396 research outputs found
Size versus fairness in the assignment problem
When not all objects are acceptable to all agents, maximizing the number of objects
actually assigned is an important design concern. We compute the guaranteed size ratio
of the Probabilistic Serial mechanism, i.e., the worst ratio of the actual expected size to
the maximal feasible size. It converges decreasingly to 1 − 1 e 63.2% as the maximal size
increases. It is the best ratio of any Envy-Free assignment mechanism
Efficiency of Truthful and Symmetric Mechanisms in One-sided Matching
We study the efficiency (in terms of social welfare) of truthful and
symmetric mechanisms in one-sided matching problems with {\em dichotomous
preferences} and {\em normalized von Neumann-Morgenstern preferences}. We are
particularly interested in the well-known {\em Random Serial Dictatorship}
mechanism. For dichotomous preferences, we first show that truthful, symmetric
and optimal mechanisms exist if intractable mechanisms are allowed. We then
provide a connection to online bipartite matching. Using this connection, it is
possible to design truthful, symmetric and tractable mechanisms that extract
0.69 of the maximum social welfare, which works under assumption that agents
are not adversarial. Without this assumption, we show that Random Serial
Dictatorship always returns an assignment in which the expected social welfare
is at least a third of the maximum social welfare. For normalized von
Neumann-Morgenstern preferences, we show that Random Serial Dictatorship always
returns an assignment in which the expected social welfare is at least
\frac{1}{e}\frac{\nu(\opt)^2}{n}, where \nu(\opt) is the maximum social
welfare and is the number of both agents and items. On the hardness side,
we show that no truthful mechanism can achieve a social welfare better than
\frac{\nu(\opt)^2}{n}.Comment: 13 pages, 1 figur
Random Matching and assignment under dichotomous preferences
We consider bilateral matching problems where each person views those on the other side of the market as either acceptable or unacceptable: an acceptable mate is preferred to remaining single, and the latter to an unacceptable mate all acceptable mates are welfare-wise identical. Using randomization, many efficient and fair matching methods define strategyproof revelation mechanisms. Randomly selecting a priority ordering of the participants gives a simple example. Equalizing as much as possible the probability of getting an acceptable mate accross all participants stands out for its normative and incentives properties: the profile of probabilities is Lorenz dominant, and the revelation mechanism is groupstrategyproof for each side of the market. Our results apply to the random assignment problem as well.
Collective Choice under Dichotomous Preferences
Agents partition deterministic outcomes into good or bad. A direct revelation mechanism selects a lottery over outcomes - also interpreted as time-shares. Under such dichotomous preferences, the probability that the lottery outcome be a good one is a canonical utility representation. The utilitarian mechanism averages over all deterministic outcomes "approved" by the largest number of agents. It is efficient, strategy-proof and treats equally agents and outcomes. We reach the impossibility frontier if we also place the lower bound 1/n on each agent's utility, where n is the number of agents; or if this lower bound is the fraction of good outcomes to feasible outcomes. We conjecture that no ex-ante efficient and strategy-proof mechanism guarantees a strictly positive utility to all agents at all profiles, and prove a weaker version of this conjecture.
Dividing bads under additive utilities
We compare the Egalitarian rule (aka Egalitarian Equivalent) and the Competitive rule (aka Comeptitive Equilibrium with Equal Incomes) to divide bads (chores). They are both welfarist: the competitive disutility profile(s) are the critical points of their Nash product on the set of efficient feasible profiles. The C rule is Envy Free, Maskin Monotonic, and has better incentives properties than the E rule. But, unlike the E rule, it can be wildly multivalued, admits no selection continuous in the utility and endowment parameters, and is harder to compute. Thus in the division of bads, unlike that of goods, no rule normatively dominates the other
Competitive division of a mixed manna
A mixed manna contains goods (that everyone likes) and bads (that everyone dislikes),
as well as items that are goods to some agents, but bads or satiated to others.
If all items are goods and utility functions are homogeneous of degree 1 and concave
(and monotone), the competitive division maximizes the Nash product of utilities
(Gale–Eisenberg): hence it is welfarist (determined by the set of feasible utility profiles),
unique, continuous, and easy to compute.
We show that the competitive division of a mixed manna is still welfarist. If the zero
utility profile is Pareto dominated, the competitive profile is strictly positive and still
uniquely maximizes the product of utilities. If the zero profile is unfeasible (for instance,
if all items are bads), the competitive profiles are strictly negative and are the
critical points of the product of disutilities on the efficiency frontier. The latter allows
for multiple competitive utility profiles, from which no single-valued selection can be
continuous or resource monotonic.
Thus the implementation of competitive fairness under linear preferences in interactive
platforms like SPLIDDIT will be more difficult when the manna contains bads
that overwhelm the goods
The most ordinally-egalitarian of random voting rules
Aziz and Stursberg propose an “Egalitarian Simultaneous Reservation” rule (ESR), a generalization of Serial rule, one of the most discussed mechanisms in the random assignment problem, to the more general random social choice domain. This article provides an alternative definition, or characterization, of ESR as the unique most ordinally egalitarian one. Specifically, given a lottery p over alternatives, for each agent i the author considers the total probability share in p of objects from her first k indifference classes. ESR is shown to be the unique one which leximin maximizes the vector of all such shares (calculated for all i, k). Serial rule is known to be characterized by the same property. Thus, the author provides an alternative way to show that ESR, indeed, coincides with Serial rule on the assignment domain. Moreover, since both rules are defined as the unique most ordinally egalitarian ones, the result shows that ESR is “the right way” to think about generalizing Serial rule
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
Social Welfare in One-sided Matching Markets without Money
We study social welfare in one-sided matching markets where the goal is to
efficiently allocate n items to n agents that each have a complete, private
preference list and a unit demand over the items. Our focus is on allocation
mechanisms that do not involve any monetary payments. We consider two natural
measures of social welfare: the ordinal welfare factor which measures the
number of agents that are at least as happy as in some unknown, arbitrary
benchmark allocation, and the linear welfare factor which assumes an agent's
utility linearly decreases down his preference lists, and measures the total
utility to that achieved by an optimal allocation. We analyze two matching
mechanisms which have been extensively studied by economists. The first
mechanism is the random serial dictatorship (RSD) where agents are ordered in
accordance with a randomly chosen permutation, and are successively allocated
their best choice among the unallocated items. The second mechanism is the
probabilistic serial (PS) mechanism of Bogomolnaia and Moulin [8], which
computes a fractional allocation that can be expressed as a convex combination
of integral allocations. The welfare factor of a mechanism is the infimum over
all instances. For RSD, we show that the ordinal welfare factor is
asymptotically 1/2, while the linear welfare factor lies in the interval [.526,
2/3]. For PS, we show that the ordinal welfare factor is also 1/2 while the
linear welfare factor is roughly 2/3. To our knowledge, these results are the
first non-trivial performance guarantees for these natural mechanisms
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