44,843 research outputs found
Linear Regression from Strategic Data Sources
Linear regression is a fundamental building block of statistical data
analysis. It amounts to estimating the parameters of a linear model that maps
input features to corresponding outputs. In the classical setting where the
precision of each data point is fixed, the famous Aitken/Gauss-Markov theorem
in statistics states that generalized least squares (GLS) is a so-called "Best
Linear Unbiased Estimator" (BLUE). In modern data science, however, one often
faces strategic data sources, namely, individuals who incur a cost for
providing high-precision data.
In this paper, we study a setting in which features are public but
individuals choose the precision of the outputs they reveal to an analyst. We
assume that the analyst performs linear regression on this dataset, and
individuals benefit from the outcome of this estimation. We model this scenario
as a game where individuals minimize a cost comprising two components: (a) an
(agent-specific) disclosure cost for providing high-precision data; and (b) a
(global) estimation cost representing the inaccuracy in the linear model
estimate. In this game, the linear model estimate is a public good that
benefits all individuals. We establish that this game has a unique non-trivial
Nash equilibrium. We study the efficiency of this equilibrium and we prove
tight bounds on the price of stability for a large class of disclosure and
estimation costs. Finally, we study the estimator accuracy achieved at
equilibrium. We show that, in general, Aitken's theorem does not hold under
strategic data sources, though it does hold if individuals have identical
disclosure costs (up to a multiplicative factor). When individuals have
non-identical costs, we derive a bound on the improvement of the equilibrium
estimation cost that can be achieved by deviating from GLS, under mild
assumptions on the disclosure cost functions.Comment: This version (v3) extends the results on the sub-optimality of GLS
(Section 6) and improves writing in multiple places compared to v2. Compared
to the initial version v1, it also fixes an error in Theorem 6 (now Theorem
5), and extended many of the result
Regression games
The solution of a TU cooperative game can be a distribution of the value of the grand coalition, i.e. it can be a distribution of the payo (utility) all the players together achieve. In a regression model, the evaluation of the explanatory variables can be a distribution of the overall t, i.e. the t of the model every regressor variable is involved. Furthermore, we can take regression models as TU cooperative games where the explanatory (regressor) variables are the players. In this paper we introduce the class of regression games, characterize it and apply the Shapley value to evaluating the explanatory variables in regression models. In order to support our approach we consider Young (1985)'s axiomatization of the Shapley value, and conclude that the Shapley value is a reasonable tool to evaluate the explanatory variables of regression models
Cheating for Problem Solving: A Genetic Algorithm with Social Interactions
We propose a variation of the standard genetic algorithm that incorporates
social interaction between the individuals in the population. Our goal is to
understand the evolutionary role of social systems and its possible application
as a non-genetic new step in evolutionary algorithms. In biological
populations, ie animals, even human beings and microorganisms, social
interactions often affect the fitness of individuals. It is conceivable that
the perturbation of the fitness via social interactions is an evolutionary
strategy to avoid trapping into local optimum, thus avoiding a fast convergence
of the population. We model the social interactions according to Game Theory.
The population is, therefore, composed by cooperator and defector individuals
whose interactions produce payoffs according to well known game models
(prisoner's dilemma, chicken game, and others). Our results on Knapsack
problems show, for some game models, a significant performance improvement as
compared to a standard genetic algorithm.Comment: 7 pages, 5 Figures, 5 Tables, Proceedings of Genetic and Evolutionary
Computation Conference (GECCO 2009), Montreal, Canad
Variance Allocation and Shapley Value
Motivated by the problem of utility allocation in a portfolio under a
Markowitz mean-variance choice paradigm, we propose an allocation criterion for
the variance of the sum of possibly dependent random variables. This
criterion, the Shapley value, requires to translate the problem into a
cooperative game. The Shapley value has nice properties, but, in general, is
computationally demanding. The main result of this paper shows that in our
particular case the Shapley value has a very simple form that can be easily
computed. The same criterion is used also to allocate the standard deviation of
the sum of random variables and a conjecture about the relation of the
values in the two games is formulated.Comment: 20page
Learning the Structure and Parameters of Large-Population Graphical Games from Behavioral Data
We consider learning, from strictly behavioral data, the structure and
parameters of linear influence games (LIGs), a class of parametric graphical
games introduced by Irfan and Ortiz (2014). LIGs facilitate causal strategic
inference (CSI): Making inferences from causal interventions on stable behavior
in strategic settings. Applications include the identification of the most
influential individuals in large (social) networks. Such tasks can also support
policy-making analysis. Motivated by the computational work on LIGs, we cast
the learning problem as maximum-likelihood estimation (MLE) of a generative
model defined by pure-strategy Nash equilibria (PSNE). Our simple formulation
uncovers the fundamental interplay between goodness-of-fit and model
complexity: good models capture equilibrium behavior within the data while
controlling the true number of equilibria, including those unobserved. We
provide a generalization bound establishing the sample complexity for MLE in
our framework. We propose several algorithms including convex loss minimization
(CLM) and sigmoidal approximations. We prove that the number of exact PSNE in
LIGs is small, with high probability; thus, CLM is sound. We illustrate our
approach on synthetic data and real-world U.S. congressional voting records. We
briefly discuss our learning framework's generality and potential applicability
to general graphical games.Comment: Journal of Machine Learning Research. (accepted, pending
publication.) Last conference version: submitted March 30, 2012 to UAI 2012.
First conference version: entitled, Learning Influence Games, initially
submitted on June 1, 2010 to NIPS 201
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