social choice theory Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, soc ...

, a solid coalition or

voting bloc A voting bloc is a group of voting, voters that are strongly motivated by a specific common concern or group of concerns to the point that such specific concerns tend to dominate their voting patterns, causing them to vote together in elections. ...

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 A political faction is a group of people with a common political purpose, especially a subgroup of a political party that has interests or opinions different from the rest of the political party. Intragroup conflict between factions can lead to ...

or voting bloc, allowing social choice theorists to study how

electoral system An electoral or voting system is a set of rules used to determine the results of an election. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, nonprofit organizations and inf ...

s 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 Southern Democrats are members of the U.S. Democratic Party who reside in the Southern United States. Before the American Civil War, Southern Democrats mostly believed in Jacksonian democracy. In the 19th century, they defended slavery in the ...

in the 20th century).

Definition

A solid coalition is a group of voters N together with some set of candidates C such that each voter ''v'' in N prefers all candidates in C to all candidates outside of C.

Example

Consider the following example, taken from American politics of the 1800s: 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 The single transferable vote (STV) or proportional-ranked choice voting (P-RCV) is a multi-winner electoral system in which each voter casts a single vote in the form of a ranked ballot. Voters have the option to rank candidates, and their vot ...

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

Proportional representation

One important use of solid coalitions is in defining

proportional representation Proportional representation (PR) refers to any electoral system under which subgroups of an electorate are reflected proportionately in the elected body. The concept applies mainly to political divisions (Political party, political parties) amon ...

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 .

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.

References

{{Reflist Voting theory Electoral systems