Fair Division of Goods with Market Values

Inheritances, divorces or liquidations of companies require that a common asset is divided among the entitled parties. Legal methods usually consider the market value of goods, while fair division procedures take into account the parties' preferences expressed as cardinal utilities. We combine the two practices to define two procedures that optimally allocate goods with market values to people with preferences.Comment: 29 pages, 1 figure, 17 table

The Strong Maximum Circulation Algorithm: A New Method for Aggregating Preference Rankings

We present a new optimization-based method for aggregating preferences in settings where each decision maker, or voter, expresses preferences over pairs of alternatives. The challenge is to come up with a ranking that agrees as much as possible with the votes cast in cases when some of the votes conflict. Only a collection of votes that contains no cycles is non-conflicting and can induce a partial order over alternatives. Our approach is motivated by the observation that a collection of votes that form a cycle can be treated as ties. The method is then to remove unions of cycles of votes, or circulations, from the vote graph and determine aggregate preferences from the remainder. We introduce the strong maximum circulation which is formed by a union of cycles, the removal of which guarantees a unique outcome in terms of the induced partial order. Furthermore, it contains all the aggregate preferences remaining following the elimination of any maximum circulation. In contrast, the well-known, optimization-based, Kemeny method has non-unique output and can return multiple, conflicting rankings for the same input. In addition, Kemeny's method requires solving an NP-hard problem, whereas our algorithm is efficient, based on network flow techniques, and runs in strongly polynomial time, independent of the number of votes. We address the construction of a ranking from the partial order and show that rankings based on a convex relaxation of Kemeny's model are consistent with our partial order. We then study the properties of removing a maximal circulation versus a maximum circulation and establish that, while maximal circulations will in general identify a larger number of aggregate preferences, the partial orders induced by the removal of different maximal circulations are not unique and may be conflicting. Moreover, finding a minimum maximal circulation is an NP-hard problem.Comment: 22 pages, 4 figure

Redividing the Cake

A heterogeneous resource, such as a land-estate, is already divided among several agents in an unfair way. It should be re-divided among the agents in a way that balances fairness with ownership rights. We present re-division protocols that attain various trade-off points between fairness and ownership rights, in various settings differing in the geometric constraints on the allotments: (a) no geometric constraints; (b) connectivity --- the cake is a one-dimensional interval and each piece must be a contiguous interval; (c) rectangularity --- the cake is a two-dimensional rectangle or rectilinear polygon and the pieces should be rectangles; (d) convexity --- the cake is a two-dimensional convex polygon and the pieces should be convex. Our re-division protocols have implications on another problem: the price-of-fairness --- the loss of social welfare caused by fairness requirements. Each protocol implies an upper bound on the price-of-fairness with the respective geometric constraints.Comment: Extended IJCAI 2018 version. Previous name: "How to Re-Divide a Cake Fairly