Axiomatic Characterizations of a Proportional Influence Measure for Sequential Projects with Imperfect Reliability

We define and axiomatically characterize a new proportional influence measure for sequential projects with imperfect reliability. We consider a model in which a finite set of players aims to complete a project, consisting of a finite number of tasks, which can only be carried out by certain specific players. Moreover, we assume the players to be imperfectly reliable, i


Introduction
Projects are omnipresent in society.Whether it concerns an improvised explosive device (IED) that needs to be developed, moved and placed (see Lindelauf (2011) for a more detailed breakdown of the tasks in a typical IED-project) or a project involving architecture, engineering and a CentER, Department of Econometrics & Operations Research, Tilburg University, P.O.Box 90153, 5000 LE Tilburg, The Netherlands b Corresponding author.E-mail address: a.j.vanbeek@tilburguniversity.educonstruction (AEC).Generally speaking, projects consist of tasks, and each task can only be carried out by certain players, leading to a so-called task structure.
A significant part of quantitative research on projects is concerned with project planning.
For example, Brown et al. (2009) develop interdiction actions that maximally delay completion of a (nuclear) weapons project.Estévez-Fernández et al. (2007) use so-called project games to analyze projects in which certain tasks can be delayed or expedited, leading to costs or rewards, respectively, that need to be allocated.The distribution of shared costs in delayed projects is also analyzed by Bergantiños and Sánchez (2002) and Brânzei et al. (2002).In this paper, we do not focus on project planning, but on measuring the relative control of each player in the completion of a project.
In the context of organized crime, an intelligence agency may be interested in determining who is the most important player for the completion of a project, in order to determine who should be eliminated or apprehended.Regarding AEC projects, completing the project may lead to some (monetary) reward allocation that should fairly reflect the contribution of the various players.As a common feature, the question at hand is which players are most influential for the completion of a project.To this aim, we define and axiomatically characterize a new proportional influence measure based on the task structure of a project.
Before elaborating on the new influence measure and its axiomatic characterizations, it is important to discuss what type of projects we are analyzing.The general definition of a task structure concerns projects for which a finite player set attempts to carry out a finite number of tasks.We assume that for the completion of a project, each task must be carried out successfully.Each task can only be carried out by a certain set of players, called the task group.
The tasks of a project can be carried out sequentially.For example, this implies that if a player is a member of all task groups, this player could attempt to complete the project alone.We emphasize 'attempt to' here; when a player attempts to carry out a task, we generally do not assume this player is guaranteed to carry out the task successfully.Concretely, each player-task combination has a certain success probability, also referred to as the reliability of the player for that specific task.Depending on the context, reliability can reflect e.g., operational risk, trustworthiness, or quality of a player in a broader sense.While the reliability of players is often not incorporated in the literature, it is a key aspect of our model.Thus, we focus mainly on sequential projects with imperfect (player) reliability.
In sequential projects with imperfect reliability, the success probabilities of the individual players in each task group also lead to success probabilities of the tasks, which in turn yields a success probability of the project as a whole.For the latter, we assume that the successes of different tasks are independent.In our model, all players in a task group can attempt to carry out the corresponding task.The probability that a task is not carried out successfully equals the probability that all players fail to carry out this task.This can be interpreted in two ways.First, it is possible that all players in a task group attempt to carry out the task in parallel, where the task is considered successful if at least one player manages to carry out the task.Alternatively, a project might be such that some (random) player in the task group attempts to carry out the task, after which another (random) player in the task group attempts the task only if the first player has failed.This process continues until the task is either carried out successfully, or all players (in the task group) have failed.We do not consider, e.g., additional set-up or delay costs associated with attempting to carry out the task multiple times.
We define a proportional influence measure which allocates the final success probability of a project over the players, to determine to which extent each player contributes to the completion of a project.For each task players can carry out, the increase in value of their influence measure corresponds to their relative success probability within the task group.Hence, the allocated value per task increases with the success probability of the player (for this task), but decreases with the 'total' success probability of the other players in the corresponding task group.This essentially reflects a balance between a player's quality and a player's replaceability.
We axiomatically characterize our proportional influence measure by proving it is the only allocation mechanism for sequential projects with imperfect reliability that satisfies two sets of logically independent properties.The first characterization is on the domain of projects with a fixed player set and its main property concerns the task decomposability of a project.It describes the relationship between the influence measure of a project and the measures of influence one would obtain if one divides the tasks of the project over multiple independent smaller projects.For the second characterization, we consider the domain of projects with a varying set of players.The characterization is mainly based on the property invariance under replacement.
This property relates projects with different player sets.In particular, it prescribes the relation when in a certain task group a specific player is replaced by a new player who was not in the original player set.We also observe that both characterizations still work on the smaller domain of sequential projects with perfect reliability.
A potential application lies within the framework of construction projects.Matthews and Howell (2005) propose an Integrated Project Delivery (IPD) method that emphasizes coopera-tion between various parties who share risk and reward.Despite the fact that IPD is regarded as an effective method, Teng et al. (2019) point out that the number of construction projects using IPD is limited, in part due to the lack of a fair mechanism to allocate profit.For IPD projects that fit the assumptions of our framework, the proportional influence measure could be used as such a mechanism.
An alternative way to analyze sequential projects with imperfect reliability is to define an appropriate cooperative game and use existing game-theoretic solution concepts.We sketch a path for future research in this direction in the final section of the paper.
Section 2 formally introduces projects with imperfect reliability and the proportional influence measure, as well as their counterparts in case of perfect reliability.Section 3 discusses the first characterization, based on decomposing the tasks of a project.The second characterization, based on replacing players, is covered in Section 4. Section 5 concludes.

Projects and the Proportional Influence Measure
Let N denote the finite set of players and T the finite set of tasks to be carried out by these players.For each k ∈ T , let p k ∈ [0, 1] N denote a probability vector such that p k i , i ∈ N , represents the success probability of player i to carry out task k.A sequential project with imperfect reliability P can thus be summarized by the tuple We assume that N k ̸ = ∅ for all k ∈ T .Otherwise, the project cannot be completed.
The class of all projects is denoted by P. If we restrict to the class of projects with some fixed player set N , we emphasize this in the notation using P N .This distinction of domains becomes relevant in the characterizations of the proportional influence measure later on.
All players in a task group can attempt to carry out the corresponding task, and all task must be carried out successfully in order to complete the project.The probability that a task is not carried out successfully is equal to the probability that all players fail to carry out this task.Assuming independence between the success of tasks, the success probability of a project can then be found by simply multiplying the success probabilities of each individual task.We denote the probability that a project P ∈ P is completed successfully as q(P ).This can be seen as the 'quality' of the player set (with respect to carrying out the tasks in the project), or simply as the success probability of the project.Formally, we define the success probability of a project as a function q : P → [0, 1] such that for any P = N, T, {p k } k∈T ∈ P.
We now define a solution concept for sequential projects with imperfect reliability.We call f : P N → R N a solution concept on P N .For i ∈ N , f i (P ) represents a measure of influence of player i on the completion of the project.Similarly, f is a solution concept on P if it assigns a vector in R N to any P ∈ P N for any finite player set N .We propose a proportional influence measure for sequential projects with imperfect reliability, denoted by ρ, in which the total probability of success q(P ) is allocated among the players in the following way.First, by the nature of a project, q(P ) is shared equally among the tasks.Second, for each task k ∈ T , q(P )/|T | is allocated to players proportional to their individual task-specific success probabilities provided by p k .
Definition 2.1 The proportional influence measure ρ : P N → [0, 1] N is defined by setting for all P = N, T, {p k } k∈T ∈ P N and any i ∈ N .
Note that we define ρ as a solution concept on P N here.For the second characterization, we define properties of a solution concept on P. In this case, we also interpret ρ as a solution concept on P, simply by using the definition above for any finite N .
Example 2.1 Consider a project P = N, T, {p k } k∈T that consists of two tasks, to be carried out by a set of three players, with N = {1, 2, 3}, T = {a, b}, and p a = (0.8, 0.9, 0) and p b = (0.8, 0, 1).Clearly, A project P = N, T, {p k } k∈T ∈ P N is called a project with perfect reliability if p k i ∈ {0, 1} for all k ∈ T and i ∈ N .The subclass of P N of all projects with perfect reliability is denoted by S N .Similarly, S denotes the corresponding subclass of P. Since N k ̸ = ∅ for all k ∈ T , note that we have q(P ) = 1 for all P ∈ S. Clearly, for a project P = N, T, {p k } k∈T ∈ S N with perfect reliability, we have

Characterization Using the Decomposition of Tasks
In this section, we present our first axiomatic characterization of the proportional influence measure.The most prominent property of a solution concept in this characterization considers the effect of dividing the tasks of a project over multiple smaller projects which we 'solve' independently.In particular, it relates the solution of this project to the solutions of the smaller projects.
We use the following four properties of a solution concept f : P N → R N to axiomatically characterize ρ on P N .
We say that f satisfies efficiency if f allocates the total probability that a project is completed.
Efficiency (EFF) f satisfies EFF on P N if i∈N f i (P ) = q(P ) for any P ∈ P N .
For any P = N, T, {p k } k∈T ∈ P N we define Z(P ) = N \ k∈T N k as the set of null players with respect to P .These null players do not have a positive success probability for any task.
We say that f satisfies the null player property if all null players with respect to P are allocated zero value.
Null Player (NUL) f satisfies NUL on P N if f i (P ) = 0 for all P ∈ P N with i ∈ Z(P ).
Proportionality only applies to projects with a single task.For such projects, this property states that the values players get allocated are proportional to their success probabilities.
for all P = N, T, {p k } k∈T ∈ P N such that T = {l} and all i, j ∈ N such that p l i > 0 and p l j > 0.
The final property of task decomposability only applies to projects P = N, T, {p k } k∈T ∈ P N with |T | > 1.For such projects, this property describes the relationship between the solution of this project and the solutions of two projects P T 1 and P T 2 over which the tasks in T of the original project are divided into two disjoint sets T 1 and T 2 .In general, we denote smaller projects with P S = N, S, {p k } k∈S ∈ P N for any S ⊆ T .Note that for all P S , P S 1 , P S 2 ∈ P N with S ⊆ T , S 1 ∪ S 2 = S and S 1 ∩ S 2 = ∅, we have q P S = q P S 1 q P S 2 .
We say that f satisfies task decomposability if the value allocated by f for the original project can be written as a certain weighted average of the values allocated by f for the two corresponding smaller projects.These weights contain the relative number of tasks to be carried out and the success probabilities of the smaller projects.In particular, the higher the relative number of tasks, the higher the weight of that project.Further, the weight increases as the success probability of the other smaller project increases.The intuition behind this is that the original project is only completed if both smaller projects are completed, meaning completion of one smaller project should only count if the other smaller project is also successful.
Task decomposability (DEC) f satisfies DEC on P N if for all P = N, T, {p k } k∈T ∈ P N such that |T | > 1 and all P T 1 , P T 2 ∈ P N with |T 1 | ≥ 1, Example 3.1 Reconsider the project P = N, T, {p k } k∈T with N = {1, 2, 3}, T = {a, b}, p a = (0.8, 0.9, 0) and p b = (0.8, 0, 1), as described in Example 2.1.We decompose this project into two smaller projects P T 1 and P T 2 with T 1 = {a} and T 2 = {b}, and note that q P T 1 = 0.98 and q P T 2 = 1.Task decomposability is satisfied by ρ in this example, since Next, we show that ρ is the only solution concept for sequential projects with imperfect reliability that satisfies the four properties defined above.To do so, we first derive a consequence of the DEC property towards decomposing a project into single-task projects.

Lemma 3.1
Let f be a solution concept on P N that satisfies DEC.Then, for all P = N, T, {p k } k∈T ∈ P N such that |T | > 1.
Proof.Let P = N, T, {p k } k∈T ∈ P N with |T | > 1.We give a proof by induction on |T |.
First, consider the base case T = {k, l} with k ̸ = l.By DEC, we have Next, assume the induction hypothesis that equation ( 1) holds for all P = N, T, {p k } k∈T ∈ P N such that |T | = t for a given natural number t ≥ 2.Then, let P = N, T, {p k } k∈T ∈ P N with |T | = t + 1 and let l ∈ T .We get where we use the fact that f satisfies DEC in the first equality, the induction hypothesis in the second equality, and that q P T \{k} = q P T \{k,l} q P {l} for any k ∈ T \ {l} in the third equality.

Theorem 3.2
Let f be a solution concept on P N .Then, f = ρ if and only if f satisfies EFF, NUL, PROP, and DEC.
Proof.We first show that ρ satisfies the four properties.EFF and NUL are obvious from the definition of ρ.
PROP: Consider P = N, T, {p k } k∈T ∈ P N such that T = {l} and let i, j ∈ N be such that p l i > 0 and p l j > 0.Then, T and T1 ∩ T 2 = ∅ and consider P T 1 and P T 2 .Then, Next, let f : P N → R N satisfy the four properties.We show that f (P ) = ρ(P ) for all P ∈ P N .
We first focus on projects with one task.Let P = N, T, {p k } k∈T ∈ P N with T = {k}.Let i ∈ N such that p k i = 0.By NUL, we have f i (P ) = 0 = ρ i (P ).Next, let i ∈ N such that p k i > 0. Since f satisfies PROP, we know that for any j ∈ N with p k j > 0, we have Using EFF, we get From this, we may conclude Next, let P = N, T, {p k } k∈T ∈ P N such that |T | > 1.By Lemma 3.1, for any i ∈ N , which concludes the proof.
Finally, we observe that the four characterizing properties mentioned in the theorem are logically independent.For any subset of three properties, it suffices to find an alternative solution concept f with f ̸ = ρ that satisfies 1 these properties.
No EFF: Consider f (P ) = 2ρ(P ) for all P ∈ P N .
No NUL: Consider f : P N → R N defined by for all P ∈ P N . 2 No PROP: Let 2 N denote the collection of subsets of N .Fix a representation function for all P = N, T, {p k } k∈T ∈ P N .
No DEC: Consider f : P N → R N defined by for all P = N, T, {p k } k∈T ∈ P N and any i ∈ N .
To conclude this section, we analyze the results for the special case of sequential projects with perfect reliability.Despite the fact that the class S N of projects becomes significantly smaller than P N , the proportional influence measure ρ is still the only solution concept on this subdomain that satisfies the four properties of Theorem 3.2.
For the sake of completeness, we provide the explicit reformulation of the four properties on Efficiency (EFF) f satisfies EFF on S N if i∈N f i (P ) = 1 for all P ∈ S N .
Null Player (NUL) f satisfies NUL on S N if f i (P ) = 0 for all P ∈ S N with i ∈ Z(P ).
2 Here, e1 denotes the unit vector of length |N | of which the first element equals one, or, more generally, ei denotes such a unit vector of which the i-th element equals one.
Partition proportionality (PAP) f satisfies PAP on P if for all P = N, T, {p k } k∈T ∈ P such that {N k } k∈T is a partition of N \ Z(P ) and all i, j ∈ N and l, m ∈ T such that p l i > 0 and p m j > 0.
The final property of invariance under replacement states that when in a certain task group a specific player is replaced by exactly one player who was not in the original player set and who has the same success probability for that task, this does not affect any of the other non-null players.Before formally defining the property itself, we first introduce the general definition of a replicate project Pi ∈ P corresponding to a project P = N, T, {p k } k∈T ∈ P, in which a player i ∈ N is replaced by a 'new' player in one task group.3Definition 4.1 Let P = N, T, {p k } k∈T ∈ P be a project and let i ∈ N , l ∈ T such that p l i > 0.Then, a replicate project Pi ∈ P with replica repl(i) for player i is defined by Pi = ( N , T, {p k } k∈T ), with N = N ∪ {repl(i)}, pl repl(i) = p l i , pl i = 0, pl j = p l j for all j ∈ N \ {i}, and, for all k ∈ T \ {l}, pk repl(i) = 0 and pk i = p k i for all i ∈ N .
Invariance under replacement (IUR) f satisfies IUR on P if for all P = N, T, {p k } k∈T ∈ P, all i ∈ N \ Z(P ) and j ∈ N \ (Z(P ) ∪ {i}), and all replicate projects Pi .
Note that we also have ρ 1 ( P1 ) + ρ 4 ( P1 ) = ρ 1 (P ).△ In the example, the value allocated to the player who is replaced (in one task group) in the project is equal to the sum of the values allocated to this player and the replica in the replicate project.In fact, this holds for any solution concept f that satisfies EFF, NUL, and IUR on P.
Lemma 4.2 Let f be a solution concept on P that satisfies EFF, NUL, and IUR.Then, for all P = N, T, {p k } k∈T ∈ P, all i ∈ N \ Z(P ), and all replicate projects Pi with replica repl(i) for player i.
Proof.Let P = N, T, {p k } k∈T ∈ P, let i ∈ N and l ∈ T be such that p l i > 0, and let Pi = ( N , T, {p k } k∈T ) ∈ P be the replicate project in which replica repl(i) replaces player i for task l.First, note that the success probabilities of P and Pi are equal, since where we use pk repl(i) = 0 for all k ∈ T \ {l} in the third equality and pl i = 0 in the fourth equality.It follows that where we use the fact that f satisfies EFF and NUL in the first and third equality, and we use q(P ) = q( Pi ) and the fact that f satisfies IUR in the second equality.
We now show that ρ is the only solution concept for sequential projects with imperfect reliability satisfying the four properties defined above.

Theorem 4.3
Let f be a solution concept on P.Then, f = ρ if and only if f satisfies EFF, NUL, PAP, and IUR.
for all P = N, T, {p k } k∈T ∈ P and any i ∈ N .
To conclude this section, we note that similar to the first characterization, the proportional influence measure ρ is still the only solution concept on the subdomain S of sequential projects with perfect reliability that satisfies the four properties.Here, we omit the (direct) reformulations of the properties on S.

Theorem 4.4
Let f be a solution concept on S.Then, f = ρ if and only if f satisfies EFF, NUL, PAP, and IUR.

Project Games
The proportional influence measure is a solution concept based directly on the task structure of a project.This solution concept takes into account the success probabilities of all players in N , but does not explicitly take into account the ability of subsets of N (coalitions) to complete the project.To analyze this interesting topic, we can model the situation as a project game 5 , in which appropriate values for coalitions are quantified.Based on such a project game, the influence of all players in N can then be measured using a game-theoretic solution concept, like the Shapley value (Shapley, 1953).
To give an impression of how to define an appropriate associated project game, let P = N, T, {p k } k∈T be a sequential project with imperfect reliability.One possible corresponding game v P could be defined by setting the value of a coalition equal to the success probability of that coalition by means of players in S only: for any coalition S. Note that v P (N ) = q(P ).
Example 5.1 Reconsider the project P = N, T, {p k } k∈T of Example 2.1, with N = {1, 2, 3}, T = {a, b}, p a = (0.8, 0.9, 0) and p b = (0.8, 0, 1).The project game v P is given in Table 1.In the allocation of v P (N ), it cannot be guaranteed that a coalition S is allocated (irrespective of coalition N \ S) at least its value, as demonstrated by v P ({1}) and v P ({2, 3}).Notwithstanding this modeling drawback, the project game is a consistent representation from which we can derive an allocation of influence.For example, the Shapley value of this game is given by (0.504, 0.234, 0.242).Note that this allocation is largely similar to the proportional influence measure ρ(P ) ≈ (0.45, 0.26, 0.27).△ Let P = N, T, {p k } k∈T ∈ P N be such that |T | > 1 and let i ∈ N .Let T 1 and T 2 be 98.