Sensitivity to an Interaction Term in the Value Function 119 

Multi-attribute methodologies require analysts to assume (or to define) each attribute to be independent of the others. The metrics capturing waste and cost attributes in Chapter 4 were defined as much as possible to preserve independence. The cost metric, for example, does not include the price of waste management and disposal: this is assumed to be part of the “social” cost which is captured in non-dollar denominations by the waste metric. Yet the assumption of complete independence between these two attributes is not perfectly accurate. For example, the cost of the nuclear fuel cycle system may affect public opinion of nuclear power overall, which in turn will affect the social disutility for nuclear waste.

Clemen (1996) suggests several ways to adjust value functions so that they account for the non-independence of attributes. One option, which is relatively straightforward for two-

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attribute functions, is to define a utility surface over the two-dimensional space of the attributes. The advantage of this method is that it does not require any assumptions whatsoever about attribute independence. A significant disadvantage, however, is that properly eliciting the structure of the utility surface is computationally intensive, requiring a decision maker to establish preference-indifference curves by comparing a large number of attribute lotteries.

If the two attributes are determined to be mutually utility independent, attribute utility functions are separable and the value function can be expressed as:

, 1

where U(x,y) is the value of having the attributes at level x and level y, and kx and ky are

weighting constants. Determining the appropriate utility equations for a separated utility function is simpler than determining a joint utility surface.

Attributes are mutually utility independent if a decision maker determines that his or her preferences for uncertain choices involving different levels of Y are independent of the value of X, and vice versa. A thought experiment helps to determine whether cost and waste are utility independent in the context of the U.S. nuclear fuel cycle. If cost is utility independent of waste, the decision maker should have a certainty equivalent for e.g. a lottery where there is a 50% chance the fuel cycle will produce 2 repositories worth of waste over the century, and a 50% chance the cycle will produce enough to fill only one repository: this certainty equivalent should not depend on whether the fuel cycle is expensive or cheap. For some decision makers, this will be true. The decision maker might prefer the certainty of 1.5 repositories’ worth of waste to the lottery above, regardless of the nuclear system cost. Similarly, a lottery over costs does not necessarily depend on the amount of waste the system generates: lower cost is always preferred. Political decision makers, however, might be more risk averse about waste if nuclear system costs are high: justifying the need for a new repository site might be made more difficult if nuclear power is expensive compared to other sources of electricity. Further work should explore the true utility independence of these attributes for different decision makers, and could explore whether decisions change substantially given different utility surfaces. Here, mutual utility independence is assumed (and indeed may hold for some decision makers) in order to see the effect of adding one layer of complexity to the value function.

The waste metric is calculated as before, using an additive multi-attribute function for the waste types. The waste utility function is still applied to each type individually, such that

121 The overall value function will now be:

1

Note that we no longer require that wC + wW = 1 (and indeed, if wC and wW satisfy this

condition, the equation simplifies to the original form with no interaction term). If the coefficient of the interaction term is positive, the two attributes are considered to be complements (such that increasing the disutility of the attributes entails an extra penalty: having high cost and a large amount of waste, in a sense, is thus highly catastrophic). A negative coefficient, however, is more likely. A negative coefficient is reasonable if the attributes are substitutes, such that higher cost-scenarios are more tolerable if they produce less waste. The “true” weights could be elicited formally from a decision maker, in order to determine which characterization of the attributes makes more sense. TreeAge, however, allows exploration of an extensive range of the weights, so that we do not need to assume either case and can instead present decision results as a sensitivity to a range of weights.

The figures below show that we receive very little new insight by adding an interaction term to the value function. The decision result depicted in Figure 6-5 is calculated for the five- option tree, which allows the decision maker to build TFRs or EUFRs at 10% of their allowed amount in the first period (compare to the results of section 5.3). All probabilities for growth and cost are set to 0.5, and wC and wW are each varied from 0 to 1. This means that in the upper- right region of the graph, the attributes act as complements, while in the lower-left, they act as substitutes.

The results are fairly intuitive: if cost weights are low and waste weights are high, fast reactors are desirable. The graph does show that for this range of cost and waste weights, the decision is more sensitive to the cost weight than to the waste weight. One possible explanation relates to an insight from chapter five: waiting to build TFRs significantly reduces cost, because of discounting, whereas strong waste benefits can still be gained even if TFRs are built later.

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Figure 6-5: Cost and waste weight tradeoff for a value function with a cost-weight interaction term

Figure 6-5 does not change substantially when other parameters are shifted. For example, increasing the probability of high growth in period one simply makes TFRs more attractive, shifting the pink space to the right (as we would expect, given that for the 10% TFR decision, higher growth makes TFRs more attractive). Similarly, graphs comparing the sensitivity of wC to P1 with wW set at various values look almost exactly the same as the graph of the basic 10% decision presented in section 5.3, with the wW determining the thickness of the 10% TFR desirability band.

Overall, the simple addition of an interaction term does not significantly impact the results presented in Chapter 5. It may be, however, that consideration of a wildly different utility surface over the attributes would change the results more substantially (but this surface would need to accurately represent the desires of decision makers, and would not likely depart far from the linear or exponential curves explored in this thesis).

Rather than concentrate too heavily on the independence of the attributes and structure of the value function, analysts conducting further work would probably do better to examine the effect of a completely different value function with new metrics. For example, if decisions were made that rendered volume to be the only relevant waste metric (e.g. because deep “hot”

123 borehole disposal were deemed the best pathway for geologic management), this might

substantially alter the results. Alternatively, adding metrics or replacing waste or cost with other attributes deemed to be more important also could have a large effect. Safety, for example, is paramount in the public’s post-Fukushima thoughts on nuclear power; further work could examine the effect of incorporating a safety metric into the value function.

6.2 Key Takeaways from the Sensitivity Analysis on the Addition of an Interaction Term Adding an interaction term to the value function does not change the qualitative outcomes observed so far. We do see that when we separate the waste and cost preference weights so that they are independent of one another, the results are somewhat more sensitive to the cost weight. The decision results generally respond to uncertain parameters as before.