What is the core definition of Proportionality for Solid Coalitions (PSC)?

Proportionality for Solid Coalitions (PSC) is a criterion used to evaluate the proportionality of electoral systems, specifically those that utilize ranked voting where voters can directly support candidates. It adapts the concept of the quota rule to scenarios lacking official party lists.

To which specific types of electoral systems is the Proportionality for Solid Coalitions (PSC) criterion primarily applied?

PSC is primarily applied to ranked voting systems, particularly those where voters express preferences for individual candidates rather than relying on pre-defined party lists. This contrasts with party-list proportional representation systems.

Who is recognized as the originator of the Proportionality for Solid Coalitions (PSC) criterion?

The criterion was first proposed by Michael Dummett, a British philosopher and logician known for his extensive work in logic and the philosophy of mathematics.

What fundamental electoral principle does Proportionality for Solid Coalitions (PSC) adapt for systems without official party lists?

PSC is an adaptation of the quota rule, which is a principle commonly employed in proportional representation systems to allocate seats based on vote thresholds or minimum vote counts.

What is the essential condition a group of voters must meet to be considered a "solid coalition" for a set of candidates?

A group of voters forms a solid coalition for a specific set of candidates if every voter within that group consistently ranks all candidates from the specified set ahead of any candidate who is not part of that set.

How does the concept of a "solid coalition" function as an analogue to political parties in electoral systems that do not use official party lists?

In electoral systems lacking formal party lists, a solid coalition serves as the equivalent of a party by representing a group of voters whose preferences are consistently aligned towards a particular set of candidates, thereby ensuring proportional representation for that cohesive voter bloc.

What does it mean for a voter to be "solidly supporting" or "solidly committed to" a particular set of candidates?

A voter is considered to be solidly supporting a set of candidates if their ranking of preferences consistently places all candidates within that chosen set above any candidates outside of it, demonstrating a unified preference.

What is the most basic demonstration of a voter's solid support for a candidate within the framework of PSC?

The most fundamental way a voter demonstrates solid support for a candidate is by ranking that candidate as their first preference, indicating a clear and primary choice.

How can solid coalitions be structured or "nested" in relation to each other, particularly concerning voters organized along a political spectrum?

Solid coalitions can be nested, meaning a smaller, more specific group of voters might be a subset of a larger, broader group. However, solid coalitions cannot span across distinct factions because preferences might not align consistently across those divisions.

Provide an example of why certain groups, such as moderate factions on a political spectrum, might fail to form a solid coalition.

For instance, if voters are spread across a political spectrum, moderate groups might not form a single solid coalition. This is because some voters in a center-right group might prefer a far-right candidate over a center-left candidate, thus breaking the consistent preference required for a solid coalition.

What are the variables commonly used in defining quota rules for PSC, and what do they represent?

The variables typically used are 'n' for the total number of voters, 'k' for the number of seats to be filled, and 'j' which is a positive integer used in specific quota calculations to determine the number of guaranteed seats.

How is the k-PSC, also known as Hare-PSC, formally defined in relation to electoral quotas?

Hare-PSC is defined using the Hare quota (n/k), stating that if a solid coalition accumulates at least 'j' Hare quotas, then a minimum of 'j' candidates from that coalition must be elected.

What is the specific quota used when calculating the Hare-Proportionality for Solid Coalitions (Hare-PSC)?

The Hare quota, calculated as the total number of voters (n) divided by the number of seats to be filled (k), is used for the Hare-PSC criterion.

What is the guarantee provided by the Hare-PSC criterion for a solid coalition that achieves a certain threshold of Hare quotas?

The Hare-PSC criterion guarantees that if a solid coalition secures at least 'j' Hare quotas, then a minimum of 'j' candidates from that coalition will be elected.

What is the outcome for candidates if their solid coalition has fewer members than the number of seats required by the Hare-PSC criterion?

If a solid coalition's set of candidates is smaller than the number 'j' specified by the criterion, meaning they have fewer than 'j' candidates, then all candidates within that set must be elected.

Who is credited with proposing the specific criterion known as Hare-PSC?

The Hare-PSC criterion was proposed by Michael Dummett.

In the context of single-winner elections, how does the Hare-PSC criterion relate to the unanimity criterion?

In single-winner elections (where k=1), the Hare quota (n/1 = n) encompasses all voters. Thus, Hare-PSC becomes equivalent to the unanimity criterion, ensuring that if all voters support a candidate, that candidate wins.

How is the (k+1)-PSC, also referred to as Droop-PSC, defined using electoral quotas?

Droop-PSC is defined similarly to Hare-PSC but uses the Droop quota, calculated as n/(k+1). It stipulates that if a solid coalition accumulates 'j' Droop quotas, then at least 'j' candidates from that coalition must be elected.

What assurance does the Droop-PSC criterion offer regarding the representation of solid coalitions that command a majority of votes?

Droop-PSC assures that any solid coalition possessing a majority of votes will always be able to elect at least half of the available seats.

What is identified as a significant drawback or consequence of using Droop proportionality in electoral systems?

A major drawback of Droop proportionality is its tendency to create substantial "seat bias," which disproportionately favors larger parties in the allocation of seats.

How does the seat bias associated with Droop proportionality potentially disadvantage smaller parties that collectively form a majority?

Seat bias means that even if a coalition of smaller parties wins a majority of the total votes, they might still fail to secure a majority of seats in the legislature due to the way seats are allocated by the system.

Can you name some electoral methods that are considered examples of quota-proportional methods?

Examples of electoral methods that employ quota-proportional principles include the expanding approvals rule, the method of equal shares, and the single transferable vote (STV).

What property did Aziz and Lee define that extends the concept of PSC to handle weaker forms of voter preference rankings?

Aziz and Lee defined properties called "generalized PSC" and "inclusion PSC," which are applicable even when voters express weak rankings, meaning they may be indifferent between certain candidates.

Which specific electoral rule is mentioned as satisfying the generalized PSC properties defined by Aziz and Lee?

The expanding approvals rule is cited as a method that satisfies these generalized versions of PSC, including those that accommodate weak rankings.

What fairness property was defined by Brill and Peters, and how does it differ from standard PSC in terms of voter coalitions?

Brill and Peters defined a property called Rank-PJR+, which applies to weak rankings and provides proportionality guarantees not only to solid coalitions but also to coalitions that are only partially solid.

Which electoral system is identified as satisfying the Rank-PJR+ property defined by Brill and Peters?

The expanding approvals rule is mentioned as a system that satisfies the Rank-PJR+ property.

Which electoral system is noted for violating the Rank-PJR+ property as defined by Brill and Peters?

The single transferable vote (STV) is identified as an electoral system that violates the Rank-PJR+ property.

Is it computationally feasible to determine whether a given committee satisfies the Rank-PJR+ criterion?

Yes, it has been determined that it is possible to efficiently decide in polynomial time whether a given committee adheres to the Rank-PJR+ fairness property.

Describe the image presented in the article's sidebar.

The image in the sidebar is an SVG graphic depicting interlocking gears. This visual metaphor symbolizes the complex and interconnected nature of various electoral systems and their underlying mechanisms.

What is the primary purpose of the Proportionality for Solid Coalitions (PSC) criterion?

The primary purpose of PSC is to ensure a degree of proportionality in electoral outcomes, particularly in systems where voters rank candidates directly. It aims to guarantee representation for cohesive groups of voters.

How does the definition of a solid coalition ensure that voters' preferences are consistently aligned towards a specific group of candidates?

The definition requires a strict ordering: all candidates within the coalition's set must be preferred over all candidates outside it. This ensures a unified and consistent preference structure for the coalition.

What is the significance of the phrase "no official party lists" for the application of PSC?

The phrase "no official party lists" signifies that PSC is designed for electoral systems where voters vote directly for candidates, and these candidates may or may not be formally affiliated with parties. Proportionality is assessed based on voter preferences, not party registration.

What is the potential consequence of voters ranking candidates from different parties together in the context of PSC?

If voters rank candidates from different parties together, they do not form a "solid coalition" as defined by the criterion. This means their collective preferences might not guarantee proportional representation under PSC.

What is the difference between the Hare-PSC and Droop-PSC in terms of the quota used and the potential outcome for coalitions?

Hare-PSC uses the Hare quota (n/k), while Droop-PSC uses the Droop quota (n/(k+1)). The Droop quota is generally smaller, potentially leading to different seat allocations and possibly favoring larger parties more due to seat bias.

What does the term "quota rule" imply about the minimum number of votes needed to secure representation in proportional systems?

The quota rule establishes a threshold, typically calculated based on the number of votes and seats, that a party or coalition must meet or exceed to be entitled to representation. It's a mechanism for ensuring a minimum level of support translates to seats.

What is the significance of the reference to "ranked voting systems" in the definition of PSC?

It signifies that PSC is designed for electoral systems where voters express preferences in order (e.g., 1st, 2nd, 3rd choice), rather than simply choosing a single candidate or party. This ordered preference is crucial for defining solid coalitions.

What is the role of the integer 'j' in the definitions of Hare-PSC and Droop-PSC?

The integer 'j' represents the number of candidates from a solid coalition that must be elected if the coalition meets the relevant quota threshold (either Hare or Droop). It scales the proportionality guarantee based on the coalition's strength.

What does the citation to Dummett's "Voting procedures" suggest about the academic background of PSC?

The citation indicates that PSC is a concept rooted in formal studies of voting systems and social choice theory, originating from academic research in logic and political science.

What is the relationship between a voter's first preference and the concept of a solid coalition?

A voter's first preference is the most basic form of solid support for a candidate. It represents a clear preference for that candidate over all others, forming the foundation for more complex solid coalitions.

What is the core difference between "party-list proportional representation" and the systems PSC is designed for?

Party-list systems allocate seats based on the total votes received by a party's list of candidates. PSC, however, applies to systems where voters rank individual candidates, and proportionality is assessed based on cohesive groups of voters supporting those candidates, regardless of formal party affiliation.

What is the implication of the phrase "no official party lists" for how proportionality is assessed in the context of PSC?

It means that proportionality is not measured by the vote share of registered political parties, but rather by the collective preferences of voters who form cohesive groups. The focus is on voter solidarity rather than party structure.

What is the potential consequence of a solid coalition having *exactly* 'j' Hare quotas?

If a solid coalition has exactly 'j' Hare quotas, the Hare-PSC criterion guarantees that at least 'j' candidates from that coalition will be elected, ensuring a minimum level of representation.

What is the difference between the Hare quota and the Droop quota in terms of their calculation?

The Hare quota is calculated as n/k (total voters divided by the number of seats), while the Droop quota is calculated as n/(k+1) (total voters divided by the number of seats plus one).

How does the Droop quota generally compare to the Hare quota in terms of size?

The Droop quota (n/(k+1)) is generally smaller than the Hare quota (n/k). This means it typically requires fewer votes to meet the Droop quota compared to the Hare quota.

What is the term used for the phenomenon where a voting system disproportionately favors larger parties in seat allocation?

This phenomenon is referred to as "seat bias," where the distribution of seats does not perfectly mirror the distribution of votes, often benefiting larger parties.

What is the implication of "seat bias" for the representation of smaller parties in an electoral system?

Seat bias can lead to smaller parties receiving fewer seats than their vote share would suggest, or conversely, larger parties receiving more seats than their vote share strictly warrants.

What is the "expanding approvals rule" and how does it relate to PSC?

The expanding approvals rule is an electoral method that satisfies generalized versions of PSC. It is noted for its ability to accommodate weak rankings and even partially solid coalitions.

What is the significance of "weak rankings" in the context of generalized PSC properties defined by Aziz and Lee?

Weak rankings allow voters to express indifference between certain candidates. Generalized PSC properties are designed to function correctly and provide proportionality guarantees even when voters do not provide strict, complete rankings.

What does it mean for a voting system to satisfy "inclusion PSC"?

Inclusion PSC is a property defined by Aziz and Lee that applies to weak rankings. It ensures proportionality for coalitions that are subsets of larger, cohesive groups, extending the concept of solid support.

Proportionality For Solid Coalitions Wiki: The Calculus of Consensus: Proportionality in Ranked Voting

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Introduction to PSC

Defining Proportionality

Proportionality for Solid Coalitions (PSC) is a criterion used to evaluate the fairness of ranked voting systems, particularly those without explicit party lists. It adapts the fundamental concept of the quota rule to scenarios where voters directly support candidates, forming groups known as "solid coalitions."

Dummett's Contribution

This criterion was first articulated by the distinguished British philosopher and logician, Michael Dummett. His work sought to establish robust mathematical frameworks for assessing the proportionality of electoral outcomes, especially in systems where voter preferences are ranked.

Michael Dummett's extensive research in logic and philosophy extended into the field of social choice theory and electoral systems. His seminal work, "Voting Procedures," laid much of the groundwork for understanding the mathematical properties of various voting methods, including proportionality criteria like PSC.

Minimal Definition

PSC is considered a relatively minimal definition of proportionality. It primarily ensures that if a group of voters consistently ranks a specific set of candidates above all others (forming a solid coalition), then at least a proportional number of seats should be allocated to candidates from that set.

Understanding Solid Coalitions

The Essence of a Solid Coalition

In electoral systems that lack formal party lists, the concept of a "solid coalition" serves as the analogue to a political party. A solid coalition is defined as a group of voters who, when ranking candidates, consistently prefer any candidate within a particular set over any candidate not in that set.

Commitment and Preference

When voters are part of a solid coalition, they are considered to be "solidly supporting" or "solidly committed to" the candidates within that set. This implies a strong, unified preference structure among these voters for their chosen candidates.

Boundaries and Factions

Crucially, solid coalitions cannot arbitrarily cut across different factions or preference groups. For instance, on a political spectrum, moderate voters from the center-left and center-right might not form a single solid coalition if some center-right voters rank a far-right candidate higher than a center-left candidate. This segmentation is key to how PSC operates.

Quotas and PSC

Hare-PSC

The Hare-PSC criterion utilizes the Hare quota, calculated as the total number of voters ( $n$ ) divided by the number of seats ( $k$ ), i.e., $n/k$ .

It mandates that if a solid coalition commands at least $j$ Hare quotas, then at least $j$ candidates from that coalition must be elected.

In the context of a single-winner election ( $k=1$ ), the Hare quota equals the total number of voters ( $n$ ). A solid coalition with $j=1$ Hare quota (i.e., all voters) is guaranteed to elect one candidate, making it equivalent to the unanimity criterion in single-winner elections.

Droop-PSC

Droop-PSC employs the Droop quota, calculated as the total number of voters ( $n$ ) divided by the number of seats plus one ( $k+1$ ), i.e., $n/(k+1)$ .

This variant ensures that if a solid coalition commands at least $j$ Droop quotas, at least $j$ candidates from that coalition must be elected.

Droop-PSC is a generalization of the majority criterion. A significant advantage is that any solid coalition achieving a majority of votes is guaranteed to secure at least half of the available seats. However, this method can introduce a "seat bias," potentially favoring larger parties and disadvantaging smaller coalitions even if their combined vote share is substantial.

Related Methods

Several electoral methods are known to satisfy quota-proportional criteria, including PSC. Notable examples include the Expanding Approvals Rule, the Method of Equal Shares, and the widely used Single Transferable Vote (STV) system.

Advanced Concepts & Generalizations

Generalized and Inclusion PSC

Researchers Aziz and Lee have extended the PSC concept. They introduced Generalized PSC and Inclusion PSC, which are applicable even when voters employ "weak rankings" (allowing for indifference between candidates). Their Expanding Approvals Rule is noted for satisfying these more comprehensive versions of PSC.

Rank-PJR+

Brill and Peters proposed Rank-PJR+, another fairness property designed for weak rankings. This criterion specifically addresses coalitions that are only "partially solid." The Expanding Approvals Rule adheres to Rank-PJR+, whereas the Single Transferable Vote (STV) does not. Importantly, determining if a committee satisfies Rank-PJR+ can be achieved efficiently through polynomial-time computation.

Related Concepts

Justified Representation

A concept analogous to proportional representation for electoral systems that utilize approval ballots (where voters can approve of multiple candidates without ranking them) is known as Justified Representation. This highlights the broader theoretical landscape of ensuring fair outcomes in diverse voting mechanisms.

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References

D. R. Woodall: Monotonicity of single-seat preferential election rules. Discrete Applied Mathematics 77 (1997), p. 83â84.

Tideman N.: Collective Decisions and Voting. Ashgate Publishing Ltd, Aldershot, 2006, p. 268â269.

A full list of references for this article are available at the Proportionality for solid coalitions Wikipedia page

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This content has been generated by an Artificial Intelligence and is intended for educational and informational purposes only. It is based on publicly available data and may not be exhaustive or entirely up-to-date.

This is not professional advice. The information provided herein does not constitute expert advice in political science, mathematics, or electoral system design. Readers are encouraged to consult primary sources and qualified professionals for in-depth analysis and specific applications.

The creators of this page are not responsible for any errors or omissions, or for any actions taken based on the information provided.

The Calculus of Consensus

💡 Dive in with Flashcard Learning! 💡

Introduction to PSC ⚖️

🎯 Defining Proportionality

🧠 Dummett's Contribution

📜 Minimal Definition

Understanding Solid Coalitions 🤝

🧩 The Essence of a Solid Coalition

➡️ Commitment and Preference

🚧 Boundaries and Factions

Quotas and PSC 🔢

📊 Hare-PSC

⚖️ Droop-PSC

🔗 Related Methods

Advanced Concepts & Generalizations 💡

🔬 Generalized and Inclusion PSC

🛡️ Rank-PJR+

Related Concepts 🔗

🌟 Justified Representation

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📜 Important Notice

Dive in with Flashcard Learning!

Introduction to PSC

Defining Proportionality

Dummett's Contribution

Minimal Definition

Understanding Solid Coalitions

The Essence of a Solid Coalition

Commitment and Preference

Boundaries and Factions

Quotas and PSC

Hare-PSC

Droop-PSC

Related Methods

Advanced Concepts & Generalizations

Generalized and Inclusion PSC

Rank-PJR+

Related Concepts

Justified Representation

Teacher's Corner

True or False?

Test Your Knowledge!

Gamer's Corner

Discover other topics to study!

References

Feedback & Support

Disclaimer

Important Notice