Solid coalition

In social choice theory, a solid coalition or voting bloc is a group of voters who support a given group of candidates over any opponent outside the group. Solid coalitions formalize the idea of a political faction or voting bloc, allowing social choice theorists to study how electoral systems behave when there are ideological divisions, without having to make explicit reference to political parties. This definition is useful even in the absence of party labels, or when labels do not accurately reflect ideological divisions in the electorate (as in the cleavages between Northern and Southern Democrats in the 20th century).

Definition

Let L be a subset of candidates. A solid coalition in support of L is a group of voters who strictly prefer all members of L to all candidates outside of L. In other words, each member of the solid coalition ranks their least-favorite member of L higher than their favorite member outside L. In other words, the voters are strict partisans, who always prefer members of the coalition to non-members.

A voter is part of the solid coalition for a group of candidates—sometimes called clones or copartisans—if they rank every member inside the group higher than every member outside the group. In other words, their least-preferred candidate inside the coalition must be ranked higher than their most-preferred candidate outside the coalition.

Example

Consider the following example, taken from American politics of the 1800s:

Share:	25%	30%	20%	25%
Clay	2	1	4	3
Webster	1	2	3	4
van Buren	3	3	1	2
Jackson	4	4	2	1

In this election, the Whig faction creates a solid coalition with 55% of the vote, because 55% of voters rank both Clay and Webster over both van Buren and Jackson. Similarly, the remaining 45% of voters form a Democratic coalition.

Note that solid coalitions can be nested within each other. For example, the solid coalition consisting only of Jackson has support from 25% of voters (the voters ranking him first). However, there cannot be overlapping, non-nested, solid coalitions. This fact underlies the tendency of systems like the single transferable vote to become disproportional when voters are not cleanly divided into homogenous political parties, but instead face cross-cutting cleavages (as can happen if racial and ethnic groups do not consistently vote for the same party).

Applications

Descending Coalitions

Main article: Descending solid coalitions

Proportional representation

Main article: Proportionality for solid coalitions

One important use of solid coalitions is in defining proportional representation systems that do not rely on party labels. A voting system is proportional for solid coalitions (PSC) if it always elects a number of candidates from each solid coalition that is proportional to its size. For instance, if there are 100 voters and 10 seats, and a solid coalition of 20 voters supports candidates A, B, and C, then a PSC voting system should elect at least 2 candidates from {A, B, C}.

Cooperative game theory

Solid coalitions can be used to model coalition formation in cooperative game theory, where individuals can communicate and behave strategically in their group's interests.