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Joshua T. Knight1

Department of Naval Architecture and

Marine Engineering,University of Michigan,

Ann Arbor, MI 48109

e-mail: [email protected]

David J. SingerAssistant Professor,

Department of Naval Architecture and

Marine Engineering,

University of Michigan,

Ann Arbor, MI 48109

Prospect Theory-basedReal Options Analysis forNoncommercial Assets

When an engineering system has the ability to change or adapt based on a future choice,then flexibility can become an important component of that systems total value. However,evaluating noncommercial flexible systems, like those in the defense sector, presents manychallenges because of their dynamic nature. Designers intuitively understand the impor-tance of flexibility to hedge against uncertainties. In the naval domain, however, they oftendo not have the tools needed for analysis. Thus, decisions often rely on engineering ex-

perience. As the dynamic nature of missions and new technological opportunities pushthe limits of current experience, a more rigorous approach is needed. This paper describesa novel framework for evaluating flexibility in noncommercial engineering systems called

prospect theory-based real options analysis (PB-ROA). While this research is motivated bythe unique needs of the U.S. Navy ship design community, the framework abstracts the

principles of real options analysis to suit noncommercial assets that do not generate cashflows. One contribution of PB-ROA is a systematic method for adjusting agent decisionsaccording to their risk tolerances. The paper demonstrates how the potential for losscan dramatically affect decision making through a simplified case study of a multimissionvariant of a theoretical high-speed connector vessel. [DOI: 10.1115/1.4026398]

Keywords: prospect theory, loss aversion, real options analysis, stochastic modeling, risk

1 Introduction

For many modern engineering systems, flexibility is an impor-tant measure of performance. This is especially true for naval war-ships; the use of modular mission packages on the littoral combatship (LCS), commercial standards on the joint high-speed vessel(JHSV), and vertical launch systems (VLS) on the Arleigh Burke-class destroyers (DDG-51) are but three examples of design fea-tures that lend some form of flexibility to the asset. But flexibility

is more than a physical design feature. In this research, true flex-ibility is viewed as the ability to make a choice, or the capabilityto re-evaluate decisions in the future. Flexibility is a commonattribute of many engineering systems, though its form will vary.So no matter the physical form of flexibility, the challenge thenbecomes how to value the freedom to make a choice in a navalcontext.

Static budgetary techniques and net present value (NPV) analy-sis underestimate the value of managerial and operational flexibility[1], and the embedded optionality of many flexible design fea-tures, like modularity and design-for-upgradability. Currently, deci-sions made concerning U.S. Navy systems with a high degree ofoptionality are largely based on anecdotal evidence [2], or engineer-ing experience. This is because designers intuitively understand thevalue of flexibility, and the need to hedge against uncertainty. How-

ever, a rigorous mathematical framework for evaluating flexibilityfor the U.S. Navy is necessary, particularly as designs are pushed tobe increasingly adaptable, have longer service lives, and exceed thelimits of current engineering experience. The use of real optionsanalysis (ROA) has been proposed for such a framework [3]; how-ever, the theory is not universally applicable to the naval domainbecause of at least three key assumptions.

Firstly, traditional ROA assumes that the underlying asset gen-erates cash flows. This critical assumption has many implications.For instance, value is defined in terms of some currency. Also,

risk, orvolatility, is defined by the variance of the cash flows. Navalassets do not generate cash flows, so nonmonetary measure(s) ofvalue and risk are necessary.

Secondly, ROA also requires a market to accurately price assets.Expected value (i.e., price) is calculated using an equivalent mar-tingale measure, also known as the risk-neutral measure. In ad-dition to ensuring that the price of the option permits no arbitrageopportunities, using the risk-neutral measure incorporates decision-makersrisk aversion into the price of the option. The market pro-vides the risk-neutral measure; however, a market does not exist fornaval assets. While the concept of arbitrage may not be relevantfor naval assets, risk aversion certainly is. Another mechanism forcapturing risk aversion is required for naval applications.

Thirdly, most ROA assumes that the payoff (or penalty) to theoption owner is exclusive. The market is tacitly assumed to be com-posed of many rivals, each sufficiently small that no action taken byany one agent will have a measurable effect on market behavior.This assumption is invalid in many naval applications involvinginteractive decision making between agents where the purchaseor creation of an option may influence other agentsstrategies andhence change the operating environment, or market behavior. Somenaval options may be leveraged to induce change in the market(i.e., their operating environment). Such game-changing optionscannot be evaluated by traditional real options, but promising ave-nues of research exist in the field of options game theory.

The preceding paragraphs explain why a traditional real optionsapproach to flexibility will not be suitable for many naval designapplications. A new framework is required. A high-level outline ofthe authorspath to transitioning from real options for commercialapplications to naval applications is shown in Fig. 1. Utility is usedto define value in the absence of cash flows, and prospect theoryprovides a consistent mechanism for risk-adjustment without a mar-ket. Game theory abstracts the framework for applications where theoption may have feedback on its environment.

While noncommercial assets are frequently assigned monetaryvalues, the authors believe that using utility is more appropri-ate for military applications. For example, it can be difficult or

1Corresponding author.Manuscript received April 3, 2014; final manuscript received November 19, 2014;

published online February 27, 2015. Assoc. Editor: Bilal M. Ayyub.

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems,Part B: Mechanical Engineering

MARCH 2015, Vol. 1 / 011004-1Copyright 2015 by ASME

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its explicit treatment of failure regions (loss). Info-gap robustnessanalysis gives decision makers a measure of how uncertain theirestimates can be and still achieve a desirable outcome. From com-peting options, the most preferred is that with the highest robust-ness: the highest tolerance to error or uncertainty. However, theinfo-gap does not assign probability distributions to random varia-

bles. For this reason, prospect theory is more amenable to a realoptions formulation.

A typical financial option does not consider the interdependen-cies of agent decision making. This is because such interdependen-cies either do not exist or are considered negligible according to theefficient market hypothesis [20]. The works of Kifer [21], Smit andAnkum [22], Lukas et al. [23], Villani [24], Smit [25], and Smit andTrigeorgis [26] are notable exceptions where multiple decisionmakers impact the value of the option and game theory is used inthe analysis. Smit and Trigeorgis [27] give a good summary of howto use combined game option theory. However, where others haveinvestigated the impact of the environment on the option, this frame-

work considers the impact of the option on the environment.

3 Prospect Theory-Based Real Options Analysis

Figure 3 shows the flowchart for the eight steps of the newframework. To build a new framework for valuing real optionsfor the U.S. Navy, let us begin by identifying the set ofrisk factorsto which a particular naval option is exposed. These risk factors arerandom variables, and may also have a time component

Xt fX1t; : : : ;Xntg

LetXtbe the set of all risk factors. The individual risk factors,Xit, may be continuous or discrete. These risk factors may also bethought of as states of the world, and will affect the utility of a de-

sign. The set of relevant risk factors will depend on the application.For instance, the technology readiness level (TRL) may be a

Fig. 2 Hypothetical weighting function in prospect theory,(taken from Ref. [17], p. 283)

Fig. 3 Process flowchart for PB-ROA framework


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relevant risk factor for an option involving new technologicalinnovations. The prices of different types of fuel may be relevantrisk factors for a duel-fuel engine design. In practice, some risk fac-tors, like fuel prices, will be easier to model than others, like TRL.There may be risk factors that simply do not lend themselves toquantitative modeling. This research addresses such issues on anindividual basis.

Letpt; Xt denote the joint probability density function as-sociated with the set of risk factors. This will often be referred tosimply as the real probability measure, P.

Next, a system design is broken down into a set of designfeatures. For the purposes of this framework, design features aredefined to be those physical attributes of a system that contributeto, or dictate, its performance. The specific definition of a designfeature will likely change depending on the system or applicationunder consideration. But, for example, a design feature may be asgranular as the web and flange dimensions of a stiffener, or as largeas a davit for launching autonomous vehicles. Letd be the set of alldesign features that influence the systems performance

d fd1; : : : ;dmg

From the sets of risk factors and design features, this frameworkdefines a systems capability. A system may have one or manycapabilities. For instance, the capability of a pump design, whosesole function is water delivery, could be measured in gallons per

minute (gpm). However, the latest variant Aegis combat systemon the DDG-51 Arleigh Burke-class guided missile destroyer (asystem of systems) has multiple capabilities, such as radar trackingand guidance for cruise missiles (offensive capability), and ballisticmissile defense (defensive capability). Its offensive and defensivecapabilities may even be measured differently

cd;Xt fc1d;Xt; : : : ; crd;Xtg

Let cd;Xtbe the systems set of capabilities. The capabil-ities will depend not only on the design features, d, but also on thestate of the risk factors, Xt. For example, a radar systems capa-bilities may depend on the weather conditions (the risk factor).

From the set of design features, the authors also define acom-plexity metricfor the system,d. In this framework, complexity is

treated as a known quantity, resulting directly from the set of designfeatures. Complexity is not random. Complexity is important fordifferentiating between alternative designs with similar capabilities.A good example of complexity is the metric by Rigterink et al. [28]for the design of stiffened panels. The construction complexity of apanel design is assessed based on a number of factors such as weld-ing access, bracketing, and steps in plate thickness. It is believedthat lower construction complexity leads to lower life costs.

The authors also assume that a utility function may be quantifiedfor each system under the option analysis. Utility is a measure ofthe total value of a system, and is a function of the systemsdesign complexity and capabilities. Utility may also have a timecomponent

u ut; ;c

Letut; ; cbe the systems utility function, wheretis time,isthe complexity metric, and cis the capability set. Althoughcmaybe a vector, utility is a scalar function. Utility can have differentunits depending on the application, or may be unitless. The authorsbelieve that the elasticity of a measure like utility has many advan-tages over rigidly defined measures such as currency, which willallow more meaningful analysis of flexible systems for the Navy,where the cost of such systems is only one consideration. In contrastto most financial analyses, cash flow and cost are not the only waysto measure value for noncommercial assets like Navy ships. Othermeasures of value may be mission effectiveness, security, systemreliability, etc. Such considerations may be expressed in a utilityscore, which is another reason why utility theory was chosen to

form this new framework. Utility will often be denoted simplyby uc.

With risk factors modeled, design features identified, capabilityand complexity quantified, and the utility function defined, it isfinally possible to value an option. A European-style naval optionhas present value, v , of

vt;Xt EQXTjFt 2

whereXTis the payoff of the option at terminal time T, giventhe prevailing state of the risk factors, XT, and EQ signifies that

the expectation is being calculated using the risk-adjusted probabil-ity measure, Q. Ft is the filtration

1 on the risk-adjusted probabilityspace, Q, up to time t .

Similarly, the present value of an American-style naval optionwith payoff function, X, is

vt;Xt supt;TEQjFt 3

where is the optimal stopping time for the option.At first glance, these equations seem the same as the typical

equations found in financial option valuation literature. However,there are two highly important differences. The first is that any dis-counting due to the time value of the naval asset must be taken intoconsideration through the utility function. In other words, if a fixedquantity of a naval asset is worth more today than it will be in the

future (as with money), then the utility function for the asset mustperform the discounting. Secondly, and most importantly, the prob-ability measure, Q, which is used for the valuation, is not the samerisk-neutral measure as in standard options analyses. Typically,the risk-neutral measure is provided by the market thanks to a no-arbitrage argument. For naval options analysis, the authors advocatethat a risk-adjusted measure, also denoted by Q, should be used andis formulated by renormalizing the product of the real probabilitymeasure, P, and the marginal utility. For a system with a scalarcapability function (i.e., the vector c contains only one element),the risk-adjusted probability measure is given by

qx px u 0cx 4

whereqxis the risk-adjusted probability density for eventx,pxis the real probability density for eventx, and u 0cxis the mar-

ginal utility for eventx. The marginal utility is

u 0cx u

c

ccx

5

Much like the nonlinear decision weights in prospect theory[17], the intuition behind the risk-adjusted probability measure inEq. (4) is that it measures the impact of events on the desirabilityof outcomes, and not merely the likelihood of those outcomes. Anevent may be impactful for two reasons. The first, obvious reason isbecause it has high utility (positive or negative). However, the sec-ond reason is because it has high marginal utility; it lies in a rapidlychanging region of preferences.

For a system with a vector capability function, like the Aegisexample, the marginal utility weight is a function of the magnitude

of the gradient of the utility function

kuck

u

c1

2

u

cr

212ccx

6

However, for our purposes, rescaling of capability measures maybe necessary for applications where the magnitudes of individualcapabilities differ greatly. To illustrate, consider an asset with twocapabilities. The first, being related to deck space, is measured insquare feet. The second, related to transit speed, is measured inknots. Deck areas are commonly on the order of104 square feet,

1In simple terms, a filtration collects all of the information from time zero to thepresent, allowing conditional probability calculations to be performed.

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or more, for Navy assets, but speed is commonly below 40 knots.Due to the drastic difference in scale, the marginal utility weight inEq. (6) would be dominated by the speed capability, as the marginalutilityper square footof deck area is negligible by comparison. Forthis reason, each component of the gradient is normalized accordingto its maximum value

u 0cx

1

dc1

u

c1

2

1

dcr

u

cr

21=2ccx

7

where

dci supu

ci8

Despite the complications of vector capability functions, thegeneral approach of using the marginal utility function to formulatea risk-adjusted measure for option valuation is supported by Davis[13], Nau and McCardle [15], Hugonnier et al. [16], and Beja [14],but is also unique among the literature on valuing real options fornaval applications. It is similar to the use of nonlinear weightingfunctions in prospect theory [17]. Hence, this framework is referredto as the prospect theory-based real options analysis framework(PB-ROA). Also in accordance with prospect theory, the payofffunction,, is measured in the same units as the utility function,but the payoff is expressed relativeto a reference point. The refer-

ence point will be specific to each application.

3.1 Constraints on Utility Functions.The current frameworkonly allows for fully ordered utility curves (i.e., non-negative mar-ginal utility). This constraint is a result of the risk-adjustment pro-cess. If the marginal utility in Eq. (4) were ever negative, then therisk-adjusted measure, q, would also be negative and violate therequirements of a probability measure. Preliminary research indi-cates this is not a significant constraint in practice.

3.2 Elicitation of Utility Curves. One of the practical hurdlesto institutionalizing the PB-ROA framework is how one determinesthe shape of the utility curve(s). The subject of utility elicitation isnot the focus of this paper; however, a brief treatment is merited

since utility is a core concept of PB-ROA.In the naval options thesis by Page [5], which makes use of

multi-attribute utility functions, an analytical hierarchy process(AHP) is used to rank design alternatives. AHP is commonly usedin the U.S. Navy as a decision-making aid to weigh the relative im-portance of many factors at once. Attributes of designs (i.e., main-tainability) and their relation to objectives (i.e., reduced cost)are weighted in pairwise comparisons against other attributes andobjectives. Then, design alternatives are ranked according to theweighted sum of their performance in each attribute-objective cat-egory. AHP may be a useful tool for mapping design features to setsof capabilities, and then weighing those capabilities against eachother in a pairwise manner to determine an overall utility score.However, care should be taken because many AHP applications as-sume constant objective weights over the entire design space [29],

which may lead to a linear utility surface implying that decisionmakers are risk-neutral. Since this is not true, modifications to theAHP process may be necessary.

A common method to elicit utility, from the area of experimentaleconomics, is to present test subjects with a series of gambles, andsystematically determine the subjects certainty equivalent for thatgamble. This means that subjects are presented with a gamble, forexample, a 50% chance at gaining $1 and 50% chance of nothing.Then, the subjects are offered certain payments (i.e., guaranteedpayments), for example, a certain payment of $0.25. This certainamount can be varied experimentally to determine the thresholdat which the subject is indifferent between taking the certain pay-ment and taking the gamble. If a parametric form is assumed forthe shape of the utility function (such as a power function), then

a least-squares regression from subject data can be used to esti-mate the parameters. When using this approach assuming prospecttheory, however, things are more complicated. Since prospecttheory states that peoples decisions are made with a combinationof utility and nonlinear weighting of the probability of risky out-comes, the process of certainty equivalent testing is slightly moreinvolved. Abdellaoui [30] and Abdellaoui et al. [31] give excellentoverviews of how the process works under a prospect theoryassumption. Their process involves separate testing of the gains andlosses regions. What is significant about their work, however, is that

they propose ways to elicit the utility curves without resorting toparametric assumptions.

3.3 Selecting Options and Optimization. The PB-ROAframework described in the preceding sections allows one to valueindividual options on noncommercial assets. Valuation of individualoptions is based on expected value calculated under the risk-adjusted probability measure. This is supported by a long line ofresearch that rational decision makers are expected utility maximiz-ers, although the expectation may be adjusted for factors like lossaversion or point of reference [17]. This is akin to reliability-basedoptimization in its focus on mean values. When selecting betweenmultiple options or optimizing option value, however, it should benoted that there may also be a place for robustness-based optimi-zation. In the process of valuation, uncertainty is the result of the

stochastic risk factors (refer to Fig. 3). If, however, there is uncer-tainty over other parameters (the design features, or capabilities, forinstance), then the process of choosing between multiple options oroptimizing option value may be augmented with robustness-basedoptimization. If faced with constraints, for instance, the decisionmaker may choose to sacrifice some option value (expectation) inexchange for greater confidence in the outcome (robustness). In-deed, this would be no different than robust optimization in otherdisciplines, like manufacturing [32]. The scope of this effort, how-ever, is the valuation process.

3.4 Naval Options with Interdependent Decision Making:Games. The new framework discussed so far has only consideredoptions with a one-way dependence on their environment, whatthe authors termreactionary options. Once a reactionary option is

purchased, or created, the owner observes what changes occur in theenvironment as time passes, and then decides (at the appropriatetime) whether or not to exercise the option. The assumption withreactionary options is that the existence of the option has no feed-back effect on developments in the environment. The dependence isone way. This is the standard assumption with financial options, aswell as virtually all ROA, where no single agent is capable of mov-ing the market, in accordance with the efficient market hypothesis.

If, on the other hand, the value of the option relies on the inter-dependent decision making of multiple agents, then the precedingframework will not yield an accurate option value. A modified,game-theoretic approach is required. Such is the case with manynaval options.

PROPOSITION. If a naval option exists in the presence of inter-dependent decision making of multiple agents, then the value of the

option is equal to the change in the value of the Nash equilibria ofthe games before and after the introduction of the option.This may be illustrated through the following theoretical exam-

ple. Consider that the status quo (before the introduction of the op-tion to the environment) is reflected by the game in Fig. 4. This is, ofcourse, very nearly the famous prisoners dilemma game. If bothplayers were to choose strategy A, then the payoff to both wouldbe1. However, Player 2 has an incentive to change to strategy B, ifthey believe the other will play A. This leads both players to finallyplay strategy B, which is the Nash equilibrium of this game despitebeing suboptimal.

Now suppose that Player 1, in the process of devising a way tobe able to safely play strategy A, is considering purchasing an op-tion that would give her some form of leverage over Player 2 in the


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event that Player 2 did not also play strategy A. The option works as

follows. Player 1 will begin by playing strategy A. If Player 2 alsoplays strategy A, then Player 1 will not exercise the option, whichwill result in a payoff of1for both players. However, if Player 2chooses strategy B, then Player 1 will exercise the option, resultingin payoffs of3, and 1.5for Players 1 and 2, respectively. This isshown in Fig. 5.

This game, after the introduction of Player 1s option, now hasan optimal Nash equilibrium. According to this framework, thevalue of the option in this example, v, is v 1 2 1.

The critical reader may argue that this research is simply chang-ing the structure of the game. This is true. However, it is exactly thisgame-changing attribute of many naval options that one is unableto value using standard ROA. One of the original contributions ofPB-ROA is this perspective on naval options that may be gamechanging. A full treatment of the theory of naval game options

is outside the scope of this paper but is the subject of a futurepublication.

4 Example: Hospital Variant of a High-Speed

Connector

This section presents a simplified example intended to demon-strate how the PB-ROA framework can be used to generate usefulinsight for decision makers in the design of a vessel. The vesselconsidered in this example is a theoretical high-speed connector(HSC) vessel.

The concept of operations for this HSC includes both wartimeand peacetime roles for personnel and material transport, such ashumanitarian aid and disaster relief (HADR). In addition to its pri-

mary role as a HSC, a secondary mission as a fast-response medicalsupport ship is being proposed. For this mission, the HSC wouldtemporarily install modular medical facilities in its cargo area.The important design consideration is how to best implement sucha hospital variant, as the necessary medical equipment will haveconsiderable space and power generation impacts on the HSC.

For this example, let us consider that two design variants arebeing proposed. One variant only has enough power-generating fa-cilities for the connector mission, requiring extra generator modulesin the event of deployment on the hospital mission. The other vari-ant has surplus power for the hospital mission demands built in,creating additional space for beds. For the purposes of this example,the only significant differences between variants are the number ofbeds and installed electrical power.

Which design alternative should the decision maker choose? Forthis example, the relevant risk factor, Xt, is the number of HSCsthat will be configured for the medical mission at any given time. Ifwe letnbe the existing HSC fleet size (of variant 1 type), and mbethe number of new HSCs being acquired, then it is possible tomodel the risk factor using the binomial distribution

px; nm

x

x1 nmx 9

whereis the probability of any one vessel being configured for the

medical mission. This is the real probability measure, P. It is furtherassumed that

1. Variant 1 has capacity ofc1 150 beds, and has complexitymetric, 1 1.

2. Variant 2 has capacity of c2 200 beds, and has complex-ity metric, 2 1.05 (because of added embedded powergeneration.)

3. The U.S. Navys utility functions for each design alternative, inthis theatre, are captured by

u1cx 1

1 eac1x 10

u2cx

c2

c1 12e

ac2x 11

wherea 1.7 103 andxis the volume of beds provided bythe fleet of HSCs. These are exponential utility curves, and arefrequently used in economics literature. The parametera con-trols the steepness of the utility curve, and is chosen somewhatarbitrarily for this example. This value ofameans that 15 ves-sels will provide approximately 98% of the maximum possibleutility of HSC medical capabilities. The U.S. Navys true uti-lity curve may be different. What is important for this exampleis that the utility curves capture the diminishing marginal uti-lity of bed capacity.

As can be seen in Fig. 6, variant 2 has the greater maximumutility. However, it also has negative utility when not in use, reflect-ing the increased cost and maintenance of a more complex systemthat is unused. The risk-adjusted measure,Q, for each variant is thenfound by

q1x nm

x

x1 nmx aeac1x 12

q2x nm

x

x1 nmx

ac2x

c1x eac2x 13

where it is necessary to renormalize each q i such that it integratesto one. Finally, it is possible to calculate the value (the expected

Fig. 4 Status quo game with suboptimal Nash equilibrium

Fig. 5 Game with option: game now has optimal Nashequilibrium

Fig. 6 Utility function for HSC hospital design alternatives



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utility), under the risk-adjusted measure, of the expanded fleet ofHSCs for each of the proposed variants

vi EQviciXt 14

where the value function is given by

vix uicix Uref;i 15

Uref;i EQi u1c1Xt 16

The reference point is, in other words, the Q-expected utilityof the starting fleet of vessels. Such calculations can be performedfor all values of 0; 1, and it is possible to find the criticalprobability,, at which the decision of which variant to choosechanges. This is shown in Fig. 7, for the case ofn 0 and m 1,where the critical point is approximately 50%.

This means that if it is believed that the probability of an HSCbeing configured for the medical mission at any given time is lessthan 50%, then variant 1 should be chosen (the variant without thepreinstalled generators). However, if it is believed to be greater than50%, then variant 2 should be chosen.

How the value of changes as a function of total fleet size isshown in Fig. 8. Of course, this surface will change based on theshape of the utility curves. The results of this demonstration do not

apply to any actual HSC program. The results of this highly sim-plified example are intended solely for demonstration purposes.However, it is possible to use this example to make important ob-servations about the PB-ROA valuation framework. For instance,within the PB-ROA framework, it is possible to analyze how onesdecisions regarding flexible assets might vary with initial resources,the number of assets being proposed to acquire, and assumptionsabout operating conditions.

Since one of the original contributions of this framework is theinclusion of loss aversion through prospect theory, then it is inter-esting to compare the results from PB-ROA to what would be hadfrom a traditional expected utility approach. One major differencebetween the approaches is that expected utility methods use the realprobability measure, P, instead of the risk-adjusted measure. In anexpected utility approach, all risk aversion is assumed to be cap-tured in the shape of the utility curve. A cross section of the surfacein Fig.8is taken for the case ofn 0 (current HSC fleet size equalto zero). Figure 9shows how the critical value ofvaries depending

on the number of HSCs acquired. Both methods result in the samegeneral trend; the critical value decreases exponentially as the fleetsize increases. However, the threshold value is always higher forPB-ROA by a magnitude of 310%. The difference between thecurves reflects theloss premiumthat decision makers require. BothPB-ROA and expected utility approaches capture aversion to uncer-tainty. But only PB-ROA, by its use of prospect theory to adjust theprobability measure, captures the added concern of loss aversion.

To simplify future analyses, the critical reader may suggest thatsome margin simply be added to the results of an expected utilityanalysis to account for loss aversion. Since PB-ROA is a more com-plicated analysis framework than expected utility, such an approachmight be appealing if it were possible. However, PB-ROA does notalways result in the same trend as expected utility analyses, as it didin this example. In cases with severe loss aversion, PB-ROA andexpected utility may yield divergent results. To illustrate this pos-

sibility, the analysis of the hospital variant of the HSC will be re-peated with a different assumption about how vessels are deployed.

In the preceding results, it was assumed that vessels may be de-ployed on the medical mission on an individual basis. In this way,the relevant risk factor was the number of vessels needed at anygiven time, which was modeled by a binomial distribution. Let thisassumption be known as the variable deployment assumption.Forthe sake of illustration, let us now assume that vessels must be de-ployed together, regardless of the fleet size. This assumption will beknown as the all-or-nothing assumption.Under the all-or-nothingassumption, the relevant risk factor is reduced to a Bernoulli randomvariable; either the fleet is deployed on the medical mission or itis not.

If the real probability of deployment is p, then to get therisk-adjusted measure, first define intermediate variables for each

variant

tu1 p u01c1nm p a e

ac1nm 17

td1 1 p u010 1 p a e

0 18

Fig. 7 Expected utility of HSC fleet for the medical mission;n0, m 1

Fig. 9 Comparison of the variation in probability decisionthreshold, , with fleet size between PB-ROA and an expectedutility approach, for n0

Fig. 8 Variation in probability decision threshold, , with fleetsize using PB-ROA


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and

tu2 p u02c2nm p a

c2

c1eac2nm 19

td2 1 p u020 1 p a

c2

c1e0 20

where the superscriptu denotes when the medical capabilities areutilized, and d denotes de-utilized. Then, the Q-measures for eachvariant become

q1 tu1tu1 t

d1

21

q2 tu2

tu2 td2

22

Under the all-or-nothing assumption, the decision threshold ap-proaches 100% asymptotically in bothnandm, as shown in Fig.10

(left). What is significant to note is that this positive asymptotic re-lationship does not express itself in a standard expected utilityanalysis. The probability decision threshold according to an ex-pected utility analysis is shown in Fig. 10 (right). In this case,the threshold increases approximately linearly in n, and is slightlyquadratic in m.

To highlight the differences between the standard expected util-ity method and the PB-ROA framework in more detail, Fig. 11shows the cross section of the surfaces in Fig. 10 for the case ofn 0. The two approaches no longer result in the same trend, asthey did under the variable deployment assumption. In fact, the re-sults are completely divergent.

The explanation for this is that the all-or-nothing assumption hasgreater perceived loss than the variable deployment assumption. Inthis example, loss is the perceived loss of underutilizing an asset.The perceived loss is even greater when underutilizing the upgradedvariant. Under the variable deployment assumption, any upgradedvariants in the fleet could be deployed first in order to speed-uprecoupment of the investment in the upgrades. For example, iftwo ships are required in 2015 on the medical mission, and the fleetconsists of four variant 1 vessels, and two variant 2 vessels, then thevariant 2 vessels could be deployed to exploit their upgrades. Ifthree vessels were required, they could be augmented with anothervariant 1 vessel. In other words, it is possible to increase utilizationof the upgraded variants under the variable deployment assumption,

Fig. 10 Variation in probability decision threshold using PB-ROA (left), and expected utilitytheory (right), under the all-or-nothing assumption

Fig. 11 Variation in probability decision threshold over m forPB-ROA and expected utility methods, n0

Fig. 12 A hypothetical decision weight in prospect theory, from [17] (left) and therisk-adjusted measure, q, for variant 2 (n0, m1) from PB-ROA (right)



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which will mitigate some of the effects of loss aversion. However,under the all-or-nothing assumption, such mitigation is impossible.If the fleet is not deployed on the medical mission, the perceivedloss is high because allof the vessels are underutilized simultane-ously. It is not the point of this paper to argue the validity or fidelityof either the variable or all-or-nothing assumptions. But they areimportant from a research perspective because they highlight a sig-nificant difference between the PB-ROA framework and existingexpected utility methods. One reason PB-ROA has such potentialbenefit as a decision tool is because the implications of loss aversion

may not be known a priori.It is plain to see that these two methods may lead to drasticallydifferent conclusions. This is especially true for applications whereloss aversion is prevalent. Finally, to highlight the similarity withprospect theory, Fig.12compares the shapes of the weighting func-tion from prospect theory with the PB-ROA framework (under theall-or-nothing assumption). The differences at the endpoints arisefrom the constraint in PB-ROA thatq must be a true probabilitymeasure.

5 Conclusion

This paper presents a novel framework for evaluating flexiblenaval assets called prospect theory-based real options analysis(PB-ROA). The framework abstracts the principles of real options

analysis (ROA) to suit a wider variety of applications in the navaldomain than could previously be accommodated. Rather than beingconstrained to any physical instantiation of flexibility, the full po-tential of this new framework is in viewing flexibility as having thefreedom to make a choice and providing the mathematical means toexpress the value of having that freedom. With PB-ROA, analysesare no longer constrained to expressing value in terms of currency,which the authors believe could enable more meaningful analyses ofengineering tradeoffs where cost is not the only consideration.Although the framework is motivated by challenges in naval engi-neering, the authors believe PB-ROA may be of interest to otherfields where there is considerable value in flexibility that is not as-sociated with cash flows, such as urban planning, hospital design,and public works projects.

A simplified example was given to demonstrate the process of

using PB-ROA. The example evaluated the relative worth of twocandidate designs for a theoretical high-speed connector (HSC)vessel. PB-ROA was used to elucidate the conditions under whicha particular design is preferred, based on the value of the optionto convert to a medical response mission. The example also high-lighted important differences between PB-ROA and existing ex-pected utility methods. PB-ROA incorporates loss aversion, whichwas shown to have the potential to dramatically impact decisions.

This paper focused on what the authors call reactionary options,assets that react to changing exogenous demands. This representsone-half of the PB-ROA framework. The other half of the frame-work deals with naval game options, which were only briefly men-tioned in this paper. Naval game options represent those assets thatthe Navy can leverage to influence the behavior of other agentsin their environment, or exert a feedback effect. A full discussion

of the theory to handle such game options should be given infuture work.

Acknowledgment

This research has been generously supported by Kelly Cooperand the Office of Naval Research (ONR).

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