Consensus through respect: A model of rational group decision-making

Decision-making groups frequently face the task of reaching consensus regarding the value of one or more numerical variables. These values may represent the final outcome of deliberation (as in the case of an allocation of funds among competing projects), or they may play an intermediate role in the analysis of a decision problem (as in the case of probability and utility estimates required for computation of expected utilities). At the outset of the deliberative process there will typically be disagreement about the values in question, and thus it is Of interest to develop both descriptive and normative models for the process by which consensus is, or may be, achieved. The first steps toward the development of a descriptive model appear in French [1] and Harary [3]. Initial individual opinions are entered in a matrix A = (ai/), where aij represents individual i's opinion as to the most appropriate value of variable ]. This matrix is regarded as undergoing modification through repeated multiplication by a 'power matrix' P = (p/j), where Pi] is a numerical measure of the power of individual ] to influence the opinion of individual i. According to this model, individuals revise their opinions over discrete units of time, holding fast (in a temporary unstable equilibrium) to their latest opinions until the forces quantified in the power matrix once again move them closer to consensus. The mathematical theory of Markov chains is used to identify a particular class of power relationships which result in convergence of opinion. Results here depend strongly upon the assumption that the power parameters Pi] derive from a more or less enduring relationship between individuals, and thus remain constant over time. Individuals may, in fact, be unaware of the exact values of these parameters, or if aware, resentful of them. In recent years Lehrer [5, 6, 7] has resurrected the use of Markov chains as a model of rational decision-making by communities of distinterested experts. Here, by contrast, individuals freely and consciously grant weight to the opinions of others on the sole basis of respect for their expertise and