An Expository Development of a Mathematical Model of the Electoral Process

TLDR

The electoral process aggregates citizens’ preferences into a collective choice, a concept first formalized by Downs and later scrutinized and refined within spatial models of party competition. The paper aims to develop, explain, and extend a non‑technical mathematical model of the electoral process, clarifying how it addresses the democratic problem of determining a social preference. The model treats the electoral mechanism as a multidimensional spatial competition in which candidates affect turnout and voter perceptions, with the social choice being the policy package of the winner, and it analyzes equilibrium under assumptions that electorate preferences are captured by one (or two) functions.

Abstract

The fundamental process of politics is the aggregation of citizens' preferences into a collective—a social—choice. We develop, interpret, and explain non-technically in this expository essay the definitions, assumptions, and theorems of a mathematical model of one aggregative mechanism—the electoral process. This mechanism is conceptualized here as a multidimensional model of spatial competition in which competition consists of candidates affecting turnout and the electorate's perception of each candidate's positions, and in which the social choice is a policy package which the victorious candidate advocates. This approach, inaugurated by Downs's An Economic Theory of Democracy , and falling under the general rubric “spatial models of party competition,” has been scrutinized, criticized, and reformulated. To clarify the accomplishments of this formulation we identify and discuss in section 2 the general democratic problem of ascertaining a social preference. We review critically in section 3 the definitions and assumptions of our model. We consider in sections 4 and 5 the logic of a competitive electoral equilibrium. We assume in section 4 that the electorate's preferences can be summarized and represented by a single function; the analysis in section 5 pertains to competition between two organizational structures or two opposed ideologies (i.e., when two functions are required to summarize and represent the electorate's preference). Finally, we suggest in section 6 a conceptualization of electoral processes which facilitates extending and empirically testing our model.