Minimum Winning Coalitions: Why Bigger Sometimes is Better

As the final part of a course I am taking on social choice theory, I wrote the below brief essay.


In understanding the politics of parliamentary democracies with multiple parties, an interesting idea is the theory of political coalitions developed by William Riker. The logic is elegant and the assumptions are simple. Given a parliament where no single party has a majority of seats, smaller parties will build coalitions that allow them to form a majority and thus form a government. In order to maximize the spoils or benefits, there is an incentive to form minimum winning coalitions—which are the coalitions in a given situation with the minimum number of members necessary to achieve at least a simple majority. This is because the per-capita average “spoils” or α is  , where N is the number of members in the coalition. The members of the coalition have a reason to minimize N to maximize their share of the ministries. The problem with this model is that empirically this does not always happen. In fact, coalitions in most European parliaments are over-sized. In this paper I hope to explain this empirical phenomenon by exploring structural incentives that would promote deviation from the goal of a minimum winning coalition. Specifically I will address the influence location, preference similarity, and party leadership play in the game of coalition building.

The first such incentive is that created by location. One such form of the “spoils” of government membership are public works projects (or pork barrel spending). Given that these types of programs are critical components of party platforms, it is safe to say that they also factor in to the coalition formation game.  In order to illustrate the effect districts have on coalition formation incentives, let us consider the following fictional country with a 100 member unicameral parliament, with the assumption that the party with a plurality will always be in the winning coalition (this is not an unlikely assumption as rules governing government formation often give the plurality winning party agenda setting power over forming a government):

Party A Party B Party C Party D
19 members 30 members 40 members 11 members

As shown, the MWC is a union of Party C and Party D. Not only is this the MWC of this universe, it is the best possible MWC in any universe where there is no single party with a plurality because it has the very barest minimum number of members required (51). If only size mattered, there is no doubt that this coalition would form. However, consider the situation where this fictional country has four legislative districts all of equal population size and thus equal numbers of seats per district. Let us further assume that all members are elected by a system where voters vote for a party and a candidate. Within their district the seats are divided proportionally between the parties, and the seats are distributed to the party candidate with the most votes first. Below is a fictional distribution of parties divided into four districts:

District 1

Party A: 0 members

Party B: 8 members

Party C: 17 members

Party D: 0 members

District 2

Party A: 5 members

Party B: 8 members

Party C: 12 members

Party D: 0 members

District 3

Party A: 6 members

Party B:  8 members

Party C: 5 members

Party D: 6 members

District 4

Party A: 8 members

Party B:  6 members

Party C: 6 members

Party D: 5 members

Several things can be deduced. First, in this example, the plurality winner is Party C, and it is concentrated in District 1. The most likely reason for this concentration is that the party has a supporter base that is regional. Examples of a party that could exhibit a regional concentration include ethnic parties, industry based parties, and others. Having such a base makes public works programs and military bases especially attractive prizes to be won. For the sake of this thought experiment, it will be assumed that Party C advocated building a freeway system in District 1 during the recent election. Because it is a freeway, it can generally assume that if each district is sufficiently large, this freeway system is effectively useless to nonresidents.( Alternatively, a different public works product that is more excludable like a dam or an aquaduct can be used). Just at a glance, it is clear why Party C’s leadership would pursue such a goal, as half of its base lives in the district. Assuming equal taxation, a total cost that equals Ω and a total payoff that equals P, each of the parties will see the following net profit or loss based on party distribution:

Party A:    Party B: Party C: Party D:

From the above functions, it is evident that Party A and D cannot possibly benefit from the project, and they will certainly oppose it. If the freeway’s benefit is at least worth the cost, Party C makes away with a hefty profit. The interesting function is that of Party B. Party B has a sizeable group of members from District 1, and it just so happens to have just enough members so as to make the freeway a net profit for it as a party as long as the freeway is worth the total cost. With all this in mind, it is possible to see why an over-sized coalition (70 members) consisting of Party B and C might form. Such a coalition increases in likelihood as the importance of the freeway as a campaign promise increases.

The logic behind a regional principle also holds true in terms of ideology and socioeconomic similarity. Just as there is a spatial distribution of parties, there is also an ideological distribution, with parties being composed of people from similar socioeconomic backgrounds and with ties to different industries. In this distribution, it is also true that some parties will be ideologically and compositionally more similar to some parties than others. If the above map is considered again in metaphorical terms with the 2-dimensional space representing policy preference instead of location, than Party B and C could very well form an over-sized coalition around shared redistribution goals or shared tax credits for certain industries. The principles and framework in this case remain the same.

The last institution that incentivizes oversized coalitions is party leadership.  The leader of the plurality-winning party is likely to be picked as the prime minister at least in the initial round of coalition government construction.  This is because in most cases either the rules demand it, someone within his party is the formateur, or he himself is the formateur. As a result, the leader has a reason to make the first government coalition as stable as possible. The location of this equilibrium point is at the smallest Nash equilibrium, or the first point where no other party can unilaterally deviate to another coalition and change the result.

Returning to the previous example, the smallest coalition that achieves this goal is one that includes Party A, C, and D (70 members). This is clearly not a least minimum winning coalition, but it is optimal for the leader of party C in that if either Party A or D individually leaves the coalition it can still maintain a majority. Thus the status quo is “stable.” This can be proven using the following game, where both must deviate in order to form a new coalition, but if one deviates but the other does not, the deviation only results in them getting kicked out of the winning coalition, and the loyal party’s share of the ministries increases.

Party B
Party A Stay in Winning Coalition Deviate
Stay in Winning Coalition 2,2

(Nash Equilibrium)

3,-2
Deviate -2,3 3,3

(This is only a Nash with these payoffs if the other parties can somehow make a coalition without C)

Forcing this game of cooperation onto the other member parties makes the leader of Party C more secure as prime minister. Of course, this will only have a relevant effect when parties are strong and centralized. Any systems with independents and more pervasive single member districts will erode this explanation’s effectiveness as a reason for over-sized coalitions.

It is these factors, among others,that lead to the empirical result of over-sized coalitions. Incorporating party leadership, regional concentration, and demographic similarities into the model of minimum winning coalitions brings necessary nuance. My new theory then predicts that while in general parties do strive to minimize coalitions, they do so with respect to incentives.  Under certain regional distributions public works projects like freeways can pull parties to form over-sized coalitions. Demographics can similarly push parties to align with other parties that have socioeconomic overlap or that are beholden to similar industries. Finally, in countries with strong centralized parties, formateur rules can prioritize stability for the plurality-winning party. Stability can necessitate coalitions large enough to prevent unilateral deviation.

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