Abstract
We present an axiomatic characterization of the Owen–Shapley spatial power index for the case where issues are elements of two-dimensional space. This characterization employs a version of the transfer condition, which enables us to unravel a spatial game into spatial games connected to unanimity games. The other axioms include two conditions concerned particularly with the spatial positions of the players, besides spatial versions of anonymity and dummy. The last condition says that dummy players can be left out in a specific way without changing the power of the other players. We show that this condition can be weakened to requiring dummies to have zero power if we add a condition of positional continuity. We also show that the axioms in our characterization(s) are logically independent.
Highlights
Voting power in political bodies can be represented by simple games, which identify the winning and losing coalitions: a winning coalition can enforce laws, amendments, etc
We show that the Owen–Shapley spatial power index is uniquely characterized by five axioms: a version of the well known transfer condition similar to the one in Einy and Haimanko
Compared to Theorem 3.1, in the following characterization of the Owen–Shapley spatial power index for fixed player set N, the dummy property is replaced by the weak dummy property and positional continuity
Summary
Voting power in political bodies can be represented by simple games, which identify the winning and losing coalitions: a winning coalition can enforce laws, amendments, etc. See Owen and Shapley (1989) for details and a proof of this result Another relation between the Owen–Shapley spatial power index and the evaluation of positions using Euclidian distance has recently been obtained by Martin et al (2014). They show, by using a limit argument, that if pivotalness is based on closeness in terms of Euclidian distance and all possible points in Rm are regarded as ‘issues’, the Owen–Shapley spatial power index again results They clarify the difference and overlap between the original concepts of Owen (1971) and Shapley (1977).
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