Centroids of the core of exact capacities: a comparative study

Abstract

Capacities are a common tool in decision making. Each capacity determines a core, which is a polytope formed by additive measures. The problem of eliciting a single probability from the core is interesting in a number of fields: in coalitional game theory for selecting a fair way of splitting the wealth between the players, in the transferable belief model from evidence theory or for transforming a second order into a first order model. In this paper, we study this problem when the goal is to determine the centroid of the core of a capacity, and we compare four approaches: the Shapley value, the average of the extreme points, the incenter with respect to the total variation distance and the limit of a procedure of uniform contraction. We show that these four centroids do not coincide in general, we give some sufficient conditions for their equality, and we analyse their axiomatic properties. We also discuss how to define a notion of centrality measure indicating the degree of centrality of an additive measure in the core. Finally, we also analyse these four centroids in the more general context of imprecise probabilities.

1 Introduction

A problem that naturally arises in many branches of operations research, such as decision making (Huntley & Troffaes, 2012; Keith & Ahner, 2021; Troffaes, 2007) or expected utility theory (Gilboa & Schmeidler, 1989; Klibanoff et al., 2005; Sarin & Wakker, 1992), is that of determining the probability measure modelling the underlying uncertainty. Due to a number of factors (missing data, conflicting sources of information, etc.), it is sometimes difficult, or even impossible, to elicit such probability measure with minimal guarantees. A possible approach in those cases is to consider instead a capacity (Grabisch, 2016), that in turns determines a polytope of associated additive measures: its core. When the capacity is normalised, the core is formed by probability measures.

Capacites appear in many different contexts and with different interpretations: (i) in decision making, the values of the capacity measure the beliefs supporting the occurrence of each event; (ii) in coalitional game theory, each event represents a coalition of players, the value of the capacity is interpreted as the minimum reward guaranteed by the coalition, and the core contains the distributions of the rewards compatible with these constraints; and (iii) in imprecise probability theory (Augustin et al., 2014), a capacity gives lower bounds for the real but unknown values of the probability measure underlying the experiment.¹ The core, also called credal set (Levi, 1980), is formed then by the probability measures that are candidates for being the unknown probability measure. Among the many applications of capacities, we refer for instance to Grabisch (2013) and Shapley (1953) for some applications in game theory, and to Angilella et al. (2016) and Destercke (2017) in ordinal classification.

In any of these contexts, the problem of selecting a probability measure from the core of the normalised capacity is quite common. For instance, within coalitional game theory we may consider the solution of a game as a way to fairly divide the wealth it represents among the players; within imprecise probability theory, we may also consider transformations between imprecise and precise probability models (Klir & Parviz, 1992; Smets, 2005); in addition, we may also look for the element of the core maximising the entropy (Abellán & Moral, 2003; Jaffray, 1995) or establish procedures for assigning relevance degrees to the different features in machine learning problems (Kumar et al., 2020; Lundberg & Lee, 2017).

Our goal here is also to select a probability measure from the core of a normalised capacity, but the interpretation of the output of the process is different to the cases mentioned above: we seek to determine the center of the core of the capacity, similarly to the point with greatest data depth (Cascos, 2009; Tukey, 1975) in a data cloud; thus, our final result should be an element in the interior of the core, whenever the latter is non-empty. This already rules out methods based on maximising the entropy or minimising the Kullback Leibler divergence. Pursuing this objective, in this paper we analyse four centers: (i) the Shapley value, that first appeared in coalitional game theory (Shapley, 1953; ii) the average of the extreme points of the core; (iii) the incenter with respect to a distance (in our case, the total variation distance, due to its good properties within the imprecise probability framework (Montes et al., 2020b)); and a center that is obtained following a procedure of uniform contraction of the core.

The investigation of the notion of center of a core leads naturally to that of a measure of centrality with respect to a given set of probability measures. We propose an axiomatic definition and analyse several examples, both based on a choice of a centroid or not.

The remainder of the paper is organised as follows. After introducing some preliminary notions in Sect. 2, in Sect. 3 we discuss the four possible notions of centroids mentioned above and study the relationships between them. In Sect. 4, we make a further comparison in terms of the axiomatic properties they satisfy and in Sect. 5 we discuss the notion of centrality measure. Finally, in Sect. 6 we show that the centroids can be defined in the more general context of coherent lower previsions. Some additional comments are given in Sect. 7. To ease the reading, proofs have been relegated to the Appendix.

A preliminary version of this paper was presented at ECSQARU 2021 Conference (Miranda & Montes, 2021). This extended version contains a deeper discussion of the four centroids, additional results, proofs and other comments stemming from the discussions carried out at the conference.

2 Preliminary concepts

Let us introduce the main concepts we shall use in this paper. We refer to Grabisch (2016) for more details.

Consider a finite possibility space $\mathcal {X}=\{x_1,\ldots ,x_n\}$, let $\mathbb {P}(\mathcal {X})$ be the set of all the probability measures on $\mathcal {X}$ and let $\mathbb {P}^{*}(\mathcal {X})$ be the probability measures assigning strictly positive probabilities to all the non-empty events.

A capacity is a function $\mu :\mathcal {P}(\mathcal {X})\rightarrow \mathbb {R}$ that is monotone: $A\subseteq B$ implies $\mu (A)\le \mu (B)$ and satisfies $\mu (\emptyset )=0$. A capacity is normalised when in addition $\mu (\mathcal {X})=1$. Throughout this paper, we will always consider normalised capacities. The conjugate of a capacity $\mu$, denoted by $\bar{\mu }$, is defined by $\bar{\mu }(A)=1-\mu (A^c)$ for any $A\subseteq \mathcal {X}$. Also, a capacity $\mu$ determines the core, defined by:

$$\begin{aligned} \text{ core }(\mu )=\{ P\in \mathbb {P}(\mathcal {X})\mid P(A)\ge \mu (A) \ \forall A\subseteq \mathcal {X}\}. \end{aligned}$$

When the core is non-empty, the capacity is called balanced and it holds that $\mu \le \bar{\mu }$. A balanced capacity is called exact when $\mu (A)=\min _{P\in \text{ core }(\mu )}P(A)$ for every $A\subseteq \mathcal {X}$.

As we mentioned in the introduction, capacities can be interpreted in many different ways. In this paper, we will focus mostly on the decision making or game theoretic interpretations (cases (i) and (ii) from Sect. 1). The interpretation as imprecise probability models (case (iii)) will be explored in more detail in Sect. 6.

The core of a capacity is a closed and convex subset of $\mathbb {P}(\mathcal {X})$, and since it is determined by a finite number of restrictions, it is a polytope that can be characterised by a finite number of extreme points. Recall that $P\in \text{ core }(\mu )$ is an extreme point of the core if $P=\lambda P_1+(1-\lambda )P_2$ for some $\lambda \in (0,1)$ and $P_1,P_2\in \text{ core }(\mu )$, then $P_1=P_2=P$. The set of extreme points shall be denoted by $\text{ ext }\big (\text{ core }(\mu )\big )$.

A property that an exact capacity may satisfy is that of supermodularity, also called convexity in coalitional game theory, which means that:

$$\begin{aligned} \mu (A\cup B)+\mu (A\cap B)\ge \mu (A)+\mu (B) \quad \forall A,B\subseteq \mathcal {X}. \end{aligned}$$

When the capacity is supermodular, the set of extreme points of the core is given by $\{P_{\sigma } \mid \sigma \in S_n\}$ (Shapley, 1971), where $S_n$ denotes the set of permutations of $\{1,\ldots ,n\}$, and given $\sigma \in S_n$, $P_{\sigma }$ is determined by:

$$\begin{aligned} P_{\sigma }\big ( \{x_{\sigma (1)},\ldots ,x_{\sigma (i)}\}\big )=\mu \big ( \{ x_{\sigma (1)},\ldots ,x_{\sigma (i)} \} \big ) \quad \forall i=1,\ldots ,n. \end{aligned}$$

(1)

A capacity can be equivalently represented in terms of its Möbius inverse, which is the function $m:\mathcal {P}(\mathcal {X})\rightarrow \mathbb {R}$ given by:

$$\begin{aligned} m(A)=\sum _{B\subseteq A} (-1)^{\mid A\setminus B \mid }\mu (B) \quad \forall A\subseteq \mathcal {X}. \end{aligned}$$

From the Möbius inverse we can retrieve the capacity by means of:

$$\begin{aligned} \mu (A)=\sum _{B\subseteq A}m(B) \quad \forall A\subseteq \mathcal {X}. \end{aligned}$$

The Möbius inverse m takes values in $\mathbb {R}$. When it is non-negative for every event, the capacity $\mu$ is called belief function and its conjugate $\bar{\mu }$ is called plausibility function, creating a bridge with Evidence Theory (Shafer, 1976). Any belief function is a supermodular capacity.

3 Center points of an exact capacity

Next, we introduce the different notions of centroid of the core of a capacity we shall compare in this paper. We shall consider four possibilities: the Shapley value, the average of the extreme points, the incenter with respect to the total variation distance and the contraction centroid. In general, we shall use the notation $\Phi ^{\mu }$ to denote a centroid of a capacity $\mu$. Moreover, we shall assume throughout that the capacity $\mu$ is exact.

3.1 The Shapley value

One of the most popular notions of centroid of a capacity is the Shapley value. It was introduced by Shapley (1953, 1971) in the framework of coalitional game theory, as a “fair” procedure to distribute some wealth between the players. Later on, it was rediscovered in the context of non-additive measures (Dubois & Prade, 1980) and popularised by Smets as the pignistic transformation of a belief function (Smets & Kennes, 1994).

Definition 1

Given an exact capacity $\mu$, its Shapley value is defined as the probability measure associated with the following distribution:

$$\begin{aligned} \Phi _1^\mu (\{x\})=\sum _{A\mid x\notin A}\frac{\mid A\mid !(n-\mid A\mid -1)!}{n!}\big ( \mu (A\cup \{x\})-\mu (A) \big ) \quad \forall x\in \mathcal {X}. \end{aligned}$$

(2)

When $\mu$ is a belief function with Möbius inverse m, it was proven in Smets (2005) that $\Phi _1^\mu$ can be equivalently computed as

$$\begin{aligned} \Phi _1^\mu (\{x\})=\sum _{A\mid x\in A}\frac{m(A)}{\mid A\mid } \quad \forall x\in \mathcal {X}. \end{aligned}$$

(3)

More generally, when $\mu$ is supermodular (and in particular when it is a belief function), the extreme points of $\text{ core }(\mu )$ are given by Eq. (1), and the Shapley value can be computed as

$$\begin{aligned} \Phi _1^\mu (\{x\})=\frac{1}{n!}\sum _{\sigma \in S_n}P_{\sigma }(\{x\}) \quad \forall x\in \mathcal {X}. \end{aligned}$$

(4)

Even if the expressions in Eqs. (3) and (4) were only established for belief functions and supermodular capacities, respectively, it follows from basic combinatorial analysis that they can be extended to arbitrary capacities:²

Proposition 1

Let $\mu$ be an exact capacity with Möbius inverse m, and let $\Phi _1^\mu$ be given by Eq. (2). Then

$$\begin{aligned} \Phi _1^{\mu }(\{x\})=\sum _{A\mid x\in A}\frac{m(A)}{\mid A\mid }=\sum _{\sigma \in S_n} \frac{P_{\sigma }(\{x\})}{n!} \quad \forall x\in \mathcal {X}, \end{aligned}$$

(5)

where $P_{\sigma }$ is given by Eq. (1).

It is worth remarking that, even if m(A) may be negative in some events A, Proposition 1 implies that the aggregation by means of Eq. (5) always produces a non-negative value.

While the Shapley value seems like a reasonable choice as a central point, it has one important drawback: it is only guaranteed to belong to the core of $\mu$ (i.e., we can only assure that $\Phi _1^\mu \ge \mu$) when the capacity $\mu$ is supermodular. In those cases, it follows immediately from Eq. (4) and the fact that $P_{\sigma }$ dominates $\mu$ for every permutation $\sigma \in S_n$.

More generally, when the exact capacity $\mu$ is not supermodular, we can only assure that $\Phi _1^\mu$ dominates $\mu$ for small cardinalities, as showed by Baroni and Vicig (2005, Prop.5). We refer to Miranda and Montes (2018) for a study of the connection between the Shapley value and the core of the capacity.

3.2 Average of the extreme points

The second possibility we consider in this paper is the average of the extreme points of the core of the capacity:

Definition 2

Let $\mu$ be an exact capacity, and denote by $P_1$, ..., $P_k$ the extreme points of its core. The average of the extreme points, also called vertex centroid (Elbassioni & Tiway, 2012), is defined as

$$\begin{aligned} \Phi _2^\mu (\{x\})=\frac{1}{k}\sum _{i=1}^{k} P_i(\{x\}) \quad \forall x\in \mathcal {X}. \end{aligned}$$

(6)

It follows from Definition 2 that $\Phi _2^\mu$ always belongs to $\text{ core }(\mu )$; considering the comments in the previous section, this implies that it need not coincide with the Shapley value when supermodularity is not satisfied. As we shall see later on, they need not coincide either even if the capacity is supermodular: while the set of extreme points of the core is $\{P_{\sigma }\mid \sigma \in S_n\}$, a key difference is that in the computation of Shapley value in Eq. (4) we are allowing for repetitions of the same extreme point, while in Definition 2 we do not.

It is also important to clarify that the average of the extreme points does not generally coincide with the center of gravity of the core, defined as the expectation of the set over a uniform probability distribution.

Example 1

Consider a 3-element possibility space and the exact capacity $\mu$ given by:

$$\begin{aligned} \begin{array}{r|c c c c c c c} A &{} \{x_1\} &{} \{x_2\} &{} \{x_3\} &{} \{x_1,x_2\} &{} \{x_1,x_3\} &{} \{x_2,x_3\} &{} \mathcal {X}\\ \hline \mu (A) &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0.5 &{} 1 \end{array} \end{aligned}$$

This capacity is supermodular³ and the extreme points of $\text{ core }(\mu )$ are given by:

$$\begin{aligned} \begin{array}{c|c c c} &{} \ x_1 \ &{} \ x_2 \ &{} \ x_3 \ \\ \hline P_1 &{} 0 &{} 0 &{} 1\\ P_2 &{} 0 &{} 1 &{} 0 \end{array} \qquad \begin{array}{c|c c c} &{} \ x_1 \ &{} \ x_2 \ &{} \ x_3 \ \\ \hline P_3 &{} \nicefrac {1}{2} &{} \nicefrac {1}{2} &{} 0\\ P_4 &{} \nicefrac {1}{2} &{} 0 &{} \nicefrac {1}{2} \end{array} \end{aligned}$$

The average of the extreme points is given by:

$$\begin{aligned} \Phi _2^{\mu }(\{x_1\})=0.25, \quad \Phi _2^{\mu }(\{x_2\})=\Phi _2^{\mu }(\{x_3\})=0.375, \end{aligned}$$

while the expectation over the core with respect to the uniform distribution E is:

$$\begin{aligned} E(\{x_1\})=\nicefrac {2}{9}, \quad E(\{x_2\})=E(\{x_3\})=\nicefrac {7}{18}. \end{aligned}$$

Therefore, both concepts do not coincide in general. Intuitively, if we assume that the mass is uniformly distributed over $\text{ core }(\mu )$, we can see in Fig. 1 that there is more mass for values of $x_1$ closer to 0 than closer to $\nicefrac {1}{2}$, whence $E(\{x_1\})$ must be smaller than $\Phi _2^\mu (\{x_1\})=\nicefrac {1}{4}$.

In fact, this example also shows that the center of gravity does not coincide with the Shapley value either, even in this case where $\mu$ is supermodular, because:

$$\begin{aligned} \Phi _1^\mu (\{x_1\})=\nicefrac {1}{6}, \quad \Phi _1^\mu (\{x_2\})=\Phi _1^\mu (\{x_3\})=\nicefrac {5}{12}. \quad \blacklozenge \end{aligned}$$

Fig. 1

Graphical representation of the core of $\mu$ in Example 1 (in gray) together with $\Phi _2^\mu$ (in blue) and the center of gravity E (in red)

While the center of gravity has the advantage of being applicable over any closed and convex set in $\mathbb {P}(\mathcal {X})$, and not only on polytopes, it also has the drawback of being computationally more expensive (see for example (Elbassioni & Tiway, 2012)). For this reason, we have left this approach out of our study.

3.3 Incenter

Next we consider the incenter, that corresponds to the center (or centers) of the largest balls included in the interior of the core of the capacity.⁴ To make this notion precise, we must specify the distance under which the balls are defined. In this respect, there are several possibilities, such as the Euclidean or the $L_1$ distances for example, or even the Kullback-Leibler divergence. We have considered in this paper the total variation distance (Levin et al., 2009), which is the one associated with the supremum norm:

$$\begin{aligned} d_{TV}(P,Q)=\max _{A\subseteq \mathcal {X}}\mid P(A)-Q(A)\mid \qquad \forall P,Q\in \mathbb {P}(\mathcal {X}). \end{aligned}$$

Our choice of this distance is due to the fact that the closed balls it induces are always polytopes, unlike the case of the Euclidean distance or the Kullback-Leibler divergence, and those closed balls correspond to exact capacities satisfying supermodularity (Montes et al. 2020b, Sec.2). Although the $L_1$ distance also induces a polytope, it does not correspond to the core of an exact capacity and our analysis in Destercke et al. (2022), Montes et al. (2020b) shows that its use is rather complex. In what follows, for the sake of notational simplicity, the total variation distance will be simply denoted by d. Moreover, we shall denote

$$\begin{aligned} B_c^{\alpha }(P_0)=\{P\in \mathbb {P}(\mathcal {X})\mid d(P,P_0)\le \alpha \}, \ \ B_o^{\alpha }(P_0)=\{P\in \mathbb {P}(\mathcal {X})\mid d(P,P_0)<\alpha \}\end{aligned}$$

the closed and open balls centered on $P_0$ and with radius $\alpha$, respectively.

This leads us to the following definition:

Definition 3

Let $\mu$ be an exact capacity. The incenter radius of $\text{ core }(\mu )$ is defined as

$$\begin{aligned} \alpha _{I}=\sup \big \{\alpha >0\mid \exists P_0\in \text{ core }(\mu ) \text{ such } \text{ that } B_{o}^{\alpha }(P_0)\subseteq \text{ core }(\mu )\cap \mathbb {P}^*(\mathcal {X}) \big \}. \end{aligned}$$

(7)

Any $P_0\in \text{ core }(\mu )$ such that $B_{o}^{\alpha _{I}}(P_0)\subseteq \text{ core }(\mu )$ is called an incenter of $\mu$.

It may be surprising to see that in Eq. (7) we are requiring the inclusion of the open ball $B_{o}^{\alpha }(P_0)$ into the intersection $\text{ core }(\mu )\cap \mathbb {P}^*(\mathcal {X})$. The reason is that if we simply require $B_{o}^{\alpha }(P_0)\subseteq \text{ core }(\mu )$ then we may obtain centers that are in the boundary of the core, something in our view counterintuitive. This is illustrated in the following example.

Example 2

Let $\mathcal {X}=\{x_1,x_2,x_3\}$ and consider the exact capacity $\mu$ given by:

$$\begin{aligned} \begin{array}{r|c c c c c c c} A &{} \{x_1\} &{} \{x_2\} &{} \{x_3\} &{} \{x_1,x_2\} &{} \{x_1,x_3\} &{} \{x_2,x_3\} &{} \mathcal {X}\\ \hline \mu (A) &{} 0.5 &{} 0.1 &{} 0 &{} 0.75 &{} 0.5 &{} 0.1 &{} 1 \end{array} \end{aligned}$$

The value $\alpha _{I}$ determined by Eq. (7) is $\alpha _{I}=0.125$, and any convex combination of the probability measures $Q_1$ and $Q_2$ given by:

$$\begin{aligned} Q_1(\{x_1\})&=0.65, \quad Q_1(\{x_2\})=0.225, \quad Q_1(\{x_3\})=0.125,\\ Q_2(\{x_1\})&=0.625, \quad Q_2(\{x_2\})=0.25, \quad Q_2(\{x_3\})=0.125. \end{aligned}$$

is an incenter of $\text{ core }(\mu )$.

Besides, if we do not require the open ball to be included in $\mathbb {P}^{*}(\mathcal {X})$, we obtain:

$$\begin{aligned} \sup \{\alpha >0\mid \exists P_0\in \text{ core }(\mu ) \text { such that } B_o^{\alpha }(P_0)\subseteq \text{ core }(\mu )\}=0.2, \end{aligned}$$

and the only $P_0\in \text{ core }(\mu )$ satisfying $B_{o}^{\tiny 0.2}(P_0)\subseteq \text{ core }(\mu )$ is given by:

$$\begin{aligned} P_0(\{x_1\})=0.7, \quad P_0(\{x_2\})=0.3, \quad P_0(\{x_3\})=0. \end{aligned}$$

This probability measure belongs to the boundary of $\text{ core }(\mu )$, which leads us to believe that it does not adequately represent the idea underlying the notion of incenter.

This example also shows the necessity of taking the open ball rather than the closed in the definition of the incenter radius in Eq. (7). The graphical representation of $P_0, Q_1$ and $Q_2$ can be seen in Fig. 2. $\blacklozenge$

Fig. 2

Graphical representation of the core of $\text{ core }(\mu )$ in Example 2, the incenters $Q_1$ and $Q_2$ (in blue) and the probability $P_0$ (in red)

On the other hand, when the core of the capacity is included in the interior of the simplex, then we immediately have that $B_o^{\alpha }(P)\subseteq \text{ core }(\mu ) \cap \mathbb {P}^*(\mathcal {X})$ if and only if $B_c^{\alpha }(P)\subseteq \text{ core }(\mu )$, whence Eq. (7) can be rewritten as

$$\begin{aligned} \alpha _{I}=\sup \big \{\alpha >0\mid \exists P_0\in \text{ core }(\mu ) \text{ such } \text{ that } B_{c}^{\alpha }(P_0)\subseteq \text{ core }(\mu )\big \}. \end{aligned}$$

(8)

A first question arising naturally from the definition of incenter is whether it always exists. Our next result shows that this is indeed the case.

Proposition 2

Consider an exact capacity $\mu$ such that $\text{ core }(\mu )$ has a non-empty interior. Then the value $\alpha _{I}$ given by Eq. (7) is a maximum. As a consequence, the incenter of $\mu$ always exists.

On the other hand, a capacity may have more than one incenter; while this was already showed by Example 2, we next give another example where the core of the capacity is included in $\mathbb {P}^*(\mathcal {X})$:

Example 3

Let $\mathcal {X}=\{x_1,x_2,x_3\}$, and consider the capacity $\mu$ given by:

$$\begin{aligned} \begin{array}{r|c c c c c c c c} A &{} \{x_1\} &{} \{x_2\} &{} \{x_3\} &{} \{x_1,x_2\} &{} \{x_1,x_3\} &{} \{x_2,x_3\} &{} \mathcal {X}\\ \hline \mu (A) &{} 0.5 &{} 0.1 &{} 0.1 &{} 0.65 &{} 0.75 &{} 0.2 &{} 1 \end{array} \end{aligned}$$

The value $\alpha _{I}$ determined in Eq. (7) is given by:

$$\begin{aligned} \alpha _{I}=\max \big \{ \alpha \mid \exists P_0\in \text{ core }(\mu ) \text{ such } \text{ that } B_{o}^{\alpha }(P_0)\subseteq \text{ core }(\mu )\cap \mathbb {P}^{*}(\mathcal {X})\big \}=0.075. \end{aligned}$$

To see that there is more than one $P_0\in \text{ core }(\mu )$ such that $B_{o}^{\alpha _I}(P_0)\subseteq \text{ core }(\mu )\cap \mathbb {P}^{*}(\mathcal {X})$, note that the set of such $P_0$ is given by the probability measures $Q_1,Q_2$ defined below as well as any convex combination $Q_{\beta }=\beta Q_1+(1-\beta ) Q_2$ for $\beta \in [0,1]$:

$$\begin{aligned} Q_1(\{x_1\})&=0.575, \quad Q_1(\{x_2\})=0.175, \quad Q_1(\{x_3\})=0.25.\\ Q_2(\{x_1\})&=0.65, \quad Q_2(\{x_2\})=0.175, \quad Q_2(\{x_3\})=0.175. \end{aligned}$$

Figure 3 gives the graphical representation of the core of $\mu$ as well as the balls $B_{c}^{\alpha _I}(Q_1)$, $B_{c}^{\alpha _I}(Q_2)$ and $B_{c}^{\alpha _I}(Q_{\beta })$ for $\beta =0.5$. $\blacklozenge$

Fig. 3

Graphical representation of the core of $\mu$, the incenters $Q_1$ and $Q_2$ with the ball they induce (left hand-side figure) and the incenter $Q_{\beta }$ for $\beta =0.5$ with the ball it induces (right hand-side figure)

Due to the lack of uniqueness, we shall denote by $\Psi _3^\mu$ the non-empty set of incenters of the capacity $\mu$.

A dual approach to the one above is to consider the circumcenter of the capacity, which is the center (or centers) of the smallest balls that include the core. Formally, we may consider

$$\begin{aligned} \alpha _{C}=\inf \big \{\alpha >0\mid \exists P_0\in \mathbb {P}(\mathcal {X}) \text{ such } \text{ that } B_{c}^{\alpha }(P_0)\supseteq \text{ core }(\mu ) \big \}, \end{aligned}$$

and then consider the set of those P such that $B_{c}^{\alpha _{C}}(P)\supseteq \text{ core }(\mu )$. However, this approach has the two drawbacks we have discussed so far: not only it need not lead to a unique solution, but also it may produce values outside the core, as showed in Bader et al. (2012); this last issue may be overcome by considering instead

$$\begin{aligned} \alpha '_{C}=\inf \big \{\alpha >0\mid \exists P_0 \in \text{ core }(\mu ) \text{ such } \text{ that } B_{c}^{\alpha }(P_0)\supseteq \text{ core }(\mu ) \big \}, \end{aligned}$$

and by calling circumcenter those probabilities $P\in \text{ core }(\mu )$ such that $B_d^{\alpha '_C}(P)\supseteq \text{ core }(\mu )$. However, this second approach does not prevent us from obtaining circumcenters that lie in the boundary of the core, leading to the same counterintuitive situation we discussed in Example 2.

Example 4

Consider $\mathcal {X}=\{x_1,x_2,x_3\}$ and the capacity $\mu$ given by:

$$\begin{aligned} \begin{array}{r|c c c c c c c} A &{} \{x_1\} &{} \{x_2\} &{} \{x_3\} &{} \{x_1,x_2\} &{} \{x_1,x_3\} &{} \{x_2,x_3\} &{} \mathcal {X}\\ \hline \mu (A) &{} 0.3 &{} 0.2 &{} 0.2 &{} 0.7 &{} 0.6 &{} 0.5 &{} 1 \end{array} \end{aligned}$$

This capacity is exact and the extreme points of its core are given by:

$$\begin{aligned} \begin{array}{c|c c c} &{} \ x_1 \ &{} \ x_2 \ &{} \ x_3 \ \\ \hline P_1 \ {} &{} 0.3 &{} 0.4 &{} 0.3\\ P_2 \ {} &{} 0.5 &{} 0.2 &{} 0.3 \end{array} \qquad \begin{array}{c|c c c} &{} \ x_1 \ &{} \ x_2 \ &{} \ x_3 \ \\ \hline P_3 \ {} &{} 0.4 &{} 0.4 &{} 0.2\\ P_4 \ {} &{} 0.5 &{} 0.3 &{} 0.2 \end{array} \end{aligned}$$

It follows that $\alpha '_C=0.1$ and that the only circumcenter is $P_0$ given by $P_0(\{x_1\})=0.4$, $P_0(\{x_2\})=P_0(\{x_3\})=0.3$. The graphical representation of the core of $\mu$ and the ball $B_c^{\alpha '_C}(P_0)$ can be seen in Fig. 4. $\blacklozenge$

Fig. 4

Graphical representation of the core of $\text{ core }(\mu )$ (in gray) in Example 4, as well as the ball $B_c^{\alpha '_C}(P_0)$

3.4 Contraction centroid

Our fourth and last approach is motivated by the lack of uniqueness that has been illustrated in the case of the incenter.

Given an exact capacity $\mu$ with conjugate $\bar{\mu }$, we can split the events in $\mathcal {X}$ into $\mathcal {L}^{=}$ and $\mathcal {L}^{>}$ such that:

$$\begin{aligned} \mu (A)=\bar{\mu }(A) \quad \forall A\in \mathcal {L}^{=} \ \text{ and } \ \mu (A)< \bar{\mu }(A)\quad \forall A\in \mathcal {L}^{>}. \end{aligned}$$

Using this notation, the core of $\mu$ can be expressed as:

$$\begin{aligned} \text{ core }(\mu )=\big \{ P\in \mathbb {P}(\mathcal {X})\mid P(A)=\mu (A) \ \forall A\in \mathcal {L}^{=}, \ P(A)\ge \mu (A) \ \forall A\in \mathcal {L}^{>} \big \}. \end{aligned}$$

(9)

Note that when $\mathcal {L}^>$ is empty we obtain that $\mu (A)=\bar{\mu }(A)$ for any $A\subseteq \mathcal {X}$, meaning that $\mu$ is additive and that its core contains one single element.

The idea in this approach is to contract the core in a uniform manner as long as we can, and then proceed in the same way by reducing the number of constraints. More specifically, we increase the value of the capacity in a constant amount $\alpha$ in all the events $A\in \mathcal {L}^{>}$. In this respect, we wonder whether there is some value $\alpha$ small enough such that this approach gives rise to a non-empty core, and also which is the maximum/supremum we can consider. Our next result gives the answer to both questions.

Proposition 3

Consider an exact capacity $\mu$, and let us express its core as in Eq. (9) using the sets $\mathcal {L}^{=}$ and $\mathcal {L}^{>}$. For a given $\alpha >0$, let us define:

$$\begin{aligned} \text{ core }(\mu )_{\alpha }=\big \{ P\in \mathbb {P}(\mathcal {X})\mid P(A)=\mu (A) \ \forall A\in \mathcal {L}^{=}, \ P(A)\ge \mu (A)+\alpha \ \forall A\in \mathcal {L}^{>} \big \}, \end{aligned}$$

and let $\Lambda =\{\alpha >0\mid \text{ core }(\mu )_{\alpha }\ne \emptyset \}$. It holds that:

• (a) $\Lambda \ne \emptyset$.

• (b) The set $\Lambda$ has a maximum $\alpha _{S}$.

• (c) There exists some $A\in \mathcal {L}^{>}$ such that P(A) is constant for any $P\in \text{ core }(\mu )_{\alpha _S}$.

This result assures that we can uniformly increase the capacity in the events whose value is imprecise (i.e., such that $\mu (A)<\bar{\mu }(A)$) and that when the process stops the size of $\mathcal {L}^{>}$ has decreased in the sense that, given the exact capacity $\mu _{\alpha _S}$ that is the lower envelope of the set

$$\begin{aligned} \text{ core }(\mu )_{\alpha _{S}}=\big \{ P\in \mathbb {P}(\mathcal {X})\mid P(A)\ge \mu (A)+\alpha _{S} \ \ \forall A\in \mathcal {L}^{>}\big \} \end{aligned}$$

and its conjugate $\bar{\mu }_{\alpha _S}$, then $\{A \mid \mu _{\alpha _S}(A)<\bar{\mu }_{\alpha _S}(A)\}\subsetneq \mathcal {L}^{>}$.

This means that we may apply the same procedure to the capacity $\mu _{\alpha _S}$, and after iterating it a finite number of times, we obtain a set formed by a single element that we shall call the contraction centroid. In other words, the procedure leads to the values $\alpha _{S_1}=\max \Lambda _1,\ldots ,\alpha _{S_l}=\max \Lambda _l$ and the chain of nested sets⁵

$$\begin{aligned} \text{ core }(\mu )\supset \text{ core }(\mu )_{\alpha _{S_1}}\supset \ldots \supset \text{ core }(\mu )_{\alpha _{S_l}}=\big \{ \Phi _4^\mu \big \}. \end{aligned}$$

(10)

$\Phi _4^\mu$ is the contraction centroid of the capacity $\mu$.

Let us illustrate this procedure with an example.

Example 5

Consider again the exact capacity from Example 3. There, the set $\mathcal {L}^{=}$ only contains the trivial events $\emptyset$ and $\mathcal {X}$, while $\mathcal {L}^{>}$ contains the six non-trivial events:

$$\begin{aligned} \begin{array}{r|c c c c c c c c} A &{} \{x_1\} &{} \{x_2\} &{} \{x_3\} &{} \{x_1,x_2\} &{} \{x_1,x_3\} &{} \{x_2,x_3\} &{}\mathcal {X}\\ \hline \mu (A) &{} 0.5 &{} 0.1 &{} 0.1 &{} 0.65 &{} 0.75 &{} 0.2&{} 1\\ \bar{\mu }(A) &{} 0.8 &{} 0.25 &{} 0.35 &{} 0.9 &{} 0.9 &{} 0.5&{} 1 \end{array} \end{aligned}$$

Let us see that $\alpha _{S_1}=\max \Lambda =0.075$. On the one hand, $\text{ core }(\mu )_{\alpha _{S_1}}$ is non-empty because it includes for instance the probability measure P given by:

$$\begin{aligned} P(\{x_1\})=0.625, \quad P(\{x_2\})=0.175, \quad P(\{x_3\})=0.2. \end{aligned}$$

To see on the other hand that this is the maximum value of $\Lambda$, note that if we increase $\mu$ in $\alpha >0$, to keep exactness it should be that:

$$\begin{aligned} 1\ge \big (\mu (\{x_2\})+\alpha \big )+\big (\mu (\{x_1,x_3\})+\alpha \big )=0.85+2\alpha , \end{aligned}$$

whence $\alpha \le 0.075$. Therefore, $\alpha _{S_1}=0.075$, and this gives rise to the following core:

$$\begin{aligned} \text{ core }(\mu )_{\alpha _{S_1}}=\big \{ P\in \mathbb {P}(\mathcal {X})\mid P(A)\ge \mu (A)+\alpha _{S_1} \ \ \forall A\ne \emptyset ,\mathcal {X}\big \}. \end{aligned}$$

The exact capacity $\mu _1$ determined as the lower envelope of $\text{ core }(\mu )_{\alpha _{S_1}}$ and its conjugate $\overline{\mu }_1$ are given by:

$$\begin{aligned} \begin{array}{r|c c c c c c c} A &{} \ \{x_1\} \ {} &{} \ \{x_2\} \ {} &{} \ \{x_3\} \ {} &{} \ \{x_1,x_2\} \ {} &{} \ \{x_1,x_3\} \ {} &{} \ \{x_2,x_3\} \ {} &{} \mathcal {X}\\ \hline \mu _1(A) &{} \ 0.575 \ {} &{} \ 0.175 \ {} &{} \ 0.175 \ {} &{} \ 0.75 \ {} &{} \ 0.825 \ {} &{} \ 0.35 \ {} &{} 1 \\ \overline{\mu }_1(A) &{} 0.65 &{} 0.175 &{} 0.25 &{} 0.825 &{} 0.825 &{} 0.425 &{} 1 \end{array} \end{aligned}$$

Let us apply now the procedure to this capacity. The sets $\mathcal {L}_1^{>}$ and $\mathcal {L}_1^{=}$ are given by:

$$\begin{aligned} \mathcal {L}_1^{=}=\big \{\{x_2\},\{x_1,x_3\},\emptyset ,\mathcal {X}\big \}, \ \text{ and } \ \mathcal {L}_1^{>}=\big \{ \{x_1\},\{x_3\},\{x_1,x_2\},\{x_2,x_3\}\big \}, \end{aligned}$$

i.e., there are two non trivial events whose value is now fixed, $\{x_2\}$ and $\{x_1,x_3\}$.

Repeating the same steps, we obtain $\alpha _{S_2}=0.0375$, and in this case $\text{ core }(\mu )_{\alpha _{S_2}}$ is formed by a single probability measure, that is therefore the contraction centroid $\Phi _4^{\mu }$. It is given by:

$$\begin{aligned} \Phi _4^{\mu }(\{x_1\})=0.6125, \quad \Phi _4^{\mu }(\{x_2\})=0.175,\quad \Phi _4^{\mu }(\{x_1\})=0.2125. \end{aligned}$$

(11)

In Fig. 5 we have depicted the sets $\text{ core }(\mu )_{\alpha _{S_1}}$ (in blue) and $\text{ core }(\mu )_{\alpha _{S_2}}$ (in red), as well as the initial core of $\mu$. $\blacklozenge$

Fig. 5

Graphical representation of the core of $\mu$, as well as the sets $\text{ core }(\mu )_{\alpha _{S_1}}$ (in blue) and $\text{ core }(\mu )_{\alpha _{S_2}}$ (in red)

In the above procedure, it is worth mentioning that the lower envelope of the set $\text{ core }(\mu )_{\alpha }$ does not necessarily coincide with the capacity $\mu '$ given by

$$\begin{aligned} \mu '(A)={\left\{ \begin{array}{ll} \mu (A)+\alpha &{}\text { if } A\in \mathcal {L}^{>} \\ \mu (A) &{}\text { otherwise}. \end{array}\right. } \end{aligned}$$

While by construction it dominates this capacity, they may not agree on some events because $\mu '$ need not be exact. This can be seen in Example 5, where given $A=\{x_1,x_2\}$, it holds that $\mu '(A)=\mu (A)+\alpha _{S_1}=0.65+0.075<0.75=\mu _1(A)$.

Proposition 3 assures that there exists a maximum value $\alpha _S=\max \Lambda$ giving rise to a non-empty set. This naturally leads us to the problem of computing more efficiently the value of $\alpha _S$. Our next result gives an explicit formula for $\alpha _S$ for a particular case of exact capacities which only coincide with their conjugate for the trivial cases of $\emptyset ,\mathcal {X}$.

Definition 4

Let $\mu$ be an exact capacity with conjugate $\bar{\mu }$. We shall call $\mu$ maximally imprecise when $\mu (A)<\bar{\mu }(A)$ for every $A\ne \emptyset ,\mathcal {X}$.

Let us define

$$\begin{aligned} \mathbb {A}(\mathcal {X})=\bigg \{\mathcal {A}=\big (A_i\big )_{i=1,\ldots ,k}\mid \exists \beta _{\mathcal {A}}\in \mathbb {N} \text{ such } \text{ that } \sum _{i=1}^k I_{A_i}=\beta _{\mathcal {A}} \bigg \}. \end{aligned}$$

In other words, $\mathbb {A}(\mathcal {X})$ is the class of all finite families of subsets of $\mathcal {X}$ such that every $x\in \mathcal {X}$ belongs to the same number of elements in the family. Note that in each of these families $\mathcal {A}$ there may be repeated elements, i.e., we may consider for instance the family $\mathcal {A}=\big (\{x_1\},\{x_1\},\{x_2\},\{x_2,x_3\},\{x_3\}\big )$ on $\mathcal {X}=\{x_1,x_2,x_3\}$. We consider on $\mathbb {A}(\mathcal {X})$ the partial order determined by the inclusion, i.e., we say that $\mathcal {A}_1\subseteq \mathcal {A}_2$ when each element in the family $\mathcal {A}_1$ also belongs to the family $\mathcal {A}_2$.

Theorem 4

Let $\mu$ be a maximally imprecise exact capacity. Then

$$\begin{aligned} \alpha _S=\min _{\mathcal {A}\in \mathbb {A}(\mathcal {X})} h_{\mathcal {A}}, \text { where } h_{\mathcal {A}}:=\frac{1}{\mid \mathcal {A}\mid }\left( \beta _{\mathcal {A}}-\sum _{A\in \mathcal {A}}\mu (A) \right) . \end{aligned}$$

(12)

Let us return to the running Example 5. In that case, $\alpha _S$ satisfies Eq. (12) for $\mathcal {A}=\big (\{x_2\},\{x_1,x_3\}\big )$ and $\beta _{\mathcal {A}}=1$, giving that

$$\begin{aligned} \frac{1}{\mid \mathcal {A}\mid }\big ( \beta _{\mathcal {A}}-\mu (\{x_2\})-\mu (\{x_1,x_3\}) \big )=\frac{1}{2}(1-0.1-0.75)=0.075, \end{aligned}$$

which is indeed the value we obtained in Example 5.

The computation of $\alpha _S$ in Eq. (12) requires the computation of the value $h_{\mathcal {A}}$ for all the families $\mathcal {A}$. Our next result shows a more tractable expression when the capacity is supermodular. For this aim, we denote by $\mathbb {A}^{*}(\mathcal {X})$ the subclass of $\mathbb {A}(\mathcal {X})$ formed by the partitions of $\mathcal {X}$.

Theorem 5

Let $\mu$ be a maximally imprecise supermodular capacity with conjugate $\bar{\mu }$. Then:

$$\begin{aligned} \alpha _S=\min _{\mathcal {A}\in \mathbb {A}^{*}(\mathcal {X})}\left\{ \frac{1-\sum _{A\in \mathcal {A}}\mu (A)}{\mid \mathcal {A}\mid }, \frac{\sum _{A\in \mathcal {A}}\bar{\mu }(A)-1}{\mid \mathcal {A}\mid } \right\} . \end{aligned}$$

(13)

This means that, under supermodularity, it suffices to focus on partitions of $\mathcal {X}$, which simplifies considerably the computation of Eq. (12).

Example 6

Consider again our running Example 3. The capacity in that example is supermodular because any exact capacity in a 3-element space is, so Theorem 5 is applicable. Next table summarises the values associated with each partition in Eq. (13):

$$\begin{aligned} \begin{array}{r|c|cc} \mathcal {A}\in \mathbb {A}^{*}(\mathcal {X}) &{} \mid \mathcal {A}\mid &{} \nicefrac {\left( 1-\sum _{A\in \mathcal {A}}\mu (A)\right) }{\mid \mathcal {A}\mid } &{} \nicefrac {\left( \sum _{A\in \mathcal {A}}\bar{\mu }(A)-1\right) }{\mid \mathcal {A}\mid } \\ \hline \{x_1\}, \{x_2\}, \{x_3\} &{} 3 &{} \nicefrac {\left( 1-(0.5+0.1+0.1)\right) }{3}=0.1 &{} \nicefrac {\left( 0.8+0.25+0.35-1\right) }{3}=0.1\overline{3} \\ \hline \{x_1,x_2\}, \{x_3\} &{} 2 &{} \nicefrac {\left( 1-(0.65+0.1)\right) }{2}=0.125 &{} \nicefrac {\left( (0.9+0.35)-1\right) }{2}=0.125 \\ \hline \{x_1,x_3\}, \{x_2\} &{} 2 &{} \nicefrac {\left( 1-(0.75+0.1)\right) }{2}=0.075 &{} \nicefrac {\left( (0.9+0.25)-1\right) }{2}=0.075 \\ \hline \{x_2,x_3\}, \{x_1\} &{} 2 &{} \nicefrac {\left( 1-(0.2+0.5)\right) }{2}=0.15 &{} \nicefrac {\left( (0.5+0.8)-1\right) }{2}=0.15 \\ \hline \end{array} \end{aligned}$$

The minimum of these values is $\alpha _S=0.075$ and it is attained in the partition $\mathcal {A}$ formed by $\{x_1,x_3\}$ and $\{x_2\}$, both for the capacity and its conjugate. This is in line with our comments in Example 5. $\blacklozenge$

Let us show that the result in Theorem 5 does not generalise to the case where $\mu$ is an exact capacity but not supermodular:

Example 7

Consider $\mathcal {X}=\{x_1,x_2,x_3,x_4\}$ and the capacity $\mu$, with conjugate $\bar{\mu }$, given by:

$$\begin{aligned} \begin{array}{r|c c} A &{} \mu (A) &{} \bar{\mu }(A)\\ \hline \{x_1\} &{} 0 &{} 0.1\\ \{x_2\} &{} 0.1 &{} 0.2\\ \{x_3\} &{} 0.5 &{} 0.6\\ \{x_4\} &{} 0.2 &{} 0.3\\ \{x_1,x_2\} &{} 0.2 &{} 0.3\\ \{x_1,x_3\} &{} 0.5 &{} 0.6\\ \{x_1,x_4\} &{} 0.2 &{} 0.4 \end{array} \qquad \begin{array}{r|c c} A &{} \mu (A) &{} \bar{\mu }(A)\\ \hline \{x_2,x_3\} &{} 0.6 &{} 0.8\\ \{x_2,x_4\} &{} 0.4 &{} 0.5\\ \{x_3,x_4\} &{} 0.7 &{} 0.8\\ \{x_1,x_2,x_3\} &{} 0.7 &{} 0.8\\ \{x_1,x_2,x_4\} &{} 0.4 &{} 0.5\\ \{x_1,x_3,x_4\} &{} 0.8 &{} 0.9\\ \{x_2,x_3,x_4\} &{} 0.9 &{} 1 \end{array} \end{aligned}$$

The extreme points of its core are given by:

$$\begin{aligned} \begin{array}{c|c c c c} &{} \ x_1 \ {} &{} \ x_2 \ {} &{} \ x_3 \ {} &{} \ x_4 \ \\ \hline P_1 &{} \ 0.1 \ {} &{} \ 0.2 \ {} &{} \ 0.5 \ {} &{} \ 0.2 \ \\ P_2 &{} \ 0 \ {} &{} \ 0.2 \ {} &{} \ 0.6 \ {} &{} \ 0.2 \ \end{array} \qquad \begin{array}{c|c c c c} &{} \ x_1 \ {} &{} \ x_2 \ {} &{} \ x_3 \ {} &{} \ x_4 \ \\ \hline P_3 &{} \ 0.1 \ {} &{} \ 0.1 \ {} &{} \ 0.5 \ {} &{} \ 0.3 \ \\ P_4 &{} \ 0 \ {} &{} \ 0.2 \ {} &{} \ 0.5 \ {} &{} \ 0.3 \ \end{array} \end{aligned}$$

It follows that $\mu$ is an exact capacity, and that it is maximally imprecise too. Note that $\mu$ is not supermodular, since

$$\begin{aligned} 0.6=\mu (\{x_1,x_2\})+\mu (\{x_2,x_4\})\not \le \mu (\{x_1,x_2,x_4\})+\mu (\{x_2\})=0.5. \end{aligned}$$

The value of $\alpha _S$ is attained with $\mathcal {A}=\big ( \{x_3\},\{x_2,x_4\},\{x_1,x_2\},\{x_1,x_3,x_4\} \big )$, that produces:

$$\begin{aligned} \alpha _S=\frac{1}{4}\big (2-\mu (\{x_3\})-\mu (\{x_2,x_4\})-\mu (\{x_1,x_2\})-\mu (\{x_1,x_3,x_4\})\big )=0.025. \end{aligned}$$

To see this, note any $P\in \text{ core }(\mu )_{0.025}$ satisfies:

$$\begin{aligned} 2= & {} \beta _{\mathcal {A}} \ge \big (\mu (\{x_3\})+0.025\big )+\big (\mu (\{x_2,x_4\})+0.025\big )\\{} & {} +\big (\mu (\{x_1,x_2\})+0.025\big )+\big (\mu (\{x_1,x_3,x_4\})+0.025\big )=2, \end{aligned}$$

which implies that $P(A)=\mu (A)+0.025$ for every $A\in \mathcal {A}$, and also that $\text{ core }(\mu )_{\alpha }=\emptyset$ for every $\alpha >0.025$. Moreover, the above restrictions imply that

$$\begin{aligned} P(\{x_3\})&=0.525.\\ P(\{x_2,x_4\})&=0.425 \Rightarrow P(\{x_1,x_3\})=0.575, \text{ whence } P(\{x_1\})=0.05.\\ P(\{x_1,x_2\})&=0.225 \Rightarrow P(\{x_2\})=0.175.\\ P(\{x_2,x_4\})&=0.425 \Rightarrow P(\{x_4\})=0.25. \end{aligned}$$

Thus, $\text{ core }(\mu )_{0.025}$ only includes the probability mass function (0.05, 0.175, 0.525, 0.25). It is easy now to verify that this probability measure satisfies $P(A)\ge \mu (A)+0.025$ for any non-trivial event A.

Moreover, the bounds determined by the partitions in $\mathbb {A}^{*}(\mathcal {X})$ are the following:

$$\begin{aligned} \begin{array}{r|c|c c} \text{ Partition } \mathcal {A} &{} \mid \mathcal {A}\mid &{} \nicefrac {(1-\sum _{A\in \mathcal {A}}\mu (A))}{\mid \mathcal {A}\mid } &{} \nicefrac {(\sum _{A\in \mathcal {A}}\bar{\mu }(A)-1)}{\mid \mathcal {A}\mid }\\ \hline \{x_1\},\{x_2\},\{x_3\},\{x_4\} &{} 4 &{} \nicefrac {(1-0.8)}{4} &{} \nicefrac {(1.2-1)}{4}\\ \{x_1,x_2\}, \{x_3\},\{x_4\} &{} 3 &{} \nicefrac {(1-0.9)}{3} &{} \nicefrac {(1.2-1)}{3}\\ \{x_1,x_3\}, \{x_2\},\{x_4\} &{}3 &{} \nicefrac {(1-0.8)}{3} &{} \nicefrac {(1.1-1)}{3}\\ \{x_1,x_4\}, \{x_2\},\{x_3\} &{} 3 &{} \nicefrac {(1-0.8)}{3} &{} \nicefrac {(1.2-1)}{3}\\ \{x_2,x_3\}, \{x_1\},\{x_4\} &{} 3 &{} \nicefrac {(1-0.8)}{3} &{} \nicefrac {(1.2-1)}{3}\\ \{x_2,x_4\}, \{x_1\},\{x_3\} &{} 3 &{} \nicefrac {(1-0.9)}{3} &{} \nicefrac {(1.2-1)}{3}\\ \{x_3,x_4\}, \{x_1\},\{x_2\} &{} 3 &{} \nicefrac {(1-0.8)}{3} &{} \nicefrac {(1.1-1)}{3}\\ \{x_1,x_2\},\{x_3,x_4\} &{} 2 &{} \nicefrac {(1-0.9)}{2} &{} \nicefrac {(1.1-1)}{2}\\ \{x_1,x_3\},\{x_2,x_4\} &{} 2 &{} \nicefrac {(1-0.9)}{2} &{} \nicefrac {(1.1-1)}{2}\\ \{x_1,x_4\},\{x_2,x_3\} &{} 2 &{} \nicefrac {(1-0.8)}{2} &{} \nicefrac {(1.2-1)}{2}\\ \{x_1\},\{x_2,x_3,x_4\} &{} 2 &{} \nicefrac {(1-0.9)}{2} &{} \nicefrac {(1.1-1)}{2}\\ \{x_2\},\{x_1,x_3,x_4\} &{} 2 &{} \nicefrac {(1-0.9)}{2} &{} \nicefrac {(1.1-1)}{2}\\ \{x_3\},\{x_1,x_2,x_4\} &{} 2 &{} \nicefrac {(1-0.9)}{2} &{} \nicefrac {(1.1-1)}{2}\\ \{x_4\},\{x_1,x_2,x_3\} &{} 2 &{} \nicefrac {(1-0.9)}{2} &{} \nicefrac {(1.1-1)}{2} \end{array} \end{aligned}$$

We obtain the minimum value $\frac{0.1}{3}=0.0\overline{3}$, strictly greater than $\alpha _S=0.025$. We conclude that Theorem 5 does not hold without the hypothesis of supermodularity. $\blacklozenge$

3.5 Relationships between the centroids

So far, we have introduced four different notions of the center of an exact capacity. Let us begin by showing that these four notions are indeed different:

Example 8

Consider the capacity defined in Example 3; there, we gave the set of incenters $\Psi _3^\mu$, while the contraction centroid $\Phi _4^\mu$ was given in Example 5. The extreme points of the core of $\mu$ are given by:

From this, we conclude that the average of the extreme points of $\text{ core }(\mu )$ is the probability measure with mass function

$$\begin{aligned} \Phi _2^\mu (\{x_1\})=0.6, \quad \Phi _2^\mu (\{x_2\})=0.17, \quad \Phi _2^\mu (\{x_3\})=0.23. \end{aligned}$$

(14)

On the other hand, the Shapley value is given by

$$\begin{aligned} \Phi _1^\mu (\{x_1\})=0.6\overline{3}, \quad \Phi _1^\mu (\{x_2\})=0.158\overline{3}, \quad \Phi _1^\mu (\{x_3\})=0.208\overline{3}. \end{aligned}$$

This can be derived also using Proposition 1, noting that the permutations lead to the following extreme points:

$$\begin{aligned} \begin{array}{l l|c c c} &{} &{} x_1 &{} x_2 &{} x_3\\ \hline \sigma _1=(1,2,3) &{} \ \ P_{\sigma _1} \ {} &{} 0.5 &{} 0.15 &{} 0.35\\ \sigma _2=(1,3,2) &{} \ \ P_{\sigma _2} \ {} &{} 0.5 &{} 0.25 &{} \ 0.25 \ \\ \sigma _3=(2,1,3) &{} \ \ P_{\sigma _3} \ {} &{} 0.55 &{} 0.1 &{} 0.35 \\ \sigma _4=(2,3,1) &{} \ \ P_{\sigma _4} \ {} &{} 0.8 &{} 0.1 &{} 0.1 \\ \sigma _5=(3,1,2) &{} \ \ P_{\sigma _5} \ {} &{} \ 0.65 \ {} &{} \ 0.25 \ {} &{} 0.1 \\ \sigma _6=(3,2,1) &{} \ \ P_{\sigma _6} \ {} &{} 0.8 &{} 0.1 &{} 0.1 \\ \end{array} \end{aligned}$$

Here, $P_{\sigma _4}=P_{\sigma _6}$, so this extreme point gets twice the weight of the others in the computation of Shapley value, hence the difference with $\Phi _2^\mu$. These centroids are represented in Fig. 6. $\blacklozenge$

Fig. 6

Graphical representation of the centroids of the exact capacity in Example 3: the average of the extreme points (in blue), the Shapley value (in brown), the set of incenters (in red) and the contraction centroid (in black)

Even if the four approaches do not lead to the same solution in general, in Examples 5 and 8 we have seen an example where the contraction centroid $\Phi _4^\mu$ belongs to the set of incenters $\Psi _3^\mu$. This leads us to investigate whether there is a connection between these two approaches. Our next result shows that, under some conditions, the set we obtain in the first step of the contraction approach coincides with the set of incenters.

Proposition 6

Let $\mu$ be a maximally imprecise exact capacity with conjugate $\bar{\mu }$ satisfying $\mu (A)>0$ for every $A\ne \emptyset$, and let $\alpha _S$ be the coefficient defined in Proposition 3. Given $P_0\in \text{ core }(\mu )$ and $\alpha \le \alpha _S$,

$$\begin{aligned} P_0\in \text{ core }(\mu )_{\alpha } \Leftrightarrow B_{c}^{\alpha }(P_0)\subseteq \text{ core }(\mu ). \end{aligned}$$

As a consequence, $\text{ core }(\mu )_{\alpha _S}=\Psi _3^{\mu }$.

On the other hand, an advantage of using the contraction procedure is that the existence of $\Phi _4^\mu$ is guaranteed even if the interior of $\text{ core }(\mu )$ is empty.

4 Properties of the centroids

Next we compare the different centroids in terms of the axiomatic properties they satisfy. There exist several axiomatic characterisations of Shapley value in the context of coalitional game theory; arguably the most important one is that as the unique additive measure $\Phi ^{\mu }$ satisfying the following axioms:

Efficiency:

$\sum _{i=1}^n \Phi ^{\mu }(\{x_i\})=\mu (\mathcal {X})$.

Symmetry:

$\mu \big (A\cup \{x_i\}\big )=\mu \big (A\cup \{x_j\}\big )$ for any $A\subseteq \mathcal {X}\setminus \{x_i,x_j\}$ implies that $\Phi ^{\mu }(\{x_i\})=\Phi ^{\mu }(\{x_j\})$.

Linearity:

$\Phi ^{\lambda _1 \mu _1+\lambda _2\mu _2}=\lambda _1 \Phi ^{\mu _1}+\lambda _2 \Phi ^{\mu _2}$ for any $\lambda _1,\lambda _2\in \mathbb {R}$ and every $\mu _1,\mu _2$.

Null player:

$\mu \big (A\cup \{x_i\}\big )=\mu (A)$ for any $A\subseteq \mathcal {X}\setminus \{x_i\}$ implies $\Phi ^{\mu }(\{x_i\})=0$.

Throughout this paper, we are restricting ourselves to normalised capacities, which implies that (i) the efficiency axiom simplifies to $\sum _{i=1}^n \Phi ^{\mu }(\{x_i\})=1$; and (ii) in order to guarantee that $\lambda _1\mu _1+\lambda _2\mu _2$ is again a normalised capacity, we must have $\lambda _2=1-\lambda _1$, and therefore the linearity axiom implies that $\Phi ^{\lambda \mu _1+(1-\lambda )\mu _2}=\lambda \Phi ^{\mu _1}+(1-\lambda )\Phi ^{\mu _2}$ for any $\lambda \in \mathbb {R}$ and any normalised capacities $\mu _1,\mu _2$ whenever $\lambda \mu _1+(1-\lambda )\mu _2$ is a normalised capacity too.

Next we investigate to which extent these properties are satisfied by the other centroids proposed in this paper. In this respect, note that in the framework of this paper, any center of a capacity shall be a probability measure, whence the efficiency property is trivially satisfied. Note also that when analysing the behaviour of the set of incenters, we shall say that it satisfies a property if and only if any of its elements does.

As we have already mentioned, the Shapley centroid does not necessarily satisfy feasibility, meaning that $\Phi _1^{\mu }$ may not belong to the core of $\mu$. By construction, the vertex, incenter and contraction centroids do satisfy feasibility. With respect to the other properties, it is not difficult to establish the following:

Proposition 7

$\Phi _2^{\mu }$, any $\Phi _3^{\mu }\in \Psi _3^{\mu }$ and $\Phi _4^{\mu }$ satisfy the symmetry and null-player properties.

To see that they do not satisfy linearity in general, whence their difference with Shapley value, consider the following example:

Example 9

Consider $\mathcal {X}=\{x_1,x_2,x_3\}$, the exact capacities $\mu _1,\mu _2$ and their average $\mu :=0.5\mu _1+0.5\mu _2$, given in the following table:

$$\begin{aligned} \begin{array}{r|c c c c c c c} A &{} \ \{x_1\} \ {} &{} \ \{x_2\} \ {} &{} \ \{x_3\} \ {} &{} \ \{x_1,x_2\} \ {} &{} \ \{x_1,x_3\} \ {} &{} \ \{x_2,x_3\} \ &{} \mathcal {X}\\ \hline \mu _1(A) &{} \ 0.1 \ {} &{} \ 0.2 \ {} &{} \ 0.3 \ {} &{} \ 0.35 \ {} &{} \ 0.55 \ {} &{} \ 0.75 \ {} &{} 1 \\ \mu _1(A) &{} \ 0.3 \ {} &{} \ 0.2 \ {} &{} \ 0.1 \ {} &{} \ 0.75 \ {} &{} \ 0.55 \ {} &{} \ 0.35 \ {} &{} 1\\ \mu (A) &{} \ 0.2 \ {} &{} \ 0.2 \ {} &{} \ 0.2 \ {} &{} \ 0.55 \ {} &{} \ 0.55 \ {} &{} \ 0.55 \ {} &{} 1 \end{array} \end{aligned}$$

Because of the symmetry property, it is easy to see that for $\mu$ we obtain that

$$\begin{aligned} \Phi _2^{\mu }=\Phi _3^{\mu }=\Phi _4^{\mu }=\left( \nicefrac {1}{3},\nicefrac {1}{3},\nicefrac {1}{3}\right) . \end{aligned}$$

On the other hand, $\Phi _2^{\mu _1}=(0.17,0.31,0.52)$ and $\Phi _2^{\mu _2}=(0.52,0.31,0.17)$, whence $0.5\cdot \Phi _2^{\mu _1}+0.5\cdot \Phi _2^{\mu _2}=(0.345,0.31,0.345)$.

With respect to the contraction centroid, in this case $\Phi _4^{\mu _1}=(0.175,0.325,0.5)$ and $\Phi _4^{\mu _2}=(0.5,0.325,0.175)$, whence

$$\begin{aligned} 0.5\Phi _4^{\mu _1}+0.5\Phi _4^{\mu _2}=(0.3375,0.325,0.3375). \end{aligned}$$

Finally, with respect to the incenters, $\Psi _3^{\mu _1}$ is the set of convex combinations of $\{(0.175,0.375,0.45), (0.175,0.275,0.55)\}$ and $\Psi _3^{\mu _2}$ is the set of convex combinations of $\{(0.45,0.375,0.175), (0.55,0.275,0.175)\}$. Thus, $\Psi _3^{\mu }$ does not coincide with the set $\big \{0.5Q_1+0.5Q_2 \mid Q_1\in \Psi _3^{\mu _1}, Q_2\in \Psi _3^{\mu _2}\big \}$, because this latter set includes for instance (0.3375, 0.325, 0.3375). As a consequence, none of three centroids satisfies linearity. $\blacklozenge$

Next we consider other desirable properties of a centroid.

Definition 5

Let $\Phi ^\mu$ be a centroid of an exact capacity $\mu$. We say that it satisfies:

• Continuity if for any $\varepsilon >0$, there exists $\delta >0$ such that $d(\mu _1,\mu _2):=\max _{A\subseteq \mathcal {X}}\mid \mu _1(A)-\mu _2(A)\mid <\delta$ implies $d\big (\Phi ^{\mu _1},\Phi ^{\mu _2}\big )<\varepsilon$.

• Ignorance preservation if $\text{ core }(\mu )=\mathbb {P}(\mathcal {X})$, then $\Phi ^\mu$ is the uniform distribution.

When dealing with the incenter, the previous properties should be slightly rewritten due to its lack of uniqueness: the incenter satisfies ignorance preservation if $\text{ core }(\mu )=\mathbb {P}(\mathcal {X})$ implies that $\Psi _{3}^{\mu }$ only contains the uniform distribution; and it satisfies continuity when for any $\varepsilon >0$ there exists some $\delta >0$ such that $d(\mu _1,\mu _2)<\delta$ implies that $d(\nu _1,\nu _2)<\varepsilon$, where $\nu _1$ and $\nu _2$ are the lower envelopes of $\Psi _3^{\mu _1}$ and $\Psi _3^{\mu _2}$.

Proposition 8

• (a) $\Phi _1^\mu$ satisfies continuity and ignorance preservation.

• (b) $\Phi _2^\mu$ satisfies ignorance preservation.

• (c) $\Psi _3^\mu ,\Phi _4^\mu$ satisfy continuity and ignorance preservation.

To see that the average of the extreme points does not satisfy continuity, consider the following example:

Example 10

Consider $\mathcal {X}=\{x_1,x_2,x_3\}$, $\epsilon \in (0,0.05)$ and the exact capacity $\mu$ given by

$$\begin{aligned} \begin{array}{r|c c c c c c c} A &{} \ \{x_1\} \ {} &{} \ \{x_2\} \ {} &{} \ \{x_3\} \ {} &{} \ \{x_1,x_2\} \ {} &{} \ \{x_1,x_3\} \ {} &{} \ \{x_2,x_3\} \ {} &{} \mathcal {X}\\ \hline \mu (A) &{} \ 0.1 \ &{} \ 0.2 \ {} &{} \ 0.3 \ {} &{} \ 0.35 \ {} &{} \ 0.55-2\epsilon \ {} &{} \ 0.75+\varepsilon \ {} &{} 1 \end{array} \end{aligned}$$

Then $\mu$ is supermodular, and the extreme points of $\text{ core }(\mu )$ are given by:

$$\begin{aligned} \begin{array}{l l|c c c} &{} &{} x_1 &{} x_2 &{} x_3\\ \hline \sigma _1=(1,2,3) &{} P_{\sigma _1} &{} 0.1 &{} 0.25 &{} 0.65\\ \sigma _2=(1,3,2) &{} P_{\sigma _2} &{} 0.1 &{} 0.45+2\varepsilon &{} 0.45-2\varepsilon \\ \sigma _3=(2,1,3) &{} P_{\sigma _3} &{} 0.15 &{} 0.2 &{} 0.65 \\ \sigma _4=(2,3,1) &{} P_{\sigma _4} &{} 0.25-\varepsilon &{} 0.2 &{} 0.55+\varepsilon \\ \sigma _5=(3,1,2) &{} P_{\sigma _5} &{} \ 0.25-2\varepsilon \ {} &{} \ 0.45+2\varepsilon \ {} &{} 0.3 \\ \sigma _6=(3,2,1) &{} P_{\sigma _6} &{} 0.25-\varepsilon &{} 0.45+\varepsilon &{} 0.3 \\ \end{array} \end{aligned}$$

All these extreme points are different, and their average is given by

$$\begin{aligned} \Phi _2^{\mu }=\left( \frac{1.1-4\varepsilon }{6},\frac{2+5\varepsilon }{6},\frac{2.9-\varepsilon }{6}\right) . \end{aligned}$$

On the other hand, when $\varepsilon =0$ we obtain that $P_{\sigma _5}=P_{\sigma _6}$, and, as we have seen in Example 9, the average of the extreme points becomes $\Phi _2^{\mu }=(0.17,0.31,0.52)\ne \lim _{\varepsilon \rightarrow 0}(\frac{1.1-4\varepsilon }{6},\frac{2+5\varepsilon }{6},\frac{2.9-\varepsilon }{6})$. $\blacklozenge$

Another desirable property would be that the centroid preserves the same preferences as $\mu$, in the sense that $\mu (A)\ge \mu (B)\Rightarrow \Phi ^{\mu }(A)\ge \Phi ^{\mu }(B)$. Since $\Phi ^{\mu }$ is an additive model, we shall only require this property on the singletons: otherwise the capacity should satisfy

$$\begin{aligned} \mu (A)\ge \mu (B)\Rightarrow \mu (A\cup C)\ge \mu (B\cup C) \quad \forall C\subseteq (A\cup B)^c, \end{aligned}$$

which need not hold. Unfortunately, none of the centroids considered in this paper satisfies the above property, as the following example shows:

Example 11

Consider $\mathcal {X}=\{x_1,x_2,x_3\}$ and let $\mu$ be the exact capacity, with associated Möbius inverse m, given by

$$\begin{aligned} \begin{array}{r|c c c c c c c} A &{} \ \{x_1\} \ {} &{} \ \{x_2\} \ {} &{} \ \{x_3\} \ {} &{} \ \{x_1,x_2\} \ {} &{} \ \{x_1,x_3\} \ {} &{} \ \{x_2,x_3\} \ {} &{} \ \mathcal {X}\ \\ \hline \mu (A) &{} \ 0.18 \ {} &{} \ 0.2 \ {} &{} \ 0.22 \ {} &{} \ 0.54 \ {} &{} \ 0.45 \ {} &{} \ 0.6 \ {} &{} \ 1 \ \\ m(A) &{} \ 0.18 \ {} &{} \ 0.2 \ {} &{} \ 0.22 \ {} &{} \ 0.16 \ {} &{} \ 0.05 \ {} &{} \ 0.18 \ {} &{} \ 0.01 \ \end{array} \end{aligned}$$

that satisfies $\mu (\{x_3\})>\mu (\{x_2\})>\mu (\{x_1\})$. Using Eq. (3), we obtain that $\Phi _1^{\mu }=(0.288\overline{3},0.37\overline{3},0.338\overline{3})$, for which $\Phi _1^{\mu }(\{x_2\})>\Phi _1^{\mu }(\{x_3\})>\Phi _1^{\mu }(\{x_1\})$. Moreover, in this case all permutations of $\mathcal {X}$ produce a different extreme point, whence $\Phi _2^{\mu }=\Phi _1^{\mu }$.

With respect to the incenter, it can be checked that the largest value $\alpha$ such that $\text{ core }(\mu )_\alpha \ne \emptyset$ is $\alpha =0.11$, from which we deduce that $\Psi _3^{\mu }$ is given by all the convex combinations of (0.29, 0.36, 0.35) and (0.29, 0.38, 0.33), all of which produce the same order between $x_1,x_2$ and $x_3$ as $\Phi _1^{\mu }$. This also implies that $\Phi _4^{\mu }=(0.29,0.37,0.34)$. Hence, none of the centroids keeps the same preferences as the original capacity. $\blacklozenge$

5 Centrality measures

More generally, instead of determining which element of the core can be considered its center, we may define a centrality measure, that allows us to quantify how far inside the core an element is.

Fig. 7

Graphical representation of the core of $\mu$ in Example 3

Consider for instance the same capacity as in Example 3, whose core is depicted in Fig. 7. Intuitively, given the probability measures $Q_1$ and $Q_2$ defined as:

$$\begin{aligned} Q_1(\{x_1\})&=0.75, \ Q_1(\{x_2\})=Q_1(\{x_3\})=0.125, \nonumber \\ Q_2(\{x_1\})&=0.65, \ Q_2(\{x_2\})=0.15, \ Q_3(\{x_3\})=0.2, \end{aligned}$$

(15)

and emphasised in red in Fig. 7, $Q_2$ should have a greater centrality degree than $Q_1$.

This simple example suggests the following definition of centrality measure.

Definition 6

Given an exact capacity $\mu$ whose core satisfies $\mid \text{ core }(\mu ) \mid >1$, a centrality measure is a function $\varphi :\mathbb {P}(\mathcal {X})\rightarrow [0,1]$ satisfying the following properties:

CM1:

$\varphi (P)=0$ for every $P\notin \text{ core }(\mu )$.

CM2:

If $P\in \text{ ext }\big (\text{ core }(\mu )\big )$, then $\varphi (P)=0$.

CM3:

There exists a unique $P_0\in \text{ core }(\mu )$ satisfying $\varphi (P_0)=1$. Such $P_0$ is called central point in $\text{ core }(\mu )$ with respect to $\varphi$.

CM4:

Consider $P \in \text{ ext }\big (\text{ core }(\mu )\big )$, $P_0$ the probability given in the previous item and $\lambda ,\beta \in [0,1]$ such that $\lambda \ge \beta$. Given $P_1=\lambda P+(1-\lambda )P_0$ and $P_2=\beta P+(1-\beta )P_0$, it holds that $\varphi (P_1)\le \varphi (P_2)$.

The idea underlying these properties is the following: CM1 tells us that an element outside the core should have degree of centrality zero; from CM2, the same should hold for the extreme points of the core; CM3 means that there is a unique probability $P_0$ with degree of centrality 1; finally, property CM4 represents the idea that the closer a probability is to $P_0$, the greater its degree of centrality is. We should mention that in Definition 6, and also in the remainder of this section, we are not considering the case where the core is a singleton, $\text{ core }(\mu )=\{P_0\}$, because in that case we can trivially assign a centrality degree 1 to $P_0$ and 0 to any other probability measure.

We next discuss two possible strategies for defining a centrality measure. The first one consists in considering a centroid in the interior of the core, and to measure the distance with respect to it. It requires to specify both the centroid and the distance. Out of the options considered in the previous section, we would reject $\Phi _1^\mu$ because it may not belong to the core and $\Phi _3^\mu$ because of non-uniqueness. With respect to the distance, we consider here the total variation, although as argued before it would also be possible to consider other options such as the $L_1$ or the Euclidean distances, or even the Kullback Leibler divergence.

In this sense, if we let $\Phi ^\mu$ be our centroid of choice and take⁶

$$\begin{aligned} \beta =\min \Big \{d\big (\Phi ^\mu ,P_i\big )\mid P_i\in \text{ ext }\big (\text{ core }(\mu )\big )\Big \}, \end{aligned}$$

(16)

then we can define

$$\begin{aligned} \varphi _1(P)=1-\min \left\{ \frac{d(P,\Phi ^\mu )}{\beta },1\right\} \quad \forall P\in \text{ core }(\mu ) \end{aligned}$$

(17)

and $\varphi _1(P)=0$ otherwise.

A second approach would consist in defining directly a chain $\{\text{ core }(\mu )_{\alpha }\}_{\alpha \in [0,1]}$ of sets such that $\text{ core }(\mu )_0=\text{ core }(\mu )$, $\text{ core }(\mu )_1$ is a singleton determining the centroid $\Phi$ and where $\text{ core }(\mu )_\alpha$ is included in the interior of $\text{ core }(\mu )$ for any $\alpha >0$, and letting

$$\begin{aligned} \varphi _2(P)=\sup \Big \{\alpha \in [0,1] \mid P\in \text{ core }(\mu )_\alpha \Big \}. \end{aligned}$$

(18)

The chain $\{\text{ core }(\mu )_\alpha \}_{\alpha \in [0,1]}$ of sets could be defined, for example, as:

$$\begin{aligned} \text{ core }(\mu )_{\alpha }=CH\big (\big \{(1-\alpha )P+\alpha \Phi ^\mu \mid P\in \text{ ext }(\text{ core }(\mu ))\big \}\big ) \quad \forall \alpha \in [0,1], \end{aligned}$$

(19)

where CH denotes the convex hull. Let us show that both approaches lead to a centrality measure.

Proposition 9

Let $\mu$ be an exact capacity, and let $\varphi _1,\varphi _2$ be given by Eqs. (17) and (18). Then $\varphi _1,\varphi _2$ satisfy conditions (CM1)–(CM4).

Example 12

Consider again our running Example 3. The extreme points of $\text{ core }(\mu )$ were given in Example 8. Taking as centroid the average of the extreme points $\Phi _2^\mu$, given in Eq. (14), we obtain the following distances:

$$\begin{aligned} \begin{array}{c|c c c c c} &{} P_1 &{} P_2 &{} P_3 &{} P_4 &{} P_5\\ \hline d\big (P_i,\Phi _2^\mu \big ) &{} \ 0.12 \ {} &{} \ 0.12 \ {} &{} \ 0.1 \ {} &{} \ 0.13 \ {} &{} \ 0.2 \ \end{array} \end{aligned}$$

Eq. (16) gives $\beta =0.1$, whence $\varphi _1(P)=1-\min \big \{ 10 d\big (P,\varphi _2^\mu \big ),1 \big \}$. Considering the probabilities $Q_1$ and $Q_2$ in Eqs. (15), (17) produces the following centrality degrees:

$$\begin{aligned} \varphi _1(Q_1)&=1-\min \Big \{10 d\big (Q_1,\varphi _2^\mu \big ),1\Big \}=1-1=0,\\ \varphi _1(Q_2)&=1-\min \Big \{10 d\big (Q_2,\varphi _2^\mu \big ),1\Big \}=1-0.5=0.5, \end{aligned}$$

using that $d\big (Q_1,\varphi _2^\mu \big )=0.15$ and $d\big (Q_2,\varphi _2^\mu \big )=0.05$.

If we choose instead as centroid $\Phi _4^\mu$, given in Eq. (11), we obtain the following distances to the extreme points:

$$\begin{aligned} \begin{array}{c|c c c c c} &{} P_1 &{} P_2 &{} P_3 &{} P_4 &{} P_5\\ \hline d\big (P_i,\Phi _4^\mu \big ) &{} \ 0.1375 \ {} &{} \ 0.1375 \ {} &{} \ 0.1125 \ {} &{} \ 0.1125 \ {} &{} \ 0.1875 \ \end{array} \end{aligned}$$

Thus, in this case $\beta =0.1125$ and $\varphi _1(P)=1-\min \left\{ \frac{d(P,\varphi _4^\mu )}{0.1125},1\right\}$. Considering again the probability measures $Q_1$ and $Q_2$ from Eq. (15) we obtain:

$$\begin{aligned} \varphi _1(Q_1)&=1-\min \left\{ \frac{d\big (Q_1,\varphi _4^\mu \big )}{0.1125},1 \right\} =1-1=0.\\ \varphi _1(Q_2)&=1-\min \left\{ \frac{d\big (Q_2,\varphi _4^\mu \big )}{0.1125},1 \right\} =1-\frac{0.0375}{0.1125}=\frac{2}{3}. \end{aligned}$$

We can see that the centrality degree of $Q_1$ is zero in both cases (for the centroids $\Phi _2^\mu$ and $\Phi _4^\mu$) but for $Q_2$, the centrality degree is slightly greater when considering the contraction centroid $\Phi _4^\mu$.

Fig. 8

Centrality measure $\varphi _1$ defined by Eq. (17) using $\Phi _2^\mu$ (left hand side) and $\Phi _4^\mu$ (right hand side) as centroids, with the curves with centrality degree 0, 0.2, 0.5 and 0.8

Figure 8 shows the curves with centrality degree 0, 0.2, 0.5 and 0.8 for $\varphi _1$ when considering as centroid $\varphi _2^\mu$ (left hand side figure) and $\varphi _4^\mu$ (right hand side figure). We can also consider the centrality measure $\varphi _2$ defined using the chain of sets in Eq. (19) defined using the extreme points. In that case, taking the average of the extreme points $\Phi _2^\mu$ as centroid, the centrality degrees of the probability measures $Q_1$ and $Q_2$ are $\varphi _2(Q_1)\approx 0.1923$ and $\varphi _2(Q_2)\approx 0.7142$. In contrast, if we use the contraction centroid, we obtain $\varphi _2(Q_1)=\nicefrac {2}{9}$ and $\varphi _2(Q_2)=\nicefrac {2}{3}$.

It is worth mentioning here that for this second approach, any $P\in \textrm{int}\big (\text{ core }(\mu )\big )$ has a strictly positive centrality degree. This is in contrast with the centrality measure $\varphi _1$, that assigns zero centrality degree for some probability measures in the interior of the core of $\mu$, as for example for $Q_1$.

Figure 9 shows the curves of centrality degree 0.2, 0.5 and 0.8 for $\varphi _2$ when considering as centroid $\Phi _2^\mu$ (left hand side) and $\Phi _4^\mu$ (right hand side). $\blacklozenge$

Fig. 9

Centrality measure $\varphi _2$ from Eq. (19) using $\Phi _2^\mu$ (left hand side) and $\Phi _4^\mu$ (right hand side) as centroids, with the curves with centrality degree 0, 0.2, 0.5 and 0.8

It is also possible to define a centrality measure by considering the chain of sets from Eq. (10). For this, note that for each $P\in \text{ core }(\mu )$ there is $j\in \{1,\ldots ,l\}$ such that $P\in \text{ core }(\mu )_{\alpha _{S_{j-1}}}\setminus \text{ core }(\mu )_{\alpha _{S_j}}$, where $\text{ core }(\mu )_{\alpha _0}=\text{ core }(\mu )$. Also, there is $\alpha \in \Lambda _{j-1}$ such that $P\in \big (\text{ core }(\mu )_{j-1}\big )_{\alpha }$, but $P\notin \big (\text{ core }(\mu )_{j-1}\big )_{\alpha +\varepsilon }$ for any $\varepsilon >0$. Then we let:

$$\begin{aligned} \varphi _3(P)=\frac{\alpha _{S_1}+\cdots +\alpha _{S_{j-1}}+\alpha }{\alpha _{S_1}+\cdots +\alpha _{S_l}} \ \text{ if } P\in \text{ core }(\mu ), \end{aligned}$$

(20)

and $\varphi _3(P)=0$ if $P\notin \text{ core }(\mu )$.

Proposition 10

The function $\varphi _3$ defined in Eq. (20) satisfies conditions (CM1)–(CM4).

Example 13

In our running Example 3, and in particular in Example 5, we have seen that $\alpha _{S_1}=0.075$ and $\alpha _{S_2}=0.0375$, so in this case we have $\text{ core }(\mu )_{\alpha _{S_1}}$ and $\text{ core }(\mu )_{\alpha _{S_2}}$, where the latter is a singleton formed by $\Phi _4^\mu$. It holds that $Q_1\in \big (\text{ core }(\mu )_{\alpha _{S_0}}\big )_{0.025}$ and $Q_1\notin \big (\text{ core }(\mu )_{\alpha _{S_0}}\big )_{0.025+\varepsilon }$, whence:

$$\begin{aligned} \varphi _3(Q_1)=\frac{0.025}{\alpha _{S_1}+\alpha _{S_2}}=\frac{0.025}{0.075+0.0375}=\frac{2}{9}. \end{aligned}$$

On the other hand, $Q_2\in \big (\text{ core }(\mu )_{\alpha _ {S_0}}\big )_{0.05}$ but $Q_2\notin \big (\text{ core }(\mu )_{\alpha _{S_0}}\big )_{0.05+\varepsilon }$, whence:

$$\begin{aligned} \varphi _3(Q_2)=\frac{0.05}{\alpha _{S_1}+\alpha _{S_2}}=\frac{0.05}{0.075+0.0375}=\frac{4}{9}. \end{aligned}$$

Figure 10 depicts the curves of centrality degree 0.2, 0.5 and 0.8 for $\varphi _3$. $\blacklozenge$

Fig. 10

Centrality measure $\varphi _3$ from Eq. (20) with the curves with centrality degree 0, 0.2, 0.5 and 0.8

6 Centroids from the perspective of imprecise probabilities

We conclude this paper by considering the centroid problem within the framework of imprecise probabilities, that include capacities as a particular case and that are capable of modelling more general scenarios. Before we do this, we make a number of clarifications about the terminology.

Within imprecise probability theory, a (exact) capacity $\mu$ is usually denoted by $\underline{P}$, and it is called (coherent) lower probability while its conjugate function, called (coherent) upper probability, is denoted by $\overline{P}$. $\underline{P}$ and $\overline{P}$ may be understood as functions giving lower and upper bounds to a real but unknown probability $P_0$, meaning that all we know about $P_0$ is that $\underline{P}(A)\le P_0(A)\le \overline{P}(A)$ for any event $A\subseteq \mathcal {X}$. Following this interpretation, the core of a lower probability $\underline{P}$ is called credal set (Levi, 1980) and it is denoted by $\mathcal {M}(\underline{P})$; it may be interpreted as the set of probability measures that are compatible with the information given by the lower probability. Finally, in this context the property of supermodularity is usually called 2-monotonicity.

The correspondence⁷ between the terminology used in decision making, game theory and imprecise probabilities can be seen in Table 1.

Table 1 Summary of the connection between the terminologies employed in coalitional game theory, decision making and imprecise probability theory

In this section, we shall recall the basics from the more general theory of (coherent) lower previsions, and show that the four centroids analysed before can be also considered in this context.

6.1 (Coherent) lower previsions

In the theory of imprecise probabilities from Walley (1991), rather than giving lower bounds for the values taken by some unknown probability measure on events, we give lower bounds for the values taken by its expectation operator. This is done by means of a lower prevision, a function $\underline{P}:\mathcal {L}(\mathcal {X})\rightarrow \mathbb {R}$, where $\mathcal {L}(\mathcal {X})$ denotes the set of random variables, or gambles, defined on $\mathcal {X}$. Its conjugate upper prevision is defined by $\overline{P}(f)=-\underline{P}(-f)$ for any $f\in \mathcal {L}(\mathcal {X})$. One underlying interpretation is that there exists a probability measure $P_0$ modeling our uncertainty, and all we know about it is that $\underline{P}(f)\le E_{P_0}(f)\le \overline{P}(f)$ for any $f\in \mathcal {L}(\mathcal {X})$. Associated with a lower prevision we can define a credal set by:

$$\begin{aligned} \mathcal {M}(\underline{P})=\{P\in \mathbb {P}(\mathcal {X})\mid P(f)\ge \underline{P}(f) \quad \forall f\in \mathcal {L}(\mathcal {X})\}, \end{aligned}$$

(21)

using for simplicity the same symbol P to denote a probability measure and its associated expectation operator: $P(f)=E_P(f)$. The lower prevision $\underline{P}$ is called coherent when $\underline{P}(f)=\min _{P\in \mathcal {M}(\underline{P})}P(f)$ for any $f\in \mathcal {L}(\mathcal {X})$. The credal set associated with a coherent lower prevision is a closed and convex subset of $\mathbb {P}(\mathcal {X})$ but it may not be a polytope. In fact, there is a one-to-one correspondence between coherent lower previsions and closed and convex subsets of $\mathbb {P}(\mathcal {X})$. This allows us to understand the extent of the generality of this theory: while coherent lower probabilities (or exact capacities) give rise to credal sets (or cores) that are polytopes, coherent lower previsions induce closed and convex sets of probabilities that need not be a polytope.

One particular situation where the credal set in Eq. (21) is a polytope is when the coherent lower prevision $\underline{P}$ satisfies 2-monotonicity (Walley, 1981):

$$\begin{aligned} \underline{P}(f\vee g)+\underline{P}(f\wedge g)\ge \underline{P}(f)+\underline{P}(g) \quad \forall f,g\in \mathcal {L}(\mathcal {X}), \end{aligned}$$

where $\vee$ and $\wedge$ denote the pointwise maximum and minimum. When this property is satisfied, the 2-monotone lower prevision determines the same credal set as its restriction to events, which is a 2-monotone lower probability $\underline{P}'$ defined by $\underline{P}'(A):=\underline{P}(I_A)$; and the latter determines the values of the coherent lower prevision $\underline{P}$ by means of the Choquet integral (Choquet, 1953).

6.2 Centroids for coherent lower previsions

Let us discuss how the different notions of centroids we have analysed in this paper may be applied on arbitrary credal sets or, equivalently, on coherent lower previsions.

Shapley value

The Shapley value can be straightforwardly defined by considering the coherent lower probability associated to the lower prevision (its restriction to indicators of events) and applying any of the equivalent representations of the Shapley value given in Sect. 3.1.

Nevertheless, an important drawback here is that two different coherent lower previsions with the same restriction to events will have the same Shapley value, and so will be indistinguishable in this respect. This is illustrated in our next example:

Example 14

Consider $\mathcal {X}=\{x_1,x_2,x_3\}$ and the coherent lower previsions $\underline{P}_1$ and $\underline{P}_2$ inducing the following credal sets:⁸

$$\begin{aligned} \mathcal {M}(\underline{P}_1)&=\big \{ P\in \mathbb {P}(\mathcal {X})\mid 0.5\ge P(\{x_2\})\ge P(\{x_3\}) \big \}.\\ \mathcal {M}(\underline{P}_2)&=\big \{ P\in \mathbb {P}(\mathcal {X})\mid 0.5\ge P(\{x_3\})\ge P(\{x_2\}) \big \}. \end{aligned}$$

By taking lower envelopes, we obtain that both induce the same coherent lower probability $\underline{P}$:

$$\begin{aligned} \begin{array}{r|c c c c c c c} A &{} \ \{x_1\} \ {} &{} \ \{x_2\} \ {} &{} \ \{x_3\} \ {} &{} \ \{x_1,x_2\} \ {} &{} \ \{x_1,x_3\} \ {} &{} \ \{x_2,x_3\} \ {} &{} \mathcal {X}\\ \hline \ \underline{P}(A) \ {} &{} 0 &{} 0 &{} 0 &{} 0.5 &{} 0.5 &{} 0 &{} 1 \end{array} \end{aligned}$$

Hence, the Shapley value is the same for both $\underline{P}_1$ and $\underline{P}_2$, and it is given by $\Phi _1^{\underline{P}_1}=\Phi _1^{\underline{P}_2}=(0.5,0.25,0.25)$. Note that the Shapley value does not belong to the interior of the credal sets for neither $\underline{P}_1$ nor $\underline{P}_2$. $\blacklozenge$

Average of the extreme points

Whenever the credal set $\mathcal {M}(\underline{P})$ is a polytope (i.e., it has a finite number of extreme points), the average of the extreme points can be computed using Eq. (6). While this definition imposes a restriction on the credal set, it is applicable for those associated with a lower probability $\underline{P}$ that is coherent (Wallner, 2007), and therefore also in the particular cases of 2-monotonicity, belief functions or p-boxes (Montes & Destercke, 2017). In addition, it is also applicable to some models of coherent lower previsions that are not determined by their restrictions to events, such as those associated with comparative probabilities (Miranda & Destercke, 2015).⁹

Incenter

As we did in Sect. 3.3, we can find the (set of) incenters. In this case, the definition of incenter can be straightforwardly given: the incenter radius of $\mathcal {M}(\underline{P})$ is given by

$$\begin{aligned} \alpha _I=\sup \big \{ \alpha >0 \mid \exists P_0\in \mathcal {M}(\underline{P}) \text{ such } \text{ that } B_o^\alpha (P_0)\subseteq \mathcal {M}(\underline{P})\cap \mathbb {P}^{*}(\mathcal {X}) \big \}, \end{aligned}$$

and any $P_0\in \mathcal {M}(\underline{P})$ such that $B_0^{\alpha _I}(P_0)\subseteq \mathcal {M}(\underline{P})$ is called incenter of $\underline{P}$.

Proposition 11

Let $\underline{P}$ be a coherent lower prevision whose credal set has non-empty interior. Then the value $\alpha _I$ is a maximum. As a consequence, the incenter of $\underline{P}$ always exists.

Contraction centroid

The only centroid whose extension to coherent lower previsions is not straightforward is the contraction centroid. Assuming again that $\mathcal {M}(\underline{P})$ is a polytope, we know that it is determined by a finite number of constraints. This means that there are two (disjoint) set of gambles $\mathcal {L}^{>}$ and $\mathcal {L}^{=}$ such that the coherent lower prevision $\underline{P}$ and its conjugate $\overline{P}$ satisfy:

$$\begin{aligned} \underline{P}(f)=\overline{P}(f) \quad \forall f\in \mathcal {L}^{=} \ \text{ and } \ \underline{P}(f)< \overline{P}(f)\quad \forall f\in \mathcal {L}^{>}, \end{aligned}$$

and that the credal set can be expressed as:

$$\begin{aligned} \mathcal {M}(\underline{P})=\big \{ P\in \mathbb {P}(\mathcal {X})\mid P(f)=\underline{P}(f) \ \forall f\in \mathcal {L}^{=}, \ P(f)\ge \underline{P}(f) \ \forall f\in \mathcal {L}^{>} \big \}. \end{aligned}$$

(22)

Note that we may assume without loss of generality that these constraints include the indicator functions of the proper events: $\{I_A\mid \emptyset \ne A\subset \mathcal {X}\}\subseteq \mathcal {L}^>\cup \mathcal {L}^=$. In that case, when $\mathcal {L}^>$ is empty we obtain that $\underline{P}(I_A)=\overline{P}(I_A)$ for any $A\subseteq \mathcal {X}$ (or $\underline{P}(A)=\overline{P}(A)$, if we use this abuse of notation), meaning that $\mathcal {M}(\underline{P})$ contains one single probability measure. Moreover, using the properties of coherent lower and upper previsions, we can also assume without loss of generality that $0\le \min f<\max f=1$ for every $f\in \mathcal {L}^{>}\cup \mathcal {L}^{=}$.

The idea in this approach is the same explained in Sect. 3.4: we contract the credal set in a uniform manner as long as possible, increasing the value of the lower prevision in a constant amount $\alpha$ in all the gambles $f\in \mathcal {L}^{>}$, and then proceed in the same way by reducing the number of constraints. As we showed in Proposition 3, there exists a value $\alpha$ small enough such that this approach produces a non-empty credal set and there is a maximum value satisfying this property.

Proposition 12

Let $\underline{P}$ be a coherent lower prevision whose credal set is a polytope that can be expressed as in Eq. (22). For a given $\alpha >0$, let:

$$\begin{aligned} \mathcal {M}(\underline{P})_{\alpha }=\big \{ P\in \mathbb {P}(\mathcal {X})\mid P(f)=\underline{P}(f) \ \forall f\in \mathcal {L}^{=}, \ P(f)\ge \underline{P}(f)+\alpha \ \forall f\in \mathcal {L}^{>} \big \}. \end{aligned}$$

(23)

Consider also the set $\Lambda =\{\alpha >0\mid \mathcal {M}(\underline{P})_{\alpha }\ne \emptyset \}$. It holds that:

• (a) $\Lambda \ne \emptyset$.

• (b) The set $\Lambda$ has a maximum $\alpha _{S}$.

• (c) Given the set $\mathcal {M}(\underline{P})_{\alpha _S}$, there exists some $f\in \mathcal {L}^{>}$ such that P(f) is constant for any $P\in \mathcal {M}(\underline{P})_{\alpha _S}$.

Moreover, as we explained before, when the coherent lower prevision satisfies 2-monotoninicty, it is determined by its restriction to events. Hence, Theorem 4 also applies in this context, where we simply need to understand $\underline{P}(A)$ and $\overline{P}(A)$ as the lower and upper previsions of the indicator $I_A$.

Finally, the connection between the set of incenters and the first step of the process determining the contraction centroid also holds for coherent lower previsions.

Proposition 13

Let $\underline{P}$ be a coherent lower prevision whose associated credal set $\mathcal {M}(\underline{P})$ is included in $\mathbb {P}^{*}(\mathcal {X})$ and such that $\mathcal {L}^{=}=\emptyset$. If $\alpha _S$ is the incenter radius, then for any $P_0\in \mathcal {M}(\underline{P})$ and any $\alpha \le \alpha _S$:

$$\begin{aligned} P_0\in \mathcal {M}_{\alpha }(\underline{P})\Leftrightarrow B_c^{\alpha }(P_0)\subseteq \mathcal {M}(\underline{P}). \end{aligned}$$

6.3 Particular cases

We have seen that the four centroids can be defined to coherent lower previsions. In this subsection we analyse them for some particular families of imprecise models. We start with probability intervals.

A probability interval (de Campos et al., 1994) is an uncertainty model $\mathcal {I}$ that gives lower and upper bounds to the probability of the singletons:

$$\begin{aligned} \mathcal {I}=\big \{[l_i,u_i] \mid l_i\le u_i \ \ \forall i=1,\ldots ,n\big \}. \end{aligned}$$

It determines a credal set given by:

$$\begin{aligned} \mathcal {M}(\mathcal {I})=\big \{P\in \mathbb {P}(\mathcal {X}) \mid l_i\le P(\{x_i\})\le u_i \ \ \forall i=1,\ldots ,n\big \}. \end{aligned}$$

This credal set is non-empty if and only if $\sum _{i=1}^{n} l_i \le 1 \le \sum _{i=1}^{n} u_i$, and we say that the probability interval avoids sure loss. Then, taking lower and upper envelopes of $\mathcal {M}(\mathcal {I})$ we obtain a lower and an upper probability, and we say that the probability interval is coherent when $\underline{P}(\{x_i\})=l_i$ and $\overline{P}(\{x_i\})=u_i$ for every $i=1,\ldots ,n$. In that case, $\underline{P}$ is 2-monotone and the values of $\underline{P}$ and $\overline{P}$ for any event $A\subseteq \mathcal {X}$ can be computed as de Campos et al. (1994):

$$\begin{aligned} \underline{P}(A)=\max \Bigg \{ \sum _{x_i\in A}l_i,1-\sum _{x_i\notin A}u_i \Bigg \}, \quad \overline{P}(A)=\min \Bigg \{ \sum _{x_i\in A}u_i, 1-\sum _{x_i\notin A}l_i \Bigg \}. \end{aligned}$$

For the particular case of coherent probability intervals, we can give an explicit formula for the value $\alpha _S$ in the contraction method.

Proposition 14

Let $\underline{P}$ and $\overline{P}$ be the coherent lower and upper probability determined by a coherent probability interval $\mathcal {I}=\{[l_i,u_i]\mid \forall i=1,\ldots ,n\}$. Consider:

$$\begin{aligned} \mathcal {I}^{>}=\big \{i\in \{1,\ldots ,n\}\mid l_i<u_i\big \}, \quad \mathcal {I}^{=}=\big \{i\in \{1,\ldots ,n\}\mid l_i=u_i\big \}. \end{aligned}$$

Then:

• (a) The value $\alpha _S=\max \Lambda$ is given by:

  $$\begin{aligned} \alpha _S=\min \left\{ \frac{1}{\mid \mathcal {I}^{>}\mid }\left( 1-\sum _{i=1}^nl_i\right) ,\frac{1}{\mid \mathcal {I}^{>}\mid }\left( \sum _{i=1}^nu_i-1\right) ,\frac{1}{2}\min _{i\in \mathcal {I}^{>}}(u_i-l_i) \right\} . \end{aligned}$$

  (24)

• (b) The credal set $\mathcal {M}(\underline{P})_{\alpha _S}$ determined by means of Eq. (23) is a probability interval avoiding sure loss.

• (c) If $\alpha _S=\frac{1}{\mid \mathcal {I}^{>}\mid }\big ( 1-\sum _{i=1,\ldots ,n}l_i\big )$ or $\alpha _S=\frac{1}{\mid \mathcal {I}^{>}\mid }\big ( \sum _{i=1,\ldots ,n}u_i-1\big )$, then $\mathcal {M}(\underline{P})_{\alpha _S}=\big \{\Phi _4^{\underline{P}}\big \}$.

The value $\alpha _S$ obtained in Eq. (24) is consistent with that from Theorem 5 for the particular case of maximally imprecise probability intervals. This shows also that in that case Eq. (13) can be simplified, in the sense that we do not need to consider all partitions of $\mathcal {X}$, but only the partitions $\{x_1\},\ldots ,\{x_n\}$ and $\{x_i\},\mathcal {X}\setminus \{x_i\}$ for every $i=1,\ldots ,n$.

From Proposition 14 we can also deduce an explicit formula for the value $\alpha _S$ for the Linear Vacuous (LV) and the Pari Mutuel Model (PMM), which constitute particular instances of distortion models (Montes et al., 2020a, b) or nearly linear models (Corsato et al., 2019). The PMM (Montes et al., 2019; Pelessoni et al., 2010; Walley, 1991) is determined by the coherent lower probability:

$$\begin{aligned} \underline{P}_{PMM}(A)=\max \big \{(1+\delta )P_0(A)-\delta ,0\big \} \quad \forall A\subseteq \mathcal {X}, \end{aligned}$$

where $P_0\in \mathbb {P}(\mathcal {X})$ is a given probability measure and $\delta >0$. Similarly, the LV (Walley, 1991) is defined by the coherent lower probability

$$\begin{aligned} \underline{P}_{LV}(A)=(1-\delta )P_0(A) \ \ \forall A\subset \mathcal {X}\ \text { and } \ \underline{P}_{LV}(\mathcal {X})=1, \end{aligned}$$

where $P_0\in \mathbb {P}(\mathcal {X})$ and $\delta \in (0,1)$.

Both the PMM and the LV are instances of probability intervals, where:

$$\begin{aligned} \mathcal {I}_{PMM}&=\Big \{ \big [\max \{(1+\delta )P_0(\{x\})-\delta ,0\},\min \{(1+\delta )P_0(\{x\}),1\}\big ] \ \mid \ x\in \mathcal {X}\Big \},\\ \mathcal {I}_{LV}&=\Big \{ \big [(1-\delta )P_0(\{x\}),\min \{(1-\delta )P_0(\{x\})+\delta ,1\}\big ] \ \mid \ x\in \mathcal {X}\Big \}. \end{aligned}$$

This means that we can apply Proposition 14 for computing the value $\alpha _S=\max \Lambda$. In fact, when both $P_0$ and the lower probability only take the values 0 and 1 for trivial events (the impossible and sure events), the formula for $\alpha _S=\max \Lambda$ can be simplified and the procedure of contracting the credal set finishes in only one step.

Corollary 15

Consider a PMM $\underline{P}_{PMM}$ or a LV $\underline{P}_{LV}$ determined by a probability measure $P_0\in \mathbb {P}(\mathcal {X})$ and a distortion parameter $\delta$. Assume that $\underline{P}_{PMM}(A)$ and $\underline{P}_{LV}(A)$ belong to (0, 1) for every $A\ne \emptyset ,\mathcal {X}$. Then:

• (a) For the PMM, $\alpha _S=\max \Lambda =\frac{\delta }{n}$ and $\mathcal {M}_{\alpha }=\big \{\Phi _4^{\underline{P}_{PMM}}\big \}$ where $\Phi _4^{\underline{P}_{PMM}}(\{x_i\})=(1+\delta )P_0(\{x_i\})-\frac{\delta }{n}$ for any $i=1,\ldots ,n$.

• (b) For the LV, $\alpha _S=\max \Lambda =\frac{\delta }{n}$ and $\mathcal {M}_{\alpha }=\big \{\Phi _4^{\underline{P}_{LV}}\big \}$ where $\Phi _4^{\underline{P}_{LV}}(\{x_i\})=(1-\delta )P_0(\{x_i\})+\frac{\delta }{n}$ for any $i=1,\ldots ,n$.

• (c) For both the PMM and LV there is a unique incenter and $\Phi _1=\Phi _2=\Phi _3=\Phi _4$.

In this respect, it is worth remarking that (i) the good behaviour of these two distortion models is in line of other desirable properties they possess, as discussed in Destercke et al. (2022), Montes et al. (2020a) and Montes et al. (2020b); and (ii) the centroid of a distortion model originated by a probability measure $P_0$ does not coincide with $P_0$, because the distortion is not done uniformly in all directions of the simplex. This was already observed in Miranda and Montes (2018 for the particular case of the Shapley value.

6.4 Properties

We consider now the properties of the centroids considered in Sect. 4.

Proposition 16

Let $\underline{P}$ be a coherent lower prevision. Then, the properties in Proposition 7 and 8 still hold.

We have already mentioned that two different lower previsions may have the same restriction to events. For this reason, in addition to the aforementioned properties, it would be desirable that the center of a coherent lower prevision $\underline{P}$ does not necessarily coincide with the center of its restriction to events. In this respect, it is not difficult to show that $\Psi _3^{\underline{P}},\Phi _4^{\underline{P}}$ are capable of distinguishing between lower previsions and lower probabilities, and so does $\Phi _2^{\underline{P}}$ (when $\mathcal {M}(\underline{P})$ is a polytope). On the other hand, Shapley value is only defined via the lower probability, so it does not distinguish between lower probabilities and lower previsions as we showed in Example 14.

6.5 Centrality measures

Finally, we may try to define centrality measures for the credal set determined by a coherent lower prevision. For this aim, we can consider exactly the definition of centrality measure as in Definition 6, and the centrality measures $\varphi _1,\varphi _2$ and $\varphi _3$ defined in Eqs. (17), (18) and (20), respectively. Note that in the case of $\varphi _1$ and $\varphi _3$ we need to restrict ourselves to coherent lower previsions whose credal set is a polytope to assure that the minimum in Eq. (16) is strictly positive and that the contraction approach finishes in a finite number of steps, respectively.

Proposition 17

Given a coherent lower prevision $\underline{P}$, the function $\varphi _2$ is a centrality measure. Moreover, if $\mathcal {M}(\underline{P})$ is a polytope, $\varphi _1$ and $\varphi _3$ are centrality measures too.

7 Conclusions

We have performed a comparative analysis of four alternatives for defining a center of an exact capacity: the Shapley value, the average of the extreme points, the incenter for the total variation distance and the limit of uniform contractions. Our results show that these four approaches may lead to different results, and also illustrate some of the properties each of them satisfies: a summary can be seen in Table 2. Note that our goal with this paper is not to take the stance that one of them is better than the others. Instead, we intend to provide some assistance to the practitioner in her choice of a centroid: if for instance she considers that linearity is an essential property, she must pick the Shapley value; if she wants to ensure that the center belongs to the interior of the core of the capacity, she must select one of the others; and so on. Also, we have seen that these centroids can also be applied in the more general framework of coherent lower previsions. Since coherent lower previsions are in one-to-one correspondence with closed and convex set of probabilities, they include the particular case of polytopes. We have seen that the results we have proved for exact capacities can also be extended to coherent lower previsions.

Table 2 Summary of the properties satisfied by the four centroids

Let us recall that the center in this paper is understood as a probability measure that is in the interior of the core of the capacity and that can play the role of its representative. For this reason, we have left out of our study other approaches based, for example, on maximising the entropy or minimising the Kullback Leibler divergence, that in our context may produce counterintuitive results.

In addition to the comparison performed between the four centroids, some comments regarding their computation in practice must be done:

• First of all, the computation of the Shapley value is known to be a hard problem that exponentially increases with $\mid \mathcal {X}\mid$, since it requires the computation of $\mu (A)$ for the $2^{n}-1$ non-trivial events.

• Secondly, the average of the extreme points is simple as long as these are known. Under the assumption of supermodularity, the extreme points coincide with the probability measures $P_{\sigma }$ defined in Eq. (1). Even more, there are particular situations where their computation is even simpler: for example, when $\mu$ is minitive there are at most $2^{n-1}$ (Miranda et al., 2003) and when $\mu$ corresponds to a p-box the maximal number of extreme points is at most the n-th Pell number (Montes & Destercke, 2017). The problem is more challenging when the capacity is not supermodular: even if the number of extreme points is at most n! (Wallner, 2007), there is no simple procedure for their computation.

• Thirdly, computing the contraction centroid may be an extremely complicated problem. Even if Theorem 4 gives a formula for computing the value $\alpha _S$ under fairly general conditions, it requires the computation of all the families $\mathcal {A}$ in $\mathbb {A}(\mathcal {X})$, whose number is extremely large. The complexity is significantly reduced for supermodular capacities, because Theorem 5 gives a simpler expression for $\alpha _S$ that depends on the partitions of $\mathcal {X}$. Still, the problem is rather complex because the number of partitions in a n element possibility space coincides with the n-th Bell number. Nevertheless, we have seen that for particular models that may arise in practice such as probability intervals or some distortion models, the computation of the four centroids is much simpler.

• Finally, under fairly general conditions the set of incenters coincide with the first step of the contraction approach (Proposition 6), hence both approaches are equivalent from the computational viewpoint.

While our results give some overview of the properties of the centroids of a capacity, there is still much work to be done in order to have a full picture of this problem. First and foremost, it would be interesting to extend our approaches to infinite possibility spaces. While this seems immediate in the case of the incenter or the vertex centroid (see also footnote 4), in the case of the Shapley value we should consider the generalisations carried out in Neyman (2002), and in the case of the contraction centroid we should verify that the process stabilises in a finite number of steps. In addition, we could consider other possibilities in the context of game solutions, such as the Banzhaf value (Banzhaf, 1965) or more generally probabilistic solutions (Weber, 1988), or other alternatives to the total variation distance, such as the Euclidean distance or the $L_1$ distance. It would also be interesting to obtain further conditions for the equality between some of these centroids. And finally, a deeper study of centrality measures and their axiomatic properties would be of interest.

Appendix A: Proofs

Proof of Proposition 1

From Eq. (2), using the definition of the Möbius inverse, we obtain that

$$\begin{aligned} \Phi _1^\mu (\{x\})&=\sum _{A\mid x\notin A}\frac{\mid A\mid !(n-\mid A\mid -1)!}{n!}\big ( \mu (A\cup \{x\})-\mu (A) \big ) \\&=\sum _{A\mid x\notin A}\frac{\mid A\mid !(n-\mid A\mid -1)!}{n!}\left( \sum _{B\subseteq A\cup \{x\}} m(B)-\sum _{B\subseteq A} m(B)\right) \\&=\sum _{A\mid x\notin A}\frac{\mid A\mid !(n-\mid A\mid -1)!}{n!}\Bigg ( \sum _{x \in B\subseteq A\cup \{x\}} m(B)\Bigg ). \end{aligned}$$

Now, in the sum above any event B that includes x appears associated with sets A that include $B\setminus \{x\}$ and do not include x, and which are thus of the form $(B\setminus \{x\})\cup C$ for some $C\subseteq B^c$. This means that in the sum above m(B) is multiplied by

$$\begin{aligned}&\left( \sum _{k=0}^{n-\mid B\mid } \frac{(\mid B\mid -1+k)!(n-(\mid B\mid -1+k)-1)!}{n!}\right) {n-\mid B\mid \atopwithdelims ()k}\\&\quad =\frac{(n-\mid B\mid )!}{n!}\sum _{k=0}^{n-\mid B\mid } \frac{(\mid B\mid -1+k)!}{k!}=\frac{(n-\mid B\mid )!(\mid B\mid -1)!}{n!}\sum _{k=0}^{n-\mid B\mid } {\mid B\mid -1+k \atopwithdelims ()k}\\&\quad =\frac{(n-\mid B\mid )!(\mid B\mid -1)!}{n!} {n-\mid B\mid +\mid B\mid -1+1 \atopwithdelims ()n-\mid B\mid }\\&\quad =\frac{(n-\mid B\mid )!(\mid B\mid -1)!}{n!} {n \atopwithdelims ()n-\mid B\mid }=\frac{1}{\mid B\mid }, \end{aligned}$$

where the third equality follows from the hockey-stick identity. This shows that the Shapley value coincides with the pignistic transformation.

To see the equality with the average of the $P_{\sigma }$, note that for a fixed permutation,

$$\begin{aligned} P_{\sigma }(\{x_{\sigma (i)}\})&=\mu (\{x_{\sigma (1)},\dots ,x_{\sigma (i)}\})-\mu (\{x_{\sigma (1)},\dots ,x_{\sigma (i-1)}\})\\ {}&=\sum _{x_{\sigma (i)}\in B \subseteq \{x_{\sigma (1)},\dots ,x_{\sigma (i)}\}} m(B), \end{aligned}$$

whence m(B) is included in the sum that determines the probability of $x_{\sigma (i)}$ only in those permutations when B is a subset of $\{x_{\sigma (1)},\dots ,x_{\sigma (i)}\}$ that includes $x_{\sigma (i)}$. Reasoning as above, the proportion of such permutations is $\frac{1}{\mid B\mid }$. Thus, the average of the $P_{\sigma }$ coincides with the pignistic transformation, and as a consequence also with the Shapley value. $\square$

Proof of Propositions 2 and 11

Since capacities are particular cases of coherent lower previsions, we prove the statement in Proposition 11, from which we immediately deduce Proposition 2.

Consider a coherent lower prevision $\underline{P}$. If the topological interior of $\mathcal {M}(\underline{P})$ is non-empty, we consider the set

$$\begin{aligned} \Lambda _1=\big \{\alpha \in [0,1]\mid \exists P \in \mathcal {M}(\underline{P}) \text { such that } B_o^\alpha (P)\subseteq \mathcal {M}(\underline{P})\cap \mathbb {P}^*(\mathcal {X})\big \}. \end{aligned}$$

This set is non-empty because

$$\begin{aligned} \textrm{int}\big (\mathcal {M}(\underline{P})\big )\ne \emptyset \Leftrightarrow \textrm{int}\big (\mathcal {M}(\underline{P})\big )\cap \mathbb {P}^*(\mathcal {X})\ne \emptyset \end{aligned}$$

and given $P\in \textrm{int}\big (\mathcal {M}(\underline{P})\big )\cap \mathbb {P}^*(\mathcal {X})$ there exists some $\alpha >0$ such that $B_o^\alpha (P)\subseteq \mathcal {M}(\underline{P})\cap \mathbb {P}^*(\mathcal {X})$. Besides being non-empty, the set $\Lambda _1$ is directed ($\alpha \in \Lambda _1\Rightarrow \alpha '\in \Lambda _1\ \forall \alpha '\in [0,\alpha ]$). Let $\alpha _1:=\sup \Lambda _1$. This means that for every $n\in \mathbb {N}$ there exists some $P_n\in \mathcal {M}(\underline{P})$ such that $B_o^{\alpha _1-\nicefrac {1}{n}}(P_n)\subseteq \mathcal {M}(\underline{P})\cap \mathbb {P}^*(\mathcal {X})$. The sequence $(P_n)_n$ is included in the compact set $\mathcal {M}(\underline{P})$, that is as a consequence also sequentially compact. Therefore, there exists a subsequence $(P_{n'})_{n'}$ that converges to some P. Note that for this subsequence we also have that $B_o^{\alpha _1-\nicefrac {1}{n'}}(P_{n'})\subseteq \mathcal {M}(\underline{P})\cap \mathbb {P}^*(\mathcal {X})$. Since $\mathcal {M}(\underline{P})$ is closed, $P\in \mathcal {M}(\underline{P})$. Moreover, for any $\epsilon >0$ there is some $n_\epsilon$ such that $d(P_{n'},P)<\epsilon$ for every $n'\ge n_\epsilon$. Take $\epsilon =\frac{1}{n}$ for a fixed n. Then for any $Q\in B_o^{\alpha _1-\nicefrac {2}{n}}(P)$ and any $m\ge n_\epsilon$ it holds that

$$\begin{aligned} d(Q,P_{m'})\le d(Q,P)+d(P,P_{m'})< \alpha _1-\frac{1}{n}. \end{aligned}$$

Since we can take m arbitrarily large, we deduce that

$$\begin{aligned} B_o^{\alpha _1-\nicefrac {2}{n}}(P)\subseteq B_o^{\alpha _1-\nicefrac {1}{n}}(P_{m'})\subseteq B_o^{\alpha _1-\nicefrac {1}{m'}}(P_{m'})\subseteq \mathcal {M}(\underline{P})\cap \mathbb {P}^*(\mathcal {X}). \end{aligned}$$

Since this holds for every n, we deduce that $B_o^{\alpha _1}(P)=\cup _n B_o^{\alpha _1-\frac{1}{n'}}(P)\subseteq \mathcal {M}(\underline{P})\cap \mathbb {P}^*(\mathcal {X})$. Therefore, $\alpha _1$ belongs to $\Lambda _1$. $\square$

Proof of Propositions 3 and 12

Again, we prove Proposition 12 and, since exact capacities are particular cases of coherent lower previsions, Proposition 3 immediately follows.

(a):

Let us denote $\mathcal {L}^{>}=\{f_1,\dots ,f_k\}$. For each $i=1,\dots ,k$, it follows by coherence that there is some $P_i\in \mathcal {M}(\underline{P})$ such that $P_i(f_i)=\overline{P}(f_i)>\underline{P}(f_i)$. If we now let $P:=\frac{P_1+\dots +P_k}{k}$, it belongs to the convex set $\mathcal {M}(\underline{P})$ and by construction it satisfies $P(f_i)>\underline{P}(f_i)$ for all $i=1,\dots ,k$. Given $\alpha =\min _{i=1,\dots ,k}\big (P(f_i)-\underline{P}(f_i)\big )>0$, we conclude that $P\in \mathcal {M}(\underline{P})_{\alpha _i}$ and therefore $\Lambda$ is non-empty.

(b):

Since $\mathcal {X}$ is finite, the topology of $\mathcal {M}(\underline{P})$ is equivalent to the topology associated with the Euclidean distance. The definition of $\mathcal {M}(\underline{P})_\alpha$ implies then that, for every $\alpha \in \Lambda$, the set $\mathcal {M}(\underline{P})_{\alpha }$ is a closed subset of $\mathcal {M}(\underline{P})$, and it is therefore compact. Thus, $(\mathcal {M}(\underline{P})_{\alpha })_{\alpha \in \Lambda }$ is a decreasing sequence of compact subsets of $\mathcal {M}(\underline{P})$, and as a consequence their intersection $\mathcal {M}^*$ is non-empty. But this intersection $\mathcal {M}^*$ must coincide with $\mathcal {M}(\underline{P})_{\alpha _S}$ for $\alpha _S=\sup \Lambda$, and this implies that this supremum is a maximum.

(c):

Let $\underline{P}_{\alpha _S},\overline{P}_{\alpha _S}$ denote the lower and upper envelopes of $\mathcal {M}(\underline{P})_{\alpha _S}$. Assume ex-absurdo that $\underline{P}_{\alpha _S}(f_i)<\overline{P}_{\alpha _S}(f_i)$ for every $f_i\in \mathcal {L}^{>}$. Then reasoning as in the first statement we can find some $P\in \mathcal {M}(\underline{P})_{\alpha _S}$ such that $P(f_i)>\underline{P}_{\alpha _S}(f_i)$, and this contradicts that $\alpha _S$ is the maximum value of $\Lambda$.

$\square$

Proof of Theorem 4

We are looking for the maximum $\alpha$ such that the set:

$$\begin{aligned} \text{ core }(\mu )_{\alpha }=\big \{ P\in \mathbb {P}(\mathcal {X})\mid P(A)\ge \mu (A)+\alpha \ \ \forall A\ne \emptyset ,\mathcal {X}\big \} \end{aligned}$$

is non-empty. This is equivalent to requiring that the capacity $\nu$ given by:

$$\begin{aligned} \nu (A)={\left\{ \begin{array}{ll} 0, &{} \text{ for } A=\emptyset \\ \mu (A)+\alpha , &{} \text{ for } A\ne \emptyset ,\mathcal {X}\\ 1, &{} \text{ for } A=\mathcal {X}\end{array}\right. } \end{aligned}$$

is balanced. Since the possibility space $\mathcal {X}$ is finite, this means (Walley 1991, Lemma 2.4.4) that for any $l\in \mathbb {N}$ and $A_1,\ldots ,A_l$, we should have

$$\begin{aligned} \max _{x\in \mathcal {X}}\sum _{i=1}^l \big ( I_{A_i}(x)-\nu (A_i)\big ) \ge 0. \end{aligned}$$

(A1)

Note that we can assume that the events $A_1,\ldots ,A_l$ are proper subsets of $\mathcal {X}$, given that $I_{\mathcal {X}}-\nu (\mathcal {X})=1-1=0$ and $I_{\emptyset }-\nu (\emptyset )=0-0=0$. This allows us to rewrite Eq. (A1) as

$$\begin{aligned} \max _{x\in \mathcal {X}}\sum _{i=1}^l \big ( I_{A_i}(x)-\mu (A_i)-\alpha \big ) \ge 0 \end{aligned}$$

for any $l\in \mathbb {N}$ and any proper subsets $A_1,\ldots ,A_l$ of $\mathcal {X}$.

Now, given such sets $A_1,\ldots ,A_l$, the gamble $f:=I_{A_1}+\ldots +I_{A_l}$ can be rewritten as $f:=a_1I_{B_1}+\ldots +a_m I_{B_m}$ for non-negative integers $a_1>\ldots>a_{m-1}>a_m\ge 0$ and where $\{B_1,\dots ,B_m\}$ is a partition of $\mathcal {X}$. Let us define the sets $C_i^j:=B_i$ for $i=2,\dots ,m$ and for $j=1,\dots ,a_1-a_i$. It holds that

$$\begin{aligned} \mathcal {A}=\big ( A_1,\ldots ,A_l,C_2^1,\ldots ,C_2^{a_1-a_2},\ldots ,C_m^1,\ldots ,C_m^{a_1-a_m}\big ) \end{aligned}$$

belongs to the class $\mathbb {A}(\mathcal {X})$ and $\beta _{\mathcal {A}}=a_1$. Then:

$$\begin{aligned}&\max _{x\in \mathcal {X}}\sum _{i=1}^l \big (I_{A_i}(x)-\mu (A_i)-\alpha \big )=\max _{x\in \mathcal {X}}\bigg (\sum _{i=1}^l\big ( I_{A_i}(x)-\mu (A_i)\big )\\&\qquad +\sum _{i=2}^m \sum _{j=1}^{a_1-a_i} \big ( I_{C_i^j}(x)-\mu (C_i^j) \big )-\sum _{i=2}^m \sum _{j=1}^{a_1-a_i} \big ( I_{C_i^j}(x)-\mu (C_i^j) \big )-l \alpha \bigg )\\&\quad =\beta _{\mathcal {A}}-\sum _{i=1}^l\mu (A_i)-\sum _{i=2}^m \sum _{j=1}^{a_1-a_i}\mu (C_i^j)-\min _{x\in \mathcal {X}}\bigg ( \sum _{i=2}^m \sum _{j=1}^{a_1-a_i}\big ( I_{C_i^j}(x)-\mu (C_i^j)\big )\bigg )-l \alpha . \end{aligned}$$

This means that

$$\begin{aligned} \max _{x\in \mathcal {X}}\sum _{i=1}^l \big (I_{A_i}(x)-\mu (A_i)-\alpha \big )\ge 0 \end{aligned}$$

if and only if

$$\begin{aligned} \alpha \le \frac{1}{l}\bigg ( \beta _{\mathcal {A}}-\sum _{i=1}^l\mu (A_i)-\sum _{i=2}^m \sum _{j=1}^{a_1-a_i}\mu (C_i^j) \bigg )-\min _{x\in \mathcal {X}}\frac{1}{l}\sum _{i=2}^m \sum _{j=1}^{a_1-a_i}\big ( I_{C_i^j}(x)-\mu (C_i^j)\big ). \end{aligned}$$

(A2)

Now, instead of considering the events $A_1,\ldots ,A_l$, consider $A_1,\ldots ,A_l,C_2^1$. We get:

$$\begin{aligned}{} & {} \frac{1}{l+1}\bigg ( \beta _{\mathcal {A}}-\sum _{i=1}^l\mu (A_i)-\sum _{i=2}^m \sum _{j=1}^{a_1-a_i}\mu (C_i^j) \bigg )\\{} & {} \qquad -\min _{x\in \mathcal {X}}\bigg [\frac{1}{l+1}\Big (\sum _{i=2}^m \sum _{j=1}^{a_1-a_i}(I_{C_j}(x)-\mu (C_j))-(I_{C_2^1}(x)-\mu (C_2^1)\Big )\bigg ]\\{} & {} \quad \le \frac{1}{l}\bigg ( \beta _{\mathcal {A}}-\sum _{i=1}^l\mu (A_i)-\sum _{i=2}^m \sum _{j=1}^{a_1-a_i}\mu (C_i^j)\bigg )-\min _{x\in \mathcal {X}}\frac{1}{l}\sum _{i=2}^m \sum _{j=1}^{a_1-a_i}\big ( I_{C_i^j}(x)-\mu (C_i^j)\big ). \end{aligned}$$

In fact, observe that $\beta _{\mathcal {A}}-\sum _{i=1}^l\mu (A_i)-\sum _{i=2}^m \sum _{j=1}^{a_1-a_i}\mu (C_i^j)\ge 0$ since $\mu$ is an exact capacity, and that the minimum in each equation is given by $-\sum _{i=2}^m \sum _{j=1}^{a_1-a_i}\mu (C_i^j)+\mu (C_2^1)$, and $-\sum _{i=2}^m \sum _{j=1}^{a_1-a_i}\mu (C_i^j)$ respectively, in both cases considering $x\in B_1$.

Iterating the procedure, the minimum value in the right-hand side part of Eq. (A2) is attained for $A_1,\ldots ,A_l,C_2^1,\ldots ,C_2^{a_1-a_2},\ldots ,C_m^1,\ldots ,C_m^{a_1-a_m}$. With this, we deduce that $\mathcal {M}_{\alpha }(\underline{P})$ is non-empty if and only if $\alpha \le \min _{\mathcal {A}\in \mathbb {A}}\frac{1}{\mid \mathcal {A}\mid } \bigg (\beta _{\mathcal {A}}-\sum _{i=1}^l\mu (A_i)\bigg )$, and the proof is complete. $\square$

In order to prove Theorem 5, we must establish first a couple of auxiliary lemmas.

Lemma 18

Let $\mu$ be a maximally imprecise exact capacity, and consider $\mathcal {A}\in \mathbb {A}(\mathcal {X})$ such that there exists $\mathcal {A}_1\in \mathbb {A}(\mathcal {X})$ with $\mathcal {A}_1\subset \mathcal {A}$. Then $\mathcal {A}_2=\mathcal {A}\setminus \mathcal {A}_1\in \mathbb {A}(\mathcal {X})$, and $\min \{h_{\mathcal {A}_1},h_{\mathcal {A}_2}\}\le h_{\mathcal {A}}$.

Proof

Let $\mathcal {A}\in \mathbb {A}$ and assume that there exists $\mathcal {A}_1\in \mathbb {A}$ such that $\mathcal {A}_1\subset \mathcal {A}$. This means that $\sum _{A\in \mathcal {A}_1}I_A=\beta _{\mathcal {A}_1}$, with $\beta _{\mathcal {A}_1}<\beta _{\mathcal {A}}$. Take $\mathcal {A}_2=\mathcal {A}\setminus \mathcal {A}_1\subset \mathcal {A}$. It holds that:

$$\begin{aligned} \sum _{A\in \mathcal {A}_2}I_A=\sum _{A\in \mathcal {A}}I_A-\sum _{A\in \mathcal {A}_1}I_A=\beta _{\mathcal {A}}-\beta _{\mathcal {A}_1}\in \mathbb {N}, \end{aligned}$$

which implies that $\mathcal {A}_2\in \mathbb {A}$ and also that $\beta _{\mathcal {A}}=\beta _{\mathcal {A}_1}+\beta _{\mathcal {A}_2}$ and $\mid \mathcal {A}\mid =\mid \mathcal {A}_1\mid +\mid \mathcal {A}_2\mid$. Hence

$$\begin{aligned}&\frac{1}{\mid \mathcal {A}\mid }\Bigg (\beta _{\mathcal {A}}-\sum _{A\in \mathcal {A}}\mu (A)\Bigg )\\&\quad =\frac{1}{\mid \mathcal {A}_1\mid +\mid \mathcal {A}_2\mid }\left( \Big ( \beta _{\mathcal {A}_1}-\sum _{A\in \mathcal {A}_1}\mu (A) \Big )+\Big ( \beta _{\mathcal {A}_2}-\sum _{A\in \mathcal {A}_2}\mu (A) \Big ) \right) \\&\quad =\frac{\mid \mathcal {A}_1\mid }{\mid \mathcal {A}_1\mid +\mid \mathcal {A}_2\mid }\frac{1}{\mid \mathcal {A}_1\mid }\left( \beta _{\mathcal {A}_1}-\sum _{A\in \mathcal {A}_1}\mu (A)\right) \\&\qquad +\frac{\mid \mathcal {A}_2\mid }{\mid \mathcal {A}_1\mid +\mid \mathcal {A}_2\mid }\frac{1}{\mid \mathcal {A}_2\mid }\left( \beta _{\mathcal {A}_2}-\sum _{A\in \mathcal {A}_2}\mu (A)\right) . \end{aligned}$$

Thus, $h_{\mathcal {A}}$ is a convex combination of $h_{\mathcal {A}_1}$ and $h_{\mathcal {A}_2}$, and as a consequence $\min \{h_{\mathcal {A}_1}, h_{\mathcal {A}_2}\}\le h_{\mathcal {A}}$. $\square$

Lemma 19

Let $\mu$ be a maximally imprecise exact capacity, and let $\mathcal {A}=\big (A_i\big )_{i\in I}\in \mathbb {A}(\mathcal {X})$ be a family where the minimum in Eq. (12) is attained.

• (a) If $\beta _{\mathcal {A}}=1$, then $\mathcal {A}$ is a partition of $\mathcal {X}$.

• (b) If $\beta _{\mathcal {A}}=\mid \mathcal {A}\mid -1$, then $\mathcal {A}^c=\big ( A_i^c \big )_{i\in I}$ is a partition of $\mathcal {X}$.

• (c) If $1<\beta _{\mathcal {A}} <\vert \mathcal {A}\vert -1$ and for every $A,B\in \mathcal {A}$ at least one of $A\cap B, A\setminus B$ and $B\setminus A$ is empty, then there exists $\mathcal {A}_1\in \mathbb {A}^{*}(\mathcal {X})$ such that $\mathcal {A}_1\subset \mathcal {A}$.

Proof

• (a) First of all, assume that $\beta _{\mathcal {A}}=\sum _{A\in \mathcal {A}}I_A=1$. This means that each $x\in \mathcal {X}$ belongs to one and only one event $A\in \mathcal {A}$, hence $\mathcal {A}$ is a partition of $\mathcal {X}$.

• (b) Secondly, assume that $\beta _{\mathcal {A}}=\sum _{A\in \mathcal {A}}I_A=\mid \mathcal {A}\mid -1$. This means that

  $$\begin{aligned} \mid \mathcal {A}\mid -1=\sum _{A\in \mathcal {A}} I_A=\sum _{A\in \mathcal {A}}\big [ 1-(1-I_A) \big ]=\mid \mathcal {A}\mid -\sum _{A\in \mathcal {A}}(1-I_A)=\mid \mathcal {A}\mid -\sum _{A\in \mathcal {A}}I_{A^c}, \end{aligned}$$

  whence $\sum _{A\in \mathcal {A}}I_{A^c}=1$ and applying (a), $\mathcal {A}^c=\big (A_i^c\big )_{i\in I}$ is a partition of $\mathcal {X}$.

• (c) From the condition, given two different elements $A,B\in \mathcal {A}$ either they are disjoint or one of them is included in the other. If we then consider the partial order $\preceq$ on ${\mathcal A}$ given by set inclusion, then we can find a subfamily ${\mathcal A}_1\subset {\mathcal A}$ of maximal elements in the order that are pairwise disjoint. Since any element of $\mathcal {X}$ must be included in some maximal element, it follows that the subfamily ${\mathcal A}_1$ is a partition of $\mathcal {X}$.

Proof of Theorem 5

Let $\mathcal {A}=\big (A_i\big )_{i=1,\ldots ,k}$ be the element in $\mathbb {A}(\mathcal {X})$ where the minimum in Eq. (12) is attained. According to Lemma 19, if $\beta _{\mathcal {A}}=1$, then $\mathcal {A}$ is partition so $\mathcal {A}\in \mathbb {A}^{*}(\mathcal {X})$ hence Eq. (13) holds. If $\beta _{\mathcal {A}}=\mid \mathcal {A}\mid -1$, then $\mathcal {A}^c=\big (A_i^c\big )_{i=1,\ldots ,k}$ is a partition, so it belongs to $\mathbb {A}^{*}(\mathcal {X})$. Also:

$$\begin{aligned}\small \alpha _S&=\frac{1}{\mid \mathcal {A}\mid }\bigg ( (\mid \mathcal {A}\mid -1)-\sum _{A\in \mathcal {A}}\mu (A) \bigg ) =\frac{1}{\mid \mathcal {A}\mid }\bigg ( (\mid \mathcal {A}\mid -1)+\sum _{A\in \mathcal {A}}(1-\mu (A)-1) \bigg )\\&=\frac{1}{\mid \mathcal {A}\mid }\bigg ( (\mid \mathcal {A}\mid -1-\mid \mathcal {A}\mid )+\sum _{A\in \mathcal {A}}\bar{\mu }(A^c) \bigg )=\frac{1}{\mid \mathcal {A}\mid }\bigg ( \sum _{A\in \mathcal {A}}\bar{\mu }(A^c)-1 \bigg )\\&=\frac{1}{\mid \mathcal {A}\mid }\bigg ( \sum _{A\in \mathcal {A}^c}\bar{\mu }(A)-1 \bigg ), \end{aligned}$$

hence Eq. (13) holds.

Assume now that $1<\beta _{\mathcal {A}}<\mid \mathcal {A}\mid -1$, and let us prove that it is possible to find another ${{\mathcal {A}}}^*\in \mathbb {A}(\mathcal {X})$ such that $\beta _{\mathcal {A}^*}<\beta _{{\mathcal {A}}}$ and where $\alpha _S$ is attained.

From item (c) in Lemma 19 we deduce that either there is $\mathcal {A}^{*}\in \mathbb {A}(\mathcal {X})$ with $\mathcal {A}^{*}\subset \mathcal {A}$ or there are two different $A_i,A_j\in \mathcal {A}$ with $A_i\cap A_j\ne \emptyset$, $A_i\setminus A_j\ne \emptyset$ and $A_j\setminus A_i\ne \emptyset$. In this second case, applying 2-monotonicity with the sets $A_i$ and $A_j$ above we deduce that:

$$\begin{aligned}\small h_{\mathcal {A}}&=\frac{1}{\mid \mathcal {A}\mid }\Big ( \beta _{{\mathcal {A}}}-\sum _{A\in \mathcal {A}}\mu (A) \Big )=\frac{1}{\mid \mathcal {A}\mid }\left( \beta _{{\mathcal {A}}}-\sum _{A\in \mathcal {A}\setminus \{A_i,A_j\}}\mu (A)-\mu (A_i)-\mu (A_j) \right) \\&\ge \frac{1}{\mid \mathcal {A}\mid }\left( \beta _{{\mathcal {A}}}-\sum _{A\in \mathcal {A}\setminus \{A_i,A_j\}}\mu (A)-\mu (A_i\cap A_j)-\mu (A_i\cup A_j) \right) =h_{\mathcal {A}_1}, \end{aligned}$$

where $\mathcal {A}_1=\big (\mathcal {A}\setminus \{A_i,A_j\}\big )\cup \big (A_i\cap A_j,A_i\cup A_j\big )$, using that $\beta _{\mathcal {A}_1}=\beta _{\mathcal {A}}$. Thus, $\alpha _S=h_{\mathcal {A}_1}$.

Now, if in $\mathcal {A}_1$ it is possible to find two different events $B_i,B_j$ with $B_i\cap B_j\ne \emptyset$, $B_i\setminus B_j\ne \emptyset$ and $B_j\setminus B_i\ne \emptyset$ a similar reasoning shows that $\mathcal {A}_2=\mathcal {A}_1\cup \big (B_i\cup B_j,B_i\cap B_j \big )\setminus (B_i,B_j)$ also satisfies $\beta _{\mathcal {A}_2}=\beta _{\mathcal {A}_1}$ and $h_{\mathcal {A}_2} =h_{\mathcal {A}_1}=\alpha _S$. Iterating the procedure, we find after a finite number of steps that there are no different events C and D in the family $\mathcal {A}_k$ such that $C\cap D\ne \emptyset$, $C\setminus D\ne \emptyset$ and $D\setminus C\ne \emptyset$. But then, applying Lemma 19 we deduce that there is $\mathcal {A}^{*}\in \mathbb {A} ^{*}(\mathcal {X})$ with $\mathcal {A} ^{*}\subset \mathcal {A}_k$. Applying Lemma 18 that either $h_{\mathcal {A}_k}=h_{\mathcal {A}^{*}}$ or $h_{\mathcal {A}_k}=h_{\mathcal {A}_k\setminus \mathcal {A}^{*}}$. Since both $\beta _{\mathcal {A}^{*}}$ and $\beta _{\mathcal {A}_k\setminus \mathcal {A}^{*}}$ are strictly smaller than $\beta _{\mathcal {A}_k}$, we deduce that we can find another element of $\mathbb {A} (\mathcal {X})$ where the value $\alpha _S$ is attained and with a smaller value of $\beta _{\mathcal A}$. If we repeat this process we end up with a family $\mathcal {A}'\in \mathbb {A}(\mathcal {X})$ such that $\beta _{\mathcal {A}'}=1$, and where $\alpha _S$ is attained, at which point we apply the first part of the proof.

Proof of Propositions 6 and 13

We start proving Proposition 13. From it, Proposition 6 trivially follows just noting that, by hypothesis, $\mathcal {L}^=$ would be empty and $\mathcal {L}^{>}$ would be formed by all the proper events of $\mathcal {X}$ (because $\mu$ is by hypothesis maximally imprecise), and this allows to apply Proposition 13 to a lower prevision were $\mathcal {L}^{>}$ contains the indicator functions of the proper events.

$(\Rightarrow )$:

Assume that $P_0(f)\ge \underline{P}(f)+\alpha$ for every $f\in \mathcal {L}^{>}$, and consider $Q\in B_{c}^{\alpha }(P_0)$. Since we can assume without loss of generality that $0=\min f<\max f=1$ for every $f\in \mathcal {L}^{>}$, it follows that $\mid Q(f)-P_0(f)\mid \le \alpha$, whence $Q(f)\ge P_0(f)-\alpha \ge \underline{P}(f)$ for every $f\in \mathcal {L}^{>}$. Since by assumption $\mathcal {L}^{=}=\emptyset$, this implies that $Q\in \mathcal {M}(\underline{P})$.

$(\Leftarrow )$:

Consider a probability measure $P_0$ such that $B_{c}^{\alpha }(P_0)\subseteq \mathcal {M}(\underline{P})$, and let us prove that $P_0(f)\ge \underline{P}(f)+\alpha$ for every $f\in \mathcal {L}^>$. Assume ex-absurdo that $P_0(f)-\alpha <\underline{P}(f)$ for some gamble $f\in \mathcal {L}^>$, and let us show that there exists some $Q\in B_{c}^{\alpha }(P_0)$ such that $Q(f)<\underline{P}(f)$. To see that this is indeed the case, note that since $B_{c}^{\alpha }(P_0)$ is included in $\mathbb {P}^{*}(\mathcal {X})$, it must be $P_0(\{x_i\})>\alpha$ for every $x_i\in \mathcal {X}$. If we now take $x_m,x_M\in \mathcal {X}$ such that:

$$\begin{aligned} 0=\min f=f(x_m), \quad 1=\max f=f(x_M), \end{aligned}$$

and define Q by means of the mass function

$$\begin{aligned} Q(\{x_{m}\})=P_0(\{x_{m}\})+\alpha , \quad Q(\{x_{M}\})=P_0(\{x_{M}\})-\alpha , \end{aligned}$$

and $Q(\{x\})=P_0(\{x\})$ for any other $x\ne x_{m},x_{M}$, it follows that $\mid Q(B)-P_0(B)\mid \le \alpha$ for every $B\subseteq \mathcal {X}$, whence $Q\in B_{c}^{\alpha }(P_0)\subseteq \mathcal {M}(\underline{P})$. However:

$$\begin{aligned} Q(f)&=\sum _{x\in \mathcal {X}} Q(\{x\})f(x)=\sum _{x\ne x_m,x_M}Q(\{x\})f(x)+Q(\{x_m\})f(x_m)+Q(\{x_M\})f(x_M)\\&=\sum _{x\ne x_m,x_M}P_0(\{x\})f(x)+\big (P_0(\{x_m\})+\alpha \big )f(x_m)+\big (P_0(\{x_M\})-\alpha \big )f(x_M)\\&=P_0(f)-\alpha \big ( f(x_M)-f(x_m) \big )=P_0(f)-\alpha <\underline{P}(f), \end{aligned}$$

hence $Q\notin \mathcal {M}(\underline{P})$, we obtain a contradiction.

The second part of Proposition 6 is an immediate consequence of the first once we realise that, since $\mathcal {M}\subseteq \mathbb {P}^{*}(\mathcal {X})$, we can compute $\alpha _I$ by means of Eq. (8). $\square$

Proof of Propositions 7, 8 and 16

We start proving the properties of the centroids for coherent lower previsions (Proposition 16). Since exact capacities are particular cases of coherent lower previsions, Propositions 7 and 8 can be regarded as a corollary. Throughout this proof, for simplicity we use the notation $\underline{P}(A):=\underline{P}(I_A)$ and $\overline{P}(A)=\overline{P}(I_A)$.

It $x_i$ is a null-player, $\underline{P}(\mathcal {X})=1=\underline{P}(\mathcal {X}\setminus \{x_i\})$, whence $\overline{P}(\{x_i\})=0$. Since $\Phi _2^{\underline{P}},\Phi _3^{\underline{P}},\Phi _4^{\underline{P}}$ belong to the credal set, any of these centroids gives probability zero to $x_i$.

With respect to symmetry, let $\sigma _{i,j}$ denote the permutation of $\mathcal {X}$ that exchanges $x_i$ and $x_j$ and leaves the other elements fixed, and let $\underline{P}_{\sigma _{i,j}}$ be given by $\underline{P}_{\sigma _{i,j}}(f)=\underline{P}(f \circ \sigma _{i,j})$.

If $P$ is an extreme point of $\mathcal {M}(\underline{P})$, then $P_{\sigma _{i,j}}$ is an extreme point of $\mathcal {M}(\underline{P}_{\sigma _{i,j}})$. If we now assume that by symmetry $\underline{P}_{\sigma _{i,j}}=\underline{P}$, it follows that $\Phi _2^{\underline{P}}=\Phi _2^{\underline{P}_{\sigma _{i,j}}}$. But on the other hand it must be $\Phi _2^{\underline{P}}(\{x_i\})=\Phi _2^{\underline{P}_{\sigma _{i,j}}}(\{x_j\})$, and as a consequence $\Phi _2^{\underline{P}}(\{x_i\})=\Phi _2^{\underline{P}}(\{x_j\})$.

Concerning the set of incenters, if $P$ satisfies that $B_{o}^{\alpha }(P)\subseteq \mathcal {M}(\underline{P})\cap \mathbb {P}^*(\mathcal {X})$, then $B_{o}^{\alpha }(P_{\sigma _{i,j}})\subseteq \mathcal {M}(\underline{P}_{\sigma _{i,j}})\cap \mathbb {P}^*(\mathcal {X})=\mathcal {M}(\underline{P})\cap \mathbb {P}^*(\mathcal {X})$. Therefore,

$$\begin{aligned} \Big \{P(\{x_i\})\mid P\in \Psi _3^{\underline{P}}\Big \}=\bigg \{P(\{x_i\})\mid P\in \Psi _3^{\underline{P}_{\sigma _{i,j}}}\bigg \}=\Big \{P(\{x_j\})\mid P\in \Psi _3^{\underline{P}}\Big \}. \end{aligned}$$

Finally, for the contraction centroid note that, for any gamble $f\in \mathcal {L}^{>}$ and any $\alpha >0$, it holds that

$$\begin{aligned} P(f)\ge \underline{P}(f)+\alpha \Leftrightarrow P_{\sigma _{i,j}}(f)\ge \underline{P}_{\sigma _{i,j}}(f)+\alpha =\underline{P}(f)+\alpha , \end{aligned}$$

whence $\mathcal {M}(\underline{P})_\alpha$ is invariant under $\sigma _{i,j}$. From here we deduce that $\Phi _4^{\underline{P}}(\{x_i\})=\Phi _4^{\underline{P}}(\{x_j\})$.

With respect to ignorance preservation, in the case of $\Phi _1^{\underline{P}}$ it follows immediately from Eq. (3) because the Möbius inverse of $\underline{P}$ satisfies $m(\mathcal {X})=1$ and zero for any other $A\ne \mathcal {X}$. In the case of $\Phi _2^{\underline{P}}$, it suffices to note that the extreme points of $\mathcal {M}(\underline{P})$ are the degenerate distributions; and for $\Phi _3^{\underline{P}},\Phi _4^{\underline{P}}$ it suffices to use that the uniform distribution $P_0$ is the only one in $\mathcal {M}(\underline{P})$ for which $B_{o}^{\frac{1}{\mid \mathcal {X}\mid }}(P_0)$ is included in $\mathcal {M}(\underline{P})\cap \mathbb {P}^*(\mathcal {X})$.

Finally, with respect to continuity, in the case of $\Phi _1^{\mu }$ it follows directly from Eq. (2); while in the case of $\Phi _3^{\mu },\Phi _4^{\mu }$ it follows trivially because we are using in their definition the topology of the total variation. $\square$

Proof of Propositions 9, 10 and 17

We first prove that $\varphi _1,\varphi _2$ and $\varphi _3$ are centrality measures for a coherent lower prevision $\underline{P}$ (assuming that $\mathcal {M}(\underline{P})$ is a polytope for $\varphi _1$ and $\varphi _3$), proving hence Proposition 17. Since any exact capacity is a particular type of coherent lower prevision whose core is a polytope, Propositions 9 and 10 trivially follow as a corollary.

Let us begin by showing that $\varphi _1$ is a centrality measure:

CM1:

This holds by definition.

CM2:

By construction, if $P\in \text{ ext }(\mathcal {M}(\underline{P}))$ then $d(P,\Phi ^{\underline{P}})\ge \beta$, hence $\varphi (P)=1-1=0$.

CM3:

By definition $\varphi _1(P)=1$ iff $d(P,\Phi ^{\underline{P}})=0$, and since d is a distance this holds iff $P=\Phi ^{\underline{P}}$.

CM4:

This follows if and only if for any $P\in \text{ ext }\big (\mathcal {M}(\underline{P})\big )$ and any $\lambda ,\beta \in [0,1]$ such that $\lambda \ge \beta$ it holds that

$$\begin{aligned} d\big (\lambda P+(1-\lambda )\Phi ^{\underline{P}},\Phi ^{\underline{P}}\big )\le d\big (\beta P+(1-\beta )\Phi ^{\underline{P}},\Phi ^{\underline{P}}\big ). \end{aligned}$$

(A3)

If we denote $P_1:=\lambda P+(1-\lambda )\Phi ^{\underline{P}}$ and $P_2:=\beta P+(1-\beta )\Phi ^{\underline{P}}$, there is some $a\in (0,1)$ such that $P_2=a P_1+(1-a) \Phi ^{\underline{P}}$. As a consequence, for any event A it holds that

$$\begin{aligned}{} & {} \mid P_2(A)-\Phi ^{\underline{P}}(A)\mid \le a\mid P_1(A)-\Phi ^{\underline{P}}(A)\mid +(1-a) \mid \Phi ^{\underline{P}}(A)-\Phi ^{\underline{P}}(A)\mid \\{} & {} \quad =a \mid P_1(A)-\Phi ^{\underline{P}}(A)\mid , \end{aligned}$$

from which Eq. (A3) follows.

Let us prove now that $\varphi _2$ is a centrality measure.

CM1,CM3:

These follow immediately from the definition of $\mathcal {M}(\underline{P})_0,\mathcal {M}(\underline{P})_1$.

CM2:

This holds because any extreme point P of $\mathcal {M}(\underline{P})$ does not belong to $\mathcal {M}(\underline{P})_\alpha$ for any $\alpha >0$.

CM4:

Consider $P_1=\lambda P+(1-\lambda ) P_0$ and $P_2=\beta P+(1-\beta ) P_0$ for $\lambda \ge \beta$, where P is an extreme point of $\mathcal {M}(\underline{P})$ and $P_0$ is the unique element of $\mathcal {M}(\underline{P})_1$. Then (CM4) holds if an only if for any $\gamma \in (0,1)$ such that $P_1\in \mathcal {M}(\underline{P})_\gamma$, also $P_2\in \mathcal {M}(\underline{P})_\gamma$. This is a consequence of the convexity of the set $\mathcal {M}(\underline{P})_\gamma$, given that $P_2=a P_1+(1-a)P_0$ for some $a\in [0,1]$.

Finally, let us see that $\varphi _3$ satisfies the four properties in Definition 6.

CM1:

For any $P\notin \mathcal {M}(\underline{P})$, $P\notin \mathcal {M}(\underline{P})_{\alpha }$ for any $\alpha >0$, whence $\varphi _3(P)=0$.

CM2:

If $P\in \text{ ext }\big (\mathcal {M}(\underline{P})\big )$, then $P(f)=\underline{P}(f)$ for some gamble f. This implies that $P(f)\not \ge \underline{P}(f)+\alpha$, and therefore $P\notin \mathcal {M}(\underline{P})_{\alpha }$ for any $0<\alpha \le \alpha _{S_1}$, which implies that $\varphi _3(P)=0$.

CM3:

By definition of the procedure, $P_0=\Phi _4^{\underline{P}}$ is the unique probability in $\mathcal {M}_{\alpha _{S_l}}(\underline{P})$, so it is the unique probability with centrality degree 1.

CM4:

Take $\lambda \ge \beta$, $P\in \text{ ext }(\mathcal {M}(\underline{P}))$ and define $P_1=\lambda P+(1-\lambda )P_0$ and $P_2=\beta P+(1-\beta )P_0$. Let us see that if $P_1(f)\ge \underline{P}(f)+\alpha$, then $P_2(f)\ge \underline{P}(f)+\alpha$ too, for every $f\in \mathcal {L}^{>}$. Consider two cases:

• 1. Case 1: $P(f)\ge P_0(f)$. In that case, it holds that $P(f)\ge P_0(f)\ge \underline{P}(f)+\alpha$, hence:

  $$\begin{aligned} P_2(f)=\beta P(f)+(1-\beta )P_0(f)\ge \beta P_0(f)+(1-\beta )P_0(f)=P_0(f)\ge \underline{P}(f)+\alpha . \end{aligned}$$

• 2. Case 2: $P(f)<P_0(f)$. Then $P_1(f)\ge \underline{P}(f)+\alpha$ is equivalent to:

  $$\begin{aligned} \lambda P(f)+(1-\lambda )P_0(f)\ge \underline{P}(f)+\alpha , \end{aligned}$$

  which is equivalent to:

  $$\begin{aligned} \lambda \big (P(f)-P_0(f)\big ) \ge \underline{P}(f)+\alpha -P_0(f). \end{aligned}$$

  Using that $P(f)-P_0(f)<0$, the previous inequality is equivalent to:

  $$\begin{aligned} \lambda \le \frac{\underline{P}(f)+\alpha -P_0(f)}{P(f)-P_0(f)}. \end{aligned}$$

  Since $\beta \le \lambda$, following the reversed steps, if follows that:

  $$\begin{aligned} P_2(f)=\beta P(f)+(1-\beta )P_0(f)\ge \underline{P}(f)+\alpha . \end{aligned}$$

With this, we conclude that if $P_1(f)\ge \underline{P}(f)+\alpha$, $P_2(f)\ge \underline{P}(f)+\alpha$ too, for every $f\in \mathcal {L}^{>}$. Hence, if the centrality degree of $P_1$ is $\varphi _3^{\underline{P}}(P_1)$, the centrality degree of $P_2$ should be at least $\varphi _3^{\underline{P}}(P_1)$. $\square$

Proof of Proposition 14

Since $\underline{P}$ is associated with a probability interval,

$$\begin{aligned} \mathcal {M}(\underline{P})=\{P\mid l_i \le P(\{x_i\})\le u_i \ \forall i=1,\dots ,n\}, \end{aligned}$$

meaning that, while we can assume without loss of generality that $\mathcal {L}^{=}\cup \mathcal {L}^{>}$ include the indicators of proper events, in this case only the indicators of the singletons are necessary, and $\mathcal {L}^{=}$ and $\mathcal {L}^{>}$ reduce to ${{\mathcal {I}}}^{=}$ and $\mathcal {I}^{>}$, respectively.

Let $\alpha$ be the minimum in Eq. (24), and let us see that $\alpha =\alpha _S$. We consider the following cases:

• 1. Assume that $\alpha =\frac{1}{\mid \mathcal {I}^{>}\mid }\left( 1-\sum _{i=1}^n l_i \right)$. Define $P_0$ by:

  $$\begin{aligned} P_0(\{x_i\})={\left\{ \begin{array}{ll} l_i+\alpha , &{} \text{ if } i\in \mathcal {I}^{>}.\\ l_i, &{} \text{ if } i\in \mathcal {I}^{=}. \end{array}\right. } \end{aligned}$$

  $P_0$ satisfies the following properties:

  • (i) It is a probability measure because it is non-negative and

    $$\begin{aligned} \sum _{i=1}^nP_0(\{x_i\})&=\sum _{i\in \mathcal {I}^{>}} (l_i+\alpha )+\sum _{i\in \mathcal {I}^{=}} l_i\\&=\sum _{i\in \mathcal {I}^{>}}l_i+\mathcal {I}^{>}\alpha +\sum _{i\in \mathcal {I}^{=}} l_i=\sum _{i=1}^n l_i+\big ( 1-\sum _{i=1}^n l_i \big )=1. \end{aligned}$$

  • (ii) $P_0(\{x_i\})\in [l_i,u_i]$. On the one hand, if $i\in \mathcal {I}^{=}$, $P_0(\{x_i\})=l_i$. On the other hand, if $i\in \mathcal {I}^{>}$, $P_0(\{x_i\})=l_i+\alpha >l_i$. Also, by definition of $\alpha$ it holds that $\alpha \le \frac{1}{2}(u_i-l_i)$, hence:

    $$\begin{aligned} P_0(\{x_i\})=l_i+\alpha<l_i+\frac{1}{2}(u_i-l_i)=\frac{1}{2}(l_i+u_i)<u_i. \end{aligned}$$

  • (iii) To see that $\mathcal {M}(\underline{P})_{\alpha }=\{P_0\}$, and as a consequence that $\alpha =\alpha _S$, note that, by construction,

    $$\begin{aligned} \mathcal {M}(\underline{P})_{\alpha }&=\{P\mid P(\{x_i\})\in [\underline{P}(\{x_i\}+\alpha ,\overline{P}(\{x_i\})-\alpha ] \ \forall i \in {{\mathcal {I}}}^{>}, \\&\qquad \qquad P(\{x_i\})=\underline{P}(\{x_i\})=\overline{P}(\{x_i\}) \ \forall i \in {{\mathcal {I}}}^{=}\} \\&=\{P\mid P(\{x_i\})\in [l_i+\alpha ,u_i-\alpha ] \ \forall i\in {{\mathcal {I}}}^{>},P(\{x_i\})=l_i=u_i \ \forall i \in {{\mathcal {I}}}^{=}\}. \end{aligned}$$

    It follows then that by definition $P_0\in \mathcal {M}(\underline{P})_{\alpha }$. Since moreover

    $$\begin{aligned}\sum _{i=1}^{n} l_i+ \mid \mathcal {I}^{>}\mid \alpha =1,\end{aligned}$$

    we deduce that any $Q\in \mathcal {M}_{\alpha }(\underline{P})$ must coincide with $P_0$ and that $\mathcal {M}_{\alpha '}(\underline{P})=\emptyset$ for any $\alpha '>\alpha$.

  This implies that $\mathcal {M}(\underline{P})_{\alpha }=\{P_0\}$ is formed by a unique probability measure $P_0$, which coincides with $\Phi _4^{\underline{P}}$.

• 2. Assume that $\alpha =\frac{1}{\mid \mathcal {I}^{>}\mid }\left( \sum _{i=1}^n u_i-1 \right)$. Define $P_0$ by:

  $$\begin{aligned} P_0(\{x_i\})={\left\{ \begin{array}{ll} u_i-\alpha , &{} \text{ if } i\in \mathcal {I}^{>}.\\ u_i, &{} \text{ if } i\in \mathcal {I}^{=}. \end{array}\right. } \end{aligned}$$

  Following the same steps as in the previous case, we obtain that $\mathcal {M}(\underline{P})_{\alpha }=\{P_0\}$, where $P_0=\Phi _4^{\underline{P}}$.

• 3. Assume that $\alpha =\frac{1}{2}(u_i-l_i)$ for some $i\in \{1,\ldots ,n\}$. We define a new probability interval $\mathcal {I}^{*}$ given by:

  $$\begin{aligned} {[}l_i^{*},u_i^{*}]={\left\{ \begin{array}{ll} {[}l_i,u_i] &{} \text{ if } i\in \mathcal {I}^{=}.\\ {[}l_i+\alpha ,u_i-\alpha ] &{} \text{ if } i\in \mathcal {I}^{>}. \end{array}\right. } \end{aligned}$$

  Let us prove some interesting properties of this probability interval:

  • (i) $\mathcal {M}(\mathcal {I}^{*})\ne \emptyset$: on the one hand,

    $$\begin{aligned} \sum _{i=1}^n l_i=\sum _{i\in \mathcal {I}^{>}}(l_i+\alpha )+\sum _{i\in \mathcal {I}^{=}}l_i=\sum _{i=1}^nl_i+\mid \mathcal {I}^{>}\mid \alpha \le 1, \end{aligned}$$

    since by hypothesis $\alpha \le \frac{1}{\mid \mathcal {I}\mid }\big (1-\sum _{i=1}^nl_i \big )$ and, since $\mathcal {I}$ avoids sure loss, $\sum _{i=1}^nl_i\le 1$. Similarly:

    $$\begin{aligned} \sum _{i=1}^n u_i=\sum _{i\in \mathcal {I}^{>}}(u_i-\alpha )+\sum _{i\in \mathcal {I}^{=}}u_i=\sum _{i=1}^nu_i-\alpha \mid \mathcal {I}^{>}\mid \ge 1, \end{aligned}$$

    since by hypothesis $\alpha \le \frac{1}{\mid \mathcal {I}\mid }\big (\sum _{i=1}^nu_i-1\big )$ and, since $\mathcal {I}$ avoids sure loss, $\sum _{i=1}^nu_i\ge 1$. We therefore conclude that $\mathcal {I}^{*}$ avoids sure loss.

  • (ii) By construction, $\mathcal {M}(\mathcal {I}^{*})=\mathcal {M}(\underline{P})_{\alpha }$. Moreover, given $\alpha '>\alpha$, if follows that there is some $i=1,\dots ,n$ such that $l_i+\alpha '>u_i-\alpha '$, because $\alpha '>\min _i \frac{u_i-l_i}{2}$, and therefore $\mathcal {M}(\underline{P})_{\alpha '}=\emptyset$. Thus, $\alpha =\alpha _S$.

  It only remains to see that $\alpha =\alpha _S$, but this straightforwardly follows from the fact that the chosen $\alpha$ saturates the probability of at least one of the events.

$\square$

Proof of Corollary 15

Consider first the PMM. Since we are assuming that $\underline{P}_{PMM}(A)\in (0,1)$ for non-trivial events A, it corresponds to the probability interval $\mathcal {I}_{PMM}$ given by:

$$\begin{aligned} l_i=(1+\delta )P_0(\{x_i\})-\delta , \quad u_i=(1+\delta )P_0(\{x_i\}) \quad \forall i=1,\ldots ,n. \end{aligned}$$

Applying Eq. (24) to this probability interval, we obtain:

$$\begin{aligned} \alpha _S&=\min \left\{ \frac{1}{n}\Big (1-\sum _{i=1}^nl_i \big ),\frac{1}{n}\Big ( \sum _{i=1}^nu_i-1 \Big ),\frac{1}{2}\min _{i=1,\ldots ,n}(u_i-l_i) \right\} \\&=\min \left\{ \frac{1}{n}\big (1-(1+\delta -n\delta )\big ),\frac{1}{n}\delta ,\frac{1}{2}\delta \right\} =\frac{\delta }{n}. \end{aligned}$$

Following the steps in the previous proposition, we deduce that $\mathcal {M}(\underline{P}_{PMM})_{\alpha _S}=\big \{\Phi _4^{\underline{P}_{PMM}}\}$, where $\Phi _4^{\underline{P}_{PMM}}$ is given by:

$$\begin{aligned} \Phi _4^{\underline{P}_{PMM}}=u_i-\alpha _S=(1+\delta )P_0(\{x_i\})-\frac{\delta }{n} \quad \forall i=1,\ldots ,n. \end{aligned}$$

Similarly, for the LV we are assuming that $\underline{P}_{LV}(A)\in (0,1)$ for non-trivial events A, so the LV corresponds to a probability interval $\mathcal {I}_{LV}$ given by:

$$\begin{aligned} l_i=(1-\delta )P_0(\{x_i\}), \quad u_i=(1-\delta )P_0(\{x_i\})+\delta \quad \forall i=1,\ldots ,n. \end{aligned}$$

Applying again Eq. (24):

$$\begin{aligned} \alpha _S&=\min \left\{ \frac{1}{n}\Big (1-\sum _{i=1}^nl_i \big ),\frac{1}{n}\Big ( \sum _{i=1}^nu_i-1 \Big ),\frac{1}{2}\min _{i=1,\ldots ,n}(u_i-l_i) \right\} \\&=\min \left\{ \frac{\delta }{n},\frac{1}{n}(-\delta +n\delta ),\frac{\delta }{2} \right\} =\frac{\delta }{n}. \end{aligned}$$

From Proposition 14, we obtain $\mathcal {M}(\underline{P}_{LV})_{\alpha _S}=\big \{\Phi _4^{\underline{P}_{LV}}\big \}$, where $\Phi _4^{\underline{P}_{LV}}$ is given by:

$$\begin{aligned} \Phi _4^{\underline{P}_{LV}}(\{x_i\})=l_i+\alpha _S=(1-\delta )P_0(\{x_i\})+\frac{\delta }{n}. \end{aligned}$$

It follows from the results in Miranda and Montes (2018) that the values obtained for the PMM and the LV coincide with the Shapley value. Also, we have seen that $\mathcal {M}(\underline{P})_{\alpha _S}$ coincides with the set of incenters with respect to the total variation distance (see Proposition 13). Since in this case $\mathcal {M}(\underline{P})_{\alpha _S}$ is a singleton (for both the PMM and the LV), the incenter is unique and it coincides with the contraction center. Finally, to see that these centers also coincide with the average of the extreme points, we just need to note (see Montes et al. 2019, Sec.3.1 and Montes et al. 2020a, Sec.5.1) that using the approach based on the permutations, each extreme point appears in the same number of permutations, so the average of the extreme points coincides with the Shapley value. $\square$