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 , where is any set, called the set of proposals, and is a finite set of preference relations on , called the set of voters, and is an integer between (inclusive) and , called the quota.
For each voter and two proposals , we denote “ prefers to ” by
A proposal is a defeat of if
denoted as (despite this notation, is not necessarily a preference relation on because it is not transitive generally, which is actually a well-known example of irrationality).
The core of the voting system is the set of such element : does not have any defeat other than itself (non-trivial defeat).
Pareto sets are common concepts in economics. To clarify, I also give the mathematical definition of them here.
Let be a set and be a family of preference relations on . Then, is called a (weak) -Pareto improvement of if , denoted as (despite the notation, is not necessarily a preference relation on ).
The Pareto set is the set of all such element : does not have any -Pareto improvement other than itself (non-trivial -Pareto improvement).
Here is the main result. For a voting system ,
Proof. To prove this, we need to show that does not have any non-trivial Pareto improvement for any voters iff does not have any non-trivial defeat.
To prove the forward direction, suppose that does not have any non-trivial Pareto improvement for any voters. Let such that , and the goal is to prove that is not a defeat of .
Then, is a -Pareto improvement of , so we have (because otherwise there is a subset of with voters for which is a Pareto improvement of ). Therefore, is not a defeat of .
To prove the backward direction, suppose that has a non-trivial -Pareto improvement, where and . Denote the improvement as . Let
because is a -Pareto improvement of , we have . Therefore, . Therefore, is a defeat of .
Specially, we have
Here is an example. Suppose we have 5 voters, and the set of proposals is . 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: