Diverse group facing stable vs. disjointed decision paths.

Can Group Decisions Be Too Divided? The Surprising Truth About the Banks Set and Bipartisan Set

Reese Marlowe in Mind & Education June 2025 • 4 min read.

"Exploring how different methods of reaching consensus might lead to completely opposing outcomes in decision-making."

Imagine trying to decide where to go for dinner with a group of friends. Everyone has different preferences, and finding a place that everyone agrees on can be a challenge. This everyday scenario reflects the complexities studied in social choice theory, a field that explores how groups make decisions. Tournament solutions play a vital role in this field, offering mathematical frameworks for understanding how collective choices emerge from individual preferences.

Among the various concepts within social choice theory, the Banks set and the bipartisan set stand out as two distinct approaches to identifying viable options. The Banks set focuses on finding alternatives that are 'maximal' within transitive subsets, essentially seeking options that can't be beaten within smaller, consistent groups. On the other hand, the bipartisan set considers the support each option receives across the entire group, aiming for a compromise that balances different viewpoints.

However, what happens when these two seemingly reasonable approaches lead to completely different outcomes? Can the Banks set and the bipartisan set ever be disjoint, meaning they identify entirely separate sets of choices? This is the intriguing question that has puzzled researchers in social choice theory for years. Recent research has demonstrated the surprising possibility that these two sets can indeed be disjoint, revealing a fundamental tension in how we think about group decision-making.

The Banks Set vs. the Bipartisan Set: Why Can't We All Just Agree?

Diverse group facing stable vs. disjointed decision paths.

To understand why the Banks set and the bipartisan set can clash, let's delve deeper into their definitions. The Banks set, named after political scientist Jeffrey Banks, seeks out alternatives that are 'stable' in a specific sense. Imagine organizing the options into smaller, internally consistent groups. An option belongs to the Banks set if it's the best choice within one of these groups and can't be overturned by any other option.

In contrast, the bipartisan set takes a more holistic view. It considers the 'support' each option receives from the entire group, aiming to find a compromise that satisfies as many people as possible. This approach often involves complex mathematical calculations to determine the 'equilibrium' point where no individual or subgroup has a strong incentive to deviate.

Banks Set: Focuses on maximal elements within transitive subsets, identifying stable options within smaller groups.
Bipartisan Set: Aims to find a compromise that balances different viewpoints across the entire group, often through complex mathematical calculations.
Disjoint Sets: Occur when the two methods identify entirely separate sets of choices, highlighting a fundamental tension in group decision-making.

The key to the disjointedness lies in the fact that the Banks set prioritizes internal consistency within subgroups, while the bipartisan set emphasizes overall compromise. In certain scenarios, the option that best balances everyone's preferences (the bipartisan set) might not be the most stable option within any particular subgroup (the Banks set), and vice versa. This creates a situation where the two sets diverge, highlighting the inherent challenges in aggregating individual preferences into a collective decision.

What Does This Mean for Real-World Decisions?

The discovery that the Banks set and the bipartisan set can be disjoint has significant implications for how we approach group decision-making in various contexts. It suggests that different methods of reaching consensus can lead to fundamentally different outcomes, and that no single approach is guaranteed to identify the 'best' solution. Instead, we need to be aware of the potential biases and limitations of each method and consider a range of perspectives to arrive at a more informed and balanced decision. This could apply anywhere from corporate boardrooms to political elections.

About this Article -

This article was crafted using a human-AI hybrid and collaborative approach. AI assisted our team with initial drafting, research insights, identifying key questions, and image generation. Our human editors guided topic selection, defined the angle, structured the content, ensured factual accuracy and relevance, refined the tone, and conducted thorough editing to deliver helpful, high-quality information.See our About page for more information.

This article is based on research published under:

DOI-LINK: https://doi.org/10.48550/arXiv.2308.01881,

Title: The Banks Set And The Bipartisan Set May Be Disjoint

Subject: econ.th

Authors: Felix Brandt, Florian Grundbacher

Published: 03-08-2023

Everything You Need To Know

What is the core difference between the Banks set and the bipartisan set in group decision-making?

The fundamental distinction lies in their approaches to finding viable options. The Banks set identifies alternatives that are 'maximal' within transitive subsets, focusing on stability within smaller, consistent groups. Conversely, the bipartisan set aims for a compromise that balances viewpoints across the entire group, often employing complex calculations to find an equilibrium where no individual or subgroup has a strong incentive to deviate. This difference in perspective can lead to conflicting outcomes in group decisions.

Can the Banks set and the bipartisan set ever produce completely different outcomes? What does it mean?

Yes, the Banks set and the bipartisan set can be disjoint, meaning they identify entirely separate sets of choices. This is a significant finding in social choice theory. It reveals a fundamental tension in group decision-making because the option preferred by the Banks set (internal consistency) might not align with the one preferred by the bipartisan set (overall compromise). This disjointedness highlights the challenge in aggregating individual preferences and suggests that different consensus methods can lead to very different outcomes.

How does the concept of 'transitive subsets' relate to the Banks set, and why is it important?

The Banks set focuses on 'maximal' alternatives within 'transitive subsets.' A transitive subset is a smaller, internally consistent group of options where preferences are clear and don't contradict each other. The Banks set identifies the best choice within each of these subgroups. The importance lies in finding stable options; an option in the Banks set can't be beaten by any other option within the subgroup. This approach prioritizes options that are robust within specific, consistent preference structures.

What real-world implications arise from the fact that the Banks set and the bipartisan set can be disjoint?

The discovery of disjoint Banks and bipartisan sets has wide-ranging implications. It shows that different methods of reaching consensus can lead to completely different outcomes. It suggests that no single decision-making approach guarantees the 'best' solution. This means decision-makers need to be aware of the potential biases and limitations of each method. They should consider multiple perspectives to make more informed and balanced decisions. This awareness is critical in contexts ranging from corporate boardrooms to political elections.

What role does 'social choice theory' play in understanding the Banks set and the bipartisan set, and why is it relevant?

Social choice theory provides the framework for understanding how groups make decisions, and it is essential for studying the Banks set and the bipartisan set. This field explores how collective choices emerge from individual preferences. It provides mathematical models and concepts to analyze decision-making processes. The Banks set and the bipartisan set are just two approaches within this broader field. Understanding social choice theory is relevant because it helps us understand the complexities of group decision-making, the potential for different outcomes, and the inherent challenges in aggregating individual preferences to reach a collective choice.