The Owen–Shapley Spatial Power Index in Three-Dimensional Space

Inspired by Owen’s (Nav Res Logist Quart 18:345–354, 1971) previous work on the subject, Shapley (A comparison of power indices and a non-symmetric generalization. Rand Corporation, Santa Monica, 1977) introduced the Owen–Shapley spatial power index, which takes the ideological location of individuals into account, represented by vectors in the Euclidean space $${\mathbb {R}}^{m}$$ R m , to measure their power. In this work we study the Owen–Shapley spatial power index in three-dimensional space. Peters and Zarzuelo (Int J Game Theory 46:525–545, 2017) carried out a study of this index for individuals located in two-dimensional space, but pointed out the limitation of the two-dimensional feature. In this work focusing on three-dimensional space, we provide an explicit formula for spatial unanimity games, which makes it possible to calculate the Owen–Shapley spatial power index of any spatial game. We also give a characterization of the Owen–Shapley spatial power index employing two invariant positional axioms among others. Finally, we calculate this power index for the Basque Parliament, both in the two-dimensional and three-dimensional cases. We compare these positional indices against each other and against those that result when classical non-positional indices are considered, such as the Shapley–Shubik power index (Am Polit Sci Rev 48(3):787–792, 1954) and the Banzhaf-normalized index (Rutgers Law Rev 19:317–343, 1965).