121 research outputs found
Cost Sharing over Combinatorial Domains: Complement-Free Cost Functions and Beyond
We study mechanism design for combinatorial cost sharing models. Imagine that multiple items or services are available to be shared among a set of interested agents. The outcome of a mechanism in this setting consists of an assignment, determining for each item the set of players who are granted service, together with respective payments. Although there are several works studying specialized versions of such problems, there has been almost no progress for general combinatorial cost sharing domains until recently [S. Dobzinski and S. Ovadia, 2017]. Still, many questions about the interplay between strategyproofness, cost recovery and economic efficiency remain unanswered.
The main goal of our work is to further understand this interplay in terms of budget balance and social cost approximation. Towards this, we provide a refinement of cross-monotonicity (which we term trace-monotonicity) that is applicable to iterative mechanisms. The trace here refers to the order in which players become finalized. On top of this, we also provide two parameterizations (complementary to a certain extent) of cost functions which capture the behavior of their average cost-shares. Based on our trace-monotonicity property, we design a scheme of ascending cost sharing mechanisms which is applicable to the combinatorial cost sharing setting with symmetric submodular valuations. Using our first cost function parameterization, we identify conditions under which our mechanism is weakly group-strategyproof, O(1)-budget-balanced and O(H_n)-approximate with respect to the social cost. Further, we show that our mechanism is budget-balanced and H_n-approximate if both the valuations and the cost functions are symmetric submodular; given existing impossibility results, this is best possible. Finally, we consider general valuation functions and exploit our second parameterization to derive a more fine-grained analysis of the Sequential Mechanism introduced by Moulin. This mechanism is budget balanced by construction, but in general only guarantees a poor social cost approximation of n. We identify conditions under which the mechanism achieves improved social cost approximation guarantees. In particular, we derive improved mechanisms for fundamental cost sharing problems, including Vertex Cover and Set Cover
On the Inefficiency of the Uniform Price Auction
We present our results on Uniform Price Auctions, one of the standard
sealed-bid multi-unit auction formats, for selling multiple identical units of
a single good to multi-demand bidders. Contrary to the truthful and
economically efficient multi-unit Vickrey auction, the Uniform Price Auction
encourages strategic bidding and is socially inefficient in general. The
uniform pricing rule is, however, widely popular by its appeal to the natural
anticipation, that identical items should be identically priced. In this work
we study equilibria of the Uniform Price Auction for bidders with (symmetric)
submodular valuation functions, over the number of units that they win. We
investigate pure Nash equilibria of the auction in undominated strategies; we
produce a characterization of these equilibria that allows us to prove that a
fraction 1-1/e of the optimum social welfare is always recovered in undominated
pure Nash equilibrium -- and this bound is essentially tight. Subsequently, we
study the auction under the incomplete information setting and prove a bound of
4-2/k on the economic inefficiency of (mixed) Bayes Nash equilibria that are
supported by undominated strategies.Comment: Additions and Improvements upon SAGT 2012 results (and minor
corrections on the previous version
Paradoxes in Social Networks with Multiple Products
Recently, we introduced in arXiv:1105.2434 a model for product adoption in
social networks with multiple products, where the agents, influenced by their
neighbours, can adopt one out of several alternatives. We identify and analyze
here four types of paradoxes that can arise in these networks. To this end, we
use social network games that we recently introduced in arxiv:1202.2209. These
paradoxes shed light on possible inefficiencies arising when one modifies the
sets of products available to the agents forming a social network. One of the
paradoxes corresponds to the well-known Braess paradox in congestion games and
shows that by adding more choices to a node, the network may end up in a
situation that is worse for everybody. We exhibit a dual version of this, where
removing available choices from someone can eventually make everybody better
off. The other paradoxes that we identify show that by adding or removing a
product from the choice set of some node may lead to permanent instability.
Finally, we also identify conditions under which some of these paradoxes cannot
arise.Comment: 22 page
Undominated Groves Mechanisms
The family of Groves mechanisms, which includes the well-known VCG mechanism
(also known as the Clarke mechanism), is a family of efficient and
strategy-proof mechanisms. Unfortunately, the Groves mechanisms are generally
not budget balanced. That is, under such mechanisms, payments may flow into or
out of the system of the agents, resulting in deficits or reduced utilities for
the agents. We consider the following problem: within the family of Groves
mechanisms, we want to identify mechanisms that give the agents the highest
utilities, under the constraint that these mechanisms must never incur
deficits.
We adopt a prior-free approach. We introduce two general measures for
comparing mechanisms in prior-free settings. We say that a non-deficit Groves
mechanism {\em individually dominates} another non-deficit Groves mechanism
if for every type profile, every agent's utility under is no less than
that under , and this holds with strict inequality for at least one type
profile and one agent. We say that a non-deficit Groves mechanism {\em
collectively dominates} another non-deficit Groves mechanism if for every
type profile, the agents' total utility under is no less than that under
, and this holds with strict inequality for at least one type profile. The
above definitions induce two partial orders on non-deficit Groves mechanisms.
We study the maximal elements corresponding to these two partial orders, which
we call the {\em individually undominated} mechanisms and the {\em collectively
undominated} mechanisms, respectively.Comment: 34 pages. To appear in Journal of AI Research (JAIR
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