An Application of Power Indices for the Family of Weighted Majority Games in Partition Function Form

An application of power indices for the family of weighted majority games in partition function form J. M. Alonso-Meijide1,5, L. M. Armijos-Toro2,3,5, B. Casas-M´endez3,5, and M. A. Mosquera4,5 1MODESTYA Research Group, Department of Statistics, Mathematical Analysis and Optimisation, Faculty of Sciences, University of Santiago de Compostela, Campus de Lugo, 27002 Lugo, Spain. 2Departamento de Ciencias Exactas, Universidad de las Fuerzas Armadas ESPE, Sangolqu´ı, Ecuador. 3MODESTYA Research Group, Department of Statistics, Mathematical Analysis and Optimisation, Faculty of Mathematics, University of Santiago de Compostela, Campus Vida, 15782 Santiago de Compostela, Spain. 4Universidade de Vigo, Departamento de Estat´ıstica e Investigaci´on Operativa, 32004 Ourense, Spain. 5CITMAga, 15782 Santiago de Compostela, Spain. Abstract Based on Holler (1982), Colomer and Mart´ınez (1995), and Armijos-Toro et al. (2021) we propose two power indices to measure the influence of the players in these class of weighted majority games in partition function form. We compare the new power indices with their original versions on the class of games in characteristic function form. Finally, we use both new power indices and the two power indices for games in partition function form studied in Alonso-Meijide et al. (2017) to study the distribution of power in the National Assembly of Ecuador that emerged after the elections of February 7, 2021. Key words: Weighted majority games, Partition function form games, Minimal winning embedded coalitions, Deegan-Packel power index, Colomer-Mart´ınez power index, Public Good power index. Math. Subj. Class. (2020): 91A12, 91A80 1 1 Introduction Game theory is a mathematical discipline that is dedicated to the study of decision problems in which various players interact. In a cooperative game the players have mechanisms that allow them to take binding agreements. One of the main lines of research in cooperative game theory is the study of simple games and power indices. In a simple game, the worth of a coalition is 1 or 0, and then, there are two types of coalitions: winning and losing. Simple games are supposed to be monotonic, in the sense that the enlargement of a winning coalition cannot cause it to become a losing one. Frequently, simple games model voting situations. A power index provides a measure of a voter’s ability to change the outcome of a voting. Many different power indices have been defined and studied. The most well known are the Shapley-Shubik power index (Shapley and Shubik, 1954) and the Banzhaf power index (Banzhaf, 1964). Both power indices assign to a player a measure that is based on the contributions the player makes by joining other coalitions. A winning coalition is minimal if it becomes a losing one if any of its members cease to be part of it, that is, all the players of a minimal winning coalition are decisive. Deegan and Packel (1978) and Holler (1982) proposed two power indices that assign a measure of power to a player taking into account only the minimal winning coalitions to which he belongs: the Deegan-Packel power index and the Public Good power index, respectively. The Deegan-Packel power index is based on the assumption that all minimal winning coalitions are equally likely, and that within a coalition all its players are equally important. The Public Good power index does not take into account the number of players who are part of the minimal winning coalitions, but the cardinality of the set of minimal winning coalitions to which a player belongs. Other power indices that follow similar arguments are those proposed in Johnston (1978) and ´ Alvarez-Mozos et al. (2015). The Johnston power index takes into account only those winning coalitions in which there are some decisive players. The power index proposed by ´ Alvarez-Mozos et al. (2015) only takes into account winning coalitions that do not contain any null player. A particular class of simple games is the family of weighted majority games. A simple game is a weighted majority if there is a set of weights for the players and a quota, such that a coalition is winning if and only if the sum of the weights of the players of the coalition (that is, the weight of the coalition) is not less than the quota. Colomer and Mart´ınez (1995) proposed a power index that is specific for the family of weighted majority games. The Colomer-Mart´ınez power index, like the Deegan-Packel or the Public Good power indices, is also based on minimal winning coalitions. This power index has in common with the Deegan-Packel power index that all the minimal winning coalitions are considered equally 2 likely, but it does not consider that all the players are equally important. The relevance of the players of a minimal winning coalition is directly related with their weights. Barua et al. (2005) defined a different power index for the family of weighted majority games. Although this power index is based on the Banzhaf power index, when absolute majority is required, the power index defined in Barua et al. (2005) assigns power in a proportional way to the weight of each player, analogous to the way the Colomer-Mart´ınez power index acts. In Armijos-Toro et al. (2021), a new power index for the family of weighted majority games was defined and axiomatically characterized. This new power index assigns to a player an amount that is defined considering ideas of both the Public Good and Colomer-Mart´ınez power indices, because it takes into account: i) the cardinality of the minimal winning coalitions to which he belongs and ii) his weight in the majority game. In addition to the simple games discussed above, two other relevant types of cooperative games are as follows: games in characteristic function form and games in partition function form. In a game in characteristic function form a coalition gets an amount regardless of how the rest of the players organize. The characteristic function assigns a value to each subset of players. Note that simple games are a subclass of the games in characteristic function form. However, in the more general model of games in partition function form (Thrall and Lucas, 1963), what a coalition gets depends on the arrangement of the rest of the players. The partition function assigns a value to each embedded coalition, that is, to each pair formed by a coalition (called active coalition) and a partition of the players outside the active coalition. Returning to the voting situations modeled by the simple games, the model of games in partition function form has proven to be useful to represent situations with plurality rules. In these cases, if no candidate is supported by a qualified majority, that which gets more votes gets the government. Thus, in a weighted majority game in partition function form, an embedded coalition is considered winning if the weight of the active coalition is greater than or equal to the weight of any of the coalitions that constitute the partition of the rest of the players. If two or more coalitions have the same greatest weight, that is, in case there are draws, to determine the active coalition for the winning embedded coalition a breaking rule (number of votes for example) has to be used. In games in characteristic function form, a solution concept is intended to distribute the value of the total coalition among all players and two of the most prominent solution concepts are the so-called Shapley (Shapley, 1953) and Banzhaf values, respectively. Myerson (1977) studied a generalization of the Shapley value for games in partition function form. Dutta et al. (2010) studied a family of values that generalizes the Shapley value. de Clippel and Serrano (2008) proposed the externality-free Shapley value. Bolger (1983) and Bolger (1990) studied generalizations of the Banzhaf value for games in partition function form. In ´ Alvarez-Mozos et al. (2017), the 3 restriction of the externality-free Shapley value (the externality-free Shapley-Shubik power index) was axiomatically characterized. In ´ Alvarez-Mozos and Tejada (2015), an alternative generalization of the Banzhaf value (the so called ordinal Banzhaf power index) was proposed for the family of simple games in partition function form. In Alonso-Meijide et al. (2017) a new class of simple games in partition function form was defined. Moreover, generalizations of the Deegan-Packel and Public Good power indices for these simple games in partition function form are proposed and characterized. To define these power indices, a notion of inclusion among embedded coalitions and a related property of monotonicity are introduced. Roughly speaking, an embedded coalition is larger when the active coalition does not decrease and the partition of the rest of players becomes finer. Carreras and Maga˜na (2008) used a similar idea even though in a different setting. There is also a large body of literature in which the methodology described above is applied to real problems, particularly in the field of politics, to measure the power of the different political parties represented in parliaments or other organizations. In Alonso-Meijide et al. (2017), the new power indices for simple games in partition function form there introduced, were used to study the distribution of power in the Parliament of Andalusia, an autonomous community in Spain, that emerged after the elections of March 22, 2015. In Ar´evalo-Iglesias and ´ Alvarez-Mozos (2020), some other well-known power indices extended to simple games in partition function form were applied to the Parliament of another autonomous community in Spain, the Basque Country. Armijos-Toro et al. (2021) presented an application of their new power index for the family of weighted majority games to the results of the elections held in 2021 for the formation of a new National Assembly of Ecuador. In this last case, the so-called legislative benches, also known as blocks, are the official political groupings within the Assembly, with the right to have authority within the different legislative commissions. If initially, after the elections, there is a configuration of these blocks, it may happen that several additional parliamentary groups are formed during the course of the legislative period, due to splits within the legislative benches, which raises the interest in analyzing the evolution of power within this Assembly over the course of a single legislative period. In this paper, the Colomer-Mart´ınez power index and the power index proposed in Armijos-Toro et al. (2021), are generalized for the model of weighted majority games in partition function form. They are also applied to the analysis of power within the National Assembly of Ecuador. To do this, we analyze minimal winning embedded coalitions in partition function form (plurality rule). Under plurality voting, the winning option is the one that received the highest number of votes, although this group has not necessarily the majority. Among the countries that use the plurality rule for elections are Canada and Great 4 Britain. We use the National Assembly of Ecuador as a real-world numerical example to illustrate the computation of the proposed indices and compare their results. The rest of the paper is organized as follows. In Section 2 we introduce some preliminaries and the basic notation for weighted majority games in partition function form. In Section 3, we generalize to this context of games in partition function form the Colomer-Mart´ınez power index and the power index proposed in Armijos-Toro et al. (2021). Finally, in Section 4 a real example from the political field, the National Assembly of Ecuador, is used to illustrate the new power indices. To complete our study, we compute several power indices from the literature and compare the results. 2 Simple games in characteristic and partition function form In this section we introduce the notation and preliminary concepts necessary for our study focusing on weighted majority games in partition function form. 2.1 Simple games in characteristic function form A game in characteristic function form is a pair (N, v) where Nis the finite set of players and v: 2N−→ Ris the characteristic function of the game satisfying v(∅) = 0. Non-empty subsets of Nare called coalitions. The number v(S) is to be understood as the worth of coalition Sregardless of how the player of N\Sorganize. A game (N, v) is monotone when v(S)≤v(T) for all S⊆T⊆N. A simple game is a monotone game such that v(N) = 1 and, for every S⊆N,v(S) = 0 or v(S) = 1. We denote by SI the class of simple games. Let (N, v)∈ SI then, a coalition S⊆Nis a winning (resp. losing) coalition when v(S)=1 (v(S) = 0). W(v) denotes the set of winning coalitions for the game (N, v). A simple game (N, v) is called decisive whenever S∈W(v) if, and only if, N\S /∈W(v) for each S⊆N. A winning coalition S⊆Nis a minimal winning coalition if there exists no T⊂Ssuch that Tis a winning coalition. We denote by M(v) the set of minimal winning coalitions of the game (N, v) and let Mi(v) = {S∈M(v) : i∈S}for each i∈N. Let Nbe a set of players and S⊆N,S=∅. The game (N, uS)∈ SI (called unanimity game of S) is the (non-decisive) simple game such that M(uS) = {S}, that is, for every T⊆N, uS(T) =      1 if S⊆T 0 otherwise. (1) 5 It is well-known that (N, uS) : S∈2N\∅constitutes a basis of the vector space of games in characteristic function form with set of players N. Let (N, v),(N, v′)∈ SI. Their union is the game (N, v ∨v′)∈ SI such that, for all S⊆N,S∈W(v∨v′) if S∈W(v) or S∈W(v′); or in an equivalent way, (v∨v′)(S) = max{v(S), v′(S)}, for every S⊆N. Two games (N, v),(N, v′)∈ SI are mergeable if for all pairs of coalitions S, T ⊆Nsuch that S∈M(v) and T∈M(v′), it holds that S⊈Tand T⊈S. If (N, v),(N, v′)∈ SI are mergeable, then M(v)∩M(v′) = ∅and M(v∨v′) = M(v)∪M(v′). A power index for the family of simple games in characteristic function form is a mapping, f, that assigns to every simple game (N, v)∈ SI a vector f(N, v)∈RN, where fi(N, v) describes the power of agent iin the game (N, v). Throughout this paper, we only consider efficient power indices, that is, Pi∈Nfi(N, v)=1. The definitions of two power indices, the Deegan-Packel power index (Deegan and Packel, 1978) and the Public Good power index (Holler, 1982) that have in common that they only take into account the minimal winning coalitions of the game are presented below. Definition 2.1. The Deegan-Packel power index (DP ) is the power index defined for every (N, v)∈ SI and i∈Nby DPi(N, v) = 1 |M(v)|X S∈Mi(v) 1 |S|. Definition 2.2. The Public Good power index (PG) is the power index defined for every (N, v)∈ SI and i∈Nby PGi(N, v) = |Mi(v)| P j∈N |Mj(v)|. Although both power indices use only minimal winning coalitions, for the computation of the DP power index the number of players that are part of each minimal winning coalition is taken into account, while for the PG power index only the number of minimal winning coalitions to which a player belongs is used, regardless of their size. A null player in a game (N, v)∈ SI is a player i∈Nsuch that Mi(v) = ∅. Two players i, j ∈Nare symmetric in a game (N, v)∈ SI if, for all coalition S⊆N\{i, j}such that S /∈W(v), S∪{i} ∈ W(v) if and only if S∪{j} ∈ W(v). A power index satisfies the property of null player if fi(N, v) = 0 when iis a null player in (N, v). A power index satisfies the property of symmetry if fi(N, v) = fj(N, v) when i, j ∈Nare symmetric in (N, v). Both power indices, DP and PG, satisfy the null player and symmetry properties. Moreover, DP and PG power indices of a union game (N, v ∨v′) of two mergeable games (N, v) and (N, v′) can be computed as a weighted sum of the corresponding power indices of the games (N, v) 6 and (N, v′), with different weights in each case (see Deegan and Packel, 1978; Holler and Packel, 1983). For every pair of mergeable simple games (N, v),(N, v′)∈ SI it holds that: DP(N, v ∨v′) = |M(v)|DP (N, v) + |M(v′)|DP(N, v′) |M(v∨v′)|and PG(N, v ∨v′) = Pj∈N|Mj(v)|PG(N, v) + Pj∈N|Mj(v′)|PG(N, v′) Pj∈N|Mj(v∨v′)|. In this paper, we study the family of weighted majority games, a particular class of simple games. Definition 2.3. A simple game (N, v)∈ SI is a weighted majority game if there exist a non-negative vector of weights, w= (w1, w2, ..., wn), where wiis the weight of player i∈N, and a quota q > 0such that a coalition S∈W(v)if and only if w(S) = Pi∈Swi≥q. We denote by SIW the class of weighted majority games. We identify a weighted majority game by the tuple [q;w1, w2, ..., wn], or [q;w] when no confusion is possible. A majority game is a weighted majority game such that wi= 1, for all i∈N. If nis an odd number then the majority game [(n+ 1)/2; 1, ..., 1] is decisive. Colomer and Mart´ınez (1995) defined a power index for the family of weighted majority games. This power index could be considered as a non-symmetric generalization of the Deegan-Packel power index. Definition 2.4. The Colomer-Mart´ınez power index (CM) is the power index defined for every [q;w]∈ SIW and i∈Nby CMi(q;w) = 1 |M(q;w)|X S∈Mi(q;w) wi Pj∈Swj . In Armijos-Toro et al. (2021) the HCM power index, a new power index for weighted majority games, was proposed. The definition of the HCM power index shares ideas with the Colomer-Mart´ınez and Public Good power indices. Like the Public Good power index, the total number of minimal winning coalitions to which a player belongs is considered, and like the Colomer-Mart´ınez power index, his weight is taken into account to define the power index. Definition 2.5. The Holler-Colomer-Mart´ınez power index is the power index defined for every [q;w]∈ SIW and i∈Nby HCMi(q;w) = |Mi(q;w)|wi Pj∈N|Mj(q;w)|wj . 7 The HCM power index computes the power of each player as the proportion, balanced by its weights in the game, of the number of minimal winning coalitions to which he belongs. An alternative expression for the HCM power index of a player i∈Nin a simple game [q;w]∈ SIW is HCMi(q;w) = PS∈Mi(q;w)wi Pj∈NPS∈Mj(q;w)wj =X S∈Mi(q;w) wi PT∈M(q;w)Pj∈Twj . When the Deegan-Packel and the Public Good power indices are restricted to the family of weighted majority games, the main difference between these indices and the Colomer- Mart´ınez and the HCM power indices is that the last two ones are not symmetric. In a simple game with a unique minimal winning coalition S, the Deegan-Packel and the Public Good power indices assign the same power to all players in S, but the Colomer-Mart´ınez and the HCM power indices assign to each player of Sa power directly related with his weight. Formally, given (N, v)∈ SIW, determined by [q;w], and S∈2N\∅ with M(v) = {S}, then DPi(N, v) = PGi(N, v)=1/|S|for every i∈S, meanwhile CMi(q;w) = HCMi(q;w) = wi/Pj∈Swj. Moreover, for every game (N, v)∈ SIW, being M(v) = {S}, every i, j ∈S are symmetric and then DPi(N, v) = PGi(N, v) = DPj(N, v) = PGj(N, v), CMi(q;w)wj=CMj(q;w)wi=HCMi(q;w)wj=HCMj(q;w)wi. So, the following property is satisfied by the Colomer-Mart´ınez and the HCM power indices. A power index on SIW satisfies the property of weighted symmetry if wjfi(q;w) = wifj(q;w) when i, j ∈Sand M(q;w) = {S}. Moreover, Colomer-Mart´ınez and the HCM power indices also satisfy some merging properties adapted to the class of weighted majority games (see Armijos-Toro et al., 2021). 2.2 Simple games in partition function form A partition of a finite set Nis a collection {P1, P2, . . . , Pm}such that Pi⊆Nfor all i∈N, Si∈NPi=N, and Pi∩Pj=∅for all pair i, j ∈Nwith i=j. We denote by P(N) the set of partitions of a finite set N. We assume that the empty set is an element of every partition, that is, ∅ ∈ Pfor every P∈ P(N). A partition Pis coarser than Qif every block B∈Qis included in some block A∈P, i.e. P⊇Q. Then (P(N),⊇) is a lattice, called the partition lattice. With this ordering, the bottom element of the lattice is the finest partition {{1}, ..., {n}}, while the top element is the coarsest partition {N}. An embedded coalition of Nis a pair (S;P) where P∈ P(N) and S∈P, is the active coalition in P. We denote by ECNthe set of embedded coalitions of N, i.e. ECN={(S;P) : P∈ P(N) and S∈P}. 8 We say that a player i∈Nparticipates in an embedded coalition (S;P)∈ECNif player i belongs to S. We simplify and write S∪iand S\iinstead S∪ {i}and S\ {i}, respectively. Given P∈ P(N) and i∈N, we denote by P(i) the element of Pthat contains i, i.e., P(i)∈P and i∈P(i). A game in partition function form is a pair (N, v), where Nis the finite set of players and v:ECN→Ris the partition function of the game satisfying v(∅;P) = 0 for every P∈ P(N). The number v(S;P) is to be understood as the worth of coalition Swhen the players are organized according to P. In a game in partition function form it is possible that v(S;P)=v(S;P′), for two pairs of partitions P,P′∈ P(N) with S∈Pand S∈P′. The set of games in partition function form with common set of players Nis denoted by GNand the set of games in partition function form with an arbitrary set of players is denoted by G. Is is easy to notice that GNis a vector space over R. Indeed, de Clippel and Serrano (2008) devised a basis of the vector space that generalizes the basis of games in characteristic function form that consists of unanimity games defined in Eq. (1). Given (S;P)∈ECN, with S=∅, let N, e(S;P)∈ G be defined for every (T;Q)∈ECNby e(S;P)(T;Q) =      1 if S⊆Tand ∀T′∈Q−T,∃S′∈Psuch that T′⊆S′ 0 otherwise, (2) where Q−T∈ P(N\T) denotes the partition Q\{T}. de Clippel and Serrano (2008) showed that N, e(S;P): (S;P)∈ECNand S=∅constitutes a basis of GN. In this paper we are concerned with a subclass of Gthat generalizes simple games in characteristic function form as introduced by von Neumann and Morgenstern (1944). For doing so, Alonso-Meijide et al. (2017) developed a concept of monotonicity for games in partition function form. The intuition behind monotonic games is that the enlargement of a coalition cannot cause a decrease in its worth. Therefore, in order to generalize this idea, Alonso-Meijide et al. (2017) used a notion of inclusion for embedded coalitions that will be of key importance for its results and that it is implicitly formulated in Eq. (2). Definition 2.6. Let Nbe a finite set and (S;P),(T;Q)∈ECN. We define the inclusion among embedded coalitions as follows: (S;P)⊑(T;Q)⇐⇒ S⊆Tand ∀T′∈Q−T,∃S′∈Psuch that T′⊆S′. For instance, ({1};{1},{2,3,4})⊑({1,2};{1,2},{3,4}) and ({1,2};{1,2},{3,4})⊑({1,2};{1,2},{3},{4}). Nevertheless ({1};{1},{2},{3},{4}) and ({1,2};{1,2},{3,4}) are not comparable. Note that whenever S=∅, (S;P)⊑(T;Q) if and only if e(S;P)(T;Q) = 1. According to the above 9 we will use the criterion that, in the event of a tie, none of the coalitions is considered the winner. This criterion is used in many voting systems: in the event that there is no single most voted option, the voting has to be repeated. In the case of the National Assembly of Ecuador there are few ties and either of the above two options for breaking ties lead to very similar results. 4.1 Original formation of the National Assembly of Ecuador: May 2021 First, we analyze the initial situation of the National Assembly of Ecuador in May 2021. In the case of ties, the vote received by each of the parties in the February 2021 general elections is used. In Table 3, we observe the 34 minimal winning embedded coalitions for the National Assembly of Ecuador in May 2021. Table 4 shows the two ties found in determining the winning embedded coalitions. In this table, the last four columns show the number of seats of the different coalitions that make up the partition. As mentioned above, the active coalition for these partitions will be the one that, having the highest number of assembly members, has also obtained the highest number of votes in the February 2021 general elections. In the initial situation of the National Assembly of Ecuador in May 2021, we observe that UNES is present in 22 of the 34 active coalitions. Therefore, UNES is the party with the most power in the National Assembly of Ecuador in May 2021. UNES is also the only party capable of having active coalitions of cardinality one (9 of 34). Likewise, UNES is present in all active coalitions of cardinality two (12). The party of the president of Ecuador, CREO, is present in 10 of the 34 active coalitions. Table 5 presents the results of the calculation of different power indices for the game in partition function form (plurality rule) in the National Assembly of Ecuador in May 2021. UNES is the party with the most power in the National Assembly of Ecuador in May 2021. Note that, although MUPP has more assembly members than ID and PSC, according to the Deegan-Packel power index, ID and PSC have more power than MUPP. This is because ID and PSC are present in 3 of the 12 active coalitions of cardinality two, while MUPP is only present in 1 of these 12 active coalitions. MUPP, ID, and PSC are each present in 11 of the 34 minimal winning embedded coalitions. Of the four proposed power indices, the Public Good power index gives the least power to UNES, but it still has the most power. This is because only the cardinality of the set of minimal winning embedded coalitions to which each player belongs is taken into account. Recall that UNES is present in 22 out of 34 minimal winning embedded coalitions, while the rest of the legislative benches are present in 10 or 11 of them. 16 Table 3: Minimal winning embedded coalitions - National Assembly of Ecuador in May 2021. Active coalition Partition {UNES} {ID, PSC, CREO},{MUPP, IND},{UNES} {UNES} {ID, CREO, IND},{MUPP, PSC},{UNES} {UNES} {PSC, CREO, IND},{MUPP, ID},{UNES} {UNES} {MUPP, CREO},{ID, IND},{UNES},{PSC} {UNES} {MUPP, CREO},{ID, PSC},{UNES},{IND} {UNES} {MUPP, CREO},{PSC, IND},{UNES},{ID} {UNES} {ID, IND},{PSC, CREO},{UNES},{MUPP} {UNES} {ID, PSC},{CREO, IND},{UNES},{MUPP} {UNES} {ID, CREO},{PSC, IND},{UNES},{MUPP} {UNES, MUPP} {ID, PSC, CREO, IND},{UNES, MUPP} {UNES, ID} {MUPP, PSC, IND},{UNES, ID},{CREO} {UNES, ID} {MUPP, CREO, IND},{UNES, ID},{PSC} {UNES, ID} {MUPP, PSC, CREO},{UNES, ID},{IND} {UNES, PSC} {MUPP, ID, IND},{UNES, PSC},{CREO} {UNES, PSC} {MUPP, ID, CREO },{UNES, PSC},{IND} {UNES, PSC} {MUPP, CREO, IND},{UNES, PSC},{ID} {UNES, CREO} {MUPP, ID, IND},{UNES, CREO},{PSC} {UNES, CREO} {ID, PSC, IND},{UNES, CREO},{MUPP} {UNES, CREO} {MUPP, PSC, IND},{UNES, CREO},{ID} {UNES, IND} {MUPP, PSC, CREO},{UNES, IND},{ID} {UNES, IND} {MUPP, ID, CREO},{UNES, IND},{PSC} {UNES, CREO, IND} {UNES, CREO, IND},{MUPP, ID, PSC} {MUPP, ID, PSC} {MUPP, ID, PSC},{UNES, CREO},{IND} {MUPP, ID, PSC} {MUPP, ID, PSC},{UNES, IND},{CREO} {MUPP, ID, PSC} {MUPP, ID, PSC},{CREO, IND},{UNES} {MUPP, ID, CREO} {MUPP, ID, CREO},{PSC, IND},{UNES} {MUPP, PSC, CREO} {MUPP, PSC, CREO},{ID, IND},{UNES} {MUPP, ID, IND} {MUPP, ID, IND},{PSC, CREO},{UNES} {MUPP, PSC, IND} {MUPP, PSC, IND},{ID, CREO},{UNES} {MUPP, CREO, IND} {MUPP, CREO, IND},{ID, PSC},{UNES} {ID, PSC, IND} {ID, PSC, IND},{MUPP, CREO},{UNES} {MUPP, PSC, CREO, IND} {MUPP, PSC, CREO, IND},{UNES, ID} {MUPP, ID, CREO, IND} {MUPP, ID, CREO, IND},{UNES, PSC} {ID, PSC, CREO, IND} {ID, PSC, CREO, IND},{UNES},{MUPP} 17 Table 4: Partitions that result in ties - National Assembly of Ecuador in May 2021. Partition Seats P1Seats P2Seats P3Seats P4 {ID, PSC, IND},{MUPP, CREO},{UNES}49 39 49 – {ID, PSC, IND},{UNES},{MUPP},{CREO}49 49 27 12 Table 5: Power indices - National Assembly of Ecuador in May 2021. Power indices UNES MUPP ID PSC CREO IND DP 0.4510 0.1078 0.1176 0.1176 0.1054 0.1005 PG 0.2933 0.1467 0.1467 0.1467 0.1333 0.1333 CM 0.5496 0.1415 0.0944 0.0944 0.0572 0.0630 HCM 0.5334 0.1470 0.0980 0.0980 0.0594 0.0643 4.2 Formation of the National Assembly of Ecuador: from Jun 2021 to Dec 2021 Table 6 shows the 37 minimal winning embedded coalitions for the National Assembly of Ecuador in June 2021. Let us note that there are five ties in determining the winning embedded coalitions (see Table 7). As mentioned above, there is no winning embedded coalition in any of these partitions. UNES is present in 28 of the 37 minimal embedded coalitions and it is the only legislative bench in active coalitions of cardinality one. Note that, although BAN and MUPP have more assembly members than PSC and ID, the latter appear in more minimal winning embedded coalitions (9 and 11, respectively) than the former (8 each). Table 6: Minimal winning embedded coalitions - National Assembly of Ecuador in June 2021. Active coalition Partition {UNES} {MUPP, IND},{BAN, PSC},{UNES},{ID} {UNES} {MUPP, IND},{BAN, ID},{UNES},{PSC} {UNES} {MUPP, IND},{ID, PSC},{UNES},{BAN} {UNES} {MUPP, ID},{BAN, IND},{UNES},{PSC} {UNES} {MUPP, ID},{BAN, PSC},{UNES},{IND} {UNES} {MUPP, ID},{PSC, IND},{UNES},{BAN} {UNES} {MUPP, PSC},{BAN, IND},{UNES},{ID} {UNES} {MUPP, PSC},{BAN, ID},{UNES},{IND} {UNES} {MUPP, PSC},{ID, IND},{UNES},{BAN} {UNES} {BAN, IND},{ID, PSC},{UNES},{MUPP} {UNES} {BAN, ID},{PSC, IND},{UNES},{MUPP} 18 Table 6: (continued) Active coalition Partition {UNES} {BAN, PSC},{ID, IND},{UNES},{MUPP} {UNES} {ID, PSC, IND},{UNES},{MUPP},{BAN} {UNES, BAN} {MUPP, ID, PSC, IND},{UNES, BAN} {UNES, MUPP} {BAN, ID, PSC, IND},{UNES, MUPP} {UNES, PSC} {MUPP, BAN, IND},{UNES, PSC},{ID} {UNES, ID} {MUPP, BAN, IND},{UNES, ID},{PSC} {UNES, PSC} {MUPP, ID, IND},{UNES, PSC},{BAN} {UNES, IND} {UNES, IND},{MUPP, BAN},{ID, PSC} {UNES, ID} {UNES, ID},{MUPP, BAN},{PSC, IND} {UNES, PSC} {UNES, PSC},{MUPP, BAN},{ID, IND} {UNES, ID} {BAN, PSC, IND},{UNES, ID},{MUPP} {UNES, ID} {MUPP, PSC, IND},{UNES, ID},{BAN} {UNES, PSC} {BAN, ID, IND},{UNES, PSC},{MUPP} {UNES, IND} {MUPP, ID, PSC},{UNES, IND},{BAN} {UNES, IND} {BAN, ID, PSC},{UNES, IND},{MUPP} {MUPP, BAN} {ID, PSC, IND},{MUPP, BAN},{UNES} {UNES, ID, IND} {UNES, ID, IND},{MUPP, BAN, PSC} {UNES, PSC, IND} {UNES, PSC, IND},{MUPP, BAN, ID} {MUPP, BAN, PSC} {MUPP, BAN, PSC},{UNES, IND},{ID} {MUPP, BAN, ID} {MUPP, BAN, ID},{UNES, PSC},{IND} {MUPP, BAN, ID} {MUPP, BAN, ID},{UNES, IND},{PSC} {MUPP, ID, IND} {MUPP, ID, IND},{BAN, PSC},{UNES} {MUPP, ID, PSC} {MUPP, ID, PSC},{BAN, IND},{UNES} {BAN, ID, IND} {BAN, ID, IND},{MUPP, PSC},{UNES} {BAN, ID, PSC} {BAN, ID, PSC},{MUPP, IND},{UNES} {MUPP, BAN, PSC, IND} {MUPP, BAN, PSC, IND},{UNES, ID} We could observe changes in the minimal winning embedded coalitions for each game from June to December 2021 in the National Assembly of Ecuador. In July 2021, two of the partitions that resulted in ties in June 2021 become minimal winning embedded coalitions, namely {{MUPP, PSC, IND},{BAN, ID},{UNES}} with active coalition {MUPP,PSC, IND}, and {{BAN, PSC, IND},{MUPP, ID},{UNES}} with active coalition {BAN, PSC, IND}. The cardinality of the set of minimal winning embedded coalitions is then 39. Table 8 shows the unique tie found in determining the winning coalitions in this case. 19 Table 7: Partitions that result in ties - National Assembly of Ecuador in June 2021. Partition Seats P1Seats P2Seats P3Seats P4 {BAN, MUPP, PSC},{UNES, ID},{IND}64 64 9 – {BAN, PSC, IND},{MUPP, ID},{UNES}48 41 48 – {MUPP, PSC, IND},{BAN, ID},{UNES}48 41 48 – {BAN, PSC, IND},{UNES},{MUPP},{ID}48 48 25 16 {MUPP, PSC, IND},{UNES},{BAN},{ID}48 48 25 16 Table 8: Partition that results in tie - National Assembly of Ecuador in July 2021. Partition Seats P1Seats P2Seats P3 {BAN, MUPP, PSC},{UNES, ID},{IND}63 63 11 On 12 October 2021, the minimal winning embedded coalitions are 40. The winning embedded coalition {{UNES, ID, PSC},{MUPP, BAN, IND}} with active coalition {UNES, ID, PSC}and the winning embedded coalition {{BAN, MUPP, PSC},{UNES, ID},{IND}} with active coalition {BAN, MUPP, PSC}(this partition resulted in tie in July 2021, see Table 8) become minimal winning embedded coalitions. Moreover, with respect to July 2021, the minimal winning embedded coalitions associated with partitions {{MUPP, BAN, IND}, {UNES, PSC},{ID}} and {{MUPP, BAN, IND},{UNES, ID},{PSC}} change their active coalitions to {MUPP, BAN, IND}in both cases. Lastly, the embedded coalition {{MUPP, BAN, PSC, IND},{UNES, ID}} with active coalition {MUPP, BAN, PSC, IND}ceases to be minimal since it contains the winning embedded coalition {{MUPP, BAN, IND},{PSC}, {UNES, ID}} with active coalition {MUPP, BAN, IND}. The minimal winning embedded coalitions on 26 October 2021 are the same as on 12 October 2021. There are no ties in either case. Finally, we observe only one change in December 2021. The minimal winning embedded coalition {{BAN, ID, PSC},{UNES, IND},{MUPP}} with active coalition {UNES, IND}is out since its partition results in tie (see Table 9). So, we have 39 minimal winning embedded coalitions in December 2021. Table 10 shows the distribution of power of the different legislative benches in the National Assembly of Ecuador from June to December 2021. In the period of analysis, we observe that UNES is the legislative bench with the most power in the Assembly. The Deegan-Packel and Public Good power indices do not change the power assigned to MUPP Table 9: Partition that results in tie - National Assembly of Ecuador in December 2021. Partition Seats P1Seats P2Seats P3 {BAN, ID, PSC},{UNES, IND},{MUPP}56 56 25 20 Table 10: Power indices - National Assembly of Ecuador. Benches Power indices Jun 21 Jul 21 12 Oct 21 26 Oct 21 Dec 21 UNES DP 0.5450 0.5171 0.4875 0.4875 0.4872 PG 0.3889 0.3590 0.3293 0.3293 0.3250 CM 0.6560 0.6179 0.5808 0.5821 0.5782 HCM 0.6346 0.5990 0.5618 0.5618 0.5547 MUPP DP 0.0788 0.0833 0.1000 0.1000 0.1026 PG 0.1111 0.1154 0.1341 0.1341 0.1375 CM 0.0889 0.0943 0.1167 0.1164 0.1188 HCM 0.0944 0.0983 0.1217 0.1217 0.1248 BAN DP 0.0788 0.0833 0.1000 0.1000 0.1026 PG 0.1111 0.1154 0.1341 0.1341 0.1375 CM 0.0889 0.0973 0.1167 0.1201 0.1301 HCM 0.0944 0.1024 0.1217 0.1266 0.1398 ID DP 0.1171 0.1111 0.1042 0.1042 0.1068 PG 0.1528 0.1410 0.1341 0.1341 0.1375 CM 0.0791 0.0752 0.0645 0.0645 0.0660 HCM 0.0831 0.0801 0.0682 0.0682 0.0699 PSC DP 0.0968 0.1090 0.1042 0.1042 0.1068 PG 0.1250 0.1410 0.1341 0.1341 0.1375 CM 0.0546 0.0668 0.0645 0.0645 0.0660 HCM 0.0595 0.0701 0.0682 0.0682 0.0699 IND DP 0.0833 0.0962 0.1042 0.1042 0.0940 PG 0.1111 0.1282 0.1341 0.1341 0.1250 CM 0.0326 0.0485 0.0567 0.0525 0.0409 HCM 0.0340 0.0501 0.0584 0.0536 0.0409 and BAN in spite of changes observed in the structure of the legislative benches. Nevertheless, the Colomer-Mart´ınez and HCM power indices show variations according to these changes. Next, we provide some general comments. First, UNES is allocated with about half or more of the total power at almost all considered periods, with the exception of the Public Good power index. This may be due to the fact that, in the formation of winning embedded coalitions, the participation of UNES is preferable since it is the legislative bench with the highest number of votes. Second, with respect to the Deegan-Packel and Public Good power indices when the simple majority rule is used, these power indices do not show variation for the legislative benches in the period of analysis (see Armijos-Toro et al., 21 2021). This is because these power indices consider as valid the minimal winning coalitions, which do not change over the analyzed period. Whereas, when using the plurality rule, the minimal winning embedded coalitions change. Thus, the Deegan-Packel and Public Good power indices present updates in the distribution of power in the National Assembly of Ecuador. Third, the Colomer-Mart´ınez and HCM power indices, specifically defined for weighted majority games, better represent the alterations made in the Assembly structure over the period of analysis. Finally, another notable result is that ID, using the Deegan-Packel and Public Good power indices, has greater power than BAN or MUPP. This is despite the fact that ID has fewer assembly members than BAN or MUPP. However, ID participates in a larger number of minimal winning embedded coalitions than BAN or MUPP. As noted, the Deegan- Packel and Public Good power indices distribute power equally among minimal winning embedded coalitions. However, the Colomer-Mart´ınez and HCM power indices consider in their calculation the number of assembly members of each legislative bench. 5 Conclusions While the study of games in partition function form and simple games in partition function form is more abundant, it is less so for the case of weighted majority games in partition function form. In this paper we extend to this context two power indices axiomatically characterized in Armijos-Toro et al. (2021) in the context of classical weighted majority games, one of them introduced in Colomer and Mart´ınez (1995) and the other defined in Armijos-Toro et al.’s own work by combining the ideas of Colomer and Mart´ınez (1995) and Holler (1982). Both extensions are illustrated by means of the case of the National Assembly of Ecuador in 2021 and the results obtained are compared with two power indices proposed and axiomatically characterized in Alonso-Meijide et al. (2017) in the context of simple games in partition function form that turn out to be extensions of the Deegan-Packel power index (Deegan and Packel, 1978) and the Public Good power index (Holler, 1982) defined on the family of simple games. In the case considered and under the adoption of the plurality rule, the two new power indices proposed turn out to be sensitive to the different relevant ingredients present, such as the resulting minimal winning embedded coalitions and the number of representatives of each party or political grouping. They are also capable of reacting to the measurement of power in a changing and complex situation such as the National Assembly of Ecuador. It is worthwhile to further investigate these new proposals of power indices for weighted majority games in partition function form in the future. 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