Fair division

Fair division is a mathematical and economic concept that deals with dividing a set of resources or goods among individuals or parties in such a way that each participant believes they have received their fair share. This can involve tangible items, such as land or goods, as well as intangible resources, such as time or opportunities. The principles of fair division can be applied in various contexts, including: 1. **Dividing Chores or Tasks**: Splitting household responsibilities among family members or roommates.

Cake-cutting refers to a problem and methodology in fair division, particularly in the context of allocating resources among multiple parties. It is often illustrated with the analogy of dividing a cake (or any divisible good) among several individuals in a way that each person believes they have received a fair share. The main goals of cake-cutting are to ensure fairness and avoid conflicts during the division process.

Cake Theory, often referred to in the context of economics and social sciences, is a metaphor used to illustrate the complexities of resource distribution and allocation among different entities, such as individuals or groups. The concept can be exemplified through various scenarios where a "cake" represents a limited resource that is to be divided among parties with possibly conflicting interests.

Efficient cake-cutting is a concept from game theory and fair division that deals with the equitable and efficient distribution of resources among multiple parties, often illustrated through the metaphor of dividing a cake. The main objective is to divide the cake (or resource) in such a way that all parties involved feel they have received a fair share, while also maximizing the total value of the resource being divided. ### Key Concepts: 1. **Fairness**: The division should be perceived as fair by all participants.

Egalitarian cake-cutting refers to a method of dividing a resource (often represented metaphorically as a cake) among multiple parties in a way that aims to be fair and equitable for all participants. The goal is to ensure that each person feels they are receiving a fair share, with an emphasis on minimizing envy and maximizing perceived fairness. The concept typically involves procedures or protocols that allow participants to express their preferences and agree on how to divide the resource.

Envy-free cake-cutting is a concept from fair division and game theory, often used in contexts where multiple parties need to divide a resource (often referred to metaphorically as "cake") in a fair manner. The aim is to ensure that all parties involved feel satisfied with their portion and do not envy the portions received by others.

Equitable cake-cutting refers to a concept in fair division that deals with dividing resources in a way that ensures each participant feels they have received their fair share. The term "cake-cutting" is often used metaphorically to describe the division of a divisible resource (the "cake"), whether it's physical (like a real cake) or abstract (like time, money, or property).

Exact division refers to the process of dividing one number by another where the result is a whole number without any remainder.

Fair cake-cutting is a mathematical approach and framework for dividing a resource (often conceptualized as a "cake") among multiple parties in a way that is perceived as fair. The goal is to ensure that each participant feels they have received their fair share of the resource. This concept is often applied in economics, game theory, and conflict resolution, and is particularly relevant in scenarios where resources are limited and need to be allocated among competing individuals or groups.

An "individual pieces set" typically refers to a collection of items that are sold or packaged separately rather than as a single unit or complete set. This term can apply to various contexts, such as in board games, collectibles, furniture, or art. For example: 1. **Board Games**: Individual pieces from a game like chess where players might buy separate pieces to replace lost ones or customize their game.

The moving-knife procedure is a method used in economics, particularly in the context of fair division problems. It is a way to allocate resources or goods among individuals in a manner that is considered fair based on their preferences. The method is often applied in situations where indivisible goods are involved, or where parties have different valuations of the items being divided.

Piecewise-constant valuation refers to a method of valuing an asset or a function by defining its value over distinct intervals, where the value remains constant within each interval but may change at the boundaries. This approach is particularly useful in situations where a variable or asset behaves differently over different ranges or conditions.

Proportional cake-cutting refers to a method of fairly dividing a resource—often represented as a "cake"—among multiple parties (or "players"), such that each player receives a piece they consider to be at least a certain fraction of the total value of the cake. The aim is to ensure that each participant is satisfied with their share and feels that it is a fair division according to their own preferences.

In measure theory, the Radon–Nikodym theorem is a fundamental result that provides conditions under which one measure is absolutely continuous with respect to another. Specifically, it deals with the existence of a density function, known as the Radon–Nikodym derivative, that describes how one measure can be expressed in terms of another. A **Radon–Nikodym set** typically refers to a measurable set that is relevant in the context of the Radon–Nikodym theorem.

The Stromquist–Woodall theorem is a result in combinatorial mathematics, particularly in the area of combinatorial geometry. It deals with the relationship between sets of points in space, often focusing on properties such as convex hulls, visibility, and other geometric configurations. To be more specific, the theorem addresses the conditions under which a certain configuration of points in a finite-dimensional space can be partitioned or arranged in a particular way, and it often involves the properties of convex sets.

Super envy-freeness is a concept in the field of fair division, which is a branch of mathematics and economics focused on how to divide resources among several parties in a way that is considered fair. In traditional envy-freeness, a division of goods is considered envy-free if no individual prefers another individual's allocation over their own. In other words, each person feels satisfied with what they have, such that they do not envy anyone else's share.

Symmetric fair cake-cutting refers to a method of dividing a "cake" (or any divisible resource) in such a way that all participants perceive the division to be fair and equitable, ensuring symmetry in their allocations. The concept stems from the fields of economics and game theory, where fairness in resource allocation is crucial. In symmetric fair cake-cutting: 1. **Symmetry** means that if two participants start with the same preferences and information, they should receive identical portions of the cake.

Utilitarian cake-cutting refers to a method of dividing a resource (in this case, a cake) among multiple parties in a way that aims to maximize overall utility or satisfaction. The concept comes from the broader principles of utilitarianism, which emphasizes the greatest good for the greatest number. In cake-cutting scenarios, the goal is to allocate pieces of cake among individuals so that each person feels they have received a fair share, ideally maximizing their happiness or utility.

Fair division is a branch of mathematics and economics that explores how to divide a set of resources or goods among individuals in a way that is considered equitable or just. Researchers in this field study various methods and algorithms for achieving fair allocation, taking into account different criteria and preferences of the involved parties.

Anna Bogomolnaia is a mathematician known for her work in the fields of combinatorics and discrete mathematics. She has contributed to various topics, including game theory, matching theory, and algorithms. Her research often focuses on the mathematical foundations of problems related to optimization and decision-making.

Ariel D. Procaccia is a prominent researcher in the fields of computer science and artificial intelligence, particularly known for his work on algorithmic game theory, computational social choice, and auction design. He has made significant contributions to understanding how algorithms can be used to solve complex problems in social settings, such as voting and resource allocation. Procaccia has published extensively on topics such as fairness in algorithms, the mechanisms of decision-making processes, and the mathematical foundations of social choice theory.

Edith Elkind is a prominent computer scientist known for her work in artificial intelligence, particularly in the areas of multi-agent systems, computational social choice, and algorithms. Her research often involves topics such as game theory, social choice theory, and the interaction of algorithms in social contexts. Elkind has contributed significantly to the understanding of how computational methods can be applied to problems in economics and social science.

Edwin Spanier is known primarily for his contributions to the field of mathematics, particularly in the areas of topology and functional analysis. He authored several influential texts and research papers throughout his career, helping to advance mathematical understanding in his areas of expertise. One of his notable works is "Algebraic Topology," which is used in many academic curricula. Additionally, Spanier has been recognized for his teaching and influence in the mathematical community.

Francis Su is a prominent mathematician known for his work in the fields of mathematical economics, applied mathematics, and education. He is a professor of mathematics at Harvey Mudd College, where he has been involved in various mathematical research and educational initiatives. Su is particularly recognized for his contributions to the study of fair division, game theory, and the mathematics of voting.

Hal Varian is an American economist known for his work in microeconomics, information economics, and the economics of technology. He is particularly recognized for his role as the Chief Economist at Google and for his contributions to the field of economics through his research and teaching. Varian has written several influential textbooks, one of the most notable being "Intermediate Microeconomics: A Modern Approach," which is widely used in economics courses.

Lester Dubins is a notable figure in the field of mathematics, particularly known for his work in probability theory, statistics, and related areas. He has contributed to various topics, including the theory of random processes, statistical inference, and combinatorial problems. Dubins is also known for the "Dubins' problem," which deals with the optimal strategies in certain stochastic models.

Ted Hill is an American mathematician known for his work in various areas of mathematics, including probability theory and combinatorics. He is particularly recognized for his contributions to the field of mathematical education and for his research on random processes and combinatorial structures. Hill's work has involved exploring the mathematical underpinnings of randomness and is often associated with concepts in both pure and applied mathematics. He has also been active in discussing the philosophy of mathematics and the pedagogical aspects of teaching mathematics.

Fair item allocation refers to the process of distributing goods or resources among multiple agents or participants in a way that is considered fair according to certain criteria or principles. This concept is often discussed in fields like economics, game theory, and computational social choice. The notion of "fairness" can be interpreted in various ways depending on the context and the specific fairness criteria applied.

The "17-animal inheritance puzzle" is a classic genetic puzzle that involves determining the inheritance patterns of certain traits in a group of animals, often used as an educational tool in genetics or introductory biology courses. The puzzle typically outlines a scenario where a certain trait is passed down through generations of animals, and participants must use information about the traits of the parents and offspring to deduce which animals carry specific traits.

Course allocation refers to the process of assigning students to specific courses or classes within an educational institution. This process can vary widely depending on the institution, educational level, and specific guidelines or policies in place. Here are some key aspects of course allocation: 1. **Student Enrollment**: Course allocation often begins with student enrollment, where students express their preferences for courses based on their educational goals, major requirements, or personal interests.

Efficient Approximately Fair Item Allocation is a concept from the field of resource allocation, particularly in economics and computer science. The idea revolves around the fair distribution of items among multiple participants in a way that is both efficient and approximately fair. ### Key Concepts: 1. **Efficiency**: An allocation is considered efficient if there are no other possible distributions of items that would make at least one participant better off without making someone else worse off.

Egalitarian item allocation is a principle or framework in the field of resource distribution that aims to ensure fairness and equality among individuals or groups when allocating items or resources. The primary goal of egalitarian allocation is to minimize disparities in the distribution of goods or services, thereby promoting an equitable outcome for all participants involved.

Envy-free item allocation is a concept in resource allocation and fair division, particularly in economics and game theory, where the goal is to distribute a set of items (or resources) among a group of individuals in such a way that no individual prefers the items allocated to someone else over their own. In other words, each participant feels satisfied with what they received and does not desire what others have, leading to a perception of fairness.

Envy-free matching is a concept often discussed in the context of fair allocation and matching theory, particularly in economics and game theory. It describes a situation where a set of agents or participants is matched to a set of items (or other agents) in such a way that no agent prefers another agent's allocation over their own. To break it down: 1. **Agents**: These are the participants who have preferences for various items or other participants.

Fair allocation of items and money refers to the process of distributing resources, goods, or funds among a group of individuals in a manner that is perceived to be just, equitable, and appropriate based on certain criteria or principles. Fair allocation aims to ensure that everyone involved receives a share that reflects their needs, contributions, or rights. This concept can be applied in various contexts, including economics, ethics, and decision-making in collective settings.

Fair random assignment is a method used to allocate resources, opportunities, or treatments to individuals or groups in a manner that is both equitable and unbiased. This approach is often utilized in various fields, including education, experimental research, job placement, and social programs. The goal is to ensure that each participant has an equal chance of receiving any particular outcome, thereby minimizing any potential biases that could arise from the selection process.

Proportional item allocation is a method used to distribute resources, items, or benefits among different recipients or categories in a way that ensures each recipient receives a quantity that is proportional to a specific criterion or variable. This approach is commonly used in various fields, including economics, finance, project management, and resource distribution. ### Key Features: 1. **Proportionality**: Recipients receive items based on their relative share or importance.

Fairness criteria refer to a set of standards or principles used to evaluate and ensure equitable treatment and outcomes in various contexts, particularly in areas such as machine learning, data science, public policy, and social justice. The goal of applying fairness criteria is to mitigate bias, promote equity, and ensure that decisions and predictions are just and do not discriminate against any particular group based on characteristics such as race, gender, age, socioeconomic status, or other attributes.

Apportionment is the process of distributing a fixed resource, such as seats in a legislature or representatives, to different groups based on specific criteria. The criteria for apportionment methods can vary, but some key principles generally guide these methods: 1. **Fairness**: The apportionment method should be fair, ensuring that each group receives a number of representatives that reflect its size relative to other groups. The goal is to represent populations accurately.

Coherence, in the context of fairness, generally refers to the consistency and logical alignment of judgments, policies, or actions regarding fairness across different situations or individuals. In areas such as ethics, law, and machine learning, coherence involves ensuring that similar situations yield similar judgments or that rules applied in one context are also applicable in another, without bias or contradiction.

Egalitarian equivalence refers to a concept in economics and social theory that seeks to establish a fair distribution of resources, opportunities, or outcomes among individuals or groups, with particular emphasis on equality. The term is often associated with theories of equity and fairness, highlighting the importance of treating individuals equally or ensuring that any disparities in treatment are justified and reasonable.

Egalitarianism is a philosophical and political principle that advocates for equality among all people, particularly in terms of social, political, and economic rights and opportunities. The central idea is that all individuals should be treated as equals and have equal access to resources and opportunities, regardless of factors such as race, gender, socioeconomic status, or other characteristics.

Envy-freeness is a concept from the field of fair division and resource allocation, primarily used in economics and game theory. It refers to a condition in which each participant in a division of goods (e.g., resources, assets, or cake) prefers their own share over the shares allocated to others. In simpler terms, a division is envy-free if no individual envies another individual's portion.

Equitability generally refers to the quality of being fair and impartial. In various contexts, it emphasizes the importance of justice, fairness, and equal treatment, often within social, economic, and legal frameworks. The concept can be applied in different fields, including economics, education, and social justice.

In economics, equity refers to the concept of fairness or justice in the distribution of resources, opportunities, and benefits within a society. It often contrasts with equality, which emphasizes uniform distribution regardless of individual circumstances. Equity recognizes that different individuals or groups may require different levels of support or resources to achieve fair outcomes due to their varying needs, backgrounds, and barriers.

Group envy-freeness is a concept that arises in the context of fair division, particularly in resource allocation among multiple agents or participants. It is an extension of the individual envy-freeness concept. **Individual Envy-freeness:** An allocation is said to be envy-free for an individual if no participant prefers another participant's share over their own. Simply put, each person believes they have received at least as good a share as anyone else in the group.

Leximin order is a method of ordering or comparing multi-dimensional vectors based on a lexicographic criterion, which considers individual elements of the vectors in a specific sequence. Specifically, the term "leximin" combines elements of two concepts: "lexicographic order" and "minimum." In the context of multi-dimensional vectors, the leximin order prioritizes the worst (minimum) element of each vector.

Population monotonicity is a concept in social choice theory and mechanism design that pertains to the allocation of resources or goods among a group of individuals as the population of individuals changes. Specifically, an allocation method is said to be population monotonic if the allocation to any existing individuals does not decrease when the population increases, assuming that the preferences and resources available remain constant.

Proportional division is a method of dividing something, such as resources, assets, or benefits, among different parties in a way that reflects their respective shares or entitlements. This concept is often applied in various contexts, including finance, law, business, and even social or political settings.

Resource monotonicity is a concept primarily used in the field of computational economics and mechanism design, particularly in the context of allocating resources efficiently among agents with varying preferences or types. The basic idea behind resource monotonicity is that if the total amount of resources available to be allocated increases, then the allocation to each agent should not decrease. In other words, if the total resources are augmented, the allocation strategy should ensure that no individual agent receives less than they would have with fewer resources.

The Dubins-Spanier theorems are results in the theory of stochastic processes, particularly concerning the behavior of certain classes of random walks, especially in relation to the concepts of ergodicity and recurrence. ### Key Concepts: 1. **Random Walks**: A type of stochastic process that describes a path consisting of a succession of random steps. Random walks can be one-dimensional or multidimensional and are foundational in probability theory.

Envy-free pricing is a concept that originates from the field of fair division, primarily used in economics and game theory. It refers to a pricing mechanism that ensures that no individual prefers the bundle of goods or services allocated to another individual over their own, based on their valuation. In other words, a pricing scheme is considered envy-free if every participant feels satisfied with their allocation and has no desire to exchange their allocation for someone else's.

Fair division among groups refers to the principles and methods used to allocate resources or assets among multiple parties in a way that is perceived as fair and equitable. This concept is particularly important in situations where resources are limited, and parties have different preferences or claims to those resources. Key aspects of fair division include: 1. **Equitability**: Ensuring that each party feels they have received a fair share relative to what others have.

Fair division experiments are studies or exercises aimed at understanding how resources can be divided among individuals or groups in a way that is perceived as fair by all parties involved. The concept of fair division is important in various fields, including economics, game theory, mathematics, and social sciences, and it addresses issues of equity, efficiency, and conflict resolution. Key aspects of fair division experiments include: 1. **Mathematical Foundation**: Fair division often employs mathematical methods and algorithms to create fair outcomes.

Fair division of a single homogeneous resource refers to the process of allocating a divisible and uniform resource—such as land, money, or goods—among multiple recipients in a way that is perceived as fair by all involved parties. The goal is to ensure that each participant receives a share that is equitable based on certain criteria or preferences.

Fair river sharing refers to the equitable distribution and management of water resources from rivers among different users, stakeholders, or regions. It encompasses legal, social, and technical measures to ensure that all parties—such as agriculture, industry, municipalities, and ecosystems—receive a fair allocation of water based on their needs, rights, and contributions to sustainability.

Free disposal is an economic concept that refers to the ability to dispose of goods or resources without incurring any costs, restrictions, or penalties. In practical terms, it means that individuals or firms can freely discard unwanted or surplus items without having to pay disposal fees or withstand legal or regulatory barriers. In terms of production and resources, free disposal is often assumed in economic models that analyze supply and demand.

Fair division is a field in mathematics and economics that deals with dividing goods or resources among individuals in a way that is considered fair. There are several open questions and unsolved problems in this area, which researchers continue to explore.

Map segmentation is a process used in geographic information systems (GIS), image processing, and various fields of computer vision to divide a map or an image into distinct regions or segments based on specific criteria. The goal of map segmentation is to facilitate analysis, interpretation, and understanding of spatial data by reducing complexity and enhancing relevant features.

No-justified-envy matching is a concept from the field of economics and game theory that deals with matching markets, such as job markets or school assignments, where individuals (such as workers or students) are matched to positions (such as jobs or schools) based on preferences and some form of evaluation or ranking. The idea of "no-justified-envy" refers to a condition where an individual cannot justify their envy towards another individual's match.

Online fair division refers to the problem of allocating resources or dividing goods among agents in a dynamic environment where the agents arrive and make requests over time. In contrast to traditional fair division, where all agents and items are present from the beginning, online fair division must consider situations where agents show up sequentially, and decisions need to be made without the knowledge of future arrivals or requests.

The "Pie Rule" is a general term that can refer to various concepts depending on the context. In the context of mathematics or optimization, it might refer to methods of equitable distribution or sharing resources, akin to slicing a pie into equal pieces. In other contexts, it could refer to behavioral rules regarding sharing or fairness, such as those seen in negotiations or game theory.

The "price of fairness" is a concept derived from economics and game theory that refers to the potential costs that individuals or groups incur when they prioritize fairness or equity in decision-making processes over their own self-interest or the most efficient outcomes. In various scenarios, particularly in negotiations, business settings, or resource allocation, the pursuit of fairness can lead to suboptimal results or inefficiencies.

The "Problem of the Nile" typically refers to the historical and ongoing disputes over the management and use of the waters of the Nile River, particularly among the countries that rely on it for their water supply. The Nile is one of the longest rivers in the world and flows through multiple countries, including Uganda, Sudan, and Egypt.