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Fishery Bulletin 90(3), 1992 



Gushing exponent q is constrained to fall between 

 and 1. In the limiting case of q = 0, recruitment is con- 

 stant, while in the other limiting case of q= 1, recruit- 

 ment is proportional to biomass. 



Differentiating Equation (1) with respect to F and 

 setting the resulting expression equal to zero gives the 

 following equation for Fmsy • 



F' 



MSY = 



(q+ 1) K" + 1 + V(q+1)2K"2 + (6q- 2)K" + 1 

 2q 



(2) 



1, 



where F'msy = Fmsy/M. 



A common rule of thumb is that F'msy should equal 

 1. The locus of parameter values for which this rule 

 holds precisely is given by 



1 

 K" = - - 2. 



q 



(3) 



where L[z(F, q)] represents the losses resulting from 

 selection of a particular value of F given a particular 

 value of q, and E(L[z(F, q)]} is the expected value of 

 L[z(F, q)] (the "risk," DeGroot 1970). The minimum 

 value of E (L [z(F, q)] } is referred to as the "Bayes risk" 

 (DeGroot 1970). The integral is taken over the inter- 

 val to 1 because the Gushing stock-recruitment rela- 

 tionship constrains q to that range. 



The Bayes decision can be derived by differentiat- 

 ing E {L[z(F, q)]} with respect to F and solving for the 

 value that sets the derivative equal to zero. The valid- 

 ity of this procedure requires that all parameter values, 

 including those describing P(c[), remain constant into 

 the future. The solution corresponding to such an 

 assumption is sometimes known as a "myopic Bayes" 

 solution (Ludwig and Walters 1982, Mangel and Glark 

 1983, Mangel and Plant 1985, Parma 1990). A more 

 general alternative is to allow for the possibility that 

 parameter estimates will be updated in the future, 

 but this approach is vastly more difficult (Glark et al. 

 1985, Mangel and Plant 1985, Walters and Ludwig 

 1987). 



Analyzing the model 

 in a Bayesian framework 



Parameter estimates in any model are by definition 

 associated with some degree of uncertainty. For ex- 

 ample, parameters governing the stock-recruitment 

 relationship are particularly difficult to estimate 

 precisely (Larkin 1973, Paulik 1973, Ludwig and 

 Walters 1981, Walters and Ludwig 1981 and 1987, 

 Shepherd 1982, Glark 1985, Glark et al. 1985, Roths- 

 child and Mullen 1985, Shepherd and Gushing 1990). 

 In the presence of such uncertainty, a Bayesian ap- 

 proach would use the probabilities associated with 

 different parameter values to weight the losses (i.e., 

 opportunity costs to society) associated with choosing 

 a particular fishing mortality rate. Following similar 

 studies by Ludwig and Walters (1982), Glark et al. 

 (1985), and Walters and Ludwig (1987), the present 

 analysis will focus on the uncertainty surrounding a 

 single parameter, in this case the stock-recruitment 

 exponent q. This uncertainty takes the form of a prob- 

 ability density function (pd^ P(q) which describes the 

 relative credibility of alternative q values. 



To simplify notation, define z(F, q) as the ratio of 

 Y(F) to MSY for an arbitrary value of q drawn from 

 P(q). Then, the "Bayes decision" (DeGroot 1970) is the 

 value of F that minimizes 



E{L[z(F, q)]} 



X' 



L[z(F, q)l P(q) dq, (4) 



Minimizing risl< under 



a logarithmic loss function 



Of course, specification of the functions L and P is 

 crucial to this problem. Following Lord (1976) and Lud- 

 wig and Walters (1982), one possible choice is to assume 

 that L is a linear function of z(L(z)=l-z). Another 

 common form is the quadratic L(z) = (l -z)-. which has 

 been used in the fisheries literature by Walters (1975), 

 Hightower and Grossman (1987), and Gharles (1988). 

 One of the oldest alternatives is the logarithmic loss 

 function, L(z)= -ln(z), dating back to the work of Ber- 

 noulli in 1738 (transl. 1954). Logarithmic loss (or, con- 

 versely, utility) seems first to have been used in the 

 fisheries literature by Gleit (1978), followed by Lewis 

 (1981, 1982), Mendelssohn (1982), Opaluch and Bock- 

 stael (1984), Ruppert et al. (1984, 1985), Deriso (1985), 

 Walters (1987), Walters and Ludwig (1987), Getz and 

 Haight (1989), Hightower and Lenarz (1989), High- 

 tower (1990), Parma (1990), and Parma and Deriso 

 (1990). 



Linear, quadratic, and logarithmic loss functions are 

 compared in Figure 1. As Figure 1 indicates, the 

 logarithmic loss function corresponds to a "preserva- 

 tionist" viewpoint, in which extinction of the stock is 

 absolutely unacceptable (i.e., the loss corresponding to 

 extinction is infinite). Because the logarithmic loss func- 

 tion is clearly identifiable as a risk-averse alternative 

 function (see Discussion), it is a good candidate for il- 

 lustrating how a Bayesian approach can differ from 

 more traditional approaches which do not incorporate 

 uncertainty in an explicit fashion. 



