Mathematical equivalence of geometric mean fitness with probabilistic optimization under environmental uncertainty

Natural selection can be considered as optimising fitness. Because ‘mean’ fitness is maximized with respect to the genotypes of carriers, traditional theory can be viewed as a statistical theory of natural selection. Probabilistic optimisation is a way to incorporate such uncertainty into optimality analyses of natural selection, where environmental uncertainty is expressed as a probability distribution. Its canonical form is a weighted average of fitness with respect to a given probabilistic distribution. This concept should be applicable to three different levels of uncertainty: (1) behavioural variations of an individual, (2) individual variations within a generation, and (3) temporal change over generations (geometric mean fitness). The former two levels are straightforward with many empirical evidences, but the last category, the geometric mean fitness, has not well understood. Here we studied the geometric mean fitness by taking its logarithm, where the log growth rates become the fitness value. By further transforming the log growth rates, the fitness of log growth rates becomes its linear function. Therefore, a simple average of these distributions becomes the fitness measure across generations and consideration of variance discount or the entire probability distributions becomes unnecessary. We discuss some characteristic features of probabilistic optimization in general. Our view is considered a probabilistic view of natural selection, in contrast with the traditional statistical view of natural selection.     

Planning for alkali land reclamation under rainfall uncertainty

A chance-constrained linear programming model with rainfall as stochastic input has been developed to plan optimal land and water use in alkali soils under reclamation. The model is based on the water balance of a typical alkali catchment in the command area of the Western Jamuna Canal in Haryana (India). The zero-order decision rule has been used to obtain the deterministic equivalent of chance constraint and optimal solutions have been obtained for four levels of rainfall probability. Available water supplies at low rainfall probabilities were found to be inadequate to sustain the high cost reclamation technology. Exploitation of the full potential of agricultural land calls for the augmentation of available water supplies.     

Setting conservation priorities for migratory networks under uncertainty

Conserving migratory species requires protecting connected habitat along the pathways they travel. Despite recent improvements in tracking animal movements, migratory connectivity remains poorly resolved at a population level for the vast majority of species, thus conservation prioritization is hampered. To address this data limitation, we developed a novel approach to spatial prioritization based on a model of potential connectivity derived from empirical data on species abundance and distance traveled between sites during migration. We applied the approach to migratory shorebirds of the East Asian‐Australasian Flyway. Conservation strategies that prioritized sites based on connectivity and abundance metrics together maintained larger populations of birds than strategies that prioritized sites based only on abundance metrics. The conservation value of a site therefore depended on both its capacity to support migratory animals and its position within the migratory pathway; the loss of crucial sites led to partial or total population collapse. We suggest that conservation approaches that prioritize sites supporting large populations of migrants should, where possible, also include data on the spatial arrangement of sites.     

Environmental policy choice under uncertainty

The Markovian model provides an insightful structure for analyzing environmental policy decisions. This framework is applied to a variety of conceptual issues. The analysis of reswitching phenomena shows that when multiple risks are involved, there is unlikely to be an unambiguous ranking of policies according to their future-mindedness. The discussion of irreversibilities demonstrates that irreversibilities differ only in degree from other probabilistic structures; they create no special analytic complications. In the final section, the Markovian decision framework is extended to incorporate the impact of budgetary constraints on optimal decisions.     

Structurally-consistent estimation of use and nonuse values for landscape-wide environmental change

We address the problem of estimating the use and nonuse value derived from a landscape-wide programme of environmental change. Working in the random utility framework, we develop a structural model that describes both demand for recreational trips to the landscape's quality-differentiated natural areas and preferences over different landscape-wide patterns of environmental quality elicited in a choice experiment. The structural coherence of the model ensures that the parameters of the preference function can be simultaneously estimated from the combination of revealed and stated preference data. We explore the properties of the model in a Monte Carlo experiment and then apply it to a study of preferences for changes in the ecological quality of rivers in northern England. This implementation reveals plausible estimates of the use and nonuse parameters of the model and provides insights into the distance decay in those two different forms of value.     

Commercial fisheries under price uncertainty

The deterministic models applied in economics of fisheries are extended to comprise price uncertainty and risk aversion among the fishing units. It is proved that in the open-access fishery both the total fishing effort and the number of fishing units are reduced as the variance of the price increases; that the total fishing effort may be smaller in the open-access fishery than in the optimal fishery at a high variance; that only a fixed producer price system can create a first-best optimum, and that a tax on revenue is more efficient than both fishing unit quotas or tax on catch.     

Emission control policies under uncertainty

The paper deals with the problem of optimal environmental policy under uncertainty. Usually, when an environmental policy is considered, only the expected values of the parameters of the marginal benefit and marginal cost functions associated with the policy are known. Thus a relevant question is: In the presence of uncertainty, what is the optimal policy mean for achieving the environmental objective? The study addresses itself to the specific objective of improving air quality although the analysis is generally applicable. The policy means are emission taxes and emission quotas. It is shown that under uncertainty neither of these means is generally optimal, and that specific parameter values of the costs and benefits relations and their distributions determine the optimal policy for each situation.     

Resolving structural uncertainty in natural resources management using POMDP approaches

In recent years there has been a growing focus on the uncertainties of natural resources management, and the importance of accounting for uncertainty in assessing management effectiveness. This paper focuses on uncertainty in resource management in terms of discrete-state Markov decision processes (MDP) under structural uncertainty and partial observability. It describes the treatment of structural uncertainty with approaches developed for partially observable resource systems. In particular, I show how value iteration for partially observable MDPs (POMDP) can be extended to structurally uncertain MDPs. A key difference between these process classes is that structurally uncertain MDPs require the tracking of system state as well as a probability structure for the structure uncertainty, whereas with POMDPs require only a probability structure for the observation uncertainty. The added complexity of the optimization problem under structural uncertainty is compensated by reduced dimensionality in the search for optimal strategy. A solution algorithm for structurally uncertain processes is outlined for a simple example in conservation biology. By building on the conceptual framework developed for POMDPs, natural resource analysts and decision makers who confront structural uncertainties in natural resources can take advantage of the rapid growth in POMDP methods and approaches, and thereby produce better conservation strategies over a larger class of resource problems.     

Robust population management under uncertainty for structured population models.

Structured population models are increasingly used in decision making, but typically have many entries that are unknown or highly uncertain. We present an approach for the systematic analysis of the effect of uncertainties on long-term population growth or decay. Many decisions for threatened and endangered species are made with poor or no information. We can still make decisions under these circumstances in a manner that is highly defensible, even without making assumptions about the distribution of uncertainty, or limiting ourselves to discussions of single, infinitesimally small changes in the parameters. Suppose that the model (determined by the data) for the population in question predicts long-term growth. Our goal is to determine how uncertain the data can be before the model loses this property. Some uncertainties will maintain long-term growth, and some will lead to long-term decay. The uncertainties are typically structured, and can be described by several parameters. We show how to determine which parameters maintain long-term growth. We illustrate the advantages of the method by applying it to a Peregrine Falcon population. The U.S. Fish and Wildlife Service recently decided to allow minimal harvesting of Peregrine Falcons after their recent removal from the Endangered Species List. Based on published demographic rates, we find that an asymptotic growth rate lambda > 1 is guaranteed with 5% harvest rate up to 3% error in adult survival if no two-year-olds breed, and up to 11% error if all two-year-olds breed. If a population growth rate of 3% or greater is desired, the acceptable error in adult survival decreases to between 1% and 6% depending of the proportion of two-year-olds that breed. These results clearly show the interactions between uncertainties in different parameters, and suggest that a harvest decision at this stage may be premature without solid data on adult survival and the frequency of breeding by young adults.     

Uncertainty and allocation decisions involving unique environmental resources

The implications of the treatment of uncertainty in the definition of optimal investment criteria for irreversible allocations of unique environmental resources are considered herein. This analysis argues that the single-project, orientation of conventional cost-benefit analysis can lead to inconsistent decisions because it fails to take account of the potential for risk pooling across projects. In the models discussed in this paper uncertainty arises because the planner must estimate the net benefits associated with a mix of developed and preserved natural environments. The results suggest that the criteria for optimal investment plans will be affected by those factors influencing this uncertainty. Both the scale and timing of optimal investments may be altered from those prescribed with the conventional (Fisher-Krutilla-Cicchetti rule) framework when the potential for risk pooling is introduced. Thus, it is important to consider as a part of the specification of the objective function the factors which give rise to uncertainty.     

Entitlements, missing markets, and environmental uncertainty: Reply

At one level Dore and Ward seem intent on immunizing received doctrine. They say that I did not prove that rights violated Nozick's principles of justice, and that I did not prove that missing markets were the problem. More telling, they allege I did not prove that my story was better than the particular story favored by Dore and Ward. That is, they argue that : “ … there is an alternative approach, based on the theory of public economics, which is indeed operational.”³ They then use the bulk of their comment to explain and defend this alternative model. They hope to convince the reader that the model of “public economics” survives intact in the face of my alternative model. To the extent that their model allows one to define the problem away, it is not difficult to see why they find it so compelling.More seriously, Dore and Ward fail to make a coherent argument about the concept of rights and duties and the logic of a particular rights structure that happens to prevail. They seem concerned that my article was tampering with some sacred social construct—the status quo rights structure—and that my article threatened the received wisdom of both “public economics” and Paretian-Walrasian bliss. Indeed, theirs seems to be a genuine fear that if economists release their grip on the certitude of received orthodoxy the floodgates will be open to all manner of bumbling regulators seeking to shift rights and wealth positions across generations. My article is apparently threatening because it seems to suggest that the received wisdom may, at times, be misleading.     

Dynamic aspects of effluent taxation under uncertainty

The optimal adjustment process for an effluent charge is derived when the firm's response to the charge is uncertain. The optimal process is shown to exhibit the dual effects of caution and probing. Because of uncertainty, society should be risk averse or cautious when setting the charge. However, because of the possibility of generating information for improved future control, this caution is balanced by the incentive to probe for additional information. Finally, if the uncertain parameters of the firm's response function are nonautonomous, their change over time must be explicitly considered to avoid bias in estimation.     

Option value under income and price uncertainty

We present a rigorous examination of the sign of option value under income and price uncertainty. We state and prove three theorems on the sign of option value when preferences are state independent. If income is uncertain, the sign of option value is opposite the sign of the income elasticity of the good in question. The analysis of price uncertainty is in two parts: one covers the case of own price uncertainty and the other the case where the price of some other good is uncertain. In both cases the sign of option value is generally ambiguous.     

Making conservation decisions under uncertainty for the persistence of multiple species.

Population models for multiple species provide one of the few means of assessing the impact of alternative management options on the persistence of biodiversity, but they are inevitably uncertain. Is it possible to use population models in multiple-species conservation planning given the associated uncertainties? We use information-gap decision theory to explore the impact of parameter uncertainty on the conservation decision when planning for the persistence of multiple species. An information-gap approach seeks robust outcomes that are most immune from error. We assess the impact of uncertainty in key model parameters for three species, whose extinction risks under four alternative management scenarios are estimated using a metapopulation model. Three methods are described for making conservation decisions across the species, taking into account uncertainty. We find that decisions based on single species are relatively robust to uncertainty in parameters, although the estimates of extinction risk increase rapidly with uncertainty. When identifying the best conservation decision for the persistence of all species, the methods that rely on the rankings of the management options by each species result in decisions that are similarly robust to uncertainty. Methods that depend on absolute values of extinction risk are sensitive to uncertainty, as small changes in extinction risk can alter the ranking of the alternative scenarios. We discover that it is possible to make robust conservation decisions even when the uncertainties of the multiple-species problem appear overwhelming. However, the decision most robust to uncertainty is likely to differ from the best decision when uncertainty is ignored, illustrating the importance of incorporating uncertainty into the decision-making process.     

An inexact-stochastic with recourse model for developing regional economic-ecological sustainability under uncertainty

Effective planning of resources management is important for facilitating socio-economic development and eco-environmental sustainability. Such a planning effort is complicated with a variety of uncertain, dynamic and nonlinear factors as well as their interactions. In this study, an inexact-stochastic quadratic programming with recourse (ISQP-R) method is developed for reflecting dynamics of system uncertainties based on a complete set of scenarios as well as tackling nonlinearities in the objective function to reflect the effects of marginal utility on system benefits and costs. Moreover, since penalties are exercised with recourse against any infeasibility, the ISQP-R can support the analysis of various policy scenarios that are associated with different levels of economic consequences when the promised targets are violated. The developed method is applied to a case study of planning resources management and developing regional ecological sustainability. The results have been generated and are helpful for decision makers in not only identifying desired resources-allocation strategies but also gaining insight into the tradeoff between economic objective and eco-environment violation risk.     

Losses from effluent taxes and quotas under uncertainty

Recent theoretical papers by Adar and Griffin (J. Environ. Econ. Manag.3, 178–188 (1976)), Fishelson (J. Environ. Econ. Manag.3, 189–197 (1976)), and Weitzman (Rev. Econ. Studies41, 477–491 (1974)) show that,different expected social losses arise from using effluent taxes and quotas as alternative control instruments when marginal control costs are uncertain. Key assumptions in these analyses are linear marginal cost and benefit functions and an additive error for the marginal cost function (to reflect uncertainty). In this paper, empirically derived nonlinear functions and more realistic multiplicative error terms are used to estimate expected control and damage costs and to identify (empirically) the mix of control instruments that minimizes expected losses.     

Reserve and land prices with exploration under uncertainty

Arrow and Chang (J. Econom. Environ. Manag.9, 1–10 (1982) analyze an exploration-extraction model with uncertainty about the distribution of deposits. Reserve prices do not exhibit any rising trend when the unexplored land area is big but, in order to know what happens when the unexplored area is smaller, “a probabilistic analysis not yet performed” (ibid. p. 10) is required. Their conjecture of a rising trend at less than the discount rate is confirmed. However, land prices can be expected to grow faster than reserve prices, which helps discriminate between Arrow and Chang's, and other rationalizations of the failure of resource prices to rise at the discount rate.     

Determine the most influencing stressors and the most susceptible resources for environmental integrated assessment

The paper presents a method to determine the most influencing stressors and the most susceptible resources for complex assessment problems involving multiple stressors impacting multiple resources over a region. The method is based on the concept of limiting priorities in a square matrix which capture the transmission of influence along all paths between stressors and resources in the matrix. The proposed method allows the relationship between stressors and resources to be looked at in both univariate and multivariate fashion, taking into account the interactions among the variables. Hypothetical and case study examples are given for illustration purpose. It shows that the proposed method is suitable for the determination of the most important stressors and the most susceptible resources, a common (but often uneasy) task in integrated environmental assessment.     

Control theory applied to natural and environmental resources an exposition

Four control theory models of natural and environmental resource use, drawn from the existing literature, are developed in a manner to emphasize their technical and decentralized interpretive similarity. Renewable, nonrenewable, and amenity resource use are treated as closely related problems of optimal (biological, earth material, ecological, or environmental) capital allocation over time. Thus nonrenewable resources, and the problem of exhaustion, are just limiting (zero growth) cases of renewable resources, and the problem of species extinction. Just as exhaustion can be optimal, extinction can be optimal. Waste recycling is treated as part of the problem of optimal regeneration of “sclean” environmental capital; wilderness use as a problem of managing the regeneration of ecological capital.