# Stable Winner Set

A stable winner set is a requirement on a winner set

Given a winner set S of K winners, another winner set S' containing K’ winners blocks S iff V(S,S’)/n >= K’/K. Where V(S,S’) is the number of voters who strictly prefer S’ to S and n is the number of voters. A winner set is stable if no replacement set blocks it.

There are a few points which are important to note:

- In most cases K’ < K. This means that the definition of a stable winner set is that a subgroup of the population cannot be more happy with less winners given the relavant size comparison of the Ks and the group.
- There can be more than one stable winner set. The group of all stable winner sets is referred to as
*the core*.

## Relation to Proportional Representation

Each group of voters should feel that their preferences are sufficiently respected, so that they are not incentivized to deviate and choose an alternative winner set of smaller weight. In the common scenario that we do not know beforehand the exact nature of the demographic coalitions, we adopt the robust solution concept which requires the winner set to be agnostic to any potential subset of voters deviating. This means that the requirement of a stable winner set is equivalent to but more robust than the concept of Proportional representation.

This is a more strict definition than the Hare Quota Criterion which is typically what used as a stand in for Proportional Representation in non-partisan systems since there is no universally accepted definition. The existing definitions of Proportional Representation are unclear and conflicting.

## Example

Let's look at a common example lets say we have two voting blocks group A and B. B makes up 79% of the population and A 21%. In 5 winner election with max score of 5 and 100 voters, Group A will score all the A candidates 5 and the B candidates 0. Group B will do the opposite.

The best winner set for group A is {A1,A2,A3,A4,A5}. This is the bloc voting answer and is not the proportional answer. So lets prove it is not stable

S = {A1,A2,A3,A4,A5}

A blocking set is S’ = {B1}

V(S,S’) = 21 since their total utility from S is 0 and S’ is 5.

V(S,S’)/n = 21/100 = 0.21 K’/K = 1/5 = 0.2

0.21≥ 0.2 so S’ blocks S. Therefore, S is not stable.