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In single-member constituency, simple plurality electoral systems, the disproportion between the popular vote polled by a party and the number of representatives it elects has stimulated much controversy. We review the mathematical and statistical models of two-party systems of this type which have been proposed by Kendall and Stuart, Theil, Lemieux, March, and the present authors. This leads to a detailed empirical study of series of elections in three systems: the provincial systems of Quebec and Ontario, and the Canadian federal system. In verifying a "sliding distribution" model more general than that of Kendall and Stuart, we construct the distribution of constituencies according their level of support for a given party, and hence provide an index of electoral regionalization. Less detailed seat-vote data, for longer series of elections in five systems, provide evidence for the persistence of electoral disproportions and the distinguishability of seat-vote characteristics between different systems. The extreme sensitivity of legislative proportions in Québec to small changes in popular vote proportions is shown to be a consequence not of peculiarities in the electoral map, but of the small standard deviation and large kurtosis (peakedness) of the distribution of constituencies.

We propose a simple game-theory model of single-member plurality electoral systems, two parties with unequal resources being the players. Strategies consist of allocations of resources among the n contests, and a party's payoff is the number of contests to which it has assigned more resources than the other party. Mixed strategies exist which are asymptotically optimal as n increases. Identifying a party's proportion of total resources with its total vote proportion, we predict that the swing ratio, or marginal seat proportion per vote proportion, is 2. This compares to empirical findings which range between 2 and 4, and to the hitherto unexplained cube law, which predicts 3. We suggest that the strategic problem modeled by this game accounts for the major part of the swing ratio effect. Factors which vary from system to system, such as proportion of hard-core support attached to parties, may amplify this effect.