54,081 research outputs found
Copmment on Egalitarianism under Incomplete Information
The paper aims at extending the egalitarian principle to environments with incomplete information. The approach is primarily axiomatic, focusing on the characteristic property of monotonicity: no member of the society should be worse off when more collective decisions are available. I start by showing the incompat- ibility of this property with incentive efficiency, even in quasi-linear environments. This serious impossibility result does not follow from the mere presence of incentive constraints, but instead from the fact that information is incomplete (asymmetric information at the time of making a decision). I then weaken the monotonicity property so as to require it only when starting from incentive compatible mecha- nisms at which interim utilities are transferable (in a weak sense). Adding other axioms in the spirit of Kalai's (Econometrica, 1977, Theorem 1) classical character- ization of the egalitarian principle under complete information, I obtain a partial characterization of a natural extension of the lex-min solution to problems with incomplete information. Next, I prove that, in each social choice problem, there is a unique way of rescaling the participants' interim utilities so as to make this solu- tion compatible with the ex-ante utilitarian principle. These two criteria coincides in the rescaled utilities exactly at the incentive ecient mechanisms that maxi- mize Harsanyi and Selten's (Management Science, 1972) weighted Nash product. These concepts are illustrated on classical examples of profit-sharing, public good production and bilateral trade. The richness of the topic of social choice under in- complete information is illustrated by considering two alternative extensions of the egalitarian principle { one based on an idea of equity from the point of view of the individuals themselves (given their private information) instead of an uninformed third party (social planner or arbitrator), and another notion based on the idea of
A Combinatorial Approach to Nonlocality and Contextuality
So far, most of the literature on (quantum) contextuality and the
Kochen-Specker theorem seems either to concern particular examples of
contextuality, or be considered as quantum logic. Here, we develop a general
formalism for contextuality scenarios based on the combinatorics of hypergraphs
which significantly refines a similar recent approach by Cabello, Severini and
Winter (CSW). In contrast to CSW, we explicitly include the normalization of
probabilities, which gives us a much finer control over the various sets of
probabilistic models like classical, quantum and generalized probabilistic. In
particular, our framework specializes to (quantum) nonlocality in the case of
Bell scenarios, which arise very naturally from a certain product of
contextuality scenarios due to Foulis and Randall. In the spirit of CSW, we
find close relationships to several graph invariants. The recently proposed
Local Orthogonality principle turns out to be a special case of a general
principle for contextuality scenarios related to the Shannon capacity of
graphs. Our results imply that it is strictly dominated by a low level of the
Navascu\'es-Pironio-Ac\'in hierarchy of semidefinite programs, which we also
apply to contextuality scenarios.
We derive a wealth of results in our framework, many of these relating to
quantum and supraquantum contextuality and nonlocality, and state numerous open
problems. For example, we show that the set of quantum models on a
contextuality scenario can in general not be characterized in terms of a graph
invariant.
In terms of graph theory, our main result is this: there exist two graphs
and with the properties \begin{align*} \alpha(G_1) &= \Theta(G_1),
& \alpha(G_2) &= \vartheta(G_2), \\[6pt] \Theta(G_1\boxtimes G_2) & >
\Theta(G_1)\cdot \Theta(G_2),& \Theta(G_1 + G_2) & > \Theta(G_1) + \Theta(G_2).
\end{align*}Comment: minor revision, same results as in v2, to appear in Comm. Math. Phy
Le Cam meets LeCun: Deficiency and Generic Feature Learning
"Deep Learning" methods attempt to learn generic features in an unsupervised
fashion from a large unlabelled data set. These generic features should perform
as well as the best hand crafted features for any learning problem that makes
use of this data. We provide a definition of generic features, characterize
when it is possible to learn them and provide methods closely related to the
autoencoder and deep belief network of deep learning. In order to do so we use
the notion of deficiency and illustrate its value in studying certain general
learning problems.Comment: 25 pages, 2 figure
Characterization of complex networks: A survey of measurements
Each complex network (or class of networks) presents specific topological
features which characterize its connectivity and highly influence the dynamics
of processes executed on the network. The analysis, discrimination, and
synthesis of complex networks therefore rely on the use of measurements capable
of expressing the most relevant topological features. This article presents a
survey of such measurements. It includes general considerations about complex
network characterization, a brief review of the principal models, and the
presentation of the main existing measurements. Important related issues
covered in this work comprise the representation of the evolution of complex
networks in terms of trajectories in several measurement spaces, the analysis
of the correlations between some of the most traditional measurements,
perturbation analysis, as well as the use of multivariate statistics for
feature selection and network classification. Depending on the network and the
analysis task one has in mind, a specific set of features may be chosen. It is
hoped that the present survey will help the proper application and
interpretation of measurements.Comment: A working manuscript with 78 pages, 32 figures. Suggestions of
measurements for inclusion are welcomed by the author
Some recent results in the analysis of greedy algorithms for assignment problems
We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems
Fair social decision under uncertainty and belief disagreements
This paper aims to address two issues related to simultaneous aggregation of utilities and beliefs. The first one is related to how to integrate both inequality and uncertainty considerations into social decision making. The second one is related to how social decision should take disagreements in beliefs into account. To accomplish this, whereas individuals are assumed to abide by Savage model’s of subjective expected utility, society is assumed to prescribe, either to each individual when the ex ante individual well-being is favored or to itself when the ex post individual well-being is favored, acting in accordance with the maximin expected utility theory of Gilboa and Schmeidler (J Math Econ 18:141–153, 1989). Furthermore, it adapts an ex ante Pareto-type condition proposed by Gayer et al. (J Legal Stud 43:151–171, 2014), which says that a prospect Pareto dominates another one if the former gives a higher expected utility than the latter one, for each individual, for all individuals’ beliefs. In the context where the ex ante individual welfare is favored, our ex ante Pareto-type condition is shown to be equivalent to social utility taking the form of a MaxMinMin social welfare function, as well as to the individual set of priors being contained within the range of individual beliefs. However, when the ex post individual welfare is favored, the same Pareto-type condition is shown to be equivalent to social utility taking the form of a MaxMinMin social welfare function, as well as to the social set of priors containing only weighted averages of individual beliefs
Distributed convergence to Nash equilibria in two-network zero-sum games
This paper considers a class of strategic scenarios in which two networks of
agents have opposing objectives with regards to the optimization of a common
objective function. In the resulting zero-sum game, individual agents
collaborate with neighbors in their respective network and have only partial
knowledge of the state of the agents in the other network. For the case when
the interaction topology of each network is undirected, we synthesize a
distributed saddle-point strategy and establish its convergence to the Nash
equilibrium for the class of strictly concave-convex and locally Lipschitz
objective functions. We also show that this dynamics does not converge in
general if the topologies are directed. This justifies the introduction, in the
directed case, of a generalization of this distributed dynamics which we show
converges to the Nash equilibrium for the class of strictly concave-convex
differentiable functions with locally Lipschitz gradients. The technical
approach combines tools from algebraic graph theory, nonsmooth analysis,
set-valued dynamical systems, and game theory
Bargaining and the theory of cooperative games: John Nash and beyond
This essay surveys the literature on the axiomatic model of bargaining formulated by Nash ("The Bargaining Problem," Econometrica 28, 1950, 155-162).Nash's bargaining model, Nash solution, Kalai-Smorodinsky solution, Egalitarian solution
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