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Figure 14.7. Comparison of actual and predicted next-period state variables. Actual values from GBFSM (lines), Direct-method predictions (Empty symbols) and Meta-modelling predictions (solid symbols).

the mean predicted values for surplus and each of the state variables with the values that actually were found in the simulation model along the direct method policy path in Figure 14.5.6 The results of these comparisons are presented in Figures 14.7 and 14.8. As seen in Figure 14.7, either method is reasonably precise in the prediction of the next period state variables and neither method is preferred. Both methods err significantly in the prediction of the adult stock in the first few periods, but after four periods none of the prediction of the next period's stock differed by more than five percent and the largest error in the prediction of the catch per unit effort was just over eight percent. The re-

6The results for the meta-method path are qualitatively similar to those from the direct-method path, but are less interesting because of the relatively constant optimal TAC policies used along that path.

Figure 14.8. Comparison of actual and predicted surplus. Actual values from GBFSM (line), Direct-method predictions (Empty symbols) and Meta-modelling predictions (solid symbols).

sults for surplus in each period, however, show important differences between the two methods. The direct method is quite accurate in its prediction of ut, never deviating by more than four percent from the value actually predicted by GBFSM. This indicates either that the model is quite good at predicting the disaggregated stocks (as is suggested by Figure 14.2 above) or that the distribution of the predicted returns in a year is not very sensitive to the distribution of the cohorts, or both. In contrast, the meta-modelling approach is quite inaccurate in its prediction of returns in a year, particularly in the first several years when the optimal TACs are low. In the first several years, the prediction of ut are as much as 23% below the value predicted by GBFSM. This poor prediction of the surplus when the TAC and stocks are low, is probably an important part of the reason that low TACs are avoided in the predicted optimal path found using the meta-modelling approach as shown in Figure 14.5.

6. Conclusions

This chapter has presented an approach to DPSim modelling, conducting dynamic optimization with a large simulation model. When analysts have sought to unify a simulation model to dynamic optimization, they have typically taken the meta-modelling approach. We offer an alternative, which we call the direct method. The meta-modelling approach has an important advan tage in that its computational burden is significantly less than for the direct method. But we find that this benefit may have costs - in our application the direct method provided better results. First, the value function found using the direct method was much more plausible and did not violate monotonic-ity as does the meta-modelling approach. Secondly, the policy path seemed intuitively more reasonable, with low TACs initially, and higher TACs once the stock has recovered. Finally, and most importantly, we found that the direct method's prediction of annual surplus was much better than the prediction from the meta-modelling approach.

As was found in WWG, we believe that optimal TAC management for the Gulf of Mexico's red snapper fishery will involve reductions in the TAC in the short term, followed by expansion in the TAC in the long-term. This policy recommendation follows from the results of the direct method. If we instead used the meta-modelling approach, the policy recommendation would be quite different: greater harvests in the short term and lower harvests in the long term. The solution method makes a difference.

We wish to emphasize two important points. First, in DPSim analysis the optimal simulation runs are carried out with the full simulation model; regardless of the approach taken, the DP models' results are used only to calculate the value of future stocks so that optimal policies at each point in time can be identified. Secondly, although the direct method was preferred here, either approach is a plausible way to work around the curse of dimensionality. Unless general results can be found on which of the two methods is preferred under what conditions, we recommend that analysts use and compare both approaches when attempting to carry out dynamic optimization linked to large simulation models.

Chapter 15

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