# Balanced 2-subsets

## TL;DR

This paper classifies 2-element balanced families from cooperative game theory and explores their applications in generalizing Sperner and Tucker lemmas.

## Contribution

It provides a new classification of balanced 2-subsets and extends classical combinatorial lemmas using these balanced sets.

## Key findings

- Classification of balanced 2-subset families
- Generalizations of Sperner and Tucker lemmas
- Insights into nonempty core conditions in cooperative games

## Abstract

Balanced sets appeared in the 1960s in cooperative game theory as a part of nonempty core conditions. In this paper we present a classification of balanced families containing only 2-element subsets. We also discuss generalizations of the classical Sperner and Tucker lemmas using balanced sets.