III. SUDANO'S PROBABILITIES
Recently, Sudano has proposed interesting alternatives denoted P rPl, P rNPl, P raPl, **P rBel** and P rHyb to **BetP**,
all defined in DST framework [15]. Sudano uses different kinds of mappings either proportional to the plausibility, to the normalized plausibility, to all plausibilities, to the belief or a hybrid mapping. **P rPl** and **P rBel** are defined4for all X 6= ∅ ∈ Θ by:

$$PrPl(X)=Pl(X)\cdot\sum_{Y\in2^{\Theta},X\subseteq Y}\frac{m(Y)}{CS[Pl(Y)]}\tag{4}$$  $$PrBel(X)=Bel(X)\cdot\sum_{Y\in2^{\Theta},X\subseteq Y}\frac{m(Y)}{CS[Bel(Y)]}\tag{5}$$  where the compound-to-sum of singletons (CS) operator of 

any function5 f(.) is defined by [12]:
CS[f(Y )] ,X Yi∈2Θ,|Yi|=1,∪iYi=Y f(Yi) P rNPl, P raPl and P rHyb are given by [12], [15]:
- a mapping proportional to the *normalized* plausibility

 ${\Pr N P l(X)=\frac{1}{\Delta}\sum_{Y\in2^{\Theta},Y\cap X\neq\emptyset}m(Y)=\frac{1}{\Delta}\cdot P l(X)}$ (6)  where ${\Delta}$ is a normalization factor. 
- a mapping proportional to all plausibilities

$$PraPl(X)=Bel(X)+\epsilon\cdot Pl(X)\tag{7}$$  with $\epsilon\triangleq(1-\sum_{Y\in2^{\Theta}}Bel(Y))/(\sum_{Y\in2^{\Theta}}Pl(Y).$ * a hybrid transformation $$PrHyb(X)=PraPl(X)\cdot\sum_{\begin{subarray}{c}Y\in2^{\Theta}\\ X\subseteq Y\end{subarray}}\frac{m(Y)}{CS[PraPl(Y)]}$$ (8) IV. Cuzzolin's Intersection Probability * **1007**
In 2007, a new transformation has been proposed in [1]
by Cuzzolin in the framework of DST. From a geometric interpretation of Dempster's rule, an *Intersection Probability* measure was proposed from the proportional repartition of the Total Non Specific Mass6(TNSM) by each contribution of the non-specific masses involved in it. For notation convenience, we will denote it **CuzzP** in the sequel. **CuzzP**(.) is defined on any finite and discrete frame Θ = {θ1**, . . . , θ**n}, n ≥ 2, satisfying Shafer's model, by

satisfying Slater's model, by $$CuzzP(\theta_i)=m(\theta_i)+\frac{\Delta(\theta_i)}{\sum_{j=1}^n\Delta(\theta_j)}\times TNSM\tag{9}$$ with $\Delta(\theta_i)\triangleq Pl(\theta_i)-m(\theta_i)$ and $$TNSM=1-\sum_{j=1}^n m(\theta_j)=\sum_{A\in2^{\Theta_i}|A|>1}m(A)\tag{10}$$ to compute the $n$-th order of $\theta_i$. 
CuzzP is however not appealing for the following reasons:
1) Although (9) does not include explicitly Dempster's rule, its geometrical justification [1], [2] is strongly conditioned by the acceptance of Dempster's rule as the fusion operator for belief functions. This is a dogmatic point of view we disagree with since it has been recognized since many years by different experts of AI community, that other fusion rules can offer better performances, especially for cases where high conflicting sources are involved.

2) Some parts of the masses of partial ignorance, say A,
involved in the TNSM, are also transferred to singletons, say θi ∈ Θ which are not included in A (i.e. such that {θi} ∩ A = ∅). Such transfer is not good and does not make sense in our point of view. To be more clear, let's take Θ = {**A, B, C**} and m(.) defined on its power set with all masses strictly positive. In that case, m(A ∪ B) > 0 does count in TNSM and thus it is a bit redistributed back to C with the ratio ∆(C)
∆(A)+∆(B**)+∆(**C)
through **T NSM >** 0. There is no solid reason for committing partially m(A ∪ B) to C since, only A and B are involved in that partial ignorance. Similar remark holds for the partial redistribution of m(A ∪ C) > 0.

3) **CuzzP** is not defined when m(.) is a probabilistic mass because one gets 0/0 indetermination. This remark is important only from the mathematical point of view.

## V. A New Generalized Pignistic Transformation

Our new mapping, denoted **DSmP** is straight, different from Sudano's and Cuzzolin's mappings which are more refined but less interesting in our opinions than what we present here. The basic idea of **DSmP** consists in a new way of proportionalizations of the mass of each partial ignorance such as A1 ∪A2 or A1 ∪(A2 ∩A3) or (A1 ∩A2)∪(A3 ∩A4), etc. and the mass of the total ignorance A1 ∪ A2 ∪ **. . .** ∪ An, to the elements involved in the ignorances. This new transformation takes into account both the values of the masses and the cardinality of elements in the proportional redistribution process. We first present the general formula for this new transformation and the numerical examples and comparisons with respect to other transformations are given in next sections.

## A. The Dsmp Formula

Let's consider a discrete frame Θ with a given model
(free DSm model, hybrid DSm model or Shafer's model),
the **DSmP** mapping is defined by7 **DSmP**ǫ(∅) = 0 and
∀X ∈ GΘ **\ {∅}** by

$$DSmP_{\epsilon}(X)=\sum_{Y\in G^{\Theta}}\frac{\sum_{\substack{Z\subseteq X\cap Y\\ \mathcal{C}(Z)=1\\ Z\subseteq Y\\ \mathcal{C}(Z)=1}}m(Z)+\epsilon\cdot\mathcal{C}(Y)}{\sum_{\substack{Z\subseteq Y\\ \mathcal{C}(Z)=1}}m(Z)+\epsilon\cdot\mathcal{C}(Y)}m(Y)\tag{11}$$  3. 
7The formulation of (11) for the case of singletons θi of Θ is given in [8].