2 research outputs found
The complexity of cake cutting with unequal shares
An unceasing problem of our prevailing society is the fair division of goods.
The problem of proportional cake cutting focuses on dividing a heterogeneous
and divisible resource, the cake, among players who value pieces according
to their own measure function. The goal is to assign each player a not
necessarily connected part of the cake that the player evaluates at least as
much as her proportional share.
In this paper, we investigate the problem of proportional division with
unequal shares, where each player is entitled to receive a predetermined
portion of the cake. Our main contribution is threefold. First we present a
protocol for integer demands that delivers a proportional solution in fewer
queries than all known algorithms. Then we show that our protocol is
asymptotically the fastest possible by giving a matching lower bound. Finally,
we turn to irrational demands and solve the proportional cake cutting problem
by reducing it to the same problem with integer demands only. All results
remain valid in a highly general cake cutting model, which can be of
independent interest
Fair and Square: Cake-Cutting in Two Dimensions
We consider the classic problem of fairly dividing a heterogeneous good
("cake") among several agents with different valuations. Classic cake-cutting
procedures either allocate each agent a collection of disconnected pieces, or
assume that the cake is a one-dimensional interval. In practice, however, the
two-dimensional shape of the allotted pieces is important. In particular, when
building a house or designing an advertisement in printed or electronic media,
squares are more usable than long and narrow rectangles. We thus introduce and
study the problem of fair two-dimensional division wherein the allotted pieces
must be of some restricted two-dimensional geometric shape(s), particularly
squares and fat rectangles. Adding such geometric constraints re-opens most
questions and challenges related to cake-cutting. Indeed, even the most
elementary fairness criterion --- proportionality --- can no longer be
guaranteed. In this paper we thus examine the level of proportionality that can
be guaranteed, providing both impossibility results and constructive division
procedures.Comment: Journal versio