The dynamics of consensus in group decision making: investigating the pairwise interactions between fuzzy preferences.

In this paper we present an overview of the soft consensus model in group decision making and we investigate the dynamical patterns generated by the fundamental pairwise preference interactions on which the model is based. The dynamical mechanism of the soft consensus model is driven by the minimization of a cost function combining a collective measure of dissensus with an individual mechanism of opinion changing aversion. The dissensus measure plays a key role in the model and induces a network of pairwise interactions between the individual preferences. The structure of fuzzy relations is present at both the individual and the collective levels of description of the soft consensus model: pairwise preference intensities between alternatives at the individual level, and pairwise interaction coefficients between decision makers at the collective level. The collective measure of dissensus is based on non linear scaling functions of the linguistic quantifier type and expresses the degree to which most of the decision makers disagree with respect to their preferences regarding the most relevant alternatives. The graded notion of consensus underlying the dissensus measure is central to the dynamical unfolding of the model. The original formulation of the soft consensus model in terms of standard numerical preferences has been recently extended in order to allow decision makers to express their preferences by means of triangular fuzzy numbers. An appropriate notion of distance between triangular fuzzy numbers has been chosen for the construction of the collective dissensus measure. In the extended formulation of the soft consensus model the extra degrees of freedom associated with the triangular fuzzy preferences, combined with non linear nature of the pairwise preference interactions, generate various interesting and suggestive dynamical patterns. In the present paper we investigate these dynamical patterns which are illustrated by means of a number of computer simulations.

Mathematical Structures in Group Decision-Making on Resource Allocation Distributions.

Optimal decisions on the distribution of finite resources are explicitly structured by mathematical models that specify relevant variables, constraints, and objectives. Here we report analysis and evidence that implicit mathematical structures are also involved in group decision-making on resource allocation distributions under conditions of uncertainty that disallow formal optimization. A group's array of initial distribution preferences automatically sets up a geometric decision space of alternative resource distributions. Weighted averaging mechanisms of interpersonal influence reduce the heterogeneity of the group's initial preferences on a suitable distribution. A model of opinion formation based on weighted averaging predicts a distribution that is a feasible point in the group's implicit initial decision space

Collective learning and optimal consensus decisions in social animal groups.

Published onlineJournal ArticleResearch Support, Non-U.S. Gov'tResearch Support, U.S. Gov't, Non-P.H.S.Learning has been studied extensively in the context of isolated individuals. However, many organisms are social and consequently make decisions both individually and as part of a collective. Reaching consensus necessarily means that a single option is chosen by the group, even when there are dissenting opinions. This decision-making process decouples the otherwise direct relationship between animals' preferences and their experiences (the outcomes of decisions). Instead, because an individual's learned preferences influence what others experience, and therefore learn about, collective decisions couple the learning processes between social organisms. This introduces a new, and previously unexplored, dynamical relationship between preference, action, experience and learning. Here we model collective learning within animal groups that make consensus decisions. We reveal how learning as part of a collective results in behavior that is fundamentally different from that learned in isolation, allowing grouping organisms to spontaneously (and indirectly) detect correlations between group members' observations of environmental cues, adjust strategy as a function of changing group size (even if that group size is not known to the individual), and achieve a decision accuracy that is very close to that which is provably optimal, regardless of environmental contingencies. Because these properties make minimal cognitive demands on individuals, collective learning, and the capabilities it affords, may be widespread among group-living organisms. Our work emphasizes the importance and need for theoretical and experimental work that considers the mechanism and consequences of learning in a social context.This research was supported by a National Science Foundation Graduate Research Fellowship and National Science Foundation Doctoral Dissertation Improvement Grant 1210029 to ABK, a National Sciences and Engineering Research Council of Canada Fellowship to NM, and National Science Foundation Award PHY-0848755 and EAGER Grant IOS-1251585, Office of Naval Research Award N00014-09-1-1074, Army Research Office Grant W911NG-11-1-0385, and Human Frontiers Science Program Award RGP0065/2012 to IDC

Parkinson's Law Quantified: Three Investigations on Bureaucratic Inefficiency

We formulate three famous, descriptive essays of C.N. Parkinson on bureaucratic inefficiency in a quantifiable and dynamical socio-physical framework. In the first model we show how the use of recent opinion formation models for small groups can be used to understand Parkinson's observation that decision making bodies such as cabinets or boards become highly inefficient once their size exceeds a critical 'Coefficient of Inefficiency', typically around 20. A second observation of Parkinson - which is sometimes referred to as Parkinson's Law - is that the growth of bureaucratic or administrative bodies usually goes hand in hand with a drastic decrease of its overall efficiency. In our second model we view a bureaucratic body as a system of a flow of workers, which enter, become promoted to various internal levels within the system over time, and leave the system after having served for a certain time. Promotion usually is associated with an increase of subordinates. Within the proposed model it becomes possible to work out the phase diagram under which conditions bureaucratic growth can be confined. In our last model we assign individual efficiency curves to workers throughout their life in administration, and compute the optimum time to send them to old age pension, in order to ensure a maximum of efficiency within the body - in Parkinson's words we compute the 'Pension Point'.Comment: 15 pages, 5 figure

Consensus problems in networks of agents with switching topology and time-delays

In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results

Dynamics of opinion formation in a small-world network

The dynamical process of opinion formation within a model using a local majority opinion updating rule is studied numerically in networks with the small-world geometrical property. The network is one in which shortcuts are added to randomly chosen pairs of nodes in an underlying regular lattice. The presence of a small number of shortcuts is found to shorten the time to reach a consensus significantly. The effects of having shortcuts in a lattice of fixed spatial dimension are shown to be analogous to that of increasing the spatial dimension in regular lattices. The shortening of the consensus time is shown to be related to the shortening of the mean shortest path as shortcuts are added. Results can also be translated into that of the dynamics of a spin system in a small-world network.Comment: 10 pages, 5 figure

Deriving mesoscopic models of collective behaviour for finite populations

Animal groups exhibit emergent properties that are a consequence of local interactions. Linking individual-level behaviour to coarse-grained descriptions of animal groups has been a question of fundamental interest. Here, we present two complementary approaches to deriving coarse-grained descriptions of collective behaviour at so-called mesoscopic scales, which account for the stochasticity arising from the finite sizes of animal groups. We construct stochastic differential equations (SDEs) for a coarse-grained variable that describes the order/consensus within a group. The first method of construction is based on van Kampen's system-size expansion of transition rates. The second method employs Gillespie's chemical Langevin equations. We apply these two methods to two microscopic models from the literature, in which organisms stochastically interact and choose between two directions/choices of foraging. These `binary-choice' models differ only in the types of interactions between individuals, with one assuming simple pair-wise interactions, and the other incorporating higher-order effects. In both cases, the derived mesoscopic SDEs have multiplicative, or state-dependent, noise. However, the different models demonstrate the contrasting effects of noise: increasing order in the pair-wise interaction model, whilst reducing order in the higher-order interaction model. Although both methods yield identical SDEs for such binary-choice, or one-dimensional, systems, the relative tractability of the chemical Langevin approach is beneficial in generalizations to higher-dimensions. In summary, this book chapter provides a pedagogical review of two complementary methods to construct mesoscopic descriptions from microscopic rules and demonstrates how resultant multiplicative noise can have counter-intuitive effects on shaping collective behaviour.Comment: Second version, 4 figures, 2 appendice

Distributed Decision Through Self-Synchronizing Sensor Networks in the Presence of Propagation Delays and Asymmetric Channels

In this paper we propose and analyze a distributed algorithm for achieving globally optimal decisions, either estimation or detection, through a self-synchronization mechanism among linearly coupled integrators initialized with local measurements. We model the interaction among the nodes as a directed graph with weights (possibly) dependent on the radio channels and we pose special attention to the effect of the propagation delay occurring in the exchange of data among sensors, as a function of the network geometry. We derive necessary and sufficient conditions for the proposed system to reach a consensus on globally optimal decision statistics. One of the major results proved in this work is that a consensus is reached with exponential convergence speed for any bounded delay condition if and only if the directed graph is quasi-strongly connected. We provide a closed form expression for the global consensus, showing that the effect of delays is, in general, the introduction of a bias in the final decision. Finally, we exploit our closed form expression to devise a double-step consensus mechanism able to provide an unbiased estimate with minimum extra complexity, without the need to know or estimate the channel parameters.Comment: To be published on IEEE Transactions on Signal Processin