Stabilizing Consensus with Many Opinions

We consider the following distributed consensus problem: Each node in a complete communication network of size $n$ initially holds an \emph{opinion}, which is chosen arbitrarily from a finite set $\Sigma$. The system must converge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be \emph{valid}, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of an adaptive, malicious adversary who can modify the opinions of a bounded number of nodes in every round. We consider the \emph{3-majority dynamics}: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let $k$ be the number of valid opinions. We show that, if $k \leqslant n^{\alpha}$, where $\alpha$ is a suitable positive constant, the 3-majority dynamics converges in time polynomial in $k$ and $\log n$ with high probability even in the presence of an adversary who can affect up to $o(\sqrt{n})$ nodes at each round. Previously, the convergence of the 3-majority protocol was known for $|\Sigma| = 2$ only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for $|\Sigma| > 2$

ADAPTIVE MAJORITY PROBLEMS FOR RESTRICTED QUERY GRAPHS AND FOR WEIGHTED SETS

Suppose that the vertices of a graph G are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists), if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by m(G). It was shown by Saks and Werman that m(K-n) = n - b(n) where b(n) is the number of 1's in the binary representation of n. In this paper we initiate the study of the problem for general graphs. The obvious bounds for a connected graph G on n vertices are n - b(n) <= m(G) <= n - 1. We show that for any tree T on an even number of vertices we have m(T) = n - 1, and that for any tree T on an odd number of vertices, we have n - 65 <= m (T) <= n - 2. Our proof uses results about the weighted version of the problem for K-n, which may be of independent interest. We also exhibit a sequence G(n) of graphs with m(G(n)) = n - b(n) such that the number of edges in G(n) is O(nb(n))

Playing off-line games with bounded rationality

We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payo is the average of a one-shot payo over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First we dene the complexity of a sequence by its smallest period (a non-periodic sequence being of innite complexity) and study the maxmin of the game where player 1 is restricted to strategies with complexity at most n and player 2 is restricted to strategies with complexity at most m. We study the asymptotics of this value and a complete characterization in the matching pennies case. We extend the analysis of matching pennies to strategies with bounded recall.We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payo is the average of a one-shot payo over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First we dene the complexity of a sequence by its smallest period (a non-periodic sequence being of innite complexity) and study the maxmin of the game where player 1 is restricted to strategies with complexity at most n and player 2 is restricted to strategies with complexity at most m. We study the asymptotics of this value and a complete characterization in the matching pennies case. We extend the analysis of matching pennies to strategies with bounded recall.Refereed Working Papers / of international relevanc

The TOMS model of social entrepreneurship: The new way of harnessing capitalism to turn a social profit

College students aren’t oblivious to philanthropy; in fact, many studies report that millennials comprise one of the largest demographics in favor of philanthropic giving. The James Madison University (JMU) campus has recently seen increased student interest in TOMS Shoes, a company that donates a pair of shoes to a child in need for every pair purchased. The aim of this study was to localize where this interest in charity is coming from in a population with little disposable income. Through online surveys collected from thirty-nine TOMS-owning JMU students, this study explored the factors that have led millennials (also referred to as Generation Y) to support TOMS, and furthermore, what led these students to donate their money to charitable organizations. This research is useful to the field of nonprofit studies in that it critically analyzed what is and isn’t effective in the marketing world when targeting college students ranging in age from 18 to 23. This topic is important to social work because by harnessing capitalism and tying it to social causes, this additional consumer motivation makes philanthropic giving more powerful. With this new practice of connecting charitable causes to consumer wants, social workers and those who work in nonprofit management can consider the ways in which their ethical responsibility to be advocates for the less-fortunate can combine with social entrepreneurship to provide a new model with which to solve social problems

Majority problems of large query size

We study two models of the Majority problem. We are given n balls and an unknown coloring of them with two colors. We can ask sets of balls of size k as queries, and in the so-called General Model the answer to a query shows if all the balls in the set are of the same color or not. In the so-called Counting Model the answer to a query gives the difference between the cardinalities of the color classes in the query. Our goal is to show a ball of the larger color class, or prove that the color classes are of the same size, using as few queries as possible. In this paper we improve the bounds given by De Marco and Kranakis for the number of queries needed.Comment: We cut the non-adaptive results from the first version to publish separatel

Post modern identity : "in between" real and virtual

The article focus on the phenomena of the "radical change" (transformation) brought by new ICT technologies, associated with the information and communication revolution, both on the level of the collective: bringing disruptive changes within economic, social and cultural sphere, and on the level of the individual, evoking fundamental, yet subtle, changes within our psyche impacting our identity (ies). The multiple scientific discourses, when analyzing the impact of new technologies, usually focus on changes in the economic, social or cultural sphere, defining them within the context of its semantic field (resulting different explanatory models built around different theoretical concepts, together with the accompanying different methodology). The main hypothesis of the "radical change" (transformation) brought by new technologies, usually described in reference to new paradigm change, refers to constantly increasing impact (direct mediation) of ICT technologies in almost all spheres of our lives: social, economical and cultural, but hardly ever discuss the extremely subtle reconfiguration of our psychological space made under the influence of new technologies. As such the article focuses mainly the impact of new technologies upon the psyche and post-modern identity, trying to fully grasp and understand both the visible (direct) and the invisible (subtle) changes, from the perspective of psychological approach, with special reference to Jung’s analytical psychology. The core element (novelty) is the attempt to fully grasp (understand) the phenomena of moving (living) ‘in between’ real and virtual (identity/ environment), mainly from the point view (implications) of psychological as well as philosophical (ontological), not as in majority of cases (discourses) from the technological, economical, sociological perspective. Cultural anthropology evokes the concept of liminality to denominate the boundaries between two different states : functioning within the existing normative (institutional) governance and stepping outside or aside of it, meaning suspension of the existing norms and standards (and break in or pause within the existing culture). The post-modern individual is somehow forced to move ‘in between’ and experience two different environments simultaneously - the physical environment, embedded in real space and time continuum, where we live at the very moment and digital environment created by new technologies (virtual and/or digital space). As such this continuous transition from reality to virtuality evokes the characteristics (attributes) of the liminal experience. Critical analysis of the defined phenomena implies the need of interdisciplinary approach based on the comparative methodology, both from the point view of theoretical discourses as well as more empirical approach, based mainly on the interdisciplinary approach of Jungian Analytical Psychology, as the outlined theme moves ‘in between’ new technologies, culture (as well as economy or social science) and psychology