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The Swing Ratio and Game Theory

Published online by Cambridge University Press:  01 August 2014

David Sankoff  and
Koula Mellos
David Sankoff
Affiliation:
University of Montreal
Koula Mellos
Affiliation:
University of Ottawa
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Abstract

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.

Type
Articles
Information
American Political Science Review , Volume 66 , Issue 2 , June 1972 , pp. 551 - 554
Copyright
Copyright © American Political Science Association 1972

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References

1 Kendall, M. G. and Stuart, A., “The Law of the Cubic Proportion in Election Results,” British Journal of Sociology, 1 (09, 1950) 183–197 CrossRefGoogle Scholar.

2 May, J. G., “Party Legislative Representation as a Function of Election Results,” Public Opinion Quarterly, 21 (Winter, 1957–1958), 521–542 Google Scholar.

3 Kendall and Stuart, “The Law of the Cubic Proportion …”

4 A recent study which assumes the cube law is Theil, Henri, “The Cube Law Revisited,” Journal of the American Statistical Association, 65 (09, 1970) 1213–1219 CrossRefGoogle Scholar.

5 March, “Party Legislative Representation …”

6 Owen, G., Game Theory (Philadelphia: W. B. Saunders, 1968), pp. 88–93 Google Scholar.

7 We would like to thank Professors Mark Kac, Jean-Jacques Moreau, and Frank Stenger for advice, and Professor Guillermo Owen for information about this type of game. As far as possible we shall keep to the terminology and notation of Owen, Game Theory.

8 Optimal strategies exist if

in which case v is called the value of the game.

9 Thiel, “The Cube Law Revisited,” shows one way of doing this.

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