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 \({\mathbb {R}}^{m}\), 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).

受 Owen (Nav Res Logist Quart 18:345–354, 1971) 先前关于该主题的工作的启发，Shapley（功率指数和非对称概括的比较。兰德公司，圣莫尼卡，1977）介绍了 Owen-Shapley 空间权力指数，考虑个人的意识形态位置，用欧几里德空间中的向量表示\({\mathbb {R}}^{m}\)，来衡量他们的实力。在这项工作中，我们研究了三维空间中的 Owen-Shapley 空间力量指数。Peters 和 Zarzuelo（Int J Game Theory 46:525–545, 2017）针对二维空间中的个体对该指数进行了研究，但指出了二维特征的局限性。在这项专注于三维空间的工作中，我们为空间一致性博弈提供了一个明确的公式，这使得计算任何空间博弈的 Owen-Shapley 空间权力指数成为可能。我们还使用两个不变的位置公理等对 Owen-Shapley 空间功率指数进行了表征。最后，我们在二维和三维情况下计算巴斯克议会的这个权力指数。