General Elections Modelled with Infinitely Many Voters
by
Maria Ekes
Warsaw School of Economics, Al. Niepodleglosci 162, 02-554 Warsaw, Poland

General Elections Modelled with Infinitely Many Voters The paper examines a model of general elections (such as referendum or presidential elections) with electorate composed of infinitely many voters classified into a finite number of types. The electorate has to choose, by voting, one of a fixed number of options, possibly one of them being abstention. If there is an option (not abstention) which was chosen by the largest number of voters, this option is called the winner. Otherwise we say that the elections end up with a draw. As usual, members of the electorate should have some preferences (formally, preference-indifference relations, in this paper assumed to be pre-ordering relations), which do not apply only to the results of the elections but also to their individual behavior, so each member of the electorate has a pre-ordering relation on the set being the product of the set of all options and the set of all outcomes. The number of possible pre-ordering relations is very large, even if the number of options is small. Most of these relations are strange and contradictory in the common sense, so even if dealing with all preference-indifference relations present among the electorate were technically possible, the results obtained would be hardly readable. Therefore, in the present paper we assume that only a few ''reasonable'' preference-indifference relations are represented in the electorate. Hence, the whole electorate is divided into a finite number of types, differing in their preferences. We define an equilibrium of such model of elections by reducing this model to a game with infinitely many players, classified into a finite number of types, characterized by their preference-indifference relations instead of payoff functions. In the rest of the paper we focus on the case of elections among two candidates, we give a full characterization of equilibria in such models in the case of eight different types of voters. We also classify equilibria with respect to their stability. We show that although there exist an equilibrium resulting in a draw, it is not stable and it is rather not possible to occur in reality, since an arbitrary small disturbance of voters' decisions can alter the outcome of elections. Hence, in reality we may expect that there occur equilibria at which one of the candidates is the winner.

Date received: June 13, 2000

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