Risk Assessment under Ambiguity: Precautionary Learning vs. Research Pessimism

Transcription

1 Risk Assessment under Ambiguity: Precautionary Learning vs. Research Pessimism Daniel Heyen Grantham Research Institute London School of Economics Boris Wiesenfarth Independent Researcher September 14, 2017 Timo Goeschl Department of Economics Heidelberg University Abstract Agencies charged with regulating complex risks frequently need to take decisions on risk assessment and risk management under conditions of ambiguity. What mandates should society write for such agencies? Two approaches stand out: Costbenefit analysis (CBA) based on expected utility theory and the Precautionary Principle (PP) based on maxmin expected utility (MEU). Using a model in which the agency can decide on the precision of a signal which provides noisy information on a payoff-relevant parameter, we compare CBA and PP. Contrary to intuition, a PP mandate will often give rise to a less informed regulator, making regulatory mistakes more likely. Keywords: scientific uncertainty; ambiguity; learning; risk assessment; precautionary principle; active information acquisition; regulatory mandates. JEL codes: D81, D83, Q58. We are grateful to Gregor Betz, Valentina Bosetti, Simon Dietz, Adam Dominiak, Juergen Eichberger, Itzhak Gilboa, Simon Grant, Antony Millner, Juan Moreno-Cruz, Joerg Oechssler, Frank Riedel, Bernard Sinclair-Desgagné, Christian Traeger, Nicolas Treich, and participants at the conferences AU- ROE 2012 Bern, Spring Meeting of Young Economists 2012 Mannheim, AERE 2012 Asheville, EAERE 2012 Prague, Workshop on Irreversible Choices 2012 in Brescia, the Harvard Geoengineering Summer School 2013 and the workshop and seminars at the Grantham Research Institute at LSE, the Department of Economics Heidelberg, FEEM, IASS, ZEW, Heidelberg Geoengineering Forum for their helpful comments and suggestions. The first two authors acknowledge the support of the German Research Foundation, grant no. GO 1604/2. Daniel Heyen also acknowledges support through the German Research Foundation s Postdoctoral Research Fellowship HE 7551/1-1. 1

2 1 Introduction In the early summer of 2011, the European Union experienced an outbreak of Shigatoxin producing Escherichia coli (STEC). 1 More than 3,100 cases of bloody diarrhoea and more than 850 of haemolytic uremic syndrome (HUS), a serious condition that can lead to kidney failure, were reported during the outbreak. deaths (EFSA 2012). There were 53 confirmed The outbreak of STEC was linked through epidemiological research to the consumption of fresh salad vegetables (BfR 2012). Because research found toxin producing E.coli on cucumbers from Spain, European Commission and German officials issued alerts and effectively required Spanish cucumbers to be withdrawn from the market (BBC 2011; BfR 2012). These events saddled Spanish vegetable growers, among others, with economic damages of several hundred millions of Euros (BBC 2011). After more research commissioned by the European Food Safety Authority (EFSA) and the German Interior Ministry, however, it became clear that officials had in all likelihood misidentified the source: Bean sprouts from a German farm, rather than contaminated Spanish cucumbers, carried the dangerous strain that gave rise to potentially lethal HUS among consumers (BfR 2012). While acknowledged as the most likely source, an entirely conclusive result on the cause of the outbreak has never been established. The STEC incident from 2011 typifies an important recurring problem for regulators. Here, a regulator was called upon to make decisions affecting both public mortality and morbidity risk and economic livelihoods on the basis of ambiguous evidence on the source of risk. Not only did the regulator have to make a risk management decision, viz. to decide on which produce to ban in order to eliminate the source of risk, thus imposing damages of several hundred million Euros on its producers. The regulator also had to decide on the scale of risk assessment, viz. collecting and screening thousands of samples in order to reduce the risk of erroneously banning harmless produce while allowing harmful produce to continue to be sold. Situations in which a regulator needs to take a highly consequential decision based on poor, but improvable knowledge, are commonplace in today s world (Kaebnick et al. 2016; Sunstein 2005a; Randall 2011; Graham 2001). This raises the question of what rules should govern these decisions. Differently put, under what mandate should a regulator reach its decisions on assessment and management of risks? Two approaches for writing this mandate stand out in the current discussion. One is the use of a traditional welfare-economic approach based on expected utility theory. This is the approach that underpins most forms of conventional cost-benefit analysis (CBA) (Viscusi et al. 2005). 2 The other is the Precautionary Principle (PP). Despite lacking a clear definition and being criticized on grounds of internally inconsistency and logical incoherence (Peterson 2006; Sunstein 2005b), the PP has been adopted by the European Commission (Eu- 1 The outbreak was first incorrectly classified as EHEC and has become widely known under this label. 2 This traditional approach is prone to subtleties as well, cf. Martin and Pindyck (2015). 2

3 ropean Commission 2000) and gained significance of a general principle of EU law (Recuerda 2006). The PP has several connotations, all of them rooted in the presence of fundamental uncertainties that challenge traditional risk assessments. One of these interpretations of the PP requests the regulator to build a margin of safety and avoid harm when the causal chain is subject to scientific uncertainty, and thus to prepare against unfavourable events (Zander 2010; Sunstein 2005c). A related, yet distinct, interpretation of the PP directly targets the level of information under which the regulator has to make her decision. In this widespread view, regulatory mandates based on the PP would lead to more science (Tickner 2002) and thus a better informed regulatory decision than conventional CBA mandates (Cranor 2005; Myers and Raffensberger 2005; Martuzzi 2007; Bourg and Whiteside 2009). How to meaningfully compare the implications of CBA and PP? Over the last decades, the literature on decision-making under uncertainty has offered alternatives to the subjective expected utility framework (Savage 1954) which underpins traditional CBA (Dasgupta and Pearce 1972; Shaw and Woodward 2008). These non-expected utility frameworks provide several coherent rationalizations of ambiguity averse preferences (Gilboa and Schmeidler 1989; Klibanoff et al. 2005; Chateauneuf et al. 2007), forming the decision-theoretic foundations of the PP in applications (Asano 2010; Athanassoglou and Xepapadeas 2012; Barrieu and Sinclair-Desgagné 2006; Basili et al. 2008; Gollier and Treich 2003; Lemoine and Traeger 2012; Millner et al. 2013; Treich 2010; Courbage et al. 2013; Vardas and Xepapadeas 2010). Despite being important steps into a solid economic foundation of the PP, these contributions do not focus on informational choices and are hence not capable of shedding light on the interplay of regulatory decision and research effort that was at the heart of the risk regulatory challenge during the STEC outbreak. Other contributions that put informational choices centre-stage (Gollier et al. 2000; Gollier and Treich 2003) maintains standard preferences that cannot capture the ambiguous state of knowledge that complicates many regulatory problems. The purpose of the present paper is to fill this gap and analyse the interplay of the PP with investments to improve the risk assessment precision. For this purpose, we study a model of active learning under ambiguity averse preferences. 3 We operationalize the PP as maxmin expected utility preferences (meu) (Gilboa and Schmeidler 1989), arguably the best-known axiomatization of ambiguity aversion. The maxmin rule is conceptually simple and, by avoiding some of the reservations against the idea of precaution, provides a relatively modest operationalization of the PP: Like other ambiguity averse decision-rules, meu offers a transparent framework for assessing different policy options that explicitly balances the costs and benefits of regulation. Also, and often a source of confusion, the maxmin rule in this form is sensitive to levels of uncertainty and infor- 3 The reader familiar with decision theory may be surprised to see the terms risk and ambiguity often appear next to each other in the present paper. We use the term risk here in the broad sense used in the risk regulation literature to describe all types of uncertain, potentially detrimental societal outcomes. 3

4 mation since choices are based on the worst probability scenario, not the worst possible outcome. 4 We model learning under ambiguity by relying on the framework of Epstein and Schneider (2007). A key innovation, and a contribution to the decision-theoretic literature on the value of information under ambiguity (Al-Najjar and Weinstein 2009; Siniscalchi 2009; Snow 2010; Hoy et al. 2014), is to give the meu decision-maker control over the precision of a noisy signal. Jointly, these steps demonstrate that intuition leads in many cases to a mistaken prediction about the relative research efforts expended by a regulator operating under a conventional CBA mandate and one operating under a PP mandate. In many settings conventional CBA, not the PP, leads to a greater effort to understand the true state of the world. The paper proceeds in three steps. By introducing two specific stylized examples of regulatory decision-making situations, we first illustrate numerically that in one of them the standard intuition holds, while in the other one it fails. This discrepancy of outcomes, in which the PP sometimes leads to more research, sometimes to less, justifies the development of a simple conceptual model that explains the nature of these effects. We develop such a model and demonstrate that there are two effects at work, a Precautionary Learning Effect that makes a PP regulator value risk assessment precision more highly than a CBA regulator, and a Research Pessimism Effect that has the opposite effect. Which effect dominates depends on the specific features of the decision-making situation. This means that no mandate will ensure that the regulatory decision is always better informed, irrespective of the circumstances. The insight that the choice of the mandate does not have a uniform impact on the regulator s level of information is not only important in its own right. The identification of the two countervailing effects also has immediate implications for the design of regulatory institutions. We will show that writing separate mandates for the two risk regulatory tasks can reconcile the PP with its notion of better informed decision-making. We proceed as follows. Section 2 numerically works out two stylized examples, the STEC example from above and the approval decision for a new pesticide by the Environmental Protection Agency. Despite being structurally very similar, the examples exhibit sharply different ramifications of the PP on the research effort. Section 3 develops the fundamental decision-theoretic model that embraces both examples and leads to a general analysis of risk management decisions and the choice of risk assessment precision in section 4 and section 5, respectively. The reader more interested in the formal analysis may thus skip section 2 and go straight to section 3. Section 6 demonstrates the existence of two countervailing effects of the PP on information acquisition and, by comparing their relative dependency on the payoff-structure, provides a compelling understanding of the contrarian findings in section 2. Section 7 concludes. 4 The set of probability scenarios combines the decision-maker s knowledge and preferences (Gilboa and Marinacci 2016) and is hence flexible enough to capture very extreme forms of the PP (when the set of considered probability scenarios includes the one which puts full weight on the worst outcome) as well as the CBA (when the set of scenarios contains only one medium element) 4

5 2 Two leading examples This section presents a motivating juxtaposition of two stylized numerical examples. The first example is a version of the STEC outbreak introduced earlier. The second example describes the approval decision by the Environmental Protection Agency (EPA) for a novel pesticide. The examples are very similar in structure. Yet, basing risk regulation on the PP has sharply divergent implications for the regulator s risk assessment choices. These differences both motivate and anticipate the formal analysis in later sections. 2.1 Example 1: STEC outbreak and Food Safety The risk regulatory challenge at the heart of the STEC incident was uncertainty as to the origin of the epidemic outbreak. Several vegetables were considered, among them cucumbers and sprouts. As a simple caricature, suppose that the set of potential origins of STEC outbreak has been narrowed down to these two vegetables. The risk management decision is then which vegetable to remove from the market. 5 Table 1 gives a stylized specification of societal outcomes that arise under the two possible states of the world (origin of the outbreak) and the two different risk management decisions by the European authorities. The numerical values are illustrative but in the order of magnitude of the actual decision problem back in Table 1: Societal outcomes in millions EUR of the STEC risk management decision. Sprouts contaminated Cucumbers contaminated Ban sprouts Ban cucumbers Table 1 captures the fact that a product ban that removes a vegetable from the market results in losses to the agricultural sector that can be approximated by the market value of the product. Assume these losses are 500 million EUR for either one of the vegetables. Health damages caused by a STEC incident are assumed to amount to 1000 million EUR. The baseline for calculating the payoffs in table 1 is the STEC incident without any intervention. Against this baseline, a ban of either cucumber or sprouts delivers a positive payoff of = +500 million EUR if the banned vegetable is the source of STEC contamination. If the banned vegetable is not the contaminated one, then the ban only generates market value losses of 500 million EUR. 6 Which of the 5 The simultaneous occurrence of the dangerous strain in cucumbers and sprouts is extremely unlikely due to separate production lines and can thus be ignored. 6 There are de facto more risk management options, namely a ban of both products or of none. Both actions would result in payoffs of 0 EUR relative to the baseline, irrespective of the true state of the world. Once however the regulator has observed evidence that, say, sprouts are more likely to be the cause of the outbreak, banning sprouts strictly dominates banning both as well as banning none product. Vice versa for cucumbers. The actions ban both and ban none can hence be ignored right from the 5

6 two risk management options will be chosen depends on the regulator s attitude towards uncertainty and her state of knowledge regarding the source of the outbreak. Improving this state of knowledge is crucial for making a better decision. In the jargon of risk regulation this is called risk assessment and, in the present case, means devoting resources to sampling cucumbers and sprouts from different regions and testing them for the specific E.coli strain. Here we emphasize two aspects of such risk assessment. One is that information acquired over the course of testing is not just costly, it is also necessarily imprecise: The misleading detection of E.coli on Spanish cucumbers and the resulting misguided ban on cucumber sales are a case in point. The second aspect is that the degree of precision is a choice variable of risk assessment. The risk regulator, rather than being a passive recipient of research results, can decide how many resources to allocate to risk assessment research such as sampling and testing. This decision on risk assessment precision itself requires the regulator to weigh the costs and benefits of improved knowledge. With the payoff table and the choice of precision in mind, we now provide a comparison of the research decision of a risk regulator that follows a standard Cost Benefit Analysis (CBA) based on subjective expected utility (seu) and that of a risk regulator that employs an operational version of the Precautionary Principle (PP). The decisions share common assumptions about reasonable functional specifications on the likelihood of research outcomes and costs of different research precision levels (see appendix A). We start with the CBA risk regulator, who uses seu with a uniform initial prior over the two possible sources of the outbreak. The (noisy) information gained in the risk assessment transforms this initial prior on the source of the outbreak to a posterior view, thus informing the risk management decision. Given the payoffs and the functional specifications, the regulator can determine, through backward induction, the expected benefits and costs of research precision. 7 The results of this exercise are displayed in Table 2. Table 2: Risk assessment precision choice in the STEC example with a CBA mandate. Numbers are in millions of Euros. Precision level Expected benefits Costs Net benefits Low Medium High There are many familiar elements to Table 2: Marginal expected (gross) benefits are decreasing in research precision; marginal costs are non-decreasing. The expected gross benefits of choosing a high precision level approach 500 million EUR. This is intuitive: Perfect information, if attainable, would enable the correct risk management decision to be made without error. Weighing the expected gross benefits and costs in Table start. 7 The formal framework for calculating these numbers is developed in section 3. 6

7 2, the risk regulator equipped with the CBA decision rule chooses a medium precision level. Choosing a higher level would increase gross benefits, but not enough to justify the additional costs. Treating uncertainty in risk regulation on the basis of CBA has given rise to criticism. The argument is that risk regulatory challenges like the STEC outbreak typically go beyond the specific probabilistic requirements needed for a CBA based on seu. Proponents of the Precautionary Principle (PP) argue for acknowledging such deeper form of uncertainty and using robust decision rules ensuring a margin of safety (Sunstein 2005c; Zander 2010; Randall 2011; Wiener et al. 2013). A prominent example for such precautionary decision making is to describe the initial knowledge by a set of probability distributions and to base decision making on the worst probability scenario among them (maxmin expected utility, meu). 8 How does the STEC risk regulation change under this form of the PP? A set of initial priors implies that the ex-post knowledge after observing research results also forms a set of probability distributions (the Bayesian updates of all single priors). 9 The risk management decision, i.e. the decision which vegetable to take off the market, will be based on the worst of these ex-post beliefs. To determine the expected benefits and costs of research precision, the PP risk regulator determines the optimal risk assessment precision also through backward induction, i.e. by combining the value of the anticipated risk management decision with the expected research outcomes (evaluated under meu as well). Table 3: Risk assessment precision choice in the STEC example with a PP mandate. Numbers are in millions of Euros. Expected benefits are calculated based on the worst probability scenario. Precision level Expected benefits Costs Net benefits Low Medium High Table 3, the counterpart of table 2, presents the outcome of balancing research costs and benefits of research under the PP. First note that costs are the same, irrespective of the regulatory mandate. This is intuitive given that the cost side involves no uncertainty. The second commonality is that marginal expected gross benefits are decreasing in the level of precision. Third, expected benefits under the PP also approach the value of perfect information. The difference is in levels: Being calculated based on the worst probability scenario, expected benefits are systematically lower than their CBA counterparts. 8 Thus meu is typically less extreme than focusing on worst possible outcomes. For more discussion about the decision rule see section 3.3 and the concluding discussion in section 7. 9 For this example we consider the symmetric set of prior scenarios M 0 = [0.35, 0.65] around the uniform prior ρ 0 = 1/2. 7

8 As a result of the difference between the expected gross benefits, optimal precision levels under CBA and PP differ in a consequential way: Under PP, the risk regulator pursues a high level of precision in risk assessment; the CBA risk regulator pursues only a medium level of precision. The PP leads to more science and better informed decisions. Relative to the CBA mandate, higher information costs are tolerated in order to decrease the chance of further STEC infections and hence to improve the risk management decision. 2.2 Example 2: The EPA decides on a novel pesticide Like risk regulators elsewhere, the US Environmental Protection Agency (EPA) is charged with deciding on the approval of novel pesticides. In our example, the novel pesticide relies on an entirely new mechanism whose superior efficacy against a variety of pests has already been demonstrated. What has not been researched, however, is whether the new pesticide poses any threat to human health. Whether this is the case or not is critical for the EPA s approval decision, and the risk management task is substantially complicated by the lack of experience with this specific substance. The similarities to the first example are striking. While the authorities in example 1 were facing uncertainty as to whether sprouts or cucumbers were responsible for a serious STEC outbreak, the EPA is uncertain whether a novel pesticide has severe side-effects to human health or not. Intimately linked are two possible regulatory actions: To try and stop the STEC outbreak, the sale of either sprouts or cucumbers can be banned; likewise, the EPA can either approve or not approve the pesticide to prevent harm. Table 4 gives a stylized specification of the societal outcomes that arise under the two possible states of the world (harmfulness) and the two risk management decisions available to the EPA. We assume (uncertain) negative health impacts from a harmful pesticide to be 2000 million USD and (certain) agricultural benefits to be 1000 million USD. Relative to the baseline of non-approval, approving the pesticide gives rise to societal gains of million USD for a harmless and losses of = 1000 million USD for a harmful substance. Table 4: Societal outcomes in millions USD of EPA s risk management decision. Pesticide harmless Pesticide harmful Approval Non-approval 0 0 As in the STEC example, the EPA has an active role in determining the precision of risk assessment. Pesticides are routinely tested in animal organisms such as mice, rabbits or apes to assess their health impacts on humans. These tests involve a tradeoff: The reliability of toxicological tests increases the more similar the model organism 8

9 is to humans. At the same time, greater similarity to humans involves higher costs. This trade-off illustrates one of the ways in which research precision is a choice variable to the EPA. To compare CBA and PP, we again start with the precision choice under a CBA mandate and assume the same informational structure (both in terms of the likelihoods of erroneous research results as well as precision costs) as in the STEC example in 2.1. Expected benefits and costs are displayed in Table 5. A quick comparison with the STEC example (cf. table 2) shows that the numbers in both tables perfectly match. The reason is that on average, due to CBA s uniform prior over both states of the world, the prospect of gaining 500 in either state is equivalent with gaining 1000 in the good and 0 in the bad state of the world. In particular, trading off benefits and costs of research under the CBA mandate leads again to a medium level of risk assessment precision. Table 5: Risk assessment precision choice in the pesticide example with a CBA mandate. Numbers are millions of USD. Precision level Expected benefits Costs Net benefits Low Medium High We now compare this choice with risk regulation under a PP mandate. As in the STEC case, proponents of such a mandate could argue against CBA-based regulation on the grounds that the societal stakes implicit in the pesticide approval decision are high, and that the uncertainty in face of a novel substance is inevitably deep. We analyse the margin of safety regulation based on meu (with the same set of priors as in the STEC example to make results comparable). Adopting the PP in this form alters the risk management decision insofar as a positive approval decision now requires stronger evidence that the pesticide is harmless. It also changes the way the regulator assesses the value of different risk assessment precision levels. Table 6 presents the relevant information trade-off under the PP. As before, expected (gross) benefits are increasing in the level of precision with decreasing marginal benefits. It is also intuitive that the meu s worst probability focus gives rise to consistently lower expected benefits (table 5). Table 6: Risk assessment precision choice in the pesticide example with a PP mandate. Numbers are millions of USD. Expected benefits are calculated based on the worst probability scenario. Precision level Expected benefits Costs Net benefits Low Medium High The surprising observation is the final precision choice under the PP mandate. In 9

10 the STEC example, the consequences of adopting the PP followed the precautionary learning logic: The level of risk assessment precision increased relative to the CBAbased assessment. Here, the PP induces the EPA to reduce her expenditures for risk assessment relative to a CBA mandate. In other words, the EPA accepts a rise in wrong pesticide approval decisions in order to save lab testing costs. This choice under the PP clearly conflicts with the notion of the PP as leading to more science and better informed decisions. The divergent effects we have identified are puzzling given how similar the two examples are. Why would the PP lead to such disparate outcomes in terms of risk assessment precision? The following section develops a simple decision-theoretic framework that embraces both examples and thus opens them to a joint analysis. With this framework at hand it will be possible to verify the findings of the two examples in a general algebraic way and to identify two countervailing forces that together determine the value of information under meu. These forces relative strength under different payoff structures will facilitate a thorough understanding why the PP can give rise to such different and counterintuitive choices of risk assessment precision. 3 The general framework In order to shed light on the interplay of regulatory mandates and information acquisition, this section develops a parsimonious framework combining two building blocks. The first building block is a two states-two actions model with a noisy signal structure whose precision is a choice variable to the decision-maker. It is similar to Moscarini and Smith (2001), but instead of an optimal stopping problem in continuous time we focus on a one-shot signal structure. Around this active learning model, we build maxmin expected utility preferences as the second building block. From a decision-theoretic viewpoint our framework thus analyses active learning under ambiguity aversion and is, to our knowledge, the first model to do so. 10 The framework analyses the interplay of what the economic theory of uncertainty and information distinguishes as informational and terminal actions (Hirshleifer and Riley 1992). While this interplay is of general theoretical interest, we study it in the societal relevant domain of risk regulation where the interplay and appropriate design of risk assessment and risk management is a central matter of discussion (Sunstein 2002; Haimes 2015). 3.1 Timeline Figure 1 presents the time structure of the model. First, nature chooses a payoff relevant parameter θ from two possible values, θ and θ +. parameter is the true θ. The regulator is uncertain which In the STEC example in section 2.1, the risk regulator is 10 Related papers dealing with the value of information udner ambiguity aversion, but without the precision as a choice variable, are Snow (2010) and Hoy et al. (2014). 10

11 uncertain whether sprouts (θ = θ + ) or cucumbers (θ = θ ) is the origin of the STEC outbreak; in the second example in section 2.2, the EPA is uncertain whether a new pesticide is harmless and beneficial (θ = θ + ) or has severe side-effects to human health and the environment (θ = θ ). In the second stage, the regulator decides on the precision of risk assessment. The risk assessment outcome, modelled as a one-shot noisy signal, becomes available in stage three. These research results can be either inconclusive or they might provide evidence for one of the two parameter values being the true θ. As usual, however, there is some likelihood that even conclusive research results are wrong. The more resources the regulator invests into research precision (stage 2), the less likely get those erroneous findings. This option to influence the precision of the risk assessment process corresponds with the decision-theoretic concept of active learning, also known as active information acquisition. After research results from stage 3 have been observed and processed to a better, yet incomplete understanding of the decision problem, the regulator has to make the final risk management decision a (stage 4). The regulator chooses among (randomizations over) two actions, a and a +, where the first is optimal if θ = θ and the second if θ = θ +. In the examples, the actions available to the risk regulator are the ban of sprouts (a = a + ) or the ban of cucumbers (a = a ); the EPA can either approve the new pesticide (a = a + ) or not approve it (a = a ). In both examples, no action dominates the other: The optimal decision depends on the true state of the world θ. 1. Nature 2. RA precision 3. RA results 4. RM decision Figure 1: Time structure of the model. The regulator chooses how much to invest into the precision of risk assessment (stage 2) and makes the final risk management decision (stage 4). Both decisions have to be taken under incomplete (and typically ambiguous) knowledge because the regulator is uncertain which payoff relevant parameter θ was chosen by nature in stage 1. Risk assessment results in stage 3 however provide noisy information about θ. The precision of this information depends on the regulator s investment decision in stage 2. In the following subsections we will explain the missing components of the framework, 11

12 namely the payoff structure (3.2), the decision rules and updating procedures (3.3) and a specific signal structure that will later allow us to get sharp analytical results (3.4). 3.2 Payoff structure A risk regulatory setting is characterised by the societal outcomes under different combinations of state of the world θ and regulatory action a. Let π + denote the payoff if the true state is θ and the regulator chooses action a + ; all other notations accordingly. The assumptions π > π+ and π+ + > π + then reflect that there is no dominant action. We also assume that the difference in payoffs under correct and incorrect decision be independent of the states, π + + π + = π π+ =: a, and call a the error cost. It measures how large the cost of erroneous decision-making by the regulator is. The second parameter we define is the payoff asymmetry parameter π = π + + π = π + π+. This parameter captures, orthogonal to the interpretation of a, how asymmetric the regulatory setting is in terms of the unknown parameter θ. Parameter π is non-negative if we assume, without loss of generality, that π + + π. As a further simplification, we normalise payoffs such that π + π + = π+ + + π+ = 0. We will see in section 4 that this is equivalent with normalising the CBA regulator s value of the final risk management decision in the absence of information to 0. The initially four dimensional payoff space is now fully described by the error cost a and the payoff asymmetry π. Figure 2 gives a graphical illustration. Figure 2: The parameter a and π fully describe the payoff structure. The error cost a captures how wide the span between correct and wrong decision is. The payoff asymmetry π measures the extent to which payoffs differ across the two realizations of the unknown parameter θ. Table 7 summarizes how the two examples fit into the general framework. payoff asymmetry parameter π is zero in the STEC example as the final payoffs are symmetric over both parameter values θ and θ + ; the payoff only depends on whether the risk regulator is able to identify the contaminated vegetable. In contrast to that, π is positive in the pesticide example because the payoffs associated with a harmless pesticide, θ = θ +, are consistently higher than those associated with a harmful pesticide, The 12

13 θ = θ. We will see in section 6 that π is a central driver of the value of risk assessment precision under the PP. Table 7: The two examples from section 2 in the language of the general framework. Interpretation and Numbers Parameter STEC example (Sec. 2.1) Pesticide example (Sec. 2.2) θ + Sprouts contaminated Pesticide harmless θ Cucumbers contaminated Pesticide harmful a + Ban sprouts Approve pesticide a Ban cucumbers Deny pesticide approval π π π π a π Prior knowledge, Information, and Decision Rules This section describes how a regulator s prior knowledge is transformed to posterior knowledge upon observing the noisy, yet informative test results ( signal ) gained in the risk assessment. Any signal structure is characterised by a signal space S of possible test results and a likelihood function l : Θ (S) that describes how likely different test results s S are if θ is the true state of the world. For better readability we write l + (s) for l(θ + )(s). Because Θ has only two elements, every belief prior and posterior can be expressed as a single number in the unit interval. CBA regulation. How does the CBA regulator come to decisions on risk management and risk assessment? The basis for the standard CBA mandate is the maximisation of subjective expected utility (seu, Savage 1954). Integral part of this decision rule is the formation of a single belief ρ over Θ that best reflects the decision-maker s knowledge. Based on this belief, the optimal action in a set of alternatives A is found by maximising subjective expected utility, max E ρu(θ, a), (1) a A where u is a utility function. For simplicity we assume risk neutral preferences throughout. Depending on the context u either represents the outcomes under different risk management decisions or expected net benefits of different levels of risk assessment precision. Let the regulator initially hold ρ 0 [0, 1], her ex-ante belief that θ = θ +. If she 13

14 observes the signal s S, Bayesian updating transforms the prior to the posterior ρ 1 (s, ρ 0 ) = ρ 0 l + (s) ρ 0 l + (s) + (1 ρ 0 )l (s). (2) When the regulator initially has no information about the true θ Θ, she would usually follow the Principle of Insufficient Reason (see for instance Gilboa 2009) and hold the uniform prior ρ 0 = 1/2. With that, formula (2) simplifies further, ρ 1 (s, 1/2) = l + (s)/(l + (s) + l (s)). PP regulation. In light of the high stakes involved, the PP regulator may want to both acknowledge the poor level of knowledge typically available and make cautious decisions. Schmeidler 1989), Both aspects are captured in maxmin expected utility (meu, Gilboa and max min E µu(θ, a). (3) a A µ M The PP regulator bases her decision on a set of beliefs M and the optimal action is one that maximises utility based on the worst belief, where what is the worst belief may be action dependent. It is important to note that this decision-rule is less extreme than it may appear. The set M is not exogenously given but part of the preference structure (Gilboa and Schmeidler 1989; Etner et al. 2012). It hence combines scientific knowledge and the regulator s attitude towards uncertainty. In fact, in our stylized regulatory model neither regulator has any scientific knowledge in the first place. Recalling that we can express any belief about Θ on the unit interval, an accurate scientific description of the state of knowledge is hence the full set [0, 1]. Possible ways of dealing with this (lack of) scientific knowledge range from selecting the best single belief (the CBA approach) to an extreme form of precaution that uses (3) with the full set, i.e. preparing against the worst possible outcome. In analogy to the Principle of Insufficient Reason we assume that the initial prior set M 0 is symmetric around the uniform distribution, M 0 = [1/2 δ, 1/2 + δ] with 0 δ 1/2. We call δ the regulator s level of precaution, where the CBA regulator is the corner case δ = 0. We will see shortly that the extreme case δ = 1/2 is ruled out (see 3.4) so that we consider in this paper a moderate form of the PP. How to extend the static preferences in (3) to dynamic settings like ours? Epstein and Schneider (2007) provide an application-friendly framework of dynamic meu preferences that respects backward induction (recursive preferences) and beliefs are updated priorby-prior with the Bayesian updating formula (2), M 1 (s) = {ρ = ρ 1 (s, ρ 0 ) ρ 0 M 0 }. (4) 14

15 In other words, we assume full Bayesian updating of multiple priors. 11 This updating process has very intuitive features. For instance, a non-informative signal s with l (s) = l + (s) results in no learning, M 1 (s) = M 0. A maximal informative signal structure, on the other extreme, transforms M 1 into a singleton reflecting subjective certainty; for instance, observing a signal s that can only be observed in case of θ = θ +, l (s) = 0, gives rise to ex-post certainty that θ + must be the payoff relevant parameter, M 1 (s) = {1}. 3.4 The simple signal space with three elements In terms of the signal structure, a natural choice is to consider real-valued, normally distributed signals where l and l + have the same variance but different means. With decreasing variance (i.e. increasing precision, the reciprocal of the variance) the signal gets more informative as it becomes easier to infer whether the true parameter is θ or θ +. The main drawback of this approach, however, is its lack of analytical tractability, especially in the meu case. We thus design a discrete signal space that is rich enough to hold the non-trivial impact of meu on the value of information and at the same time allows closed form solutions. 12 The final justification for choosing this signal structure will be given in appendix B where we demonstrate numerically that the paper s main findings do not depend on the signal structure. Consider the signal space with three elements, S = {s, s?, s + }. Their interpretation is straightforward: The signals s and s + represent research results that suggest θ = θ and θ = θ + respectively, while s? models inconclusive research outcomes. A signal structure requires a specification of the likelihood of all signals under all parameters θ Θ. Assuming symmetry over θ and θ +, the signal structure is fully determined by specifying the likelihood of correct, erroneous and inconclusive research results P corr = l (s ) = l + (s + ), P mist = l (s + ) = l + (s ), P inconcl = l (s? ) = l + (s? ). (5) The likelihoods P corr, P mist and P inconcl are functions of the signal structure s precision τ R 0 and sum to one for all τ. We further assume that P corr, P mist and P inconcl are 11 For simplicity and to ensure dynamic stable preferences (Heyen 2014), we abstain from the ex-post rejection of beliefs that is allowed for in Epstein and Schneider (2007). 12 The simplest binary signal structure does not pass this test as the difference between precision choices under CBA and PP collapse. 15

16 twice continuously differentiable in τ and that for all precision level τ lim P corr(τ) = 1, P τ corr(τ) > 0 lim P mist(τ) = 0, P τ mist(τ) < 0 lim P inconcl(τ) = 0, P τ inconcl (τ) < 0 Assumptions (6a) to (6c) are straightforward: (6a) (6b) (6c) (1 2δ)P corr (τ) > (1 + 2δ)P mist (τ) (6d) (P corr P mist ) (τ) < 0. (6e) The noisiness of the signal structure decreases in the precision of the signal structure and vanishes in the limit of perfect information. Equivalent with (6c) is P corr + P mist > 0. Assumption (6d) is equivalent with ρ 1 (s + ) > 1/2 and ρ 1 (s ) < 1/2 and thus ensures that PP is moderate in the sense that conclusive research results result in clear beliefs and hence clear actions; otherwise information trivially has no value. In particular, assumption (6d) precludes the extreme regulator preferences δ = 1/2 right from the start. Assumption (6d) rules out, for given 0 δ < 1/2, small precision levels. A restriction on small precision levels also comes from the technical assumption (6e) which unambiguously holds in the limit because P corr P mist is non-negative and P corr P mist 0 for τ. Thus, assumptions (6d) and (6e) basically translate into the existence of a smallest feasible precision level. For the discrete signal structure just defined we can give simple expressions for the updating formula (2). Let ρ 0 be an initial prior. Then ρ 1 (s, ρ 0 ) = ρ 0 P mist ρ 0 P mist + (1 ρ 0 )P corr, ρ 1 (s?, ρ 0 ) = ρ 0, ρ 1 (s +, ρ 0 ) = ρ 0 P corr ρ 0 P corr + (1 ρ 0 )P mist. (7) From the inconclusive research result s? no information can be gained. Observation of s transforms the initial prior towards ρ 1 = 0, and this movement is stronger the higher the research precision τ. Observing s + induces a similar push to the right towards ρ 1 = 1. The formulas in (7) are of use both in the CBA and the PP analysis. For CBA, ρ 0 = 1/2. For the PP, every ρ 0 M 0 is updated in that way, together forming the set of posteriors M 1. This completes the exposition of the framework components. We move on to solving the model. In line with backward induction we begin with the risk management decision in section 4 and then analyse the risk assessment precision choice in Section 5. 16

17 4 Risk management decision The terminal risk management decision depends on the payoff structure, the confidence gained in the risk assessment process, and the regulatory mandate. This section will explain which risk management decisions are made under different mandates and, crucial for the subsequent analysis, how the value of these optimal decision depends on the risk regulator s ex-post confidence, where the level of ex-post confidence is directly linked to the precision of the just obtained risk assessment outcome. We begin with the general case of arbitrary signal and payoff structures (section 4.1) and then specialize to the two examples (section 4.2). 4.1 Risk management: The general case We analyse the risk management decision as a function of the risk regulator s then up-todate state of knowledge, i.e. the posterior ρ 1 (CBA regulator) and the set of posteriors M 1 (PP regulator). In focusing on the ex-post knowledge we do not have to make any assumptions on the signal structure that gave rise to this posterior knowledge. CBA regulation. The optimal risk management decision maximises expected terminal payoffs (cf. section 3.3), max a [0,1] ρ 1 ( aπ (1 a)π +) + (1 ρ1 ) ( aπ + + (1 a)π ). (8) Here, a is a randomization over the two actions a and a +, with a = 1 (a = 0) corresponding to the pure action a + (a ). Because the error cost a = π + + π + = π π+ does not depend on θ (cf. section 3.2), it is easy to show that the regulator strictly prefers the pure action a + (a ) if and only if ρ 1 > 1/2 (ρ 1 < 1/2); under inconclusive knowledge ρ 1 = 1/2 the regulator regards all randomizations a [0, 1] as equally reasonable. In line with the standard backward approach for recursively defined preferences we introduce the value of the risk management decision as a function of the posterior knowledge, ρ 1 V (ρ 1 ). 13 We express all payoffs in terms of error cost a and payoff asymmetry π and get for the CBA value of the risk management decision V (ρ 1 ) = ( 1 2 ρ 1)(a π ) ρ 1 < 1/2 0 ρ 1 = 1/2 (ρ )(a + π ) ρ 1 > 1/2. As a result of the normalisation π + π + = π+ + + π+ = 0 (see section 3.2) the value of inconclusive knowledge V (1/2) is zero. When the regulator s posterior knowledge 13 This has been used for the analysis of information acquisition problems before, see for instance Mirman et al. (1993); Grossman et al. (1977). (9) 17

18 ρ 1 approaches subjective certainty, ρ 1 = 0 and ρ 1 = 1, the value V (ρ 1 ) converges to π = (a π )/2 and π + + = (a + π )/2, respectively. 14 PP regulation. The test results have transformed the PP regulator s set of prior beliefs M 0 = [1/2 δ, 1/2 + δ] to the set of posteriors M 1 = [ρ 1, ρ 1 ]. The optimal risk management decison is (see 3.3) max a [0,1] ( min ρ 1 aπ (1 a)π ) + + (1 ρ1 ) ( aπ + + (1 ) ρ 1 M a)π. (10) 1 The optimal action a (M 1 ), again a randomization over actions a and a +, depends on the set of posteriors M 1. PP regulator chooses the pure action a + (a ). Only if all posteriors are higher (lower) than 1/2, the If however 1/2 M 1, the regulator randomizes over actions in a non-trivial way that depends on the payoff structure (see appendix C): The probability of choosing a + is 1/2 for symmetric settings π = 0 and decreases linearly as π increases; it is zero for π a. In particular, regulatory settings of high payoff asymmetry π a are characterised by the PP risk manager choosing a as long as the posterior knowledge remains ambiguous, 1/2 M 1. This is a direct consequence of the maxmin rule and an expression of precautionary decisionmaking. We will come back to this in the pesticide example below. What is the value of the risk management decision under the PP regulation? Based on the profile of optimal decisions we get a π ( 1 2 ρ 1)(a π ) ρ 1 < 1/2 V (M 1 ) = 0 1/2 M 1 (ρ )(a + π ) ρ 1 > 1/2. (11a) a < π ( 1 2 ρ 1 )(a π ) ρ 1 < 1/2 V (M 1 ) = ( 1 2 ρ 1 )(a π ) 1/2 M 1 (ρ )(a + π ) ρ 1 > 1/2. (11b) A first observation is that, due to the maxmin rule, the value of the risk management under PP (11) is never larger than the CBA counterpart (9). One difference to (9) is that the PP value depends on the shape of the payoff structure. While the value of ambiguous knowledge, 1/2 M 1, is zero (as in the CBA case) for modest payoff asymmetry, π a, it is a function of the lowest posterior ρ 1 and turns negative for high payoff asymmetry π > a (The pesticide example is the corner case π = a ). Also, what the worst posterior is, ρ 1 or ρ 1, depends on the context. It is noteworthy that the value in the high payoff asymmetry case is characterised by the single posterior ρ 1, while the value of the risk management decision in the low payoff asymmetry case cannot be expressed with a single posterior. 14 For π > a we have V (0) < V (1/2) so that the movement towards subjective certainty ρ 1 = 0 decreases value; this reduction however is over-compensated by the simultaneous value increase of those signal realizations that suggest θ +. 18

19 4.2 Risk management in the STEC and pesticide example We now apply the general results to the two examples and the simple signal structure with three possible test results. We begin with the optimal risk management decision and then discuss the associated value. Thanks to assumption (6d), conclusive risk assessment outcomes s + (s ) result in (non-randomized) risk management decision a + (a ), irrespective of the setting and mandate. This is different for inconclusive test results s?. Recall that, after obtaining s?, the CBA regulator is indifferent between all randomizations irrespective of the regulatory setting. The PP regulator, in contrast, finds a specific randomization optimal and this optimal decision depends on the setting. In the symmetric STEC example the PP regulator randomizes over banning cucumbers and sprouts with equal probabilities. For the asymmetric pesticide example, however, the PP regulator takes inconclusive test results s? as the sufficient reason for non-approval of the pesticide. This is expression of the PP s margin of safety notion. (a) STEC example (b) Pesticide example Figure 3: The value of the risk management decision of the two examples as a function of the posterior knowledge ρ 1 and evaluated for the three risk assessment outcomes s, s? and s +. The dark and light dots represent the CBA regulator (δ = 0) and PP regulator (δ > 0), respectively. The value of the risk management decision is depicted in Figure 3. The solid red lines show the value of the risk management decision as a function of the ex-post knowledge ρ 1 and is hence a graphical representation of (9). The dark dots accordingly represent the CBA regulator s value under the test results s, s? and s + that transform the initial knowledge ρ 0 = 1/2 to test result specific posteriors ρ 1 (s). Similarly, the light points represent, based on (11), the PP regulator s value. The set of priors M 0 = [1/2 δ, 1/2 + δ] is updated to test specific sets of posteriors M 1 (s, δ) and the worst of these posteriors determines the value V (s, δ). Due to the maxmin rule the PP value never exceeds the CBA value. The figure is drawn for a certain level of precision τ. Higher levels of precision would push the posteriors under s to the left and those under s + to the right, overall increasing the value associated with the risk management decision. 19