Bounded Disagreement

A well-known generalization of the consensus problem, namely, set agreement (SA), limits the number of distinct decision values that processes decide. In some settings, it may be more important to limit the number of "disagreers". Thus, we introduce another natural generalization of the consensus problem, namely, bounded disagreement (BD), which limits the number of processes that decide differently from the plurality. More precisely, in a system with n processes, the (n, l)-BD task has the following requirement: there is a value v such that at most l processes (the disagreers) decide a value other than v. Despite their apparent similarities, the results described below show that bounded disagreement, consensus, and set agreement are in fact fundamentally different problems. We investigate the relationship between bounded disagreement, consensus, and set agreement. In particular, we determine the consensus number for every instance of the BD task. We also determine values of n, l, m, and k such that the (n, l)-BD task can solve the (m, k)-SA task (where m processes can decide at most k distinct values). Using our results and a previously known impossibility result for set agreement, we prove that for all n >= 2, there is a BD task (and a corresponding BD object) that has consensus number n but can not be solved using n-consensus and registers. Prior to our paper, the only objects known to have this unusual characteristic for n >= 2 (which shows that the consensus number of an object is not sufficient to fully capture its power) were artificial objects crafted solely for the purpose of exhibiting this behaviour

Agreeing to disagree in a countable space of equiprobable states

An example is given in which agents agree to disagree, showing that Aumann's (1976) Agreement Theorem does not extend to a countable space of equiprobable states of nature. Even in this unorthodox setting, if the sets of the information partitions are intervals, an agreement theorem holds. A result that describes the margin for disagreement is also obtained.Agreeing to disagree, Interactive epistemology, Bounded rationality.

Group Decision-Making in the Shadow of Disagreement

A model of group decision-making is studied, in which one of two alternatives must be chosen. While group members differ in their valuations of the alternatives, everybody prefers some alternative to disagreement. Our model is distinguished by three features: private information regarding valuations, varying intensities in the preference for one out-come over the other, and the option to declare neutrality in order to avoid disagreement. We uncover a variant on the “tyranny of the majority": there is always an equilibrium in which the majority is more aggressive in pushing its alternative, thus enforcing their will via both numbers and voice. However, under very general conditions an aggressive minority equilibrium inevitably makes an appearance, provided that the group is large enough. This equilibrium displays a “tyranny of the minority": it is always true that the increased aggression of the minority more than compensates for smaller number, leading to the minority outcome being implemented with larger probability than the majority alternative. In all cases the option to remain neutral ensures that the probability of disagreement is bounded away from one (as group size changes), regardless of the supermajority value needed for agreement, as long as it is not unanimity.Collective decision-making, Groups, Disagreements, Decision rules

On the benefits of saturating information in consensus networks with noise

In a consensus network subject to non-zero mean noise, the system state may be driven away even when the disagreement exhibits a bounded response. This is unfavourable in applications since the nodes may not work properly and even be faulty outside their operating region. In this paper, we propose a new control algorithm to mitigate this issue by assigning each node a favourite interval that characterizes the nodes desired convergence region. The algorithm is implemented in a self-triggered fashion. If the nodes do not share a global clock, the network operates in a fully asynchronous mode. By this algorithm, we show that the state evolution is confined around the favourite interval and the node disagreement is bounded by a simple linear function of the noise magnitude, without requiring any priori information on the noise. We also show that if the nodes share some global information, then the algorithm can be adjusted to make the nodes evolve into the favourite interval, improve on the disagreement bound and achieve asymptotic consensus in the noiseless case

Behavioural Anomalies, Bounded Rationality and Simple Heuristics

The use of bounded rationality in explaining economic phenomena has attracted growing attention. In spite of this, there is still considerable disagreement regarding the meaning of bounded rationality. Basov (2005) argues that when modeling boundedly rational behaviour it is desirable to start with an explicit formulation of the learning process. A complete understanding of the boundedly rational decision-making process requires development of an evolutionary-dynamic model which can give rise to such learning processes. Evolutionary dynamics implies that individuals use heuristics to adjust their choices in light of past experiences, moving in the direction that appears most beneficial, where these adjustment rules are assumed ‘hardwired’ into human cognition through the process of biological evolution. In this paper we elaborate on the latter point by building a model of evolutionary selection relevant to heuristics. We show that in addition to explaining the origin of learning rules this approach also sheds light on some well documented preference anomalies.Bounded Rationality;Heuristics;Replicator Dynamics

Disagreement percolation for Gibbs ball models

We generalise disagreement percolation to Gibbs point processes of balls with varying radii. This allows to establish the uniqueness of the Gibbs measure and exponential decay of pair correlations in the low activity regime by comparison with a sub-critical Boolean model. Applications to the Continuum Random Cluster model and the Quermass-interaction model are presented. At the core of our proof lies an explicit dependent thinning from a Poisson point process to a dominated Gibbs point process.Comment: 23 pages, 0 figure Correction, from the published version, of the proof of Section

Disagreement percolation for the hard-sphere model

Disagreement percolation connects a Gibbs lattice gas and i.i.d. site percolation on the same lattice such that non-percolation implies uniqueness of the Gibbs measure. This work generalises disagreement percolation to the hard-sphere model and the Boolean model. Non-percolation of the Boolean model implies the uniqueness of the Gibbs measure and exponential decay of pair correlations and finite volume errors. Hence, lower bounds on the critical intensity for percolation of the Boolean model imply lower bounds on the critical activity for a (potential) phase transition. These lower bounds improve upon known bounds obtained by cluster expansion techniques. The proof uses a novel dependent thinning from a Poisson point process to the hard-sphere model, with the thinning probability related to a derivative of the free energy