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Coalition building and the power index of parties in this process
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First, it takes all coalitions as equally probable to be built, which is in reality usually not the case. Secondly, in the denominator, one coalition can be included several times as it may comprise of more than one pivotal player. It neither takes into account any of the incentives that make parties cooperate with each other much more often than just a random probability (similar political views, for example).
I will illustrate these queries in an example from a real life.
Slovak parliamentary elections 2002 :
Party Votes [%] Seats Seats [%] Banzhaf index [%]
HZDS 19.50 36 24.00 28.0
SDKÚ 15.09 28 18.66 18.4
Smer 13.46 25 16.66 16.8
SMK 11.16 20 13.33 10.4
KDH 8.25 15 10.00 10.4
ANO 8.01 15 10.00 10.4
KSS 6.32 11 7.33 5.6
Together 81.79 150 100 100
As we can see from the table, in the situation of Slovak 2002 election results, at least three parties were needed to build a coalition. We may also conclude a rule that if party A has more seats in parliament it has higher or the same Banzhaf index as a party with less seats. This is known as the Local Monotonocity principle. It allows, therefore, that party A (here SMK) can have more seats in parliament than party B (here KDH and ANO) and nevertheless has the same value of Banzhaf index – same power in coalition building. Holler (2002) adds that Local Monotonocity is not always present in power indexes , but he doesn’t show any example of such a paradox. Political scientists, however, often find the nonmonotonocity of power as a contradiction to democracy.
Slovak example also nicely visualizes the deficiencies of policy-blind indexes. In this case, the greatest coalition-building potential should have HZDS. In fact, every other party refused to negotiate with HZDS because of it’s negative national as well as international reputation. Actual coalition was built among three parties that took part in governing in the previous term (SDKÚ, SMK and KDH) plus one newly created party (ANO).
This leads us to another possible formulation of the Banzhaf index that incorporates the probability of each coalition being built. After all, it’s much more plausible to expect that the power of a party with certain amount of votes depends not only on the numbers of coalitions in which it is critical but also upon the probabilities by which these various coalitions arise.
If we assign each coalition a probability that it would be formed, with three players we will get something like this:
In this formula, the equals to the sum of probabilities of all coalitions where A is the pivotal voter.
Zdroje: 1. Bilal, Sanoussi, Paul Albuquerque and Madeleine O. Hosli. 2001. “The Probability of Coalition Formation: Spatial Voting Power Indices”. Retrieved: April 4, 2003.
, 2. Felsenthal, D. and Moshe Machover. 1998. The Measurement of Voting Power. Theory and Practice, Problems and Paradoxes. Cheltenham: Edward Elgar., 3. Holler, M. 2002. “How to sell power indices”. Retrieved: April 7, 2003. , 4. Lijphart, A. 1999. Patterns of democracy government forms and performance in thirty-six countries. New Haven: Conn. London Yale University Press., 5. Pajala, A. “The Voting Power and Power Index Website”. Retrieved: April 11, 2003. , 6. Saari, D. 2001. Chaotic Elections! – A Mathematician Looks at Voting. American Mathematical Society., 7. Shapley, Lloyd S. and Martin Shubik. 1954. “A Method for Evaluating the Distribution of Power in a Committee System” American Political Science Review, 48, 787-92., 8. Strom, K. “Coalition building”. In: Encyclopedia of Democracy: 255-258.