145 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 ā 1 e 63.2% as the maximal size
increases. It is the best ratio of any Envy-Free assignment mechanism
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.
The most ordinally-egalitarian of random voting rules
Aziz and Stursberg propose an āEgalitarian Simultaneous Reservationā 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 āthe right wayā to think about generalizing Serial rule
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.
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
(GaleāEisenberg): 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
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
The egalitarian sharing rule in provision of public projects
In this note we consider a society that partitions itself into disjoint
jurisdictions, each choosing a location of its public project and a taxation
scheme to finance it. The set of public project is multi-dimensional, and their
costs could vary from jurisdiction to jurisdiction. We impose two principles,
egalitarianism, that requires the equalization of the total cost for all agents
in the same jurisdiction, and efficiency, that implies the minimization of the
aggregate total cost within jurisdiction. We show that these two principles
always yield a core-stable partition but a Nash stable partition may fail to
exist.Comment: 7 page
Random Matching 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 strategy-proof revelation mechanisms. Randomly selecting a priority ordering of the participants is a simple example. Equalizing as much as possible the probability of getting an acceptable mate across all participants stands out for its normative and incentives properties: the profile of probabilities is Lorenz dominant, and the revelation mechanism is group-strategy-proof for each side of the market. Our results apply to the random assignment problem as well.
- ā¦