<i>Ex Ante</i> and <i>ex Post</i> Voting Power: A Method for Calculating Party Power in Party Government

doi:10.4236/sm.2013.32022

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2013. Vol.3, No.2, 149-155

Published Online April 2013 in SciRes (http://www.scirp.org/journal/sm) http://dx.doi.org/10.4236/sm.2013.32022

Ex Ante and ex Post Voting Power: A Method for Calculating

Party Power in Party Government

Jan-Erik Lane, Alexander M. Preker

Department of Political Science, University of Freiburg, Freiburg im Breisgau, Germany

Email: [email protected]m, [email protected]

Received December 30th, 2012; revised February 3rd, 2013; accepted February 22nd, 2013

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

As political power tends to be wielded in the form of voting power in the national assembly, especially

under the institutions of party government, one needs a method to calculate the voting power of political

parties, both longitudinally and for a cross-section of European democracies. This paper suggests such a

method, derived from the power index approach in cooperative game theory. The application of the

method on the history of democracy in the two German nations results in party scores—mandates, ex ante

and ex post voting power—that are much in congruence with the standard interpretations of the interwar

period and the post-war politics in these two countries.

Keywords: Party Government; Government Coalitions and Coalitions in the Legislature; Ex Ante and ex

Post Voting Power; Penrose-Banzhaf-Coleman Framework; Banzhaf Numbers or Scores;

Double Banzhaf Games; Party Government in the Two German Speaking Nations

Introduction

Political power under a regime with party government in a

multi-party system derives from the capacity of parties to enter

into winning coalitions, either ad hoc in relation to the voting

on decision issues in Parliament, or as a permanent member of

a coalition in government. The Penrose-Banzhaf-Coleman

model of coordination in N-person games—coalitions—offers a

method to estimate this form of political power. In short: The

cooperative game theory model measures the capacity of a

party to contribute to the success of a coalition.

Here we wish to show how the power index method can be

employed for the systematic and comparative analysis of the

power of political parties in the typical regime of party gov-

ernment. We give as example a small study of party govern-

ment in Germany—Weimar and FRG—as well as Austria—1st

and 2nd Republic (1919-1933 and 1945-today). The research

question can be posed as such: Have the various political par-

ties been able to exercise voting power in proportion to their

electoral support, as measured by the seats of the party in the

nation’s representative assembly?

Logic of Party Government: Double Banzhaf

Games

In order to estimate the power of political parties under the

regime of party government one needs to calculate Banzhaf

numbers for the players involved in the games of party gov-

ernment, i.e. the political parties. In addition, one has to calcu-

late the Banzhaf power for governments, whether majority or

minority ones. All voting power derives from the capacity to

form winning coalitions in the national assembly.

Banzhaf Voting Power

The Banzhaf voting power index (Banzhaf, 1965; Coleman,

1971; Felsenthal & Machover, 1988, 2002, 2004, 2005) models

the marginal contribution of a player to the success of a coali-

tion, given a social choice mechanism. Looking here at the so-

called Penrose-Banzhaf-Coleman framework for solving one

kind of N-person games, namely so-called simple games, one

starts from the characteristic function of a game. Basically, one

considers the set 2N of all possible coalitions S employing the

assumption of equally likely coalitions. From these 2N coali-

tions, the Banzhaf score of player i is calculated as the number

of coalitions in which i is critical according to:

(1)













SSi



















that is, the number of coalitions that player i is able to swing.

The Penrose-Banzhaf power index β of player i is defined by

the ratio of swings η of i to the number of coalitions not con-

taining the player i. Thus, we have:

(2) 1







.

The employment of the Banzhaf formula results in voting

power scores, either absolute or relative ones (normalised).

Since we compare these scores with the percentage number of

seats or mandates in Parliament, we use relative Banzhaf scores

below. All data contained in the Tables below have been calcu-

lated with the help of a computer algorithm for voting power

analysis by David Leech. Online:

http://homepages.warwick.ac.uk/~ecaae/#Progam_List [last re-

view: 29 January 2013].

J.-E. LANE, A. M. PREKER

Below political parties will be looked upon as organised

players making coalitions involving a perfect degree of indi-

vidual party member discipline.

Ex Ante and ex Pos t Voting Power

The representative assembly makes a huge number of deci-

sions concerning for instance government formation and sup-

port, the budget and taxation as well as regulation in the form

of legislation and policies. Additional functions of parliament

include controlling government and articulating relevant prob-

lems as well as electing government. Under party government,

the actors in collective decisions are the political parties that

tend to vote with one voice, based on their differential number

of mandates. Both the constitutional rules of decision-making

and the strategy of coalition formation are decisive for the out-

comes of parliamentary decision-making.

Most decisions in Parliament require simple majorities,

which is why a minimum winning coalition is enough. Some-

times certain decisions of constitutional relevance are singled

out requiring a heavier support or qualified majority. To prevail,

a winning coalition would then have to be oversized. In the

normal business of government, simple majority suffices in

most parliamentary countries.

A coalition that is minimum winning does not need to be a

minimum sized coalition. Actually, coalitions may be of many

kinds, but the essential thing is that they win the voting se-

quences. Coalitions may be temporary, focusing on one voting

sequence, or they may be permanent over an election period,

such as for instance 4 years. Government coalitions are formed

for the purpose of exercising power over a longer period, but

government coalitions may break up prematurely, leading either

to a new government or to new elections.

A political party maximizes its voting power by actively par-

ticipating in the formation of coalitions. If it enters a govern-

ment coalition, then it shares the voting power of the govern-

ment with its governing partners. It may also exercise voting

power by participating in temporary coalitions that achieve the

minimum-winning format. Political power is mainly exercised

through the capacity to influence voting in the national assem-

bly. Typical for continental European democracies is the multi-

party system, meaning that the parliamentary arena is com-

prised of three or more political parties, where no party has a

majority position.

One may enquire after the differences in seats and voting

power for the major political parties under the regime of party

government by calculating two power index scores: ex ante

power measures based upon the election outcomes before gov-

ernment formation, and ex post power measures based upon the

pattern of government formation: minority, simple majority,

oversized, etc., after government formation. The following dis-

tinctions can be made in this new method for the calculation the

power of parties under party government:

a) The calculation of ex ante power scores is the straightfor-

ward solution of the Banzhaf game for parties in the legislature

before the formation of a government or government coalition.

b) The calculation of the ex post power scores involves a

double Banzhaf game for the parties after government forma-

tion, where the parties in a government coalition share whatever

power government has in the legislature, on the basis of a una-

nimity game between the government coalition partners.

One may in principle distinguish between two parliamentary

situations: a majority coalition and a minority coalition. The

voting power score of the government coalition will here be

assumed to be equally shared among the coalition parties on the

basis of an imagined second Banzhaf game within the govern-

ment—a unanimity game among equal coalition partners. Thus,

we have:

1) Banzhaf number or score for a simple majority govern-

ment = the party(ies) in government constitute the winning

coalition in legislative voting (Banzhaf score = 1/number of

parties in the government coalition).

2) Banzhaf number or score for a minority government = the

party(ies) in government enter some of the winning coalitions

in the legislature (Banzhaf score 0 < 1/number of coalition par-

ties). When the coalition government does not control the vot-

ing in the legislature, it is forced to make ad hoc coalitions with

the other parties—see Appendix for an example.

When there is a minority coalition government, other parlia-

mentary actors like the opposition parties will receive Banzhaf

scores as well, depending on whether they are decisive in coali-

tions.

c) One may further calculate the parties’ Banzhaf power in-

dex numbers for both minority and majority coalitions for every

legislative period and weighted for every single year, thus ar-

riving at aggregated scores for legislative periods, and longer

time spans.

d) By relating the ex post Banzhaf scores to the relative size

of the parties’ representation in the legislative assembly (seats

or mandates), it can be shown how the parties’ voting power

differ from their sizes in Parliament, because some parties tend

to dominate the political game, whereas others do not play a big

role despite their relative strength.

Let us proceed to a couple of examples from the spectacular

political history of democracy in Germany and Austria.

Example: Party Government in the Two

German Speaking Nations

Germany

Proportional election formulas have been utilized in Ger-

many to establish a legislative assembly. In compliance with

parliamentarism, main power lies with the Kanzler. Two kinds

of governments prevail: simple majority governments (FRG)

and minority governments/coalitions (Weimar). There have

only been a few consociational governments. In the Weimar

Republic, the disproportion between the size of representation

of the left-wing (seats, mandates) and its ex post Banzhaf vot-

ing power is striking. In the FRG, an effort was made to elimi-

nate this effect, but has it been successful?

The Weimar Republic

In the Weimar Republic the smaller and right wing parties

often augmented their voting power in comparison to their

share of seats in the Reic hstag by joining government. This

discrepancy and disproportionality can be demonstrated by

subtracting the parties’ relative share of seats from its ex post

Banzhaf points: Especially DDP and BVP enlarged their influ-

ence and voting power in comparison to their share of seats

(DDP + 6.7 per cent, BVP + 5.3 per cent). But also larger con-

servative parties, that is Zentrum and DVP, increased their vot-

ing power by entering the mainly minority governments. Ap-

parently, party coalitions, first minority governments, then

150

J.-E. LANE, A. M. PREKER

shifting majority coalitions, were the key to exercise power in

the Weimar Republic, highly successfully accomplished by

DVP’s Gustav Stresemann between 1923-1929 (Table 1).

On the contrary, the Social Democratic party (SPD) and the

Table 1.

Banzhaf power distribution in the weimar republic 1919-1933.

Party Modified aggregation Deviations from proportionality (power – seats)

SPD seats (rel.) 0.254736842

ex ante 0.291411 0.0366742

ex post 0.179639259 −0.0750976

U SPD seats (rel.) 0.154

(1919-1924) ex ante 0.139715 −0.014285

ex post 0.1057972 −0.0482028

DNVP seats (rel.) 0.138947368

ex ante 0.126354053 −0.0125933

ex post 0.113363011 −0.0255843

Zentrum seats (rel.) 0.139635857

ex ante 0.129780053 −0.0098558

ex post 0.155244942 0.0156091

BVP seats (rel.) 0.037010234

ex ante 0.034639444 −0.0023708

ex post 0.090371939 0.0533617

DVP seats (rel.) 0.084253906

ex ante 0.077607053 −0.0066469

ex post 0.136149469 0.0518955

DDP seats (rel.) 0.058270387

ex ante 0.052078947 −0.00619144

ex post 0.124839478 0.066569091

NSDAP seats (rel.) 0.132896772

(1924-1933) ex ante 0.148824857 0.015928085

ex post 0.116784334 −0.016112438

Wirtschafts- seats (rel.) 0.024752883

partei ex ante 0.023595211 −0.001157672

ex post 0.029837938 0.005085055

KPD seats (rel.) 0.096936858

ex ante 0.094663278 −0.00227358

ex post 0.045465496 −0.051471362

Dt.-Hannov. seats (rel.) 0.006994286

Partei ex ante 0.006280333 −0.000713953

ex post 0.001587514 −0.005406772

Christl.-nat. seats (rel.) 0.020770765

Landvolk (1924-1933) ex ante 0.018560071 −0.002210694

ex post 0.012855946 −0.007914819

CSVD seats (rel.) 0.015807965

(1930-1933) ex ante 0.014741333 −0.001066632

ex post 0.013180917 −0.002627048

Konservative seats (rel.) 0.001299827

Volkspartei ex ante 0.0034224 0.002122573

(1930-1932) ex post 0.013798129 0.012498302

Deutsche seats (rel.) 0.010404347

Bauernpartei ex ante 0.008974556 −0.001429791

(1928-33) ex post 0.003109667 −0.00729468

Others seats (rel.) 0.006491473

ex ante 0.00438 −0.002111473

ex post 0.001472844 −0.005018629

Abbr.: SPD = Sozialdemokratische Partei Deutschlands (social democratic); U SPD = Unabhängige Sozialdemokratische Partei Deutschlands (independent, socialist);

DNVP = Deutschnationale Volkspartei (nationalist, conservative); Zentrum = Center party (catholic); BVP = Bayerische Volkspartei (Bavarian, conservative); DVP =

Deutsche Volkspartei (liberal); DDP = Deutsche Demokratische Partei (social liberal); NSDAP = Nationalsozialistische Deutsche Arbeiterpartei; Wirtschaftspartei =

liberals; KPD = Kommunistische Partei Deutschlands (communists); Deutsch-Hannoversche Partei (local conservative party); Christl.-nat. Landvolk = Christlich-Nationale

Bauern- und Landvolkpartei (agrarian, conservative); CSVD = Christlich sozialer Volksdienst (protestant conservative); Konservative Volkspartei = small conservative

party; Deutsche Bauernpartei (agrarian).

J.-E. LANE, A. M. PREKER

Communist Party (KPD) respectively were often unable to turn

their mandate strength in parliament effectively into ex post

influence or voting power after the broad “Weimar coalition”.

Social democrats became frequently the largest party in parlia-

ment after elections. Additionally, their ex ante Banzhaf score

was greater than their relative mandate strength. But even so,

they seldom joined government. A loss of power can be stated

from the ex ante to the ex post stage of government formation,

which can be explained by a lack of coalition options from the

mid-1920s onwards.

The nationalist DNVP was a large right-wing opposition

party, which often supported one of the minority governments.

But, in contrast to SPD or KPD, the DNVP shows only a small

loss of power when comparing its relative proportion of seats

and its voting power. Thus, only a small difference can be

stated between the DNVP with its nearly equal ex ante and its

post Banzhaf scores and the government parties.

The Federal Republic (FRG)

At first sight, the results for the FRG seem to differ from the

findings for the Weimar republic: The Federal Republic devel-

oped a 2.5 party system (Poguntke, 1999), comprising two

“catch all” parties (Kirchheimer, 1965), namely CDU respec-

tively CSU and SPD, and the FDP as a 0.5 “pivotal” party

(Keman, 1994).

But when applying the Banzhaf-Coleman approach to the

distribution of power, we thus find a parallel between the Wei-

mar republic and the FRG: Again, the political left shows a

tendency of underrepresentation in governments on the federal

level, also when the party system changed since the 1980s and

1990s.

As can be seen in Table 2, only the FDP has been able to in-

crease its voting power in comparison to its relative mandate

strength. The “Union”, consisting of CDU and CSU, shows

only small losses of power in the ex ante and ex post govern-

ment estimates. The FDP has played a powerful role by means

of its pivotal status. Taking the discrepancy between the rela-

tive share of seats and the ex post Banzhaf scores into account,

it can be noted that only the liberal FDP (+28 per cent) and to a

lesser degree the Greens (+4.6 per cent) are the parties whose

Banzhaf ex post power scores increased between 1949 and 2012,

compared with their share of the seats (mandates).

The Social Democrats have often been unable to transfer

their profound mandate strength into power through an ex post

government formation. Their voting power is small compared

with their representation. Thus, the loss of power in ex post

government formation is nearly twice as much (−18.1 per cent)

as the Christian democratic loss (−9.5 per cent). Yet, the lack of

power of the left-wing on the federal level is compensated by

participation in several Länder governments and their power in

the Bundesrat, although Die Linke is still regarded as “regi-

erungsunfähig” (incapable of governing) by the other four par-

ties at the federal level.

Austria

When looking at the First Republic in Austria, the right-wing

parties were heavily overrepresented in comparison to their

relative share of seats in the Nationalrat. On the other hand, in

Table 2.

Banzhaf power distribution in the FRG 1949-2012.

Party Modified aggregation Deviations from proportional ity (power – seats)

CDU/CSU seats (rel.) 0.52482205

ex ante 0.501040618 −0.023781432

ex post 0.429696945 −0.095125105

SPD seats (rel.) 0.435363492

ex ante 0.301324545 −0.134038947

ex post 0.254545455 −0.180818037

FDP seats (rel.) 0.111550297

ex ante 0.242562455 0.131012158

ex post 0.395151491 0.283601194

Grüne (since 1983) seats (rel.) 0.070312448

ex ante 0.153269967 0.082957519

ex post 0.116666667 0.046354219

PDS (since 1991)/ seats (rel.) 0.055674471

Linkspartei.PDS / ex ante 0.072640727 0.016966256

Die Linke ex post 0.000000000 −0.055674471

Others seats (rel.) 0.01402697

(1949-1961) ex ante 0.01501596 0.00098899

ex post 0.01301044 −0.00101653

Abbr.: CDU/CSU = Christdemokratische Partei Deutschlands/Christlichsoziale Partei Bayerns (Christian democratic); SPD = Sozialdemokratische Partei Deutschlands

(Social democratic); FDP = Freie Demokratische Partei (liberal); Grüne (ecological); PDS = Partei des demokratisches Sozialismus, Linkspartei. PDS, Die Linke (left

socialist).

152

J.-E. LANE, A. M. PREKER

the Second republic (1945-today) with its consociational gov-

ernments, an opposing finding can be made: Both the Austrian

social democrats (SPÖ) and the Christian democratic (ÖVP) are

able to augment voting power from the ex ante to the ex post

government formation. In contrast to the German system, the

smaller parties (FPÖ, Greens, BZÖ) play a minor role in par-

liament and government.

The First Republic (1919-1933)

Between 1919 and 1933 especially three political blocks

dominated the Austrian party system: a socialist (SDAP), a

Christian-catholic (Christlichsoziale) and a German nationalist

block, which included some smaller parties (Pelinka, 2004: p.

535).

Neglecting the two years, in which SDAP and Christlich-

soziale formed a grand coalition before the constitution was

passed in 1920, it was the Christian social party and the smaller

German nationalist party group, which was participating in

government on a regular basis. As can be seen in the Banzhaf

power distribution in Table 3, the SDAP was heavily underrep-

resented in comparison to its relative share of seats (−25.9 per

cent). The German nationalist party group, in contrast, is mas-

sively overrepresented with a plus of 31 per cent in relation to

the mandate strength won in elections.

The Second Republic (1945-Today)

Germany is often described as the “Grand coalition state”

(Schmidt, 1996). But, when comparing the number of grand

coalitions in the FRG (2) and Austria (10), it has to be empha-

sized that this concept would fit Austria better. This is reflected

in the Banzhaf numbers for the parties ex post government for-

mation: in Austria, it is only the two “catch all” parties that

augment their power by frequently forming government (SPÖ +

4.2 per cent, ÖVP + 2.7 per cent) while all other parties incur

losses.

For a long time, the Austrian party system was a 2.5 party

system like the German one, consisting of SPÖ, ÖVP and FPÖ.

But in contrast to the FRG, the Austrian FPÖ never gained the

“pivotal” status as the FDP could in Germany, because of the

major parties exercising power in an oversized coalition (Table

4).

Conclusion

The Banzhaf-Coleman approach may be developed into a

tool for the systematic and comparative analysis of the power of

political parties under the institutions of party government.

Voting power is wielded over legislation, budgets and taxation,

policies and regulations. It derives from winning by simple

majority in the legislative assembly, either as a single party or

as a coalition partner in government, with majority or minority

status.

Applying this new method to the political history of democ-

racy in the two German nations, we observe quite a remarkable

fit between the stylised images of the politics of these countries

and the power scores for the political parties. Thus, the interwar

period comprised highly skewed party government, with dis-

Table 3.

Banzhaf power distribution in Austria 1919-1933 (1st Republic).

Party Modified aggregation

Deviations from proportionali ty (power – seats)

modified a ggregati o n

SDAP seats (rel.) 0.326533333

ex ante 0.333333333 0.0068

ex post 0.066666673 −0.259866666

Christlichsoziale seats (rel.) 0.462751467

ex ante 0.333333333 −0.129418134

ex post 0.466666666 0.003915199

Deutschnationale seats (rel.) 0.13902

ex ante 0.3333333 0.1943133

ex post 0.449999967 0.310979967

Landbund seats (rel.) 0.14

ex ante 0 −0.14

ex post 0 −0.14

Heimatbund/Heimwehr seats (rel.) 0.04424

ex ante 0 −0.04424

ex post 0.01777767 −0.02646233

Others seats (rel.) 0.012

ex ante 0 −0.012

ex post 0 −0.012

Abbr.: SDAP = Sozialdemokratische Arbeiterpartei (social democratic); Christlichsoziale = Christlichsoziale Partei (christian-conservative); Deutschnationle = Deutsch-

Nationale Bewegung (nationalist); Landbund (agrarian); Heimatbund/Heimwehr (paramilitary, nationalist).

J.-E. LANE, A. M. PREKER

Table 4.

Banzhaf power distribution in Austria after 1945 (2nd Republic).

Party Modified a g gregation De v ia tions from proportionality (p ower – seats)

KPÖ (1945-1959) seats (rel.) 0.024675318

ex ante 0.035714293 0.011038975

ex post 0 −0.024675318

Greens (since 1986) seats (rel.) 0.074190836

ex ante 0.058608031 −0.015582805

ex post 0 −0.074190836

SPÖ seats (rel.) 0.433602479

ex ante 0.391257975 −0.042344504

ex post 0.475124378 0.041521899

ÖVP seats (rel.) 0.418770529

ex ante 0.366382331 −0.052388198

ex post 0.445273631 0.026503102

VdU/FPÖ (since 1949) seats (rel.) 0.104511154

ex ante 0.212018132 0.107506978

ex post 0.084656084 −0.01985507

Liberales Forum (1994-1999) seats (rel.) 0.055737705

(1994-2002) ex ante 0 −0.055737705

ex post 0 −0.055737705

BZÖ (since 2006) seats (rel.) 0.089253188

ex ante 0.142857 0.053603812

ex post 0 −0.089253188

Abbr: KPÖ = Kommunistische Partei Österreichs (communist), Greens = Grünen – Die grüne Alternative (ecological); SPÖ = Sozialdemokratische Partei Österreichs

(social democratic); ÖVP = Österreichische Volkspartei (Christian democratic); VdU/FPÖ = Verband der Unabhängigen (1945-1955)/since 1955: Freiheitliche Partei

Österreichs (right-wing-demagogic, populist); Liberales Forum (liberal); BZÖ = Bündnis Zukunft Österreichs (nationalist-right-wing demagogic, economically liberal).

proportionate power of the centre (Germany) and the right-wing

(Austria), indicating democratic instability. The post-war shows

considerably more of balance between the right and the left,

especially in Austria, where consociationalism provides the left

with more power than the simple majority governments in the

FRG.

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Appendix

Calculation of ex ante and ex post voting power in a 5 player legislature with parties of the following size: P1 = 35%, P2 = 25%, P3 = 18%, P4 = 12%

and P5 = 10%.

P1 P2 P3 P4 P5

Ex ante voting power: 0.38 0.23 0.23 0.077 0.077

Ex post voting power:

Majority gov P1 + P2: 1/2 1/2 0 0 0

Minority gov with P1 + P5: 0.5/2 0.17 0.17 0.17 0.5/2