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Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of inheritance, partnership dissolutions, divorce settlements, electronic frequency allocation, airport traffic management, and exploitation of Earth observation satellites. It is an active research area in mathematics, economics, dispute resolution, etc. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a mediator but certainly not an arbiter as only the players really know how they value the goods.
An envy-free cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation.
Divide and choose is a procedure for fair division of a continuous resource, such as a cake, between two parties. It involves a heterogeneous good or resource and two partners who have different preferences over parts of the cake. The protocol proceeds as follows: one person cuts the cake into two pieces; the other person selects one of the pieces; the cutter receives the remaining piece.
Entitlement in fair division describes that proportion of the resources or goods to be divided that a player can expect to receive. In many fair division settings, all agents have equal entitlements, which means that each agent is entitled to 1/n of the resource. But there are practical settings in which agents have different entitlements. Some examples are:
Fair cake-cutting is a kind of fair division problem. The problem involves a heterogeneous resource, such as a cake with different toppings, that is assumed to be divisible – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be unanimously fair - each person should receive a piece that he or she believes to be a fair share.
In the theory of fair division, the price of fairness (POF) is the ratio of the largest economic welfare attainable by a division to the economic welfare attained by a fair division. The POF is a quantitative measure of the loss of welfare that society has to take in order to guarantee fairness.
Equitable (EQ) cake-cutting is a kind of a fair cake-cutting problem, in which the fairness criterion is equitability. It is a cake-allocation in which the subjective value of all partners is the same, i.e., each partner is equally happy with his/her share. Mathematically, that means that for all partners i and j:
Fair item allocation is a kind of a fair division problem in which the items to divide are discrete rather than continuous. The items have to be divided among several partners who value them differently, and each item has to be given as a whole to a single person. This situation arises in various real-life scenarios:
Envy-freeness, also known as no-envy, is a criterion for fair division. It says that, when resources are allocated among people with equal rights, each person should receive a share that is, in their eyes, at least as good as the share received by any other agent. In other words, no person should feel envy.
A proportional cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the proportionality criterion, namely, that every partner feels that his allocated share is worth at least 1/n of the total.
Utilitarian cake-cutting is a rule for dividing a heterogeneous resource, such as a cake or a land-estate, among several partners with different cardinal utility functions, such that the sum of the utilities of the partners is as large as possible. It is a special case of the utilitarian social choice rule. Utilitarian cake-cutting is often not "fair"; hence, utilitarianism is often in conflict with fair cake-cutting.
Envy-free (EF) item allocation is a fair item allocation problem, in which the fairness criterion is envy-freeness - each agent should receive a bundle that they believe to be at least as good as the bundle of any other agent.
Strategic fair division is the branch of fair division in which the participants are assumed to hide their preferences and act strategically in order to maximize their own utility, rather than playing sincerely according to their true preferences.
Truthful cake-cutting is the study of algorithms for fair cake-cutting that are also truthful mechanisms, i.e., they incentivize the participants to reveal their true valuations to the various parts of the cake.
Truthful resource allocation is the problem of allocating resources among agents with different valuations over the resources, such that agents are incentivized to reveal their true valuations over the resources.
Envy-free pricing is a kind of fair item allocation. There is a single seller that owns some items, and a set of buyers who are interested in these items. The buyers have different valuations to the items, and they have a quasilinear utility function; this means that the utility an agent gains from a bundle of items equals the agent's value for the bundle minus the total price of items in the bundle. The seller should determine a price for each item, and sell the items to some of the buyers, such that there is no envy. Two kinds of envy are considered:
In computer science, the Robertson–Webb (RW) query model is a model of computation used by algorithms for the problem of fair cake-cutting. In this problem, there is a resource called a "cake", and several agents with different value measures on the cake. The goal is to divide the cake among the agents such that each agent will consider his/her piece as "fair" by his/her personal value measure. Since the agents' valuations can be very complex, they cannot - in general - be given as inputs to a fair division algorithm. The RW model specifies two kinds of queries that a fair division algorithm may ask the agents: Eval and Cut. Informally, an Eval query asks an agent to specify his/her value to a given piece of the cake, and a Cut query asks an agent to specify a piece of cake with a given value.
Online fair division is a class of fair division problems in which the resources, or the people to whom they should be allocated, or both, are not all available when the allocation decision is made. Some situations in which not all resources are available include:
Fair division among groups is a class of fair division problems, in which the resources are allocated among groups of agents, rather than among individual agents. After the division, all members in each group consume the same share, but they may have different preferences; therefore, different members in the same group might disagree on whether the allocation is fair or not. Some examples of group fair division settings are:
References
- ↑ Chen, Yiling; Lai, John K.; Parkes, David C.; Procaccia, Ariel D. (2013-01-01). "Truth, justice, and cake cutting". Games and Economic Behavior. 77 (1): 284–297. doi:10.1016/j.geb.2012.10.009. ISSN 0899-8256.
- ↑ Bei, Xiaohui; Huzhang, Guangda; Suksompong, Warut (2020). "Truthful fair division without free disposal". Social Choice and Welfare. 55 (3): 523–545. arXiv: 1804.06923 . doi:10.1007/s00355-020-01256-0. PMC 7497335 . PMID 33005068.
- ↑ Segal-Halevi, Erel; Hassidim, Avinatan; Aumann, Yonatan (2016). "Waste Makes Haste". ACM Transactions on Algorithms. 13: 1–32. arXiv: 1511.02599 . doi:10.1145/2988232. S2CID 11358086.
- ↑ Aziz, Haris; MacKenzie, Simon (2016). "A discrete and bounded envy-free cake cutting protocol for any number of agents". FOCS 2016. arXiv: 1604.03655 . Bibcode:2016arXiv160403655A.
- ↑ Arzi, Orit; Aumann, Yonatan; Dombb, Yair (2016-04-01). "Toss one's cake, and eat it too: partial divisions can improve social welfare in cake cutting". Social Choice and Welfare. 46 (4): 933–954. doi:10.1007/s00355-015-0943-y. ISSN 1432-217X. S2CID 29132296.
- ↑ Feldman, Jon; Korula, Nitish; Mirrokni, Vahab; Muthukrishnan, S.; Pál, Martin (2009). Leonardi, Stefano (ed.). "Online Ad Assignment with Free Disposal". Internet and Network Economics. Lecture Notes in Computer Science. Springer Berlin Heidelberg. 5929: 374–385. doi:10.1007/978-3-642-10841-9_34. ISBN 978-3-642-10841-9.