Elections are one of the most important processes in representative democracies and build the basis for a government’s legitimacy [1] [2]. With their votes, electors transfer the people’s sovereignty to representatives in parliaments. How elections proceed and in which way voters decide to give their votes to candidates or parties are therefore of critical importance for democracies. For the voters it is especially critical that they are aware of the consequences their votes have and how they influence policy outcomes by voting for one or another alternative. In this vein, multiple authors have outlined the importance of electoral systems that are easy to understand (e.g. [3] [4] ). However, there are also factors beyond the electoral system that can render voting in one's best interest a complicated task. Even in the simplest form of Proportional Representation (PR) systems, voters have to consider not only the political positions and potential strengths of parties, but also which parties are or are not likely to form coalitions with each other and which policies the various coalitions would probably implement [5]. While the potential of voting strategically presents a chance for voters to bring outcomes closer to their own preferences, incentives for strategic voting can pose severe normative problems for at least two reasons. First, the need for strategic calculations can overburden voters so that they might not be able to detect which decision is in their best interest [6]. This problem grows if certain groups of voters are left behind and sophisticated voting factually becomes an elite privilege. Second, strategic voting can lead to inner conflicts for voters. If there are incentives to vote strategically, voters can either vote in an expressive way for their preferred party or maximize their expected utility over policy outputs―but not both at the same time like in situations without strategic incentives.

It is thus important to map out under which conditions strategic incentives appear more or less frequently. Tackling the puzzle of strategic voting under PR rules, the existing literature has focused on voters’ calculations based on policy positions, party strengths, and the resulting policy expectations (e.g. [7] [8] ) in order to explain how often and for whom strategic incentives arise. In this way, the prevalence of strategic voting in PR systems can be explained to a considerable degree. However, existing research typically ignores varying styles of coalition formation. This is problematic because incentives to vote strategically are expected to depend on the way in which parties take sizes and policy positions into account when negotiating over what will be the eventual government coalition which will then decide on policies. Depending on the―office and policy―considerations which determine coalition formation by the parliamentary parties, voters should be faced with varying incentives to vote strategically. Hence, the way a government coalition is typically formed should affect the prevalence of strategic incentives for the electorate at large―a critical characteristic of an election, both from a practical and from a normative perspective. Therefore, the question we seek to investigate in this paper is how different rationales of coalition formation affect the prevalence of incentives for strategic voting.

We contribute to an answer by first outlining how existing research has largely omitted the potential variance with respect to how coalition governments form and how different rationales of government coalitions affect voters’ strategic calculations. Second, we present a model to identify rational choices for complex voting decisions including the anticipation of coalition building processes and the legislative stage. We then apply this model and use simulations in order to explore in how far different rationales of coalition formation in combination with characteristics of the party system affect the prevalence of strategic incentives. We conclude by mapping out the key implications of our investigation.

While the elder literature on strategic voting has been focused almost exclusively on plurality systems, newer contributions have started to overturn “the view that voters don’t vote strategically in PR elections” [8] (p. 955). Some analyses which try to measure the share of strategic voters even estimate that strategic voting occurs to a similar degree under pure PR electoral systems and under plurality systems [9] [10]. Whereas, in plurality systems, incentives to vote strategically derive primarily from the attempt not to waste the ballot by voting for a chanceless candidate [11] , in PR systems, it is especially the coalition formation process that provides potential for strategic votes [12] (pp. 90-91).

In explaining this initially puzzling phenomenon of strategic voting in PR systems, scholars have developed multi-stage models of strategic voting under PR rules (for an overview, see [8] ). Already Downs argued that rational voters in PR systems must consider the following steps of government formation and legislative decision-making in order to maximize the expected utility over their voting decision [5]. Figure 1 visualizes how voters must anticipate what happens after the elections in order to optimally influence policy outputs with their votes. Accordingly, Figure 1 shows that multiple factors potentially complicate voting decisions.

Yet, aforementioned calculations are made fairly complex by the fact that PR electoral systems usually do not see a single party gaining a majority of the parliamentary seats and, hence, coalition formation becomes necessary. Voters base their choices not only on party but on coalition preferences [13]. Furthermore, multiple researchers have highlighted how voters―since they base their choices on expected government policy―try to engage in policy-balancing and vote strategically in order to steer the expected coalition policy in their direction [7] [14]. Critically, voters’ expectations regarding the likelihood of different potential coalitions play a vital role for their decision, and these expectations are based on past behavior of the parties as well as coalition signals [15] [16] [17].

What follows is that not only electoral systems but the whole electoral process including all its consequences can render the identification of the optimal vote a complicated challenge. In a laboratory experiment, [6] have shown that the

question of which kinds of pre-electoral coalitions form heavily influences the share of voters being able to identify what would be their rational choice. Illustrating the importance of taking the stage of coalition formation into account when explaining the prevalence of strategic incentives, [8] (p. 965) has emphasized how the rationale of coalition formation should lead to rather different strategic incentives, concerning both their prevalence as well who is mostly affected by them.

Yet, while existing studies have stretched the importance of voters’ expectations regarding specific coalitions [17] , they have neglected the impact of variance with respect to the overall rationale that guides parties as they form a coalition government, the general style of government formation in a political system. Existing research largely focuses on stages one and three as depicted in Figure 1 while the coalition formation stage is taken into account only in so far as one specific logic of coalition formation is assumed in order to determine the eventual government coalition―or is consciously left unspecified like in Linhart’s model [21] [22].

So far it is almost uniformly assumed that parties care strongly about policy when considering coalition options [7] [23] [24] [25]. However, the role of parties that also may follow rather office-oriented or mixed-motivated rationales of coalition formation is understudied. In the following sections, we will thus explore the impact of different rationales of coalition formation on the prevalence and distribution of strategic incentives among the electorate.

For our purpose, we choose to follow Linhart’s decision theoretic model¹ [21] [22] which is more flexible in varying the stage of coalition formation than are later models by [8] or [17] ―which have other strengths. This model is an expansion of Riker and Ordeshook’s approach for plurality systems [26]. It is assumed that rational voters build preferences over policy outputs o, not (only) over parties P. This means that for the calculus it does not matter how much a voter likes or dislikes a certain party, but what the election of this party means for the eventual policy output. Formally speaking, u(P) is not a meaningful expression in this model, but u(o) is (cf. also [27] ).

Let n be the number of electable parties, then a voter has n alternatives, namely giving her vote to one of the parties k (A_k) (see Table 1).² Riker and Ordeshook do not model this situation in a game theoretic way where all voters would have to react to each other but as a decision under risk [26]. This means

that the collective choice of all voters but ego is considered as a state of the world sotw. We call the set of all states of the world SOTW. Each sotw can be interpreted as (and formally looks like) a preliminary election result in which only the vote of ego is missing. The combination of a state of the world sotw and one of the alternatives A then leads to a final electoral result and a corresponding seat distribution S. The expected utility Eu of an alternative A then equals the sum of all utility values resulting from A weighted by the probability p of the respective states of the world. The rational choice is the alternative with the highest expected utility.

In the standard model for plurality systems only two parties have a realistic chance to form the government.³ This makes stages 2 and 3 in Figure 1 trivial: The majority party can form a government (stage 2) and implement her preferred policy (stage 3). Consequently, utilities over seat distributions S and over policy outputs o converge, and only two policy outputs are possible: u ( S ) ≡ u ( o ) = u ( y 1 ) or u ( y 2 ) , where y_k denotes the ideal position of a party P_k, and P₁ and P₂ are the largest parties.

In situations with more than two parties, typically emerging under PR or mixed electoral systems, the model is much more complicated. In general, there is no party with an absolute majority, i.e. two or more parties must form a coalition C. The coalition government has to compromise on a policy y_C which is unlikely to be any party’s ideal position. Hence, stages 2 and 3 of the political process become non-trivial.

The extension for any electoral system and party systems of any structure by Linhart works as follows [21] [22]. Since the rational voter’s utility refers to policy outputs o, the three-stage model must be solved from the end via backward induction. Starting there, preferences over policy outputs are measured by distances in a policy space O [5] in which oftentimes squared distances are used [28] [29] :

u i ( o ) = – ( y i – o ) 2 for all oÎO. (1)

If π_C(o) is a probability function which describes the likelihood that a government C implements a policy o, then the expected utility of voters over coalition governments can be estimated as

E u i ( C ) = ∑ o ∈ O π C ( o ) ⋅ u i ( o ) = – ∑ o ∈ O π C ( o ) ⋅ ( y i – o ) 2 (2)

for finite policy spaces O, or

E u i ( C ) = ∫ o ∈ O π C ( o ) ⋅ u i ( o ) d o = − ∫ o ∈ O π C ( o ) ⋅ ( y i − o ) 2 d o (2’)

for infinite policy spaces O, respectively.

A reasonable simplification of formulas (2) and (2’) can be found, for example, in Bandyopadhyay and Oak’s work [30] (see also [31] ) where a coalition’s policy is estimated as the mean policy of its members, weighted by the parties’ sizes, i.e.

y C = ∑ P k ∈ C y k ⋅ s k / s C , (3)

where s_k denotes a party P_k’s size and s_C the cumulative size of all parties in C. In terms of formula (2) and (2’) this means that π C ( y C ) = 1 , while π C ( o ) = 0 for all o ≠ y C . As a consequence,

u i ( C ) = – ( y i – y C ) 2 . (4)

We follow formulas (3) and (4) because of their simplicity and straightforwardness (cf. [24] ). This way is the easiest possibility to estimate policy outputs depending on the coalition C and the sizes of C’s members.

Going back one stage and denoting by q_S(C) the probability that, given a seat distribution S, a government C forms, it is possible to construct expected utility functions for voters over seat distributions:

E u i ( S ) = ∑ C q S ( C ) ⋅ u i ( C ) = – ∑ C q S ( C ) ⋅ ( y i – y C ) 2 . (5)

With help of formula (5), the entries of all cells in Table 1 can be computed, and the Riker-Ordeshook model can be applied also to multi-party systems. Equation (5) highlights that the question of how likely a coalition will form, q_S(C), strongly influences the rational calculus. As it is our research question which coalition building procedures raise or confine incentives for strategic voting, we abstain from fixing q_S(C) here but compare effects of different alternatives (i.e. different rationales of coalition formation) in our simulation.

In order to completely solve the utility maximization problem, we have to specify the p(sotw) values needed for the computation of the alternatives’ expected utilities. According to the decision theoretic standard approach we would have to fix a number of voters, combine all possible voters’ decisions (except ego’s decision) to states of the worlds and choose a meaningful probability function p. This standard procedure is problematic for two reasons. First, any chosen probability function would be ad hoc, and second, for a reasonable number of voters, there are more states of the world to consider than standard computers can deal with. For this reason, we do not compute expected utility values according to the standard approach but use an approximation. Technical details are described in Appendix I.

In order to explore the consequences of different general rationales of coalition formation and derive empirical implications, we simulate party systems and detect strategic incentives using the model introduced above. In this way we are able to generate the variation of cases needed in order to fully map out the consequences of varying styles of coalition formation (for similar approaches see [32] [33] [34] [35] ).

Our simulation is designed as follows. We let the number of parties run between 3 and 7 and simulate 2000 party systems for each number of parties.⁴ Since we research the effect of coalition formation procedures, one- and two-party-systems are not of interest. Further, an inspection of our simulated data shows that seven-party-systems are highly fragmented (see Appendix II), so that simulations with eight or more parties are not assumed to deliver additional insights. We allocate party positions y_k and party sizes s_k to every party by random choice with help of the random function in Mathematica©. The party positions are drawn from a uniform distribution over the [0, 1] interval. Party sizes are simulated as follows. Each party is randomly assigned a number of seats between 1 and 101―the latter is the size of our simulated parliaments and therefore the maximum possible number of seats for a party. The party sizes then are normalized by division through the total of all party seats so that the parliament sizes equal 101 seats. Before a simulated party system becomes part of our data set, we test it for two conditions: First, every party must hold at least one seat after normalization, so that the party system is a real n-party-system, not actually an (n − 1)-party-system. Second, no party may hold an absolute majority of the seats since, then, government formation becomes trivial. If one of these conditions does not hold, we delete the case and re-draw. While alternative mechanisms make sense, too, but might lead to different party systems, it is noteworthy that we finally verified that our simulated party systems resemble factually existing party systems. The respective summary statistics can be found in Appendix II.

We base our exploration on established theories of coalition formation in order to cover a broad range of possible rationales. For each party system, we apply nine different coalition building procedures following standard coalition theories (see below for details) which vary in their assumptions about how much parties care about gaining offices and influencing policy respectively.⁵ We order these procedures beginning with coalition formation procedures with strict emphasis on offices, neglecting any role of policy. Step by step, policy becomes more and offices become less relevant, ending with scenarios in which coalition formation is solely depending on policy aspects. This thread allows us to systematically show how and which office considerations influence incentives for strategic voting compared to policy considerations.

Following theories which assume office seeking parties, our first bloc of procedures considers minimal winning coalitions only [36]. The strictest form of office-orientation is the deterministic formation of the smallest size coalition (based on parliamentary seats), independent of all further factors [37] [38]. We model this procedure in

C1: q S ( C ) = 1 if C is smallest size; q S ( C ) = 0 otherwise. If two or more coalitions are smallest size, one of them is selected randomly.

In order to add an element of uncertainty (beyond tie-breaking rules) to the process of coalition formation, we soften strictly deterministic rules: C1 is thus contrasted with a similar procedure in which coalition bargaining is more flexible, i.e. no coalition comes into office with a probability of 1:

C2: q S ( C ) = 0.5 if C is smallest size or second-smallest size; q S ( C ) = 0 otherwise.

This variation implies more leeway given to parties actually bargaining over various coalitions, which is i) more realistic, ii) makes more parties potentially relevant to influence policy outputs, and iii) softens the strict assumption that office seeking parties will necessarily form the smallest size coalition.

Alternatively, Leiserson proposes to minimize the number of coalition parties instead of the parties’ cumulative seat share―a coalition formation logic which he labels the “bargaining proposition” [39]. We follow this idea in scenarios 3 and 4:

C3: q S ( C ) = 1 if C minimizes the number of parties; q S ( C ) = 0 otherwise. If two or more coalitions minimize the number of parties, one of them is selected randomly.

Likewise C2, C4 softens the rule’s determinism:

C4: q S ( C ) = 1 / # C ∗ if C minimizes the number of parties; q S ( C ) = 0 otherwise. #C* denotes the number of coalitions which minimize the number of parties.

In order to consider findings of policy oriented coalition theories, we follow Leiserson’s approach of minimal range coalitions [40] (see also [41] ) and combine it with the bargaining proposition in scenario 5:

C5: q S ( C ) = 1 if C minimizes the number of parties; q S ( C ) = 0 otherwise. If two or more coalitions minimize the number of parties, the one with the smallest range is selected. If still two or more coalitions remain, one of them is selected randomly.

Formally, it looks as if the policy aspect mattered very rarely only in special cases of parity regarding the bargaining proposition. This would be true if the office rule were “smallest size”, where the seat minimizing coalition usually is unique. In most of the cases, however, there are two or more coalitions with a minimal number of parties, so that the policy aspect comes into operation.

Making one more step towards policy orientation, we turn away from the strictest forms of office seeking theories and formulate as a baseline condition only that coalitions should be minimal winning and choose the minimal range coalition out of this set. Although this scenario still includes office aspects, the focus clearly lies on policy. This formation procedure functions analogous to De Swaan’s policy seeking modeling approach in which he, too, considered minimal winning coalitions only [41]. Formally, we model this scenario as:

C6: q S ( C ) = 1 if C is minimal winning and minimizes the range; q S ( C ) = 0 otherwise. If two or more minimal winning coalitions minimize the range, one of them is selected randomly.

The respective less deterministic scenario is

C7: q S ( C ) = 0.5 if C is minimal winning and has the smallest or second-smallest range; q S ( C ) = 0 otherwise.

Giving up office considerations completely, we conclude with scenarios 8 and 9 which reflect the idea that party competition consists of two competing blocs of parties along the left-right spectrum [42]. From Black’s Median-voter theory we know that the median position beats any other proposal in a pairwise vote, i.e. the median position is a Condorcet winner [43]. The median player―if she exists―gets support for her ideal position from players to her right against proposals to her left and vice versa. This means that she has the choice to form a coalition with either the players to her right or to her left. In coalition theory, we often find other denominations like “pivotal player” [41] or “central player” [44] [45] which are very close or identical to this concept. Denoting the party farthermost left by p_L, the party farthermost right by p_R, and the median party by p_M, scenario 8 is formally defined as

C8: q S ( C ) = 1 for C = { p L , ⋯ , p M } ; q S ( C ) = 0 otherwise.

The choice of the coalition left of the median follows a without-loss-of-generality assumption. This means that results would not change if we replaced the coalition by that right of the median ( C = { p M , ⋯ , p R } ). Scenario 9, finally, is the non-deterministic pendant to scenario 8. Here, both the left and the right coalition occur with a probability of 0.5.

C9: q S ( C ) = 0.5 for C = { p L , ⋯ , p M } and for C = { p M , ⋯ , p R } ; q S ( C ) = 0 otherwise.

Table 2 summarizes the coalition formation scenarios and shows attributes of

Notes: the column-heading “minimal winning” implies that a scenario will certainly lead to a minimal winning coalition where this criterion applies, but not that a minimal winning coalition cannot occur where this criterion does not apply; the same holds for the column-heading “connected”.

the different procedures. Most importantly, we see that from 1 to 9, office consideration become less, and policy aspects more important. For scenarios 1 to 7, any coalition government is minimal winning, while this is not guaranteed in scenarios 8 and 9. This does not mean that minimal winning coalitions cannot be formed in these scenarios, but surplus coalitions are possible, too, since dummy players―which are not necessary for a coalition’s majority―may be located between p_L and p_M (or between p_M and p_R). On the other hand, the connectedness of coalitions (cf. [46] ) is guaranteed in the last two scenarios only. If surplus coalitions are assumed not to form (like in scenarios 1 to 7), the resulting exclusion of dummy players might distort a coalition’s connectedness. This does not mean that policy distances do not matter at all. On the contrary, besides scenarios 8 and 9, aspects of policy distance play a role in three further scenarios (5 to 7).

The right column points to an aspect beyond office and policy. It indicates whether or not coalition formation is completely deterministic or, formally, whether there is one certain coalition with q S ( C ) = 1 . Table 2 shows that every strict deterministic scenario has a respective counterpart. An exception is scenario 5, in which the probabilistic component of scenario 4 is replaced by the integration of policy aspects.

Having specified the coalition building rules, the formal model as described above is complete. The rational calculus of voting can be computed for each of the 90,000 simulated cases (2000 drawings for 5 different numbers of parties times 9 coalition building scenarios). As concerns the output variable, we primarily seek to investigate how different rationales of coalition formation affect the incentives for strategic voting for the whole policy space. In order to derive an aggregate measure reflecting the overall characteristics of an electoral situation, we go through all policy positions between 0 and 1 in steps of 0.01. For each position, we both identify the closest party to this position (sincere choice)⁶ and calculate the party that emerges as the result of a rational calculus (rational choice). The share of positions for which the rational choice does not coincide with a vote for the closest party (i.e. a sincere choice) is the dependent variable in our analyses―we deem this the strategic share of an electoral situation. The higher this share, the more incentives for strategic voting can be found in the scenario, and the more this scenario is concerned with the normative problems of strategic voting as outlined in the introduction.

The structure of our results section is as follows. First, we derive implications with respect to the different coalition formation procedures. Second, we focus on effects of party system characteristics and present comparative statics including all variables. Finally, we explore at which positions incentives to vote strategically occur more or less frequently and thus focus on the distribution of strategic incentives under different circumstances.

In summary, our simulation study underscores the importance of taking variation with respect to the general rationale of coalition formation seriously. Both the manner of coalition building and party system characteristics affect the prevalence of strategic incentives in voting scenarios. As we argued in the introduction, incentives for strategic voting can be seen as problematic from a normative view. Firstly, they produce situations in which voters cannot vote both rationally with regard to policy outputs and expressively for their first preference but must make a choice between ratio and emotion. Secondly, at least some voters might not be able to detect the rational vote and thus do not vote for their own best even if they would like to do so.

In terms of implications, our simulation detects three key factors that should reduce this undesirable effect, two of them coming from coalition formation and one from party system characteristics. First, the higher the policy orientation of parties during coalition formation (and thus the smaller their office orientation), the fewer policy positions are concerned with the problem of strategic incentives. Parties, thus, can reduce the problem by predominantly signaling and building connected coalitions or at least coalitions with parties that are not too far away from their own policy position instead of focusing on office perks. Second, strict deterministic rules raise the strategic share. Therefore, parties can further minimize that issue by leaving the coalition formation process open and not excluding other parties generally as coalition partners. The more parties have a chance to be part of the government, the fewer voters have to vote strategically in order to influence policy for their own best.¹³ However, a more open stage of coalition formation could be considered undesirable on the grounds that it blurs the voter-government link by transferring the power of government-making from voters to parties. This clearly highlights that trade-offs emerge as we compare different (informal) institutional mechanisms of government formation. Third, incentives for strategic voting rise with party system fragmentation. While the stage of coalition bargaining typically only becomes unnecessary in plurality electoral systems, our results show that fragmentation continues to make a great difference among multi-party systems and therefore moderate party system fragmentation is preferable to highly fragmented party systems. As the latter are often associated with pure PR electoral systems, mixed electoral rules appear as a fruitful alternative. Said mixed electoral systems are also hoped to reach interparty efficiency via inducing the competition of two political blocs (see [42] ). Applying the coalition building logic of bloc-competition our analysis has shown that it should also lead to a further decrease in strategic incentives while also distributing strategic incentives evenly across the policy space. Coalition formation according to strict office incentives or including policy considerations via the formation of minimal range coalitions should both lead to a more unequal distribution of strategic incentives across the policy space, the former placing the burden of higher strategic incentives on center positions, the latter on positions at the extremes of the policy space.

Linhart, E. and Raabe, J. (2018) Different Rationales of Coalition Formation and Incentives for Strategic Voting. Applied Mathematics, 9, 836-860. https://doi.org/10.4236/am.2018.97058

For the reasons we outlined introducing our model, we estimate ego’s expected utility values E u e g o ( A k ) with help of approximations E u ^ e g o ( A k ) . Results for Eu and E u ^ are almost identical; errors do not occur systematically. This approximation works as follows. We first identify the seat distribution according to the simulated poll by applying the Sainte-Laguë PR method. We assume the uncertainty of the poll being exactly so large that a party could win or lose not more than one seat. For the majority of the states of the worlds, ego’s vote does not change the seat distribution at all, independent of which party she is voting for. Since the respective utility values are constant over all alternatives, they can be excluded from the subsequent computation. Further, as we use a strict PR rule, the numbers of states of the world in which ego is pivotal when voting for a party P_k are roughly of equal size for all parties. This means that we do not have to care about how often ego is pivotal because this number is equal for all parties. If then, one of the parties P_k gains one additional seat, one of the other parties loses one. For the same reason as discussed above, we consider all other parties as equally probable to be this seat loser. Given a seat distribution S = ( s 1 , s 2 , ⋯ , s k , ⋯ , s n ) , we therefore compute

E u ^ e g o ( A k ) = [ ( s 1 − 1 , s 2 , ⋯ , s k + 1 , ⋯ , s n ) + ( s 1 , s 2 − 1 , ⋯ , s k + 1 , ⋯ , s n )     + ⋯ + ( s 1 , s 2 , ⋯ , s k + 1 , ⋯ , s n − 1 ) ] / ( n − 1 ) . E u e g o ( A k ) and E u ^ e g o ( A k ) , certainly, take very different values because of different (or lacking, respectively) normalizations. The alternatives A_k, however, which solve the maximization problems are the same aside from very rare cases.

For each number of parties, the table shows the average fragmentation, the average polarization,¹⁴ the respective standard deviations and the distribution to types of party systems. For the latter, we use Laver and Benoit’s categorization system [52]. It is not surprising that party system fragmentation increases with the number of parties. At the same time the effective number of parties is, on average, lower than the pure number of parties for each N, reflecting unequal size distributions―which is of course reflective of the situation in real cases. On the other hand, we see that party system polarization is almost independent of the number of parties while there is sufficient variation within each group. This feature of the simulated data will make it easier for us to cleanly estimate the effects of fragmentation and polarization.

We further see that the more parties we include, the fewer party systems fall into the rather concentrated categories top-three (with at least three two-party winning coalitions) and “strongly dominant party” (with at least two two-party winning coalitions) and the more correspond to more fragmented top-two (with one two-party winning coalition only) or “open” party systems (without any two-party winning coalition). Reassuringly, these patterns of our simulated data match those identified for actual data from European legislative elections by Laver and Benoit, although there is a greater tendency for the simulated party systems to be of the open type as the number of parties increases―likely caused by the random assignment of party sizes from a uniform distribution [52]. However, our simulated data seems to be sufficiently similar to actual data in order to generalize our results.

Notes: The extreme minimum and maximum values for C8 stem from the fact that this scenario always sees a bloc of parties consisting of the median party and all parties left of it forming the government. While this formation rule can be used without loss of generality for all other analyses in this paper this does not hold for the summary statistics reported here. An adapted rule randomly choosing the left or right bloc deterministically would lead to statistics similar to those of C9.