USING MARKOV DECISION MODELS AND RELATED TECHNIQUES 



FOR PURPOSES OTHER THAN SIMPLE OPTIMIZATION: 



ANALYZING THE CONSEQUENCES OF POLICY ALTERNATIVES 



ON THE MANAGEMENT OF SALMON RUNS 



Roy Mendelssohn' 



ABSTRACT 



The mathematics of Markov decision processes and related techniques are used to analyze a model 

 relevant to salmon management. It is shown that the choice of grid can have a significant effect on the 

 results obtained. Optimal policies that maximize total expected discounted return may be too variable. 

 Smoothing costs are included to trade off long-run total return against the smoothness of the year-to- 

 year fluctuations in the allowed harvest. Simpler, approximate policies that have a smoothing effect 

 are also found. Preliminary analysis suggests the results are robust against misspecification of the 

 parameters of the model. Concepts such as maximum sustainable yield would seem to impute a very 

 high smoothing cost and are probably not practical for fish populations with a significant degree of 

 randomness. 



The history of most managed natural populations 

 is one of sizable, nondeterministic variations in 

 the dynamics of the population. This observed 

 variation tends to have two sources: The first 

 source is actual randomness in the system, such as 

 that due to environmental variability, which will 

 exist no matter how accurate our models become; 

 and the second source is the inaccurate or incom- 

 plete specification of the transition probabilities 

 themselves. Standard production models 

 (Schaefer 1954; Pella and Tomlinson 1969; Fox 

 1970, 1971, 1975) assume deterministic dynamics, 

 as do most recent bioeconomic analyses, as in 

 Clark (1976) or Anderson (1977). For randomly 

 varying populations, at best only extremely low 

 harvests may be sustainable year to year, and it is 

 not difficult to develop realistic scenarios where 

 policies that are sustainable in a deterministic 

 model would cause possible depletion in a stochas- 

 tic model. 



In this paper, the latest tools from stochastic 

 optimization, particularly in the area of Markov 

 decision problems (MDFs) are used to analyze a 

 model relevant to salmon management. The view- 

 point taken is that of the analyst, who must 

 analyze trade offs and provide a decision maker 

 with as few policies as possible that contain the 



'Southwest Fisheries Center Honolulu Laboratory, National 

 Marine Fisheries Service, NOAA, Honolulu, HI 96812. 



maximum amount of information, rather than 

 that of the decision maker, who ultimately decides 

 if a particular concern or trade off is worthwhile. 

 The salmon model is used as an example — the goal 

 is to gain insight into managing randomly varying 

 populations. 



Ricker (1958) appears to be the first to examine 

 the effects of variability on management. He used 

 intuition and simulation to arrive at policies that 

 are of the same general form as many of the 

 policies to be discussed in this paper. However, 

 Ricker presented no systematic way of developing 

 optimal policies and made the incorrect assump- 

 tion that the long-run stochastic behavior will 

 have a mean equal to the deterministic equilib- 

 rium yield, with noise around this mean. 



Reed ( 1974) derived qualitative properties of op- 

 timal policies if the random variable has a mean of 

 1, if it affects the population dynamics in a multi- 

 plicative manner, and if it has costs when the 

 system is shut down (no harvesting) and then 

 started up again (resumption of harvesting). 

 Reed's results are not relevant to the model dis- 

 cussed in this paper, since he assumed the deter- 

 ministic population model is concave, while the 

 models examined in what follows are pseudocon- 

 cave. A more complete treatment of one dimen- 

 sional stochastic growth models can be found in 

 Mendelssohn and Sobel (in press). 



Walters (1975) and Walters and Hilborn (1976, 

 1978) discussed a variety of topics as the concerns 



Manuscript accepted September 1979. 

 FISHERY BULLETIN: VOL. 78, NO. 1, 1980. 



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