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)(P,V,q), where PP is any set, called the set of proposals, and VV is a finite set of preference relations on PP, called the set of voters, and qq is an integer between (inclusive) 00 and V\left|V\right|, called the quota.

For each voter vVv\in V and two proposals x,yPx,y\in P, we denote “vv prefers xx to yy” by

xvy.x\succeq_vy.

A proposal xPx\in P is a defeat of yPy\in P if

{vV|xvy}q,\left|\left\{v\in V\,\middle|\,x\succeq_vy\right\}\right|\geq q,

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

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


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

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

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


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

C(P,V,q)=QV,Q=qP(P,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 xPx\in P does not have any non-trivial Pareto improvement for any qq voters iff xx does not have any non-trivial defeat.

To prove the forward direction, suppose that xPx\in P does not have any non-trivial Pareto improvement for any qq voters. Let yPy\in P such that yxy\ne x, and the goal is to prove that yy is not a defeat of xx.

Let

Y{vV|yvx}.Y\coloneqq\left\{v\in V\,\middle|\,y\succeq_vx\right\}.

Then, yy is a YY-Pareto improvement of xx, so we have Y<q\left|Y\right|<q (because otherwise there is a subset of YY with qq voters for which yy is a Pareto improvement of xx). Therefore, yy is not a defeat of xx.

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

Y{vV|yvx}.Y\coloneqq\left\{v\in V\,\middle|\,y\succeq_vx\right\}.

because yy is a QQ-Pareto improvement of xx, we have QYQ\subseteq Y. Therefore, YQ=q\left|Y\right|\geq\left|Q\right|=q. Therefore, yy is a defeat of xx. \square


Specially, we have

C ⁣(P,V,V)=P(P,V).\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 R2\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:

The core of the example