Kristina Stefanova & Michal Lehuta

International University Bremen
14 April 2003
1 Introduction
Coalition Building or “the process of uniting different political actors or organizations in the pursuit of some common goal ” constitutes one of the most fundamental political problems. Political coalitions might consist of individual legislators, parties that want to control of the executive branch, or states united in an international action. In a coalition the actors are fully committed and invest all their resources, such as votes, money or soldiers for the achievement of goals that might not be completely their own. At the end they distribute the gains among them.
Sometimes the participants in a coalition might disagree over what result they expect after organizing the collective action and therefore the question why and in which conditions would coalitions form and survive is a substantial one for predicting and understanding the behavior of various political players.
This essay concentrates mainly on interparty executive coalitions in parliamentary democracies – in other words, parties that seek to control a cabinet responsible to a parliamentary or legislative majority. In the course of describing this process we present the most famous theories on coalition building, which are accompanied with some of the incentives that make parties build certain types of coalitions. Together with showing the political side of this political process, we turn our attention also to the mathematical point of view, which describes parties’ influence in coalition building by a power index. Various people have pondered over the question what makes a party strong in relation to other political players, which lead to the emergence of numerous indices. We look at some of the most important and frequently mentioned in this respect.

When are political coalitions built?
In parliamentary systems that have two dominant parties (Conservatives and Labor in Britain), one of the parties always holds a majority of the seats in the legislature and does not need to build a coalition. In such conditions there are coalitions within the dominant parties, like the one between northern and southern democrats.
In cases where none of the parties has the majority of the parliamentary seats it must necessarily build a coalition. Sometimes the cabinet consists only of representatives of one party although members of other parties support it in debates.

Such coalitions in which cabinet responsibilities are not shared are called legislative coalitions. Parties that agree to support the cabinet without getting cabinet representation are referred to as support parties. The more common type of coalitions, however, are the one that all the members want to share the control of the executive branch – known as cabinet or executive coalitions. 2 Politics view
Coalition Theories:
Cabinets have to be formed so that they have the confidence of the parliamentary majority. Several theories have been proposed to predict what coalitions would form in parliamentary systems:

Parties A(left) B C D E(right)
Seats 8 21 26 12 33

Theories :

Minimal winning coalition ABC ADE BCD BE CE
Minimal size ADE
Bargaining position BE CE
Minimal range ABC BCD CE
Minimal connected winning ABC BCD CDE
Policy-viable coalition ABC BCD CE
1) Minimal winning coalitions:

This theory was predicted by William H. Riker who, based on his “size criterion ”, states that only those parties will unite that are minimally necessary to win the majority in the parliament. Thus a coalition between parties ABC is one of the likely to be formed because the elimination of one member would leave it with a minority and also the addition of another member would make it larger than minimal.
The assumption lying under this theory is that parties are interested in maximizing their power, so they aim to have as many cabinet seats as possible. That is why to enter the cabinet a minority party would be willing to be a part of a coalition with another party, but each of them would not want to welcome more parties than needed in the coalition because that would reduce their own share of seats in cabinet.
When there is a majority party in the parliament this theory predicts only one outcome – non-coalition cabinet formed by the majority party. If there is no majority party the resulting coalitions might be many. In the example with five parties in the table above the coalitions likely to be formed are five.
Riker’s theory also assumes that parties have full information while they bargain; that prize for the winning coalition is the same regardless of who its members are; and that each case of coalition bargaining is an independent event, in which parties do not pay attention to what has happened in the past or what they expect to happen in the future. Critiques challenge Riker’s assumption that parties are totally indiscriminate in their search for partners.

The other weaknesses of the size principle are that it does not explain real-world cabinet coalitions very well. It is inaccurate and cannot explain the emergence of the widely common minority and surplus governments.
2) Minimum size coalitions:

This theory again assumes parties would always strive for maximizing their power and thus being part of a minimal winning coalition. The difference is that according to the minimum size theory parties will always choose the most advantageous coalition for them – the one giving them the highest share of seats. That is why coalition ADE is predicted with a majority of fifty-three seats instead of some of the other coalitions that have a majority ranging from fifty-four to fifty-nine seats.

3) Coalitions with the smallest number of parties:

This theory uses one more criterion for reasoning which kind of coalition will form. Based on Michael Leiserson’s “bargaining position” it argues that parties will choose to form those minimal winning coalitions that involve the smallest number of members participating in them because “negotiations and bargaining [about the formation of a coalition] are easier to complete and hold together, other things being equal, with fewer parties.” Therefore, the theory predicts that of all the minimal winning coalitions in the table, only BE and CE will tend to form because they would be more preferred than a three-party coalition.

4) Minimal range coalitions:

In contrast to the other theories, which take into account only the size and the number of parties in coalition building, this theory considers also their policy programs and preferences. It assumes that coalitions are more likely to be formed and well maintained among parties with similar views than among those, which are far apart in this respect.
In the example above, party A is to the extreme left and party E is to the extreme right. The distance between the different parties are the “spaces” that separate them. Coalition ABC with a range of two is much more plausible than coalition ADE which includes four spaces. BCD and CE are also predicted because of the two “spaces” range.

5) Minimal connected winning coalitions

This theory was proposed by Robert Axelrod who assumes that parties will always try to coalesce with their immediate neighbors until a coalition is formed. The minimal connected winning coalitions are not necessarily minimal winning.

In the coalition CDE it is obvious that party D is redundant, but Axelrod states that it is necessary to make the coalition connected.

6) Policy-viable coalitions:

The focus of this theory is only on policy preferences, assuming that parties do not care about holding office. In this case power resides in the legislature rather than the cabinet. The most important party in the legislature is the “core” party because it is the median member of the parliament. It can actually dictate policy since the parties on its neither side have the majority to enact a policy against its will. A policy-viable coalition is one that cannot be defeated by those in the legislature who prefer different policies. If parties care only about policy, no majority can rationally agree to shift the policy position away from that of the median party. Therefore the “core” party can prevail even if it controls much less than a parliamentary majority and it should always be in the governing coalition. The problem with the theories explained above is that they always predict a minimal winning coalition of a different kind. Only Axelrod’s coalitions may be larger. This is based on majoritarian assumptions and is in conflict with all the minority and oversized cabinets in parliamentary democracies. Michael Laver and Norman Schofield classify 196 cabinets formed when there is no a majority party in the parliament in twelve European multiparty democracies from 1945 to 1987. Only 77 of them were minimal winning, 46 were oversized and 73 were minority cabinets. How can this be explained?

Incentives for formation of minority cabinets:

Minority governments are generally less stable and less effective than majority cabinets, but still are quite common in well-functioning democracies such as Denmark, Norway, Sweden, (alternating with single-party majority governments) as well as in Canada and Ireland, which were influenced by British parliamentary tradition and also in France, Italy and Spain. 1. Strom reasons that even if it is correct to assume that parties aim at holding power positions it does not mean that they would want to enter cabinets at all times. Sometimes it might be electorally advantageous for them to leave the government responsibility in the hands of the others. Thus they would remain with the possibility to acquire power position in the future. If many powers expect electoral gains from not participating in the cabinet, this creates a high probability that minority cabinets will be formed.
2. Several institutional features may favor the formation of a minority cabinet. An example is if a new cabinet can take office without the need for a parliamentary vote formally electing or approving it.

This restricts bargaining by placing a hurdle at the end of the negotiations between parties. There are many parliamentary democracies without such investiture rules like the United Kingdom and most of the former British colonies, the Scandinavian countries and the Netherlands. Incentives for formation of oversized cabinets:

Oversized coalitions are particularly common in some countries, in which population consists of several distinct ethnic, linguistic or religious groups. Coalitions that involve all or almost all of the parliamentary parties are called grand coalitions and are especially common in wartime and other emergencies. They are typical for Switzerland in everyday politics as well.

1. William Riker’s reason for the formation of larger than minimal winning cabinets is the “information effect”: “If coalition-makers do not know how much weight a specific uncommitted participant adds, then they may be expected to aim at more than a minimum winning coalition.” This means that during the negotiations before the formation of a cabinet, there is uncertainty about how loyal the any of the future coalition members may be to the proposed cabinet. For this reason additional parties may be brought to the coalition “as insurance against defections and as guarantee for the cabinet’s winning status”.
2. Parties’ preferences are another reason for the enlargement of coalitions. Each party would prefer to have parties of equal weight on both of its sides. Coalition ABC is more attractive for B while C would prefer BCD, which brings the likeliness that an oversized coalition ABCD will be formed.
3. If the first priority objective of all or most of the parties is to defend the country or the democratic regime from external or internal threats. Churchill’s war cabinet in Britain is one of the examples of the grand coalitions that have frequently occurred in wartime. Internal threats are mainly any anti-democratic parties or movements. Ian Budge and Valentine Herman found that when a country is externally or internally threatened, in 72% of the cases such broad coalitions were formed.
4. Special majorities necessary for the amendments or regular legislation may be a strong reason for the emergence of oversized cabinets. If the new cabinet members intend to make important amendments to the constitution, any special majorities needed for achieving that will most probably broaden the coalition. An example of oversized cabinets is in Belgium during its constitutional reform, which led to the establishment of a new state in 1993.

Until early 1990’s Finland also needed even five-sixths majorities for certain types of economic legislation, which account for its tendency to forming oversized cabinets.
3 Mathematics view
Trying to describe social processes with pure mathematics is a Sisyphean task. The more one tries to incorporate everything that influences the social outcome, the more it becomes difficult and problematic. This all applies also, as we will see, to such a complex problem as the coalition formation. Manfred Holler argues that there is “no obvious answer to the problem about the right power measure”. Nevertheless, the discussion over power indexes is characterized mainly by the search for the “right index”. At a first sight, coalition building can be seen only as a process of aggregating enough voting power required to legitimately make political decisions. In other words, this means aquiring the majority of votes. This basic principle is used as a premise for all mathematics attempts to formulate a power index. We come to problems, when approaching the notion of power. Is power in social context a probability, capacity, potential or just a purely theoretical concept? The latter was rejected, at least by the authors concerned with mathematical expression of the concept. Another fundamental question would be whether the power depends on preferences of players. As we will see now, the two basic indices take the probability (or the share of the total) as its’ core principle. 3.1 Banzhaf index
This index reflects the number of coalitions where the party A holds a critical position as a share of the sum of all parties’ number of critical position coalitions.
Let us assume three players (parties A, B and C) in a voting game with this profile:
[3; 2, 1, 1]
Here, the first number indicates the vote quota (in our example majority voting), the following numbers represent the amount of players’ votes. Winning coalitions are only three:
{A,B}, {A,C} and {A,B,C}
where players holding critical position (position of pivotal or swing voter) are in bold. This means, that if party in bold leaves the coalition it loses its’ majority.
Now, as we can see, party A has three critical positions, B and C one each. Therefore, the Banzhaf index will look like this:
(or 60%),
(or 20%)
and (20% as well).
Consequently, Banzhaf index in general can be expressed as:

where denotes the number of winning coalitions in which party i (i=1 to n) holds the critical position and is the sum of all this values for all parties (not only the total number of possible coalitions).

However, there can be many remarks raised against the Banzhaf index.

First, it takes all coalitions as equally probable to be built, which is in reality usually not the case. Secondly, in the denominator, one coalition can be included several times as it may comprise of more than one pivotal player. It neither takes into account any of the incentives that make parties cooperate with each other much more often than just a random probability (similar political views, for example).
I will illustrate these queries in an example from a real life.

Slovak parliamentary elections 2002 :
Party Votes [%] Seats Seats [%] Banzhaf index [%]
HZDS 19.50 36 24.00 28.0
SDKÚ 15.09 28 18.66 18.4
Smer 13.46 25 16.66 16.8
SMK 11.16 20 13.33 10.4
KDH 8.25 15 10.00 10.4
ANO 8.01 15 10.00 10.4
KSS 6.32 11 7.33 5.6
Together 81.79 150 100 100

As we can see from the table, in the situation of Slovak 2002 election results, at least three parties were needed to build a coalition. We may also conclude a rule that if party A has more seats in parliament it has higher or the same Banzhaf index as a party with less seats. This is known as the Local Monotonocity principle. It allows, therefore, that party A (here SMK) can have more seats in parliament than party B (here KDH and ANO) and nevertheless has the same value of Banzhaf index – same power in coalition building. Holler (2002) adds that Local Monotonocity is not always present in power indexes , but he doesn’t show any example of such a paradox. Political scientists, however, often find the nonmonotonocity of power as a contradiction to democracy.
Slovak example also nicely visualizes the deficiencies of policy-blind indexes. In this case, the greatest coalition-building potential should have HZDS. In fact, every other party refused to negotiate with HZDS because of it’s negative national as well as international reputation. Actual coalition was built among three parties that took part in governing in the previous term (SDKÚ, SMK and KDH) plus one newly created party (ANO).
This leads us to another possible formulation of the Banzhaf index that incorporates the probability of each coalition being built. After all, it’s much more plausible to expect that the power of a party with certain amount of votes depends not only on the numbers of coalitions in which it is critical but also upon the probabilities by which these various coalitions arise.
If we assign each coalition a probability that it would be formed, with three players we will get something like this:

In this formula, the equals to the sum of probabilities of all coalitions where A is the pivotal voter.

In the example from the beginning with the assigned probability (W) this means
.
The and would be computed analogically. In general, the normalized revised Banzhaf index looks as follows.

In the revised index, even so, comes immediately the question of how do we compute the probability that a coalition would be formed. This would require additional research as in our available literature the authors do not comment on this problem. But even pure guesses of political observers would add a lot into the approximation to reality.
3.2 Shapley-Shubik index
As we saw, the Banzhaf index counts with all possible combinations of coalitions. Here, Shapley-Shubik index counts with all possible permutations of winning coalitions. In this case the order of parties matters. Obviously, we get many more possibilities of coalitions now. In the same example as in 2.1 [3; 2, 1, 1], all possible orders (namely 3!) are:
{A,B,C}, {A,C,B}, {B,A,C}, {B,C,A}, {C,A,B}, {C,B,A}
In bold are pivotal players – i.e. players turning a loosing coalition into a winning one. Here, following Shapley-Shubik indices result as the probabilities of being randomly chosen from the set of possible outcomes:

For n players, the standard Shapley-Shubik index is defined as

where s denotes the number of players of S. The function ν has value 1 for winning coalitions and 0 for losing .
In Banzhaf index, some coalitions are counted in the denominator several times, as there can be more parties with critical position in one coalition. Whereas in the Shapley-Shubik index, only one party can be the pivotal player, because we assume their voting in an order.
To summarize, the voting power index (as by Shapley and Shubik) is given by the number of times a player is pivotal in all orderings of the players divided by the number of all such orderings .
3.3 Spatial Models
Power index need not to be expressed only as a probability of being chosen from a set of winning coalitions. The intuition can lead us to the minimization of interparty conflict achieved by ‘political closeness’. It’s plausible to assume that the closer are the parties’ preferences on a certain policy scale the more likely they are to form a coalition with each other.
One-dimensional explanation of coalition formation is formally explained by the normalized uni-dimensional scale in an Euclidean space, such that each party can be assigned a value on this interval (usually from 0 to 1).

Hence, the difference in political preferences can be measured by the distance of the two players on this scale.

Here, the parties’ distance is computed as an absolute value of the difference between the parties' preferences. Logically, the closer the preferences (the smaller the distance) between parties, the more likely they are to reach an agreement.
Now, we will apply this model to the coalition formation assuming that the probability that a coalition will be formed (W) depends solely on the distance between the extreme players of that coalition and the parameter u measuring the flexibility of negotiation between players.

The range of W, as in the case of probability should be, is: <0,1>. Understandably, the higher the flexibility of parties, the more likely an agreement is reached. The parameter, however, reflects high flexibility with low values. The higher the u is the more rigid the parties in the negotiation process are ( ). If u is unknown, we assign it value = 1. If the flexibility is fixed, the probability of reaching an argument depends only on the distance between parties’ preferences. According to this logic, as well, two players with identical preferences or infinite negotiational flexibility will always reach an argument (which is quite intuitive). At this moment, we may prove that the probability of building a coalition according to policy closeness condition depends only on the distance between the two extreme players:
W(a,b).W(b,c) = exp(-uda,b).exp(-udb,c) = exp(-u(da,b+db,c)) =
= exp(-uda,c) = W (a,c)
Similarly, these rules apply to n players. The order in which players enters a coalition does not matter.
From a political view, this approach matches reality very narrowly. As the several studies in political change demonstrate, today’s political spectrums can no longer be described by only one dimension – namely, according to the old left-right cleavage. There are already many multidimensional indexes that allow several issues to form the party’s preference, but since their understanding and delineation requires higher mathematical knowledge I will not introduce them in this paper.

4 Conclusion
Defining and measuring the power index of party in the coalition building process is a very difficult and ambiguous task. It has been approached mainly in two ways – the “policy-oriented” and the “policy-blind” (mathematical).
From a political view, several ‘laws’ can be described. Coalitions tend to be minimal winning, minimum sized, preferably with smallest number of parties and with smaller range of political preferences.

However, minority governments or oversized coalitions occur more than exceptionally, too.
Mathematical indexes reflect these rules only partially. One way of defining the index of power is the probability of being in the winning coalition represented by the Banzhaf and Shapley-Shubik indices. Another method is based on the political nearness principle measured in the Euclidean space. Power indices constitute important description of potentials in any kind of social games. In coalition building their application is still very limited. It is based on innumerable number of factors that influence the social outcome. Nevertheless, they are used in projecting assemblies consisting of subject of different size that are supposed to have the same power per vote. One example of application for all: in the United States presidential elections more votes is given to smaller states than their share of population in the union. This means that in the District of Columbia or Wyoming one presidential elector corresponds with about 60-70,000 voters, whereas in Florida, Ohio or Illinois this number is about three times higher (200-240,000). Unfair? This inproportionality was created in order to give the same power to each vote taking into account the coalition-building potential of this vote. Therefore, larger states are represented by fewer electors with the intention of lessening their high coalition formation potential. The same approach is used in the European Parliament, for example.

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Coalition building and the power index of parties in this process