Axiomatizations for partial preference orders

The domain of discernment is represented by a finite set of events $\mathcal{E}=\{E_1,\ldots,E_n\}$ (among them, $\phi$ and $\Omega$ denote the impossible and the sure event, respectively). The events are seen as the relevant propositions on which the subject of the analysis expresses his/her opinion. As mentioned, $\mathcal{E}$ does not necessarily represent a full model, i.e. it does not comprehend all elementary situations and all of their combinations. For this reason, a crucial component of partial assessments is the knowledge of the logical relationships (incompatibilities, implications, combinations, equivalences, etc.) holding among events. Such relationships are usually expressed by stating a collection $\mathcal{C}$ of constraints on the events (as well as on conjunctions and disjunctions of events). By taking into account the constraints $\mathcal{C}$ , the family $\mathcal{E}$ spans a minimal Boolean algebra $\mathcal{A}_\mathcal{E}$ containing $\mathcal{E}$ itself.

Definition 1 Let $\mathcal{A}_\mathcal{E}$ be an algebra of events. A binary relation $\preccurlyeq^{*}$ over $\mathcal{A}$ is a (total) preference order if it satisfies the following conditions:

(A1): $\preccurlyeq^{*}$ is a pre-order, i.e. it is reflexive, transitive, and total;
(A2): $\phi\preceq^{*}\Omega$ and $\neg(\Omega\preceq^{*}\phi)$ (non-triviality);
(A3): for all events , $A\subseteq B\rightarrow(A\preccurlyeq^{*}B)$ (monotonicity).

If $\preccurlyeq^{*}$ is a total preference order, $\sim^{*}$ is its symmetric factor, i.e. $\forall E_1,E_2\,(E_1\sim^{*}E_2\leftrightarrow E_1\preccurlyeq^{*}E_2 \wedge E_2\preccurlyeq^{*}E_1).$ Moreover, $\prec^{*}$ is the asymmetric factor of $\preccurlyeq^{*}$ , i.e. $\forall E_1,E_2\,(E_1\prec^{*}E_2\,\leftrightarrow\,E_1\preccurlyeq^{*}E_2\wedge \neg(E_2\sim^{*}E_1)).$

Definition 2 Let $\preccurlyeq$ and $\prec$ be binary relations over a set of events $\mathcal{E}$ , such that $E_1\prec E_2\,\rightarrow\,E_1\preccurlyeq E_2$ . The pair $\langle\preccurlyeq,\prec\rangle$ is a weak preference structure for $\mathcal{E}$ (w.p.s., for short) if it exists a total preference order $\preccurlyeq^{*}$ over $\mathcal{A}_\mathcal{E}$ such that: $\forall E_1,E_2\in\mathcal{E}\: \big((E_1\preccurlyeq E_2\,\rightarrow\,E_1\preccurlyeq^{*}E_2) \wedge (E_1\prec E_2\,\rightarrow\,E_1\prec^{*}E_2)\big)$ .

For any w.p.s. $\langle\preccurlyeq,\prec\rangle$ for $\mathcal{E}$ , the following properties hold:

Differentiation among uncertainty notions is done by considering the specific way of combining distinct pieces of information (e.g., as mentioned, for Probabilities additivity is adopted). Within the numerical context, this yields a taxonomy of numerical measures. By following [14,46,47], various qualitative preference orders have been classified in [4,5] according to their agreement with the numerical models. The correspondence between a qualitative uncertainty notion and a numerical measure is given in terms of a representability result.

Definition 3 Let $\mathcal{E}$ be a set of events. A total preference order $\preccurlyeq^{*}$ over $\mathcal{A}_\mathcal{E}$ is said to be representable by a numerical measure $f:\mathcal{A}_\mathcal{E}\rightarrow[0,1]$ if for all $E_1,E_2\in\mathcal{A}_\mathcal{E}$ it holds that $E_1\preccurlyeq^{*} E_2 \leftrightarrow f(E_1)\leqslant f(E_2)$ .

A w.p.s. $\langle\preccurlyeq,\prec\rangle$ for $\mathcal{E}$ is said to be representable by a partial uncertainty measure $g:\mathcal{E}\rightarrow[0,1]$ if it admits an enlargement $\preccurlyeq^{*}$ over $\mathcal{A}_\mathcal{E}$ which is representable by an uncertainty measure $g^{*}:\mathcal{A}_\mathcal{E}\rightarrow[0,1]$ extension of to $\mathcal{A}_\mathcal{E}$ .

We refer to any specific class of preference orders by the name of the corresponding numerical notion. The representability results we are going to present legitimate this choice.

Proposition 1 Comparative belief. Let $\langle\preccurlyeq,\prec\rangle$ be a w.p.s. for $\mathcal{E}$ . The following properties hold: $\langle\preccurlyeq,\prec\rangle$ can be extended to a total preference order $\preccurlyeq^{*}$ over $\mathcal{A}_\mathcal{E}$ representable by a belief function iff for all $X,Y,Z,W\in\mathcal{E}$ s.t. $X\subset Y$ , $Z\subset W \subset Y$ , $W\setminus Z \subseteq Y\setminus X$ it holds:

(B') $\quad X\sim Y\rightarrow \neg(Z\prec W)$ .

Comparative lower-probability. $\langle\preccurlyeq,\prec\rangle$ can be extended to a total preference order $\preccurlyeq^{*}$ over $\mathcal{A}_\mathcal{E}$ representable by a lower-probability function iff for all $A,B,C\in\mathcal{E}$ s.t. $A\subset B, C \subseteq B\setminus A$ it holds that

(L') $\quad A\sim B\rightarrow \neg(\phi\prec C)$ .

Comparative plausibility. $\langle\preccurlyeq,\prec\rangle$ can be extended to a total preference order $\preccurlyeq^{*}$ over $\mathcal{A}_\mathcal{E}$ representable by a plausibility function iff for all $A,B,C,D\in\mathcal{E}$ s.t. $A\subset B$ , $C\subset D \subset B$ , $D\setminus C = B\setminus A$ it holds that

(PL') $\quad A\prec B\rightarrow \neg(C\sim D)$ .

Comparative upper-probability. $\langle\preccurlyeq,\prec\rangle$ can be extended to a total preference order $\preccurlyeq^{*}$ over $\mathcal{A}_\mathcal{E}$ representable by an upper-probability function iff for all $A,B\in\mathcal{E}$ s.t. $A\subset B$ , it holds

(U') $\quad A\prec B\rightarrow \neg(\phi\sim B\setminus A)$ .

Comparative lower/upper-probability. $\langle\preccurlyeq,\prec\rangle$ can be extended to a total preference order $\preccurlyeq^{*}$ over $\mathcal{A}_\mathcal{E}$ representable by both a lower-probability function and an upper-probability function iff it satisfies both (L') and (U').

Proposition 2 (Comparative probability) A w.p.s. $\langle\preccurlyeq,\prec\rangle$ for $\mathcal{E}$ can be extended to a total order $\preccurlyeq^{*}$ over $\mathcal{A}_\mathcal{E}$ representable by a probability function iff
(CP) for any $A_1,\ldots,A_n,B_1,\ldots,B_n\in\mathcal{E}$ , with $B_i\preccurlyeq A_i, \forall i=1,\ldots,n$ , such that for some $r_1,\ldots,r_n>0$ $\sup\big(\sum_{i=1}^n r_i(a_i-b_i)\big)\leq 0$ , implies that $A_i\sim B_i$ , for all $i=1,\ldots,n$ ( are the indicator functions of , resp.).

Axiom (CP) involves quantitative notions (e.g., indicator functions and summations), and its verification requires numerical elaborations. Nevertheless, it is possible to (qualitatively) state a necessary, but not sufficient, condition for representability through a probability function.