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

Abstract

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).

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Correspondence to M. J. Albizuri.

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This work has been partially supported by the Ministry of Science and Innovation (PID2019-105291GB-I00), and by UPV/EHU (GIU20/019)

Appendix

Appendix

Proof of Lemma 5

By 3DPI, without a loss of generality, we can suppose that \(\measuredangle aed=\measuredangle aec=\pi /2\) (fixing e,  and moving c and d towards or outwards a). By DP and 3DPI, we have to prove

$$\begin{aligned}&\varphi _{0}\left( \left\{ 0,1,2,3\right\} ,u_{\left\{ 0,1,2,3\right\} },p_{\left\{ 0,1,2,3\right\} }\right) \\&\quad =\varphi _{0}\left( \left\{ 0,1,2,3\right\} ,u_{\left\{ 0,1,2,3\right\} },q_{\left\{ 0,1,2,3\right\} }\right) +\varphi _{0}\left( \left\{ 0,1,2,3\right\} ,u_{\left\{ 0,1,2,3\right\} },s_{\left\{ 0,1,2,3\right\} }\right) -\frac{1}{4}. \end{aligned}$$

Let \(d^{\prime }=d+(b-a)\), \(e^{\prime }=e+(b-a)\) and \(c^{\prime }=c+(b-a)\) (Fig. 13). Consider \({\widetilde{N}}=\left\{ 0,1,2,3,4,5\right\} \) and \( {\widetilde{p}},{\widetilde{q}},{\widetilde{s}}\in C ^{{\widetilde{N}}}\) such that \({\widetilde{p}}_{\left\{ 0,1,2,3\right\} }=p_{\left\{ 0,1,2,3\right\} }\) , \({\widetilde{q}}_{\left\{ 0,1,2,3\right\} }=q_{\left\{ 0,1,2,3\right\} }\), \( {\widetilde{s}}_{\left\{ 0,1,2,3\right\} }=s\), \(\left( {\widetilde{p}}_{4}, {\widetilde{p}}_{5}\right) =\left( c^{\prime },d^{\prime }\right) \), \(\left( {\widetilde{q}}_{4},{\widetilde{q}}_{5}\right) =\left( e^{\prime },d^{\prime }\right) \) and \(\left( {\widetilde{s}}_{4},{\widetilde{s}}_{5}\right) =\left( c^{\prime },e^{\prime }\right) .\)

Fig. 13
figure13

In this figure c′, d′ and e′ are added

By DP and 3DPI, the equality to prove is

$$\begin{aligned} \varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}}\right) +\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{s}}\right) - \frac{1}{4}. \end{aligned}$$
(6)

Since \(\varphi \) is a power index,

$$\begin{aligned} \sum \limits _{j=0}^{5}\varphi _{j}\left( {\widetilde{N}},u_{{\widetilde{N}} },{\widetilde{p}}\right) =1. \end{aligned}$$
(7)

By AN and RI, taking Q the plane through \(a+0.5\left( b-a\right) \) with normal vector \(b-a\), equality (7) turns into

$$\begin{aligned} 2\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) +2\varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) +2\varphi _{2}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =1. \end{aligned}$$
(8)

(i) We obtain another expression for \(\varphi _{2}\left( {\widetilde{N}},u_{ {\widetilde{N}}},{\widetilde{p}}\right) \). DP and 3DPI imply

$$\begin{aligned} \varphi _{2}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =\varphi _{2}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}}\right) \text {,} \end{aligned}$$
(9)

and since \(\varphi \) is a power index,

$$\begin{aligned} \sum \limits _{j=0}^{5}\varphi _{j}\left( {\widetilde{N}},u_{{\widetilde{N}} },{\widetilde{q}}\right) =1. \end{aligned}$$

Again, AN and RI, with Q the plane through \(a+0.5\left( b-a\right) \) with normal vector \(\left( b-a\right) \), transform this equality into

$$\begin{aligned} 2\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}}\right) +2\varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}}\right) +2\varphi _{2}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}}\right) =1. \end{aligned}$$

Taking into account that \(\left( \measuredangle {\widetilde{q}}_{0}\widetilde{q }_{1}{\widetilde{q}}_{2},\measuredangle {\widetilde{q}}_{0}{\widetilde{q}}_{1} {\widetilde{q}}_{4},\measuredangle {\widetilde{q}}_{2}{\widetilde{q}}_{1} {\widetilde{q}}_{4}\right) =\left( \pi /2,\pi /2,\pi /2\right) \), Lemma 4 implies

$$\begin{aligned} \varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}}\right) = \frac{1}{8}, \end{aligned}$$

and hence

$$\begin{aligned} 2\varphi _{2}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}}\right) = \frac{3}{4}-2\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}} \right) . \end{aligned}$$
(10)

Therefore, this equality and (9) imply

$$\begin{aligned} 2\varphi _{2}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) = \frac{3}{4}-2\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}} \right) . \end{aligned}$$
(11)

(ii) We obtain another expression for \(\varphi _{1}\left( {\widetilde{N}},u_{ {\widetilde{N}}},{\widetilde{p}}\right) \). Similarly, by DP and 3DPI it holds

$$\begin{aligned} \varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =\varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{s}}\right) , \end{aligned}$$

and reasoning as in (i) we obtain

$$\begin{aligned} 2\varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{s}}\right) = \frac{3}{4}-2\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{s}} \right) . \end{aligned}$$

And hence,

$$\begin{aligned} 2\varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) = \frac{3}{4}-2\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{s}} \right) . \end{aligned}$$
(12)

Substituting (11) and ( 12) in (8), equality (6) is obtained.

Finally, if \(\measuredangle cae=\measuredangle dae\), the equality to prove is

$$\begin{aligned} \varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =2\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}}\right) - \frac{1}{4}. \end{aligned}$$

By AN and RI, with Q the plane through e with normal vector \(c-e\), we have

$$\begin{aligned} \varphi _{2}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}}\right) =\varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{s}}\right) , \end{aligned}$$

which, together with (10) and ( 12), implies

$$\begin{aligned} \varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{q}}\right) =\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{s}}\right) . \end{aligned}$$

Substituting this equality in (6), we get the result we require. \(\square \)

Proof of Lemma 7

By 3DPI, without loss of generality, we can suppose that \(\measuredangle aed=\measuredangle aec=\pi /2\). Let \(a^{\prime }=e+(e-a)\), \(b^{\prime }=b+2(e-a)\), \(c^{\prime }=c+(b-a)\), \(d^{\prime }=d+(b-a)\) and \(e^{\prime }=e+(b-a)\) (Fig. 14), and consider \({\widetilde{N}} =\left\{ 0,1,2,3,4,5,6,7\right\} \), \({\widetilde{p}}\in C ^{{\widetilde{N}} }\) such that \({\widetilde{p}}_{\left\{ 0,1,2,3\right\} }=p_{\left\{ 0,1,2,3\right\} }\), \(\left( {\widetilde{p}}_{4},{\widetilde{p}}_{5},{\widetilde{p}} _{6},{\widetilde{p}}_{7}\right) =\left( a^{\prime },b^{\prime },c^{\prime },d^{\prime }\right) \), and \({\widetilde{q}}\in C ^{{\widetilde{N}} \backslash \{1,6\}}\) such that \({\widetilde{p}}_{{\widetilde{N}}\backslash \{1,6\}}={\widetilde{q}}_{{\widetilde{N}}\backslash \{1,6\}}\).

Fig. 14
figure14

In this figure a′, b′, c′, d′ and e′ are added

By DP and 3DPI,

$$\begin{aligned} \varphi _{2}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =\varphi _{2}\left( {\widetilde{N}}\backslash \{1,6\},u_{{\widetilde{N}} \backslash \{1,6\}},{\widetilde{q}}\right) . \end{aligned}$$
(13)

RI and AN imply

$$\begin{aligned} \varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =\varphi _{3}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =\varphi _{4}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =\varphi _{5}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) \end{aligned}$$

and

$$\begin{aligned} \varphi _{2}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =\varphi _{7}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) ,\varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =\varphi _{6}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) , \end{aligned}$$

and therefore, taking the definition of a power index into account,

$$\begin{aligned} 2\varphi _{2}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =1-4\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) -2\varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) . \end{aligned}$$
(14)

RI and AN imply \(\varphi _{2}\left( {\widetilde{N}}\backslash \{1,6\},u_{ {\widetilde{N}}\backslash \{1,6\}},{\widetilde{q}}\right) =\varphi _{7}\left( {\widetilde{N}}\backslash \{1,6\},u_{{\widetilde{N}}\backslash \{1,6\}}, {\widetilde{q}}\right) \) and

$$\begin{aligned} \varphi _{0}\left( {\widetilde{N}}\backslash \{1,6\},u_{{\widetilde{N}} \backslash \{1,6\}},{\widetilde{q}}\right)= & {} \varphi _{3}\left( {\widetilde{N}} \backslash \{1,6\},u_{{\widetilde{N}}\backslash \{1,6\}},{\widetilde{q}}\right) \\= & {} \varphi _{4}\left( {\widetilde{N}}\backslash \{1,6\},u_{{\widetilde{N}} \backslash \{1,6\}},{\widetilde{q}}\right) =\varphi _{5}\left( {\widetilde{N}} \backslash \{1,6\},u_{{\widetilde{N}}\backslash \{1,6\}},{\widetilde{q}}\right) , \end{aligned}$$

and hence, taking the definition of a power index into account,

$$\begin{aligned} 2\varphi _{2}\left( {\widetilde{N}}\backslash \{1,6\},u_{{\widetilde{N}} \backslash \{1,6\}},{\widetilde{q}}\right) =1-4\varphi _{0}\left( {\widetilde{N}} \backslash \{1,6\},u_{{\widetilde{N}}\backslash \{1,6\}},{\widetilde{q}}\right) . \end{aligned}$$
(15)

Substituting ( 14) and ( 15) in ( 13) , we get:

$$\begin{aligned} -2\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) -\varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}}\right) =-2\varphi _{0}\left( {\widetilde{N}}\backslash \{1,6\},u_{{\widetilde{N}} \backslash \{1,6\}},{\widetilde{q}}\right) . \end{aligned}$$

Since \(\varphi _{1}\left( {\widetilde{N}},u_{{\widetilde{N}}},{\widetilde{p}} \right) \ge 0\), then \(\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}}, {\widetilde{p}}\right) \le \varphi _{0}\left( {\widetilde{N}}\backslash \{1,6\},u_{{\widetilde{N}}\backslash \{1,6\}},{\widetilde{q}}\right) \). By DP and 3DPI, it holds \(\varphi _{0}\left( {\widetilde{N}},u_{{\widetilde{N}}}, {\widetilde{p}}\right) =\varphi _{0}\left( N,u_{S},p\right) \) and \(\varphi _{0}\left( {\widetilde{N}}\backslash \{1,6\},u_{{\widetilde{N}}\backslash \{1,6\}},{\widetilde{q}}\right) =\varphi _{0}\left( N,u_{S},q\right) \), and hence \(\varphi _{0}\left( N,u_{S},p\right) \le \varphi _{0}\left( N,u_{S},q\right) \). \(\square \)

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Albizuri, M.J., Goikoetxea, A. The Owen–Shapley Spatial Power Index in Three-Dimensional Space. Group Decis Negot 30, 1027–1055 (2021). https://doi.org/10.1007/s10726-021-09746-x

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Keywords

  • Power index
  • Owen–Shapley spatial index

JEL Classification

  • C71