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A Shapley Value Analysis of the Proposed Canadian Constitutional Amendment Scheme

Published online by Cambridge University Press:  10 November 2009

D.R. Miller
D.R. Miller
Affiliation:
University of Florida
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Extract

The interactions of a group of non-identical voting units may be studied by applying the concept of the Shapley value from n-person co-operative game theory. In this theory one assumes that voting units, or players, may form coalitions of various kinds in order to achieve success in the game, and one may assign a “value” to each such coalition based on what it can accomplish against arbitrary coalitions of the remaining players. The relative value of an individual player is calculated by considering how much he brings to each coalition he might join, that is, by how much the value of that coalition increases because of his membership, and summing this figure over all coalitions of which he could be a part.

Type
Notes
Information
Canadian Journal of Political Science/Revue canadienne de science politique , Volume 6 , Issue 1 , March 1973 , pp. 140 - 143
Copyright
Copyright © Canadian Political Science Association (l'Association canadienne de science politique) and/et la Société québécoise de science politique 1973

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References

1 Luce, R.D., and Raiffa, H., Games and Decisions (New York, 1957Google Scholar), section 6.8 and chap. 11. For a more recent discussion and an application similar to that presented here, see Banzhaf, J.H. III, One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College,” Villanova Law Review, 13 (1968), 304–12.Google Scholar For a critique see Sickels, R.J., “The Power Index and the Electoral College,” Villanova Law Review, 14 (1968), 92–6.Google Scholar

2 Owen, G., Game Theory (Philadelphia, 1968Google Scholar), chaps. 8 and 9.

3 Lucas, W.F., “An Overview of the Mathematical Theory of Games,” Management Science, 18 (part 2) (1972), 3–19.CrossRefGoogle Scholar

4 Owen, Game Theory, 179 ff.

5 A veto is also held by the federal government. Here, we consider only the votes of the provinces, justifying the simplification by pointing out that we only want to compare regional voting powers. The inclusion of the federal veto in some sense gives a double voting avenue to some people. The larger problem is certainly of interest, but is considerably more complicated.

6 The constraints of regional interests were always maintained. That is, requiring that some sub-coalition have a certain number of members in favour was applied only to sub-coalitions consisting of neighbouring provinces. Abandoning this constraint might lead to closer figures, but it is not clear that there would be any politically workable way of putting such a scheme into practice.

7 Toronto Globe and Mail (10 February 1971), 2.

8 This paper was prepared while the author was attending an institute on deterministic methods in operations research at Cornell University under the sponsorship of the National Science Foundation. I thank Dr William F. Lucas, Director of the Institute, for many helpful suggestions.