8.2.1 Safety Investment Option 1

The first option involves installing a computerized machine to automate the log cutting at high speed and to ensure enhanced levels of safety for operators working in the production area. With the new machine, the level of dust in the air, as well as the risk of repetitive strain injury, cuts, crushing injuries of fingers or toes, excessive noise, excessive mental problems, and lower back pain would be significantly reduced. The duration of the investment is assumed to be 10 years.

Using the new machine, the cutting process, usually performed by a team of three carpenters and a supervisor, would be performed by three operators: one technical operator would be responsible for the cutting machine, another would be responsible for the maintenance activities, and the third would be in charge of the final product design. With this new working configuration, the same workload would be performed in 60% less time. As a result of the cutting optimization, the number of defects would be reduced and the quality of the final products would also be significantly enhanced. The beneficial effect of the improved quality of the product has also been quantified in terms of additional profits of €25 000 per year. Moreover, a team responsible for the risk assessment has estimated a yearly saving in the production operations of €135 000 due to waste reduction and a reduction in raw materials. On the investment side, the initial purchase price of the machine is €280 000. Additional costs due to training, redesign of the layout in the production area, feasibility studies are to be initially sustained. The company has conducted an economic analysis associated with the investment, and quantified the main benefits and costs, as shown in Tables 8.1 and 8.2. The initial investment and yearly benefits and costs associated with safety investment option 1 are shown in Table 8.3.

The costs and the hypothetical benefits summarized in Tables 8.1 and 8.2 can be further distinguished in terms of initial costs/benefits and yearly recurring costs/benefits. Initial costs/benefits are supposed to be sustained in year 0 (in which the evaluation is made). Yearly recurring costs/benefits are sustained during the time horizon in which the investment is evaluated. Figure 8.2 shows a graphical representation of the evolution of the net yearly cash flow. Yearly cash flows are given by the difference between costs (negative cash flows) and benefits (positive cash flows) from year 0 to year 10. Analyzing the nature of costs and benefits it can be seen that the total initial investment required by the company is €0.477 million.

The well-known formula in Eq. (8.1) (see also Chapter 5, Eq. (5.1)) has been used to compute the net present value (NPV) associated with the investment, assuming for each year t the previously mentioned negative cash flows (C_t) and positive cash flows (B_t) and the discount factor :

8.1

The total NPV associated with the investment in this case is equal to €702 501. As this value is greater than zero, the investment is profitable for the company. The PBP is equal to 3.33. This means that after 3 years and 4 months the investment will cover the costs and it will start producing profits for the firm as shown in Figure 8.3.

Another indication that is often used by decision-makers to assess the profitability of an investment is the internal rate of return (IRR), as also mentioned in Chapter 5. This measure represents the discount factor that makes the NPV equal to zero, or, in other words, the value of the interest rate at which an investment reaches a breaks-even point. In this case, the IRR is equal to 24.93% as shown in Figure 8.4.

8.2.2 Safety Investment Option 2

The second investment option is to drastically improve the ventilation system and thus enhance the working conditions in the production area. Due to the cutting operation, the level of dust in the production area is high. The exposure of the personnel to dust is very close to the maximum value allowed by law. In addition, dust and wood particles also reduce the visibility conditions during the cutting operations, exposing the personnel to a risk of significant work-related injury. Safety conditions are legally respected even if the wellness of the personnel is currently not being fully addressed, with high levels of illness due to employees suffering the effects of elevated levels of wood dust. In the past, to improve the quality of the air, the company introduced longer breaks to allow personnel to rest in areas with a lower amount of dust.

The company is currently considering buying a dedicated aspiration and ventilation system, which uses cyclotron technology; it will significantly reduce the level of dust and thereby improve the working conditions in the production area. As a side-effect, the company's productivity could be increased by 25% and the quality of the final products slightly improved. The health of the personnel would be significantly improved and the turnover reduced. Moreover, as the dust and the wood particles are automatically expelled by the new system and compacted into special bags, the waste could be sold to a specialized company for recycling. The equipment inside the production area, due to a lower level of dust, would require less maintenance and cleaning activities. Moreover, the downtime would be reduced by 30%, with a significant impact on the production costs. After a thorough assessment, a team of experts has estimated the benefits and costs associated with the investment; these are summarized in Tables 8.4 and 8.5, respectively.

The costs and benefits can be further analyzed to distinguish between the initial investments and the yearly recurring costs/benefits; these are shown in Table 8.6.

As done for safety investment option 1, the NPV is computed considering a time horizon of 10 years. As the time horizon is the same as the one used to evaluate the first investment option, the comparison between the alternative investments will be simplified, as it will use the same criteria. In addition, the amount of money required for the initial investment (€0.479 million) is comparable to that in the previous investment option.

In this case, the NPV is equal to €490 884, which is lower than the €702 501 associated with the previous investment option. In addition, the PBP shown in Figure 8.5 is slightly longer than 3.3 years. Therefore, this investment option seems to be inferior to investment option 1. Given the company's limited safety budget, safety investment option 1 is to be preferred.

If additional resources were found, investment option 2 should also be considered and implemented as its profitability in the medium to long term is guaranteed by a positive NPV and a relatively limited PBP.

The IRR associated with the second safety investment option is computed as shown in Figure 8.6. The IRR in this case is equal to 18.89%. Therefore, in this case too, investment option number 2 is clearly worse than the new cutting machine.

8.3.1 Scenario Thinking Approach

One way to develop a decision analysis tree is by using scenario thinking. For example, possible scenarios are identified using specific risk assessment techniques, such as event tree analysis. Using the event tree analysis approach (for more information, see, e.g., Greenberg & Cramer [2], the Centre for Chemical Process Safety [3] and Meyer & Reniers [4]), a graph such as that in Figure 8.7 is obtained. The illustrative example of a decision analysis tree for a runaway reaction event is given. All costs and probabilities are assumed to be yearly.

Based on the event tree of Figure 8.7, an analysis of safety investment decisions can be made. The costs of the safety measures can be collected and displayed in the figure (e.g., in the case of this illustrative runaway reaction event, a total prevention cost of €250 000 is obtained), as well as the expected total costs of the event. The expected total costs (which can be seen as the expected hypothetical benefits) can then be compared with the prevention costs. In this illustrative example, the expected total costs are equal to , and should be compared with €250 000 when formulating a decision recommendation. In this way, a decision can be made regarding this safety investment portfolio composed of three barriers (i.e., the cooling system, the operator manual intervention, and the automatic inhibition intervention).

8.3.2 Cost Variable Approach

It is possible to draft a tree where the cost of each decision is presented as a variable named “cost” (see also Muennig [5]). The cost is a running total of the costs of each event in a given event pathway. At each box in the tree, the running total is increased by the cost assigned to the box. Figure 8.8 presents an illustrative example.

8.7.1 Constructing a Bayesian Network

A Bayesian network (BN) is a network composed of nodes and arcs, where the nodes represent variables and the arcs represent causal or influential relationships between the variables. Hence, a BN can be viewed as an approach to obtain an overview of relationships between causes and effects of a system. In addition, each node/variable has an associated so-called conditional probability table (CPT). Thus, BNs can be defined as directed acyclic graphs, in which the nodes represent variables, arcs signify direct causal relationships between the linked nodes, and the so-called CPTs assigned to the nodes specify how strongly the linked nodes influence each other. The key feature of BNs is that they enable us to model and reason about uncertainty.

The nodes with no arcs directed into them are called “root nodes,” and they are characterized by marginal (or unconditional) prior probabilities. All other nodes are intermediate nodes and each one is assigned a CPT. Among intermediate nodes, the nodes having arcs directed into them are called “child nodes” and the nodes having arcs directed away from them are called “parent nodes.” Each child has an associated CPT, given all combinations of the states of its parent nodes. Nodes with no children are called “leaf nodes.” There are two possible approaches to constructing a BN. The first approach develops a network topological structure to quantitatively determine the correlation and causal relationships between the nodes. The second approach builds a CPT to quantitatively determine the conditional probability of every node in the network.

The probabilities that have to be assigned to the different states of the variables can be determined in different ways. One way is to use historic data and statistical information (e.g., observed frequencies) to calculate the probabilities. Another way, especially when there is no or insufficient statistical information, is to use expert opinion. Hence, both probabilities based on objective data and subjective probabilities can be used in BNs.

Bayesian networks are based on Bayes' theorem and Bayes' theory to update initial beliefs or prior probabilities of events using data observed from the event studied. Bayes' theorem can be expressed using the well-known Bayes' formula:

In this equation, P(A) is the prior probability of an event, P(B|A) is the likelihood function of the event, P(B) is the probability of information/data observed (commonly called evidence), and P(A|B) is the posterior probability of the event.

Hence, a prior probability of an event is the starting point of knowledge, but the goal is to find out what the posterior probability of this event is, given the evidence “B.” In the case where A is a variable with n states, according to the total probability formula, the following equation holds:

Hence, Bayes' formula transforms into:

For multivariable, multinode BNs, considering the independence and separation theorems of BNs, the joint probability P(X₁, X₂, …, X_n) can be expressed as the product of edge probabilities of each node:

where X_i is the i-th node of the BN, and parent(X_i) is the corresponding i-th parent node.

As Fenton and Neil [14] explain, BNs offer several important benefits as compared with other available techniques. In BNs, causal factors are explicitly modeled. In contrast, in regression models, historical data alone are used to produce equations relating dependent and independent variables. No expert judgment is used when insufficient information is available, and no causal explaining is carried out. Similarly, regression models cannot accommodate the impact of future changes. In short, classical statistics (e.g., regression models) are often good for describing the past, but poor for predicting the future. A BN will update the probability distributions for every unknown variable whenever an observation or piece of evidence is entered into any node. Such a technique of revised probability distributions for the cause nodes as well as the effect nodes is not possible in any other approach. Moreover, predictions are made with incomplete data. If no observation is entered then the model simply assumes the prior distribution. Another advantage is that all types of evidence can be used: objective data as well as subjective beliefs.

This range of benefits, together with the explicit quantification of uncertainty and ability to communicate arguments easily and effectively, makes BNs a powerful solution for all types of risk assessment, as well as assessments involving safety investment decisions.

8.7.2 Modeling a Bayesian Network to Analyze Safety Investment Decisions

As shown in the event tree of Figure 8.11, an initiating event (IE) could result in four consequences, c1, c2, c3 and c4, based on the function/failure (presence/absence) of two safety barriers, SB1 and SB2. The safety barriers SB1 and SB2 have a cost of €1000 and €4000, respectively. Also, the monetary values of losses, given c1, c2, c3, and c4, are equal to €1000, €10 000, €100 000, and €1 000 000, respectively. The occurrence probability of IE is equal to 0.01 while the failure probabilities of SB1 and SB2 have been estimated as 0.2 and 0.1, respectively. The decision-maker wants to decide: “Given a specific amount of budget and an acceptable (tolerable) level of risk, what would be the best (optimal) allocation of safety barriers? SB1, SB2, both, or none?” The technique of BNs can be employed to answer this question.

The event tree shown in Figure 8.11 can be mapped into a BN, as shown in Figure 8.12. The CPT of the node “consequence” is given in Table 8.10.

IE, initiating event; SB, safety barrier.

Figure 8.13 shows an extended version of the BN in Figure 8.12 in order to account for the effect of safety barriers on both the required budget (node “budget”) and the amount of risk. The CPTs for nodes “SB1,” “SB2,” and “budget” are presented in Tables 8.11–8.13, respectively.

SB, safety barrier.

SB, safety barrier.

SB, safety barrier.

It should be noted that in Table 8.11, if SB1 is installed, it can work or fail (the second column in Table 8.11). However, if SB1 is not installed, the effect on the consequences would be the same as that of the failure of SB1. In other words, it has been assumed that the absence and failure of SB1 would result in the same consequences (the third column in Table 8.11); the same rationale has been used to assign the conditional probabilities for SB2 in Table 8.12.

Figure 8.13 shows the same BN as in Figure 8.12 in which the node “budget” has been given the value of to €5000, implying that both SB1 and SB2 can be afforded. As a result, the cost (required budget) would be €5000, while the value of the risk (according to the node “risk” in Figure 8.14) would be:

Similarly, the values of cost and risk can be calculated for different combinations of safety barriers, as shown in Table 8.14.

SB, safety barrier.

Given the constraints regarding the available budget (to purchase safety barriers) and the level of acceptable risk, multi-criteria decision-making can be employed to find the optimal allocation of safety barriers.

The BN of Figure 8.12 can alternatively be extended as shown in Figure 8.15 to incorporate the uncertainty in the available budget on the selection of safety barriers. Figure 8.15 shows a case in which the probability distribution of budget is P(€5000, €4000, €1000) = (0.3, 0.5, 0.2). Also, it has been assumed that in case of a budget deficit, SB1 is given priority over SB2. The CPTs of SB1 and SB2 are given in Tables 8.15 and 8.16, respectively.

SB, safety barrier.

SB, safety barrier.

According to Figure 8.15, the expected value of the budget can be calculated as . Moreover, the estimated value of the risk (with respect to the node “risk” in Figure 8.14) would be equal to . Following the same procedure, the values of risk for other distributions of the budget can be calculated. Similar to the previous approach, a multi-criteria decision-making technique can be used to determine the most cost-effective safety barrier to use.

8.11.1 An Introduction to Game Theory

People make decisions all the time. In an organizational setting, these decisions typically are interdependent with decisions made by others. Hence, there is a kind of multi-personal interaction. A central feature of multi-personal interaction is the potential for the presence of strategic interdependence, meaning that a person's well-being (e.g., measured as utility or profit, or as non-accident) depends not only on their own actions but also on the actions of other individuals involved into the situation with strategic interdependence. The actions that are best for the person may depend on actions that other individuals have already taken, or expected to be taken at the same time, and even on the future actions other individuals may take, or decide not to take, as a result of the current actions.

The mathematical tool that is used for analyzing interactions with strategic interdependence is non-cooperative game theory. The term “game” highlights the theory's central feature: the persons (also called “players,” “decision-makers,” or “agents,” as persons can also represent entire organizations or nations, for example) under study are concerned with strategy and winning (in the general sense of expected utility maximization). The agent will have some control over the situation, as their choice of strategy will influence it in some way. However, the outcome of the game is not determined by their choice alone, but also depends on the choices of all other players. This is where conflict and cooperation come into the play. It should be emphasized that the term “non-cooperative game theory” does not mean that non-cooperative theory is unable to explain cooperation within groups of individuals. Rather, it focuses on how cooperation may emerge from rational objectives of the players, given that rationality is a common knowledge, and in the absence of any possibility of making a binding agreement.

It is useful to notice that in the decision-making theory and, consequently, in game theory, individual rationality should not be associated with dispassionate reasoning. It is assumed that while making an optimal decision, an individual will be able to observe the courses of action that are available and determine the outcomes he or she can receive as a result of the interaction with others. An individual should be able to rank those outcomes in terms of preferences, so that the system of preference is internally consistent and can form a basis for choice. Thus a rational player is defined as one who has consistent preferences concerning outcomes and will attempt to achieve preferred outcomes. Rational play will involve complicated individual decisions about how to choose a strategy that will produce a favorable outcome, knowing that other players are trying to choose strategies that will also produce an outcome favorable to them. It will also involve social decisions about how and with whom a player should try to cooperate.

Game theory is a logical analysis of situations of conflict and cooperation. More specifically, a game is defined to be any situation in which:

• There are at least two players. A player may be an individual, but may also be a more general entity, such as a company, an institution, a country, and so on.
• Each player has a number of possible strategies, courses of action that the player may choose to follow.
• The strategies chosen by each player determine the outcome of the game.
• Associated with each possible outcome of the game is a collection of numerical payoffs, one to each player. These payoffs present the value of the outcome to the different players.

Game theory can generally be divided into non-cooperative game theory and cooperative game theory. Furthermore, a distinction can be made between “static games” and “sequential move games.” A static game is one in which a single decision is made by each player, and each player has no knowledge of the decision made by the other players before making their own decision. Sometimes such games are referred to as “simultaneous decision games,” because any actual order in which the decisions are taken is irrelevant. A sequential game is a game where one player chooses their action before the others choose theirs. Importantly, the later players must have some information of the first player's choice, otherwise the difference in time would have no strategic effect. Another distinction that can be made between the game-theoretic models concerns the information available to the players: games of perfect information should be solved differently from games of imperfect information. In many economically important situations, the game may begin with some player disposing of private information about a relevant issue with respect to their decision-making. These are called “games of incomplete information” or “Bayesian games.” Incomplete information should not to be confused with imperfect information in which players do not perfectly observe the actions of other players. Although any given player does not know the private information of an opponent, he or she will have some beliefs about what the opponent knows, and the assumption that these beliefs are common knowledge can be made. Traditional applications of game theory attempt to find equilibria in these games. In an equilibrium, each player of the game has adopted a strategy that they are unlikely to change. Many equilibrium concepts have been developed (in particular, the Nash equilibrium) in an attempt to capture this idea. These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide.

It should be noted that this methodology is not without criticism, and debates continue concerning the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally. For instance, game theory should not be considered as a theory that will prescribe the best course of actions in any situation of conflict and cooperation. First of all, the real-world situations are performed in a quite simplified form of a game. In real life it may be hard to say who the players are, to delineate all conceivable strategies and to specify all possible outcomes, and it is not easy to assign payoffs. What is typically done is to develop a simple model which incorporates some important features of the real situation. Thus building such a model and its analysis may give insights into the original situation. The second obstacle is that game theory deals with play that is rational. Each player logically analyses the best way to achieve their goals, given that the other players are logically analyzing the best way to achieve their own goals. In this way, rational play assumes rational opponents. It means that players are able to tell which outcomes are preferable to others and that they are able to align them in some kind of preference relationship order. However, experiments show that in some cases rationality may fail. Game theory results should thus always be interpreted with caution. For example, game theory does not give a unique prescription for games with more than two players. What game theory offers is a variety of interesting examples, analysis, suggestions, and partial prescriptions for certain situations. Practitioners using game theory should be aware of this.

In summary, a game is a situation where, for two or more individuals, their choice of action or behavior has an impact on others. The outcome of the game is expressed in terms of the strategy combinations that are most likely to achieve the players' goals given the information available to them. Games are often characterized by the way or order in which players move. Games in which players move at the same time or in which the players' moves are hidden from others until the end of the game, are called “simultaneous-move” games or “static” games. Games in which moves are made in some kind of predetermined order are referred to as “sequential move” games or “dynamic” games.

Furthermore, a distinction is made between zero-sum games and non-zero-sum games. In the former case, the payoffs for each outcome add to zero, and the interests of the players are strictly opposed. In the latter case, there is no strict conflict and the payoffs do not add up to zero.

8.11.2 The Prisoner's Dilemma Game

One classic game theory example, called Prisoner's Dilemma, is particularly interesting. The Prisoner's Dilemma game is a two-person non-cooperative non-zero-sum game. The Prisoner's Dilemma, as applied to this book, i.e., economics in combination with operational safety in an organization, can be understood as follows. Assume two workers in an organization, having to decide whether to focus on safety and production, or production over safety. If they both focused on production over safety, they would experience a type I accident once in a while, and their payoff function would be two payoff units for both. If they were both to decide to place emphasis on safety as well as production, no accidents would occur, there would still be a high production rate, and both their payoffs would end up in 3 units. If one of the workers (worker 1) decided to focus on production over safety, and the other worker (worker 2) decided to focus on both safety and production, there would still be an occasional (type I) accident in the organization (due to worker 1, not necessarily affecting worker 1). However, due to the fact that production figures are, for example, deemed more important than safety figures in this particular organization (many organizations still stress production over safety), worker 1 would be more rewarded and receive 4 payoff units, compared with worker 2, who would only receive 1 payoff unit.

The interest in this game arises from the following observation. Both players, by following their individual self-interest, end up worse off than if they had not followed their self-interest, and followed the organizations' interest instead. One could argue that the workers should follow company rules, which place the emphasis on both production and safety, no matter what. However, when each worker has no absolute assurance that the other will follow this rule, the equilibrium outcome would be (P, P).

The conditions for the game Nash equilibria and Pareto optimal solution result in the following conditions for players' payoffs in the case of a Prisoner's Dilemma game (see Figure 8.22):

An organization wishing to avoid this rather unpleasant sub-optimal equilibrium outcome has to intervene in the payoffs of the workers (not necessarily expressed in monetary terms; they can also be expressed in utils) in such a way that these conditions are not met.

8.11.3 The Prisoner's Dilemma Game Involving Many Players

The Prisoner's Dilemma can also be played with more than two players simultaneously. This game is known as the “Tragedy of the Commons.” With the help of an illustrative example, using the concept of hypothetical benefits (see earlier), the effect on safety when this game is played in an organizational context can be explained. Assume that four workers (e.g., in a shift) in one single organization receive 10 hypothetical benefit units each (i.e., the company looks after their safety by making safety investments), and, of course, that they themselves have the choice between placing emphasis on safety and production and focusing on production over safety. If they focus on safety and production, there is a pot of joint hypothetical benefit which is even enforced thanks to their personal commitment, and the joint pot is multiplied by 1.5 – this hypothetical benefit profit stems from not having any accidents due to the safety focus of the workers. The total hypothetical benefit can then be divided evenly among all the workers, as the workplace safety is uniform across the company. Obviously, if all four workers decided to pool their hypothetical benefit units and all placed the focus on safety and production, the pool would contain 40 units. Multiplied by 1.5 this leads to 60 units being divided among the four workers. Hence, each worker ends up with 15 hypothetical benefit units. Full cooperation and focus on safety as well as production thus lead to a hypothetical benefit profit of 5 units for every worker.

However, there is a catch. Assume that one worker refuses to focus on safety and production at the same time, and places emphasis instead on production over safety and thus does not put any hypothetical benefit unit in the joint pot? In other words, one worker relies only on his 10 company-given hypothetical benefit units without sharing them. If the other three workers still focused on safety as well as production, the joint hypothetical benefit pot would contain 30 units at this point. Hence, 30 × 1.5 delivers 45 units to be divided among four workers (as the fourth worker also benefits from the uniform workplace safety created by the other three), thus giving rise to 11.25 hypothetical benefit units per worker. In this situation, while three of the workers have made a profit of only 1.25 units, one worker has made a profit of 11.25 units, and is as safe as the others while putting the focus on production over safety. There is even an extra incentive to deviate from cooperation for the non-sharing worker: he not only receives safety from the company and from his co-workers for free, but he himself focuses on production, and therefore if his production figures are better than the others, he will be rewarded for them too. Of course, such a process is difficult to see and/or to quantify or prove, especially as hypothetical benefits are invisible. Nevertheless, from a theoretical perspective, the Tragedy of the Commons is easy to understand and clearly occurs in real industrial settings.

Most people, also workers, are of course initially optimistic. If a hypothetical benefit investment of 10 units is made, this would lead to a profit of 5 units – that is, if every worker does the same. But clearly there might always be one worker who holds back on investing (does not put the focus on safety) and takes advantage of the others' generosity – a so-called “free rider.” When the return on investment is less than 15 hypothetical benefit units, if they could calculate such hypothetical benefits, the workers would realize that someone had not invested his or her fair share. Of course, in that case, other workers would be tempted to hold back as well. In summary, there is a strong incentive to let everyone else focus on safety so that one worker, by placing his or her focus on production and gaining by doing so, and being relatively safe due to the company's investments and other workers' safety focus, can reap the benefits of both focus on production and not having many accidents. If everyone were to think this same way, nobody would focus adequately on safety, and nobody would make the extra hypothetical benefits that can be made by such a safety focus. The rational course of this game is thus not to put too much focus on safety in the company. This is the Tragedy of the Commons for industrial safety.

There is some good news, however. Based on game theoretical experiments surrounding climate change, the Tragedy of the Commons game can indeed be influenced in the right direction, i.e., having the workers cooperate and focus on safety and production, instead of acting only out of self-interest [19]. The first ingredient of cooperation is information. Workers need correct information as regards safety aspects of decisions and there needs to be transparency – as much as possible. The second ingredient of cooperation appears to be reputation. The effect of reputation is surprisingly strong. People really do like to be seen doing the right thing. In industrial safety terminology, this boils down to social control and management by walking around having a positive effect.

8.12.1 The Problem of Choice

The Bayesian framework is now applied to a problem of choice in which a decision-maker must decide how much he or she is willing to invest in order to reduce the probability of a type II risk event occurring. The two decisions under consideration in this simple scenario are:

1. D₁ = keep the status quo,
2. D₂ = improve barrier for type II event.

The possible outcomes in the risk scenario remain the same under either decision, and therefore are not dependent upon the particular decision taken. These outcomes are:

1. O₁ = catastrophic type II event occurs,
2. O₂ = no type II event.

The hypothetical damages associated with these outcomes are:

8.4

and the investment costs associated with the additional improvement of the type II event barriers are expressed by the parameter:

8.5

The decision on whether to improve the type II event barriers or not is of influence in the probabilities of the respective outcomes. Under the decision to make no additional investments in the type II event barriers and keep the status quo, D₁, the probabilities of the outcomes will be, say:

8.6

Under the decision to improve the type II event barriers, D₂, the probability of the catastrophic type II event will be decreased by, say:

8.7

where . Stated differently, the proposed barrier improvements will decrease the chances of the catastrophic type II event by a factor of .

In what follows, the solution of this problem of choice will be given for expected outcome theory, expected utility theory, and Bayesian decision theory. These solutions will be given in terms of variables x, θ, and φ, respectively (Eqs. 8.6 and 8.7).

8.12.2 The Expected Outcome Theory Solution

The prosperous merchants in seventeenth-century Amsterdam bought and sold expectations as if they were tangible goods. It seemed obvious to many that a person acting out of pure self-interest should always behave so as to maximize his expected profit [20]:

8.8

Combining Eqs. (8.4)–(8.7), one may construct the outcome probability distributions under the decisions D₁ and D₂:

8.9

and

8.10

where in Eq. (8.10) one may explicitly conditionalize on the investment parameter I, which is to be to estimated. The expected outcomes of these probability distributions are, respectively [21]:

8.11

and

8.12

The decision theoretical equality

8.13

represents the equilibrium situation, where it will be undecided whether to keep the status quo, D₁, or invest in additional barrier improvements, D₂. Now, if one solves for I in Eq. (8.13), by way of Eqs. (8.11) and (8.12):

8.14

then we find that investment where one will remain undecided.

Stated differently, any investment cost smaller than Eq. (8.14) will turn Eq. (8.13) into an inequality, where D₂ becomes more attractive than D₁. It follows that the equilibrium investment (Eq. 8.14) is also the maximal investment one will be willing to make in order to improve the type II event barriers, or, equivalently, Eq. (8.14) is the hypothetical benefit of the type II event barrier improvement.

8.12.3 The Expected Utility Solution

For a rich man €100 is an insignificant amount of money. So, the prospect of gaining or losing €100 will fail to move the rich man, i.e., for him an increment of €100 has a utility that tends to zero. For the poor man €100 is a significant amount of money. So the prospect of gaining or losing €100 will most likely move the poor man to action. It follows that for him an increment of €100 has a utility significantly greater than zero.

In 1738 Daniel Bernoulli derived the utility function for the subjective value of objective money by way of a variance argument, in which he considered the subjective effect of a given fixed monetary increment, c, for two persons of different initial wealth. Based on this variance argument, he derived the utility function of going from an initial asset position x to the asset position :

8.15

where q is some scaling constant greater than zero [22].

In expected utility theory, the expected values of the utility probability distributions are maximized. Assuming that the decision-maker has a total wealth (i.e., an actual income and asset portfolio) of:

8.16

then, using Eq. (8.15), or, equivalently,

8.17

one may construct from Eqs. (8.9) and (8.10) the corresponding utility probability distributions as:

8.18

and

8.19

The expected outcomes of the utility probability distributions are, respectively [21]:

8.20

and

8.21

The decision theoretical equality:

8.22

represents the equilibrium situation, between the decision to keep the status quo D₁ and the decision to invest in additional barriers D₂. Now, if one substitutes Eqs. (8.20) and (8.21) into Eq. (8.22), then one obtains the closed expression for that investment value where one is indifferent between either decision:

8.23

Any investment cost smaller than the numerical solution of I in Eq. (8.23) will turn Eq. (8.22) into an inequality, where D₂ becomes more attractive than D₁. It follows that the equilibrium investment (Eq. 8.23) is also the maximal investment one will be willing to make to improve the type II event barriers, or, equivalently, Eq. (8.23) is the hypothetical benefit of the type II event barrier improvement.

8.12.4 The Bayesian Decision Theory Solution

In Bayesian decision theory the scaled sum of the confidence bounds and the expectation value of the utility probability distributions is maximized as the risk measure that captures the position of the underlying utility probability distribution (see Section 7.2.2):

8.24

where the lower confidence bound is corrected for undershooting the worst possible outcome a, (Section 7.2.1.1):

8.25

and the upper confidence bound is corrected for overshooting the best possible outcome b, (Section 7.2.1.1):

8.26

Substituting Eqs. (8.25) and (8.26) into Eq. (8.24), one obtains the risk index:

8.27

where it is noted that the first row of Eq. (8.27) corresponds with the expected utility theory criterion of choice [23], and the fourth row is a kind of adjusted Hurwitz criterion of choice, which may differentiate two probability distributions that have the same minimal and maximal values while at the same time having an opposite skewness.

In the illustrative toy problem under consideration, a simple type II risk scenario is modeled, which is typically a high-impact, low-probability scenario, i.e., both large monetary costs and small probabilities for the high-impact event, or, equivalently, on the impact side (Eq. 8.5), x ≫ 0 and, on the probability side (Eqs. 8.6 and 8.7), θ, φ ≪ 0.5. Stated differently, the utility probability distributions (Eqs. 8.18 and 8.19) under consideration will both be highly skewed to the left and, as a consequence, will lead to the third condition in Eq. (8.27):

8.28

The best possible outcome under decision D₁ is Eq. (8.18):

8.29

and the standard deviation of Eq. (8.20) is [21]:

8.30

So, the risk index under the decision to keep the status quo is, substituting Eqs. (8.20), (8.29), and (8.30) into Eq. (8.28):

8.31

The best possible outcome under decision D₂ is (Eq. 8.19):

8.32

and the standard deviation of Eq. (8.19) is [21]:

33

So, the risk index under the decision to invest in additional barriers is, substituting Eqs. (8.21), (33), and (8.23) into Eq. (8.28):

34

The decision theoretical equality:

35

represents the equilibrium situation, between the decision to keep the status quo, D₁, and the decision to invest in additional risk barriers, D₂. Now, if one substitutes Eqs. (8.31) and (34) into Eq. (35), then one obtains the closed expression for that investment value which will leave one undecided:

36

Any investment smaller than the numerical solution of I in Eq. (36) will turn Eq. (35) into an inequality, where D₂ becomes more attractive than D₁. It follows that the equilibrium investment (Eq. 36) is also the maximal investment one will be willing to make to improve the type II event barriers, or, equivalently, Eq. (8.23) is the hypothetical benefit of the type II event barrier improvement.

Note that the “Weber constant” q has fallen away in both the decision theoretical equalities, Eqs. (8.23) and (36). This will hold in general, as both the expectation values and standard deviations in Eqs. (8.28) and (8.29) are linear in the unknown constant q. It follows that one may set, without any loss of generality, q = 1.

8.12.5 A Numerical Example Comparing Expected Outcome Theory, Expected Utility Theory, and Bayesian Decision Theory

After the great Dutch floods of 1953, the “Oosterschelde Waterkering” was built. This was a movable dyke that allowed for an improved safety of from 1/100 to 1/4000 per year, while keeping the Oosterschelde connected to the North Sea. This open connection to the North Sea was decided upon in order to keep the salt-sea ecological system of the Oosterschelde river intact.

The total costs of the Oosterschelde Waterkering where about €2.5 billion. The bulk of these costs were due to the movable character of this dyke. Had the Dutch government decided to build an unmovable dyke, then the costs would only have been about €175 million.

The total value of the assets at risk at the time were about 1/20th of the then gross domestic product (GDP):

37

The wealth of the decision-maker, i.e., the Dutch government, was about 40% of the Dutch GDP at that time, aggregated over a period of 5 years, this being the total construction time of the movable Oosterschelde dyke:

38

The status quo probability of a catastrophic flood directly after the great flood was estimated to be (Eq. 8.24):

39

whereas the probability of the catastrophic flood with the improved flood defenses were estimated as (Eq. 8.25):

40

Substituting the values in Eqs. (37)–(40) into Eqs. (8.14), (8.23), and (36), one obtains the following solutions for the maximal investment willingness, or, equivalently, the hypothetical benefit, I:

• Expected outcome theory:
  1. – Any sigma level:
• Expected utility theory:
  1. – Any sigma level:
• Bayesian decision theory:
  1. – 1-sigma level:
  2. – 2-sigma level:
  3. – 3-sigma level:

It is noted here that after the great Dutch flood, the discussion was not about whether to build additional flood defenses or not, but rather whether to choose the expensive solution over the “cheap” solution, which would keep the Oosterschelde salt-sea ecosystem intact. Under expected utility theory, the cheap solution of an unmovable dyke would have been too expensive by a factor of 3, whereas under Bayesian decision theory the cheap solution was well within the 2-sigma bounds.

Let us now move to the present time. The current total value of the assets at risk in the Oosterschelde region are about 1/20th of the current GDP (Eq. 8.23):

41

The wealth of the decision-maker, i.e., the Dutch government, is about 20% of the current Dutch GDP:

42

If one assumes the current probability of a catastrophic flood to be 1/4000, and if one assumes that in the absence of any maintenance the flood defenses will have deteriorated such that the probability of a catastrophic flood will double to 1/2000 in 5 years' time, then is the implied “doubling” 1 year away from the latest maintenance round. Using this doubling factor of , the probability of a catastrophic flood becomes (Eq. 8.24):

43

If one assumes that the probability of a catastrophic flooding under the flood defence maintenance is our current probability of a catastrophic flooding, (Eq. 8.5):

44

Then one has a scenario where one wishes to prevent a current situation, which is very safe (Eq. 44), from sliding into a somewhat less safe situation (Eq. 43).

Substituting the values from Eqs. (42)–(44) into Eqs. (8.14), (8.23), and (36), one obtains the following solutions for the maximal investment willingness, or, equivalently, the hypothetical benefit, I:

• Expected outcome theory:
  1. – Any sigma level:
• Expected utility theory:
  1. – Any sigma level:
• Bayesian decision theory:
  1. – 1-sigma level:
  2. – 2-sigma level:
  3. – 3-sigma level:

It is noted here that in order to obtain the very real safety benefit of preventing the probability of a catastrophic flood of from sliding to , expected utility theory is not willing to invest more than €1.3 million, whereas Bayesian decision theory with utility transformation, under a 2-sigma safety level, is willing to invest €26.9 million for the safety maintenance of the Oosterschelde Waterkering.

So it would seem that the Bayesian decision theory solution is more commensurate with observed safety management practices, seeing that the Dutch government spends about €20 million a year to maintain the Oosterschelde Waterkering.

8.12.6 Discussion of the Illustrative (Toy) Problem – Link with the Disproportion Factor

In this section, expected outcome theory, expected utility theory, and Bayesian decision theory were compared for a simple illustrative toy problem regarding the investment willingness to avert a high-impact, low-probability event. It was demonstrated that the adjusted criterion of choice, in which scalar multiples of the sum of the lower confidence bound, expectation value, and upper confidence bound of the utility probability distributions are maximized, though mathematically trivial [23], has non-trivial practical implications for the modeled investment willingness, or, equivalently, for the modeled hypothetical benefits.

To summarize this section, based on the results from a numerical example, it might be argued that it is the insufficiency of the expectation value (Eq. 8.8) as an index of risk that forces a cost-benefit analysis to introduce disproportion factors (DFs) as an ad hoc fix-up; i.e., the DFs are needed in cost-benefit analyses because the currently computed hypothetical benefits (Eq. 8.24) may severely underestimate the actual hypothetical benefits (Eq. 36) which are computed by way of the more realistic index of risk (Eq. 8.27).

In the case where one restricts oneself to outcome probability distributions, the hypothetical benefits in a cost-benefit analysis are computed as the difference between the expectation value of the outcome under the additional safety barriers and the expectation value of the outcome under the current status quo (Eq. 8.14):

45

But under the alternative index of risk (Eq. 8.27), which not only takes into account the most likely trajectory but also the worst- and best-case scenarios, the corresponding hypothetical benefits may be computed as (cf. Eq. 36):

46

So, for the outcome probability distributions Eqs. (8.9) and (8.10), the alternative criterion of choice (Eq. 8.27) implies a theoretical DF which is the ratio of Eqs. (42) to (41):

47

In Chapter 5 a study on the DF was performed (based on Goose [24] and Rushton [25]) in which, by way of common sense considerations, it is recommended to calculate and employ DFs for which the NPV becomes zero (see also Section 5.7). Alternatively, in Eq. (43) a DF is derived for a specific risk scenario, by comparing the hypothetical benefits under the traditional criterion of choice (Eq. (8.8)) and the alternative criterion of choice (Eq. (8.27)).