FISHERY BULLETIN: VOL. 71, NO. 4 



associated with 'c; some fixed cost C{i;). The 

 quanitity ^ , which is an abstract designation 

 of the experimental design, will also appear as 

 a conditioning quantity when we consider the 

 various probability density functions associated 

 with the sampled quantities, i.e., the distribu- 

 tion of the sample estimates will depend on, 

 among other things, the manner in which the 

 data are acquired. 



Next is the loss associated with the catch. 

 Consider the start of the Ath time i)eriod where 

 1 ^ A' ^ m but otherwise arbitrary. If the eco- 

 nomic value of the catch is assumed to be linear 

 and additive, we may write 

 k-i 



^x(/0=2^<l^/(^)-^J",(£) (2) 



I = 1 



as the loss through the (k - l)st time period for 

 the catch.-' Here v. is the unit value of the fish 

 for the /th interval. Note that if 77. exceeds 

 T?,-, i.e., the actual catch exceeds the optimum 

 catch, the loss function is negative and becomes 

 a gain function. However, this apparent bene- 

 fit must be offset by the loss associated with 

 the corresponding decreased escapement. If 

 this were not the case, then this would not be 

 the correct optimum since any departure from 

 the optimum must result in a nonnegative in- 

 cremental loss. 



Finally consider the loss function for the 

 escapement. This function cannot be considered 

 to be linear or additive since the average num- 

 ber of returns per unit of spawners escaping on 

 any particular day will be a function primarily 

 of the final value of the total escapement. This 

 is a consequence of the fact that late spawners 

 may interfere with the redds of the earlier ar- 

 rivals and thus diminish the returns for this 



^ If the capacity of the cannery becomes limiting, as 

 may happen in cycle years, the loss function for the catch 

 may be written in slightly more general form as 



k - 1 

 L^(K) = E V. JT}.«.(ff) - min [^./i.(9), cap(/)l} (2') 

 1=1' ' 



where cap(/) denotes the cannery capacity for the ;th time 

 period. This relation implies that any actual catch that 

 exceeds the cannery capacity will not decrease the cor- 

 responding loss. This function is no longer linear but it is 

 still additive. Note also that no cost has been ascribed to 

 the additional economically nonproductive fishing effort. 

 Cannery capacity was one of the constraints imposed by 

 Rothschild and Balsiger in their paper. 



group. Also, excessive escapement may lead to 

 increased competition for food among the fry to 

 the general detriment of the population as a 

 whole. Thus, to a first approximation, the loss 

 function for the escapement depends only on 

 the final values of the actual and optimum es- 

 capements. Symbolically this may be written as 



L^im) = {E, E) 



where 



m 



^ = 2<i-^<>'^<<£> 



(3) 



(4a) 



/ = 1 



is the optimum total escapement while 



m 



(4b) 



i = 1 



is the actual escapement and (■,) denotes 

 some suitable functional form. An even more 

 general formulation is possible if the ])0ssibility 

 is admitted that the magnitude and timing of 

 the arrivals on the spawning beds are also sig- 

 nificant. The loss function must still be ex- 

 pressed in terms of the entire run but the func- 

 tional form would be of the type 



where £. and £. are, respectively, the optimum 

 and actual escapements for the /th time period. 

 However, the determination of the optimum 

 total level of escapement, which is necessary 

 to characterize {E, E), is a subject of current 

 research and is by no means resolved at present. 

 Thus, the characterization of a function of the 

 g^enerality of (})'{E^,E^, . . . E^; E^,E^, . . . 

 E^j ) must await further biological data. 



The Bayes risk is defined as the average or 

 expected loss where the averaging is over all 

 l^ossible outcomes and an optimum strategy 

 will be defined as that strategy that minimizes 

 the Bayes risk.^ An expression for the Bayes 

 risk will now be constructed that is api)ropriate 

 for the salmon management problem just out- 



* The specification of the minimum Bayes risk as the 

 criterion of optimalily, while a reasonable one, is some- 

 what arbitrary. Other criteria are in common use, most 

 notably the "Minimax," in which the optimum strategy is 

 that which minimizes the maximum risk. 



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