Several sources of uncertainty generate variability in fishery performance: (1) variability in abiotic factors affecting the spatio-temporal distribution and abundance of a stock; (2) effects of ecological interdependencies; (3) fluctuations in costs and product prices that determine changes in exploitation intensity and in the quantity demanded; (4) variations in fishing effort determined by fleets with different fishing power and type of gear, as well as by differential skill of the skipper and crew; and (5) variability in the behavior of policy makers due to value judgments when taking management decisions (Caddy & Mahon, 1995; FAO, 1995a, b). In this context, use of bioeconomic models in a formal decision analysis that explicitly includes uncertainty has been rarely documented in the primary literature. Because these uncertain events involve more than one possible outcome, bioeconomic evaluation of alternative management strategies requires the quantification of uncertainty and risk associated with the application of each strategy. In this Chapter, we summarize some types and sources of uncertainty and risk in fisheries. Moreover, some formal decision models directed to quantify the economic value of alternative management strategies are presented in a precautionary fishery management context.
In Chapter 1 we identified those inherent characteristics of fish stocks that result, under open access conditions, in the overexploitation syndrome, dissipation of economic rent and overcapitalization of fishing fleets. These problems could also be caused by not recognizing the high uncertainty levels that characterize most of the fisheries and the corresponding lack of caution in management (Garcia, 1996). Indeed, many fisheries experienced biological and economic overfishing as a result of risky management policies (Garcia, 1992; FAO, 1993; Garcia & Newton, 1994; Caddy & Mahon, 1995).
In June 1995, FAO and the Sweden Government organized in Lysekil (Sweden) a scientific consultation guided to establish a precautionary approach in the investigation and management of fishing resources (FAO, 1995a). The following definitions were adopted: (i) risk is the probability of occurrence of an undesirable event, (ii) uncertainty is the incompleteness of knowledge about the states or processes of nature, and (iii) statistical uncertainty is the stochasticity or error coming from several sources as described using statistical methodology.
Hilborn & Peterman (1996) identify seven sources of uncertainty in fisheries stock assessment: (1) in the estimates of fish abundance; (2) in the structure of the mathematical model of the fishery; (3) when estimating model parameters; (4) in future environmental conditions; (5) in the response of users to regulations; (6) in future management objectives; and (7) in economical, political and social conditions (see Hilborn & Peterman, 1996, for details).
Univertainty in abundance estimates.. One of the main sources of error in abundance estimates can arise from incorrect or unreliable estimates of inputs used in stock assessment models. For example, stock estimates from VPA are based on time-invariant and mean values of M estimates made in the 1950s to 1980s (Caddy, 1996), when it is known that M strongly affects estimates of current abundance by this methodology (Lapointe et al., 1989). Moreover, many stock estimates are unreliable due to incomplete information concerning catch/effort data, which often results in baised abundance indexes (CPUE)) and estimates of population structure significantly different from that observed. Variance and error structure of the entire data series might also be a source of uncertainty, and should be quantified (Schnute & Hilborn, 1993).
Uncertainty in model structure. Most of the models used in stock assessment are based on single-species population dynamics, although important ecological interdependencies have been recognized (see Chapter 3). Difficulties in data gathering and lack of robustness in abundance estimates make multispecies models quite unreliable.
Even for single-species models, uncertainty associated with model structure is often large and it is not usually reported. Results obtained by alternative models should be useful depending on which of them is most appropriate in a given situation. It may be also considered precautionary to fit several models, and for management purposes, emphasize the one that provides the most cautious approach. A detailed analysis of the basic assumptions of the model used and the estimation of uncertainty associated with the bioeconomic functions and equations fitted in a fishery model (see Chapter 4), should be a common practice to validate model structure and evaluate its performance (e.g. Caddy & Defeo, 1996). Several possible management options using the whole range of available bioeconomic models should ideally be systematically analyzed.
Uncertainty in estimation of model parameters. Although reporting the variance for bioeconomic parameters is attracting increasing attention by fisheries agencies, possible data biases and resulting estimates are rarely discussed and made explicit. Moreover, many key parameters used as inputs in stock assessment models, even in methods that purport to analyze yearly data as VPA, are assumed to be time-invariant, and variability around mean estimates are neither reported nor used to perform sensitivity analyses (Caddy, 1996). This tends to underestimate uncertainty in the parameters and, consequently, in the outcomes provided by the models.
Uncertainty in future environmental conditions. Changes in environmental conditions have an important impact on the abundance and spatio-temporal distribution of fish resources (Walters & Parma, 1996). In order to carry out dynamic forecasts of a stock, the prediction of environmental future conditions is required, especially for resources sensitive to an extreme degree to environmental changes. Three types of alternative assumptions are considered: (1) environmental constant conditions correspond to the average of the historical observations; (2) environmental conditions randomly vary conditions around the average, with a known probabilistic density function; and (3) environmental conditions show systematic patterns, e.g., periodic or linear trends. In this context, random variability about past average conditions might be considered when modelling a fishery.
Uncertainty in the behavior of resource users. Fishers and fishery policies are affected by costly, imperfect enforcement of fisheries law (Sutinen & Andersen, 1985). Therefore, the response of fishers to regulations should be considered when the expectations of implementation, execution and surveillance of a specific management strategy are evaluated (Sutinen et al., 1990). The simplest approach is to assume that the regulations will not be violated and that a free-rider behavior will not occur (Hilborn & Peterman, 1996; see also Chapter 1). However, variations in the dynamic behavior of fishers concerning their allocation of fishing effort, selection of the target species and the fishing gear, and the reliability of catch and effort data reported, could change as a response of changes in management regulations. The dynamic behavior and responses of resource users to regulations should ideally be incorporated into stock assessments in order to mitigate uncertainty levels associated with the forecasted outcomes of management actions (Hilborn & Peterman, 1996; see also Rosenberg & Brault, 1993). When the nature of the regulatory sector is such that its behavior is unpredictable (see below), fishers behavior will be seriously affected and will tend to increase fishing intensity, with the obvious effect on both stock size and profit levels.
Uncertainty in future management objectives. Management strategies should be periodically revised and adapted to the dynamic conditions of the stock and resource users, as well as to changes in the intertemporal preferences of the fishing sector. Changes in management objectives resulting from an unpredictable behavior of the regulatory sector constitute an important source of uncertainty (Anderson, 1984). Policy makers will tend to maximize their utility or satisfaction functions, which are subjected to their financial budget and political capital (Niskanen, 1971). In this context, Anderson (1984) introduces the concept of bioregunomic equilibrium, which occurs when the biological, economic and regulation components achieve equilibrium simultaneously. Anderson views regulation as endogenous to the fishery system, in which government is seen as the producer or supplier of regulation and the fishers as demanders of regulation. He describes the behavior of marginal and average costs and benefits when introducing this regulation component. According to the model, three basic points exist: open-access equilibrium, bioregunomic equilibrium and the socially optimal level of effort. These points do not necessarily occur at the same fishing effort level. Anderson considers constant prices, and describes the behavior of the regulator trying to select that combination of yield and profit that will maximize his utility, as he must satisfy the demands of several groups (e.g., consumers and producers of fish) involved in the fishery. In this sense, uncertainty in the preference functions of policy makers could be represented by a series of indifference curves of costs and profits, whose shape and position will depend upon the negotiation, political and economic power of each group of the society involved (see Anderson, 1982a, b; 1984).
Uncertainty in economic, political and social future conditions. Uncertainty in species price according to market fluctuations, as well as in fixed/variable costs of fishing effort, could influence the dynamic behavior of users. Consequently, the magnitude of catches and related fishing mortality might vary in the short-term, with obvious effect in population abundance. Changes in both local and international political conditions may also constitute a source of uncertainty and could determine different responses of fishers to regulations, especially in cases of political instability and scarcity of employment. This is extremely important in cases of shared resources among developing countries, where the costs of monitoring, control and surveillance depends on international political agreements rather than in enforcement efforts provided by a single country. Moreover, variations in welfare levels of artisanal coastal communities, caused by endogenous (i.e. variations in stock abundance) or exogenous (e.g. changes in species price in the international market) factors, could add uncertainty to the fishery system. These biological, economic or market uncertainties affect expected profits and consequently the short-term dynamic behavior of the fishing fleet. A sudden increase in fishing. mortality generally occurs as a result of uncertainty as to future modifications in the regulatory process, because fishers will not be sure of the benefits that will follow from the regulatory measures.
Three levels of risk in the attitude of the manager can be distinguished in decision theory (Pearce & Nash, 1981 Schmid, 1989). (1) A decision-maker who is satisfied to be acting on the basis of expected values, without considering the variance of the different outcomes of decisions, is referred to as risk neutral. (2) A risk averse decision maker assigns to uncertain prospects as to future costs and benefits of alternative management strategies, certainty equivalencies lower than the expected values. (3) A policy-maker that assigns to future and uncertain economic rents, certainty equivalencies greater than the expected values, is risk prone. In practice, a continuous gradient of risk aversion is observed with more than three discrete categories.
Where there are inadequate observations to assign probabilities of occurrence to dissimilar states of nature that have occurred, decision tables could be used to represent different degrees of management caution through the Maximin, Minimax and Maximax criteria. Maximin is a cautious approach that consists in selecting the management decision that involves the maximum Net Present Value (NPV) of the observed minimum outcome. It is used when the policy maker is risk averse. When he is less cautious, the Minimax regret criterion could be used. This approach selects the management action that minimizes the maximum regret, defined as the difference between the real benefit and the one that could have been obtained if the correct decision had been taken. Finally, an optimist and risk prone policy maker could use the Maximax approach, by selecting the management option with the higher NPV (see FAO, 1995a; Perez & Defeo, 1996).
The above decision criteria could be incorporated in the bioeconomic models described in Chapters 2 and 3. Thus, uncertainty in the biological, economic and social processes should be included when developing management advice. For this purpose, several states of nature (SN) could be considered for critical parameters (e.g., natural mortality rate, species price, MSY, MEY), and also for hypotheses about the behavior of many bioeconomic processes (e.g. heterogeneity in the spatial distribution of the stock, or strategies for spatial allocation of fishing intensity). This will result in different performances of biological and economic variables if alternative management decisions D1, D2…., Dξ are taken.
Illustrative example
Consider that the number of observations is insufficient to assign probabilities to SN1and SN2 (e.g., two observed recruitment levels). The risk and uncertainty associated with three alternative management decisions D1, D2, D3 are incorporated, and fishery performance as a function of the NPV for these decisions are presented for each state of the nature in a decision table without mathematical probabilities. Different degrees of management caution are highlighted through the Maximin, Minimax and Maximax criteria (Tables 7.1 to 7.3).
| Decision | NPV of states of nature | Minimum NPV | |
|---|---|---|---|
| SN1 | SN2 | ||
| D1 | -30 | 400 | -30 |
| D2 | 80 | 150 | 80 |
| D3 | 50 | 300 | 50 |
Decision criterion: Select the management strategy with the maximum NPV of the minimum outcome: D2
| Decision | Regret for each decision | Maximum Regret | |
|---|---|---|---|
| SN1 | SN2 | ||
| D1 | 80-(-30)=110 | 400-400=0 | 110 |
| D2 | 80-80=0 | 400-150=250 | 250 |
| D3 | 80-50=30 | 400-300=100 | 100 |
Decision criterion: Select the management action that minimizes the maximum regret: D3
| Decision | NPV of states of nature | MaximumNPV | |
|---|---|---|---|
| SN1 | SN2 | ||
| D1 | -30 | 400 | 400 |
| D2 | 80 | 150 | 150 |
| D3 | 50 | 300 | 300 |
Decision criterion: Select the management strategy which provides the maximum NPV: D1
Decision theory provides an excellent framework for management advice. The analysis includes stochastic outcomes for a fishery model that considers uncertainty in its parameters and variables, and also in the structure of the model. The outcomes of alternative management actions can thus be evaluated by means of a decision table containing hypotheses about parameter values or model structure and the referred management alternatives (Hilborn & Peterman, 1996).
When considering mathematical probabilities of fishery performance (Pλ) associated with λ possible states of nature SN(e.g., recruitment 20% higher or lower than its average), it is possible to estimate the expected value (EV) of the NPV resulting from different management decisions (D1, D2,…, Dξ):
In these cases, the policy maker must do a balance between the expected value and the variances of alternative management decisions(see below).
The example that follows considers again two possible states of nature corresponding to two observed recruitment levels, but now with probabilities of occurrence P1=.75, P2=.25. The risk and uncertainty associated with D1,D2,D3 are incorporated, and the NPV for these decisions, are presented for each SN in Table 7.4.
| Decision | States of nature | EV | VAR | SD | |
|---|---|---|---|---|---|
| SN1(P1=.75) | SN2(P2=.25) | ||||
| D1 | -30 | 400 | 77.5 | 34668 | 186 |
| D2 | 80 | 150 | 97.5 | 919 | 39 |
| D3 | 50 | 300 | 112.5 | 11719 | 108 |
The maximum expected value of NPV is generated by D3. However, VAR is greater for D3 than for D2 and thus the policy maker must decide between both actions, balancing EV and VAR.
In the previous section, the Minimax principle suggested that we proceed according to a distribution defined over few alternative states. However, there is no reason to expect that nature behave according to this distribution. In some situations, the decision-maker has some information about possible states of nature that allow him to judge a priori which state of nature is more likely to be true. Such information can usually be expressed as a probability distribution, considering the state of nature as a random variable. This distribution is referred as a prior distribution. Prior distribution are often subjective, and depend on the experience or intuition of an individual.
A procedure for utilizing a prior to aid in the selection of alternative management actions is the Bayesian Criterion. When prior probabilities can be assigned to different states of nature and management options, the Bayesian approach could be most appropriate for fisheries stock assessment problems (Hilborn & Mangel, 1997 and references therein). The Bayes' Theorem prompts the decision maker to select an action (Dξ) that minimizes the expected loss (LOSS (Dξ)), which is evaluted with respect to a prior distribution (P(SNλ)) defined over the possible λ states of nature (SNξ), where ξ are alternative management options.
The Bayesian approach could be applied to the above example as follows (Table 7.5):
| Decision | Loss matrix with prior probabilities | Expected Value | |
|---|---|---|---|
| SN1(P1=.75) | SN2(P2=25) | ||
| D1 | 80-(-30)=110 | 400-400=0 | 82.5 |
| D2 | 80-80=0 | 400-150=250 | 62.5 |
| D3 | 80-50=30 | 400-300=100 | 47.5 |
Decision criterion: Select the management strategy which provides the minimum expected losses: D3
Other decision tables could be built with different performance variables (e.g., biomass, yield, employment, etc.). A detailed analysis of the Bayesian approach as applied in ecology and fisheries stock assessment is beyond the scope of this book. For a complete description, the reader must refer to Hilborn & Mangel (1997) and references therein.
Even though highly recommended in the fisheries literature (e.g., Hilborn & Walters, 1992), there are few examples of the estimation of uncertainty (variance) in the management parameters of surplus production models (Kizner, 1990; Polacheck et al., 1993; Punt & Hilborn, 1996). This has substantial importance, because a precautionary approach to management specifically requires a comprehensive treatment of risk and uncertainty in critical management parameters known as references points (e.g., FMSY, ZMSY, MSY; FAO, 1993). In order to use fishery models as explicitly predictive , it is necessary to build confidence intervals that account for the variability around the mean estimates of the parameters. In the case of linear models or those that could be linearised through for example, logarithmic transformations, confidence intervals could be built based on linear classical statistics (Zar, 1984). However, most fishing models are nonlinear, and thus confidence intervals can be calculated by computer intensive methods, such as “jackknife” and “bootstrap” (Efron, 1979; 1981; 1982).
Jackknife
The “jackknife” consists of sequentially sampling with substitution a pair of data from a base of n data pairs corresponding to a series of n years of e.g., catch and effort data. Thus, n groups of n-1 data sets are generated, which provide n estimates of biological/management parameters. The confidence interval calculated by “jackknife” could be estimated using the classical procedures of linear statistics, assuming that the probabilistic density functions of the parameters are normal (“jackknife normal based” sensu Meyer et al., 1986).
Although this constitutes a potentially useful technique for building confidence intervals (Hiborn & Walters, 1992), its application has been questioned. The “jackknife” can be ineffective in providing variance estimates with the usual moderately short fisheries time series, and in such cases do not always provide a clear distribution of the management parameters (Caddy & Defeo, 1996).
Bootstrap
This is based on two different approaches (Efron, 1982; Manly, 1991): random resampling of: (1) the original values in the data set; and (2) the errors derived from the estimations vs observations used to fit a given fishery model (see Punt & Hilborn, 1996).
“Bootstrap” with original data. This technique is based on random resampling with replacement of the original set of n (yearly) data pairs corresponding to a particular fishery model (e.g., annual series of catch-effort, total catch-mortality, stock-recruitment). Thus, any of the n data pairs corresponding to a given year could be reused multiple times in successive bootstrap estimates.
“Bootstrap” based on errors. In this case, observed data is fitted to a certain fishery model, and the predictions of the dependent variable Y generated are then subtracted from the observed ones, obtaining the term of error e for each one of the n observations. These errors are randomly resampled with replacement and assigned to a certain Y predicted value, thus generating a new group of Y data. This group of data is used jointly with the independent variable X in order to refit the model.
A minimum of 500 “bootstrap” replicates is suggested to estimate confidence bounds (Punt & Hilborn, 1996). The mean and confidence intervals of each parameter could be estimated in two ways: (1) When the probabilistic density functions of the parameters are symmetric and they can be assumed to follow a normal curve, the mean and confidence intervals could be estimated following classical parametric statistics. (2) When parameter distributions are asymmetric, the percentile method is a useful tool for enclosing a given percentage of the bootstrap distribution, in order to provide an empirical confidence interval. In this case, the median is an unbiased and more informative estimate than the mean (Meyer et al., 1986).
Caddy & Defeo (1996) applied version (1) of bootstrap to equilibrium versions of the logistic and exponential versions of the yield-mortality models mentioned in Chapter 2. Confidence limits were obtained directly for the population (M, B∞ and r) and management (FMSY, ZMSY, MSY) parameters, from the pool of bootstrap estimates, using the quartile approach. Given the skewed nature of almost all distributions, they determined the median, and calculated the 10th and 90th percentile values that enclosed the central 80% of the bootstrap distribution as the empirical confidence interval.Chapter 2 also shows an additional application of bootstrapping, directed to quantify variability in the parameters of a bioeconomic yield-mortality model (see also Punt & Hilborn, 1996 for a detailed application of bootstrapping to biomass dynamic models).
Due to the asymmetric nature of almost all the parameters distribution functions generated by fishery models, resampling the residuals together with the application of the percentile theory seems to be a useful method. It preserves the “design” of the study, by using the original values of the independent variable, and also diminishes the probability of obtaining negative values (Pérez & Defeo, 1996).
Caddy & Defeo (1996) elaborated a simple approach to formulation of risk-averse management strategies, using the percentiles of a cumulative distribution of MSY estimates from bootstrapping against Z, which it is suggested would allow fishery management advice to be generated based on annual mortality and yield vectors alone. The bootstrap distribution for ZMSY obtained by equations developed in Chapter 2 suggests a simple way of using this fitting procedure in a management context, where Z values (from size frequency analysis) could be used as a management control variable, and where the risk of making a wrong management decision is a real one. A cumulative distribution of values of MSY with Z from bootstrapping provided the basis for evaluating a proposed management target. This might be either framed in terms of F (M known), or simply using the Z value expected for a given management action as the control variable which provide feedback on the impact of fishing. The probability that a given value Znow will exceed ZMSY in the current year of fishing could be determined by this procedure. This simple risk analysis could be carried out by bootstrapping the results of the different models provided in Chapter 2.
The “bootstrap” procedure provides further advantages for selecting management options in a probabilistic context. For example, taking into account that the values of M, r and management parameters from a single bootstrap fit are all cross-correlated, a manager seeking advice about what to do on the basis of a fit of the models described here, should primarily consider what would be the most precautionary parameter set to adopt, i.e., which would be the one least likely to lead to overexploitation.
Other procedures related to the estimation of risk and uncertainty for scientific advice in fisheries are beyond the scope of this document. The reader is referred to FAO (1996) and Francis & Shotton (1997) among other relevant scientific papers on the subject.
