Thompson Bayesian approach to management advice from a simple dynamic pool model 



569 



Bayesian estimate (m). These two Fmsy values bracket 

 the value of 0.155 which Walters and Wilderbuer (1988) 

 derived from a surplus production model. Regardless 

 of which Fmsy value is chosen, however, it exceeds 

 Fmelsy by a significant amount. 



Discussion 



Evaluation of assumptions 



The approach described here consists of three main 

 components: the basic model represented by Equation 

 (1), the logarithmic loss function, and the beta form for 

 P(q). These components were chosen in part because 

 they are tractable, making possible the analytic solu- 

 tion for F'melsy given by Equation (20). In addition, 

 each has some degree of theoretical support, as de- 

 scribed below. 



The basic model The basic model was evaluated by 

 Thompson (1992). In brief, the model includes terms 

 for all of the requisite features of dynamic pool models 

 (recruitment, growth, natural mortality, fishing mor- 

 tality). The distinguishing features of the model (linear 

 growth and a Gushing stock-recruitment relationship) 

 satisfy the principal theoretical requirements for 

 growth and stock-recruitment functions given by 

 Schnute (1981) and Ricker (1975), respectively. Al- 

 though the basic model is a simple one, it approximates 

 more complicated models fairly well under a wide range 

 of parameter values. 



Logarithmic loss function The logarithmic loss func- 

 tion may require a bit more discussion. As mentioned 

 earlier, this loss function is only one of several pos- 

 sibilities, two of the other most-common being the 

 linear and quadratic forms. The principal argument 

 against the linear loss function is that it implies strict 

 risk neutrality, whereas most individuals tend to be at 

 least somewhat risk-averse. Thus, if fishery managers 

 tend to be risk-averse, a linear loss function would be 

 inappropriate, except over a narrow range of yield 

 values. 



In contrast, the quadratic loss function implies a 

 degree of risk aversion. In addition, the quadratic form 

 has properties which prove convenient for a number 

 of statistical applications. However, it has also been the 

 subject of substantial criticism (Pratt 1964, Samuelson 

 1967, Box and Tiao 1973). Although the quadratic loss 

 function does fall into the "risk-averse" category, this 

 functional form manifests its risk aversion somewhat 

 perversely by exhibiting increasing absolute risk aver- 

 sion (Pratt 1964). In other words, a fishery manager 

 using a quadratic loss function would be less willing to 

 take risks as yields became higher. 



The logarithmic loss function is another risk-averse 

 alternative. It can be described as a special case of the 

 isoelastic marginal loss function defined by L(z) = 

 (l-z'^)/<t>, where ^>0 (the logarithmic case being ob- 

 tained in the limit as <t> approaches zero). Unlike the 

 quadratic loss function, isoelastic marginal loss func- 

 tions exhibit decreasing absolute risk aversion (Pratt 

 1964). Isoelastic marginal loss functions also display 

 the conveniens property of constant relative risk aver- 

 sion R(z), defined as -zL"(z)/L'(z) (Pratt 1964). Spe- 

 cifically, R(z) = l-it> for the isoelastic marginal loss 

 family. The logarithmic case, where R(z)=l, thus 

 represents a clear risk-averse alternative to the risk- 

 neutral linear loss function, where ij> = l and R(z) = 0. 



The fact that the logarithmic loss function tends 

 toward negative infinity as the resource approaches ex- 

 tinction may be viewed as problematic by some. On the 

 other hand. Smith (1985) views this behavior as a re- 

 quisite characteristic for any loss function to be used 

 in the context of renewable resources, arguing that it 

 "introduces a useful conservation motive into the deci- 

 sion making process." Opaluch and Bockstael (1984) 

 go even further, stating, "It is well known that the log 

 function exhibits the best properties of the simple func- 

 tional forms. ..." 



Beta probability density function The principal 

 justification for using the beta pdf to describe P(c|) is 

 that the beta is a natural choice for the pdf of any con- 

 tinuous variable which is constrained to fall within the 

 to 1 range. The fact that it allows for an explicit solu- 

 tion to Equation (7) is another argument in its favor. 



Unfortunately, the method presented here for esti- 

 mating the parameters of P(q) is based on a model 

 (Press 1989) which yields a truncated t distribution, 

 not a beta distribution. If this model is accepted as a 

 true description of reality, then the beta form for P(c[) 

 is only an approximation. Of course, most functional 

 forms used in modeling are only approximations, so the 

 question is whether the advantages of increased tract- 

 ability provided by the beta distribution outweigh any 

 attendant losses of accuracy. Holloway (1979) argues 

 in the affirmative after noting the difficulty of identi- 

 fying natural processes which yield the beta distribu- 

 tion as a formal result. 



In general, the effectiveness of Bayes decisions is 

 relatively insensitive to small changes in the assumed 

 pdf (DeGroot 1970). This being the case, the question 

 really is whether the difference between the truncated 

 t distribution and the beta approximation is typically 

 small. To assess the magnitude of this difference, the 

 goodness-of-fit between the truncated t and beta dis- 

 tributions was examined for a wide range of n, q, 

 and s^q values (Fig. 8). Note that R^>0.95 for a wide 

 range of parameter values, indicating that the loss of 



