Abstract
In this chapter, we view the party charged with forming a governing coalition immediately following an election as attempting to put together a coalition that will, with some compromise, promote its ability to implement its legislative agenda and to influence the legislative outcome in this direction. We thus view the problem of the coalition leader as one of maximizing its influence as measured by the Penrose measure of absolute voting power and subject this hypothesis and three variants to empirical testing using election data from nine countries. Two variants, namely: restricting the maximization process to the set of closed winning coalitions, or likewise but with a further requirement that the winning coalition selected be of minimal range, achieved levels of predictive success comparable to the Leiserson-Axelrod minimal range theory, suggesting that a closer examination of the role of a priori measures of power in political coalition formation may be useful.
This article was originally published in Homo Oeconomicus 25(2), 2008.
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In the context of simple voting games, the Shapley value is usually referred to as the Shapley-Shubik index. See Shapley and Shubik (1954).
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Briefly, on a policy scale in which any two adjacent parties are regarded as one unit distance apart, the range of a coalition is the distance between the two parties in the coalition whose positions on the policy scale are furthest apart. The minimal range theory, due to Leiserson (1966) and to Axelrod (1970), hypothesizes that the coalition formed will be of minimal range.
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See, for instance, Coleman (1971) for a statement to this effect.
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This last point is another important departure from Aumann’s approach. In Aumann’s approach, the Shapley-Shubik value of the leading party in the legislature is computed using a quota that is set equal to a simple majority of the entire legislature as opposed to a simple majority of the size of the alliance.
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At this juncture, one could argue that although the Penrose measure and the Shapley-Shubik index are not co-monotonic, departures are rare and therefore the Shapley-Shubik index will do just as well under the alternative definition of the quota used in the present investigation. This is indeed the case as our own earlier investigation has revealed. But it misses the point of our approach which emphasizes power as influence together with an associated calculus as opposed to power as prize in the game-theoretic sense with a different associated calculus.
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As Felsenthal and Machover (2002, 2008) have explained, denoting the power of party v within the alliance S by ψ v (W s ) and the power of the alliance in the assembly as ψ &s (W|& s ), then the overall power of party v which is a member of the alliance and is denoted ψ v (W∥W s) is equal to ψ v (W s ) · ψ &s (W|& s ).
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The data are taken from de Swaan (1973, p. 269). A simple majority of 71 implies that there were at that time 140 members in the Danish parliament. However, the total number of seats of the four listed parties is only 136. This gap is explained by the fact that de Swaan ignored small parties (or independents) who controlled no more than 2.5 % of the seats in parliament because such parties were very seldom included in governmental coalitions.
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The Penrose measure of a priori voting power of a voter v, denoted ψ v , is equal to the number of winning coalitions in which v is critical divided by the number of coalitions to which v belongs. A coalition is winning if it has sufficient votes to pass a decision, otherwise it is losing. A voter is critical if his defection from a winning coalition renders it losing, or if his joining a losing coalition renders it winning. For n voters there are altogether 2n coalitions (or bi-partitions) of which every voter belongs to 2n − 1 coalitions. In the above example let us consider the alliance consisting of three members whose weights are 42, 17, and 49. This is a winning alliance (coalition) because its combined votes (weights) is 108—which exceeds the quota of 71. If we assume that the internal decision rule of this alliance is a simple majority of its members’ weights (55), then there are altogether four (internal) coalitions within this alliance in which the member with weight 49 (the coalition leader) exists—{49}, {49, 17}, {49, 42}, {49, 42, 17}—of which this member is critical in the second and third coalitions. Consequently the coalition leader’s internal (direct) voting power in this case is 2/4 = 0.5. But since the alliance as a whole controls an absolute majority of the votes within the parliament—and hence is a dictator whose a priori voting power is 1—it follows that the overall (indirect) a priori voting power of the coalition leader in this case is equal to its internal voting power within the alliance multiplied by the alliance’s power within the parliament, i.e., 0.5·1 = 0.5.
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The data are taken from de Swaan (1973, p. 260) who ignored in this case small parties controlling together five seats.
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It is important to make the distinction between: (a) restricting the domain to closed coalitions and picking an arrangement that is maximal from the restricted set; and (b) picking an arrangement from the maximal set that satisfies the restriction. Although (a) and (b) may pick the same arrangement, it is the former that is employed in our analysis in order to test the hypothesis that the maximization of the leader’s voting power in forming a governmental coalition is an important consideration only (or mainly) if the coalition is closed.
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Such an approach is however, distinct from the minimum size principle and it also overcomes the objection raised by Aumann concerning coalitional stability when the minimum size principle is invoked.
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In this chapter, a winning coalition is defined as one that controls over half of the seats in the legislature.
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In a more general context, a prediction is regarded as consistent as long as the actual outcome is included in the predicted set.
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The Central Limit Theorem for independent variables states the following. Let \( X_{1} ,X_{2} ,\, \cdots \) be a sequence of independent random variables having respective means and variances \( \mu_{i} \, = \,E\left( {X_{i} } \right), \) \( \sigma_{i}^{2} \, = \,Var(X_{i} ). \) If (1) the X i are uniformly bounded; that is, for some M, P{|X i | < M} = 1 for all i, and (2) \( \sum\limits_{i = 1}^{\infty } {\mathop \sigma \nolimits_{i}^{2} } \, = \,\infty , \) then \( P\left\{ {\left. {\frac{{\sum\limits_{{i = 1}}^{n} {(X_{i} - \mu _{i} )} }}{{\sqrt {\sum\limits_{{i = 1}}^{n} {\sigma _{i}^{2} } } }} \le a} \right\} \to } \right.\frac{1}{{\sqrt {2\pi } }}\int_{{ - \infty }}^{a} {{\text{e}}^{{{\raise0.7ex\hbox{${ - {\text{x}}^{2} }$} \!\mathord{\left/ {\vphantom {{ - {\text{x}}^{2} } 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} } dx\,as\,n \to \infty . \) See Ross (2002).
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Felsenthal and Machover (2002, 2008) defined a feasible alliance as one in which the overall absolute voting power of each member of the alliance is not smaller than his voting power when no alliance exists. An expedient alliance is defined as one in which the overall absolute voting power of each member of the alliance is larger than his voting power when no alliance exists.
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Acknowledgments
The authors wish to thank Matthew Braham, Moshé Machover, and Frank Steffen for their helpful comments. While working on this chapter Dan Felsenthal was also co-director of the Voting Power and Procedures Programme at the Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science. This programme is supported by Voting Power in Practice Grant F/07 004/AJ from the Leverhulme Trust.
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Chua, V.C.H., Felsenthal, D.S. (2013). Do Voting Power Considerations Explain the Formation of Political Coalitions? A Re-Evaluation. In: Holler, M., Nurmi, H. (eds) Power, Voting, and Voting Power: 30 Years After. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35929-3_23
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