Consensus and Fragmentation of Opinions with a Focus on Bounded Confidence. Issue 8 (14th September 2019)

• Record Type: Journal Article
• Authors: Hegselmann, Rainer; Krause, Ulrich
• Is Part Of: American Mathematical Monthly. Volume 126:Issue 8(2019)
• Page Start: 700
• Page End: 716
• Publication Date: 2019-09-14
• Subjects: MSC: Primary 15B51 -- Secondary 39A10; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2019.1626685
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: During the last several years opinion dynamics, in particular the question of a consensus, have been studied by many researchers across various disciplines, ranging from physics, computer science, and electrical engineering to biology, economics, and sociology. Considering the mathematical analysis, the framework of multiagent systems has proven to be very useful. This article explores convergence behavior of d -dimensional multiagent systems x ( t + 1 ) = A ( t ) x ( t ) with particular emphasis on opinion dynamics under bounded confidence. Conditions are given that guarantee geometrically fast convergence to consensus, sharpening in particular a well-known theorem on Markov chains. A crucial condition to get consensus is what we call the principle of the third agent. In case overall consensus is not reachable, the process of fragmentation is analyzed and partial consensus among agents is proven given certain assumptions. Using these assumptions, the principle of the third agent holds for subgroups of agents.