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Research on strategic voting has mainly focused on electoral system effects but largely neglected the impact of different rationales of coalition formation. Based on a formal model of rational party choice and a simulation study, we systematically investigate this impact and explore the implications. We show that the logic of the underlying coalition formation procedure clearly affects the degree to which the electorate is exposed to strategic incentives regarding the vote choice. The key implications are that sincere voting is more often in the voter’s best interest if parties are policy-seeking and if there is increased uncertainty during the stage of coalition formation. Furthermore, we explore how different types of coalition formation affect strategic incentives across the policy space.

Elections are one of the most important processes in representative democracies and build the basis for a government’s legitimacy [

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. [

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” [

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 [

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 [

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, [

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, [

Yet, while existing studies have stretched the importance of voters’ expectations regarding specific coalitions [

So far it is almost uniformly assumed that parties care strongly about policy when considering coalition options [

For our purpose, we choose to follow Linhart’s decision theoretic model^{1} [

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 ^{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 [

sotw_{1} | sotw_{2} | … | Eu(A_{k}) | |
---|---|---|---|---|

A_{1} (election of P_{1}) | u(S_{11}) | u(S_{12}) | … | E u ( A 1 ) = ∑ s o t w j ∈ S O T W p j u ( S 1 j ) |

A_{2} (election of P_{2}) | u(S_{21}) | u(S_{22}) | … | E u ( A 2 ) = ∑ s o t w j ∈ S O T W p j u ( S 2 j ) |

… | … | … | … | |

A_{n} (election of P_{n}) | u(S_{n}_{1}) | u(S_{n}_{2}) | … | E u ( A n ) = ∑ s o t w j ∈ S O T W p j u ( S n j ) |

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 _{k} denotes the ideal position of a party P_{k}, and P_{1} and P_{2} 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 [

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 [

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. [

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 _{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 [

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 [

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” [

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 [

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 [

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 [_{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.

Scenario | Rule | minimal winning | distance relevant | connected | deterministic |
---|---|---|---|---|---|

C1 | smallest size I | x | x | ||

C2 | smallest size II | x | |||

C3 | bargaining proposition I | x | x | ||

C4 | bargaining proposition II | x | |||

C5 | bargaining proposition and minimal range | x | x | ||

C6 | minimal range I | x | x | x | |

C7 | minimal range II | x | x | ||

C8 | central player I | x | x | x | |

C9 | central player II | x | x |

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. [

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 .

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.

The relatively high values for strategic share are partially driven by the kind of modeling since our calculus of voting includes all types of strategic voting (making some coalitions more and others less likely by strengthening/weakening certain parties, intra coalition balancing of policy outputs, considering wasted votes with regard to policy outputs etc.) and is not restrained to special types. Secondly, the calculus is rigorously focused on policy outputs which are influenced by government parties only.^{7}

A first inspection of

Furthermore,

Turning to the impact of uncertainty, strategic incentives are less prevalent in less deterministic scenarios (C2, C4, C7, C9) than in their deterministic equivalents (C1, C3, C6, C8). This effect results from the openness of the coalition formation process. The more open a government formation process is, the more parties are likely to influence governmental policy, and the fewer are therefore excluded as non-rational alternatives.

^{8} If a rule guarantees the connectedness of governing coalitions, the share shrinks by 6.0 points. We find an even larger effect when comparing deterministic and non-deterministic rules. The latter’s strategic share is 7.3 percentage points smaller. In sum, policy oriented government formation appears to alleviate the problem of a strong prevalence of strategic incentives in an

Yes | No | Difference (Yes ? No) | |
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Distance relevant | 0.581 (0.187) | 0.628 (0.260) | −0.047 |

Connected | 0.555 (0.186) | 0.615 (0.232) | −0.060 |

Deterministic | 0.634 (0.213) | 0.561 (0.231) | 0.073 |

Smallest size (C1, C2) | 0.660 (0.259) | 0.585 (0.210) | 0.075 |

Bargaining proposition (C3, C4, C5) | 0.598 (0.234) | 0.603 (0.220) | −0.005 |

Minimal range (C5, C6, C7) | 0.598 (0.186) | 0.603 (0.241) | −0.005 |

Central player (C8, C9) | 0.555 (0.186) | 0.615 (0.232) | −0.060 |

electoral situation. Furthermore, increased openness of formation procedures sees sincere and rational choices coincide more often.

Breaking down the various office and policy oriented models, we again see the structure discussed above. The most rigorous office oriented formation rule (smallest size) leads to the highest differences in means (7.5 percentage points), the strictest policy oriented rule (central player) to the lowest (−6.0 percentage points). The others lie in-between with −0.5 percentage points. Yet, there are two results that do not seem to fit the general patterns outlined so far.

First, it is striking that the office oriented rule bargaining proposition exerts a, albeit small, negative effect. The slightly negative difference in means, however, does not contradict our findings with respect to the effect of different rationales of coalition formation. Since the reference category includes policy oriented rules but also the stricter office oriented smallest size rule, it simply seems to be the case that the effects of C1 and C2 are stronger than those of C6 to C9. Further, the bargaining proposition sample also includes one rule (C5) which at least partially considers policy.

Second, we might expect a strong monotonic decline of the differences in means as we move from office to policy oriented coalition building procedures, but see the same values for bargaining proposition and minimal range. This is mainly an effect of C4’s results whose outlier position can already be seen in _{S}(C) function thus makes (nearly) all minimal winning coalitions equally probable (see also footnote 9). Thus, what seems puzzling based on the general implication that office oriented government formation will drive up the strategic share is easily explained by taking the role of increased uncertainty into account. The results regarding the effect of uncertainty based on the simulated data also fit very well with recent findings that increased uncertainty about outputs should lead to more sincere voting (e.g. [

Continuing with the party system characteristics of parliamentary fragmentation and ideological polarization, ^{9} We further see that the effect of the number of parties itself is larger in some scenarios (e.g., in C1 and C2) and smaller in others (like C5 and C6).

We assess the effect of party system polarization^{10} within regression analyses concluding our primary empirical investigation (see ^{11} The regression analysis furthermore largely reproduces the results from

Model 1 | Model 2 | |
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Number of parties | 0.081*** (0.0004) | 0.081*** (0.0042) |

Polarization | 0.052*** (0.0073) | 0.049 (0.0693) |

Distance relevant | −0.042*** (0.0014) | −0.042*** (0.0137) |

Connected | −0.031*** (0.0017) | −0.032* (0.0164) |

Deterministic | 0.076*** (0.0013) | 0.076*** (0.0121) |

Constant | 0.171*** (0.0031) | 0.172*** (0.0295) |

N | 90,000 | 1000 |

R^{2} | 0.306 | 0.309 |

Notes: *p < 0.1, **p < 0.05, ***p < 0.01 standard errors in parentheses.

Based on Model 2 we examine whether or not our results would remain statistically significant if we used a more common number of cases. We therefore iteratively draw a random sample of 1000 from our dataset 1000 times and run the statistical model for each random sample (see [^{2} is the average R^{2}.

The regression analyses yield the same implications as our inspections of

Model 2 clearly exposes that there is a lot of uncertainty around the effect of polarization. For a more realistic number of cases, the effect is far from being statistically significant (p = 0.477). It is also worth mentioning that for Model 2, the variance around the effect of connectedness is comparably large (leading to p = 0.053). Yet, the policy effect is captured quite clearly by the distance variable.

After having assessed strategic incentives via a summary score for every electoral situation, it is important to further explore which positions are affected by strategic incentives. In a first step, we ask whether it is voters in the political centre or at political extremes that are most often faced with strategic incentives―and whether this varies by coalition formation procedure.

The procedures depicted in

II)―representing a policy-seeking, bloc-building logic of coalition formation―exhibits trends much more akin the office-seeking procedures discussed above but on a distinctly lower level. When political blocs compete for government, variation in strategic shares is fairly low with no pronounced difference between centre and extreme positions. However, positions at the extreme right or left are slightly less concerned with strategic incentives. These findings with respect to the policy oriented scenarios are also well in line with what Bargsted and Kedar find when assuming that governments, as voters, are driven by policy concerns [

As it is not only of interest which (absolute) positions are more or less affected by strategic incentives but also if it is positions close or farther away from the parties’ positions, we examine in a next step whether an effect of the distance to the closest party can be observed in a logit model predicting when a particular position will present a strategic incentive (i.e. the rational choice is not the sincere choice).

It is first worth mentioning that the analysis confirms our results above. Strategic incentives occur less often in scenarios with policy oriented coalition formation rules but more often if rules are deterministic (while the significance of the latter effect is questionable). More parties lead to more strategic incentives. Interpreting the impact of distance the regression estimates suggest a negative effect under most circumstances, implying that the farther away a voter’s position is

Model 1 | Model 2 | |
---|---|---|

Number of parties | 0.351*** (0.001) | 0.351*** (0.05) |

Policy scenario (C5-C9) | −0.345*** (0.002) | −0.341* (0.192) |

Deterministic scenario | 0.213*** (0.002) | 0.212 (0.191) |

Distance to closest party | −1.361*** (0.012) | −1.368 (1.183) |

Distance to closest party * Policy scenario | 0.95*** (0.013) | 0.928 (1.307) |

Distance to closest party * Deterministic scenario | 1.302*** (0.013) | 1.342 (1.305) |

Constant | −1.232*** (0.003) | −1.228*** (0.313) |

N | 9,090,000 | 1000 |

Pseudo R^{2} | 0.05 | 0.053 |

Notes: *p < 0.1, **p < 0.05, ***p < 0.01.

from the party closest to her, the less likely it is that she will face incentives to vote strategically. This negative effect is moderated in scenarios with policy seeking coalition building rules and almost equalized in scenarios with deterministic rules.^{12} The negative sign could be surprising since one might expect voters with options close to their ideal points being in more favourable positions. This is still true with regard to the expected utilities over policy outcomes but appears to be different regarding the likelihood of strategic incentives.

The analysis of distances allows us to also discuss effects of different voter distributions. Our main dependent variable, the strategic share, counts every position in the policy space equally. This is adequate as we are interested in the share of positions with strategic incentives. Additionally, however, the share of voters having incentives to vote strategically is of interest and, as voters usually are not distributed equally over the policy space, these two shares are different. Assuming that parties, as rational actors, offer policy platforms at positions where voters agglomerate, it is reasonable to expect more voters at positions close to parties’ positions than on positions farther away from the parties.

From this standpoint, a negative effect of the distance variable means that the share of voters with strategic incentives is even higher than the share of pure positions accompanied by strategic incentives, which would render the dilemma more important. While the effect of distance does not reach common levels of statistical significance when a realistic number of cases is observed and therefore our results for the share of positions should roughly also hold for the share of voters, the results depicted in

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 [

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 [

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 [

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.