Three-Valued Simple Games

In this paper we introduce three-valued simple games as a natural extension of simple games. While simple games are used to evaluate single voting systems, three-valued simple games offer the opportunity for a simultaneous analysis of two different voting systems within the same parliamentary body. This paper analyzes the core and the Shapley value of three-valued simple games. Using the concept of vital players, the vital core is constructed and we show that the vital core is a subset of the core. The Shapley value is characterized on the class of all three-valued simple games. The model is applied to evaluate the relative influence of countries within the current EU-28 Council.


Introduction
In this paper we introduce a new class of transferable utility games, called three-valued simple games. The class of three-valued simple games is a natural extension of the class of simple games, introduced by Von Neumann and Morgenstern (1944) and widely applied in the literature on voting power. In a simple game, a coalition is either 'winning' or 'losing', i.e., there are two possible values for each coalition. The concept of three-valued simple games goes one step further than simple games in the sense that there are three, instead of only two, possible values.
A simple game can, for example, serve as a model for a voting system in which a bill is pitted against the status quo. For instance, Bilbao, Fernández, Jiménez, and López (2002) considered the legislative procedure of the EU-27 Council. This legislative procedure works as follows. Usually, the EU Council acts on the basis of a proposal from the European Commission. When, on the other hand, the EU Council does not act on the basis of a proposal from the European Commission, there are more restrictions for legislation to get accepted. Thus within the EU Council two different voting systems are in place simultaneously. Bilbao et al. (2002) modelled the legislative procedure of the EU Council by two separate simple games, one for the situation in case the legislation is proposed by the European Commission and one for the case that the legislation is not proposed by the European Commission. By the introduction of three-valued simple games, it is possible to model the legislative procedure of the EU Council by means of a single TU-game. Three-valued simple games can also be applied to model a bicameral system. In a bicameral system, legislators are divided into two separate houses and a bill has to be approved by both houses.
This paper formally defines the class of three-valued simple games and focuses on analyzing the core and the Shapley value of these games. In particular, by introducing vital players, primary vital players and secondary vital pairs, we define the vital core. The vital core is shown to be a subset of the core. We show that for a specific subclass of three-valued simple games the core and the vital core coincide. Next, we provide a characterization for the Shapley value on the class of three-valued simple games in the spirit of Dubey (1975). Moreover, the model of three-valued simple games and the Shapley value are applied to analyze the relative influence of the various countries in the legislative procedure of the current EU-28 Council.
The organization of this paper is as follows. Section 2 formally introduces three-valued simple games. In Section 3 the core of three-valued simple games is investigated. Section 4 provides a characterization for the Shapley value on the class of three-valued simple games. Finally, Section 5 provides some concluding remarks on possible generalizations.

Three-valued simple games
In this section we introduce three-valued simple games, a new subclass of transferable utility games. We also provide some practical examples in which the concept of a threevalued simple game is a suitable model.
With N a non-empty finite set of players, a transferable utility (TU) game is a function v : 2 N → R which assigns a number to each coalition S ∈ 2 N , where 2 N denotes the collection of all subsets of N . By convention, v(∅) = 0. Let TU N denote the class of all TU-games with player set N .
A coalition is winning if v(S) = 1 and losing if v(S) = 0. Let SI N denote the class of all simple games with player set N . In this paper we investigate a new subclass of TU-games, the class of three-valued simple games. A game v ∈ TU N is called three-valued simple if Let TSI N denote the class of all three-valued simple games with player set N . The concept of three-valued simple games goes one step further than simple games in the sense that there are three, instead of only two, possible values. Next to the value 0, we have chosen the values 1 and 2. Of course, the relative proportion between these two values may depend on the application at hand. In Section 5 we briefly discuss such a generalization.
Three-valued simple games can be used to model a bicameral legislature, in which the legislators are divided into two separate houses, the lower house and the upper house, and a bill has to be approved by both houses. Example 2.1 shows how a three-valued simple game can be used to model the bicameral legislature in the Netherlands.
Example 2.1. In the bicameral legislature of the Netherlands, the States General of the Netherlands, the lower house is called the House of Representatives and the upper house is called the Senate. The House of Representatives consists of 150 members and the Senate consists of 75 members. The members of the Senate and House of Representatives each represent a political party. The members of the House of Representatives are elected directly by the Dutch citizens. The members of the Senate however are elected indirectly by the members of the provincial councils, who, in turn, are elected directly by the Dutch citizens.
The House of Representatives can accept, reject or revise a bill. A bill is accepted by the House of Representatives only if there is a (simple) majority in favor. When a bill is accepted by the House of Representatives it is forwarded to the Senate. The Senate does not have the right to revise a bill and thus it can only accept or reject a bill, again via (simple) majority voting. If the bill is accepted by the Senate, then the bill becomes a law. 1 Three-valued simple games offer the opportunity to model the bicameral legislature of the Netherlands into a single TU-game. Consider the three-valued simple game v ∈ TSI N where the set of players N is the set of parties. Then, the value v(S) of a coalition S ⊂ N equals: -2, if the members of all parties in the coalition form a majority in both the House of Representatives and Senate, -1, if the members of all parties in the coalition form a majority in the House of Representatives but not in the Senate, -0, otherwise.
Hence, with a i and b i denoting the number of members in the House of Representatives and Senate respectively for party i ∈ N , we have for all S ⊂ N .
As the following example illustrates, three-valued simple games also offer the opportunity for a simultaneous analysis of two different voting rules within the same parliamentary body.
Example 2.2. Some legislative procedures require that any action that may alter the rights of a minority (for example, amending the constitution) has a supermajority requirement (such as a two-thirds majority). In the legislature of the Netherlands, amending the constitution requires a two-thirds majority. In this example we model the power of the parties in the House of Representatives by taking into account the difference between a simple majority and a supermajority. We distinguish between a normal legislative procedure, in which a simple majority is needed, and the revision of the constitution, in which a two-thirds majority is required. Consider the three-valued simple game v ∈ TSI N where the set of players N is the set of parties and the value v(S) of a coalition S ⊂ N equals: -2, if the members of all parties in the coalition form a two-thirds majority in the House of Representatives, -1, if the members of all parties in the coalition form a simple majority in the House of Representatives, -0, otherwise.
As a result, with a i denoting the number of members in the House of Representatives for party i ∈ N , 3 The core of three-valued simple games In this section we investigate the core of three-valued simple games. Using the concept of vital players, the vital core is constructed. The vital core is shown to be a subset of the core.
The core C(v) of a game v ∈ TU N is defined as the set of all allocations x ∈ R N such that i∈N x i = v(N ) (efficiency) and i∈S x i ≥ v(S) for all S ⊂ N (stability). Hence, the core consists of all possible allocations of v(N ) for which no coalition has the incentive to leave the grand coalition. Consequently, if the core is empty, then it is not possible to find a stable allocation of v(N ).
We characterize the core of a three-valued simple game using the concept of vital players. Proposition 3.1 states that only the vital players of a three-valued simple game can receive a positive payoff in the core, while all other players receive zero. We now introduce the concept of vital players. For v ∈ TSI N the set of vital players is defined by Hence, the vital players are those players who belong to every coalition with value 2.
we have x ≥ 0. Then, because of efficiency and stability of x ∈ C(v), Using the concept of vital players, Proposition 3.2 provides a sufficient condition for emptiness of the core of a three-valued simple game.
From Proposition 3.2 it follows that only the set of permissible three-valued simple games may have a non-empty core, where a game v ∈ TSI N is called permissible if the following two conditions are satisfied As a consequence, from now on we focus only on the set of permissible three-valued simple games and define for every permissible three-valued simple game, a reduced game where the player set is reduced to the set of vital players. We define this reduced game in such a way that the core of a permissible three-valued simple game equals the core of the reduced game, when extended with zeros for all players outside the set of vital players (see Theorem 3.4).
For a permissible game v ∈ TSI N the reduced three-valued simple game for all S ⊂ Vit(v). The following proposition states that a reduced permissible game v r is also a three-valued simple game and, interestingly, allows for only one coalition with value 2.
Proof. From the definition of v r it immediately follows that v r ∈ T SI Vit(v) . Suppose that there exists an S Vit(v) with v r (S) = 2. Then v(S ∪ (N \Vit(v))) = 2 and consequently, using the definition of Vit(v), we have Vit(v) ⊂ S, which is a contradiction.
2 Note that the condition is only a sufficient condition and not a necessary condition. Consider for example the game v ∈ TSI N , with N = {1, 2, 3}, given by v(S) = 2 if S = N, 1 otherwise.
For a permissible game v ∈ TSI N and for an x ∈ R Vit(v) we define x 0 ∈ R N as Theorem 3.4. Let v ∈ TSI N be permissible. Then, where the inequality follows from stability of x ∈ C(v). Because of efficiency of x ∈ C(v) and due to Proposition 3.1 we have where the last equality follows from Proposition 3.3. Hence, where the first inequality follows from stability of x ∈ C(v r ) and the second inequality follows from monotonicity of v and the fact that where the penultimate equality follows from Proposition 3.3. Hence, x ∈ C(v).
Theorem 3.4 is illustrated in the following example.
The corresponding reduced three-valued simple game v r is given in Table 1. Since we have, according to Theorem 3.4, that In a reduced three-valued simple game the number of coalitions with value 2 is reduced to one, only the grand coalition, and thus Vit(v r ) = Vit(v). This property makes it more manageable to characterize the core of reduced three-valued simple games compared to non-reduced three-valued simple games. As Example 3.1 already suggests, the extreme points of the core of reduced three-valued simple games have a specific structure. The extreme points depend in particular on the set of players that belong to every coalition with value 1 or 2 in v r and the set of pairs of players such that for every coalition with value 1 or 2 at least one player of such a pair belongs to the coalition. For a permissible three-valued simple game v ∈ TSI N we define the set of primary vital players of v by and define the set of secondary vital pairs of v by Using the primary vital players and the secondary vital pairs, the vital core V C(v) of a permissible game v ∈ TSI N is defined by where e S ∈ R N , with S ∈ 2 N \{∅}, is such that The vital core is a subset of the core as is seen in the following theorem.
Theorem 3.5. Let v ∈ TSI N be permissible. Then, Proof. Due to the fact that C(v) is a convex set, it is sufficient to show that 2e {i} ∈ C(v r ) for all i ∈ PVit(v), e {i,j} for all i ∈ PVit(v) and j ∈ Vit(v)\PVit(v), and e {i,j} ∈ C(v r ) for all {i, j} ∈ SVit(v), with all vectors in R Vit(v) .
The following example illustrates that there are three-valued simple games for which the vital core is empty, but the core is non-empty. Note that Vit(v) = N , so v is permissible and the corresponding reduced three-valued simple game v r is the same as v. Since v r ({1, 2}) = 1 and v r ({3, 4}) = 1, we have PVit(v) = ∅. Moreover, for i, j ∈ N with i = j, we have v r (N \{i, j}) = 1. Hence, also SVit(v) = ∅ and consequently V C(v) = ∅. However, the core of v is non-empty since 1 2 , 1 2 , 1 2 , 1 2 ∈ C(v).
As the following example shows, the vital core and the core coincide for some threevalued simple games.
Example 3.3. Reconsider the three-valued simple game of Example 3.1. From the reduced game v r (see Table 1) it follows that Therefore, the vital core of v is given by and, using the results from Example 3.1, we have V C(v) = C(V ).
Theorem 3.6 below shows that the core and the vital core coincide for the class of double unanimity games, a specific subclass of three-valued simple games. For T 1 , T 2 ∈ 2 N \{∅}, we define the double unanimity game u T 1 ,T 2 ∈ TSI N by for all S ∈ 2 N . Note that a three-value simple double unanimity game is a natural extension of a simple unanimity game u T ∈ SI N , with T ∈ 2 N \{∅}, defined by for all S ∈ 2 N . In the proof of Theorem 3.6 we use the notion of convexity.
for all S, T ∈ 2 N , i ∈ N such that S ⊂ T ⊂ N \{i}. One readily verifies that every double unanimity game is convex. From Shapley (1971) and Ichiishi (1981) it follows that if v ∈ TU N is convex, then where Π(N ) = {σ : N → {1, . . . , |N |} | σ is bijective} is the set of all orders on N and the marginal vector m σ (v) ∈ R N , for σ ∈ Π(N ), is defined by for all i ∈ N .
Theorem 3.6. Let v ∈ TSI N be a double unanimity game. Then, Proof. Let T 1 , T 2 ∈ 2 N \{∅} be such that v = u T 1 ,T 2 (with player set N ). Observe that the reduced three-valued simple game v r ∈ TSI T 1 ∪T 2 can also be expressed by v r = u T 1 ,T 2 (with player set T 1 ∪ T 2 ). Therefore, the set of primary vital players of v is given by The set of secondary vital pairs of v is given by From Theorem 3.5 we already know that V C(v) ⊂ C(v), so we only need to prove C(v) ⊂ V C(v). Since a double unanimity game is convex, (1) implies that it suffices to show that for all σ ∈ Π(N ).
Let σ ∈ Π(N ). Since v(S) ∈ {0, 1, 2} for all S ⊂ N and v is monotonic, m σ (v) either contains one two with all other coordinates zero (Case 1) or it contains two ones with all other coordinates zero (Case 2). We thus distinguish between these two cases.
Since v(P ) = 0 we have T 1 ⊂ P and T 2 ⊂ P . From this together with T 1 ⊂ P ∪ {i} it can be concluded that i ∈ T 1 , and thus i ∈ T 1 ∩ T 2 or i ∈ T 1 \T 2 . Since v(Q) = 1 and v(Q ∪ {j}) = 2, it can be concluded that T 1 ⊂ Q, T 2 ⊂ Q and T 2 ⊂ Q ∪ {j}. This is only possible if j ∈ T 2 \T 1 , consequently m σ (v) = e {i,j} with i ∈ T 1 ∩ T 2 or i ∈ T 1 \T 2 and j ∈ T 2 \T 1 .
Example 3.4 illustrates that there exist three-valued simple games for which the vital core is a strict non-empty subset of the core.  Consequently, Note that 0, 1 2 , 1 2 , 1 2 , 1 2 , 0, 0 belongs to C(v), but does not belong to V C(v) and thus V C(v) C(v).

The Shapley value for three-valued simple games
In this section we analyze the Shapley value for three-valued simple games. In the context of simple games and three-valued simple games, a one-point solution concept like the Shapley value can be interpreted as a measure for the relative influence of each player.
A one-point solution concept f on the class G N with G N ⊂ TU N is a function f : G N → R N . The Shapley value (Shapley (1953)) is a solution concept on TU N defined by for all v ∈ TU N , where a marginal vector m σ (v) with σ ∈ Π(N ) is as defined in (2). In this section we focus on one-point solution concepts on the class TSI N . A one-point solution concept f : TSI N → R N satisfies for all S ⊂ N \{i, j}.
• the dummy property if f i (v) = v({i}) for all v ∈ TSI N and every dummy player for all S ⊂ N \{i}.
• the transfer property if for all S, T ∈ 2 N with S T .
The first four axioms are modifications of their counterparts for one-point solution concepts on the class of simple games. Dubey (1975) proved that the unique one-point solution concept on SI N that satisfies efficiency, symmetry, the dummy property and the transfer property is the Shapley value restricted to the class of simple games (Shapley and Shubik (1954)). However, these modifications are not sufficient to characterize the Shapley value on the class of three-valued simple games (see Example A.1 in Appendix A). To obtain a characterization of Φ on TSI N , we introduced an additional axiom: unanimity level efficiency.
The unanimity level efficiency axiom compares the aggregate payoff of coalition S within a double unanimity game u S,T , with S T , to half of the payoffs to S in the double unanimity game u T,T . Note that the double unanimity game u T,T is a rescaling of the unanimity game u T and half of the payoffs to S in the double unanimity game u T,T equals the payoffs to S in the unanimity game u T . Therefore, unanimity level efficiency intuitively states that the players in S can allocate a payoff of 1 between themselves, while for the remaining payoff of 1 the players in S and T \S are treated equally.
The combination of the axioms of efficiency, symmetry, the dummy property, the transfer property and unanimity level efficiency fully determines the Shapley value for a three-valued simple game.
Theorem 4.1. The Shapley value Φ is the unique one-point solution concept on TSI N satisfying the axioms efficiency, symmetry, the dummy property, the transfer property and unanimity level efficiency.
Proof. 4 We first prove that, on TSI N , the Shapley value Φ satisfies the five axioms mentioned in the theorem. From Shapley (1953) it follows that Φ satisfies efficiency, symmetry and the dummy property on TU N . Moreover, from Dubey (1975) it follows that Φ satisfies the transfer property on TU N . Hence, Φ also satisfies efficiency, symmetry, the dummy property and the transfer property on TSI N , and thus it only remains to prove that Φ satisfies unanimity level efficiency on TSI N . Note that, for S, T ∈ 2 N with S T , we have To finish the proof, we show that the five axioms exactly fix the allocation vector prescribed by the solution concept for every three-valued simple game. Let f : TSI N → R N be a one-point solution concept that satisfies efficiency, symmetry, the dummy property, the transfer property and unanimity level efficiency on TSI N . Let v ∈ TSI N and define and thus f (u N,N ) is uniquely determined.
If |MWC 1 (v)| = 0 and |MWC 2 (v)| = 1, then v = u T,T for some T N . From efficiency, symmetry, and the dummy property it follows that f (u T,T ) = 2 |T | e T and thus f (u T,T ) is uniquely determined.
On the other hand, if |MWC 1 (v)| = 1 and |MWC 2 (v)| = 0, then v = u S,N for some S N . From efficiency and symmetry together with unanimity level efficiency it follows that i∈S f (u S,N ) = 1 + 1 2 i∈S f i (u N,N ) = 1 + |S| |N | .
Then, due to efficiency and symmetry, we know f (u S,N ) = 1 |S| e S + 1 |N | e N and thus f (u S,N ) is uniquely determined. Now let v ∈ TSI N such that |MWC(v)| = m with m ≥ 2. We use induction to show that f (v) is uniquely determined. Assume that f (w) is uniquely determined for all w ∈ TSI N with |MWC(w)| < m. Set MWC 1 (v) = {S 1 , S 2 , . . . , S p } and MWC 2 (v) = {T 1 , T 2 , . . . , T q } with p + q = m. Note that from monotonicity of v it follows that for all i ∈ {1, . . . , p} and j ∈ {1, . . . , q}. We distinguish between two cases: q = 0 and q > 0. Since v = max{u S 1 ,N , . . . , u Sp,N } = max{u S 1 ,N , w} and by the transfer property, we have Since by our induction hypothesis the right hand side is uniquely determined, f (v) is uniquely determined.
Then, w, w ∈ TSI N and Since v = max{u S 1 ,N , . . . , u Sp,N , u T 1 ,T 1 , u T 2 ,T 2 , . . . , u Tq,Tq } = max{u T 1 ,T 1 , w} and by the transfer property, we have Since by our induction hypothesis the right hand side is uniquely determined, f (v) is uniquely determined too.
In the following example we model the legislative procedure of the current EU-28 Council as a three-valued simple game and we apply the Shapley value to measure the relative influence of each country.
Example 4.1. The EU Council represents the national governments of the member states of the European Union, so it sits in national delegations rather than political groups. The legislative procedure of the EU Council is as follows. Usually, the EU Council acts on the basis of a proposal from the European Commission. When, on the other hand, the EU Council does not act on the basis of a proposal from the European Commission, there are more restrictions for legislation to get accepted by the EU Council.
In this example we will illustrate the situation of the current EU-28 Council. As from 1 November 2014 the EU Council uses a voting system of double majority (member states and population) to pass new legislation, which works as follows. If the EU Council acts on the basis of a proposal from the European Commission, then it requires the support of at least 55% of the member states representing at least 65% of the EU population. When the EU Council does not act on the basis of a proposal from the European Commission, then it requires 72% of the member states representing at least 65% of the EU population. In case of EU-28, the 55% and 72% majorities of the member states correspond to at least 16 and 21 member states, respectively.   Johnston and Hunt (1977) were the first to perform a detailed analysis to the distribution of power in the EU Council. For this, both Johnston and Hunt (1977) and Widgrén (1994) assumed that the EU Council only acts on the basis of a proposal from the European Commission, which is not always the case. Bilbao, Fernández, Jiménez, and López (2000) and Bilbao, Fernández, Jiménez, and López (2002) also considered the case when the legislation is not proposed by the European Commission. By the introduction of three-valued simple games, it is possible to model the legislative procedure of the EU Council as a single TU-game that takes into account both cases.
Let N be the set of member states of EU-28, i.e., each member state of EU-28 is considered as an individual player. Define the value v(S) of a coalition S ⊂ N by: -2, if the coalition forms a double majority in case the EU Council does not act on the basis of a proposal from the European Commission, -1, if the coalition forms a double majority only in case the EU Council acts on the basis of a proposal from the European Commission, -0, otherwise.
Hence, with w i denoting the population share of member state i ∈ N compared to the total EU-28 population (see Table 2), we have v(S) =      2 if i∈S w i ≥ 0.65 and |S| ≥ 21, 1 if i∈S w i ≥ 0.65 and 16 ≤ |S| < 21, 0 if i∈S w i < 0.65 or |S| < 16, for all S ⊂ N , a three-valued simple game. In Table 3 we listed Φ(v), which can be interpreted as a measure for the relative influence of each country.

Concluding remarks
In this paper three-valued simple games are introduced, where the possible values of each coalition are 0, 1 or 2. First, in order to investigate the core of three-valued simple games, we introduced the notion of vital players. Using this notion, we constructed the vital core and showed that the vital core is a subset of the core. We also provided a characterization for the Shapley value on the class of three-valued simple games. The concept of three-valued simple games can be generalized to three-valued TUgames with coalitional values 0, 1 or β with β > 1. For such games, the notions of vital players and the vital core can be extended in a natural way. It can be shown that if β ≥ 2, then it still holds that the vital core forms a subset of the core. Moreover, by appropriately modifying the unanimity level efficiency axiom, also the characterization of the Shapley value can be generalized to any class of three-valued TU-games.
Consequently, p + q = 3 is the only constraint that follows from the transfer property. Hence, every p, q ∈ R with p + q = 3 is allowed. For example, the Shapley value takes (p, q) = (1 1 2 , 1 1 2 ). Another solution solution concept is the average of the marginal vectors that respects the coalitional values of the individuals, which in the case of N = {1, 2} boils down to . This one-point solution concept takes p = 1 and q = 2. As a result, there is no unique onepoint solution concept on TRI N satisfying the axioms efficiency, symmetry, the dummy property and the transfer property.