Redividing the Cake

Abstract

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.

Two's company, three's a crowd: Consensus-halving for a constant number of agents

“…For the case of n = 2, this problem coincides with ε-Consensus-Halving, and thus one can interpret our results for n = 2 as extensions of the results in [21] (which are only for additive valuations) to the case of monotone valuations (for which the problem is solvable with polynomially-many queries) and to the case of general valuations (for which the problem admits exponential query complexity lower bounds). Besides the aforementioned results, there is a plethora of works in computer science and artificial intelligence related to computational aspects of fair cake cutting and fair division in general, e.g., see [2,10,[14][15][16][17]23,27,35,45,49,52,62].…”

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

“…The cake serves as a metaphor for any divisible resource such as time or land, and our aim is to perform the division in a fair manner. This problem has a long and storied history that dates back over 70 years (Steinhaus, 1948;Dubins & Spanier, 1961;Stromquist, 1980;Brams & Taylor, 1995;Su, 1999;Stromquist, 2008;Edmonds & Pruhs, 2011;Aziz & Mackenzie, 2016) and has received considerable attention from artificial intelligence researchers in the past decade (Caragiannis et al, 2011;Bei et al, 2012;Aumann et al, 2013;Balkanski et al, 2014;Brânzei & Miltersen, 2015;Alijani et al, 2017;Menon & Larson, 2017;Bei et al, 2018;Segal-Halevi, 2018;Hosseini et al, 2020).…”

Counting Blanks in Polygonal Arrangements

“…Besides cutting cakes, an additional application of this question is for re-division of land [6]. There are some small lots on a piece of land.…”