Managing uncertainty in decision‐making for conservation science

Science‐based decision‐making is the ideal. However, scientific knowledge is incomplete, and sometimes wrong. Responsible science‐based policy, planning, and action must exploit knowledge while managing uncertainty. I considered the info‐gap method to manage deep uncertainty surrounding knowledge that is used for decision‐making in conservation. A central concept is satisficing, which means satisfying a critical requirement. Alternative decisions are prioritized based on their robustness to uncertainty, and critical outcome requirements are satisficed. Robustness is optimized; outcome is satisficed. This is called robust satisficing. A decision with a suboptimal outcome may be preferred over a decision with a putatively optimal outcome if the former can more robustly achieve an acceptable outcome. Many biodiversity conservation applications employ info‐gap theory, under which parameter uncertainty but not uncertainty in functional relations is considered. I considered info‐gap models of functional uncertainty, widely used outside of conservation science, as applied to conservation of a generic endangered species by translocation to a new region. I focused on 2 uncertainties: the future temperature is uncertain due to climate change, and the shape of the reproductive output function is uncertain due to translocation to an unfamiliar region. The value of new information is demonstrated based on the robustness function, and the info‐gap opportuneness function demonstrates the potential for better‐than‐anticipated outcomes.

considera la incertidumbre del parámetro, pero no la incertidumbre en las relaciones funcionales.Consideré los modelos de vacío de información en la incertidumbre funcional, usados de forma extensa fuera de las ciencias de la conservación, aplicados a la conservación de una especie genérica amenazada mediante la reubicación a una nueva región.Me enfoqué en dos incertidumbres: la temperatura en el futuro es incierta debido al cambio climático y la forma de la función del rendimiento reproductivo es incierta debido a la reubicación a una región desconocida.El valor de la nueva información queda demostrado con base en la función de la solidez y la función de la conveniencia demuestra el potencial para resultados mejores a lo esperado.

INTRODUCTION
Diverse strategies are used to assess and select among decision alternatives in conservation science.Some strategies employ scientific models of the processes involved.Other strategies use empirical relations based on historical data for forecasting future outcomes.Yet other strategies employ professional judgment of experts.These idealized strategies are sometimes combined: scientific models are calibrated with empirical data; empirical relations are based on scientific understanding; professional judgment is grounded in empirical data.
Whatever strategy one uses for decision-making, a common approach, which I refer to as putative optimization, is to select the decision that has the greatest predicted effectiveness or most cost-effectively achieves the goal.The logic behind this strategy seems sound: use the best available knowledge to assess decision alternatives and select the putatively optimal alternative.The problem is that knowledge is incomplete, and sometimes wrong, but one does not always know where or to what degree knowledge is deficient.Responsible selection between decision options must exploit knowledge while managing uncertainty.
If one confidently knows probability distributions of the relevant uncertainties, then various probabilistic and statistical tools are appropriate for managing those uncertainties (Hines & Montgomery, 1990).If one can choose a fuzzy membership function, then fuzzy logic provides a decision methodology (Klir & Folger, 1988).However, I examined situations of deep uncertainty for which probability distributions or membership functions are poorly known or entirely unknown.Thus, I applied info-gap decision theory: a nonprobabilistic methodology for modeling and managing uncertainty.An info-gap approach depends on less information than probability or fuzzy logic and is suitable for the deep uncertainties I considered.I used info-gap models for uncertainty in parameters and in functional relationships (Ben-Haim, 1999, 2000, 2005, 2006, 2010).I focused on 3 methodological problems that are highly pertinent to decision-making in conservation science.
I examined whether predicted outcomes, based on the best available knowledge, are an acceptable basis for evaluating and selecting among decision options.To apply the best available knowledge and still account for the deep uncertainty in some crucial aspects of the knowledge, I assessed robustness to uncertainty.I also examined the trade-offs between the quality of the outcome and the confidence in achieving that outcome.This trade-off underlies a sort of precautionary principle in decisionmaking: best predictions are likely wrong, so one must assess vulnerability to error and adjust the evaluation and selection of a decision accordingly.
Finally, I examined the potential for a reversal of preference among decision options by considering 2 decision options, P and Q, where P is predicted to have better outcome than Q but Q is more reliable for achieving critical outcomes.I examined when the putative preference for P is reversed to a preference for Q when considering deep uncertainty.

INFO-GAP ROBUST SATISFICING IN CONSERVATION SCIENCE
I reviewed applications of info-gap robust satisficing in 33 conservation publications (Table 1).Many important explorations of a vast range of conservation issues related to diverse biological systems have been made.However, important implications were not always included.I examined uncertainty in the shape of functions based on the info-gap approach, which has been neglected previously, although info-gap uncertainty in parameters has been addressed.
The fractional-error info-gap model of uncertainty is extremely widespread in conservation science, and it should be defined precisely.Let x be a parameter (e.g., the probability of a specific mechanism or the utility of a specific conservation strategy).Let x denote the best available estimate of this parameter.The fractional error of the estimate is The value of h is unknown and unbounded (hence h ≥ 0), and h is called the horizon of uncertainty.The concept of robustness I sought to develop is the greatest horizon of uncertainty (in one's knowledge) up to which a proposed decision would lead to an acceptable outcome.Additional constraints sometimes apply, for example, if x is a probability, then it cannot be negative or >1.Sometimes the denominator is not the estimate, x, but rather an estimated error of x, but again the structure is the same: an unbounded fractional error of an estimated parameter.
The absolute-error info-gap model employs only the numerator in Equation (1).Sometimes the error interval is asymmetric.Regan et al. (2005) applied info-gap decision theory to planning decisions for conservation of an endangered species, the Sumatran rhinoceros (Dicerorhinus sumatrensis).They provide a very accessible description of the info-gap methodology for analyzing robustness to uncertainty in parameters of the conservation model.They also considered 4 different mechanisms that may be responsible for the drastic reduction in the size of the rhinoceros population.They have estimates of the probability for each mechanism, but recognize that these estimates are highly uncertain: the fractional error of each estimate is unknown.They considered 3 different conservation strategies and estimated the utility of each combination of reduction mechanism and conservation strategy, but again these estimates are highly uncertain and the fractional error of each estimate is unknown.
The 3 conservation strategies Regan et al. (2005) studied were captive breeding of the rhinoceros population, translocation of the population, and extension or revised management of the existing reserve.They show that captive breeding is predicted to be the most successful strategy, whereas establishment of a new reserve and translocation have successively lower predicted performances.However, they demonstrate that these predicted outcomes have no robustness to uncertainty in the parameter values underlying these predictions.They explain, therefore, that prioritization of the strategies based on their predicted outcomes is unreliable and unjustified.This is a specific realization of a generic phenomenon: predictions based on one's best models and data have 0 robustness to uncertainty in that knowledge, which is known as the "zeroing property" (Ben-Haim, 2006).The zeroing property is of great importance because one might be inclined to prioritize available decision options based on best-model predictions.These predictions, however, have no robustness to uncertainty and thus do not provide a valid basis for prioritization.
Recognizing the significance of the zeroing property, Regan et al. (2005) then discuss the generic trade-off between robustness to uncertainty and performance (Ben-Haim, 2006).Specifically, they examine the trade-off between robustness and probability of survival of the endangered species for any specific conservation strategy.They show that the probability of survival increases (which is desirable) as the robustness to uncertainty decreases (which is undesirable).That is, for any specific strategy, if one allows lower probability of survival for the species, then one has greater confidence in achieving this relaxed requirement.The info-gap analysis quantifies this trade-off, elucidating the interaction between uncertainty and outcome for each alternative strategy.Regan et al. (2005) show that the trade-off is rather strong: robustness to uncertainty decreases dramatically for small increase in survival probability.
Finally, Regan et al. (2005) considered preference reversal between strategies (Ben-Haim, 2006).They demonstrate that the robust preference between alternative conservation strategies, in the Sumatran rhinoceros case, changes as one's requirement for survival probability changes.Specifically, at relatively large expected utility of survival, the most robust (and hence preferred) strategy is captive breeding.However, at lower expected utility the most robust strategy is new reserve.No single strategy is robust dominant; no single strategy is predicted to provide better outcome at all levels of uncertainty.Combined with the phenomena of zeroing and trade-off, this preference reversal between strategies supports decision-making for managing deep uncertainty.
Many other scholars have also used fractional-error or absolute-error info-gap models of parameter uncertainty in a wide range of applications in conservation.Applications include conservation of endangered species, reserve planning, species distributions, and controlling invasive non-native species (Table 1).
The ellipsoid-bound info-gap model of uncertainty (Ben-Haim, 2006) has also been used in conservation science.An ellipsoid-bound info-gap model is one in which uncertain parameters are constrained within a family of expanding ellipsoids of known shape but unknown maximum size.McDonald-Madden et al. (2008) considered the protection of endangered species when the relation between the amount of money invested and the resulting conservation benefit is highly uncertain.They developed a quantitative conservation model in which 3 interrelated parameters are uncertain: the annual probability of extinction, the budget required to halve this probability, and a parameter influencing the shape of the functional dependence of probability on investment.They used an ellipsoid-bound info-gap model to represent the unbounded uncertainty about these parameters and their interdependence.van der Burg and Tyre ( 2011) studied management of moun-tain plover populations.They used an ellipsoid-bound info-gap model to represent deep uncertainty in a vector of parameters of the dynamic model.In summary, existing applications of info-gap theory in conservation science consider uncertainty in parameters, but not in the full shape of functional relations.Other info-gap models are relevant to conservation science (see "Info-gap models of functional uncertainty").Two additional questions addressed by info-gap analysis received little or no attention: what is the value of additional information and what is the opportuneness, from uncertainty, for better-than-anticipated outcomes?

INFO-GAP MODELS OF FUNCTIONAL UNCERTAINTY
Many models in conservation science entail functional relationships, such as the relation between reserve size and biodiversity or between cost and survival probability.An estimate of the function is known, along with additional knowledge, such as that the function is nonnegative, monotonically increasing, or normalized.But the true shape of the function is unknown and may differ substantially from the best estimate.This functional uncertainty may cover the entire domain of definition of the function or may be dominant on the tails.The estimated function may be specified by parameters, such as linear slope or median value, but variation of these parameters does not reflect the true range of potential deviation of the estimated from the true function.
For example, Wintle et al. ( 2011) "assume a linear relationship between net cost and area of habitat protected."The true relationship may be monotonic but not necessarily linear.In fact, decreasing marginal benefit may strongly indicate nonlinearity, though the shape of that nonlinearity is uncertain.
Likewise, Hayes et al. (2013) state: "Paradoxically, ecological applications of IGDT [info-gap decision theory] focus almost exclusively on only 1 source of uncertainty in ecological problems, model parameter uncertainty, and typically ignore other sources, particularly model structure uncertainty and dependence between parameters, that can be just as severe." I examined 3 info-gap models for functional uncertainty (Ben-Haim, 2006): fraction error, envelope bound, and shape bound.An info-gap model should faithfully represent one's knowledge and its limits.Otherwise, either knowledge or uncertainty is ignored.
For fractional error, let f (x) denote the best estimate of a function whose unknown true form is f (x).Analogous to Equation (1), the fractional-error info-gap model for uncertainty in the shape of the function f (x) is At any horizon of uncertainty (h), the info-gap model is the set containing all functions f (x) for which fractional deviation from the estimate f (x) is no greater than h (left inequality in Equation 2) at any value of x in the range of definition of the function.The value of h is unbounded (right inequality), so the info-gap model is an unbounded family of nested sets of possible functions: it is an expanding envelope of possible functions.These sets include functions that are fundamentally different in shape from the estimate, not just parametric variations.This info-gap model represents uncertainty in assumptions underlying the estimated function and reflects the possibility of mechanisms and interactions that were not considered or even known when formulating the estimated function.
The uncertainty represented by an info-gap model is unbounded: a worst case is not known.Hence, one does not ask, how bad could things be?Rather, one asks, how large an uncertainty can be tolerated while assuring specified outcomes?The robustness function, to be defined, responds to this question.
For the envelope-bound info-gap model, Equation ( 2) can be immediately generalized to include information about the expanding envelope of uncertainty.If the function is a probability distribution function, then it is monotonic and bounded above by 1 and below by 0. If it is a measure of biodiversity, it may be bounded above by the number of possible species and below by the composition of the surrounding environment.The uncertainty-how much the true function, f (x), deviates from the estimate, f (x)-is mediated by lower and upper envelope functions w 1 (x) and w 2 (x): (3) At any h, the info-gap model is the set of all functions within the envelop, and the envelop expands as h increases.
Now consider the slope-bound info-gap model.Equations ( 2) and (3) contain both smoothly varying functions and functions that wiggle and jump.Sometimes one has further information that constrains the slope of the function.For instance, biodiversity may increase (but not decrease) as reserve size increases, or survival probability may decrease (but not increase) as tourism limits increase.The estimated function, f (x), obeys the slope constraint, and the slope-bound info-gap model reflects functional uncertainty around that constraint.Let s(x) represent the slope of f (x) as a function of x, whereas s(x) represents the slope of the unknown true function f (x).One can now implement a fractional-error or an envelop-bound info-gap model for uncertainty in the slope function.

FURTHER INSIGHTS FROM INFO-GAP THEORY
I considered 2 insights from info-gap analysis of uncertaintyvalue of information and opportuneness from uncertainty (Ben-Haim, 2006;Moilanen, Runge, et al., 2006; van der Burg & Tyre, 2011)-that have received scant attention in conservation science.

Value of information
More knowledge is better than less, but this does not determine budget-constrained allocations.Should one improve a survey to identify the next-least-common species or to improve the spatial resolution of common species?Should one invest in modeling existing reserves or in modeling new reserves?These questions reflect the value of information.Many approaches assess the worth of new information.One approach employs robustness to uncertainty.
Robustness to uncertainty usually increases as the amount of information increases (Ben-Haim, 2006).More robustness is better than less.Hence, the worthiness of alternative allocations (for acquiring information) can be prioritized by the magnitude of the robustness increment that accrues from each.It is not the predicted increment of worthiness that underlies the prioritization.Rather, prioritization is based on the robustness to uncertainty for achieving an acceptable outcome.

Opportuneness from uncertainty
A situation is opportune, due to uncertainty, if windfalls are possible even in ordinary conditions.The potential for favorable surprise motivates one to enhance the opportuneness from uncertainty.Conversely, the potential for adverse surprise motivates one to enhance robustness.Opportuneness and robustness are complementary responses to uncertainty.The opportuneness of an action is the least horizon of uncertainty at which a wonderful outcome is possible (though not guaranteed).The robustness of an action is the greatest horizon of uncertainty at which an adequate outcome is guaranteed.

ROBUSTNESS TO UNCERTAINTY
Decision-making is based on knowledge in various forms, such as quantitative scientific models, empirical correlations, or expert professional judgment.Prioritization of decision options based on that knowledge would undoubtedly result in the best possible effectiveness or cost if the knowledge were comprehensive and entirely true.However, I considered situations in which the best available knowledge is deficient-incomplete or erroneous-in important but unknown ways; comprehensive and entirely true knowledge is not available.I identified generic situations in which prioritization based on available knowledge is more robust to uncertainty than any other strategy.In these situations, the putatively optimal decision is preferred for seeking specified goals.I also identified generic situations in which a putatively suboptimal choice is more robust and thus may be preferred.I adopted no explicit model or measure of decision performance (effectiveness or cost) and made no assumptions about what constitutes performance or how it is measured, although I assumed that it can be assessed quantitatively, subject to uncertainty.
The results in this section are not new (Ben-Haim, 1999, 2000, 2005, 2006), but have received little or no attention in conservation science.My analysis is based on info-gap decision theory that has been applied in engineering, economics, medicine, national security, and other areas, as well as conservation (see info-gap.com).

Scalar performance function
I considered the situation in which the planner's knowledge is invested in a single scalar best estimate of performance of any specified decision.The strategy of putative optimization chooses the decision to optimize this estimated performance.I examined whether putative optimum is more robust to uncertainty than any other decision-making strategy.I refer to this as robust dominance of the putative optimum.I also examined an important trade-off between performance and the robustness.I made no assumptions about how performance is measured other than by assuming that it is assessed with a single scalar function.
Multiattribute performance functions are often reduced to single-attribute functions to which the results in "Scalar performance function" are relevant.The examinations here can be extended if the planner's knowledge is represented by a multiattribute evaluation of performance.
Many studies focus on a single measure of performance of a decision or action.Backus and Baskett (2021)  I considered the selection between several alternative decisions.Let d denote a decision.For any specific d , its performance is measured by a scalar function M (d ).The function M (d ) may be a probabilistic quantity, such as a mean or variance, or it may be a nonprobabilistic quantity, such as number of species protected.However, the form of this function is poorly known.The best-known estimate of this function is M (d ), but the accuracy of this estimate is highly uncertain.I assumed that M (d ) is positive.
I found that, under specific conditions, the putative optimum policy is also the policy whose robustness to uncertainty is maximal.This reassuring conclusion is based on 3 central concepts: deep uncertainty, satisficing, and robustness to uncertainty.These concepts also underlie the discussion in "Scalar effectiveness function with an error estimate," in which one sees that this finding (regarding the putative optimum) is not the robust optimum when one has slightly greater knowledge about the uncertainty.
The estimated performance function is deeply uncertain.I expressed the satisficing requirement by requiring that the actual effectiveness, M (d ), be no less than a critical value, M c .If M (d ) exceeds M c , then one asks, how much error in M (d ) can be tolerated so that the correct M (d ) still exceeds M c with decision d ?The answer is the robustness of decision d .Large robustness implies high confidence that this decision satisfies the requirement; small robustness implies low confidence.
The robustness function takes the following form (see Appendix S1): If the performance function is loss rather than effectiveness, then M (d ) must not exceed the critical value M c .The robustness function for loss is as follows (Appendix S1): or 0 if this is negative.The robust dominance of the knowledgebased strategy that minimizes the estimated loss holds in this case as well.
Other than assuming that the best-known performance function for either effectiveness or loss, M (d ), is a positive scalar function, no other assumptions are made about either the true or the estimated scalar performance functions, M (d ) or M (d ), or about the class of available decisions.Also, no explicit knowledge other than knowledge of M (d ) is assumed.This very general result-that knowledge-based decision-selection strategies (defined after Equation 4) are more robust than all other strategies-holds because of the extreme paucity of information about the scalar performance function.This is a somewhat ironic result: extreme lack of knowledge justifies the knowledgebased decision-selection strategy.From a contrarian perspective, one might say that the knowledge-based strategy is better than any other strategy simply because one knows so little about the processes involved.This ironic result is also a note of caution against overconfidence in knowledge-based optimization of decision performance.The limits of this result are examined in subsequent sections.

Scalar effectiveness function with an error estimate
One now has best estimates of 2 different functions.First, one knows a single scalar best estimate of performance of any specific decision.Second, one knows a function that assesses the error of the estimated performance, while acknowledging that greater error can occur.The strategy of putative optimization is not necessarily the most robust to uncertainty, which means there is an "innovation dilemma" (Ben-Haim et al., 2013) that arises when choosing between 2 options, A and B. Option A is predicted to be better than option B based on available knowledge.However, A is also more uncertain than B, so A may plausibly turn out worse than B; hence, the dilemma.The innovation dilemma characterizes situations in which decisions other than the putative optimum can be more robust than the putative optimum decision.The concept of robustness to uncertainty resolves the innovation dilemma.
In a continuation of the previous development, I considered further information.In addition to an estimate, M (d ), of the scalar performance function as a function of the decision, d , there is also a positive error estimate, w(d ).For instance, w(d ) could quantify contextual understanding that suggests that the estimated performance is less uncertain when making a large investment in human development and a small investment in physical infrastructure than the reverse.More generally, the error estimate w(d ) reflects relatively lower uncertainty about specific agents, organizations, or ecosystems as demonstrated by their past behavior.That is, it is thought that the true performance, M (d ), may deviate from M (d ) by as much as ±w(d ) or more.The error function w(d ) is not an upper bound or worst case of the error, but it does compare the anticipated errors for different d : some decisions have lower estimated error than other decisions.
When the performance is assessed as effectiveness, the robustness to uncertainty in the estimated performance function is as follows (Appendix S1): or 0 if this expression is negative.This reduces to Equation (4) if w(d ) = M (d ).
When the performance is measured as a loss, the robustness function becomes or 0 if this expression is negative.This reduces to Equation (5) if w(d ) = M (d ).The robustness function for effectiveness in Equation ( 6) is shown schematically in Figure 1.There are 2 properties of all info-gap robustness functions in Figure 1: zeroing and trade-off.

Consider zeroing
The estimated effectiveness of policy d , based on all available information, is M (d ), which is the horizontal intercept of the robustness curve in Figure 1.Any error in this information entails the possibility of effectiveness < M (d ).That is, the robustness (to uncertainty in the information) of the estimated effectiveness is 0. This implies that the estimated effectiveness of a decision is not a sound basis for prioritizing that decision.Consider trade-off.Robustness trades off against required effectiveness.Figure 1 shows that the robustness, ĥ(d , M c ), increases  6).
The horizontal axis is the lowest value of effectiveness, M c , one can accept with policy d .The vertical axis is the robustness (to uncertainty) for achieving this critical effectiveness with policy d , denoted as ĥ(d , M c ). (which is good) as the required effectiveness, M c , decreases (which is undesirable).The negative slope of the robustness curve in Figure 1 results from the trade-off, and the magnitude of the slope expresses a cost of robustness.If the slope is steep, then relinquishing a small increment of effectiveness entails a large increase in robustness, implying low cost of robustness.Conversely, if the slope is shallow, then the demanded effectiveness must be greatly reduced to achieve a substantial increase in robustness, implying a large cost of robustness.
Figure 2 shows robustness curves of effectiveness in Equation (6) for 2 different decisions: d k is the knowledge-based decision and d a is an alternative decision, where these decisions satisfy 2 relations and Equation ( 8) asserts that the predicted effectiveness of the knowledge-based decision is greater than the predicted effectiveness of the alternative decision, as required by the definition of d k as discussed following Equation (4).Equation ( 9) asserts that the predicted relative error of the alternative decision is less than the predicted relative error of the knowledge-based decision.Taken together, these relations mean that even though d k is predicted to be more effective than d a , this prediction is relatively more uncertain than the prediction regarding d a .Specifically, d a may reflect specific entities with past behavior that legitimately establishes lower relative uncertainty about their attributes.
Decision d a is predicted to be less effective than decision d k (Equation 8), perhaps because d a emphasizes specific capabilities.Nonetheless one is less uncertain regarding the capabilities that d a offers compared with d k (Equation 9).This highlights the distinctive difference between decision selection based on optimizing the outcome with available knowledge (d k ) and decision selection based on satisficing the outcome and optimizing the robustness to uncertainty.
The planner who must choose between these 2 decisions faces a dilemma: choose the putatively better but more uncertain option, d k , or the putatively worse but less uncertain option, d a .This innovation dilemma often arises when choosing between a new and innovative but less familiar option and a standard state-of-the-art option (Ben-Haim et al., 2013).
The innovation dilemma is revealed by the intersection between the robustness curves of these options, as seen in Figure 2. The zeroing property asserts that each robustness curve reaches the horizontal axis at the predicted performance.Thus, Equation ( 8) explains that the robustness function, ĥ(d k , M c ), reaches the horizontal axis to the right of ĥ(d a , M c ). Equation ( 9) implies that the cost of robustness of d a is lower than that of d k ; thus, ĥ(d a , M c ) versus M c is steeper.Hence, the robustness curves intersect one another.
Decision support for resolving the innovation dilemma is provided by the robustness curves.Let M × denote the value of critical effectiveness, M c , at which the robustness curves cross one another in Figure 2. If the planner's required effectiveness is greater thanM × , but < M (d k ), then the knowledge-based decision is more robust than the alternative decision.Large robustness is desirable because the planner faces deep uncertainty, so this could motivate the planner to choose the knowledge-based decision.However, if the planner can accept effectiveness <M × , then the alternative decision is more robust than the knowledge-based decision.In this case, deep uncertainty can motivate the choice of the alternative decision because, although its predicted effectiveness is lower than that of the knowledge-based decision, its robustness to uncertainty is greater.
In other words, the innovation dilemma can motivate a reversal of preference between the putative preference, d k , and the putative second choice, d a .The innovation dilemma arises from the additional information embedded in the uncertainty estimate, w(d ).

INFO-GAP ANALYSIS OF ROBUSTNESS TO UNCERTAINTY
I illustrate the info-gap analysis of robustness to uncertainty with a case study of the reproductive output of an endangered plant threated by climate change.Four decisions confront the planner: identify an endangered species, identify a region favorable to survival of that species, decide how large a reproductive output function is necessary for survival, and decide what fraction of the species to translocate to that region.Two uncertainties accompany these decisions.First, reproductive output is a function of temperature, but future temperature is uncertain due to climate change.Second, the variation of reproductive output, as a function temperature, is uncertain in the new region because few if any individuals of the species presently inhabit it.The planner faces a dilemma: larger reproductive output (implying more likely survival) is desirable, but more uncertain reproductive output (implying less confident survival) is undesirable.
Backus and Baskett (2021) developed a detailed time-varying multispecies stochastic model to support a priori assessments of the feasibility of translocating a fraction of each plant species beyond their historical domain to regions that are anticipated to be more favorable in the future as a result of climate change.Their model is built on extensive theoretical and empirical foundations.Nonetheless, uncertainties accompany the model.For instance, the reproductive output of each species, as a function of ambient temperature, combines the Gaussian distribution and its cumulative inverse (the error function), themselves embedded in an exponential function.This entails various plausible assumptions and practical simplifications, but cannot be viewed as apodictic.The functional shape of this reproductive output function could plausibly be different.However, one cannot know how and where the differences could arise.In addition, future ambient temperatures in the translocation region are uncertain due to uncertain mechanisms of climate change.I applied info-gap analysis of robustness to uncertainty of the shape of the reproductive output function and uncertainty in future temperatures.My example addresses a static situation, but info-gap theory has also been applied to dynamic decisions over time in ecological restoration (Moilanen et al., 2009) and other areas.

Reproductive output function
The reproductive output of a species is the number of offspring per individual.I considered a single species and denote its reproductive output by b(T ), which is a function of the ambient temperature T .Backus and Baskett (2021)  on earlier work by Norberg (2004) and Urban et al. (2012): where z, , r, and  denote the thermal optimum, temperature tolerance, reproductive strength, and skewness constant of this species, respectively.The variables z, , and r in this expression are themselves random variables in the model of Backus and Baskett (2021).I assigned them specific values to illustrate the analysis of robustness to uncertainty in the shape of the reproductive output function and uncertainty in the future temperature.The reproductive output function, Equation (10), with the average values of z and , is shown in Figure 3.This function has a skewed unimodal shape, the maximum of which is somewhat below the value of z due to the negative value of the skewness constant .The reproductive output slightly exceeds 6 at its maximum but falls rapidly as the temperature deviates from the mode.The width of the reproductive output function in Figure 3 is about 3 at a reproductive output of 2.

Uncertainty
I focused on 2 central uncertainties: shape of the reproductive output function, b(T ), and future ambient temperature in the region to which Backus and Baskett (2021) translocate part of the population, which I denote as T f .(For algebraic simplicity, I assumed that T f is nonnegative, although generalization to negative temperatures is straightforward.)The estimated reproductive output function (Equation 10) is denoted b(T ), but the shape of the true function, b(T ), may differ.The estimated future ambient temperature is denoted Tf .
The value of T f and the shape of b(T ) are uncertain.As discussed earlier, probabilistic information is lacking because the future is unknown and environmental and physiological mechanisms are uncertain.One might anticipate that the reproductive output function would be unimodal, even if it deviates from the precise form of Equation ( 10).However, I am considering deep uncertainty in the environmental impact of future climate change and in interactions between a species and a new region to which it has been translocated.These considerations entail the possibility of complex, unfamiliar, and conflicting mechanisms that could result in multimodal reproductive output functions.Furthermore, although various considerations support the strong Gaussian-exponential decay of the tails of Equation ( 10), one should acknowledge that the tails of this function are the least familiar parts because they represent rare or unusual circumstances.The actual shape of the far tails of the reproductive output function may differ from the Gaussian model.
I represent this uncertainty with a nonprobabilistic fractionalerror info-gap model: Each set, U (h), of this info-gap model contains all nonnegative values of future temperature, T f , and all nonnegative reproductive output functions, b(T ), of any shape, whose fractional deviations from their respective estimates are no greater than the nonnegative value h.Each set, U (h), contains the best estimates, Tf and b(T ).The value of h, called the horizon of uncertainty, is unknown and unbounded.Thus, the info-gap model is an unbounded family of nested sets of uncertain entities, implying that there is no known worst case.For h > 0, the sets U (h) contain reproductive output functions whose shape as a function of temperature differs from Equation (10), as well as the estimated function.Backus and Baskett (2021) discuss physical translocation of a fraction of the population to a region where the anticipated future temperature, Tf , is favorable.The challenge is that the future temperature and the shape of the reproductive output function are uncertain, as specified by the info-gap model of Equation ( 11).The future ambient temperature will be T f and this may differ from the anticipated value of Tf , and the true reproductive output will be b(T ), which may differ in shape from b(T ) in Equation ( 10).

Robustness to uncertainty
If b(T ) < 1, then individuals produce too few offspring to maintain the population.If b(T ) > 1, then individuals produce more than just replacing themselves, but some offspring may not reproduce sufficiently to maintain the population.The lower bound for population maintenance is at b(T ) = 1, but a larger value is needed for reliability.
Let b c denote a value of the reproductive output at which one can have confidence in survival of the species.The future repro- ductive output, at the future ambient temperature, must be no less than this critical value, b c : The uncertainties in T f and in b(T f ) motivate the robustness function, which is the greatest horizon of uncertainty up to which the condition in Equation 12is assured.The definition and derivation of the robustness function appear in Appendix S2.The robustness curve in Figure 4 reaches the horizontal axis precisely at the estimated reproductive output, which is 6.0.This 0 robustness of the estimated outcome is the zeroing property discussed earlier: predicted outcomes have no robustness to uncertainty in the knowledge underlying the prediction.In other words, translocation of the species to a region whose anticipated future temperature is Tf = 18.5 is predicted to have quite large reproductive output, namely, 6.0.However, this prediction has no robustness to uncertainty in the future temperature or in the shape of the reproductive output function.In short, this encouraging prediction is not a reliable basis for assessing the translocation.

Interpretation and operational implications of the robustness function
The negative slope of the robustness curve expresses the trade-off between robustness to uncertainty (for which a large value is desirable) and critical reproductive output (for which a small value is undesirable).
The robustness is quite low for values of the reproductive output, b c , at or above unity.For instance, the robustness is 0.030 for b c = 4, and the robustness is 0.078 at b c = 1.A robustness of 0.078 means that the uncertain entities-the value of T f and the shape of b(T )-can deviate from their estimates by at most 7.8% without causing the reproductive output to fall below the corresponding critical value of b c = 1.This is rather That is, 18.5 • C is expected to be favorable; the predicted reproductive output is 6.However, the central point here is the impact of uncertainty.Even small deviations of temperature and reproductive output function from their anticipated values can cause the reproductive output to plummet.Predicted outcomes are not a reliable basis for decision-making to conserve the species in question.The survival of this species is far more precarious-due to uncertainty-than the prediction of the best available (but highly uncertain) knowledge.
Figure 5 shows 3 robustness curves for 3 different values of the estimated future temperature Tf : 18.5, 17, and 16 • C. The reductions in predicted ambient temperature cause substantial reduction of the predicted reproductive output, from b c = 6.0 at 18.5 • C to 3.1 and 1.2 at 17 and 16 • C, respectively (horizontal intercepts).This is because the predicted temperature is moving to lower values of the predicted reproductive output function.However, the robustness of these predicted reproductive outputs is precisely 0 (the zeroing property), and only lower reproductive output has positive robustness to uncertainty (the trade-off property).
Results are similar for higher predicted temperatures (Figure 6).Once again, as the predicted temperature moves away from the peak of the predicted reproductive output function, the predicted reproductive output falls (horizontal intercepts).These predictions, however, have no robustness to uncertainty, and only lower reproductive output has positive robustness.

DISCUSSION
My focus here is methodological (how to manage deep uncertainty surrounding knowledge used in decision-making for conservation) rather than a specific taxon, ecosystem, or decision.Uncertainty is commonly modeled with probability distributions that provide powerful analytical tools if those probability distributions are reasonably accurate.However, the deep uncertainties prevalent in many applications entail unfamiliar or changing circumstances or unanticipated human intervention or qualitative verbal information.Verification and validation of probability distributions are not always feasible under these circumstances.I addressed deep uncertainty with nonprobabilistic info-gap models of uncertainty.An info-gap model includes existing knowledge (which may be probabilistic) and acknowledges the unbounded horizon of uncertainty around this knowledge.The info-gap model underlies my analysis of robustness to uncertainty.
The robustness is the greatest horizon of uncertainty (in the knowledge) up to which a proposed decision would lead to an acceptable outcome.Alternative decisions are prioritized by their robustness to uncertainty, whereas critical outcome requirements are satisficed.Robustness is optimized; outcome is satisficed.A decision with a predicted suboptimal outcome may be preferred over a predicted outcome-optimal decision if the former is more robust for achieving an acceptable outcome.
Analysis of robustness to uncertainty is based on 3 components: a measure of performance of a decision, an unbounded nonprobabilistic info-gap model of uncertainty, and specification of a least acceptable level of performance.The robustness of a decision is the greatest horizon of uncertainty up to which acceptable performance is guaranteed.I considered uncertainty in parameters and in functional shape.
The robustness function has 3 properties that provide decision support: zeroing, trade-off, and preference reversal.Consider zeroing of the robustness.The best available knowledge is used to predict the outcomes of alternative decisions.However, that knowledge contains errors and deficiencies that undermine those predictions.Outcome predictions have 0 robustness to uncertainty in the knowledge and hence cannot be relied upon.Predicted outcomes are not a reliable basis for decision-making.
Consider trade-off between robustness and outcome.Outcomes that are more modest or less ambitious than the predicted outcome have positive robustness to uncertainty.Robustness increases (which is good) as the required performance becomes less demanding (which is undesirable).Quantitative assessment of this trade-off between robustness (to uncertainty) and quality of the outcome supports the evaluation and choice among alternative decisions.
Consider preference reversal between alternatives.All alternative decisions have trade-offs between robustness and out-come, but some decisions may display less severe trade-offs than others.Considering 2 alternative decisions, 1 decision may be predicted to be worse but, at suboptimal outcomes, is nonetheless more robust than the alternative.A planner may prefer the first alternative, achieving greater robustness in exchange for reduced predicted outcome.Quantitative robustness curves support the judgments involved.
assessed effectiveness with the likelihood of persistence of a translocated species.Joshi et al. (2021) evaluated an interactive model of crop damage by wild herbivores in terms of impact on agricultural productivity.Horton et al. (2021) evaluated dynamic forecasts of bird-migration nights in terms of the number of nights needed for detecting a given fraction of migrating birds.Hylander et al. (2022) assessed the conservation of managed forests with biodiversity within the forest.The use of a single assessment of effectiveness is widespread.
) or 0 if this expression is negative, which occurs if the estimated effectiveness, M (d ), is less than the critical value, M c .Appen-dices are available from https://info-gap.technion.ac.il/files/ 2023/06/pol-sel014appendices-1.pdf.A knowledge-based strategy can employ any combination of scientific models, empirical relations, or expert judgment.A knowledge-based strategy chooses the decision, d , that maximizes the best estimate of M (d ).I denote this knowledgebased selection d k .Thus, M (d k ) is no less than M (d ) for any d .From Equation (4), one sees that the knowledge-based selection, d k , maximizes the robustness to uncertainty in the decision effectiveness because M (d k ) is no < M (d ) for any other d .That is, the robustness of d k is strictly greater than the robustness of any other d .Recall the assumption that one has no knowledge other than of the estimated effectiveness function, M (d ).

FIGURE 1
FIGURE 1Robustness function of decision effectiveness (Equation6).The horizontal axis is the lowest value of effectiveness, M c , one can accept with policy d .The vertical axis is the robustness (to uncertainty) for achieving this critical effectiveness with policy d , denoted as ĥ(d , M c ).

FIGURE 2
FIGURE 2 Robustness functions of effectiveness for 2 allocations of decision effectiveness: knowledge-based (d k ) and alternative (d a ).

Figure 4
Figure4shows a robustness function.The estimated future temperature, Tf , and the parameters of the estimated reproductive output function, b(T ) in Equation (10), are specified in the figure caption.The estimated temperature, Tf , is close to the peak of the estimated reproductive output function in Figure3.The robustness curve in Figure4reaches the horizontal axis precisely at the estimated reproductive output, which is 6.0.This 0 robustness of the estimated outcome is the zeroing property discussed earlier: predicted outcomes have no robustness to uncertainty in the knowledge underlying the prediction.In other words, translocation of the species to a region whose anticipated future temperature is Tf = 18.5 is predicted to have quite large reproductive output, namely, 6.0.However, this prediction has no robustness to uncertainty in the future temperature or in the shape of the reproductive output function.In short, this encouraging prediction is not a reliable basis for assessing the translocation.The negative slope of the robustness curve expresses the trade-off between robustness to uncertainty (for which a large value is desirable) and critical reproductive output (for which a small value is undesirable).The robustness is quite low for values of the reproductive output, b c , at or above unity.For instance, the robustness is 0.030 for b c = 4, and the robustness is 0.078 at b c = 1.A robustness of 0.078 means that the uncertain entities-the value of T f and the shape of b(T )-can deviate from their estimates by at most 7.8% without causing the reproductive output to fall below the corresponding critical value of b c = 1.This is rather

TABLE 1
Applications of info-gap theory in conservation science.

Endangered species Info-gap model of parameter uncertainty
*Range of species endangered.