LORD: OPTIMUM DATA ACQUISITION 



and its dual allows one to make inferences be- 

 yond simply the attainment of the optimum. 

 This was emphasized adequately by Rothschild 

 and Balsiger in their identification and discus- 

 sion of the various shadow prices, etc. A more 

 serious reservation concerns the very serious 

 analytical and computational difficulties to be 

 anticipated. It is a truism of dynamic program- 

 ming that many more problems may be formu- 

 lated than may be solved, and it is not at all 

 certain at present whether the salmon fisheries 

 problem falls into the soluble category. Thus, 

 the present discussion will be confined to the 

 presentation of the theory, which is self-con- 

 tained. The very difficult problems of formula- 

 tion of the loss functions and the selection of 

 optimum decision rules will be the subject of 

 subsequent investigations. 



THEORY 



It is helpful to think of the salmon run, its 

 assessment, and its management as evolution- 

 ary processes in time. Prior to the start of fish- 

 ing the management biologist has at his dis- 

 posal certain prior information, such as pre- 

 season forecasts, on which to base his early 

 management strategy. As the run proceeds, 

 additional data are gathered so that, as his 

 knowledge of the true state of nature increases, 

 he may modify his strategy to conform more 

 closely to the optimum strategy. This will now 

 be developed more formally. 



Assume that the entire run occurs over a 

 total of ni discrete nonoverlapping time inter- 

 vals. If /2 is the number of fish entering the 

 fishery on the ith day then the total run size, 

 Ntot, is given by 



N, 



tot 





(1) 



i = 1 



Define a parameter vector £ , of arbitrary 

 dimension, that is assumed to characterize all 

 relevant details of the run. As a specific ex- 

 ample, we could define 9 of the /^-dimensional 

 vector (Hj, Hg,. . . '^„, ), i-e., the ith component of 

 £ is the number of fish entering the fishery on 

 the ith day. More generally, we can leave 

 arbitrary and write n,- = n^ (6^). For each 

 known there exists some known set of op- 



timum allocation rules 77. ( )ii = 1, . . . m) 

 where 77. is the optimum fraction of the fish 

 to be allocated to the catch on the ith day. For 

 example, the linear programming formulation 

 of Rothschild and Balsiger provided a set of 

 optimum allocation rules based on a fixed total 

 run size and a two parameter time profile pro- 

 posed by Royce (1965). 



Let D he a. finite set of decision rules and let 

 5., a member of D, be the decision adopted on 

 the ith day. Typically the set D consists of such 

 management decisions as fishery opening or 

 closing, fishing area limitations, etc. For each 

 §• there will be an actual allocation 77. where, 

 in general, both 7?. and i?, will be random 

 variables. The former will depend on the true 

 (unknown) state of nature, 6 , while the latter 

 will also be a random function of 6 as well as 

 of the decision taken, 5. . As the actual and 

 optimum allocations differ, various economic 

 losses will be assumed to accrue, and the aver- 

 age or expected loss will be these losses aver- 

 aged over all possible outcomes. This will be 

 developed more formally after considering the 

 various loss functions. 



We may postulate the existence of an overall 

 loss function that reflects economic losses from 

 all sources. For our purposes we will consider 

 the loss as arising only from 1) the cost of 

 data acquisition, 2) the catch, and 3) the es- 

 capement. Since the catch and the escapement 

 are complementary quantities, their sum com- 

 prising the total run, we could consider either 

 one individually as is done in a subsequent ex- 

 ample using the Ricker (1958) spawner-return 

 relation. However, an individual treatment of 

 each permits the separate discussion of loss 

 functions that are linear, additive, and sepa- 

 rable, as for the catch, and those which are 

 nonlinear and not additive, as will be postulated 

 for the escapement. 



First consider the cost of data acquisition. 

 Generally, the sampling schemes to be used 

 and the level of effort for each are selected 

 prior to the run. There may be in-season varia- 

 tions, such as occasional stream surveys, etc. 

 but the cost associated with these is much less 

 than that allocated prior to the run. We denote 

 the experimental design symbolically by ^ 

 and, in line with the above argument, there is 



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