Parliamentary Coalitions: A Tour of Models

Abstract

The study of government and politics, like all the other social sciences, has made increasing use of mathematics in the last thirty years. This has involved not only the use of statistics and computers to increase the scope and thoroughness of empirical work, but also the development of mathematical models in political theory. Through the use of models, mathematics has come to play an important role in political science at the conceptual level. This development has not been greeted with universal enthusiasm by practitioners in the field. An academic colleagtue of one of the authors, in a department of Government, has been heard to complain with some bitterness that he can no longer read the American Political Science Review because it is full of obscure mathematical symbols. Non-enthusiasts have their point. Politics often seems far removed from the domain of precise, rational thought which mathematics epitomizes. Nevertheless, mathematics has been able to contribute insights to the study of political systems. In this article we explore the use of mathematical models in various attempts to understand one of the central concerns of politics-the ways in which politicians go about forming coalitions with other politicians. In the first section we introduce the problem of coalition formation in the context of parties trying to form a coalition government in a parliamentary system. This will be our central focus in the article. We start by considering some of the earliest and simplest models, which lead to a geometric formulation of the problem. In the second section we discuss two more complex models from the tradition of mathematical game theory. We have tried to illustrate the methods of thinking involved in these models without presenting all of the formal details (for which we give