Probabilistic Owen-Shapley spatial power indices

Games and Economic Behavior 136 (2022) 524–541 Contents lists available at ScienceDirect Games and Economic Behavior journal homepage: www.elsevier.com/locate/geb Probabilistic Owen-Shapley spatial power indices M.J. Albizuri∗, A. Goikoetxea University of the Basque Country, Faculty of Economics and Business, Department of Quantitative Methods, Bilbao, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 21 April 2021 Available online 26 October 2022 JEL classification: C71 Keywords: Power index The spatial Owen-Shapley index Probabilistic spatial power indices In this paper we study probabilistic Owen-Shapley spatial power indices, which are generalizations of the Owen-Shapley spatial power index (1977). We provide an explicit formula for calculating these spatial indices for unanimity games and give an axiomatic characterization of the family of probabilistic Owen-Shapley spatial power indices. We employ an equal power change property, a spatial dummy property, anonymity, a positional invariance property, and a positional continuity property. Some examples are also given. ©2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction Cooperative game theory has been applied successfully to measure the power of agents in voting situations, which are represented by simple games. A winning coalition is assigned a worth of one, and a losing one a worth of zero. The ShapleyShubik index (1954) and the Banzhaf index (1965) can be seen as the best known indices for measuring power. They both take into account whether the presence of an agent changes a losing coalition into a winning one, i.e. whether an agent is pivotal. In defining the Shapley-Shubik index, all possible orderings of agents are considered. Each ordering has a pivotal agent associated with it: the one whose addition to the coalition formed by the previous agents changes that coalition from losing to winning. When all orderings are equally probable, the probability of an agent being pivotal is by definition his or her Shapley-Shubik index. Orderings are not considered in defining the Banzhaf index. The index of an agent is the number of coalitions in which he or she is pivotal. A paper by Owen (1971)inspired Shapley (1977)to propose a spatial power index: the Owen-Shapley power index (see also Owen and Shapley, 1989). In this new model, ideological differences between the agents can be taken into account. It is formalized by means of a spatial game, which is a simple game together with a constellation of agent profiles, i.e., a set of vectors in the Euclidean space Rmthat represents the ideological locations of voters. The different dimensions can be seen as ideological considerations or criteria, so each position represents the “ideal point” (of highest preference) in the space. Shapley (1977)writes that the use of the Euclidean space Rm“seems to leave us ample scope for capturing many kinds of political and ideological parameters without an excess of abstraction and generality”. An issue is formalized by Shapley (1977)through a vector r∈Rm. A player in position xis more in favor of rthan a player in position yif the scalar product r·xis less than or equal to r·y. Therefore, players can be ordered from the most to the least enthusiastic with respect to an issue, which implies that one of them is pivotal in that ordering. When all issues are equally likely, the probability of a player being pivotal is his or her Owen-Shapley spatial power index. *Corresponding author. E-mail addresses: [email protected] (M.J. Albizuri), goikoetx[email protected] (A. Goikoetxea). https://doi.org/10.1016/j.geb.2022.10.004 0899-8256/©2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 Peters and Zarzuelo (2017)study the Owen-Shapley spatial power index when there are two dimensions. They provide a formula for calculating the index for unanimity games and give an axiomatic characterization by means of a transfer axiom, a dummy axiom, anonymity and two invariant positional axioms (reflection invariance and positional invariance). A natural variation of the Owen-Shapley spatial power index is to consider that not all issues are equally probable. For example, think about a regional parliament of a country, in which local issues are more relevant than the state ones (see Section 6). Barr and Passarelli (2009) consider that there is a probability distribution over issues, defined by a continuous density function. They analyze the distribution of power in the Council of the EU with two dimensions. The stances toward the EU on international issues and domestic issues are the two dimensions that are taken into account. In this paper we also consider the variation of the Owen-Shapley spatial power index with two dimensions when there is a(ny) probability distribution over issues. We call these spatial power indices probabilistic Owen-Shapley spatial power indices. We give a formula for calculating the indices for unanimity games. Therefore, the index of any spatial game can be easily calculated by means of linear combinations of indices of unanimity games. We conduct an axiomatic study and prove that the family of probabilistic Owen-Shapley spatial power indices can be obtained by means of the axioms employed by Peters and Zarzuelo (2017), dropping reflection invariance and adding continuity. Reflection invariance requires the index not to change when the constellation of agents is shifted or rotated. However, as mentioned above, in a regional parliament, where local issues are more relevant than the state ones, all issues are not regarded equally likely. Consider also the possible presence of the agenda setter effect, which influences the importance or likelihood of the different issues at stake. Within the EU, the Commission would play the role of agenda setter in the Council (see Passarelli and Barr, 2007). Thus, reflection invariance should not be imposed. On the contrary, positional invariance changes relative positions of agents and is satisfied by probabilistic Owen-Shapley spatial power indices. Therefore, a transfer axiom, a dummy axiom, positional invariance and continuity generate the family of probabilistic Owen-Shapley spatial power indices. We also give some illustrative examples, including an application to the Basque Parliament. Other spatial power indices have also been introduced over the years. For example Shenoy (1982)extends the Banzhaf index to the spatial setting when voters are represented by points in Rm. Passarelli and Barr (2007)employ the multilinear extension approach for cooperative games (Owen, 1972) to define a spatial power index when issues belong to Rm. AlonsoMeijide et al. (2011)define a spatial power index taking into account lengths of paths connecting players’ positions. Benati and Marzetti (2013)obtain a family that includes both the Shapley-Shubik index and the Owen-Shapley spatial power index modeling voters’ propensity to support an issue through a random utility function. Multinomial values are introduced by Albina-Puente and Carreras (2015)to model different tendencies of agents. Blockmans and Guerry (2015)study the impact of issue saliences and distance selection on the family introduced by Benati and Marzetti (2013). Martin et al. (2017) propose a generalization of the Owen spatial power index (1971). There are also other studies. Álvarez-Mozos et al. (2017) address the problem of extending the Shapley-Shubik index to the class of simple games with externalities. Karos and Peters (2018)develop a class of power indices for effectivity functions. An issue based power index is also introduced by Kong and Peters (2021)by means of orderings of issues. The structure of the paper is as follows. Section 2gives notation and preliminaries. Section 3presents probabilistic OwenShapley spatial power indices. Section 4calculates these spatial power indices for unanimity games. Section 5presents the axiomatic characterization of the family of probabilistic Owen-Shapley spatial power indices. We also show the independence of the axioms employed. Some examples can be found in Section 6. Finally, Section 7gives some concluding remarks and pointers for future work. 2. Preliminaries 2.1. Notation Given x, y ∈R2such that x = y, the half-line which starts at xand passes through yis denoted by [x, y, →), the segment with endpoints xand yby [x, y], and we write (x,y)=[x, y] \{x, y}. The perpendicular bisector line of [x, y]is the line perpendicular to the line through xand ythat passes through 1 2x +1 2y. Given x ∈R2and a line in R2such that x /∈, xdenotes the reflection of xwith respect to . Notice that is the perpendicular bisector line of [x, x]. If x ∈, then x=x. The projection of x ∈R2on a line in R2, i.e. 1 2x +1 2x, is denoted by ¯ x. Given x =(x1, x2)and y =(y1, y2) ∈R2, x ·y =x1y1+x2y2. Given X⊆R2, co(X)refers to the convex hull of X. 2.2. Spatial games and spatial power indices Let Ube a set, the universe of players. A coalition is a finite nonempty subset of U. A transferable utility (TU) game is a pair (N, v)such that Nis a coalition and v :2N→RN, v (∅)=0. A simple game is a TU game (N, v)such that v(S) ∈{0, 1} for all S∈2N, v(N) =1 and v(S) ≤v(T)for all S, T∈2Nwith S⊆T(that is, vis monotonic). Let (N, v)be a simple game. A coalition Sis winning in (N, v)if v(S) =1; otherwise Sis losing. A minimal winning coalition is a winning coalition with no proper winning subcoalition. If v(S) =1 and v(S\{i}) =0, player iis said to be pivotal in S. Denote by GNthe set of all TU games with set of players Nand by SNthe subset of all simple games. 525 M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 Fig. 1. Polar angle θ(r). A constellation for a set of players Nis a vector p =(pi)i∈N∈(R2)Nsuch that pi= pjfor all i, j ∈Nwith i = j. The set of all constellations for Nis denoted by CN. Given p ∈CNand S⊆N, define pS∈(R2)Sby (pS)i=pifor all i ∈S. With a slight abuse of notation, co(pS)denotes the convex hull of the set {pi}i∈S. Given a line in R2and p ∈CN, denote by pthe constellation which is the reflection of pwith respect to , i.e. p=(p i)i∈N. A spatial game with a set of players Nis a triple (N, v, p)such that (N, v) ∈SNand p ∈CN. A spatial power index on A ⊆SNis a function ϕ:A →RNsuch that for all (N, v, p) ∈Ait holds that ϕi(N, v, p) ≥0for all i ∈Nand i∈Nϕi(N, v, p) =1. The Owen-Shapley spatial power index is defined as follows. Let (N, v, p)be a spatial game and consider r∈R2such that ||r|| =1, where ||r|| denotes the Euclidean length of r. Each rrepresents a possible issue to be treated by the agents in N. At each rthe agent i ∈Nwho is pivotal in {j ∈N|r·pj≤r·pi} is calculated and iis said to be pivotal at r. Notice that iis unique except for a finite number of issues r. We assume that all issues are equally likely. The Owen-Shapley spatial power index of i, denoted by i(N, v, p), is the probability of ibeing pivotal at an issue r. We take into account the following geometrical consideration. Given an issue r, let be the line with direction vector r, and for each j ∈Nthe projection ¯ p jon . We say that j ∈Nprecedes k ∈Non if ¯ p jprecedes ¯ p kin the direction of r. Thus, i ∈Nis pivotal at rif and only if iis pivotal in the set of agents who precede him or her (including himself or herself) on . 2.3. Dummy player in spatial games Player i ∈Nis a dummy (Peters and Zarzuelo, 2017) in a spatial game (N, v, p)if pi∈co pS\{i}for every coalition Sin which iis pivotal. Observe that if a player is not pivotal in any coalition, then he or she is a dummy. And if a dummy player is pivotal in a coalition S, then he or she is surrounded, according to p, by players in S, and it can therefore be assumed that the dummy player can not take advantage of his/her pivotal power in S. Observe also that if iis a dummy, then iis never pivotal at any issue r, and therefore, i(N, v, p) =0. There can be two situations. If iis not pivotal in any coalition, iclearly cannot be pivotal at any issue r. If iis pivotal in a coalition S, given that iis a dummy, pi∈co pS\{i}. As a consequence, given any issue rand a line with direction vector r, there is always a player jin Ssuch that ¯ p iprecedes ¯ p j. And hence, icannot be pivotal in the set of the agents who precede him or her (including i). Therefore, nor is ipivotal at r. 3. Probabilistic Owen-Shapley spatial power indices Let Bbe the σ-field generated by subintervals in  =(0,2π]. The Owen-Shapley spatial power index arises when the Lebesgue probability measure λis considered on B. Indeed, there is a bijection from the set of issues, r∈R2:||r||=1, into (0,2π]that associates each issue rwith its polar angle θ(r)∈(0,2π], as depicted in Fig. 1. In defining the OwenShapley spatial power index, if α, β∈(0,2π], α<β, the probability of the issues rsatisfying α<θ(r)≤βis given by λ (α,β]=(β−α)/2π. But not all the areas for the issues might be equally probable when their (Lebesgue) measure is the same. All possible situations are represented by probability measures Pon B. Moreover, the probability of single issues ris required to be zero, i.e. Pmust be non-atomic, as happens for the Lebesgue probabilistic measure. A spatial power index ϕis said to be associated with a probability P on Bif for each spatial game (N, v, p)and i ∈N, ϕi(N, v, p)is the probability of ibeing pivotal at an issue r, when the probability for the issues r, which are represented by θ(r)∈(0,2π], is given by P. We write ϕ=Pand we say that ϕis a probabilistic Owen-Shapley spatial power index. As pointed out above, this is restricted to non-atomic probability measures. 526 M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 Fig. 2. Different positions of −−→ pipj. 4. Probabilistic Owen-Shapley spatial power indices for unanimity games First note that if idenotes the union of the angle or angles formed by issues rat which iis pivotal, then P i(N,v,p)=P(i). Let v =uS, ∅ = S⊆N, be the unanimity game on S, i.e., uS(T)=1ifT⊇S, 0otherwise. Observe that if i /∈S, P i(N, v, p) =0, since iis not pivotal at any issue r. If |S|=1, it is clear that P i(N, v, p) =1 when i ∈S. If |S|>1, let S(p)=i∈S:pi/∈co pS\{i},(1) that is, the set of non dummy players in (N, uS, p). If i ∈S\S(p), then icannot be pivotal at any issue r, and therefore P i(N, v, p) =0. For i ∈S(p)take into account the following. Let i, j ∈Sand an issue r, which can be assumed to start from pi(we can consider ras a free vector). Let be the line through piand pj, ⊥be the perpendicular line to through piand be a line in the direction of r(Fig. 2). Notice that ¯ p jprecedes ¯ p iin the direction of rif and only if ris pointing into the halfplane outside pjwith contour ⊥. Two cases are distinguished. If |S(p)|=2 and i ∈S(p), ihas three expressions according to different positions of −−→ pipj, as depicted in Fig. 2, and accordingly, P i(N,v,p)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Pθ−−→ pipj+π 2,θ−−→ pipj+3π 2if θ−−→ pipj≤π 2, P0,θ−−→ pipj−π 2+Pθ−−→ pipj+π 2,2πif π 2<θ−−→ pipj≤3π 2, Pθ−−→ pipj−3π 2,θ−−→ pipj−π 2if 3π 2<θ−−→ pipj. If |S(p)|>2, consider co pS(p), which is, by definition, the smallest convex set that contains all points pisuch that i ∈S(p). By definition of S(p), the boundary of co pS(p)is the polygon whose vertices are all points pisuch that i ∈S(p). Thus, for i ∈S(p), there are two players j, k ∈S(p)\ {i}such that pjand pkare adjacent vertices to piin co pS(p)(Fig. 3). Consider (resp.  ) to be the line through piand pj(resp. pk); ⊥(resp.  ⊥) the line perpendicular to (resp.  ) through pi, and to be a line in a direction of an issue r. It turns out that ¯ p kprecedes ¯ p iin the direction of rfor all k∈Sif and only if ris pointing outwards from co pS(p)between ⊥and  ⊥. These issues form an arc ithat has several expressions depending on the positions of vectors −−→ pipjand −−→ pipk. For the sake of simplicity we write θ=θ−−→ pipjand θ=θ−−→ pipk. Without loss of generality we assume that θ<θ . Firstly, note that θ−θ=π, since, as written above, pi, pjand pkare vertices of a polygon. Two main cases are distinguished. 1) θ−θ>π. Thus, P i(N,v,p)=Pθ+π 2,θ−π 2. 527 M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 Fig. 3. iwhen |S(p)|>2. 2) θ−θ<π. Thus, P i(N,v,p)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Pθ+π 2,θ +3π 2if θ≤π/2, P0,θ −π 2+Pθ+π 2,2πif θ>π/2andθ<3π/2, Pθ−3π 2,θ −π 2if θ>π/2andθ≥3π/2. Note that the picture in Fig. 3is case 2) when θ>π/2 and θ≥3π/2. 5. Axiomatic characterization of the family of probabilistic Owen-Shapley spatial power indices Peters and Zarzuelo (2017) characterize the Owen-Shapley spatial power index by means of five axioms: Equal Power Change, Dummy Property, Anonymity, Positional Invariance and Reflection Invariance. Probabilistic Owen-Shapley spatial power indices satisfy all but the last of them, which requires symmetry for the issues. The whole family is characterized by adding one axiom: a continuity axiom. The first is equivalent to the transfer axiom of Dubey (1975), as remarked in Dubey et al. (2005). Equal Power Change (EPC) For all set of players N, all p ∈CN, and all v, v, w, w∈SN, if v −v=w −w≥0, then ϕ(N,v,p)−ϕ(N,v,p)=ϕ(N,w,p)−ϕ(N,w,p). According to this axiom, if the same winning coalitions are added when going from vto vas when going from wto w, then the change in power for the players when going from vto vis also the same as when going from wto w. In the second axiom, if (N, v) ∈SNis such that |N|≥2 and i ∈N, then (N\{i}, v−i) ∈SN\{i}is defined by v−i(S)=v(S∪{i})if ∅=S⊆N\{i}, 0ifS=∅. Game (N\{i}, v−i)can be seen as the resulting game when player ileaves and gives consent to others, since any winning coalition in (N, v)containing icontinues being winning in (N\{i}, v−i)when player iis no longer present. Dummy Property (DP) For every spatial game (N, v, p)and every dummy iin (N, v, p), ϕj(N,v,p)=ϕj(N\{i},v−i,pN\{i}) for all j ∈N\{i}. Therefore, the presence of a dummy player does not affect the power of the other players; and the power of a dummy player is zero. Anonymity is required, i.e. power is independent of the names of the players. Given a spatial game (N, v, p)and an injective function σ:N→U, define the spatial game (σ(N), σv, σp)by σv(σ(S)) =v(S)for all S⊆Nand (σp)σ(i)=pi for all i ∈N. Anonymity (AN) For every spatial game (N, v, p)and every injective function σ:N→U, ϕσ(i)(σ(N), σv,σp)=ϕi(N,v,p), 528 M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 for all i ∈N. The following axiom, which is also satisfied by the Owen-Shapley spatial power index, is a spatial invariant axiom. If the relative positions of the agents do not change with respect to a given agent, the index of that agent does not change. Positional Invariance (PI) For all player sets Nand i ∈N, if p, p∈CNsatisfy pi=p iand pj∈[pi, pj, →)for all j ∈N\{i}, then ϕi(N,v,p)=ϕi(N,v,p). The last axiom is a continuity axiom with respect to constellations, which is introduced by Peters and Zarzuelo (2017). Positional Continuity (PC) Let (N, v, p)be a spatial game and {pm}be a sequence of constellations pm∈CNsuch that pm→p. Then, lim m→∞ϕ(N,v,pm)=ϕ(N,v,p). Theorem 1. A spatial power index ϕsatisfies EPC, DP, AN, PI and PC if and only if there exists a non-atomic probability measure Pon Bsuch that ϕ=P. It is worth noting that Peters and Zarzuelo (2017)gave a second characterization adding PC to the five aforementioned axioms, weakening DP. The weak DP axiom requires dummy players to have zero power. In the case of the probabilistic Owen-Shapley spatial power indices, DP can not be weakened, as shown by this counterexample. Let  Pbe a probability measure on Bsuch that  Pπ 4,3π 4=1. Consider the spatial power index that satisfies EPC and coincides with for unanimity games uSsuch that |S|=2 and with  Pwhen |S|>2. This spatial power index satisfies EPC, AN, PI, PC and the weak DP, and it is not a probabilistic Owen-Shapley spatial power index. We prove Theorem 1in three steps (Propositions 1, 2and 3). In the first step we employ this lemma, which is used by Peters and Zarzuelo (2017). It follows from Lemma 2.3 in Einy (1987), see also Einy and Haimanko (2011). Lemma 1. Let ϕbe a spatial power index that satisfies EPC and (N, v, p)be a spatial game such that S1, ..., Skare the minimal winning coalitions of (N, v). Then, ϕ(N,v,p)= ∅=I⊆{1,...,k} (−1)|I|+1ϕ(N,uk∈ISk,p). Proposition 1. If a spatial power index ϕsatisfies EPC, DP, AN, PI and PC, then, for every x ∈R2there exists a non-atomic probability measure Pxon Bsuch that ϕi(N,v,p)=Px i(N,v,p) if pi=x. Proof. Let ϕbe a spatial power index that satisfies EPC, DP, AN, PI and PC, and x ∈R2. A probability measure Pϕ xon B must be found such that ϕi(N,v,p)=Px i(N,v,p) if pi=x. Since Pϕ xis a probability measure, Pϕ x(∅)=0. Consider now subintervals in (0,2π]. If α, β∈(0,2π]satisfy α<β and β−α<π, let i, j, k ∈Uand p ∈C{i,j,k}such that pi=x, the polar angle θ−−→ pipjis smaller than θ−−→ pipk, θ−−→ pipj=β+π 2if β+π 2≤2π, β−3π 2if β+π 2>2π, and θ−−→ pipk=α+3π 2if α+3π 2≤2π, α−π 2if α+3π 2>2π. Fig. 4shows the case in which β+π 2≤2πand α+3π 2≤2π. Define Pϕ x(α,β]=ϕi({i,j,k},u{i,j,k},p). 529 M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 Fig. 4. The case in which β+π 2≤2πand α+3π 2≤2π. Fig. 5. Graphic for i,iand i. This is well defined. Indeed, by PI, jand kcan be located nearer to or further from iin a straight line. And by AN, this index does not depend on the names of the players. If α, β∈(0,2π]satisfy α<β and β−α≥π, then there exists γ∈(0,2π]such that α<γ<β, γ−α<πand β−γ<π, and define Pϕ x(α,β]=Pϕ x(α,γ]+Pϕ x(γ,β]. To prove that it is also well defined take γ∈(0,2π]such that α<γ<β, γ−α<πand β−γ<π, and prove that Pϕ xα,γ+Pϕ xγ,β=Pϕ x(α,γ]+Pϕ x(γ,β].(2) Assume, without loss of generality, that γ<γ; then the above equality becomes Pϕ xα,γ=Pϕ x(α,γ]+Pϕ xγ,γ.(3) By definition of Pϕ x, there exist i, j, k ∈U, and p ∈C{i,j,k}such that pi=xand Pϕ xα,γ=ϕi({i,j,k},u{i,j,k},p). Let nbe the line that passes through piand has a direction vector with polar angle γ(Fig. 5). There exist y ∈pi,pj and z∈(pi,pk)such that the line that passes through yand zis perpendicular to n. Applying ii) of Lemma 2in the Appendix, ϕi({i,j,k},u{i,j,k},p)=ϕi0({i0,i1,j},u{i0,i1,j}, q)+ϕi1({i0,i1,k},u{i0,i1,k}, q), where  q∈C{i0,i1,j}and  q∈C{i0,i1,k}satisfy  qi0=pi, qj=pj, qi1=pi+z−y, and 530 M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 Fig. 6. Graphic for x,y,z∈R2, the polar angle αand the sequences αmand pm.  qi1=pi, qk=pk, qi0=pi+y−z. And taking into account the definition of Pϕ x, equality (3)is proved. Substituting in (2)gives Pϕ xγ,γ+Pϕ xγ,β=Pϕ x(γ,β], which is true because β−γ<πand can be proved with the same reasoning as above. It can be proved similarly that if α, β∈(0,2π]satisfy α<β, then Pϕ x(α,β]= m  k=1 Pϕ x(αk,β k],(4) when (α,β]= m  k=1 (αk,β k] and the intervals (αk,β k]are pairwise disjoint. Lemma 4in the Appendix proves that Pϕ xhas a unique extension (written also Pϕ x) on Bthat is a probability measure. We now prove that Pϕ xis non-atomic, that is, Pϕ x(α)=0for all α∈(0,2π]. Take x, y, z∈R2such that x ∈(y,z)and let αbe the polar angle of a direction vector of the line that passes through x and is perpendicular to the line that passes through yand z(as depicted in Fig. 6). Let αmbe a sequence in (0,2π]such that {αm}→α, i, j, k ∈Uand pm∈C{i,j,k}be a sequence such that pm i=x,pm j=y,pm k→z and Pϕ x(αm,α]=ϕi({i,j,k},u{i,j,k},pm), (5) where αmis the polar angle of a direction vector of the line that passes through xand is perpendicular to the line that passes through xand pm k. Thus, lim m→∞ϕi({i,j,k},u{i,j,k},pm)=ϕi({i,j,k},u{i,j,k},(x,y,z))=0, where the first equality holds by PC and the second by DP. Therefore, by (5), lim m→∞ Pϕ x(αm,α]=0. Since Pϕ xis a probability measure, lim m→∞ Pϕ x(αm,α]=Pϕ x(α), and hence, Pϕ x(α)=0. Finally, we prove that ϕi(N,v,p)=Px i(N,v,p) 531 M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 if pi=x. Since ϕsatisfies EPC, by Lemma 1it is sufficient to consider the equality for unanimity games. By construction, the equality holds for (N, uS, p)if |S|>2 and there are at least three non-dummy players in S. It obviously holds when |S|=1. Therefore, by DP, it is sufficient to consider (S, uS, p)where |S|=2, i.e., S={i,j}⊆N. It must now be proved that ϕi({i,j}, u{i,j}, p)coincides with Pϕ x(i)(see Fig. 2). Let y ∈pi,pjand {ym}⊆R2be a sequence such that {ym}→y. Let k ∈N\{i,j}and pm∈C{i,j,k}be a sequence such that pm {i,j}=p{i,j}and pm k=ym. Thus, PC and DP give ϕi({i,j},u{i,j},p)=lim m→∞ϕi({i,j,k},u{i,j,k},pm). (6) By definition of Pϕ x, it follows that ϕi({i,j,k}, u{i,j,k}, pm) =Pϕ xm i, where m iis a sequence of intervals or unions of two disjoint intervals in (0,2π]whose limit is i. Since Pϕ xis a probability measure, lim m→∞ Pϕ xm i=Pϕ x(i), which, together with (6), implies the required result.  Proposition 2. If a spatial power index ϕsatisfies DP, AN, PI and PC, then, for every x, y ∈R2, Pϕ x=Pϕ y. Proof. i) Let α, β∈(0,2π]. We prove that Pϕ x()=Pϕ y()when  =(α,β]⊆(0,2π], where β−α=π, or  =(α,2π]∪ (0,β]⊆(0,2π], where 2π−α+β=π. Let i, j ∈Uand p ∈C{i,j}such that pi=x, pj=yand  =i, where iis the set formed by the polar angles of the vectors that start at piand point to the half-plane with contour ⊥that does not contain pj(see Fig. 2). Since ϕis a spatial power index, ϕi({i,j},u{i,j},p)+ϕj({i,j},u{i,j},p)=1. By Proposition 1, ϕi({i,j},u{i,j},p)=Pϕ x(i) and ϕj({i,j},u{i,j},p)=Pϕ yj, where j=(0,2π]\i. Substituting the two equalities in the first one and taking into account that Pϕ yis a probability measure on B, Pϕ x(i)=Pϕ y(i)is proved. ii) Let α, β∈(0,2π]⊆(0,2π]. Let  =(α,β]⊆(0,2π], where β−α<π, or  =(α,2π]∪(0,β]⊆(0,2π], where 2π−α+β<π. Let (see Fig. 7) i, j, k, i0, i1∈U, p ∈C{i,j,k}such that pi=x, y ∈pi,pj, z∈(pi,pk)and is the set of polar angles of the vectors that start at piand point to the intersection of the half-plane with contour ⊥that does not contain yand the half-plane with contour  ⊥that does not contain z. By i) of Lemma 2, ϕi({i,j,k},u{i,j,k},p) =ϕi0({i0,i1,j,k},u{i0,i1,j,k},q)+ϕi1({i0,i1,j,k},u{i0,i1,j,k},q), (7) where q ∈C{i0,i1,j,k}satisfies q{j,k}=p{j,k}, qi0=yand qi1=z. Moving pkcloser to yin a straight line, by PI, the first index on the right-hand side of (7)does not change. Moreover, when pk∈co y,pj,z, player kbecomes a dummy player. Thus, DP implies ϕi0({i0,i1,j,k},u{i0,i1,j,k},q)=ϕi0({i0,i1,j},u{i0,i1,j},q{i0,i1,j}). Similarly, approaching pjcloser to zin a straight line, by PI, the second index on the right-hand side of (7)does not change. And player jbecomes a dummy player when pj∈co ({y,pk,z}). Again by DP, ϕi1({i0,i1,j,k},u{i0,i1,j,k},q)=ϕi1({i0,i1,k},u{i0,i1,k},q{i0,i1,k}). Then, (7)turns into ϕi({i,j,k},u{i,j,k},p) =ϕi0({i0,i1,j},u{i0,i1,j},q{i0,i1,j})+ϕi1({i0,i1,k},u{i0,i1,k},q{i0,i1,k}). 532 M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 By definition of Pϕ x, and applying PI (approaching ymcloser to xin a straight line) and DP (i2becomes a dummy player), ϕi0N,uN,pm=Pϕ xπ,3π 2. Applying PI (moving zmcloser to xmin a straight line) and DP (i3becomes a dummy player), ϕi1N,uN,pm=ϕi1{i0,i1,i2},u{i0,i1,i2},pm {i0,i1,i2}.(16) And by PI (moving pm i0and pm i2further from pm i1in straight lines), ϕi1{i0,i1,i2},u{i0,i1,i2},pm {i0,i1,i2}=ϕi1{i0,i1,i2},u{i0,i1,i2},pi1m,(17) where pi1m i1=pm i1, pi1m i0=pm i1−(1,0)and pi1m i2=pm i1+(0,1). Since pm i1→x, it follows that pi1m→pi1, where pi1∈C{i0,i1,i2}satisfies pi1i1=x, pi1i0=x −(1,0)and pi1i2= x +(0,1). And therefore, PC implies lim m→∞ϕi1{i0,i1,i2},u{i0,i1,i2},pi1m=ϕi1{i0,i1,i2},u{i0,i1,i2},pi1, which, together with (16) and (17), implies lim m→∞ϕi1N,uN,pm=ϕi1{i0,i1,i2},u{i0,i1,i2},pi1. By definition of Pϕ x, the right-hand side of this equality equals to Pϕ x3π 2,2π, and hence, lim m→∞ϕi1N,uN,pm=Pϕ x3π 2,2π. Similarly, we have lim m→∞ϕi2N,uN,pm=Pϕ x0,π 2 and lim m→∞ϕi3N,uN,pm=Pϕ xπ 2,π. Consequently, taking limits on both sides of (15), Pϕ xπ,3π 2+Pϕ x3π 2,2π+Pϕ x0,π 2+Pϕ xπ 2,π=1. That is, by (4), Pϕ x(0,2π]=1.  Lemma 4. If ϕis a spatial power index that satisfies DP, AN, PI and PC, then, for every x ∈R2, Pϕ xis a probability measure on B. Proof. We prove that Pϕ xis a probability measure on the field of finite unions of intervals, denoted by B0. The result follows since any probability measure on a field has a unique extension that is a probability measure on the associated σ-field. Pϕ xhas to satisfy several properties in order to be a probability measure on B0. i)We prove in the previous lemma that Pϕ x(0,2π]=1. ii)Pϕ xis countably additive on B0. We prove this in two steps. •First, on the class of intervals. Let (αm,β m]be a finite or infinite sequence of pairwise disjoint intervals. It needs to be proved that if (α,β]=∞  m=1 (αm,β m], then Pϕ x(α,β]=∞  m=1 Pϕ x(αm,β m]. 539 M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 The finite case is already proven (expression (4)), so now the infinite case is considered. Since Pϕ x(αm,β m]≥0for all m ∈N, the terms in this series can be rearranged so that βm+1<β mfor all m ∈Nand β1=β(note also that {αm}→α). Hence, taking into account (4), M  m=1 Pϕ x(αm,β m]=Pϕ x(αM,β] holds for all M∈N, so the equality to be proved is Pϕ x(α,β]=lim M→∞ Pϕ x(αM,β]. We distinguish two cases. If β−α<π, let i, j, k ∈Nand p ∈C{i,j,k}such that pi=xand Pϕ x(α,β]=ϕi({i,j,k},u{i,j,k},p), (18) where α(resp. β) is the polar angle of a direction vector of the line which is perpendicular to the line that passes through piand pk(resp. pj) (Fig. 4). Let {ym}⊆R2be a sequence such that {ym}→pkand αm(for big enough m ∈N) is the above polar angle associated with piand ym. Let pm∈C{i,j,k}such that pm {i,j}=p{i,j}(observe that pm i=x) and pm k=ym. Thus, PC implies ϕi({i,j,k},u{i,j,k},p)=lim m→∞ϕi({i,j,k},u{i,j,k},pm). (19) By the definition of Pϕ x, ϕi({i,j,k},u{i,j,k},pm)=Pϕ x(αm,β], so this equality, together with (18) and (19), implies the required result. If β−α≥π, there exists γsuch that α<γ<β, γ−α<π, (and by (4)) Pϕ x(α,β]=Pϕ x(α,γ]+Pϕ x(γ,β], and Pϕ x(αm,β]=Pϕ x(αm,γ]+Pϕ x(γ,β]. Hence, the equality to be proved reduces to Pϕ x(α,γ]=lim M→∞ Pϕ x(αM,γ], which is true because this is the first case again. •Now we prove that Pϕ xis countably additive on B0. First, if A= m  k=1 Ik∈B0, where Ikare pairwise disjoint intervals, define Pϕ x(A)= m  k=1 Pϕ x(Ik). This is well defined because if A= m  l=1 I l, then A= m  k=1 m  l=1Ik∩I l, and therefore, 540 M.J. Albizuri and A. Goikoetxea Games and Economic Behavior 136 (2022) 524–541 Pϕ x(A)= m  k=1 m  l=1 Pϕ xIk∩I l, and by (4), this double addition coincides with both m  k=1 Pϕ x(Ik)and m  l=1 Pϕ xI l. To prove that Pϕ xis countably additive on B0, let Akbe a sequence of pairwise disjoint elements in B0such that A=∞  k=1 Ak∈B0. Since A ∈B0and Ak∈B0, it follows that A= m  l=1 Iland Ak= mk  l=1 Ik l, where Iland Ik lare pairwise disjoint intervals. 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