14,591 research outputs found
Deterministic, Strategyproof, and Fair Cake Cutting
We study the classic cake cutting problem from a mechanism design
perspective, in particular focusing on deterministic mechanisms that are
strategyproof and fair. We begin by looking at mechanisms that are non-wasteful
and primarily show that for even the restricted class of piecewise constant
valuations there exists no direct-revelation mechanism that is strategyproof
and even approximately proportional. Subsequently, we remove the non-wasteful
constraint and show another impossibility result stating that there is no
strategyproof and approximately proportional direct-revelation mechanism that
outputs contiguous allocations, again, for even the restricted class of
piecewise constant valuations. In addition to the above results, we also
present some negative results when considering an approximate notion of
strategyproofness, show a connection between direct-revelation mechanisms and
mechanisms in the Robertson-Webb model when agents have piecewise constant
valuations, and finally also present a (minor) modification to the well-known
Even-Paz algorithm that has better incentive-compatible properties for the
cases when there are two or three agents.Comment: A shorter version of this paper will appear at IJCAI 201
Computing Socially-Efficient Cake Divisions
We consider a setting in which a single divisible good ("cake") needs to be
divided between n players, each with a possibly different valuation function
over pieces of the cake. For this setting, we address the problem of finding
divisions that maximize the social welfare, focusing on divisions where each
player needs to get one contiguous piece of the cake. We show that for both the
utilitarian and the egalitarian social welfare functions it is NP-hard to find
the optimal division. For the utilitarian welfare, we provide a constant factor
approximation algorithm, and prove that no FPTAS is possible unless P=NP. For
egalitarian welfare, we prove that it is NP-hard to approximate the optimum to
any factor smaller than 2. For the case where the number of players is small,
we provide an FPT (fixed parameter tractable) FPTAS for both the utilitarian
and the egalitarian welfare objectives
Monotonicity and Competitive Equilibrium in Cake-cutting
We study the monotonicity properties of solutions in the classic problem of
fair cake-cutting --- dividing a heterogeneous resource among agents with
different preferences. Resource- and population-monotonicity relate to
scenarios where the cake, or the number of participants who divide the cake,
changes. It is required that the utility of all participants change in the same
direction: either all of them are better-off (if there is more to share or
fewer to share among) or all are worse-off (if there is less to share or more
to share among).
We formally introduce these concepts to the cake-cutting problem and examine
whether they are satisfied by various common division rules. We prove that the
Nash-optimal rule, which maximizes the product of utilities, is
resource-monotonic and population-monotonic, in addition to being
Pareto-optimal, envy-free and satisfying a strong competitive-equilibrium
condition. Moreover, we prove that it is the only rule among a natural family
of welfare-maximizing rules that is both proportional and resource-monotonic.Comment: Revised versio
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