LORD: DECISION THEORY APPLIED TO SALMON FISHERY 



bear. However, we can invoke the "Principle of 

 Optimality" (Bellman 1957:83) to specify that 

 En +j = £"() for all j al, i.e., all future escapements 

 are assumed to be the optimum MSY escapement. 

 This is a reasonable assumption since the principle 

 of optimality states that an optimal policy is one 

 which, given the present state of the system, 

 establishes and maintains an optimal policy for all 

 future time periods. Since £"0 represents such an 

 optimum steady state escapement it follows that 

 E„ +1 = Eq for future optimality. In this case 

 Equation (5) takes the form 



^n.n + K 



= L{E„) = (K+ DiaEoe-'^o-Eo) 

 - {aE„e~ ^^"- E„) + (terms depending 



on Eq and past escapements only). (6) 



From Equation (6) it appears that the optimiza- 

 tion will be over a total of {K + 1) seasons. This is 

 not actually the case since, as noted above, the 

 constraint E„ +^ = £"0 has been imposed and the 

 analytical procedures used in year n will be ap- 

 plicable in year {n + 1), etc. Note also the intu- 

 itively reasonable result that the loss given by 

 Equation (6) is minimized by setting E„ = Eq, the 

 optimum MSY escapement. 



The analysis thus far has assumed that all 

 quantities are deterministic. Random variables 

 will now be introduced to simulate the situation 

 actually existing in salmon fishery assessment and 

 management. Let N„ denote the run size resulting 

 from the (known) escapements {£"„ _j){j = 1, . . . 

 K) and let A',, be a random variable which, for 

 definiteness, will be assumed to have the two- 

 parameter gamma density 





/3o.Vn 



(7) 



where F denotes a gamma function. The pa- 

 rameters (oq, /Sq) are subscripted to denote that 

 they are applicable prior to the start of the run and 

 /is subscripted by one to denote that it is applica- 

 ble to the first fishing period. The quantity ijo is a 

 symbolic conditioning variable denoting the pre- 

 season information that is available for the 

 specification of (oq. I^o)- Anticipating the dynamic 

 nature of the fishery and its management the 

 probability density of A^„ will be conditioned 

 successively to reflect the data obtained after the 

 start of the run. 



We assume now that the expected value, shown 

 as E[N„] is that given by the Ricker relation, i.e., 



Rn =E[N„] = a 2 PkK-ke- 



bE 



k - 1 



(8) 



The variance of A^„ may be estimated from his- 

 torical data, e.g., smolt outmigrations, high seas 

 catches, etc. Knowledge of the mean and variance 

 is sufficient to determine the parameters (a,,, /?o). 



At this point it might be well to justify, or at 

 least explain, the assumption of a gamma density 

 for A^„ . Clearly, one cannot obtain Equation (7) on 

 the basis of biological arguments. On the other 

 hand, a gamma density does not do particular 

 violence to one's intuition concerning the dis- 

 tribution of population sizes. In particular. Equa- 

 tion (7) confines A^„ to positive values with scale 

 and location specified by («(,, Po)- In salmon 

 population estimation, it is rare that parameters 

 beyond mean and variance are available from 

 whatever source. It is in this spirit that Equation 

 (7) is introduced. Further, the gamma distribution, 

 not coincidentally, has the added virtue that it is 

 an analytically convenient function. Similar ar- 

 guments will be used to justify some of the func- 

 tions to be introduced subsequently. 



For the remainder of the analysis, only events in 

 year n will be considered so that the subscript may 

 be omitted from A^„ . The fishing season is assumed 

 to consist of m nonoverlapping time periods dur- 

 ing each of which a management decision, 8, must 

 be made. Let {8,} (i = 1, . . . m) be an arbitrary 

 sequence of decisions where each of the 8^ is a 

 member of some finite set of possible management 

 decisions.'^ Assume now that during the t'th period 

 a fraction p, of the total run enters the fishery. The 

 set {p,){i = 1, . . . m), which is assumed to be 

 known, may be obtained from such sources as the 

 almanac prepared by Royce (1965). The (ft) must 

 obviously satisfy the condition 



m 



2 ^' = 1- 



i = 1 



Corresponding to any actual realization of the 

 run, N, there exists some unique set of optimum 

 catch-escapement allocations <t], ) (i = 1, . . . m). 

 Rothschild and Balsiger (1971) used linear pro- 



'A typical set of management decisions consists of such actions 

 as opening or closing the fishery, the imposition of gear limita- 

 tions, waiting periods, etc. 



839 



