FISHERY BULLETIN: VOL. 71, NO. 4 



for the risk through the kth time period. Al- 

 though we are considering the start of the kth 

 time period, Vfe-i ^^'^ have been acquired so 

 that 6^ may be chosen on the basis of this and 

 all previous observations. Similarly, the risk 

 over the remainder of the run is given by 



^3,.(§/e.i^5,,2'...5,jyo,yi,...y,_i;C) = 



2 ^,Hf/e(^ll0'Il'-I/e-i;0 

 i = k 1 



/rf^,(r?,(£)-7?,)n,(£)g,(7?,|5,;£) (8) 



where, in an implicit sense, the decisions Sfe + i, 

 5;j+2,...S,„ will be conditional upon the prior 

 decisions 5i,^2^--^k ^^ well as upon the ob- 

 servations Yq, yi,.-^/c-i- Equations (7) and (8) 

 have a similar structure although in Equation 

 (7) we are weighting past decisions by our 

 present knowledge of the state of nature while 

 in Equation (8) the future decisions must 

 necessarily reflect only the information ob- 

 tained through the (k-l)st time period. 



The risk for the escapement is assumed to be 

 nonseparable so that the entire run must be 

 considered at once. The general expression for 

 this portion of the risk is then given by 



^4,/. (5i,52'-5, 



Y Y 



l.-i;0 = 



/cfT7,„ 0[£(O),^(T7i,772,...7?,„)l 7rg,(^,|5,;£). 



I = 1 



(9) 

 The total Bayes risk is then the sum of the risks 

 given by (6), (7), (8), and (9), i.e.. 



R 



. =S«u,. 



/ = 1 



(10) 



These equations have been derived under 

 very general conditions and assumptions with 

 little effort towards characterizing any of the 

 functions indicated. It is interesting to note, 

 however, that loss functions of the same general 

 form as those appearing in the integrands of 



(7), (8), and (9) may be obtained if we assume 

 1) a steady state spawner-return relation of the 

 type proposed by Ricker (1958) and 2) the 

 economic loss (or gain) is proportional to the 

 catch. The steady state Ricker spawner-return 

 relation is 



^tot = ^^^"'^^ 



(11) 



where the parameter vector = (0..0.^) 

 describes the run and N^^^ and E denote the 

 actual run size and the escapement respectively. 

 The corresponding catch is given by 



X=0,Ee-^''^-E 



(12) 



from which it follows that the optimum escape- 

 ment, E, is given by the solution of 



dX 

 dE 







or 



d^ ( 1 -O^E)e~^'^^ = 1. 



(13) 



The appropriate loss function in terms of the 

 catch is 



L (X, X) = \{X-X) 



(14) 



where v is the average unit value of the fish. 

 The equivalent expression 



L' {E, £) = V [0, (£e"^2^ - £e"^2^ ) - {E 



E)] 



(15) 



in terms of the escapement is easily obtained by 

 substituting (12) in (14). The term v 6 -^ 

 [exp {-d ^E) - exp {-d ^E)] is nonlinear while K 

 V [E - E) is a linear and additive function of |H 

 the daily escapements. The substitution ofJBft 

 (15) into (9) would then result in an integral 

 of the same general form as (9) and two addi- 

 tional integrals corresponding to (7) and (8). 

 As indicated earlier, the experimental design. 



1034 



