630 research outputs found
Eight Degrees of Separation
The paper presents a model of network formation where every connected couple gives a contribution to the aggregate payoff, eventually discounted by their distance, and the resources are split between agents through the Myerson value. As equilibrium concept we adopt a refinement of pairwise stability. The only parameters are the number N of agents and a constant cost k for every agent to maintain any single link. This setup shows a wide multiplicity of equilibria, all of them connected, as k ranges over non trivial cases. We are able to show that, for any N, when the equilibrium is a tree (acyclical connected graph), which happens for high k, and there is no decay, the diameter of such a network never exceeds 8 (i.e. there are no two nodes with distance greater than 8). Adopting no decay and studying only trees, we facilitate the analysis but impose worst–case scenarios: we conjecture that the limit of 8 should apply for any possible non–empty equilibrium with any decay function.Network Formation, Myerson Value
Collaborative Information Processing in Wireless Sensor Networks for Diffusive Source Estimation
In this dissertation, we address the issue of collaborative information processing for diffusive source parameter estimation using wireless sensor networks (WSNs) capable of sensing in dispersive medium/environment, from signal processing perspective. We begin the dissertation by focusing on the mathematical formulation of a special diffusion phenomenon, i.e., an underwater oil spill, along with statistical algorithms for meaningful analysis of sensor data leading to efficient estimation of desired parameters of interest. The objective is to obtain an analytical solution to the problem, rather than using non-model based sophisticated numerical techniques. We tried to make the physical diffusion model as much appropriate as possible, while maintaining some pragmatic and reasonable assumptions for the simplicity of exposition and analytical derivation. The dissertation studies both source localization and tracking for static and moving diffusive sources respectively. For static diffusive source localization, we investigate two parametric estimation techniques based on the maximum-likelihood (ML) and the best linear unbiased estimator (BLUE) for a special case of our obtained physical dispersion model. We prove the consistency and asymptotic normality of the obtained ML solution when the number of sensor nodes and samples approach infinity, and derive the Cramer-Rao lower bound (CRLB) on its performance. In case of a moving diffusive source, we propose a particle filter (PF) based target tracking scheme for moving diffusive source, and analytically derive the posterior Cramer-Rao lower bound (PCRLB) for the moving source state estimates as a theoretical performance bound. Further, we explore nonparametric, machine learning based estimation technique for diffusive source parameter estimation using Dirichlet process mixture model (DPMM). Since real data are often complicated, no parametric model is suitable. As an alternative, we exploit the rich tools of nonparametric Bayesian methods, in particular the DPMM, which provides us with a flexible and data-driven estimation process. We propose DPMM based static diffusive source localization algorithm and provide analytical proof of convergence. The proposed algorithm is also extended to the scenario when multiple diffusive sources of same kind are present in the diffusive field of interest. Efficient power allocation can play an important role in extending the lifetime of a resource constrained WSN. Resource-constrained WSNs rely on collaborative signal and information processing for efficient handling of large volumes of data collected by the sensor nodes. In this dissertation, the problem of collaborative information processing for sequential parameter estimation in a WSN is formulated in a cooperative game-theoretic framework, which addresses the issue of fair resource allocation for estimation task at the Fusion center (FC). The framework allows addressing either resource allocation or commitment for information processing as solutions of cooperative games with underlying theoretical justifications. Different solution concepts found in cooperative games, namely, the Shapley function and Nash bargaining are used to enforce certain kinds of fairness among the nodes in a WSN
A cognitive hierarchy theory of one-shot games: Some preliminary results
Strategic thinking, best-response, and mutual consistency (equilibrium) are three
key modelling principles in noncooperative game theory. This paper relaxes mutual
consistency to predict how players are likely to behave in in one-shot games before they
can learn to equilibrate. We introduce a one-parameter cognitive hierarchy (CH) model
to predict behavior in one-shot games, and initial conditions in repeated games. The CH
approach assumes that players use k steps of reasoning with frequency f (k). Zero-step
players randomize. Players using k (≥ 1) steps best respond given partially rational
expectations about what players doing 0 through k - 1 steps actually choose. A simple
axiom which expresses the intuition that steps of thinking are increasingly constrained by
working memory, implies that f (k) has a Poisson distribution (characterized by a mean
number of thinking steps τ ). The CH model converges to dominance-solvable equilibria
when τ is large, predicts monotonic entry in binary entry games for τ < 1:25, and predicts
effects of group size which are not predicted by Nash equilibrium. Best-fitting values of
τ have an interquartile range of (.98,2.40) and a median of 1.65 across 80 experimental
samples of matrix games, entry games, mixed-equilibrium games, and dominance-solvable
p-beauty contests. The CH model also has economic value because subjects would have
raised their earnings substantially if they had best-responded to model forecasts instead
of making the choices they did
A Size-Free CLT for Poisson Multinomials and its Applications
An -Poisson Multinomial Distribution (PMD) is the distribution of the
sum of independent random vectors supported on the set of standard basis vectors in . We show
that any -PMD is -close in total
variation distance to the (appropriately discretized) multi-dimensional
Gaussian with the same first two moments, removing the dependence on from
the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is
obtained by bootstrapping the Valiant-Valiant CLT itself through the structural
characterization of PMDs shown in recent work by Daskalakis, Kamath, and
Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS
for approximate Nash equilibria in anonymous games, significantly improving the
state of the art, and matching qualitatively the running time dependence on
and of the best known algorithm for two-strategy anonymous
games. Our new CLT also enables the construction of covers for the set of
-PMDs, which are proper and whose size is shown to be essentially
optimal. Our cover construction combines our CLT with the Shapley-Folkman
theorem and recent sparsification results for Laplacian matrices by Batson,
Spielman, and Srivastava. Our cover size lower bound is based on an algebraic
geometric construction. Finally, leveraging the structural properties of the
Fourier spectrum of PMDs we show that these distributions can be learned from
samples in -time, removing
the quasi-polynomial dependence of the running time on from the
algorithm of Daskalakis, Kamath, and Tzamos.Comment: To appear in STOC 201
Model free variable importance for high dimensional data
A model-agnostic variable importance method can be used with arbitrary
prediction functions. Here we present some model-free methods that do not
require access to the prediction function. This is useful when that function is
proprietary and not available, or just extremely expensive. It is also useful
when studying residuals from a model. The cohort Shapley (CS) method is
model-free but has exponential cost in the dimension of the input space. A
supervised on-manifold Shapley method from Frye et al. (2020) is also model
free but requires as input a second black box model that has to be trained for
the Shapley value problem. We introduce an integrated gradient (IG) version of
cohort Shapley, called IGCS, with cost . We show that over the
vast majority of the relevant unit cube that the IGCS value function is close
to a multilinear function for which IGCS matches CS. Another benefit of IGCS is
that is allows IG methods to be used with binary predictors. We use some area
between curves (ABC) measures to quantify the performance of IGCS. On a problem
from high energy physics we verify that IGCS has nearly the same ABCs as CS
does. We also use it on a problem from computational chemistry in 1024
variables. We see there that IGCS attains much higher ABCs than we get from
Monte Carlo sampling. The code is publicly available at
https://github.com/cohortshapley/cohortintgra
A procedure for finding Nash equilibria in bi-matrix games
Game Theory;operations research
- …