Fair division with divisible and indivisible items

📝 Abstract

In the work the fair division problem for two participants in presence of both divisible and indivisible items is considered. The set of all divisions is formally described; it is demonstrated that fair (in terms of Brams and Taylor) divisions, unlikely the case where all the items are divisible, not always exist. The necessary and sufficient conditions of existence of proportional and equitable division were found.

💡 Analysis

In the work the fair division problem for two participants in presence of both divisible and indivisible items is considered. The set of all divisions is formally described; it is demonstrated that fair (in terms of Brams and Taylor) divisions, unlikely the case where all the items are divisible, not always exist. The necessary and sufficient conditions of existence of proportional and equitable division were found.

📄 Content

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FAIR DIVISION WITH DIVISIBLE
AND INDIVISIBLE ITEMS

Alexander Rubchinsky

State University – Higher School of Economics and International University “Dubna

Moscow
State University - Higher School of Economics
2009

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Rubchinsky A. Fair division with divisible and indivisible items: Working paper WP7/2009/05. – Moscow: State University - Higher School of Economics. - 25 р.

In the work the fair division problem for two participants in presence of both divisible and indivisible items is considered. The set of all divisions is formally described; it is demonstrated that fair (in terms of Brams and Taylor) divisions, unlikely the case where all the items are divisible, not always exist. The necessary and sufficient conditions of existence of proportional and equitable division were found. Three interrelated modifications of the notion of fair division – profitably, uniformly and equitably fair divisions – were introduced. Computationally efficient algorithm for finding all of them was designed. The algorithm includes repetitive solutions of integer knapsack-type problems as its essential steps. The statements of the article are illustrated by various examples.

Alexander Rubchinsky - State University – Higher School of Economics and International University “Dubna

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1. Introduction In the middle nineties two American scientists – Steven Brams and Alan Taylor – suggested a fresh approach to widespread conflict situations. In these situations conflict consists of a family of separate disputable items (partial conflicts), and conflict resolution can be reduced to agreements about each of them. The main idea can be roughly presented as follows. Participants have their own values of importance of different items that form together a conflict. Because in most cases these values do not coincide completely, it is possible to achieve conflict resolution, such that both participants receive more than 50% of satisfaction measured in their own values.

1.1. Examples The approach is carefully exposed in very comprehensive books [1, 2]. In order to clarify the approach two examples taken from [2] are considered here. Many other examples of real and hypothetical conflicts illustrating wide applicability of the approach can be found in the above mentioned books.
Example 1. Divorse arrangement. Ann and Ben are getting a divorce. The items that Bob and Carol had to divide were as follows: A retirement account (pension), which, though substantial, will remain untouchable for several years; they are valuable for both but especially for Ann because Ben has more chance to make new account before his retirement. A four-bedroom house, located close to Ben’s job; therefore Ben values this house higher than Ann. Country cottage that can be used at any season, preferable by Ann who intends to live there.
A portfolio of investments, which has lower monetary value than the pension but is all liquid assets. Other, consisting of two cars and relatively expensive yacht highly valued by Ben.
In more detail the situation is described in [2]. Valuations of Ann and Ben of all the considered items are given in Table 1.
Table 1 Item Ann Ben Retirement account
50 40 House 20 30 Cottage 15 10 Portfolio 10 10 Other 5 10 Total 100 100 Giving to everyone entire items, it is possible to suggest the following division: For Ann: retirement account + other = 50 + 5 = 55; For Ben: house + cottage + portfolio = 30 + 10 + 10 = 50. Thus, satisfaction with this division is not less than half for both participants. The exact notion of optimal or fair division, suggested by Brams and Taylor, will be considered further.
Example 2. Mergers. Disagreements between businesses are common, especially when companies merge or are acquired. If each company cares more about different parts of an agreement, complex arrangements need to be worked out to satisfy both sides. One of the most elusive ingredients in the success of a merger is what deal makers euphemistically refer to as social issues – how power, position, and status will be allocated among the merging companies’ executives. A failure to resolve these issues often leads to the destruction of shareholder wealth and the portrayal of top executives as petty corporate titans, unable to subordinate their selfish interests to the goal of promoting shareholder well-being. Social issues concern the more ineffable matters of status, role, and prestige in the merged company, as opposed to “hard” financial factors. Even if a merger is ultimately consummated, as in the case of Boeing and McDonnel Douglas, a failure to agree on the resolution of social issues quickly wastes 4

resources and the extremely valuable time of top corporate executives. The difficu

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