Thursday, 7 October 2010

Review of "Protectionism and blocking power indices" by Francesc Carreras

A separating family
Most of the voting power literature has focussed on the ability to make or change decisions. This is surprising given the way the voting rules  in many of the well-known applications, such as the European Union Council of Ministers have been designed. Indeed the attention has been more on how to prevent change rather than how to foster it. This paper is an exception.


Carreras (2009) divides coalitions into four groups: decisive winning coalitions, conflicting winning coalitions, blocking coalitions and strictly losing coalitions depending on whether a coalition or its complement is winning or losing. The focus is then on blocking coalitions that can prevent, although themselves cannot make changes. A separating family of coalitions is a set of coalitions, such that for each maximal inclusive chain of coalitions contains an element.

While it is common to define voting games in terms of winning coalitions, Carreras studies games defined in terms of blocking coalitions. The blocking coalitions Q (for Q bar) of a voting game defined from blocking coalitions Q is Q precisely when Q is a separating family. Not surprisingly a similar result holds for winning coalitions: The game defined from the blocking coalitions in W will have W as winning coalitions precisely when QW is a separating family. When this does not hold, then the mapping is not unique and the number of possible blocking collections Q depends on the number of coalitions that are not separated by Q. The main result gives the implications on the multiplicities on the voting games defined from Q and prove that in case of non-degenerate multiplicity there will be both proper and improper games generated.

The ability to block is just another manifestation of power and it is therefore natural to apply power measures to blocking. Several blocking indices are introduced analogously to power indices. While these give additional information to power indices, interestingly in games with many blocking coalitions the Banzhaf strict protectionism index will be close to the Banzhaf value. Some additional properties are discussed, but an axiomatic characterisations of the newly introduced indices is not presented.

The review will be published in Mathematical Reviews.

Reference
Carreras F. (2009) Protectionism and blocking power indices, Top 17: 70-84

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