136 research outputs found
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
Redividing the Cake
A heterogeneous resource, such as a land-estate, is already divided among
several agents in an unfair way. It should be re-divided among the agents in a
way that balances fairness with ownership rights. We present re-division
protocols that attain various trade-off points between fairness and ownership
rights, in various settings differing in the geometric constraints on the
allotments: (a) no geometric constraints; (b) connectivity --- the cake is a
one-dimensional interval and each piece must be a contiguous interval; (c)
rectangularity --- the cake is a two-dimensional rectangle or rectilinear
polygon and the pieces should be rectangles; (d) convexity --- the cake is a
two-dimensional convex polygon and the pieces should be convex.
Our re-division protocols have implications on another problem: the
price-of-fairness --- the loss of social welfare caused by fairness
requirements. Each protocol implies an upper bound on the price-of-fairness
with the respective geometric constraints.Comment: Extended IJCAI 2018 version. Previous name: "How to Re-Divide a Cake
Fairly
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 Incentive Guarantees Behind Nash Welfare in Divisible Resources Allocation
We study the problem of allocating divisible resources among agents,
hopefully in a fair and efficient manner. With the presence of strategic
agents, additional incentive guarantees are also necessary, and the problem of
designing fair and efficient mechanisms becomes much less tractable. While the
maximum Nash welfare (MNW) mechanism has been proven to be prominent by
providing desirable fairness and efficiency guarantees as well as other
intuitive properties, no incentive property is known for it.
We show a surprising result that, when agents have piecewise constant value
density functions, the incentive ratio of the MNW mechanism is for cake
cutting, where the incentive ratio of a mechanism is defined as the ratio
between the largest possible utility that an agent can gain by manipulation and
his utility in honest behavior. Remarkably, this result holds even without the
free disposal assumption, which is hard to get rid of in the design of truthful
cake cutting mechanisms. We also show that the MNW mechanism is group
strategyproof when agents have piecewise uniform value density functions.
Moreover, we show that, for cake cutting, the Partial Allocation (PA)
mechanism proposed by Cole et al., which is truthful and -MNW for
homogeneous divisible items, has an incentive ratio between
and when randomization is allowed, can be turned to be truthful in expectation.
Given two alternatives for a trade-off between incentive ratio and Nash welfare
provided by the MNW and PA mechanisms, we establish an interpolation between
them for both cake cutting and homogeneous divisible items.
Finally, we study the existence of fair mechanisms with a low incentive ratio
in the connected pieces setting. We show that any envy-free cake cutting
mechanism with the connected pieces constraint has an incentive ratio of at
least
Decentralized Pricing in Minimum Cost Spanning Trees
In the minimum cost spanning tree model we consider decentralized pricing rules, i.e. rules that cover at least the efficient cost while the price charged to each user only depends upon his own connection costs. We define a canonical pricing rule and provide two axiomatic characterizations. First, the canonical pricing rule is the smallest among those that improve upon the Stand Alone bound, and are either superadditive or piece-wise linear in connection costs. Our second, direct characterization relies on two simple properties highlighting the special role of the source cost.pricing rules; minimum cost spanning trees; canonical pricing rule; stand-alone cost; decentralization
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