## Abstract

This paper presents a framework for the study of policy implementation in highly uncertain natural resource systems in which uncertainty cannot be characterized by probability distributions. We apply the framework to parametric uncertainty in the traditional Gordon–Schaefer model of a fishery to illustrate how performance can be sacrificed (traded-off) for reduced sensitivity and hence increased robustness, with respect to model parameter uncertainty. With sufficient data, our robustness–vulnerability analysis provides tools to discuss policy options. When less data are available, it can be used to inform the early stages of a learning process. Several key insights emerge from this analysis: (1) the classic optimal control policy can be very sensitive to parametric uncertainty, (2) even mild robustness properties are difficult to achieve for the simple Gordon–Schaefer model, and (3) achieving increased robustness with respect to some parameters (e.g., biological parameters) necessarily results in increased sensitivity (decreased robustness) with respect to other parameters (e.g., economic parameters). We thus illustrate fundamental robustness–vulnerability trade-offs and the limits to robust natural resource management. Finally, we use the framework to explore the effects of infrequent sampling and delays on policy performance.

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## Notes

We use the terms “policies” and “control laws” interchangeably in this paper.

LTI means that the control laws are described by linear ordinary differential equations with constant coefficients.

To compare the performance of different policies as we do, the max–min criterion could be used but it would have to be normalized in some way. Our worst-case downside sensitivity is a particular choice of such a normalization.

While the sensitivity ordering for the small IC cases was obtained via simulation, the rankings for the parameters

*p*,*c*, and*δ*were corroborated graphically using the easy-to-derive explicit small IC formula for the optimal return: \(J_o^* = \left [ \frac{ e^{-\delta t_e^*} - e^{-\delta T} }{\delta} \right ] \left [ \; p q x_e^* - c \; \right ] u_e^*\), \(t_e^* = \frac{1}{r} \ln \left [ \frac{x_e^*}{x_o} \left ( \frac{k - x_o}{k - x_e^*} \right ) \right ]\). This can be done because these parameters have no influence at all on the evolution of the state dynamics. For*r*,*k*,*q*,*x*_{ o }, limit cycles occur even for small perturbations. Given this, an explicit formula is impossible to obtain, and simulation becomes essential. Simulation is also needed for the large IC cases because an explicit formula is much harder to obtain—even for the economic parameters.This required \(4\mathord{,}605 \times 61 \times 7 \times 4 = 7\mathord{,}865\mathord{,}340\) (nonlinear) simulations and took approximately 150 h or 70 ms per simulation on a Windows XP, 3.2 GHz, 1 GB RAM, 600 MFLOP PC.

To illustrate the ideas, we considered ±30% parametric uncertainty for all bioeconomic parameters of interest. In general, assuming uniform parametric uncertainty is unrealistic. The (decreasing) parameter uncertainty ordering

*k*,*x*_{ o },*r*,*q*,*p*,*δ*, and*c*is probably more reflective of reality. Studies with such nonuniform parametric uncertainty will be important for future work.Our good controller regions (A for

*r*,*k*,*x*_{ o }, and*δ*and E for*p*,*c*, and*q*) were selected on the basis of BOE, small IC. These controllers are generally not expected to perform well for BUE. If we sufficiently restrict the controllers within regions A and E to controllers near the optimal policy, then the resulting laws will perform well for BUE.We also note that the discontinuities observed for the sensitivity in

*q*are fundamentally due to the fact that different control laws can significantly impact the convexity properties of*J*(*q*). This will receive further attention in future work.The reader should note that the claim \(\underline{S}_{c_o}^{J_o} = 0\) when

*u*= 0 does not follow from the*c*_{ o }-sensitivity formula in Eq. 54: \(\underline{S}_{c_o}^{J_o} = S_{-} = - \frac{c_o}{J_o} \; \left ( \int_{0}^{T} e^{-\delta_o \tau} sat( u(\tau) ) \; d\tau \right ) < 0\). This is because the denominator term*J*_{ o }also goes to zero as*u*goes to zero. Our sensitivity reduction discussion shows that the claim is, in fact, true.Here, the reader should interpret a “large” (or “small”) harvesting effort as one for which the exponentially weighted harvesting effort \(\int_{0}^T e^{-\delta \tau} {\mbox{sat}}(u(\tau) ) d\tau\) (roughly speaking) is large with respect to that associated with the optimal control policy. This idea is stated precisely for the economic parameters

*p*,*c*, and*δ*within Technical Supplement TS.7.It should be noted that for

*x*_{ o }, this pattern is only observed for BOE. For BUE, the pattern is more complex and requires further investigation. This will be addressed in future work.e.g., the standard one, two, and infinity norms: \({\left\Vert {\theta}_1\right\Vert} {\buildrel \rm def \over =} \sum_{i=1}^n |\theta_i|\), \({\left\Vert {\theta}_2\right\Vert} {\buildrel \rm def \over =} \sqrt{ \sum_{i=1}^n |\theta_i|^2 }\), \({\left\Vert {\theta}_{\infty}\right\Vert} {\buildrel \rm def \over =} {\rm max}_{i=1,\dots,n} |\theta_{i}|\).

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## Acknowledgements

The authors would like to thank Elinor Ostrom, William Brock, Charles Perrings, and Ann Kinzig for helpful comments on earlier drafts of this manuscript.

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The authors gratefully acknowledge financial support for this work under National Science Foundation grant BCS-0527744.

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Rodriguez, A.A., Cifdaloz, O., Anderies, J.M. *et al.* Confronting Management Challenges in Highly Uncertain Natural Resource Systems: a Robustness–Vulnerability Trade-off Approach.
*Environ Model Assess* **16**, 15–36 (2011). https://doi.org/10.1007/s10666-010-9229-z

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DOI: https://doi.org/10.1007/s10666-010-9229-z

### Keywords

- Resource management
- Uncertainty
- Robust control
- Policy implementation
- Learning
- Vulnerability