LORD: DECISION THEORY APPLIED TO SALMON FISHERY 



minimizes the Bayes risk given that all prior 

 decisions were optimum for the time periods in 

 which they were made. In other words, the 

 "hindsight" feature was not utilized to "improve" 

 a past decision— once made any decision is retained 

 through all subsequent stages. 



The mathematical machinery developed gen- 

 erally gives intuitively reasonable results. 

 Specifically, the tendency toward larger or smaller 

 run sizes results in optimum strategies that tend 

 successively toward more or fewer open periods 

 respectively. The Bayes risk generally, but not 

 always, decreases as the season progresses, largely 

 reflecting the decreasing variances in the es- 

 timates of the run size. Increases in the Bayes risk 

 can usually be attributed to past decisions that, in 

 the light of subsequent sampling, are no longer 

 optimum thus requiring corrective action. 



CONCLUSIONS 



The mathematical models assumed and 

 developed here for the objective management of a 

 typical salmon fishery, as previously noted, are 

 based on quite specific functional forms and thus 

 represent somewhat of an idealized situation. 

 However, these functions were chosen to reflect 

 the behavior of the system insofar as the knowl- 

 edge of such behavior is available. Indeed, the 

 acquisition of such detailed knowledge is an im- 

 portant area of current research and subsequent 

 refinements of the statistics will be possible as 

 more data are gathered. 



Of more concern than the accuracy of the fine- 

 scale mathematical behavior of the system is the 

 appropriateness of the basic mathematical theory 

 upon which the models are built. I feel that statis- 

 tical decision theory is a most natural framework 

 on which to base an objective management model. 

 The nomenclature lends support to this view. For 

 example, the equivalence of a management deci- 

 sion and a statistical decision is obvious.^ The term 

 risk, in the economic if not the strict Bayesian 

 sense, is frequently used in discussions of fishery 

 management. Finally, Bayes theorem provides a 

 convenient and theoretically appropriate method 

 for accommodating the combined data acquisition 

 and dynamics of the fishery. 



®This equivalence is not always evident even within decision 

 theory itself. For example, it requires a slight mental contortion 

 to treat statistical estimation as an application of decision theory 

 as the statisticians have done. 



Advantage has been taken of some powerful 

 analytical tools to characterize salmon fishery 

 management. However, any enthusiasm for these 

 quite contemporary methods should be tempered 

 somewhat by consideration of some of the specific 

 practical difl^culties likely to be encountered. One 

 of these, mentioned in Lord (1973), is the difl^culty 

 associated with multistage dynamic processes. 

 While the fishery management problem under 

 discussion falls very naturally into a class of 

 stochastic dynamic programs it is not yet obvious 

 whether the functional equation arising from the 

 imposition of the principal of optimality can be 

 formulated or solved in a useful fashion. The 

 calculations done here were more of the brute 

 force variety in which all strategy combinations, 

 optimal or not, were considered. In other words, 

 the backward recurrence scheme central to dy- 

 namic programming was not used to reduce the 

 total number of possible strategies to be con- 

 sidered. In so doing, the "Curse of Dimensionali- 

 ty" about which Bellman (1957:6) so aptly warned, 

 proved to be a limiting condition. To evaluate 

 completely the five-stage, two-decision fishery 

 considered here required from 10 to 15 min of 

 Control Data Corporation^ 6400 central processor 

 time for each set of input parameters. This is not a 

 trivial numerical effort and should give one pause 

 when considering more elaborate models. 



In conclusion I feel that advantage should be 

 taken of the appropriate analytical tools as they 

 are made available by the mathematicians or, at 

 the very least, such tools should be investigated. 

 However, the availability of such methods in no 

 way indicates their eventual practicality for any 

 specific problem. For this careful additional in- 

 vestigation is necessary. 



LITERATURE CITED 



Bellman, R. 



1957. Dynamic programming. Princeton Univ. Press, 340 p. 



BlERENS DE HaAN, D. 



1939. Nouvelles tables d'integrules definies. G. E. Stechert 

 & Co., 716 p. 

 DeGroot, M. H. 



1970. Optimal statistical decisions. McGraw-Hill Book Co., 

 Inc., 489 p. 

 Lord, G. E. 



1973. Characterization of the optimum data acquisition and 

 management of a salmon fishery as a stochastic dynamic 

 program. Fish. Bull., U.S. 71:1029-1037. 



^Reference to trade names does not imply endorsement by the 

 National Marine Fisheries Service, NOAA. 



845 



