# The core of a voting system is the intersection of Pareto sets

Voting system is a concept in political science. Here I give the mathematical definition of a voting system.

A **(binary) voting system** is a tuple $(P,V,q)$, where
$P$ is any set, called the set of **proposals**,
and $V$ is a finite set of preference relations on $P$, called the set of **voters**,
and $q$ is an integer between (inclusive) $0$ and $\left|V\right|$,
called the **quota**.

For each voter $v\in V$ and two proposals $x,y\in P$, we denote “$v$ prefers $x$ to $y$” by

\[x\succeq_vy.\]A proposal $x\in P$ is a **defeat** of $y\in P$ if

denoted as $x\succsim_{V,q}y$
(despite this notation, $\succsim_{V,q}$ is *not* necessarily a preference relation on $P$
because it is not transitive generally,
which is actually a well-known example of irrationality).

The **core** $\mathcal C(P,V,q)$ of the voting system is the set of such element $x\in P$:
$x$ does not have any defeat other than $x$ itself (non-trivial defeat).

Pareto sets are common concepts in economics. To clarify, I also give the mathematical definition of them here.

Let $P$ be a set and $Q$ be a family of preference relations on $P$.
Then, $x\in P$ is called a (weak) **$Q$-Pareto improvement** of $y\in P$ if $\forall v\in V:x\succeq_vy$,
denoted as $x\succsim_Qy$
(despite the notation, $\succsim_Q$ is *not* necessarily a preference relation on $P$).

The **Pareto set** $\mathcal P(P,Q)$ is the set of all such element $x\in P$:
$x$ does not have any $Q$-Pareto improvement other than $x$ itself
(non-trivial $Q$-Pareto improvement).

Here is the main result. For a voting system $(P,V,q)$,

\[\mathcal C(P,V,q)=\bigcap_{Q\subseteq V,\left|Q\right|=q}\mathcal P(P,Q).\]*Proof.*
To prove this, we need to show that
$x\in P$ does not have any non-trivial Pareto improvement for any $q$ voters iff
$x$ does not have any non-trivial defeat.

To prove the forward direction, suppose that $x\in P$ does not have any non-trivial Pareto improvement for any $q$ voters. Let $y\in P$ such that $y\ne x$, and the goal is to prove that $y$ is not a defeat of $x$.

Let

\[Y:=\left\{v\in V\,\middle|\,y\succeq_vx\right\}.\]Then, $y$ is a $Y$-Pareto improvement of $x$, so we have $\left|Y\right|<q$ (because otherwise there is a subset of $Y$ with $q$ voters for which $y$ is a Pareto improvement of $x$). Therefore, $y$ is not a defeat of $x$.

To prove the backward direction, suppose that $x\in P$ has a non-trivial $Q$-Pareto improvement, where $Q\subseteq V$ and $\left|Q\right|=q$. Denote the improvement as $y$. Let

\[Y:=\left\{v\in V\,\middle|\,y\succeq_vx\right\}.\]because $y$ is a $Q$-Pareto improvement of $x$, we have $Q\subseteq Y$. Therefore, $\left|Y\right|\geq\left|Q\right|=q$. Therefore, $y$ is a defeat of $x$. $\square$

Specially, we have

\[\mathcal C\!\left(P,V,\left|V\right|\right)=\mathcal P(P,V).\]Here is an example. Suppose we have 5 voters, and the set of proposals is $\mathbb R^2$. Each voter has an ideal point and prefers points nearer to the ideal point. The 5 ideal points form a convex pentagon. Then we can find the core easily by the conclusion above: