{\dot{c}_i}{c_i}=\frac{\delta_i}{\gamma_i + \delta_i x}\left[g(x)-c_{i}-x\sum_{j\neq i}\delta _{\!j} - \sum_{j\neq i}\gamma_j\right].$$

(45)

Furthermore, from the necessary conditions (7) and (8), along an interior solution,

$$\frac{\dot{c}_{i}}{c_{i}}=\frac{1}{\eta(c_{i})}\left[ g^{\prime }(x)-\sum\limits_{j\neq i}\delta _{\!j}-r_{i}\right].$$

(46)

Equalizing Eqs. 45 and 46, we find that for any utility function u(c _i ), in order for c _i  = γ _i  + δ _i x to be a best response, the following differential equation must be satisfied:

$$[\gamma_i + \delta_i x]g^{\prime }(x)- \delta_i{\eta(\gamma_i + \delta_{i}x)}g(x)=[\gamma_i + \delta_i x]\left( \sum\limits_{j\neq i}\delta_{\!j}+ r_{i}\right) - \delta_i{\eta(\gamma_i + \delta_{i}x)}\left(x\sum_{k=1}^{n}\delta_{k}+ \sum_{k=1}^{n}\gamma_{k}\right),$$

where δ _i , γ _i , the δ _j ’s and the γ _j ’s, j ≠ i remain to be determined.

With η(γ _i  + δ _i x) = θ ≠ 1, this becomes:

$$\left(\gamma _{i} + \delta _{i}x\right) g^{\prime }(x)-\theta \delta _{i}g(x)=Ax+B$$

(47)

where

$$A=\delta _{i}\left[ r_{i}-\theta \delta _{i}+\left( 1-\theta \right) \sum\limits_{j\neq i}\delta _{\!j}\right]$$

and

$$B=\gamma _{i}\left[ r_{i}+\sum\limits_{j\neq i}\delta _{\!j}\right] -\theta \delta _{i}\sum_{k=1}^{n}\gamma _{k},$$

which has as solution:

$$g(x)=\frac{1}{\delta _{i}^{2}}\left[ \frac{A\gamma _{i}-B\delta _{i}}{\theta}-\frac{A\gamma_i}{\left( \theta -1\right) }\right]-\, \frac{A\delta _{i}}{\theta - 1}x + k\left[\gamma _{i} + \delta _{i}x\right] ^{\theta },$$

k being the constant of integration.

Therefore, in order for φ _i (x(t)) = γ _i  + δ _i x(t) to be a best response to affine strategies on the part of player i’s rivals, the growth function must be of the form:

$$g(x) = \alpha x + \beta + k[\nu + \sigma x]^{\theta}.$$

Differentiating and substituting into Eq. 47 we get:

$$(1-\theta)\delta_i \beta x + \theta k \left[\sigma\left(\frac{\gamma_i + \delta_i x}{\nu + \sigma x} \right) - \delta_i \right]\times(\nu + \sigma x)^{\theta} + \gamma_i \beta - \theta \delta_i \alpha = Ax + B.$$

Choosing k = 0, constant equilibrium values of δ _i and γ _i , i = 1, ..., n are obtained by simultaneously solving:

$$\theta \delta_i + (\theta -1)\sum_{j\neq i}\delta_{\!j} - r_i - (\theta -1)\alpha = 0$$

(48)

and

$$\left[\alpha - r_i - \sum_{j\neq i}\delta_i\right]\gamma_i + \theta \delta_i \sum_{k=1}^{n}\gamma_k - \theta \delta_i \beta = 0.$$

(49)

It is easily verified from Eq. 49 that

$$\gamma_i = \frac{\beta \delta_i}{\alpha}$$

is the solution for γ _i , with δ _i obtained from Eq. 48.

The existence of an equilibrium in affine strategies therefore requires that g(x) be an affine function of the stock (β ≠ 0). To be implementable, β should be negative (δ _i being positive), as otherwise it would imply that a positive quantity can be harvested from a zero stock. As argued in the text (footnote 5), β < 0 is also necessary for g(x) to be a sensible representation of the growth of a renewable resource stock. Hence, the decision rule will be c _i  =  max {0, γ _i  + δ _i x} so as to satisfy the necessary conditions (Eq. 7). Note that the solution for δ _i is the same as in Eq. 21.

Similar calculations can be carried out to solve for an equilibrium in affine strategies in the case of θ = 1.