338 / 0022-2496/01 $35.00 © 2001 Elsevier Science (USA) All rights reserved. Journal of Mathematical Psychology 46, 338–351 (2002) doi:10.1006/jmps.2001.1391 The Probability of Collective Choice with Shared Knowledge Structures Whitman Richards D. Richards was supported by NSF 9729847. Thanks to Rajesh Kasturirangan for comments on early drafts and to the Minnesota Supercomputing Institute for providing computing time. We are indebted to Richard Schweickert for a careful reading and detailed comments. Address correspondence and reprint requests to W. Richards, Departement of Brain and Cognitive Science, Massachussetts Institute of Technology, 545 Tech square, NE43-737, Cambridge, MA 02139. E-mail: wrichards@mit.edu. Massachusetts Institute of Technology Brendan D. McKay Australian National University and Diana Richards University of Minnesota Complex decision making typically involves many agents attempting to aggregate many alternatives. If agents’ preferences are unconstrained, cyclic outcomes are highly probable. In contrast, we show that if agents share similar models of the choice domain, then a stable collective outcome occurs with about 90 % probability. These results have implications for systems of interacting agents, such as group decisions by individuals in social settings or interpretations of sense data by competing perceptual modules of a brain. © 2001 Elsevier Science INTRODUCTION There are abundant theoretical results from the field of social choice showing that there will almost surely be no single agreement when large numbers of indi- viduals must choose from a large set of alternatives (e.g., Arrow, 1963; Campbell & Tullock, 1965; Jones, Rodcliff, Taber, & Timpore, 1995; Kelly, 1986). These results have significant implications for a variety of complex systems that are engaged in collective decision making using voting-like procedures. The most obvious application is in social settings such as legislatures or committees, where a group of