3,108 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
Cake Division with Minimal Cuts: Envy-Free Procedures for 3 Person, 4 Persons, and Beyond
The minimal number of parallel cuts required to divide a cake into n pieces is n-1. A new 3-person procedure, requiring 2 parallel cuts, is given that produces an envy- free division, whereby each person thinks he or she receives at least a tied- for- largest piece. An extension of this procedure leads to a 4-person division, us ing 3 parallel cuts, that makes at most one player envious. Finally, a 4-person envy-free procedure is given, but it requires up to 5 parallel cuts, and some pieces may be disconnected. All these procedures improve on extant procedures by using fewer moving knives, making fewer people envious, or using fewer cuts. While the 4-person, 5-cut procedure is complex, endowing people with more information about others' preferences, or allowing them to do things beyond stopping moving knives, may yield simpler procedures for making envy- free divisions with minimal cuts, which are known always to existFAIR DIVISION; CAKE CUTTING; ENVY-FREENESS; MAXIMIN
The Cost of Sybils, Credible Commitments, and False-Name Proof Mechanisms
Consider a mechanism that cannot observe how many players there are directly,
but instead must rely on their self-reports to know how many are participating.
Suppose the players can create new identities to report to the auctioneer at
some cost . The usual mechanism design paradigm is equivalent to implicitly
assuming that is infinity for all players, while the usual Sybil attacks
literature is that it is zero or finite for one player (the attacker) and
infinity for everyone else (the 'honest' players). The false-name proof
literature largely assumes the cost to be 0. We consider a model with variable
costs that unifies these disparate streams.
A paradigmatic normal form game can be extended into a Sybil game by having
the action space by the product of the feasible set of identities to create
action where each player chooses how many players to present as in the game and
their actions in the original normal form game. A mechanism is (dominant)
false-name proof if it is (dominant) incentive-compatible for all the players
to self-report as at most one identity. We study mechanisms proposed in the
literature motivated by settings where anonymity and self-identification are
the norms, and show conditions under which they are not Sybil-proof. We
characterize a class of dominant Sybil-proof mechanisms for reward sharing and
show that they achieve the efficiency upper bound. We consider the extension
when agents can credibly commit to the strategy of their sybils and show how
this can break mechanisms that would otherwise be false-name proof
Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations
Cake cutting is one of the most fundamental settings in fair division and
mechanism design without money. In this paper, we consider different levels of
three fundamental goals in cake cutting: fairness, Pareto optimality, and
strategyproofness. In particular, we present robust versions of envy-freeness
and proportionality that are not only stronger than their standard
counter-parts but also have less information requirements. We then focus on
cake cutting with piecewise constant valuations and present three desirable
algorithms: CCEA (Controlled Cake Eating Algorithm), MEA (Market Equilibrium
Algorithm) and CSD (Constrained Serial Dictatorship). CCEA is polynomial-time,
robust envy-free, and non-wasteful. It relies on parametric network flows and
recent generalizations of the probabilistic serial algorithm. For the subdomain
of piecewise uniform valuations, we show that it is also group-strategyproof.
Then, we show that there exists an algorithm (MEA) that is polynomial-time,
envy-free, proportional, and Pareto optimal. MEA is based on computing a
market-based equilibrium via a convex program and relies on the results of
Reijnierse and Potters [24] and Devanur et al. [15]. Moreover, we show that MEA
and CCEA are equivalent to mechanism 1 of Chen et. al. [12] for piecewise
uniform valuations. We then present an algorithm CSD and a way to implement it
via randomization that satisfies strategyproofness in expectation, robust
proportionality, and unanimity for piecewise constant valuations. For the case
of two agents, it is robust envy-free, robust proportional, strategyproof, and
polynomial-time. Many of our results extend to more general settings in cake
cutting that allow for variable claims and initial endowments. We also show a
few impossibility results to complement our algorithms.Comment: 39 page
Counterexamples in the theory of fair division
The formal mathematical theory of fair division has a rich history dating
back at least to Steinhaus in the 1940's. In recent work in this area, several
general classes of errors have appeared along with confusion about the
necessity and sufficiency of certain hypotheses. It is the purpose of this
article to correct the scientific record and to point out with concrete
examples some of the pitfalls that have led to these mistakes. These examples
may serve as guideposts for future work.Comment: Available at http://digitalcommons.calpoly.edu/rgp_rsr/73
Communication Complexity of Cake Cutting
We study classic cake-cutting problems, but in discrete models rather than
using infinite-precision real values, specifically, focusing on their
communication complexity. Using general discrete simulations of classical
infinite-precision protocols (Robertson-Webb and moving-knife), we roughly
partition the various fair-allocation problems into 3 classes: "easy" (constant
number of rounds of logarithmic many bits), "medium" (poly-logarithmic total
communication), and "hard". Our main technical result concerns two of the
"medium" problems (perfect allocation for 2 players and equitable allocation
for any number of players) which we prove are not in the "easy" class. Our main
open problem is to separate the "hard" from the "medium" classes.Comment: Added efficient communication protocol for the monotone crossing
proble
Public Choice and Altruism
The public choice literature has paid little attention to altruism, and the few works that do deal with it usually focus on the tradeoff between selfish and unselfish preferences, assuming some shared set of unselfish preferences. This focus leaves the question open as to whether unselfish but conflicting beliefs can be the source of public choice problems. This paper examines conflicting ethical beliefs among purely altruistic individuals to show that many of the problems that appear to go away if people are altruistic (assuming notions of the public interest are shared) return if notions of the public interest conflict no matter how altruistic people may be.Altruism
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