1 Introduction

The theory of voting power is one of the fundamental concepts in the theory of social choice. The first attempt to evaluate power of players in voting game was done by Penrose (1946), who tried to express the power of a player by a probability to be in a winning coalition under the condition that other players vote randomly. Later on, Banzhaf (1965), used Penrose idea of voters’ power to discuss fairness of Nassau county board voting system. He described the power as the ratio of so-called swing voters of winning coalitions, and he demonstrated the unfairness of such a voting system to county inhabitants. This power index was applied by Coleman (1971), who added idea of normalization to power measure calculations. Now, the Banzhaf power index is often called Penrose–Banzhaf or Banzhaf–Coleman power index.

Meanwhile, the concept of Shapley value, first introduced by Shapley (1953), was another attempt to evaluate players of cooperative games. The transformation of the Shapley value to the context of simple games was done by Shapley and Shubik (1954). The so-called Shapley–Shubik power index serves as a-priori evaluation of coalitions in voting bodies. The Shapley–Shubik power index is based only on information of decision-making rule. Later Owen (1977) extended this idea to games with formed coalitions; he formulated Owen value, which gives a power evaluation to players taking into account a-priori coalition structure. The extension of the Shapley and Shubik concept, done by Shapley and Owen (1989), is based on the idea that the power of an agent depends not only on the voting rule of decision making, but also on the position of agents in political space; roughly speaking, some voters are more likely to vote similarly than others. Thus, the index is dependent on the consistency of coalitions positions based on the group preferences.

Adjustments of the Shapley–Owen index were done by Barr and Passarelli (2009) and Benati and Marzetti (2013), who tested the theory on the decision-making process in the European Union. López and Saboya (2009) analyzed relationship between Shapley value and Owen values. Godfrey (2005) proposed an algorithm for computation of Shapley–Owen index in two dimensions, and applied results to various voting bodies, e.g. US Senate committees, IMF board voting, or UN Security Council voting. Aleskerov and Otchour (2007) applied the same approach to the power distribution among political parties in the State Duma of the Russian Federation. Gambarelli and Uristani (2009) suggested an algorithm for calculation of political parties’ power indices in bicameral systems and tested it on data from Belgium, the Czech Republic, France, Italy, the Netherlands, Poland and Romania, and the EU as a whole. Bilal et al. (2001) combined probabilistic definition of power indices with model of uni-dimensional and multidimensional policy spaces, using exponential relationship.

The main aim of this article is to add the concept of additional weights as introduced in Bilal et al. (2001) to calculations of power using the Shapley–Shubik power index together with the Banzhaf power index. The power indices are calculated on real voting data, namely the data from the Chamber of Deputies of the Czech Parliament with the emphasis on the State Budget voting issues during 2006–2010 parliamentary period and compared with success in voting of political parties within the parliamentary period. This article is organized as follows: the theory to power indices and their transformations is given in the next section; the third section covers calculations and discussions, followed by conclusions.

2 Power indices and probability of coalition creation

In general, voting can be described using a concept of a simple game. Formally, let N be a set of all players, and \(S\subset N\) be a coalition of players from the set of all players. A characteristic function game is a pair [N, v] consisting of a set of players \(N=\left\{ {1,2,\ldots ,n} \right\} \) and a characteristic function v which maps every \(S\subset N\) to a nonnegative number \(v\left( S \right) \) with condition \(v\left( \emptyset \right) =0\). The simple game is a special type of a characteristic function game with a characteristic function v such that for every coalition \(S\subset N\), \(v\left( S \right) =1\) if S wins and \(v\left( S \right) =0\) when S loses. Thus, characteristic function of a simple game can obtain only values 0 or 1, or, more formally: for all \(S\subset N\); \(v\left( S \right) =0\vee v\left( S \right) =1\). The voting game with characteristic function \(v{:}\,2^{N}\rightarrow \left\{ {0,1} \right\} \) is the simple game that satisfies three conditions: (i) the empty coalition never wins: \(v\left( \emptyset \right) =0\), (ii) the grand coalition always wins: \(v\left( N \right) =1\), and (iii) any superset of winning coalition also wins.

Parliamentary voting can be described as the special type of a simple game—the weighted voting game—in which political parties are players of the game with different weights. Let \(N=\left\{ {1,2,\ldots ,n} \right\} \) be a set of n players, let \(w=\left( {w_1 ,w_2 ,\ldots ,w_n } \right) \) be a vector of players’ weights, and let q be a quota. Then the triple \(G=\left[ {N,w,q} \right] \) is called a weighted voting game. A coalition S is a subset of players \(S\subset N\). A coalition S is winning if its total weight meets or exceeds the quota q, that means \(\mathop \sum \nolimits _{i\in S} w_i \ge q\).

The Shapley–Shubik power index (Shapley and Shubik 1954) was created to a-priori evaluation of the power division among bodies in committee system. Its main advantage is in the possible application under many different circumstances, for example in simple and weighted voting games as well as in multi-cameral systems. The derivation of the member’s Shapley–Shubik power index is based on the number of cases when the member is in ordering pivotal.

Let \({\Pi }\) denotes a set of all n! permutations of N. For every permutation \(\pi \in {\Pi }\) of the form \(\pi =\left( {\pi _1 ,\pi _2 ,\ldots ,\pi _n } \right) \) there exists a unique k such that coalition \(\left\{ {\pi _1 ,\pi _2 ,\ldots ,\pi _k } \right\} \) wins and \(\left\{ {\pi _1 ,\pi _2 ,\ldots ,\pi _{k-1} } \right\} \) does not win. Player \(\pi _k \) is the pivot of permutation \({\pi }\). Denote \({\Pi }_{p}\) the set of all permutations with pivot p. Then the Shapley–Shubik power index of player p can be expressed as (Shapley and Shubik 1954):

$$\begin{aligned} \varphi _p =\frac{|{\Pi }_{p}|}{|{\Pi }|}=\frac{|{{\Pi }_p}|}{n!} \end{aligned}$$
(1)

where the cardinality of sets \({\Pi }_p ,\) and \({\Pi }\) is denoted by \(\left| {{\Pi }_p }\right| \), and \(\left| {\Pi }\right| \), respectively.

This approach is based on original Shapley idea (Shapley 1953) applied on the idea of simple games:

$$\begin{aligned} \varphi _p ={\mathop {\mathop {\mathop {\sum }\limits _{ p\in T\subset N}}\limits _{T\;\mathrm{winning}}}\limits _{ T\backslash p\;\mathrm{losing}}} {\frac{(t-1)!(n-t)!}{n!}} \end{aligned}$$
(2)

where summation is done through all winning coalitions \(T\subset N\) containing player p such that a coalition that is created from T by omission of player p (denoted \(T\backslash p)\) is losing. The cardinality of the set Tis denoted by t. That means that t denotes the position of the player p as a pivot in permutation \(\pi \) such that the winning part of permutation is the coalition T.

The idea behind the Banzhaf power index is based on calculations of swing players—players that are crucial for a coalition to be winning. Let player p be in a coalition T. For any \(T\subset N\) we say player p swings in T if coalition T is winning and coalition \(T\backslash p\) is losing. If p is a swing player of a coalition T, then \(v\left( T \right) -v\left( {T\backslash i} \right) =1\). The originally proposed Banzhaf power index (Banzhaf 1965; Coleman 1971) counts the number of swings over all possible nonempty coalitions:

$$\begin{aligned} b_p =\frac{\theta _p }{2^{n-1}} \end{aligned}$$
(3)

where \(\theta _p \) denotes the number of voter p’s swings. According to Haller (1994), the Banzhaf index satisfies linearity, the dummy player property, anonymity, and the proxy agreement property.

The standardized Banzhaf index, also called normalized Banzhaf index (Banzhaf 1965) calculates voter p’s swings to total amount of swings. The normalized Banzhaf index for voter p is obtained by dividing the sum of p’s swings (regarding all possible combinations) by the sum of all voters’ all swings. Formally voter p’s standardized Banzhaf index is calculated as

$$\begin{aligned} b_p^n =\frac{b_p }{\sum \nolimits _{j\in N} {b_j } }=\frac{\theta _p }{\theta } \end{aligned}$$
(4)

where \(\theta _p \) denotes the number of voter p’s swings, and \(\theta \) is a total number of swings of all players.

The Shapley value calculation (2) is, in general, based on an expression of probability of occurrence of coalition Tin \({\Pi }_p \):

$$\begin{aligned} Q(T,p)=\frac{(t-1)!(n-t)!}{n!} \end{aligned}$$
(5)

To incorporate the coalition forming influence, Bilal et al. (2001) proposed to consider additional weights of possible coalitions into the quantity Q(T, p):

(6)

where S runs through all coalitions in which the player p is pivotal. In one-dimensional case, the weight of coalition of two players depends on the mutual distance \(d_{i,j} \) of respective players i, j:

$$\begin{aligned} W(i,j)=e^{-d_{i,j} } \end{aligned}$$
(7)

In the case of coalition of more players, the weight of coalition is determined by a distance \(d_{p_{\min } ,p_{\max }}\) of the two most distant players of the coalition \(p_{\min } ,p_{\max }\):

$$\begin{aligned} W(C)=W(p_{\min } ,p_{\max } )=e^{-d_{p_{\min } ,p_{\max }}} \end{aligned}$$
(8)

In the original study (Bilal et al. 2001), distances of players were calculated as weighted averages of players’ points in coalition. In this study, the ex post power evaluation of players was done using normalized Manhattan distances—simple counts when two political parties not voted accordingly over number of all votes. This measure is easy to obtain and its value is not dependent on number of votes (number of used votes should vary in different time periods).

As for the Banzhaf index, Hu (2006) proposed to take into account also blocking coalitions’ power. The calculation of both winning and blocking coalitions’ power has its reason in situations when there is possibility to create a coalition with number of votes equal to quota. Let S and \(S^{*}\) denote the coalition of players who vote for, and against a bill, respectively. The asymmetric Banzhaf index is based on so-called double swing players, which means the players of T that are swing players in either winning coalitions or in blocking coalitions (Hu 2006):

$$\begin{aligned} \overline{b}_{p} =\mathop {\mathop {\mathop {\sum {P_T}}\limits _{p\in T\subset N}}\limits _{ T\;\mathrm{winning}}}\limits _{T\backslash p\;\mathrm{losing}}\quad + \quad \mathop {\mathop {\mathop {\sum {P_T^*}}\limits _{p\in T\subset N}}\limits _{T\;\mathrm{blocking}}}\limits _ {T\backslash p\;\mathrm{not\; blocking}} \end{aligned}$$
(9)

In the last equation, \(P_T =Prob\left( {S=T} \right) \) and \(P_T^*=Prob( {S^{*}=T} )\). These probabilities can be substituted by additional weights derived from Eq. (5):

$$\begin{aligned} \hat{{Q}}(T,p)=\frac{W(T)}{2^{n}} \end{aligned}$$
(10)

The calculated Banzhaf index then can be normalized with respect to (4).

In general, each vote stands for a preference relation of a respective player on specific issue. Taking into account m votes, we have preferences in m dimensions. The overall preference of each player is represented by a point in \(\hbox {R}^{m}\). Mutual distances of players in \(\hbox {R}^{m}\) play important role in their voting power.

Calculated spatial Shapley–Shubik index and normalized spatial Banzhaf index will be compared with ex post political party success. To evaluate the party success after voting, the ex post party success is constructed with respect to calculation of ex ante voter success (described in Freixas and Pons 2013) which is based on generalization of Rae index (Rae 1969). An ex post political party success can be constructed by comparing party decision with the outcome of voting. The party A success is defined as the ratio of decisions of parliamentary decisions that were the same as the party A decisions to all decisions during the parliamentary period:

$$\begin{aligned} I_{success}^A =\frac{\hbox {number}\;\hbox {of}\; \hbox {party}\;\hbox {A}\;\hbox {decisions}\; \hbox {identical}\;\hbox {with}\; \hbox {parliamentary}\;\hbox {decisions}}{\hbox {number}\;\hbox {of}\;\hbox {parliamentary}\;\hbox {decisions}} \end{aligned}$$
(11)

The political party decision is derived from the votes of party members using simple majority rule. The political party success can reach values from the interval \(\left[ {0,1} \right] \); the higher the number, the higher ratio of party decisions was the same as the whole voting body decision. The party voting success is influenced by votes of other members of the parliament. Small parties with low power index may have a large success if they vote following the majority of votes.

3 Results and discussions

This analysis is based on the data from the Czech Parliament, namely state budget voting data from the 2006–2010 electoral period. Parliamentary discussions of state budget has to be finished before the end of the preceding year; these discussions usually cover tens of amendment votes to a state budget proposal, many of them related to one specific issue. Hence, the number of votes related to state budget issues is different throughout years, namely 414 votes in 2006, 61 votes in 2007, 130 votes in 2008, and 44 votes in 2009. The outcome of every vote for every member can be “no”, “yes”, “present, abstain”, “absent”. Every bill to be passed needs at least as many “yes” votes as quota. Even though some authors (e.g. Freixas and Zwicker 2003) propose to take into account three expressed legislators’ outcomes (that means “no”, “yes”, and “present, abstain”), in this analysis the “present, abstain” outcome is reclassified to “no” outcome. In the Chamber of Deputies of the Czech Republic, the quota of voting on common issues is based on the sum of all present legislators as stated in Act. No. 90/1995 coll., part eight, § 70, (1), (2): “(1) The Chamber of Deputies forms a quorum if at least one-third of all Deputies are present. (2) Except when otherwise provided by the Constitution, in order to be valid, every resolution of the Chamber of Deputies must be voted for by a simple majority of all present Deputies.” Deputies are aware of the rules as the manipulation with quorum of voting has been discussed in media (e.g. Valkova 2013; Presidential Election 2008), where “present, abstain” has been commented the same way as “no” outcome.

During the studied period, there were five political parties in the Lower House of the Czech Parliament; three of them created governmental coalition: Civic Democratic Party (ODS), Christian and Democratic Union—Czechoslovak People’s Party (KDU-CSL), and Green Party (SZ), while other two political parties—Czech Social Democratic Party (CSSD) and Communist Party of Bohemia and Moravia (KSCM)—stayed in opposition. Basic information on the Czech Parliamentary system as well as the set of all historical votes can be found at the official web site of the Lower House of the Czech Parliament URL: www.psp.cz.

The political situation in the Chamber of Deputies during the period was quite unusual; the distribution of seats at the beginning of the 2006–2010 electoral period in the Lower House of the Czech Parliament did not allow simple government setting—both left-wing (CSSD, KSCM) and right-wing political parties (ODS, KDU-CSL, and SZ) gained 100 votes, not enough to pass any vote. Even though 2006 Elections were hold in June 2006, the government was approved after months of discussions in January 2007. After the government was set, the governmental coalition composed of ODS, KDU-CSL, and SZ worked together. However, differences in political programs and disagreement between governmental political parties ended up in a downfall of the government; on March 24, 2009, the Lower House of the Czech Parliament approved no-confidence of the government. The caretaker government was set and functioned till elections in 2010. Detailed description of the situation after 2006 Parliamentary Elections can be found in Škochová (2008).

The number of seats of political parties in the Lower House of the Czech Parliament together with a-priori power distribution at the beginning of the period measured by Shapley–Shubik power index, Banzhaf index, normalized Banzhaf index together with ex post party success during state budget voting for political parties in the Lower House of the Czech Parliament is given in Table 1.

Table 1 Seats, a-priori power indices and party success of political parties in the Lower House of the Czech Parliament during the 2006–2010 electoral period
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In Table 1, political parties are sorted with respect to number of seats. The interesting point is a fact that the value of party success is not bound with a-priori power indices; for rough comparison, the correlation coefficient of an a-priori Shapley–Shubik power index with party success is \(-\)0.51; the correlation coefficient of an a-priori Banzhaf index (as well as a-priori normalized Banzhaf index) with party success is \(-\)0,28, both correlation coefficients are not statistically significant. Even when comparing ranks of calculated Banzhaf indices with ranks of party success values, the calculated Spearman’s rank correlation coefficient gives statistically non-significant results (r \(=\) 0) for both indices. However, the surprising fact is a negative sign in the correlation coefficient value.

3.1 Distance calculations

Power calculations in this article take into account seats distribution as it was at the beginning of the period. For every parliamentary party and every relevant vote, the party decision result was calculated. For each pair of political parties, the distance between two political parties was calculated as normalized Manhattan distances—simple counts when two political parties not voted accordingly over number of all votes. The normalization was done because of possibility to compare different time periods with different number of votes. The distance between two political parties can reach the value from the interval \(\left[ {0,1} \right] \); the higher the number, the more distant political parties in the voting were. Calculated distances between political parties during respective state budget voting are given in Tables 2, 3, 4 and 5.

Table 2 Distances of political parties in 2007 state budget voting
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Table 3 Distances of political parties in 2008 state budget voting
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Table 4 Distances of political parties in 2009 state budget voting
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Table 5 Distances of political parties in 2010 state budget voting
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The interesting point is that in calculated distances the scale of results varies in different years, even though we could expect that calculated numbers should be similar. For example, mutual distances in 2007 state budget voting varied from 0.14 to 0.49, while in 2008 state budget voting varied approximately from 0.02 to 0.79. In some cases higher amount of votes might increase possibility for any pair of political party to differ. That might be the reason why mutual distances in 2009 state budget voting represented by 130 votes had higher variation (distances were from interval \(\left[ {0.01,0.90} \right] )\), while in 2010 state budget voting represented by 44 votes the variation decreased approximately from (distances were from interval \(\left[ {0.18,0.66} \right] )\).

Another interesting point is to compare mutual distances of governmental coalition political parties—ODS, KDU-ČSL, and SZ with respect to the main opposition political party—CSSD. Graphical comparison of distances of political parties during 2007–2010 state budget voting is given in Fig. 1. Figure 1 gives the distances of two main political opponents—ODS and CSSD and compares it with closeness of governmental coalition composed of ODS, KDU-CSL and SZ in the unified scale through years 2007–2010.

Fig. 1
figure 1

Distances of political during 2007–2010 state budget voting; comparison of distance of main political opponents—ODS and CSSD with closeness of governmental coalition composed of ODS, KDU-CSL and SZ. Source: Own calculations

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Calculated distances reflect situation in the Lower House of the Czech Parliament. The first, 2007 state budget voting was held in December 2006, while the government was approved after months of discussions in January 2007. The 2007 state budget voting was approved after agreement of two main rivals—ODS, and CSSD. This fact is visible in mutual distances of political parties; in 2007 state budget voting the two political parties, usually staying in opposition—ODS and CSSD, are more close to each other than any pair of later governmental political parties—ODS, KDU-CSL, and SZ (Table 2; Fig. 1).

After January 2007, the governmental coalition composed of ODS, KDU-CSL, and SZ worked together. However, differences in political programs and disagreement between governmental political parties ended up in a downfall of the government in March 2009. The political situation is visible at mutual distances between governmental political parties. In 2008 and 2009 state budget voting all governmental political parties stayed relatively closer, while the distance increased in 2010 state budget voting (Fig. 1).

3.2 Power calculations

Calculations of spatial Shapley–Shubik power index are based on all possible rankings with added distance variable from Eq. (10) to calculation of \(\hat{{Q}}(T,p)\) and to the Shapley–Shubik power index:

$$\begin{aligned} \hat{{\varphi }}_p (v)={\mathop {\mathop {\mathop {\sum }\limits _{ p\in T\subset N}}\limits _{T\;\mathrm{winning}}}\limits _{ T\backslash p\;\mathrm{losing}}} {\hat{{Q}}(T,p)} \end{aligned}$$
(12)

The situation in the Chamber of Deputies of the Czech Parliament is reflected on calculated ex post power of political parties in state budget voting 2007–2010 (Table 6).

Table 6 Calculated ex post Shapley–Shubik power of political parties in 2007–2010 state budget voting
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In 2006 (that means 2007 state budget voting), the cooperation of two political parties with the highest amount of seats (ODS and CSSD) with support of KSCM almost ruled out other two coalitional political parties—KDU-CSL and SZ—their power decreased to zero, while power of the influential political parties was distributed approximately 7:6:5. In the 2009 state budget voting, the influence of CSSD decreased. The voting discipline of governmental parties in 2009 state budget voting caused increase in power of all three governmental parties. However, the power distribution in the last studied 2010 state budget voting reveals increase influence of two main oppositional political parties—CSSD and KSCM.

Calculations of spatial Banzhaf power index are based on all possible winning coalitions as well as all possible blocking coalitions. The interesting point is the fact, that in the 2006–2010 there were fifteen winning coalitions, but seventeen blocking coalitions—the consequence of difficult situation at the beginning of electoral period. List of all winning and blocking coalitions together with swing players is given in Table 7.

Table 7 List of winning and blocking coalitions and swing players in 2006–2010 Chamber of Deputies
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Calculations of spatial Banzhaf power index are based on all possible winning and blocking coalitions with added distance variable to calculation of \(\hat{{Q}}(T,p)\) to the Banzhaf index:

$$\begin{aligned} \overline{b}_{p}=\mathop {\mathop {\mathop {\sum }\limits _{ p\in T\subset N}} \limits _{T\;\mathrm{winning}}} \limits _{T\backslash p\;\mathrm{losing}} {\frac{W(T)}{2^{n}}}\quad +\mathop {\mathop {\mathop {\sum }\limits _{ p\in T\subset N}}\limits _{ T\;\mathrm{blocking}}}\limits _{ T\backslash p\;\mathrm{not\; blocking}} {\frac{W(T)}{2^{n}}} \end{aligned}$$
(13)

The first part of Eq. (11) is based on the swings of players in winning coalitions, while the second summation is done over all coalitions with swing players in blocking coalition. The calculated spatial Banzhaf indices are given in Table 8, indices after normalization are given in Table 9. The calculated normalized Banzhaf indices are similar to the calculated Shapley–Shubik power indices with the exception that there is no apparent shift in power distribution during the studied period; results are consistent with calculated a-priori power of political parties. However, the power distribution patterns are still visible in calculated power indices.

Table 8 Calculated Banzhaf power indices of political parties in 2007–2010 state budget voting
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Table 9 Calculated normalized Banzhaf power indices of political parties in 2007–2010 state budget voting
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When comparing calculated power indices with a-priori power indices and with calculated party success, the results show high correlation between estimated and calculated results with exception of 2009 state budget voting (Table 10). This is bound with the strict coalitional behavior of governmental political parties during 2009 state budget voting. The interesting point is that this change is not visible in calculated ex post Banzhaf index.

Table 10 Correlation coefficients of estimated and calculated power indices and calculated power indices with index of party success
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Similarly as in the study of Aleskerov and Otchour (2007) on State Duma of Russian Federation, who modelled time changes in power distribution of political parties in respect with their distances in two-dimensional space, this method allows to study of power distribution patterns in a real voting body—in this case in the Chamber of Deputies of the Czech Parliament. The disadvantage of the approach is the impossibility of depiction of more than three players into two dimensional space, however this inconvenience is compensated by better distance measures. This approach does not use the ideology measures as shown in Godfrey (2005), who placed players (in his case senators), into two-dimensional space. Similarly, in the study of Barr and Passarelli (2009) these ideology measures were replaced by measures about inter- and intra-national involvement of EU in member state surveys and served as one dimension of Shapley–Oven indices calculation.

4 Conclusion

This article compares the calculated a-priori Shapley–Shubik power index and normalized Banzhaf power index with decisional power of political parties calculated using state budget voting data from the 2006–2010 electoral period. The results of calculations of mutual distances as well as results of power distribution show differences through years. The situation during 2006–2010 electoral period in the Chamber of Deputies of the Czech Parliament was evolving, votes of legislators reflected given situation. The obtained results copy political movement in parliament. For example, governmental political parties—ODS, KDU-CSL, and SZ—had highest mutual influence (measured as voting power) during 2009 state budget voting, even though the biggest parliamentary party—ODS—had higher real power index in 2007 state budget voting. After the change of the government in March 2009, the consequent state budget voting had shown increase in power of political parties staying originally in opposition—CSSD and KSCM.