FISHERY BULLETIN: VOL. 71. NO. 4 



in mind that Sjj has been uniquely determined 

 by past decisions and observations so that, in 

 an implicit sense, the  §.H/ = l,...m) are not 

 generally mutually independent. 



Consider now a typical Bristol Bay salmon 

 fishery. The usual allowable decision rules con- 

 sist of either opening or closing the fishery. In 

 addition the management biologists also have 

 the option of allowing fishing over an increased 

 or decreased area depending on whether the 

 run is larger or smaller than normal. Thus, in 

 the most general case, a total of four distinct 

 strategies is available although it is unlikely 

 that both increased and decreased fishing areas 

 would be used during a single season.'' In the 

 usual case, then, a total of three distinct strate- 

 gies is available each day from which it follows 

 that a total of 3'^' separate courses of action 

 may be pursued during a fishing season nt days 

 long. Typically ni is equal to about 20 days in 

 Bristol Bay so that the total number of allow- 

 able strategies is of the order of 10^. This is not 

 a number to be taken lightly and is an example 

 of what Bellman (1957, p. 6) refers to as "The 

 Curse of Dimensionality." 



The principle of optimality, which is particu- 

 larly useful in multistage allocation processes, 

 may be invoked in an effort to reduce this prob- 

 lem in many dimensions to a sequence of prob- 

 lems in one dimension. Assume that we are at 

 the beginning of the nith time period where the 

 state of the system is characterized by S^ where 

 S„, reflects the observations (Yq, Yi,...y„,_i) 

 as well as past decisions ( 6^, d2,-- 6^_i).Thus 

 the only decision at our disposal is 6^ and 

 presumably an optimal 6^ may be chosen as 

 a function of S^. Consider next the beginning 

 of the (m-l)st time period for which the system 

 is characterized by S^.j. For every ^m-i se- 

 lected and executed the system is transformed, 

 after the acquisition of the data Ym-i- i'''to the 

 state S^ for which an optimal decision has 

 already been obtained. Thus at this stage we 

 need optimize only with respect to 5„^_i. In 

 this manner we can proceed backward to the 

 beginning of the first time period, characterized 



* It is also possible to impose waiting periods for the 

 entry of gear into selected fisheries but this will not be 

 considered here. 



by an Sj depending on Yq at which point an op- 

 timal 5 J is selected. 



This backward recursive scheme is typical 

 of the method of attack on dynamic program- 

 ming problems. For a concise but elegant ex- 

 ample of this technique applied to a simulated, j 

 but numerical, fishery problem see Rothschild 

 (1970). The particular example he used had dis- 

 crete stages in time with a finite number of 

 strategies available for each stage. The desired 

 solution described the optimum sequence, or 

 path, in time of visiting various fisheries, for a 

 fixed total number of time periods, so that the 

 total catch was maximized. While highly ideal- 

 ized, this problem constitutes a true dynamic 

 program. However, it lacks the stochastic fea- 

 tures that are an essential feature of the pres- 

 ent discussion. 



Verbally this method of solution appears to 

 be most attractive since we have apparently 

 overcome the problem of excessive dimension- 

 ality by the recursive consideration of a se- 

 quence of problems of lower dimension. This 

 feature is emphasized in the previously cited 

 example presented by Rothschild. However, in 

 problems of larger scale, either in terms of the 

 number of stages or the number of possible 

 states per stage, a rather more subtle problem 

 of dimensionality appears. This concerns the 

 successive specifications of the states of the sys- 

 tem I S,,\{k = l,...m). We recall that Sf, is char- 

 acterized not only by 6^, the true state of nature, 

 but all prior observations (Yq, Y-^,...Y,^_-i) 



and all prior decisions (§i, S2'--^/j-i)- ^^^ 

 observation vectors are necessarily multidi- 

 mensional, each component of which represents 

 a particular piece of data or the observation of 

 a particular entity. To make matters worse, 

 the dimension of the Y- will generally increase 

 with i since new forms of data, such as catch 

 reports, tower counts, catch per unit of effort, 

 etc., will become successively available. Thus, 

 the I y, \{i = 0,...k -1) required to specify S^ will 

 bring with them their own dimensionality 

 which will soon become overwhelming unless 

 S/j can be described adequately by relatively 

 few parameters. Obviously, aside from possible 

 computer core limitations, the most useful al- 

 gorithms are those that can accommodate the 

 requisite dimensionality. In a more practical 



1036 



