[Paper Review] Fair Cake-Cutting among Groups

Erel Segal-Halevi, Shmuel Nitzan|arXiv (Cornell University)|Oct 13, 2015

Game Theory and Voting Systems3 citations

TL;DR

This paper extends fair cake-cutting to groups of agents by defining group value through six aggregation methods—four cardinal welfare functions and two ordinal preference relations—demonstrating that the choice of group value function critically affects the existence and applicability of fair division protocols.

ABSTRACT

This paper extends the classic cake-cutting problem from individual agents to groups of agents. Applications include dividing a land-estate among families or dividing disputed lands among states. In the standard cake-cutting model, each *agent* should receive an individual subset of the cake with a sufficiently high individual In our model, each *group* should receive a subset with a sufficiently high value. Six ways to define the aggregate group value based on the values of the group members are examined: four based on cardinal welfare functions and two based on ordinal preference relations. Our results show that the choice of the group value function has crucial implications on the existence and applicability of fair division protocols.

Motivation & Objective

• To address fair division of resources among groups rather than individuals, motivated by real-world applications like land division among families or states.
• To formalize group value as an aggregation of individual members' valuations using cardinal and ordinal methods.
• To investigate how different aggregation functions affect the existence and practicality of fair division protocols.
• To provide a framework for group fair division that maintains individual fairness while ensuring group-level satisfaction.

Proposed method

• Proposes six ways to aggregate individual agent valuations into a group value: four cardinal welfare functions (utilitarian, egalitarian, Nash, and generalized utilitarian) and two ordinal preference-based functions.
• Models group fairness as requiring each group to receive a subset of the cake with value at least equal to its defined group value function.
• Analyzes the existence of fair protocols under each group value function, focusing on proportionality and envy-freeness in group settings.
• Uses mathematical analysis to compare the feasibility and properties of fair division protocols under different aggregation rules.
• Applies standard cake-cutting techniques (e.g., recursive division, query-based protocols) adapted to group valuations.
• Compares theoretical guarantees across aggregation methods to identify which functions allow for practical and fair solutions.

Experimental results

Research questions

• RQ1 How can group fairness be defined when agents are organized into groups rather than acting individually?
• RQ2 Which group value aggregation functions allow for the existence of fair division protocols in group cake-cutting?
• RQ3 How do cardinal welfare functions compare to ordinal preference-based functions in enabling fair and implementable protocols?
• RQ4 What are the theoretical and practical limitations of each group value function in group fair division?

Key findings

• The choice of group value function has crucial implications for the existence and applicability of fair division protocols.
• Cardinal welfare functions such as utilitarian and egalitarian yield different fairness guarantees and feasibility outcomes.
• Ordinal preference-based functions may simplify implementation but can conflict with individual fairness if not carefully designed.
• Some aggregation functions make it impossible to achieve proportionality or envy-freeness, even in principle.
• The generalized utilitarian function offers a balanced trade-off between fairness and efficiency across diverse group compositions.
• The study identifies that not all aggregation methods support the existence of fair protocols, highlighting a fundamental trade-off in group fair division design.