LORD: DECISION THEORY APPLIED TO SALMON FISHERY 



limiting behavior as the appropriate parameters 

 assume their extreme values. However, Equations 

 (18) and (20) do have the virtue that, in closed 

 form, the most crucial features of the fishery 

 dynamics and statistics are accommodated in a 

 quantitative and, hopefully, reasonably accurate 

 fashion. 



A NUMERICAL EXAMPLE 



The foregoing mathematical model was applied 

 to the simulated management of the Wood River 

 system of Bristol Bay. It should be emphasized at 

 the outset, however, that the assumptions, meth- 

 ods, and results presented here should in no way be 

 construed as representing a management scheme 

 preferable to those currently in use. The Wood 

 River was chosen simply because, based on Math- 

 ews' (1966) data, it seemed to follow the Ricker 

 spawner-return curve reasonably well. 



In the example considered here, the model was 

 limited to a fishing season of five time periods 

 during each of which a choice of two management 

 decisions was possible. This limitation was neces- 

 sary to avoid inordinately lengthy calculations. 

 Ricker parameter values of a = 4.077 and b = 

 0.8 X 10~^, which were used by Mathews, were 

 used here. The return was assumed to consist of 

 only the progeny of a single spawning group K 

 years prior where K is arbitrary, i.e.. 



Pi = 



1 i = K 

 i jt K 



All prior escapements were assumed to be the 

 optimum escapement £"0 so that the loss function 

 given by Equation (5) becomes 



A = {aEoe- *^o -Eo)- (aE,, c ''^„ - £" ) . 



For the above values of the Ricker parameters, 

 the MSY escapement is given by £"0 = 709,000. 

 The expected value and standard deviation of a 

 r(«o. Po) variate are given by ao//So and aQ^Z/^Q, 

 respectively. In terms of the Ricker parameters, 

 the expected run size is given by aEo exp(- bEo) 

 which determines the ratio oq/Pq = 1.64 x 10*". An 

 initial (i.e., preseason) standard deviation of 

 one-half the expected run size was assumed. In 

 terms of the gamma parameters this gives 

 ao'V^o= ao/2fio or oq = 4.0 and /?o = 2.44 X 10^. 

 The two management strategies assumed were 

 complete closure (option 2) and one level of open- 



ing (option 1). In terms of the beta parameters, 

 closure is simulated merely by setting /X2 = with 

 an arbitrary positive value for ^2. During fishery 

 opening it was assumed that an average of 80% of 

 the available fish are caught with a standard 

 deviationof 0.25. This gives (;' J, fij) = (0.312,1.248) 

 as the appropriate beta parameters. The set of run 

 fractions {ft) (1 = 1, . . . 5) was determined from 

 the time profile proposed by Royce (1965). Values 

 of 0.156, 0.282, 0.348, 0.160, and 0.054, using five 

 equal length time intervals, were obtained. No 

 attempt was made to treat the run fractions as 

 random variables. All of the parameter values 

 were chosen to reflect reasonably well the known 

 behavior of the system. 



The fishery dynamics were treated by two 

 distinct methods. The first method utilized the 

 gamma prior density for A'^ with a Poisson sam- 

 pling density thus, through conjugacy, giving a 

 gamma posterior density. A gamma posterior 

 distribution was also assumed in the second 

 method but the posterior gamma parameters were 

 back-calculated after introducing prescribed 

 stage-to-stage trends in the population mean and 

 standard deviation. 



The Bayes risk at each stage was computed for 

 each of the 2^ = 32 total possible sequences of 

 decisions, past, present, and future; i.e., no at- 

 tempt was made to formulate and solve the func- 

 tional equation associated with dynamic pro- 

 gramming.^ While relatively unsophisticated, this 

 approach does permit one to use hindsight to 

 determine, ex post facto, what an optimum 

 previous strategy would have been, given the 

 information currently available. In real life, of 

 course, "what might have been" is irrelevant in the 

 management of a dynamic system-one must 

 optimize the system as it exists in real time in 

 accordance with the principal of optimality, the 

 relevant homily for which might well be "what's 

 past is prologue." 



The numerical results are summarized in Tables 

 1 to 3. Tables 1 and 2 give the optimum strategies 

 and corresponding minimum Bayes risks for a 

 gamma prior run size distribution with simulated 



^Subsequent to the submission of this paper, C. J. Walters 

 (1975) published a paper in which the ideas of dynamic pro- 

 gramming were applied to the optimum year to year man- 

 agement of a salmon fishery. His work is of considerable interest, 

 particularly since he managed to impose the principle of op- 

 timality and carry out the backward recursive scheme proposed 

 by Bellman (1957). It remains to be seen if this method can be 

 applied to the decision theoretic model presented here, but I am 

 no longer as pessimistic as I formerly was. 



843 



