
\section{Results}

We  now show some properties which hold for the premise aggregation rule $F_{IC}$ we had defined in the previous section. We start by noticing that, in the case of the aggregation of binary evaluations, there is an obvious correspondence between proposition-wise majority voting and distance minimization. This has been already observed in several contexts (see, e.g., \cite{Brams2004}), and can be generalized to the following folk theorem.

\begin{proposition}
\label{th:folk}
Let  $\underline{J} = (J_{1}, \ldots , J_{n})$ be a profile over the agenda $\Phi$. Let $J^{maj} \subset \Phi$ be a complete and consistent set.   Let it hold that for every premise $a \in J^{maj}$, $a \in J_{i,p}$ for at least $\lfloor\frac{n}{2}\rfloor+1$ premise sets in the profile $J_{p}$. Also,  for the conclusion  $x \in J^{maj}$, let it holds that $x \in J_{i,c}$ for at least $\lfloor\frac{n}{2}\rfloor+1$ conclusion sets in the profile $J_{c}$. The sum of Hamming distances from  $J^{maj}$ to the judgment sets in $\underline{J}$ is minimal. 
\end{proposition}

This means that, in the absence of a Paretian dilemma (i.e. when $J^{maj}$ satisfies the decision rule $\mathcal{R}$), proposition-wise majority voting, distance-based merging and our procedure coincide.  


\subsection{Unanimity preservation}

One of the desirable properties for a judgment aggregation procedure is the heeding of unanimity. If all the agents unanimously support an agenda item, then it is natural to expect the unanimously supported  item  will be adopted as the collective judgment. However, PBP does not necessarily preserve unanimity on the conclusion (as it was the case with the Paretian dilemma shown in Table~\ref{tab:paretian}).

 

PBP aggregates each premise independently from the other premises, but the aggregation on the conclusion depends on the collective judgments on the  premises. Therefore the unanimity on the premises will be preserved, but the unanimity on the conclusion may be violated. 

When aggregating according to the CBP, unanimity on the conclusion will always be maintained, but unanimity on the premises may be violated. However, our procedure offers the option to preserve unanimity on the premises as well, by constraining the models which do not support unanimity. 

We begin by giving a formal definition on when a premise aggregation rule $F_{IC}$ preserves unanimity.  Whether or not the unanimity on the premises is preserved by our  $F_{IC}$ depends on the rule $\mathcal{R}$ as well as the agenda $\Phi$. We show two decision rules for which the unanimity is preserved and then we use an example to show that in the case of an arbitrary  rule and agenda, the unanimity of the premises is not guaranteed.

\begin{definition}
\label{def:unanimity}
Let $\underline{J_p} = (J_{1,p}, \ldots , J_{n,p})$ be a premise profile on the agenda $\Phi$ and  $p$  a premise from the agenda. A premise aggregation rule  $F_{IC}$ preserves unanimity on the premises if and only if the following holds:\\
If $p\in J_{i,p}$ for all $i=\{1,\ldots ,n\}$ then $p~\in~F_{IC}(\underline{J_p})$.
\end{definition}

Note that, since $F_{IC}$ can select more than one premise judgment set, $p$ needs to be in all of them for unanimity to be preserved. 


The following theorem indicates two decision rules $\mathcal{R}$, and an agenda, in the presence of which  unanimity is preserved on the premises by $F_{IC}$. 

\begin{theorem}\label{Theo2}

Let $\Phi$ be an agenda in which all the elements are atoms or negations of atoms. Let  $\mathcal{R}$ be a decision rule of the form $(a_1 \wedge \ldots \wedge a_n)  \leftrightarrow x$ or of the form $(a_1 \vee \ldots \vee a_n) \leftrightarrow x$. $\{a_1, \ldots ,   a_n\} \subseteq \Phi$ are premises and $x \subseteq \Phi$ is a conclusion. $F_{IC}$ preserves unanimity on the premises for any profile $\underline{J}$ over $\Phi$ and $\mathcal{R}$.

\end{theorem}

Due to strict page limit constraints the proofs are omitted\footnote{The proofs can be found in \cite{techPST09}.}. 


Given an arbitrary agenda, a decision rule~$\mathcal{R}$ corresponding to that agenda and an arbitrary profile~$\underline{J}$, the merging operator does not necessary preserve unanimity. We show this through an example. 

 
Consider the profile presented in Table \ref{tab:counter1}. The rule $\mathcal{R}$ is such that the value of $x$ is  1 if and only if the evaluations of the premises are one of the sets in the first column of Table~\ref{tab:counter2}. For all other evaluations of premises, $x$ is 0. 


Our procedure preserves  the unanimity on the conclusion and selects $v(x)~=~1$, but  gives an aggregation for the premises which violates the unanimity on premise $p_{13}$ (Table~\ref{tab:counter2}).



\begin{table}[h]
\begin{centering}
\begin{tabular}{|c|ccccccccccccc|c|}\hline 
& \small $p_1$& \small $p_2$&\small $p_3$&  \small $p_4$& \small $p_5$&\small $p_6$& \small $p_7$& \small $p_8$& \small $p_9$&  \small $p_{10}$&  \small $p_{11}$&  \small $p_{12}$& \small $p_{13}$ & \small $x$\\ \hline
    \small  A & \small 1&\small 0&\small 0&\small 1&\small 0&\small 0&\small 1&\small 0&\small 0&\small 1&\small 0&\small 0&\small 1&\small 1\\\hline
    \small  B  & \small 0&\small 1&\small 0&\small 0&\small 1&\small 0&\small 0&\small 1&\small 0&\small 0&\small 1&\small 0&\small 1& \small 1\\\hline
    \small  C & \small 0&\small 0&\small 1&\small 0&\small 0&\small 1&\small 0&\small 0&\small 1&\small 0&\small 0&\small 1&\small 1&\small  1\\\hline
    \small Maj. & \small 0&\small 0&\small 0&\small 0&\small 0&\small 0&\small 0&\small 0&\small 0&\small 0&\small 0&\small 0&\small 1&\small  1\\\hline
\end{tabular}

\end{centering}
\vspace{0.2cm}
\caption{\small A case in which unanimity on premises will be violated by the complete CBP.}
\label{tab:counter1}
\vspace{-0.8cm}
\end{table}

\begin{table}[h]
\begin{centering}
\begin{tabular}{|c|c|c|c|l|}\hline
    &\small $J_{1,p}$  &\small $J_{2,p}$ &\small  $J_{3,p}$ &\small $\Sigma_i d()$ \\ \hline\hline
\small(1,0,0,1,0,0,1,0,0,1,0,0,1)  &\small 0  &\small  8  &\small 8 &\small 16 \\ \hline
\small(0,1,0,0,1,0,0,1,0,0,1,0,1)  &\small 8 &\small  0   &\small 8 &\small 16 \\ \hline
\small(0,0,1,0,0,1,0,0,1,0,0,1,1)  &\small 8  &\small 8 &\small  0  &\small 16 \\ \hline
\small(0,0,0,0,0,0,0,0,0,0,0,0,0)  &\small 5  &\small 5 &\small  5  &\small 15 \\ \hline
\end{tabular}

\end{centering}
\vspace{0.2cm}
\caption{\small Selection of premises for the counterexample.}
\vspace{0.2cm}
\label{tab:counter2}
\vspace{-1.0cm}
\end{table}

The preservation of unanimously held premises can be imposed by $IC$. This is done by making $IC = \mathcal{R} \wedge \hat{x} \wedge p^*$, where $p^*$ is any unanimously voted premise. Admissible outcomes for $J_p$  then are those supporting the conclusion voted by the majority and containing the premise(s) unanimously chosen.

\subsection{Manipulability}

Another property which is of interest when dealing with aggregation procedures is that of manipulability. A judgment aggregation procedure is called manipulable if an agent, who would not obtain a desired outcome by submitting her sincere premise set, can obtain a desired outcome by choosing to submit a set of premises different than her honest premise set.
Under the context of complete-conclusion based procedures, we will distinguish between full and preferred manipulability.

\textit{Full manipulability} means that we distinguish only whether the aggregated premise set entirely corresponds to an agent's judgments on premises or not. 

A procedure is fully manipulable if an agent can obtain her complete honest premise set as an output from the procedure by submitting another (insincere) premise set that supports the same conclusion.
%
Formally, let $\underline{J_p}= (J_{1,p}, \ldots,J_{i,p} \ldots, J_{n,p}) $ be a premise profile. Let $F_{IC}(\underline{J_p}) = \{J^{\circ}_{1,p}, \ldots , J^{\circ}_{m,p} \}$, i.e. the merging operator selects the premise sets $J^{\circ}_{1,p}, \ldots , J^{\circ}_{m,p}$. Let $J_{i,p}$ be the ``honest" premise set of an agent $i$.

\begin{definition}
Assume that a premise set $J^*_{i,p} \not = J_{i,p}$ exists, such that $J^*_{i,p} $ supports the same conclusion as the premise set $J_{i,p}$. The operator $F_{IC}$ is fully manipulable if $J_{i,p} \in F_{IC}(J_{1,p},\ldots,J^*_{i,p} \ldots, J_{n,p})$ but $J_{i,p} \not \in F_{IC}(J_{1,p},\ldots,J_{i,p}, \ldots, \linebreak J_{n,p})$.
\end{definition}


\begin{theorem}\label{FM}
$F_{IC}$ is not fully manipulable.
\end{theorem}




Let us now assume that an agent has a premise $p$ which she holds most important (has a strong preference on the evaluation of this premise). We say that a procedure is \textit{preferred manipulable} if an agent can ensure that the preferred projection $w(p)$ is included in the output by submitting another premise set that supports the same conclusion.
Since we do not represent the preferred premise explicitly in our framework, any premise can be the preferred one, and preferred manipulability therefore means that the agent is able to change her premise set in a way such that one premise which is not a member of the aggregated set becomes member of it.

\begin{definition}
Assume that a premise set $J^*_{i,p} \not = J_{i,p}$ exists, such that $J^*_{i,p} $ supports the same conclusion as the premise set $J_{i,p}$ and premise $p_{pref}$ is in both of the premise sets. The operator $F_{IC}$ is preferred manipulable if $p_{pref}$ is in at least one premise set $J^{\circ}_{j,p} \in F_{IC}(J_{1,p},\ldots,J^*_{i,p},\ldots, J_{n,p})$, but $\neg p_{pref}$ is in all of the premise sets selected by $F_{IC}(J_{1,p}, \ldots,J_{i,p},\ldots, J_{n,p})$.
\end{definition}

\begin{theorem}\label{Theo4}
$F_{IC}$ is preferred manipulable.

\end{theorem}

Full manipulability is a relatively weak condition, in the sense that it is fairly easy to satisfy. 
This notion of preferred manipulability seems to conflict with the intuition of the distance measure used to aggregate the premises, which does take such distinctions into account. However, preferred manipulability is a very strong condition, since it means in practice that an agent should not be able to improve any premise (since this premise may happen to be the preferred one). Other notions of manipulability could be studied, such as the improvement of a preferred premise by changing the judgment on this premise only.





