Expected utility theory as a guide to contingency allocation


As a means of compensating for the uncertainty associated with projects, contingency (in terms of both cost and time) is generally included in an initial project estimate. Cost contingency, also referred to as management reserve or allowance, is the amount that is added to a project’s cost estimate to allow for additional expenses that may likely be incurred during the project’s lifetime (such as those that result from an incomplete design at the time the project cost estimate is established). Time contingency represents the time that is added to the project’s anticipated finish date to guarantee that project’s published finish date can be achieved with some reasonable certainty. Assuming that the technical challenges of the project can be met and thus can not (or will not) jeopardize successful project completion, the remaining risks to a project are over-budget and/or late completion. Both of these risks can be mitigated (or eliminated) with a proper contingency allocation strategy and exacerbated by an improper one. In some sense then, contingency management is one of the most important aspects of good project management.

Yet, despite this importance, the cost contingency allocation problem has been sparsely addressed in academic literature. While a plethora of algorithms exist to assist project managers in assessing the health of a project and estimating its final cost, little exists in the area of optimized contingency allocation over time. The existing research (as it relates to cost contingency) is dominated by risk-based methods by which to establish a cost contingency reserve. In fact, only one mention of the importance of establishing an allocation plan has been found in existing literature. Those authors (Noor & Tichacek, 2004, p 3) propose the development of time-phased risk profiles, from which a time-phased contingency allocation profile can be developed. In this paper, we explore the relationship between expected utility theory and the contingency allocation challenge and introduce a utility-based approach for the development of a long range contingency allocation plan.

All projects follow a similar sequence of activities that begins with feasibility or conceptual design studies. These are followed by detailed design efforts. Once the designs are finalized, fabrication or construction commences. The project finishes with installation, testing, commissioning or operations activities (Morris & Hough, 1987, p5). These can be separated into two distinct stages, the ‘creativity’ stage and the ‘execution’ stage. The ‘creativity’ stage is characterized by innovative, risk-seeking behavior. Once the designs have been finalized, the project is required to ‘freeze’ the existing configuration and proceed with construction activities. After this point, the designs must be adhered to, not improved; this is considered the execution phase of a project and can be viewed as risk averse.

One of the more heavily researched areas in the project management community is that of project failures (see, for example, (Morris & Hough, 1987, p7-13). Scope creep and delayed finalization of requirements are two of the more frequently mentioned contributors. When these occur, the duration of the ‘creativity’ stage is increased and the start of the ‘execution’ stage is delayed. In practice, it is difficult to change the focus and behavior of a project team from a free-thinking, exploratory organization to one that is focused on execution of the project (as it stands in the current design). The utility-based contingency allocation plan proposed in this paper links these behavioral attributes to allocation decisions, providing a plan that offers appropriate decisions at appropriate times.

Related Research

While other researchers (Riggs, Brown & Trueblood 1994, p521) have observed that a project can be described in terms of a utility function, to the best of our knowledge, none of them have associated the behavioral aspects of expected utility theory to a project’s contingency allocation decisions. Utility theory attempts to capture the behaviors of individuals (usually referred to as gamblers) in terms of the ways in which they make decisions. In general, decision making under uncertainty has been approached via expected utility theory (Wahlström, 1975, p. 483) and Morgan and Henrion (1990, p25) discuss the use of multi-attribute utility functions to help in the decision-making process. Piney (2003, p.28) applied utility theory to risk management and notes (in talking about a project manager’s goal to make a profit for the corporation) that, '‘it is in the project manager’s interest to focus on actions that will limit the chance of budget overruns, even at the expense of abandoning the opportunity for larger potential gains.’' Bard (1990, p756-764) tackled uncertainty in the selection of inter-related R&D projects with the use of multi-attribute utility theory. Riggs et al (1994, p523) claim that, '‘In many project management situations, the intent is not to maximize profit but rather to maximize some subjective preference for achieving technical/performance criteria, staying within cost/budget limitations, and meeting schedule milestones.’' They define a project in terms of these three attributes, as u(T,C,S), where T, C and S represent achievement of the technical, cost and schedule objectives and T’, S’ and C’ represent failure to meet them. These utility functions represent the decision maker’s project success preferences (as measured by the status of the project at completion) and hence, u(T,C,S)=1 represents complete success while u(T’,C’,S’)=0 represents utter failure. Using the Analytic Hierarchy Process, they generate weighted utilties which are used to evaluate different project options (e.g. competing design or technical approaches) and they select the alternative with the maximum expected utility.

Utility Theory and Observed Violations

Modern utility theory for decision-making under uncertainty was developed, independently, by von Neumann and Morgenstern. They postulated a set of axioms about ordering and preferences so that, if the axioms are obeyed, a decision maker will select the alternative that provides the highest expected utility (Keeney,82, p806, Schoemaker,80, p13). These axioms explore the preferences associated with different opportunities (or gambles) and highlight the mathematical properties that couple these preferences. Given these axioms, the individual’s attitudes toward risk can then determined by examining the utility of the expected value of the gamble and the expected utility of the value of the gamble (Fishburn, 1988 p11-15; Schoemaker, 1980, p13-27;,VonNeumann & Morgenstern, 1944, p15-29). In utility theory a ’sure thing’ is viewed in terms of a certainty equivalent and is the amount of money that a gambler would rather have for certain, instead of taking some risk. This represents the maximum amount that individuals are willing to pay for a particular gamble and has been described as the amount at which an individual is indifferent to a gamble. In the convex (risk-seeking) case, the worth of a gamble to an individual exceeds the expected value of the gamble. Conversely, in the concave case, the gamble is worth far less than its expected value.

Empirical studies have shown that expected utility theory often violated when decision makers are asked to make intuitive judgments (see, for example (Bernstein, 1996 p269-283; Fishburn, 1988 p11-15; Kahneman & Tversky, 1979 p264-267; Luce, 2000 p35-56, 68, 254)). The Allais Paradox is a frequently cited example which shows that individuals do not consistently make choices in the same way. Kahneman and Tversky (Kahneman& Tversky,1979, p 269) explore a number of cases in which observed behavior contradicts expected utility theory. They argue that these violations occur because individuals tend to view events in a relative sense, overweighting things they believe to be certain relative to those that are only probable. They demonstrate that a risk-averse individual, when offered choices with positive outcomes, will behave in a risk-averse manner; however, when faced with a choice of negative outcomes, that same individual will behave in a risk-seeking manner. That is, in the positive domain, decision makers are risk-averse, preferring a small sure gain over a larger gain that is merely probable. In the negative domain, one behaves in a risk-seeking way, preferring a loss that is merely probable over a smaller loss that is certain. Hull and Thomas (Hull & Thomas, 1973, p245) also comment on this idea of a negative domain, noting that some people have a utility function that is discontinuous at the origin. They attribute this to the adverse emotional reaction people have to the idea of negative cash flow.

A Multi-Attribute Project Utility Function

We propose a project utility function that incorporates technical superiority (the degree to which the project delivers state-of-the-art capabilities) as well as successful delivery of that technology (the degree to which the project is completed, on time and within budget). This proposed utility function captures the characteristics of a successful project highlighted by other researchers (see, for example, Morris & Hough,1987, p12; PMI,2000 p 65, 83, 95) and is consistent with that proposed by Riggs et al. Given the existence of a project utility function, a meaningful contingency allocation decision strategy is one that maximizes this utility. Each contingency decision is evaluated in terms of its contribution to the technical, cost and schedule dimensions of the project. In these types of complex value problems in which the consequence of a decision cannot be described by a single attribute and the decision maker is evaluating ‘trade-off’s rather than explicit changes, multi-attribute utility theory provides a way of measuring outcomes (Keeney & Raiffa, 1993, p15). Even though attributes may not be comparable, a decision maker wants to chose the particular combination of attributes that maximizes his or her utility. As an example, suppose contingency request r1 is submitted. This request affects the technical attribute T and results in a consequence t1, affects the cost attribute C with a consequence of c1 and affects the schedule attribute S with a consequence of s1. The decision maker wants to be sure that, if request r1 is approved, that the utility gained from this request exceeds the utility gained from all other n choices. That is, using Riggs’ notation, the decision maker wants to be sure that,


Since there are two types of contingency, cost contingency (measured in dollars) and schedule contingency (measured in days), there are two types of contingency decisions, those that call upon cost contingency and those that draw down the schedule contingency reserve. Requests for contingency originate in response to three very general conditions, called risk sources, that threaten a project’s cost, technical or schedule baselines. These can be classified as technical risks, new opportunities and unforseen events.

An additive utility function, based on these decisions, is the desired vehicle for the contingency allocation model. Additive utility functions must satisfy utility preference and utility independence criteria. Therefore, one must be able to translate these risk sources to quantifiable utility functions that satisfy these independence conditions. To do so, we introduce a novel form of a project’s utility. This utility is comprised of a performance component (that reflects the technological superiority), a schedule component (that represents on time completion) and a scope component (that represents the degree to which the original project specifications are adhered to over the life of the project). This latter component reflects the within budget completion criteria of a successful project. The utility components are described as follows,

Performance Utility (U1), A project’s performance utility is defined by the degree to which it exceeds the technical parameters promised to the sponsor. This includes performance improvements due to incorporation of new technology or enhanced reliability due to actions such as the purchase of additional spares or incorporation of redundant systems. The decisions that affect technical utility are considered opportunities.

Schedule Utility (U2), A project’s schedule utility is defined by the degree to which it finishes on time. Schedule utility is impacted by risks as well as opportunities.

Scope Utility (U3), A project’s scope utility is defined as the degree to which the promised baseline scope is delivered to the sponsor. If a project has insufficient cost contingency to respond to all realized risks, then one alternative is to modify the attributes of the scope of work that is delivered. As an example, one could reduce or eliminate non-essential items that would have enhanced the project but are not critical to its completion (such as reducing testing time or eliminating redundant systems) or substituting lower cost alternatives that meet specifications but are not those originally included in the project plan. The decisions that affect scope utility are responses to realized risks.

Total Project Utility (UP), A project's total utility is defined in terms of the cost and schedule contingency decisions that represent the project's response to risks that are realized and opportunities that arise and is represented by the sum of the component utilities described above. Let xit and yjt represent the cost contingency dollars allocated in period t towards type i risks and type j opportunities respectively (where the opportunities and risks are defined as in Exhibit 1). Let zt denote the schedule contingency days allocated in period t. Then, a project’s utility at period t can be described by


where w1, w2 and w3 are scaling factors assigned to the individual utilities (with w1+ w2+w3=1 and wi 0 for i=1,2,3). The relationship between the different types of decisions and the individual component utilities (for a single period t ) is shown in Exhibit 1.

Project Contingency Allocation Decisions

Exhibit 1 Project Contingency Allocation Decisions

Actual Project Data Viewed from Utility Perspective

All projects have a similar spending (or baseline) plan. During the ‘creativity’ stage, most of the expenses incurred are associated with labor costs. Therefore, the plan for this period is relatively stable. Once the ‘execution’ stage begins, contracts are being awarded and subcontractor effort is being added to the project. It is in the beginning of this phase that one sees large increases in costs from period to period. Towards the end of this phase, only testing or commissioning or operations activities remain and the costs per period once again stabilize. This creates a cumulative profile which is ‘S’ shaped.

Cumulative plan data from an ongoing project is plotted in Exhibits 2 and 3. At inception, the project is 0% complete and, at the finish date is 100% complete. In Exhibit 3, project percent complete is evaluated against total project dollars. From this approach, one can view a project’s progress (the percent complete) as a measure of the utility of the dollars spent on a project (U($)).

Exhibit 2: cumullative project plan

cumullative project plan

Exhibit 3: Project Percent Complete as a Function of project Costs

Project Percent Complete as a Function of project Costs

From the utility perspective, convex functions represent risk-seeking behavior and concave functions signal risk-averseness. Therefore, the profiles shown in Exhibit 2 and 3 suggest that risk-seeking and risk-averse behaviors are both present in a project and that these behaviors occur during different project phases (that is, over time). The ‘creativity’ stage is convex while the ‘execution’ stage is concave. The ‘creativity’ phase, Phase I, is focused on the quest for technical superiority and hence is naturally risk-seeking. The decisions that are made in this phase are those that extend the state-of-the-art frontier. The initial phases of a project are largely comprised of design activities. Enthusiasm is high, the contingency pool appears unlimited and the project finish date is many years into the future. While in the risk-seeking phase, innovation and exploration are not only desired, they are encouraged. During this period, the conceptual designs that formed the basis of the original project proposal are `fleshed out’ in more detail. Numerous (and possibly parallel) paths are explored to ensure the the project is comprised of only the ‘latest and greatest’ technology and can offer ‘state of the art’ performance. In fact, it is not unfair to say that all of the technical creativity and innovation of a project occurs while the design is being generated. Karlsen and Lereim (2005, p24) remarks that “the possibility to influence and change the project is highest at the early stages because conceptual decisions have not been made or taken effect”. From a technological perspective, the remainder of the project is just execution of the decisions made in the design phase. In general, requests for contingency to support pursuit of these new and innovative concepts are rarely rejected.

As the detailed design is completed, procurement of equipment components and physical construction commences. This ‘execution’ phase, Phase II, focuses on within budget and on-time completion and is more conservative. Uncertainty dimishes as actual costs for equipment and construction (vs the estimated value) become known and flexibility (and the ability to respond to design changes) decreases. This marks the transition to the risk-averse phase. This transition is at the intersection of the convex and concave utility functions and is graphically represented by the shaded area in Exhibit 2. Changes or modifications to designs become more costly and mitigation strategies that could have provided alternative solutions (that is, provided an alternative that did not require a call upon contingency) tend to diminish with time. Incorporation of newer, possibly even greater, technology will never occur unless that technology can be incorporated at no additional cost and in the existing time frame. From a utility theory perspective, risk-averseness (in this project sense) can be represented by a conservative contingency allocation strategy. This approach is one that is focused on project completion (those requests that contribute to a project’s scope and schedule utility functions) and elects to preserve contingency as a hedge against the future rather than approve ‘discretionary’ requests. One characteristic of this second phase is an increasing demand for a decreasing supply of contingency. Once a project enters this phase, all changes have a cost impact since only two feasible alternatives remain, 1) add resources (money) to resolve the issue or 2) eliminate a portion of the project scope. The latter is a politically dangerous option that could jeopardize the entire project (and diminish the contribution of scope utility to a total project’s utility).

Data from the same project shows that this qualitative discussion of contingency allocation can be validated by project data. These data show that contingency allocation decisions are correlated with the state of the project (how much has been completed) and that failure to transition (in behavior and in action) from risk-seeking to risk-averse soon enough in a project could result in an insufficient contingency reserve to cover the uncertainties associated with the project’s remaining work. The percent of the contingency reserve that was allocated, relative to the progress on the project, is shown in Exhibit 4. By the time the project was 25% complete, 44% of its contingency had been allocated and only 25% of the original contingency reserve was still available at the 50% point.

Exhibit 4: cumulative Contingency Allocation

cumulative Contingency Allocation

Exhibit 5: Rate of Contingency Allocation

Rate of Contingency Allocation

The rate of contingency allocation, relative to work completed is shown in Exhibit 5. Three different slopes (rates of contingency allocation) are highlighted by the straight lines. The first segment represents the greatest rate of change and occurs during the project start up and preliminary design phases. In this period, the project’s utility is measured by technical superiority and the allocation of contingency dollars to state-of-the-art technologies can make a huge difference on the project’s ultimate capabilities. This is consistent with the characteristic of convex utility functions in which small changes have little impact on the utility but large changes make substantial differences. The second segment represents the detailed design effort as well as the beginning of construction and procurement activities. This is the transition phase. The last segment represents the completion of construction and installation efforts. In this phase, utility is defined in terms of project completion (delivery of the chosen technology, on time and on budget). One will note that, in this concave period, the allocation of large dollars no longer makes large differences and the project is more interested in ensuring success than pursuing additional technologies.

A Utility Based Contingency Allocation Model

The utility based contingency allocation model presented in this section links the risk-seeking and risk-averse behavioral attributes of a project to the allocation of contingency dollars. During the early phases of a project, contingency requests that enhance the project’s technical utility are highly valued. Since flexibility diminishes once procurement and construction activities have begun, technical utility has little value in the later phases of a project. During those later phases, contingency requests that improve the project’s likelihood of finishing on time become increasingly important. Scope preservation is important in both phases and it is the only utility that has the same form throughout the life of the project. These utlities can be explained in terms of contingency allocation decisions.

The impact of a contingency decision on performance utility can be determined in a number of ways. One approach is to assign a weight to each technical component of the project based upon some qualitative criteria (e.g. performance or criticality) and then use those weights to select the option that offers the greatest impact on the overall performance utility. Another is to assume that the impact can be determined solely in terms of contingency dollars where the adage ‘you get what you pay for’ is true and ‘more expensive is better’. Imagine that one is buying a computer. For a low price, one gets standard RAM and standard performance. A computer cluster is much more expensive yet provides exponentially more performance. Given these assumptions, the allocation of contingency dollars in Phase I to technical opportunities will substantially (nonlinearly) increase a project’s technical utility. In Phase II, the project has lost most of the flexibility required to incorporate technical innovations into the baseline and allocation of contingency dollars to technical opportunities will have no impact on the performance utility. However, recalling Exhibit 1, contingency decisions that enhance reliability also contribute to a project’s performance utility. This effect is linear and occurs in both project phases. While in Phase I, the project finish date appears to be infinitely far into the future and the allocation of schedule float has minimal utility. Conversely, in Phase II, additional days of schedule float have a great deal of value, however only up to some upper bound. While it is theoretically possible for one to use all of a project’s contingency to purchase schedule days, practical realities such as the number of shifts in a day and the duration over which project personnel can continue to work safely and productively in a multiple shift scenario, impose an upper bound on this utility. Scope utility is a linear function in which each contingency dollar allocated retains one dollar of original scope.

The utility-based contingency allocation model (UM) maximizes the expected utility of the contingency allocation decisions. Exhibit 1 identified the decision variables and the optimization problem is subject to applicable policy and business constraints (such as the amount of contingency and amount of funding remaining at the end of each period). Due to the limited length of this paper, details of the exact formulation are not included.

Empirical Analysis

Our proposed utility model (UM) was validated using data from an actual, ongoing project. In this analysis, the data were grouped into four periods, two in Phase I and two in Phase II. The project risk register provided a detailed list of potential contingency requests, by type and by period, over the life of the project. These values were assigned triangular distributions (as per the work of other researchers, e.g. (Bowers,94 ,p.12,Williams,92, p268)). Crystal Ball Monte Carlo simulation was used to generate 80 random sets of these parameters for each type of request for each period. A random probability of occurrence between 0 and 1 was assigned to each risk. The solutions from the UM formulation are compared with the results of a more typical, linear decision model (LDM) in Exhibit 6.

Total Contingency Dollars Allocated by Period

Exhibit 6, Total Contingency Dollars Allocated by Period

While both formulations allocated the same amount of contingency dollars over this planning horizon, the distribution of the dollars over period and over type differs. In both cases, the same optional schedule dollars were allocated. However, UM allocated these later than LDM, reflecting the increased utility of schedule float as a project nears the finish date. LDM allocated contingency dollars for new technology in periods 3 and 4, a decision that has little practical utility. The realized risk requests were completely satisfied in both formulations, for all periods. In order to determine which formulation is better, these plans were compared against 50 different, randomly simulated futures from the same risk register data base (all assumed to occur with 100% probability). The better plan is the one that is able to address more of the project’s realized risks. These data are shown in Exhibit 7. Over the critical period in a project (the last three periods), one can see that UM offers far superior results. When a realized risk occurs at the beginning of a project, there is time to pursue no-cost recovery plans. However, when risks are realized towards the end of a project, there is not enough time to pursue these alternatives and still finish the project on time. Hence, meeting the requirements of realized risks in the latter portion of a project is much more important than satisfying those that occur at the beginning of a project.

Percent of Realized Risks Satisfied by Contingency Allocation Plan

Exhibit 7, Percent of Realized Risks Satisfied by Contingency Allocation Plan


In this paper, we have shown that the behavioral attributes associated with expected utility theory are identically those exhibited on a project. Using that information we developed utility functions for the three core metrics of project success, technological excellence delivered on time and within budget. By recognizing that project behavior is dependent on the stage of the project, we proposed distinct utility functions for the project’s major phases. These utility functions are then maximized, subject to business rules and policy constraints, generating a long range contingency allocation plan which will be feasible, in an expected sense, over all uncertain futures envisioned. This not only provides the project manager with a detailed plan on which to base the irreversable contingency allocation decisions, the optimal solution is superior to a linear decision model with the objective to optimize the total contingency allocated.


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@2006, Barbara Thibadeau
Originally published as part of 2006 PMI Global Congress Proceedings-Seattle Washington



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