Abstract
Under the most basic of assumptions of Euclidean preferences, majority rule erupts into cycling in two or more dimensional space, and no alternative remains undefeated. The resulting McKelvey’s Chaos TheoremMcKelvey’s Chaos theorem (McKelvey 1976) forces scholars to reconsider basic assumptions about the rational behavior of political actors and their attempts to form coalitions. The government formation literature remains divided on how to best solve the problem. More recently, proposed models either assume cabinet ministers are virtual dictators over their policy jurisdiction (Laver and Shepsle 1996) or rely on complex game-theoretic arguments, which do not lend themselves to empirical verification (Baron 1991; Diermeier and Merlo 2000). This chapter builds on the fuzzy maximal set model developed in Chap. 4. It presents a fuzzy maximal set multi-dimensional model to predict the outcome of the government formation process. We conclude by comparing the predictions made by the model using CMP against actual governments formed after European Parliamentary elections between 1945–2002.
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- 1.
Formally, an actor \(i\) possesses thick indifference if for \(a,b\in X\), \(a\ne b \nRightarrow aP_{i}b\) or \(bP_{i}a\) where \(P_{i}\) is the strict preference relation for player \(i\).
- 2.
- 3.
In \(k\)-dimensional Euclidean space, \(A_{i}\) would be a \(k\times k\) matrix.
- 4.
The reader should note there does not exist any \(n\)-tuple in Fig. 6.3 where \(\sigma _{i}>0\) for all three players.
- 5.
See Schofield (1993) for an important exception.
- 6.
When the maximal set is empty under thick indifference, Mordeson et al. (2011) propose using a fuzzy uncovered set. Nonetheless, the fuzzy uncovered set is a less than ideal solution set because it is almost always the Pareto set, thus not effectively reducing the number of predicted coalitions.
- 7.
\(\rho _{i}(x,y)=(0.75-0.25+r)\wedge 1\,\mathrm {because}\,\sigma _{i}(x)\ge \sigma _{i}(y).\rho _{i}(y,x)=1-[(0.75-0.25+1-r)\wedge 1]=1-[(0.5+1-r)\wedge 1\,\mathrm {because}\,\sigma _{i}(y)\ge \sigma _{i}(x)\).
- 8.
See Appendix B for the formal argument.
- 9.
See Appendix B for the formal argument.
- 10.
See Appendix B for the formal argument.
- 11.
See Appendix B for the formal argument.
- 12.
See Appendix B for the formal argument.
- 13.
For a more thorough discussion on t-norms see Triangular Norms by Klement and Pap (2000).
- 14.
A party’s economic policy is constructed by the following formula: per414 \(+\) per401 \(-\) per412 \(-\) per404 \(-\) per403. A party’s foreign affairs’ policy is constructed by the following formula: per104 \(-\) per107 \(-\) per106 \(-\) per105 \(-\) per103.
- 15.
Because the bootstrapped CMP comprises discrete policy dimensions for each party, this book uses a specific design for such purposes. For further information on two-dimensional kernel density estimation see Wand and Jones (1995). In addition, for more explanation on the interaction Kernel estimation and bootstrapping, see Schucany and Polansky (1997).
- 16.
Percentages rounded to nearest tenth.
- 17.
Since neither variables are normally distributed, it may be more appropriate to use Spearman’s rho. In this case, \(r=-0.31\).
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Casey, P.C. et al. (2014). Predicting the Outcome of the Government Formation Process: A Fuzzy Two-Dimensional Public Choice Model. In: Fuzzy Social Choice Models. Studies in Fuzziness and Soft Computing, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-319-08248-6_6
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