Incentive Schemes for Resolving Parkinson’s Law in Project Management *

,


Introduction
A project is a "temporary endeavor undertaken to create a unique product or service" (Project Management Institute 2013). As much as 30% of the world's economic activity is organized as projects (Hu et al., 2015), implying an annual value of around $27 trillion. Much of this activity is by the principal. They examine the performance of adjustable incentive contracts under unforeseeable influences. By contrast, our work considers any number of agents who are internal to the project company. Kwon et al. (2010) study a project-based system that consists of a project manager and a single contractor. They show that incentive contracts based on time, and some incentive contracts based on a fixed price, can coordinate the system; whereas, contracts based on a fixed price alone cannot do so. By contrast, our work considers multiple task operators within the project company. Kwon et al. (2012) consider an assembly-type project with a manufacturer and n ≥ 2 suppliers with one task each. The tasks are identical, have no precedence requirements, and earn the same fixed payment. They compare a traditional payment scheme, under which each supplier receives payment on completion of his task, with a delayed payment scheme, under which all payments are made only on completion of the last task. Chen et al. (2015) consider a similar problem, but with nonidentical tasks and a serial precedence network. They show that a contract containing a convex time-cost tradeoff dominates a fixed price contract, as measured by project schedule and profit. By contrast, the projects we study have variable incentive payments to task operators within the project company, and a general precedence structure.
This paper describes an IC mechanism to resolve Parkinson's Law and Student Syndrome for a project that is planned and executed by internal task operators under the critical path method. We show that our mechanism is group-strategy-proof. We also design an IC mechanism to resolve Parkinson's Law for a project that is planned under CCPM. We provide sufficient conditions on this mechanism such that the incentive payments received by all task operators are typically at least as much as those under the critical path method; moreover, the minimum guaranteed payment to the project manager remains unchanged. We develop an IC mechanism for multiple repeated projects planned under CCPM. Computational testing of our IC mechanisms reveals that both makespan and incentive payments are significantly reduced relative to the benchmark scheme most commonly used in practice, and our incentive mechanism performs effectively for multiple projects. All proofs appear in an Appendix.

Preliminaries
Section 2.1 describes the project network. Section 2.2 describes the information flow in the network. Section 2.3 describes the process by which the tasks are executed. Section 2.4 describes contractual and reporting requirements on the PM and TOs. Section 2.5 describes our model of project planning under CPM and CCPM.

Precedence network
A project controlled by a project manager (PM) consists of a number of tasks that are performed by task operators (TOs) who are employees of the project company. We initially assume that each TO operates one task, but we later relax this assumption. For distinction, we refer to the PM as "she" and to a TO as "he". The overall project network, including the precedence relationships among the tasks, is modeled as a directed acyclic graph G = (N, A), where the node set N represents events and the arc set A represents the tasks. We assume G is public knowledge. Denote n = |A|.

Private information about ready time and duration
We model uncertainty in the task durations as follows. We assume that, for each task j ∈ A, its duration ∆ j satisfies an unknown discrete probability distribution with support S j (0) = {a j , a j + 1, . . . , b j }, which is public knowledge. LetT be the project completion time calculated from task duration upper bounds {b j : j ∈ A}. At time t ≤T after task j is started, a more accurate estimate for the support of ∆ j , denoted by S j (t) = {a j (t), a j (t) + 1, . . . , b j (t)} ⊆ S j (0), may be obtained as information private to TO j. We assume support S j (t) is non-increasing in t in terms of set containment, reflecting that the process of executing the task can improve but not worsen the accuracy of the estimate for the task duration ∆ j . On the other hand, for each task j ∈ A, there is a time-dependent ready time r j (t) at which the TO is ready to execute task j, unless any of his predecessor tasks in Q j are incomplete. Note that r j = r j (0) is computed from task duration upper bounds {b j : j ∈ A}. Since the TOs work for the project company, the values of a j , b j and r j are set by the PM at the planning stage. Margraf (2018) summarizes six sources of information for doing so: work breakdown, historical data, analogous tasks, expert judgement, effort involved, and units of activity. However, for any t = 1, . . . ,T , knowledge of r j (t) before task j starts is private to TO j only.
It is possible that, at some integer time point t > 0 before task j starts, TO j may become ready earlier than r j (i.e., r j (t) < r j ) to execute his task j. In this case, he can choose to report to the PM an earlier integer ready timer j (t) with r j (t) ≤r j (t) < r j . Similarly, at some time point t > 0 after task j has actually started, the TO may have less uncertainty about the duration of his task and he can choose to report to the PM a reduced maximum integer durationb j (t) with b j (t) ≤b j (t) < b j . We provide incentives for TO j to make such reports, with the requirement that the reportedr j (t) andb j (t) values are binding. Thus, the agreed r j and b j values are updated byr j (t) andb j (t), respectively, from time t onwards.
For any j ∈ A, assume an updated reportr j (t) (respectively,b j (t)) is made. We say that report r j (t) (resp.,b j (t)) is truthful if, on the one hand, r j (t) ≤r j (t) <r j (t) (resp., b j (t) ≤b j (t) <b j (t)), wheret (0 ≤t < t) is the last time TO j reports a binding ready timer j (t) (resp., duration b j (t)) or no such report has been made (i.e.,t = 0 withr j (0) = r j andb j (0) = b j ); and on the other hand, the report is the smallest estimate without the risk of future violation of the binding requirement.
Starting with the initial setting ofr j (t) = r j ,ā j (t) = a j ,b j (t) = b j for all j ∈ A and all t ∈ {0, 1, . . . ,T }, the project proceeds for t = 1, . . . ,T with possible reported updates {r j (t),b j (t)}. Then bothr j (t) andb j (t) are nonincreasing, which we will further explain in detail in Section 2.4.

Project execution
The accessible time ξ j (t) of task j at time t is the updated event time at which all predecessors Q j of task j will have been completed, based on the latest reported information, i.e., The accessible time of task j is the time at which task j can start, provided TO j is ready, and is clearly nonincreasing in t since bothr k (t) andb k (t) are nonincreasing for all k ∈ Q j . This definition identifies an information asymmetry, where the PM knows the accessible time of task j, but does not know if task j is ready until TO j reports so.
Recall that, for each event i ∈ N , we use P i ⊆ A to denote all tasks contributing to event i. Denote byC i = max j∈P i {r j + b j } the earliest time in the planned schedule, using task duration upper bounds as estimates, by which all tasks in P i will have been completed, and let (1) Equation (1) defines C i as the time at which milestone i is achieved during project execution. During the execution of task j, TO j may improve his estimated task duration from b j to b j (t) < b j . Further improvements may occur later. However, as a result of Parkinson's Law, TO j may decide not to report an earlier ready time or a shorter task duration, and this typically worsens project performance. Another situation, also within the scope of Parkinson's Law but behaviorally different, is that a TO works more slowly than necessary. However, from the perspective of the PM and the other TOs, these situations are mathematically identical, since successor tasks cannot start until the current task is reported as completed.

Contractual and reporting requirements
In addition to the precedence constraints in G = (N, A), we impose the following requirements.
1. Contractual Constraints: The initial task ready time r j = r j (0) and task duration b j = b j (0) are contractually binding on the TOs. Since the TOs are employees of the company, these upper bounds are agreed at the planning stage.
2. Certification of Completion: A TO cannot fake the completion of his task, which is verified by an independent expert. This is common in, for example, environmental projects (Alberini et al., 2005). Further, in any project, certification of completion is important for internal quality control, accurate allocation of costs and performance incentives, accurate reporting of progress to the client, and initiation of milestone payment requests.
Initially, the PM has only the original information r j (0) and b j (0). Before the whole project is completed and at each time t = 1, . . . ,T , we define task set A(t) = {j ∈ A :r j (t − 1) > t} and let the following information exchange take place: 1. Completion Reports. Each TO j is required by contract to report the completion of task j no later than time pointr j (t − 1) +b j (t − 1). This requirement is common in environmental projects (Alberini et al., 2005).
2. The PM announces updated accessible times ξ j (t − 1) to all those TOs j who have not made their Completion Reports.
3. Each TO j, who has not made a Completion Report, has the opportunity to make an update report ofr j (t) (if j ∈ A(t)) andb j (t) based on his private information of r j (t) and b j (t). If he does so, the new ready time and duration bound are binding and satisfy the following By reporting a new ready timer j , TO j accepts the requirements (a) to start task j no later than timer j (t), and (b) to complete task j no later thanr j (t) +b j (t), by the Contractual Constraint. If no update is reported, then the existing values are used:r j (t) =r j (t − 1) and/orb j (t) =b j (t − 1).

Modeling CPM and CCPM
We define project planning under CPM, consistently with Wysocki (2009) and Klastorin (2012), as follows. Historical data is collected from all the task operators within the project company. Based on this data, estimates are formed of an optimistic duration and a pessimistic duration for each task. Even for tasks with limited data, the project management literature (Golenko-Ginzburg, 1988;Kerzner, 2009) recognizes the ability of a project manager to do this. In a CPM system, TOs often overestimate the time that their tasks will take (Goldratt, 1997), in order to avoid the appearance of mismanagement and avoid blame if the task duration is longer than expected. A study by Hill et al. (2000) finds that, for small software development tasks, the estimated durations are more than 40% greater than actual durations. Bukhari and Malik (2012) find that overestimation is more frequent than underestimation. Thus, we model CPM using the 100th percentile, i.e., the smallest duration to which the TO can commit with complete confidence. Hence, using the pessimistic task durations b j , j = 1, . . . , n, the PM applies CPM to compute a deadline for each task. The project starts at time 0 with a deadline ofT , as defined in Section 2.2. Also, the ready time of each task is computed as the latest completion time of any of its predecessor tasks. Based on information collected from all the TOs within the organization, the PM informs each TO about the initial ready time and deadline of his task.
We define project planning under CCPM, consistently with Goldratt (1997), as follows. Similar to CPM, estimates of an optimistic duration a j and a pessimistic duration b j are formed for each task. In a typical implementation of CCPM, a mid-range estimate of task duration, such as the median estimate, is used to limit unnecessary slack in the project (Leach, 1999;Herroelen and Leus, 2001). Lacking a probability distribution for task duration, we use m j = (a j + b j )/2 as a proxy measure for the median. The PM makes TO j aware that the project is planned using the m j value. Recall that, under CCPM, task deadlines are not defined. Leach (1999) notes "Critical chain project plans only provide dates for the start of activity chains and the end of the project buffer".

Formal problem definition
We formulate our problem as one of mechanism design. We omit notational dependencies where they are clear from context. Let Π = (G, S, Z) be a given project, where G = (N, A) is the set of precedence constraints between tasks j ∈ A represented by task owners (TOs), S = (S j (0) : j ∈ A) provides the support S j (0) = {a j , a j + 1, . . . , b j } for an uncertain duration ∆ j of each task j ∈ A, and Z = (Z i : i ∈ M ) specifies contractual payment Z i to the PM for achieving each milestone i ∈ M ⊆ N . We wish to design a mechanism M = M(Θ), where Θ is a set of fixed parameters in a specified subspace X ⊂ R k + and k is some positive integer, for the PM to receive from every TO j ∈ A an updated report of his task ready timer j and task durationb j . The objective is to minimize the makespan ζ = ζ(z) of the project Π, where the TOs' reports z = ((r j ,b j ) : j ∈ A) are based on their private information z(t) = ((r j (t),b j (t)) : j ∈ A) as described in Section 2.2, and are subject to the constraints described in Section 2.4. The mechanism M(Θ) specifies the amount p ij (z, Θ) of payment by the PM to every TO j ∈ A, subject to the constraint that j∈A p ij (z, Θ) ≤ Z i for each milestone i ∈ M . As discussed in Section 2, this problem reduces to designing a mechanism M(Θ) that is truthful. That is, under such a mechanism, the TOs each maximize their total milestone payments by reporting the true values, as defined in Section 2.2, of their ready times and task durations.
Let ζ min (Θ) be the minimum makespan of project Π under mechanism M = M(Θ). It is clear that the value ζ min (Θ) is a constant ζ min , independent of Θ, as long as M(Θ) is truth-ful, which we establish below for all possible Θ ∈ X. As a further step, we explicitly choose the values of parameters Θ * ∈ X, so that the total of the PM's retained milestone payment, i∈M Z i − j∈A p ij (z, Θ) , is maximized at Θ * . Thus, under truthful mechanism M(Θ * ), the PM minimizes the project makespan to ζ min and, subject to achieving this objective, she retains as much of the milestone payments as possible.

Mechanisms for CPM
This section describes our two mechanisms for a project that is planned using CPM. Section 3.1 describes our mechanism, and establishes that it is incentive compatible. Section 3.2 designs a more general "weighted" scheme to incentivize TOs who own multiple dependent tasks. Section 3.3 shows how optimal parameters for both our incentive schemes can be found. The weighted incentive scheme is evaluated computationally in Section 3.4 against a benchmark scheme.

Incentives
Assume for this subsection that a TO owns one task. We consider a milestone payment from the project client, denoted by Z i for any fixed i ∈ M , and how to share it among the PM and the TOs. We let where ρ i > 0 is a constant. Observe that (C i − C i ) > 0 represents the amount of improvement in milestone completion time. Then, ρ i converts this improvement into a contractual bonus paid by the project owner, and can be viewed as a per-unit rate of payment for performance at milestone i. However, any monotonically increasing function f (C i − C i ) which ensures that earlier project completion is better suffices. A task may contribute to several milestones, in which case its shares of milestone payments are calculated separately and sum to its total payment. We describe our incentive scheme that allows flexibility based on chosen parameters λ ij and µ ij . Here, 0 < λ ij < 1 converts both a saving in task duration and an earlier ready time into a positive amount of slack, as defined below, for task j of milestone i. Also, µ ij > 1 specifies the relative importance to slack of savings in ready time and savings in duration and ensures that there is no trade-off between the two. For each task j ∈ A, our incentive scheme incentivizes earlier submission of reports of achievable reduced ready timer j (t) and task durationb j (t). For each j ∈ A, let T j ∈ {1, . . . ,T } be the reported completion time of task j, where T j ≤ r j + b j , and denote by r j =r j (T j ) and b j =b j (T j ) the last reported, if any, ready time and duration of task j, respectively. Let τ r j and τ b j denote the corresponding times, if any, when r j and b j , respectively, are reported. Thus, We use f j = (τ r j + τ b j )/(2r j + b j ) as a metric for early reporting, where 0 < f j ≤ 1.
(λ, µ)-Scheme 1. For any milestone i ∈ M with Z i > 0, fix parameters λ ij and µ ij with 0 < λ ij < 1 < µ ij . Compute the positive slack of any task j ∈ P i as 2. Denote D i = k∈P i (µ ik r k + b k ). Make an incentive payment on task j ∈ P i of where ω ≥ 0 is a chosen parameter that controls the importance of an early report relative to an early start or early completion. For example, ω = 0 indicates that it is negligible, whereas ω → +∞ indicates that it is dominant. Also, σ i ∈ (0, 1] is another chosen parameter which we discuss in detail in Section 3.3.
3. The PM keeps the remainder of the payment Z i for milestone i: End of scheme Remark 1. For any fixed milestone i ∈ M and fixed task j ∈ P i , parameter λ ij controls the sensitivity to slack s j , and hence to the amount of incentive based on it, of an earlier ready time r j and a reduced task duration b j . Also, parameter µ ij controls the relative importance of an earlier ready time to a reduced task duration.
A numerical example of the (λ, µ)-Scheme appears in Section A.1 of the Appendix.
Remark 2. In the (λ, µ)-Scheme, the PM informs TO j ∈ P i of the values of λ ij and µ ij before project execution.
The next theorem establishes bounds on the payments received by the various parties.
Theorem 1. For each milestone M i for which Z i > 0, with the incentive payments defined in the (λ, µ)-Scheme, the PM receives a fraction of the milestone payment σ i Z i between κ i λ min and λ min i = min j∈P i λ ij and λ max i = max j∈P i λ ij , while the TOs collectively receive a fraction of the total payment σ i Z i between 1 − (λ max i + ω)/(1 + ω) and 1 − κ i λ min i /(1 + ω). Further, the (λ, µ)-Scheme is budget balanced.
Theorem 1 establishes how the choice of incentive parameters modulates the shares of the available incentive received by the various parties. These shares can be adjusted by the PM, based on the employment incentives of the TOs. We now provide the main result of this section.
Theorem 2. The (λ, µ)-Scheme is incentive compatible for TO j to report his "true values" (as defined in Section 2.2) of r j and b j as early as possible, for any milestone i ∈ M , and j ∈ P i . Remark 3. The use of two parameters λ ij and µ ij , instead of one, separates the reports on r j and b j to avoid strategic trading off of the two by TO j. Further, these parameters can be used to adjust the relative payments received by each TO j and the PM, for any fixed milestone i ∈ M . Moreover, these values can be varied across different milestones i ∈ M .
Remark 4. In the (λ, µ)-Scheme, Steps 1, 2 and 3 each require O(n) time, for each i ∈ M . Hence, the overall complexity of the scheme is O(nm) time.

Extensions
In this section, we study how the result in Theorem 2 can be generalized to allow the PM to assign multiple tasks to the same TO, as occurs in many organizations. The next result provides such a generalization, under a specific condition.
Corollary 1. The (λ, µ)-Scheme is incentive compatible, even if the PM assigns multiple tasks to a TO, provided no such task is a predecessor of another.
From Corollary 1, if several tasks without dependence among them are assigned to the same TO, there is no advantage to the TO to make decisions for them jointly. This gives the PM substantial flexibility in assigning tasks to owners. However, the following example shows that the (λ, µ)-Scheme fails when a TO owns several tasks and there is dependence among them.
Example 2. Consider events {s, 1, · · · , n, t} and tasks {(s, 1), (1, 2), . . . , (1, n), (2, t), · · · , (n, t)}. All tasks have an execution time in [a, 2a]. Suppose that a TO owns all the tasks and realized durations are 2a for all tasks except (s, 1) which requires time a. In this case, he receives an incentive payment ofᾱ · a as a result of being truthful. However, by falsely reporting all task durations as 2a except for tasks (1, t), . . . , (n, t) which are reported as a, he receives an incentive payment of n ·ᾱ · a. This false reporting is possible since he can report the completion of task (s, 1) as 2a, but start the tasks of type (i, t) right after the true completion time, a, of (s, 1). Hence, false reporting provides an increase in incentive payment to the TO. However, it may also increase the project makespan since other project tasks that are successors of task (s, 1) cannot start until time 2a when the completion of that task is reported falsely late.
To resolve this problem, we describe a generalized incentive payment scheme, the Weighted (λ, µ)-Scheme. Intuitively, we provide larger incentive weights to the tasks with more successors, so that the possibility of increasing incentive payment due to dependence between the tasks is offset by the weight each task carries. Weight Assignment For each task j ∈ A, let V j denote the set of dependent tasks of j. That is, if j = (j tail , j head ) ∈ A, then V j is the set of tasks to the tail of which there is a directed path from the head j head of j. We define a weight function as follows: if V j = ∅, let w j = 1; otherwise, let w j = 1 + k∈V j w k . Observe that each task j ∈ A is assigned a weight that is strictly greater than the sum of the weights for all its successors in V j . Since the tasks form a directed acyclic graph, the weights are well defined.
Weighted (λ, µ)-Scheme 1. Compute weights {w j : j ∈ N }, and inform each TO of the weight(s) of his task(s). For any milestone i ∈ M with Z i > 0, fix parameters λ ij and µ ij with 0 < λ ij < 1 < µ ij . Let . Make a payment to TO j ∈ P i as follows: where s j , ω and σ i are as in the (unweighted) (λ, µ)-Scheme.
2. The PM keeps the remaining positive fraction of σ i Z i (see Section 3.3).

End of scheme
A numerical example of the Weighted (λ, µ)-Scheme, appears in Section A.5 of the Appendix. The next result establishes that the Weighted (λ, µ)-Scheme is incentive compatible, even if a TO owns some tasks that are predecessors of others.
Theorem 3. Assume that we are given a directed acyclic graph G = (N, A) of events and tasks, and an arbitrary assignment of tasks to TOs. Then, the Weighted (λ, µ)-Scheme is incentive compatible for any TO to report as soon as possible the "true values" (as defined in Section 2.2) r j and b j of his jobs j.
A mechanism is strongly group-strategy-proof if it induces a noncooperative game in which no group of players can collude to misreport their private information in a way that makes at least one member of the group better off without making any of the remaining members worse off. The following corollary about the strong group-strategy-proofness of the Weighted (λ, µ)-Scheme follows immediately from Theorem 3.
To see the correctness of the above corollary, assume to the contrary that there is such a colluding group in the reporting game played by all TOs. Consider another reporting game in which all tasks of the members of the colluding group are assigned to a single TO, while assignment of the other tasks remains the same. Then in this new reporting game, the "new" TO can misreport his private information to make himself better off, which contradicts Theorem 3.
The following result is evident.
Corollary 3. Since the TOs are incentivized both to start and finish their tasks as soon as possible under the (λ, µ)-Scheme and the Weighted (λ, µ)-Scheme, Theorems 2 and 3 also resolve Student Syndrome.
In our work, we include here both the weighted and unweighted versions of our (λ, µ)-Scheme. The unweighted version is easier to understand and implement. However, the weighted version enables the PM to deal with more general and flexible assignment of tasks to the TOs, while still retaining incentive compatibility. This additional flexibility can be used both to alleviate resource bottlenecks in the project, and to implement secondary criteria in rewarding the TOs.

Optimal choice of scheme parameters
From the perspective of mechanism design, the PM is the principal who designs the IC mechanisms. The PM derives disutility from a longer makespan, as a result of specifications in the contract with the project owner. Hence, her primary objective is to resolve the effect of Parkinson's Law on the project in order to minimize the makespan. As explained in Section 2.6 above, the PM has a secondary objective of retaining as much of the milestone payments as possible, while providing them in sufficient amount to incentivize the TO employees to report truthfully. That is, among all the payment schemes that achieve the minimum project makespan, she has a preference for one that retains as much of the resulting milestone payments as possible. Such a payment scheme can be found by optimizing the payment scheme parameters.

Optimal parameters in the (λ, µ)-Scheme
For simplicity but without significant modification to the problem, in the (λ, µ)-Scheme, we set λ ij = 1/2 for any milestone i and any TO j and set ω = 0 (see the ranges and interpretations of these parameters in the description of the (λ, µ)-Scheme in Section 3.1). We assume Z i > 0 and let µ ij = µ i for any j, which indicates that all contributing tasks to milestone i are treated equally in terms of sensitivity to the incentive payment of an earlier ready time r j and a reduced task duration b j . Then, from (4), we find that a one time unit improvement in ready time r j (i.e., from r j to r j = r j − 1) and in task duration b j , respectively, lead to the following fractions of Z i being paid to TO j: where R i = k∈P i r k and B i = k∈P i b k . Note that p 1 ij , p 2 ij → q ij σ i as µ i → 1, which indicates that one unit of earlier ready time has the same incentive payment as one unit of shorter task duration for all tasks j ∈ P i , where where Q i can be viewed as the total fraction of compensation to the TOs who contribute to milestone i. Clearly, 1/2 < Q i < 1. Suppose that in (3) the overall rate ρ i of payment on milestone i, which could be negotiated by the PM with the project client, is such that it is guaranteed to motivate all TOs j ∈ P i if their corresponding individual rates sum up to ρ i /2, i.e., Q i σ i = 1/2. Now, the PM finds the minimum value of parameter σ i ∈ (0, 1] to provide sufficient motivation for any TO j ∈ P i to report truthfully, given that this is guaranteed by setting σ i = 1/(2Q i ) < 1. In practice, these values can be set from industry standards, or from employment contracts and negotiations with the TOs. In general, when desirable values of parameters λ, µ and ω are fixed and an adequate value of ρ i has been negotiated, the PM minimizes parameter σ i ∈ (0, 1] subject to the condition that the payment defined in (5) provides sufficient incentive for every TO j ∈ P i to report truthfully.

Optimal parameters in the Weighted (λ, µ)-Scheme
With the same setting of ρ i , λ ij , µ ij and ω as for the unweighted (λ, µ)-Scheme, we apply the same arguments for optimizing σ i , taking into account the valuesQ i = j∈P iq ij , wherẽ .
The optimization procedure described in the last paragraph of Section 3.3.1 also applies here.

Incentive scheme performance
We compare the performance of our Weighted (λ, µ)-Scheme with that of a widely used benchmark incentive scheme described in Section 3.4.1. Our computational study is reported in Section 3.4.2.

Benchmark
During the long history of project management practice, various incentive schemes have been developed to improve project performance. These schemes can be classified as improving project performance with respect to (a) schedule, (b) cost, (c) quality, (d) safety, or (e) combinations of these (Ibbs, 1991;Bubshait, 2003). For relevance to our discussion of Parkinson's Law, we focus on schedule incentive schemes. Several studies of business practice highlight the importance of schedule incentives in improving project performance. Arditi and Yasamis (1998) study 21 Illinois Department of Transportation highway contracts, and conclude that the projects would have taken 21% longer on average to complete without schedule incentive schemes. Kog et al. (1999) use a neural network approach to study the effectiveness of schedule incentive schemes in 20 projects. They conclude that an incentive of 3%-5% of project value is effective at improving on-time performance, especially where other aspects of project support are poor. Raduescu and Heales (2005) survey 117 information systems project managers. Among the respondents, 63% believe that schedule incentive schemes improve project delivery times.
The key characteristics of schedule incentive schemes include (a) whether they are implemented through a contract price adjustment, a side payment, or a profit sharing scheme (Bower et al., 2002), (b) whether they are based on individual milestones or on overall completion time (Ibbs, 1991), and (c) whether the incentive is positive as with a bonus, or negative as with a penalty (Herten and Peeters, 1986). For comparability, we consider profit sharing schemes based on individual milestones, and allow both positive and negative incentives. The related literature (Abu-Hiljeh and Ibbs, 1989;Arditi and Yasamis, 1998;Bower et al., 2002;Bayiz and Corbett, 2005) identifies only two such schedule incentive schemes that are widely used: Scheme 1: A per diem bonus for early completion, or penalty for late completion; Scheme 2: A lump sum bonus for on-time completion.
Scheme 1 is applicable where early completion provides additional value to the project owner (Abu-Hiljeh and Ibbs, 1989;Arditi and Yasamis, 1998), as in the problem we are studying. In the schemes we propose, there is no bonus for on-time completion. This is because completion of a task is guaranteed by the contractual requirement that the duration of each task must not exceed its upper bound, which is a conservative estimate of the task duration. Therefore, in our comparison of incentive scheme performance in Section 3.4.2, we consider only Scheme 1. As shown in Example 2, Scheme 1 is vulnerable to false reporting. which we formalize in Section A.8 of the Appendix. We now study the impact of this vulnerability on project performance.

Computational study
We conduct a computational study to compare the effectiveness of the Weighed (λ, µ)-Scheme and Scheme 1, at reducing the makespan and the total payments to the TOs. Since Scheme 1 does not reward earlier ready times and earlier reports, we also ignore ready times and remove any incentives for early reporting in the Weighted (λ, µ)-Scheme. Hence, we set µ = 0 and ω = 0. Recall that in our Weighted (λ, µ)-Scheme, we require that µ > 1 and ω > 0. This is because that scheme does not ignore ready times and does provide incentives for early reporting. For simplicity, we set ρ i = σ i = 1 in the Weighted (λ, µ)-Scheme. Then, to provide a fair comparison, in the Weighted (λ, µ)-Scheme we set λ = 1/2, and in Scheme 1 we set the per diem bonus rate at α = 1/2 and the per diem penalty rate atβ = +∞. Then in both schemes the reported duration of task j does not exceed b j , and the bonus rates for early completion are the same.
We first consider the simplest scenario of only one falsely reporting TO, with all other TOs having realized task durations at their upper bound values b j . Then, we consider multiple falsely reporting TOs. For the case of single falsely reporting TO, we generate random data as follows: (a) The underlying task precedence networks are generated with RanGen (Demeulemeester et al., 2003); (b) for all tasks j, a j ∼ UI[1, . . . , n/2 + 1], b j ∼ UI[a j + 1, . . . , a j + n/2 + 1]; and (c) for one single task k of the falsely reporting TO in each instance, its realized duration is b k ∼ UI[a k , . . . , b k ]. Our computational results are summarized in Table 1, where for each row except the last one we generate 1,000 random instances. The final row shows mean results, computed over the 9,000 instances in the study. The seven columns are explained below, where Columns 4 and 5 provide more details about Column 3, and the last two columns are performance indicators averaged over the 1,000 instances.
"n" indicates the number of tasks in an instance.
"OS", which stands for Order Strength, is a measure of the density of the task precedence network (Demeulemeester et al., 2003).
"Imp" shows the number of instances out of 1,000 in which the Weighted (λ, µ)-Scheme improves on Scheme 1 with respect to at least one of the two criteria mentioned above, i.e. reduction of makespan or of total incentive payments to the TOs.
"Imp-P" shows the number of projects out of 1,000 instances that have reduction of total incentive payment under the Weighted (λ, µ)-Scheme, relative to Scheme 1.
"Imp-M" shows the number of projects out of 1,000 instances that have reduced project makespans under the Weighted (λ, µ)-Scheme, relative to Scheme 1.
"PY" shows the percentage reduction in the total incentive payment under the Weighted (λ, µ)-Scheme relative to Scheme 1, averaged over all project instances where the total payments under the two scheme differ.
"MK" shows the percentage reduction in the makespan under the Weighted (λ, µ)-Scheme relative to Scheme 1, averaged over all project instances where the makespans under the two scheme differ (i.e., all Imp-M instances). Observe that the makespan under the Weighted (λ, µ)-Scheme cannot exceed that under Scheme 1.
n OS Imp Imp-P Imp-M PY (%) MK (%) 50 0. The results shown in Table 1 provide the following conclusions. First, even with only one task j being realized at less than its worst case estimate b j , the Weighted (λ, µ)-Scheme delivers a reduction in either makespan or total payments to the TOs, or both, relative to Scheme 1, in about 83% of instances. Second, the total incentive payment to the TOs is typically about 55% less under the Weighted (λ, µ)-Scheme than under Scheme 1 when payments under the two schemes differ. Third, the Weighted (λ, µ)-Scheme typically captures nearly 6% more of the available reduction in makespan than Scheme 1, when one improved task duration reduces the makespan. Finally, all these results are robust across projects with various realistic sizes and network densities. These results collectively show that our Weighted (λ, µ)-Scheme is more effective at reducing project makespan and total incentive payments than the benchmark Scheme 1.
We also consider projects where multiple tasks, instead of only one, have realized durations that are less than their worst-case estimates. A corresponding computational study, which appears in Section A.9 in the Appendix, leads to similar conclusions. One overall conclusion from both sets of these results is that the Weighted (λ, µ)-Scheme resolves the problem of Parkinson's Law, which is in practice left unresolved by the widely used benchmark Scheme 1. A second overall conclusion is that the resulting savings in time and cost should be of significant benefit in many projects with widely varying characteristics.

Mechanism for CCPM
Advocates of CCPM (e.g., Steyn, 2001) claim that it alleviates Parkinson's Law. Nevertheless, task durations are still subject to estimates based on historical mid-range values (Herroelen and Leus, 2001). Hence, some of the inefficiencies attributable to Parkinson's Law, as discussed in Section 1, also apply under CCPM. To study this issue, in Section 4.1, we describe a mechanism for CCPM. Theorem 5 shows that this scheme is incentive compatible. In Section 4.2, Theorem 6 establishes sufficient conditions which ensure that the payments to the PM and also to each TO are the same under CCPM as under CPM. Then, we define alternative conditions under which Theorem 7 compares the payments under the two planning systems.

Incentive scheme
We propose the following incentive scheme for projects planned under CCPM. For simplicity, we assume in this section that (a) all tasks are ready to start as soon as needed, (b) there is no bonus for earlier reports of reduced task durations, and (c) each TO owns one task or multiple but independent tasks. Our scheme uses parameters α, β and γ. Here, α j (respectively, β j ) is a parameter that converts a gain in the duration of task j into a payment adjustment in the case where that duration is larger (resp., smaller) than the median duration m j , and γ is a parameter that adjusts the value of an initial payment estimate to task j.
Define m j = (a j + b j )/2. Let r j and b j denote the last reported ready times and durations.
1. LetZ i be the milestone payment based on the assumption that task j has a duration of m j for each j ∈ P i . Define as the pre-execution estimated fraction ofZ i as an incentive payment to TO j, where 0 <λ = 2γ/(1 + κ 0 ) < 1 and σ i ∈ (0, 1] is a control parameter as in the (λ, µ)-Scheme.
2. If b j ∈ [m j , b j ], then make a payment to task j of φ j e j fraction of Z i , where φ j ∈ [α j , 1) is a monotonically decreasing function of b j : If b j ∈ [a j , m j ], then make a payment to task j of ψ j e j fraction of Z i , where ψ j ∈ [1, β j ] is a monotonically decreasing function of b j : 3. The PM keeps the remaining fraction of Z i : where x j ∈ {φ j , ψ j } and hence α j ≤ x j ≤ β j .
To remove restriction (a), we first calculate the pre-execution estimate of ready timesr j using the midpoint durations m j . Then, we define an "additional bonus" for reporting a ready time smaller thanr j and a "loss of bonus" for reporting a ready time larger thanr j . To remove restriction (b), we add into the bonus payment a suitably weighted term that uses the metric f j for early reporting. To remove restriction (c), we introduce task weights as in the Weighted (λ, µ)-Scheme to prevent a TO from trading off incentive payments on earlier ready time and shorter task duration.
Our next result establishes bounds on the payments received by the various parties.
Theorem 4. The (α, β, γ)-Scheme is budget balanced, and results in payments such that the TO of task j receives a fraction between α j e j and β j e j of Z i and the PM receives a fraction between Remark 6. As noted in Remark 3, we can add a subscript i to all relevant parameters in the statement of the (α, β, γ)-Scheme and that of Theorem 4, such as to α j , β j , m j , φ j , ψ j . These generalized parameters then adjust the relative payments received by the TOs and the PM for any fixed milestone i ∈ M , and also among different milestones in M . In practice, for example, a larger payment can be used to reward a TO with a longer tenure of employment.
The main result of this section is as follows.

Comparing CPM and CCPM
We develop conditions to compare incentive payments under CPM planning with those under CCPM planning. We assume as in Section 4.1 that all tasks are ready to start as soon as needed, i.e., equalities are required to be satisfied in inequalities (2). Therefore, in our (λ, µ)-Scheme for the CPM system, there is no incentive payment for earlier ready time than r j , and hence µ = 0. Note that the usual requirement of µ > 1 is used only when the ready times are considered for incentive payments. Therefore, κ = κ 0 , as defined in (9). Now, with 0 < λ < 1, we let and, for each j ∈ P i , we let Theorem 6. Assume that all tasks are ready to start as soon as needed. In the (λ, µ)-Scheme for the CPM system (with µ = 0) and the (α, β, γ)-Scheme for the CCPM system, with the publicly known parameters λ, {α j , β j : j ∈ P i } and γ set using (16) and (17), then: (a) conditions (10) and (11) are satisfied, and (b) the same task duration reports {b j , j ∈ P i } lead to the same payments to each TO and the PM in the two incentive schemes, for any realized task durations.
Let the parameters α j , β j , γ, λ and µ be set as in Theorem 6. We say that an (α, β, γ)-Scheme for the CCPM system is enhanced relative to a (λ, µ)-Scheme for the CPM system, if α j and β j in the (α, β, γ)-Scheme are replaced respectively by α and β for any j ∈ P i , where The next result then follows immediately.
Theorem 7. Suppose an (α, β, γ)-Scheme is enhanced relative to a (λ, µ)-Scheme. Then with the same performance and reporting, each TO receives at least the same payment under the former scheme as under the latter scheme. On the other hand, the PM is guaranteed to retain a fraction 1 − β(1 − γ)σ i of the total milestone payment Z i .
Theorem 7, informs the PM about the choice of whether to use CPM or CCPM planning for a particular project. One interpretation of the theorem is that the TOs may receive a greater reward under CCPM. As a result, there is a preference for the PM to use CPM rather than CCPM for planning the project. However, this conclusion comes with two caveats. The first caveat is that the PM does not necessarily want to minimize the rewards gained by the employee TOs, for reasons of organizational morale and staff turnover. The second caveat is that the PM can always specify and achieve a guaranteed share of a milestone payment for herself under CCPM simply by adjusting the values of relevant parameters. Nonetheless, we believe that Theorem 7 provides the first analytical comparison between CPM and CCPM systems.

Optimal scheme parameters and computational study
Using Theorems 6 and 7, the PM can optimally choose the control parameters σ i and other parameters in the enhanced (α, β, γ)-Scheme, as in the corresponding (λ, µ)-Scheme with µ = 0 detailed in Section 3.3.
We now conduct a computational study of the incentive payments to the TOs specified by the (λ, µ)-and (α, β, γ)-Schemes for the CPM and CCPM planning systems, respectively. We are not aware of any previous literature that conducts such a comparison. Our study is motivated as follows. Recall from Theorems 2 and 5 that the (λ, µ)-Scheme and the (α, β, γ)-Scheme are incentive compatible under the CPM and CCPM planning systems, respectively. Hence, given identical realizations of the task durations, the project makespan values delivered by the two systems are identical. Therefore, we compare the total payments made to the TOs under the two systems in reaching the minimum achievable makespan.
As above, for simplicity we set σ i = 1. Our data consists of nine combinations of parameters, specified by n ∈ {25, 50, 100} and OS value ∈ {0.1, 0.2, 0.3}. These OS values are representative of many real project networks (Vanhoucke et al., 2016). For each parameter combination, we randomly generate 20 instances, for a total of 9 × 20 = 180 instances, using RanGen (Demeulemeester et al., 2003). Consistent with our study in Section 3.4, in the simulation model only one task j has a realized duration that is potentially smaller than its upper bound b j . Table 2 reports results for the mean, median and standard deviation of the ratio, , of payments to the TOs under the CCPM system for each instance, relative to those under the CPM system. First, consistent with Theorem 7, these ratios are at least 1 for every instance. The mean payment is on average 23.8% higher, and the median payment is on average 27.1% higher, under the CCPM scheme. The standard deviation of the payments is similarly 21.8% higher under the CCPM scheme. The ratio of mean and median payments increases slightly with n, does not vary significantly with OS, and increases strongly with λ.
These results provide the following useful insights to managers. First, the CCPM scheme provides additional flexibility in making payments to TOs. This flexibility can be useful in motivating particular individuals within the project company. Second, on average, the payments to the TOs are significantly higher under the CCPM system. However, the PM can enforce her guaranteed share of project revenue by setting λ appropriately, hence the additional payments to the TOs need not be problematic. Third, the variability of the payments to the TOs is typically greater under the CCPM scheme, however this difference is only proportional to the average payments received, so this is unlikely to be a source of jealousy among the TOs.

Repeated Projects
In this subsection, we consider whether a TO should commit to a shorter task duration in a situation where the same project occurs multiple times, under the CCPM model. As discussed in Section 1, an important motivation for Parkinson's Law is a concern about increased expectations for future performance (Quigley, 2011). To study this issue, we consider an environment with multiple repetitions of a project, where a TO can report a reduced task duration either with or without a commitment, as defined below, to honor that duration for future rounds.

Incentive scheme
We adapt our (α, β, γ)-Scheme under CCPM for use with a project that is repeated multiple times. We consider any milestone i ∈ M . We merge φ j and ψ j , as defined in (13) and (14), into a single linear function of the reported task duration b j with θ j = 1 − α j = β j − 1:  In reporting his task duration in a single project, as shown in Theorem 5, each TO j reports his "true" duration b * j . However, if the same project is repeated multiple times, this may no longer hold true, since the TO may be concerned about receiving lower payments due to a shortened contractual task duration between b * j and b j at future rounds of the project. To address this concern, we provide an option to the TO within his report in any round, say k ≥ 1, of the same project. That is, while reporting his "true" value b * j and obtaining the corresponding incentive payment, the TO may commit for future rounds of the project to a durationb j ∈ is the task duration committed to by the TO after the first k − 1 rounds of the project. By doing so, the TO contractually agrees that this reduction in the upper bound on the task duration is acceptable for all future repetitions of the project. As a special case, the TO makes no new commitment at all in any round k if he simply choosesb j = b We provide an incentive scheme that is sufficient to ensure that the payment to each TO j for any report b j with maximum new commitment is larger than the payment for the same report with any other level of commitment, including no new commitment at all.
1. In round k ≥ 1 of the project, the realized milestone reward Z (k) i is shared as follows: (a). Give a fraction R (k) to any TO j ∈ P i , where e j is defined as in (12) and where b where b is the task duration committed by the TO up to round t.
(b). The PM keeps the following fraction of Z

End of scheme
We observe that if no reduced task duration is reported as committed in round k − 1, then δ (k−1) j = 0, and hence R (k) Theorem 8. For projects with multiple identical rounds, the (θ, γ)-Scheme is incentive compatible.
Theorem 8 shows how the use of a greater incentive when a commitment is made incentivizes the TOs to report truthfully. Regarding the allocation of incentive payments under the (θ, γ)-Scheme, we have the following bounds on payments to different parties.
Theorem 9. The (θ, γ)-Scheme is budget balanced. More specifically, in any round k of the project and any fixed milestone i ∈ M , TO j (j ∈ P i ) receives a fraction between (1 − θ j )e j and (1 + θ j ) 2 e j of Z (k) i and the PM receives a fraction between 1 − (1 + θ max ) 2 (1 − γ)σ i > 0 and Remark 7. Since the (θ, γ)-Scheme is a special case of the (α, β, γ)-Scheme at each individual round, the PM can choose the scheme parameters optimally, as discussed in Section 4.3.
Remark 8. Instead of a single linear function R (k) j (b j ), we may alternatively use piecewise linear functions φ j and ψ j , as in Section 4.1, to distinguish the "additional bonus" for reporting a duration smaller than m j from the "loss of bonus" for reporting a duration larger than m j . Also, as in Remarks 3 and 6, we may differentiate the payment across different milestones.

Computational study
In this section, we study the project makespan performance under CCPM planning of our (θ, γ)-Scheme, relative to that of a benchmark scheme, for projects with multiple identical rounds. From Theorem 8, the (θ, γ)-Scheme is incentive compatible, which implies not only that all the TOs report their "true" task durations, but also that they make commitment to the shortened task durations for future rounds. Whereas, in both benchmark Schemes 1 and 2 discussed in Section 3.4.1, the TOs are unwilling to commit to reduced task durations in future, and hence these schemes are vulnerable to Parkinson's Law. Therefore, we assume as our benchmark that the TOs report their upper bounds on task durations during multiple rounds even after shorter durations are realized. The parameters considered in our data set follow those used in the comparison of our incentive schemes under CPM and CCPM planning in Sections 4.2 and 4.3.  Table 3: Improvement in project makespan over the benchmark scheme Table 3 reports results for the relative percentage improvement in project makespan over the benchmark scheme. The overall mean reduction is 6.7% and the overall standard deviation of the reduction is 3.0%. The results also show that the relative improvement decreases slightly with n, and decreases more noticeably with the OS value.
These results provide the following useful insights to managers. First, a significant improvement in project makespan can be achieved through the use of our incentive scheme. Second, this improvement is more significant in project networks with low density. Third, the variability of improvement between projects is large relative to the mean improvement; hence, it would be useful to identify types of project networks or specific project applications where reduction in makespan can reliably be achieved from our incentive mechanism.

Concluding Remarks
This paper addresses the pervasive behavioral problem of Parkinson's Law among employee TOs, which arises at the execution stage of many projects. The effect of Parkinson's Law is to waste the potential benefit of early task completion. We describe an IC mechanism to resolve Parkinson's Law for a project planned under CPM. Our mechanism also resolves Student Syndrome. Furthermore, our incentive scheme prevents strategic behavior by any group of TOs. A recent alternative to CPM is CCPM which addresses, but does not fully resolve, Parkinson's Law. We describe an IC mechanism for a project planned under CCPM. We compare the distributions of incentive payments under the mechanisms for CPM and CCPM, and show that the latter mechanism provides both advantages and disadvantages. Finally, we consider the issue of whether a task operator should commit to earlier task completion for similar future projects.
We discuss insights for practicing project managers. Within both the planning environments of CPM and CCPM, we provide a simple incentive scheme that can easily be followed by TOs within the organization. When this incentive scheme is used, it is in the interests of each TO to report his task(s) as completed, regardless of the actions of other TOs. Hence, if all the TOs act in a self-interested way, Parkinson's Law is eliminated. Consequently, early task completion times are passed through the project, resulting in improved project performance. Hence, it is possible to eliminate the negative effects of Parkinson's Law without giving up the detailed control provided by CPM. Moreover, an advantage of using CCPM under our proposed mechanism is greater flexibility in the distribution of typically higher incentive payments to the TOs. This tradeoff informs the management choice between CPM and CCPM for project planning. Finally, the use of appropriate incentives to encourage early completion of tasks should motivate greater efficiencies at the task level. We also provide specific managerial insights following our computational studies in Sections 3.4.2, 4.2 and 4.4.2. The incentive schemes we propose are easily implementable. We recognize that performance incentives are structured differently by each organization, but our work provides a flexible basis from which various practical schemes can be designed.
Several interesting problems remain open for future research. First, this paper focuses on the timely completion of projects, without the possibility of crashing tasks, which represents an important generalization to consider. Second, the incentive scheme described in Section 4.4 for CCPM projects with multiple rounds should be extended to CPM projects. Third, considering discounting could be a valuable generalization in projects of longer duration. Fourth, IC mechanisms should be developed for other project management control structures, e.g., agile (Manifesto for Agile Software Development 2001). Fifth, in practice, our incentive schemes may be extendable to external subcontractors, where there is greater information asymmetry and the possibility of additional strategic behavior. Sixth, our analysis of the long-run implications of commitment to early completion in Section 4.4 should be extended to consider similar tasks that arise in multiple dissimilar projects. Finally, the supervision and motivation of autonomously run tasks is an important topic of current interest (Linberg, 1999;Wu et al., 2014), for which the implications of our work should be studied. In conclusion, we hope that our work will encourage further research on these important issues.
Since the PM keeps the remainder of σ i Z i , the incentive scheme is budget balanced.

A.3 Proof of Theorem 2
Consider any milestone i ∈ M with Z i > 0 and any task j ∈ P i . From (1) and (3), the milestone payment Z i is decreasing in both b j and r j and in their reporting times τ r j and τ b j . To interpret this from the perspective of TO j, from (4) and (5), his payment p ij is a function of his reports (r j , b j ) and their reporting times (r j , b j ), and is decreasing in any of these variables. Hence, it is clearly beneficial for him to report as early as possible. Now, denote by (r j , b j ) and (r f j , b f j ) the "true values" and reported values, respectively, submitted by TO j. Since the TO reports only these two values, only the following scenarios can provide additional payment to TO j based on a false report: (a) r f j < r j ; and (b) b f j < b j . Case (a). Recall that TO j is contractually required to start no later than his reported ready time. Thus, if r f j < r j , then TO j cannot honor the contract, since r j is the earliest time at which task j is ready to start. Case (b). Suppose b f j < b j . Then, due to the contractual requirement in the Certification of Completion, TO j has to start task j earlier than his reported ready time r f j , hence Let δ = r f j − r j > 0. The reported false task duration b f j cannot be less than b j − δ because of the Certification of Completion requirement, i.e., Let s j denote the slack corresponding to the truthful reports (r j , b j ) and s f j the slack correspond-ing to the false reports (r f j , b f j ). Then, we have < 0, from the definitions of λ ij , δ and µ ij .
Then, from (4) and (5), false reporting by TO j reduces his payment.

A.4 Proof of Corollary 1
The result follows from Theorem 2 and the observation that the total payment that the PM makes to a TO from two independent tasks is the sum of their individual payments.
Let A 1 ⊆ A denote the set of root task(s) of the TO, i.e., those tasks that have no predecessor in A . We consider both aspects of the TO's possible strategy in misreporting r j and b j for any j ∈ A 1 : (a) trading a loss on r j for a gain on b j ; and (b) trading a loss on task j for gains on tasks k ∈ A ∩ V j . (a) In the Weighted (λ, µ)-Scheme, the relative incentives for reducing r j and reducing b j remain the same as in the unweighted (λ, µ)-Scheme, as shown in (4). Hence, the TO reports his ready time r j truthfully, as shown in the proof of Theorem 2. On the other hand, at duration b j , the TO cannot state an earlier-than-actual completion time for task j, because of the Certification of Completion requirement.
(b) We now show that the TO does not report a later-than-actual completion time for task j, say by ∆ time units, in an attempt to trade the immediate loss for a later gain from the extra unreported ∆ time units of early start for successor tasks k ∈ A ∩ V j when it comes to reporting b k . Suppose the TO makes such a false report. Then, according to the definition of incentive payments p ij and p ik in (7), the ratio between total possible gained payment and lost payment is at most ( k∈A ∩V j w k ∆)/(w j ∆) < 1, by construction of the weights. Therefore, both r j and b j are reported truthfully. We now apply this argument inductively to the root tasks A i+1 of A \(∪ 1≤t≤i A t ) for i ≥ 1 until all tasks in A have been considered. We conclude that the TO truthfully reports r j and b j , for j ∈ A .

A.7 Proof of Corollary 2
Consider any coalition of TOs. Suppose to the contrary that strategic reports of the coalition lead to an increased total payment. Then, when all the tasks owned by members of the coalition are assigned to a single TO, with the same strategic reports, the TO receives more than from truthfully reporting about his individual tasks. This contradicts Theorem 3.

A.8 Vulnerability of Scheme 1
We discuss the possibility that Scheme 1 is vulnerable to false reporting. If a task with maximum duration of b j is completed in b j time units, Scheme 1 pays TO j a bonus ofᾱ(b j − b j ) if b j < b j and requires a penalty of max{β(b j − b j ),M j } if b j > b j , whereM j is a pre-fixed limit. This scheme works fine for a task that is not a predecessor of another owned by the same TO; early completion earns a profit for a TO, but a delay incurs a penalty. Consequently, each TO reports as soon as he can complete the task. However, Example 2 shows that, when a TO has dependent tasks, he can increase his incentive payments by false reporting. We now model the decisions of the falsely reporting TO, under the following assumptions.
4. Let T denote the set of tasks for which the earliest completion times in the original graph G are decreased as a result of the shortened duration of the current task of the falsely reporting TO, where all other task durations are at their upper bounds. We assume all tasks in T belong to the TO, and form a connected graphG(T ). 5. At the time when the TO who is falsely reporting knows the actual completion time of the current task, which is earlier than its original worst case estimate, the TO forms his strategy for reporting durations of all tasks in T , based on the updated duration of the current task and the given upper bounds on the durations of all other tasks in T as well as of all the project tasks not in T .
We model the decision problem faced by a TO who may report falsely in order to maximize his incentive payment. We begin with definitions of some parameters used in the formulation. Using the upper bounds {b k } of task durations for all tasks k, both inside T and outside T , the TOs commit earliest completion times for their tasks, which we denote by {F k }. At the time of forming a reporting strategy, the TO knows reduced duration of his current task, which results in updated earliest completion times {F j : j ∈ T }, assuming the durations of all other tasks, whether or not in T , are at their given upper bounds {b j }. We define three nonnegative variables for each task j ∈ T : x j = reported amount by which the duration of task j is shortened; y j = starting time of task j contractually required by the reports; z j = reported completion time of task j, i.e., z j = y j + (b j − x j ).
Solution of the following integer program maximizes the incentive reward received by the TO: max j∈T x j subject to: for any j ∈ T , if j is the current task z j = y j − x j + b j , (by definition of the decision variables) F j ≤ z j ≤F j , (for certification of completion or commitment) where Q j = { ∈ T : is an immediate predecessor of task j}, Q j = {k ∈ T : k is an immediate predecessor of task j}.
We note that the above precedence constraints can be linearized as follows: We now provide an example of our model of false reporting.
Example 3. In Figure 1, let the set of tasks {(s, 1), (1, 2), (1, 3), (2, 3), (2, i), (3, i)} belong to the same TO. Assume that the current job is (s, 1), with an original upper bound duration of 3 and a realized duration of 2. For simplicity, we assume that the lower bound on the duration of any task is 0. The above mixed integer program gives: Maximize z = x s1 + x 12 + x 13 + x 23 + x 2i + x 3i subject to x s1 + z s1 = 3, x 12 + z 12 = z s1 + 1, x 13 + z 13 = z s1 + 4, x 23 + z 23 = z 12 + 3, x 2i + z 2i = z 12 + 5, x 3i + z 3i = max{z 13 , z 23 } + 2, This model has multiple optimal solutions; for any 0 ≤ θ ≤ 1, an optimal solution is x s1 = 0, x 12 = 0, x 13 = 1 − θ, x 23 = 1, x 2i = 1 and x 3i = θ, where z = 3. Therefore, although the TO finishes task (s, 1) at time t = 2, he can falsely report that the task finishes at time t = 3. Then, in private he can start tasks (1, 2), (1, 3) and all his other three tasks one time unit earlier than in the original schedule calculated from upper bound durations. As a result, although the actual duration of each of these five tasks is at its upper bound, the TO, choosing θ = 0, can falsely claim that he uses one time unit less to complete each of tasks (1, 3), (2, 3) and (2, i), without being discovered by the verification of completion report. Thus, the total incentive payment received by the TO is 1 based on honest reporting, and 3 under the false reporting permitted by Scheme 1 in this solution. Hence, the TO has an incentive to report falsely.

A.9 Multiple falsely reporting TOs
We consider multiple falsely reporting TOs, each owning a task of realized duration shorter than its worst-case estimate. Our computational study uses the same setting as in Section 3.4.2, except that each instance is an aggregation of five falsely reporting TOs' subnetworks of tasks and the aggregation is either in parallel or in series, each with probability 0.5. Our computational simulation results are summarized in Table 5, using the same format as in Table 1.
n OS Imp Imp-P Imp-N Imp-M PY (%) MK (%) 50 0.  Compared with the results with one single falsely reporting TO (as shown in Table 1), the following changes due to multiple falsely reporting TOs are significant: (a) in nearly all instances, the Weighted (λ, µ)-Scheme improves upon Scheme 1 in terms of reduction either in total incentive payment or in project makespan, or both; (b) the relative payment reduction of the Weighted (λ, µ)-Scheme over Scheme 1 increases by 50%, since more false reporting results in greater incentive payments; and (c) the relative project makespan reduction decreases by 50%, since aggregation of small percentage reductions in subnetwork makespans leads to even smaller percentage reduction of the overall network makespan, especially in the case of parallel subnetworks.

A.10 Proof of Theorem 4
For the TOs, the result is immediate, by construction. It remains to prove the claim about the payment to the PM, which follows from (15) and the following equation: where A i = j∈P i a j and B i = j∈P i b j . Therefore, the (α, β, γ)-Scheme is budget balanced.
A.11 Proof of Theorem 5 For each task j, parameters a j , b j and m j are predetermined, and this information, along with P i , is publicly known and fixed. Therefore, since α j , β j and γ are also publicly known and fixed, the only private information that affects the final payments to the TOs is the actual duration of task j, which we denote by b * j , for j ∈ P i . For each j ∈ P i , suppose the TO of task j reports a task duration b j . If b j < b * j , then the TO does not meet the Certification of Completion requirement. Therefore, we have b j ≥ b * j . The θ j δ (k−1) j R j (·)e j more than that for the previous round of the project. Therefore, if TO j commits to a reduced duration b (k−1) j to the greatest extent possible in the first k − 1 rounds, he maximizes his payment fraction from the next round on. Moreover, from the additivity of {δ (t) j } in the payment fraction, such commitments from TO j are made as soon as possible.
A.15 Proof of Theorem 9 It follows from (19) that 1 − θ j ≤ R j (·) ≤ 1 + θ j . Hence, at any round k, since ∞ t=1 δ (t) j ≤ 1, the payment fraction R (k) j (·) of Z (k) i to TO j satisfies the following: The bounds for the payment to the PM follow from the corresponding proof for the second part of Theorem 4. Since by construction all the available milestone payment is distributed to the TOs and the PM, the payment scheme is budget balanced.