Explaining robust additive utility models by sequences of preference swaps

Abstract

As decision-aiding tools become more popular everyday—but at the same time more sophisticated—it is of utmost importance to develop their explanatory capabilities. Some decisions require careful explanations, which can be challenging to provide when the underlying mathematical model is complex. This is the case when recommendations are based on incomplete expression of preferences, as the decision-aiding tool has to infer despite this scarcity of information. This step is key in the process but hardly intelligible for the user. The robust additive utility model is a necessary preference relation which makes minimal assumptions, at the price of handling a collection of compatible utility functions, virtually impossible to exhibit to the user. This strength for the model is a challenge for the explanation. In this paper, we come up with an explanation engine based on sequences of preference swaps, that is, pairwise comparison of alternatives. The intuition is to confront the decision maker with “elementary” comparisons, thus building incremental explanations. Elementary here means that alternatives compared may only differ on two criteria. Technically, our explanation engine exploits some properties of the necessary preference relation that we unveil in the paper. Equipped with this, we explore the issues of the existence and length of the resulting sequences. We show in particular that in the general case, no bound can be given on the length of explanations, but that in binary domains, the sequences remain short.

Proofs

1.1 Proof of theorem 2

For the sketch of the proof, we construct, for every p, a preference between $x=(0,0,0)$ and $y=(2p ,-p,-p)$. Starting from alternative (0, 0, 0), we begin with a preference swap between attributes 1 and 2 (adding value 1 on the first attribute, and subtracting 1 on the second one). Then we perform a preference swap between attributes 1 and 3 (adding value 1 on the first attribute, and subtracting 1 on the third one). We proceed then again by a preference swap between attributes 1 and 2, and so on (the sequence is depicted in Fig. 5).

Fig. 5

Description of the sequence

Proof

(Theorem 2) The proof is based on an instantiation of $\mathcal {R}$ with the necessary preference relation. This latter is inferred from information $\mathcal {P}$, and is denoted by $\mathcal {N_P}$. Let $n= 3$, $p\in \mathbb {N} ^*$. Assume that $\mathbb {X}_1 \supseteq \{0,1,2,\ldots ,2p\}$, $\mathbb {X}_2 \supseteq \{-p,-p+1,\ldots ,-1,0\}$ and $\mathbb {X}_3 \supseteq \{-p,-p+1,\ldots ,-1,0\}$. Consider the following preference information $\mathcal {P}$:

$$\begin{aligned}&\forall j\in \{0,\ldots ,p-1\} \nonumber \\&\quad \qquad (((2j)_1,(-j)_2) , ((2j+1)_1,(-j-1)_2))_{cp} \subset \mathcal {P} \end{aligned}$$

(1)

$$\begin{aligned}&\forall j\in \{0,\ldots ,p-1\} \nonumber \\&\quad \qquad (((2j+1)_1,(-j)_3 ) , ((2j+2)_1,(-j-1)_3 ))_{cp} \subset \mathcal {P} \end{aligned}$$

(2)

where (1) [resp. (2)] correspond to a ceteris paribus pair on attributes $\{1,2\}$ (resp. $\{1,3\}$). Hence, $\mathbb {D}_1 = \{0,1,2,\ldots ,2p\}$, $\mathbb {D}_2 = \{-p,-p+1,\ldots ,-1,0\}$ and $\mathbb {D}_3 = \{-p,-p+1,\ldots ,-1,0\}$.

We set $x=(0,0,0)$ and $y=(2p ,-p,-p)$. With this $\mathcal {P}$, we clearly obtain the sequence

$$\begin{aligned}&(x \, , \, (1,-1,0 )) \in \mathcal {P} \qquad \text {(by (1))}\\&((1,-1,0 ) \, , \, (2,-1,-1)) \in \mathcal {P} \ , \ \ldots \qquad \text {(by (2))} \\&((2p-2,-(p-1),-(p-1)) \, , \, (2p-1,-p,-(p-1))) \in \mathcal {P} \qquad \text {(by (1))} \\&((2p-1,-p,-(p-1)) \, , \, (2p ,-p,-p)) \in \mathcal {P} \qquad \text {(by (2))} \end{aligned}$$

so that $(x,y) \in \mathcal {R}$. This sequence is of length $2\,p$.

There remains to prove that this is the shortest explanation.

To this end, we first need to determine the form of $\Delta _2$. By Theorem 4, the necessary preference relation cannot hold outside the interval between the minimal and maximal elements of $\mathbb {D}$. Moreover, according to Theorem 5, the necessary preference relation between two alternatives $z,z'$ holds iff a linear problem involving the covector of $(z,z')$ is feasible. From these results, checking whether $(z,z')\in \mathcal {N_P}$ is equivalent to checking boundness on z and $z'$, and also checking whether $(t,t')\in \mathcal {N_P}$ where $t,t' \in \mathbb {D}$ are appropriately chosen from z and $z'$. Therefore, we need only to consider the elements in $\Delta _2$ that belong to $\mathbb {D}_1\times \mathbb {D}_2 \times \mathbb {D}_3$ (The other ones can be deduced by Pareto dominance). The preference information (1) and (2) is very specific. In particular, any value $k \in \mathbb {D}_1$ appears only in two examples—one in which k appears in the left-hand side [in (1)] and the other one where k appears in the right-hand side [in (2)]. Moreover, we notice that, in (1) and (2), the value on the first attribute is always increasing from the left-hand side to the right-hand side, and the value of the second and the third attributes is decreasing from the left-hand side to the right-hand side. Hence, the elements of $\Delta _2$ cannot be obtained by a combination of two or more preference information. They are obtained only from one preference information [(1), (2)] and Pareto dominance $\mathcal {D}$. More precisely, $\Delta _2$ is composed of the following pairs

$$\begin{aligned}&\Big ( (i,j,k) \, , \, (i',j',k') \Big ) \end{aligned}$$

where either there exists l such that $i=2l$, $j=2l+1$, $j \ge -l > -l-1 \ge j'$ and $k=k'$, or there exists l such that $i=2l+1$, $j=2l+2$, $j=j'$ and $k \ge -l > -l-1 \ge k'$. From this, one can readily see that the explanation of the preference of x over y described earlier is the shortest one. $\square$

1.2 Proof of Theorem 4 and Theorem 5

1.2.1 Proof of (1) $\iff$ (2)

The belonging of a pair of alternatives to the necessary preference relation can be expressed as a mathematical program. We have to prove that when the pair is not unbounded, its constrains and objective function are linear and can be expressed using the proposed, fixed-length covectors.

Pairs of core alternatives, and in particular, preference statements, are never unbounded. We begin by introducing $\forall (i,k)\in \mathbb {I}, \ \Delta v_{i,k} := v_i(d_{(i,k+1)})-v_i(d_{(i,k)})$ and proving covectors, when applied to such a vector $\Delta _v$ of differences in value, correctly compute the difference of value between core alternatives.

We break down the Definition 12 by criterion:

$\forall i\in N, \quad \forall x_i, y_i\in \mathbb {X}_i, \quad \text {let}\ (x_i, y_i)\in \mathbb {R} ^{|\mathbb {D}_i|-1} : \forall k\in \mathbb {N} : 1 \le k \le |\mathbb {D}_i|-1,$

$$\begin{aligned} (x_i,y_i)^{\star }_{k}:= \left\{ \begin{array}{ll} +1, &{} \quad \text{ if }\; [d_{i,k} , d_{i,k+1}] \subset [y_i , x_i] \\ -1, &{} \quad \text{ if } \; [d_{i,k} , d_{i,k+1}] \cap ]x_i , y_i[ \ne \emptyset \\ 0, &{} \quad \text{ else } \\ \end{array} \right. \end{aligned}$$

So that $\forall x,y \in \mathbb {X}, \forall (i,k)\in \mathbb {I}, \ (x,y)^\star _{(i,k)}=(x_i,y_i)^\star _k$.

Lemma 1

(expression of differences in value as a product)

$$\begin{aligned} \forall i\in N,\forall x_i,y_i \in \mathbb {D}_i, \forall V\in \mathbb {V}, \quad \ v_i(x_i)-v_i(y_i) = \displaystyle \sum _{k=1}^{|\mathbb {D}_i| -1} (x_i,y_i)^\star _{k} \Delta v_{(i,k)} \end{aligned}$$

Proof

First, we note that for any valid indexes $k_1<k_2$, $\sum \nolimits _{k=k_1}^{k_2} \Delta v_{(i,k)}= v_i(d_{i,k_2})-v_i(d_{i,k_1})$

Second, we detail $\sum \nolimits _{k=1}^{|\mathbb {D}_i| -1} (x_i,y_i)^\star _{k} \Delta v_{(i,k)}$, according to the sign of $x_i-y_i$ :

• If $x_i > y_i$, the interval $]x_i, y_i[$ is empty, so the case leading to a coefficient $(x,y)^\star _{(i,k)}=-1$ does not occur. Non-zero coefficients correspond to intervals $[d_{i,k},k_{i,k+1}[$ partitioning $[y_i, x_i[$, so that $\sum \nolimits _{k=1}^{|\mathbb {D}_i| -1} (x_i,y_i)^\star _{k} \Delta v_{(i,k)}=(+1) (v_i(x_i)-v_i(y_i))$

• If $x_i < y_i$, the interval $[y_i, x_i]$ is empty, so the case leading to a coefficient $(x,y)^\star _{(i,k)}=+1$ does not occur. Non-zero coefficients correspond to intervals $[d_{i,k},k_{i,k+1}[$ partitioning $[x_i, y_i[$, so that $\sum \nolimits _{k=1}^{|\mathbb {D}_i| -1} (x_i,y_i)^\star _{k} \Delta v_{(i,k)}=(-1) (v_i(y_i)-v_i(x_i))=v_i(x_i)-v_i(y_i)$

• If $x_i=y_i$, the interval $[x_i , y_i]$ is trivial and the interval $]x_i, y_i[$ is empty, so every coefficient $(x,y)^\star _{(i,k)}$ is equal to zero. Consequently, $\sum \nolimits _{k=1}^{|\mathbb {D}_i| -1} (x,y)^\star _{(i,k)} \Delta v_{(i,k)}=0=v_i(x_i)-v_i(y_i)$.

Thus, $\forall i\in N, \ v_i(x)-v_i(y) = \sum \nolimits _{k=1}^{|\mathbb {D}_i| -1} (x,y)^\star _{(i,k)} \Delta v_{(i,k)}$. $\square$

For any alternatives $x,y \in \mathbb {D}$, summing up these equalities over every criteria yields $V(x) - V(y) = (x,y)^\star \times \Delta _v$

Introducing $\forall x,y \in \mathbb {X}, \ \Delta V_{\inf }(x,y) := \inf _{V\in \mathbb {V}_\mathcal {P}} V(x)-V(y)\in \mathbb {R} \cup \{-\infty \}$, Definition 9 states that

$$\begin{aligned} \forall x,y \in \mathbb {X}, \ (x,y)\in \mathcal {N_P} \iff \Delta V_{\inf }(x,y) \ge 0 \end{aligned}$$

In the case of pairs of core alternatives, the objective function as well as the constraints of the minimization problem $\Delta V_{\inf }(x,y)$ can be expressed using covectors and matrix multiplication, as permitted by Lemma 1, so that $\Delta V_{\inf }(x,y)$ is a linear program.

Lemma 2

(query between core alternatives)

$$\begin{aligned} \forall x,y \in \mathbb {D},\ \Delta V_{\inf }(x,y)= \inf \ (x,y)^\star \times \Delta v \ \ \mathrm {s.t.}\ {\Delta v \in \Omega _{\mathcal {P}}\cap \Omega _{\mathcal {D}}} \end{aligned}$$

with $\Omega _{\mathcal {P}}:=\{\Delta v \in \mathbb {R} ^\mathbb {I}\ :\ \forall \pi \in \mathcal {P}, \ \pi ^{\star } \times \Delta v \ge 0\}$ and $\Omega _{\mathcal {D}}:=\{\Delta v \in \mathbb {R} ^\mathbb {I}\ :\ \forall (i,k)\in \mathbb {I},\ \delta ^\star _{(i,k)} \times \Delta v \ge 0 \}$.

Generally, with alternatives (x, y) not necessarily belonging to the core $\mathbb {D}$, it has been shown Greco et al. (2008) that minimizing $V(x)-V(y)$ over $V\in \mathbb {V}_\mathcal {P}$ is still a linear program, with additional decision variables accounting for the distinct values $\{x_i,y_i\}\notin \mathbb {D}_i$. The $v_i(x_i), v_i(y_i)$ are only constrained by the monotonicity of the marginal value functions, so the problem is separate:

$$\begin{aligned} \Delta V_{\inf }=\inf _{\Delta v \in \Omega _{\mathcal {P}}\cap \Omega _{\mathcal {D}}}\displaystyle \sum _{i\in N} \inf _{ \begin{array}{l} {\tiny v_i(x_i)\in UX_i \cap LX_i} \\ {\tiny v_i(y_i) \in UY_i \cap LY_i} \end{array}} v_i(x_i)-v_i(y_i) \end{aligned}$$

$$\begin{aligned} \mathrm {with,}\ \forall i\in N,\ \left\{ \begin{array}{l}UX_i:=\{v_i(x_i)\in \mathbb {R} : \forall z_i\in \mathbb {D}_i\cup \{y_i\}, \quad \ z_i \succsim _i x_i \Rightarrow v_i(z_i)\ge v_i(x_i)\} \\ LX_i:=\{v_i(x_i)\in \mathbb {R} : \forall z_i\in \mathbb {D}_i\cup \{y_i\}, \quad \ z_i \precsim _i x_i \Rightarrow v_i(z_i)\le v_i(x_i)\} \\ UY_i:=\{v_i(y_i)\in \mathbb {R} : \forall z_i\in \mathbb {D}_i\cup \{x_i\}, \quad \ z_i \succsim _i y_i \Rightarrow v_i(z_i)\ge v_i(y_i)\} \\ LY_i:=\{v_i(y_i)\in \mathbb {R} : \forall z_i\in \mathbb {D}_i\cup \{x_i\}, \quad \ z_i \precsim _i y_i \Rightarrow v_i(z_i)\le v_i(l_i)\} \end{array}\right. \end{aligned}$$

Thus, it is possible to circumvent this augmentation of the decision space by:

• Considering a given criterion $i\in N$ and a given vector $\Delta v \in \Omega _{\mathcal {P}}\cap \Omega _{\mathcal {D}}$;

• Directly assigning the additional decision variables to their optimal values in the inner linear program

  $$\begin{aligned} \inf _{v_i(x_i), v_i(y_i)} v_i(x_i)-v_i(y_i)\ \text {s.t.}\ \left\{ \begin{array}{l} v_i(x_i)\in UX_i \cap LX_i \\ v_i(y_i) \in UY_i \cap LY_i\end{array}\right. ; \end{aligned}$$

• Checking this optimal case is correctly represented, either by an unbounded pair or in covector form.

We begin by focusing on the case where the values of $\mathbb {D}_i\cup \{x_i,y_i\}$ are all different. We sort these values in strictly ascending order, and we detail three cases according to the position of $x_i$ and $y_i$ amongst these $|\mathbb {D}_i|+2$ values:

• The interval $[x_i,\ y_i]$ overflows the set $\mathbb {D}_i$, so that the pair $(x,y)\in \mathcal {U_P}$ is unbounded. This case actually encompasses three subcases

• $x_i$ has no predecessor, when $x_i$ is the least element of $\mathbb {D}_i\cup \{x_i,y_i\}$. There is no constraints in $LX_i= \mathbb {R}$;

• $y_i$ has no successor, when $y_i$ is the highest element of $\mathbb {D}_i\cup \{x_i,y_i\}$. There are no constraints in $UY_i= \mathbb {R}$;

• Both preceding cases are simultaneously satisfied.

In any case,

$$\begin{aligned} \inf v_i(x_i)-v_i(y_i)\ \text {s.t.}\ \left\{ \begin{array}{l} v_i(x_i)\in UX_i\cap LX_i\\ v_i(y_i)\in UY_i\cap LY_i \end{array}\right. =-\infty , \end{aligned}$$

thus $V_{\inf }(x,y)=-\infty$ and $(x,y)\notin \mathcal {N_P}$, thus proving Theorem 4;

• $y_i$ is the predecessor of $x_i$, so $x_i$ is the successor of $y_i$. In this case, the constraints $UX_i, LX_i, UY_i, LY_i$ can all be replaced by the single equality $v_i(x_i)=v_i(y_i)$, which defines a solution both feasible and where the objective function is minimized with respect to the decision variables $v_i(x_i), v_i(y_i)$. Meanwhile, we consider the coefficients $(x,y)^\star _{(i,k)},\ 1\le k < |\mathbb {D}_i|$: the interval $[y_i, x_i]$ does not contain a single core value $d_{i,k}\in \mathbb {D}_i$, hence $(x,y)^\star _{(i,k)}\ne +1$; the interval $]x_i, y_i[$ is empty, hence $(x,y)^\star _{(i,k)}\ne -1$; finally $(x,y)^\star _{(i,k)}=0$. This proves the identity:

  $$\begin{aligned} \inf v_i(x_i)-v_i(y_i)\ \text {s.t.}\ \left\{ \begin{array}{l} v_i(x_i)\in UX_i\cap LX_i\\ v_i(y_i)\in UY_i\cap LY_i \end{array}\right. =\displaystyle \sum _{k=1}^{|\mathbb {D}_i|-1}(x,y)^\star _{(i,k)} \Delta u_{(i,k)}, \end{aligned}$$

  as both sides are equal to zero.

• $x_i$ has a predecessor which is not $y_i$, and $y_i$ has a successor which is not $x_i$. First, we rewrite $\inf v_i(x_i)-v_i(y_i)\ \text {s.t.}\ \left\{ \begin{array}{l}v_i(x_i)\in UX_i\cap LX_i\\ v_i(y_i)\in UY_i\cap LY_i \end{array}\right.$ as a difference in marginal value between surrogate alternatives in the core $\mathbb {D}_i$. The predecessor $\underline{x_i}$ of $x_i$ is given by $\underline{x_i} := \max \{d\in \mathbb {D}_i, d \precsim _i x_i\}$, so that the constraints $UX_i, LX_i$ can both be replaced by the single equality $v_i(x_i)=v_i(\underline{x_i})$, which defines a solution both feasible and where $v_i(x_i)$ is minimal with respect to the decision variable $v_i(x_i)$. The successor $\overline{y_i}$ of $y_i$ is given by $\overline{y_i} := \min \{d\in \mathbb {D}_i, d \succsim _i y_i\}$, so that the constraints $UY_i, LY_i$ can both be replaced by the single equality $v_i(y_i)=v_i(\overline{y_i})$, which defines a solution both feasible and where $v_i(y_i)$ is maximal, so the objective function is minimal, with respect to the decision variable $v_i(y_i)$.

Thus,

$$\begin{aligned} \inf v_i(x_i)-v_i(y_i)\ \text {s.t.}\ \left\{ \begin{array}{l} v_i(x_i)\in UX_i\cap LX_i\\ v_i(y_i)\in UY_i\cap LY_i \end{array}\right. = v_i(\underline{x_i})-v_i(\overline{y_i}) \end{aligned}$$

Second, as both surrogate alternatives $\underline{x_i},\overline{y_i}$ belong to $\mathbb {D}_i$, Lemma 1 ensures that

$$\begin{aligned} v_i(\underline{x_i})-v_i(\overline{y_i})=\displaystyle \sum _{k=1}^{|\mathbb {D}_i|-1}(\underline{x_i},\overline{y_i})^\star _{k} \Delta u_{(i,k)} \end{aligned}$$

Third, we check that the covector coefficients for criterion i of the original pair match those of the surrogate pair, that is:

$$\begin{aligned} \forall k\in \mathbb {N} : 1\le k < |\mathbb {D}_i|,\ (x_i,y_i)^\star _{k}=(\underline{x_i},\overline{y_i})^\star _{k} \end{aligned}$$

The proof is straightforward:

• If $x_i \succ _i y_i$, then there is at least one attribute value $d\in \mathbb {D}_i$ between $x_i$ and $y_i$, so that the predecessor of $x_i$ and the successor of $y_i$ are in the same order, thus $\underline{x_i} \succsim _i \overline{y_i}$. Hence, the coefficient indexed by (i, k) of their respective covectors are in $\{0,+1\}$, with value $+1$, respectively, when $y_i \precsim _i d_{i,k} \prec _i d_{i,k+1}\precsim _i x_i$ and when $\overline{y_i} \precsim _i d_{i,k} \prec _i d_{i,k+1} \precsim _i \underline{x_i}$. The definition of the surrogate pair ensures these conditions are equivalent.

• If $x_i \prec _i y_i$, then obviously $\underline{x_i} \precsim _i \overline{y_i}$. Hence, the coefficients of their respective covectors indexed by (i, k) are in $\{0,-1\}$, with value 0, respectively, when $y_i \precsim _i d_{i,k}$ or $d_{i,k+1}\precsim _i x_i$, and when $\overline{y_i} \precsim _i d_{i,k}$ or $d_{i,k+1} \precsim _i \underline{x_i}$. The definition of the surrogate pair ensures these conditions are equivalent. Thus,

  $$\begin{aligned} \inf v_i(x_i)-v_i(y_i)\ \text {s.t.}\ \left\{ \begin{array}{l}v_i(x_i)\in UX_i\cap LX_i\\ v_i(y_i)\in UY_i\cap LY_i \end{array}\right. =\displaystyle \sum _{k=1}^{|\mathbb {D}_i|-1}(x_i,y_i)^\star _{k} \Delta u_{(i,k)} \end{aligned}$$

  The cases where $|\mathbb {D}_i \cup \{x_i, y_i\}|=|\mathbb {D}_i|+1$ are correctly handled in the discussion above: if overflow (when either $x_i \prec _i \min \mathbb {D}_i$ or $y_i \succ _i \max \mathbb {D}_i$) does not occur, the case $x_i=y_i$ extends the case where the optimal value of $v_i(x_i)-v_i(y_i)$ is zero; the case where $y_i \in \mathbb {D}_i$ leads to the introduction of $\overline{y_i} := y_i$, and the case where $x_i \in \mathbb {D}_i$ leads to $\underline{x_i}:=x_i$.

Finally, for any pair $(x,y)\in \mathbb {X}^2$, we have proven that, in every case, either the pair is unbounded and not in the relation $\mathcal {N_P}$, or it can be represented by a covector such that $\Delta V_{\inf }(x,y)= \inf _{\Delta v \in \mathbb {R} ^\mathbb {I}} \ (x,y)^\star \times \Delta v \ \ \mathrm {s.t.}\ \left\{ \begin{array}{l}\forall \pi \in \mathcal {P}, \ \pi ^{\star } \times \Delta v \ge 0 \\ \forall (i,k)\in \mathbb {I},\ \delta ^\star _{(i,k)} \times \Delta v \ge 0 \end{array}\right.$

1.2.2 Proof of (2) $\iff$ (3)

By Farkas’ lemma, the problem (2) has no solution if, and only if, the objective linear form $(x,y)^\star$ is a linear combination with non-negative coefficients of the constraint linear forms $\{\pi ^\star , \pi \in \mathcal {P}\}$ and $\{\delta ^\star _{i,k}, (i,k)\in \mathbb {I}\}$.

1.2.3 Proof of (3) $\iff$ (4)

Obviously, (4) $\Rightarrow$ (3). Conversely, as the covectors involved in (3) have integral coordinates, the non-negative coefficients $\{\lambda _\pi , \pi \in \mathcal {P}\}$ and $\{\mu _{(i,k)}, (i,k)\in \mathbb {I}\}$, if they exist, can be chosen in the field of rational numbers. Multiplying the relation by the common denominator $n\in \mathbb {N} ^\star$ of these coefficients leads to (4).

1.3 Proof of Theorem 6

We prove Theorem 6 in four steps: $(1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (4) \Rightarrow (1)$.

• $(1) \Rightarrow (2)$: Assume a statement $\sigma :=(x,y)\in \mathcal {E}_2(\mathcal {N_P})$. By Theorem and Definition 5, there is an integer n and a tuple $(e_0, e_1, \ldots , e_n)\in \mathbb {X}^n$ such that $e_0=x, e_n=y$ and $(e_j,e_{j+1})\in \mathcal {D}\cup \Delta _2$ for any integer $j<n$. This transitive chain of dominance relations and swaps of order 2 can be transformed into the covector relation sought, by induction on the length of the explanation, as described by the following lemmas:

Lemma 3

(covector representation of dominance relations)

$$\begin{aligned} \forall \rho \in \mathcal {D}, \exists q\in \{0,+1\}^I : \rho ^\star = \sum _{(i,k)\in \mathbb {I}}q_{(i,k)}\delta ^\star _{(i,k)} \end{aligned}$$

Proof

A dominance relation has no negative argument, so its covector coefficient, given by Definition 12, is in $\{0,+1\}$. $\square$

Lemma 4

(covector representation of transitivity relations)

$$\begin{aligned} \forall x,y,z \in \mathbb {X}, \exists q\in \mathbb {N} ^\mathbb {I}: (x,z)^\star = (x,y)^\star + (y,z)^\star + \sum _{(i,k)\in \mathbb {I}}q_{(i,k)}\delta ^\star _{(i,k)} \end{aligned}$$

Proof

For core alternatives $x,y,z \in \mathbb {D}$, for any separate value function $V\in \mathbb {V}$,

$$\begin{aligned}(x,z)^\star \times \Delta v&= V(x)-V(z) \\&= (V(x)-V(y))+(V(y)-V(z))\\&= (x,y)^\star \times \Delta v\ +\ (y,z)^\star \times \Delta v \\ {}&= ((x,y)^\star + (y,z)^\star ) \times \Delta v\end{aligned}$$

As the relation above stands for any vector $\Delta v \in [0,+\infty [$, it yields $(x,z)^\star = (x,y)^\star + (y,z)^\star =(x,y)^\star + (y,z)^\star + \displaystyle \sum _{(i,k)\in \mathbb {I}}q_{(i,k)}\delta ^\star _{(i,k)}$ with $q=0$.

For alternatives not necessarily in the core, and for any criterion $i\in N$, the trivial cases where $y_i\in \{x_i,z_i\}$, the case where $x_i = z_i$, or the case where $x_i, y_i, z_i$ are all distinct, divided into 6 subcases considering the order of attributes $x_i, y_i, z_i$, all lead to $(x,z)^\star \ge (x,y)^\star + (y,z)^\star$ because of the rounding down of broken intervals occurring once in the LHS and twice in the RHS. As both sides are covectors with integer coefficients, the difference $(x,z)^\star -( (x,y)^\star + (y,z)^\star )$ is a covector with non-negative integer coefficients $q_{(i,k)}$. $\square$

• $(2) \Rightarrow (3)$: Suppose there exists integer coefficients $a, \ell _1, \ldots , \ell _q$, $m_1, \ldots , m_n$ and preference swaps of order 2: $\gamma _1, \ldots , \gamma _q$ such that

  $$\begin{aligned} a{\sigma }^\star = \displaystyle \sum _k \ell _k \gamma ^\star _k + \displaystyle \sum _k m_k \delta ^\star _{(k,1)} \end{aligned}$$

  (3)

  Multiplying both sides of the covector Equation (3) by the vector $(1, \ldots , 1)$, we obtain the relation:

  $$\begin{aligned} M := a(|{\sigma }^+| - |{\sigma }^-|)=\displaystyle \sum m_k \ge 0 \end{aligned}$$

  To homogenize the right-hand side, we represent the dominance relation thanks to a dummy criterion: $N'=N\cup \{0\}$ so that $\tilde{\Delta _1} := \{(i,0), i\in N\}\subset N'^2$. Thus, relation $\mathcal {D}\cup \Delta _2$ is a graph with nodes in $N'$. Re-indexing coefficients $\ell _k$ by the positive and negative arguments of swap $\gamma _k$ (summing up duplicates if needed), and introducing $\ell _{k,0} := m_k$ :

  $$\begin{aligned} a \ {\sigma }^\star = \displaystyle \sum _{\gamma \in \tilde{\Delta _1} \cup \tilde{\Delta _2}}\ell _{\gamma ^+,\gamma ^-} \gamma ^\star \end{aligned}$$

  (4)

To complete the flow $\ell$, we introduce:

• A source s supplying flow $\ell _{s,i}=a$ to the positive arguments $i\in {\sigma }^+$;

• A sink t collecting flow $\ell _{j,t}=a$ from the negative arguments $j\in {\sigma }^-$, and $\ell _{0,t} = M$ from node 0.

Covector Equation (4) ensures $\ell$ defines a feasible flow on the graph $(N'\cup \{s,t\},\tilde{\Delta _1}\cup \tilde{\Delta _2}\cup \{s\}\times {\sigma }^+ \cup {\sigma }^- \times \{t\}\cup \{(0,t)\})$, without capacity constraints, as projection on the $i^{th}$ coordinate ensures flow conservation for node $i\in N$. Flow $\ell$ can be decomposed as a superposition of:

• Cycles, involving necessary equivalence between the nodes, and not contributing to the value of the flow;

• Paths from the source s to the sink t passing through node 0, denoting a dominance relation. Their total contribution to the value of the flow is M;

• Paths from the source s to the sink t not passing through node 0, with an overall contribution of $a \times |{\sigma }^-|$ to the value of the flow. Each of these paths links a positive argument $i_1\in {\sigma }^+$ to a negative argument $i_r \in {\sigma }^-$ through necessary preference swaps of order 2. Transitivity of the necessary preference relation entails that $i_1$ is necessarily preferred to $i_r$: the edge $(i_1,i_r)$ belongs to $\Delta _2 \cap ({\sigma }^+ \times {\sigma }^-)$.

We reduce the flow $\ell$ by ignoring the cycles and paths passing through node 0. In addition, the flow a carried by the path from source to sink $s \rightarrow i_1 \rightarrow i_2 \rightarrow \cdots \rightarrow i_r \rightarrow t$ is redirected to edge $(i_1, i_r)$. As a result, we obtain a flow of value $a|{\sigma }^-|$ on the graph of the relation $\tilde{\Delta _2}$ restricted to ${\sigma }^+ \times {\sigma }^-$. This entails the existence of a matching of cardinality $|{\sigma }^-|$ in this graph, obtained by setting an upper capacity constraint of value 1 on each edge leaving the source s and entering the sink t (as a cut of capacity C on the network with capacity constraints $c_{i,j}\in \{1,\infty \}$ is a cut of capacity $a\times C$ on the same network with capacity constraints $a \times c_{i,j}$).

• $(3) \Rightarrow (4)$ is simply a rewording.

• $(4) \Rightarrow (1)$: Let $\phi : {\sigma }^- \rightarrow {\sigma }^+$, injective, such that $\forall k \in {\sigma }^-, (\phi (k),k)\in \tilde{\Delta _2}$. Given any ordering O of the negative argument set $\sigma ^-$, we can build a sequence of alternatives of decreasing preference $e_0 := x , e_1, \dots , e_{|\sigma ^-|}\in V$ such that the $k^{th}$ statement $(e_{k-1}, e_{k})$ matches the criteria swap $(\phi (O_k),O_k)\in \tilde{\Delta _2}$:

  $$\begin{aligned} N_{(e_{k-1}, e_{k})}^{\ne } := \{\phi (O_k),O_k\} \ ;\ N_{(e_{k},y)}^= := N_{(e_{k-1}, y)}^= \cup \{\phi (O_k),O_k)\} \end{aligned}$$

  Thus, the sequence of sets $(e_k,y)^-$ decreases from $\sigma ^-$ to $\emptyset$, one element at a time, and the sequence of sets $(e_k \succsim y)^+$ also decreases from $\sigma ^+$ to $\sigma ^+ {\setminus } \phi [\sigma ^-]$, one element at a time. If the set $\sigma ^+ {\setminus } \phi [\sigma ^-]$ is empty, $e_{|\sigma ^-|} = y$, and the sequence $x = e_0, \dots , e_{|\sigma ^-|}=y$ is an explanation of $(x,y)\in \mathcal {N_P}$ by preference swaps of order 2, of length $|\sigma ^-|$. Else, $e_{|\sigma ^-|} \ne y$ but $(e_{|\sigma ^-|},y)$ is a dominance statement, as its negative argument set is empty. Thus, the sequence $x = e_0, e_1,\dots ,e_{|\sigma ^-|},y$ is an explanation of $(x,y)\in \mathcal {N_P}$ by preference swaps of order 2 and a dominance relation, of length $|\sigma ^-|+1$.