The Uncertain Dichotomous Choice Model - Some Results

Abstract: A probabilistic framework used in studying collective decision making is the uncertain dichotomous choice model. The model is essentially characterized by two components: the organizational structure and the voting process. By organizational structure, we refer to a group composed of members with common objectives and who are tasked to choose one of the two given alternatives, one of which is assumed to be correct. These symmetric alternatives do not have a priori features that could demerit any of the two options on the basis of labeling, prior probability of being correct, or perceived loss in the event of being the incorrect choice. To come up with a collective choice, the group employs some decision rule. A stringent assumption is that the individual choices are independent. The individual decisional skill is measured by the probability of choosing the correct alternative. As a decision making entity, the group is evaluated by its ability to choose the correct alternative, and the probability that the group makes the right choice is referred to as the collective decisional skill or group competence. The dissertation presents results concerning the collective decisional skill within the framework of the uncertain dichotomous choice model. In the overview, we attempt to cover some of the important studies in this area, including a discussion of the Condorcet jury theorem, the precursor of the said model. Paper 1 describes the components of the majority function for the case of the chairman decisive rule. Moreover, a recursive formula is derived to show how the disparity of the skills between the chairman and the members can affect the collective decisional skill when two new members are added to the group. In Paper 2, we evaluate the competence of the decision body composed of members whose competencies correspond to the k highest order statistics of individual skills in a random sample of size n. The comparison of collective competencies of structures of certain combinations of n and k is facilitated by the evaluation of the expected majority functions. For a specific case, the decrease in the expected group competency due to erroneous exclusion of a qualified individual is also evaluated. Paper 3 attempts to find expressions for the resulting group competence in instances where the individual decisional skills vary. The application of the resulting equation in the simple case of 3 members provides a clear picture of the effect of heterogeneous skills on the collective competence. The paper also examines the effect of heterogeneity on a certain 2-tier hierarchy, with the primary aim of finding an optimal grouping of members. Paper 4 presents various representations of the collective decisional skill that serve as useful tools in evaluating decision structures. The paper also introduces the concept of majority deficiency, providing a new computational method for the reliability of a decision structure.