Assumptions on Seat Allocations and Policy Outputs

We specify a model in which parties and voters locate along a one-dimensional positional continuum – which we label the Left–Right policy continuum – and each voter supports the party that she prefers based upon her evaluations of the parties’ policy proximities and their valence attributes, using a decision rule that we specify below. We assume that parties’ parliamentary seat shares are exactly proportional to vote shares, i.e. that the voting system is perfectly proportional. We also assume that policy outputs are determined entirely by the parliament, an assumption that is best approximated in parliamentary democracies with unicameral legislatures.Footnote 11 Finally, we specify that the number of seats in the parliament is odd, so that there exists a single location for the median legislator.

Following Adams and Merrill, we assume that there are K policy-seeking parties with preferred positions R ₁,…,R_K and policy positions (strategies) s ₁,…,s_K .Footnote 12 We define the median parliamentary party (MPP) as the party that, together with all the parties with policy positions to its left, can form a majority and that can also form a majority if, alternatively, it were combined with all the parties with policy positions to its right.Footnote 13 We further assume that the parties project that following the election the MPP dominates the policy-making process and that it is constrained to implement its pre-election policy position. Thus, a party k's utility U_k for an election outcome is equivalent to its utility for the policy position of the MPP. Defining f (s_j , R_k ) as party k's utility for party j's policy position s_j , where f (s_j , R_k ) is assumed to be concave and to peak at R_k ,Footnote 14 it follows that k's utility for an election outcome is:

$${\rm{Party }}\ k{\rm{'s \ utility}}\,{\rm{ = }}\,f{\rm{(}}{{s}_j}{\rm{,}}\ {{R}_k}{\rm{)\ if\ party }}\ j\ {\rm{ is \ the \ MPP}}{\rm{.}}$$

We note that our assumption that the MPP controls policy outputs contrasts with alternative models of policy making in parliaments – both theoretical and empirical – which emphasize the policy primacy of the parties in the governing coalition;Footnote 15 the central importance of the formateur, i.e. the party charged with forming the government;Footnote 16 the dominance of the party with jurisdiction over the relevant government ministry;Footnote 17 or a model in which all parliamentary parties exert some influence on policy outputs.Footnote 18 Choosing between these competing models is difficult, because the empirical literature on the relationship between parties’ policy positions and government policy outputs is under-developed. McDonald and Budge, however, report empirical results from a study of twenty-one post-war democracies, which is to our knowledge the only extensive, cross-national, study that analyses the links between parties’ policy positions and government policy outputs.Footnote 19 These authors analyse government policy outputs in three areas – central government spending, social spending and international policy – and find more instances of statistically significant associations with the MPP assumption than with the position of the government or with that of the relevant government ministries, and that overall support for an MPP assumption is comparable to that for these alternative assumptions.Footnote 20

In addition to the McDonald–Budge results summarized above, we note that Cho and Duggan have recently presented important theoretical results that, as legislators become arbitrarily patient, a large class of bargaining models of distributive politics collapses to the position of the median legislator.Footnote 21 This result, which runs counter to the folk theorem for repeated games that any possible division of resources can be supported as a subgame perfect equilibrium outcome, also supports our assumption of the policy primacy of the MPP. Thus, we have both theoretical and empirical reasons to believe that our model is relevant to policy making in real world democracies. At the same time, we emphasize again that our MPP-based model is only one of several plausible models of government policy outputs, and that the degree to which it captures policy-making processes in real world democracies surely varies across political contexts. On this basis we proceed.

Assumptions on Voters

We assume that voters’ party evaluations depend on their evaluations of the parties’ policy positions, plus a valence component.Footnote 22 Specifically, for each voter i with policy preference x_i , the policy distance component of i's evaluation of party j is given as ag(s_j , x_i ), where g(s_j , x_i ) represents i's utility for party j's position s_j , and a is a strictly positive parameter denoting the salience of the policy dimension relative to the valence dimension. We assume that for each voter i, g(s_j , x_i ) is concave and peaks at x_i . We assume no abstention.

The valence component of voter i's evaluation of party j is assumed to be the same for all voters and has two components: the party's measured valence characteristics, V_j , which the parties know at the time they select their policy strategies, and which we label the party's long-term valence image; and unmeasured valence characteristics ε_j , which capture valence-related events that occur between the dates when the parties select their policies and election day, and which the parties cannot anticipate at the time they choose their policy positions. We label the ε_j terms the parties’ short-term valence images. Thus:

$${\rm{Voter }}\ i{\rm{'s \ utility \ for \ party }}\ j\,{\rm{ = }}\,ag({{s}_j},{{v}_i})\, + \,{{V}_j}\, + \,{{{\epsilon}}_j}.$$

Our distinction between the measured, long-term components of valence, V_j , and the unmeasured, short-term valence components, ε_j , plausibly captures the information environment party elites confront as they devise their election strategies. At the time that parties commit to their policy strategies – which is typically well in advance of the electionFootnote 23 – political elites are likely to have formed general impressions about the parties’ comparative valence images, based upon their contacts with constituents, public opinion polls, media coverage and conversations with fellow elites. Such information forms the basis for the measured, long-term valence component, V_j . However, elites are well aware that parties’ valence images can fluctuate sharply between the time they select their policy positions and election day, in response to late-breaking political scandals, voters’ evaluations of politicians’ responses to foreign or domestic crises, changing economic conditions and how skilfully the parties conduct their election campaigns (we discuss real-world examples of such short-term valence effects below). The unmeasured, short-term, valence component ε_j captures this uncertainty.

Note that our model specifies that voters prefer the party that offers the most attractive combination of policies and valence characteristics, so that a voter may prefer a party that is less attractive on policy grounds (relative to its competitors), if this party has strong valence-related characteristics along such dimensions as competence, integrity and unity. We assume that all voters vote sincerely. The following remarks, which are proved in Adams and Merrill,Footnote 24 develop two important implications of our model, and also support our assumption of sincere voting:

1. Remark 1. When all individuals vote sincerely, then the party that is supported by the median voter will be the MPP.

2. Remark 2. The situation where all citizens’ votes reflect their sincere party preferences is also an equilibrium in outcome-oriented voters’ strategies, i.e. no voter can increase her utility for the election outcome by voting strategically.

Assumptions on How Voters Assign Responsibility for Valence-Related Events: A Generalized Extreme Value (GEV) Model

To this point, our model is identical to the one presented in Adams and Merrill.Footnote 25 However, we depart from the Adams–Merrill model in our assumptions about the joint distribution of the ε_j 's, where ε_j is the unknown, short-term component of voters’ valence evaluations of party j. In the Adams–Merrill model the ε_j 's are selected independently over parties from a type 1 extreme value distribution, an assumption that implies that voters’ choice probabilities can be represented via a logit probability function. Note, however, that independence of the ε_j terms is a very strong assumption because, substantively, this implies that there is no connection between the short-term events that influence voters’ evaluations of the focal party j's valence image, and short-term events or conditions that influence voters’ evaluations of other parties’ valence images. Yet this independence specification is problematic. Empirically, there is extensive evidence that in cases of coalition government, voters have difficulty assigning responsibility for economic conditions to any single governing party, so that governing parties tend to share the credit or blame for the state of the economy.Footnote 26 In addition, Michael Clark's analyses of valence effects in nine West European party systems concludes that when a governing party's valence image is damaged due to negative media coverage pertaining to political scandals, incompetence, or internal divisions, this event has a ‘spillover’ effect that depresses support for the party's coalition partners.Footnote 27 Significantly, Clark also identifies this effect with respect to ‘proto-coalitions’ of opposition parties, i.e. groups of opposition parties that have been in a governing coalition together in the past, and/or that are viewed as constituting a potential future government.

Two West European elections held in 2002 illustrate the impact of late-breaking, valence-related, events, and how these effects are mediated by collective responsibility. In the Dutch parliamentary elections on 15 May 2002, the governing ‘purple coalition’, consisting of the Labour Party (PvdA), the People's Party for Freedom and Democracy (VVD), and the Democrats ‘66 (D'66), suffered significant electoral losses due to valence-related events leading up to the election. Pim Fortuyn, the charismatic leader of his own party, the Lijst Pim Fortuyn (LPF), was extremely critical of the performance of the ‘purple coalition’, as portrayed by a Fortuyn pamphlet called the Purple Mess. Pim Fortuyn was assassinated nine days before the election, which served to amplify these performance-related criticisms in the public, and the difference between the results of public opinion polls just prior to the parliamentary elections and the actual outcome of the election suggests that the coalition's standing was thereby undermined, serving as an example of a late-breaking valence-related event that plausibly contributed to these parties’ unexpectedly poor election results.Footnote 28 Academic accounts suggest that the parties in the purple coalition were held jointly responsible for its performance. Furthermore, in the September 2002 German Bundestag election the Social Democratic Party and its leader, Gerhard Schroeder, waged an unexpectedly dynamic campaign that enhanced the party's valence image, and helped it achieve an unexpectedly strong election result. There is, however, little evidence that the SPD's strong campaign in the 2002 German election significantly enhanced the image of its coalition partner, the Green Party.Footnote 29

The studies and examples discussed above suggest that political parties’ valence images are not always independent of each other. In particular, at the time when governing parties select their policy strategies, they likely anticipate that the government's handling of foreign (or domestic) policy crises will have similar effects on the short-term valence images of all members of the governing coalition, as will economic fluctuations prior to Election Day.Footnote 30 ^, Footnote 31 However the example of the 2002 German election suggests that collective responsibility is not absolute – i.e. a coalition member's short-term valence image can improve (or deteriorate) unexpectedly, for instance due to skilful or inept campaigning, without affecting its coalition partners’ valence images.

To capture these effects, we employ a generalized extreme value distribution (GEV). Using the GEV specification (the details of which are described in the Appendix), we can model situations where the error terms associated with parties’ unmeasured, short-term, valence images are correlated with each other. Specifically, let the set K of parties be partitioned into n subsets, or coalitions, labelled c ₁, c ₂,…, c_n , where C = {c ₁, c ₂,…, c_n }. Within each coalition c_f , let r_f represent the common correlation between the error terms ε_j and ε_k associated with the unmeasured characteristics of the valence images of any two parties {j, k}∈c_f , j ≠ k. We label r_f the collective responsibility coefficient associated with the coalition of parties, c_f , where higher values of r_f denote greater degrees of collective responsibility. Furthermore, let the correlation between all ε_j and ε_l for which parties j and l are in different coalitions equal zero. Under these conditions, McFadden demonstrates that the probability P(k) that the median voter prefers party k∈c_f is

$$P(k)\, = \,\frac{{{{{\exp }}^{\left[ {ag({{s}_k},m)\, + \,{{V}_k}} \right]/(1{\rm{ - }}{{r}_f})}} {{{\left[ {\mathop{\sum}\limits_{j \in {{c}_f}} {{{{\exp }}^{\left[ {ag({{s}_j},m)\, + \,{{V}_j}} \right]/(1{\rm{ - }}{{r}_f})}} } } \right]}}^{({\rm{ - }}{{r}_f})}} }}{{\mathop{\sum}\limits_{{{c}_h} \in C} {{{{\left[ {\mathop{\sum}\limits_{j \in {{c}_h}} {{{{\exp }}^{\left[ {ag({{s}_j},m)\, + \,{{V}_j}} \right]/(1{\rm{ - }}{{r}_h})}} } } \right]}}^{(1{\rm{ - }}{{r}_h})}} } }},$$

where m denotes the median voter's position.Footnote 32 We label the voting model described above the collective responsibility model. For this model, party k's expected policy utility U_k is

$${{U}_k}\, = \,\mathop{\sum}\limits_{j\, = \,1}^K {P(j)f} ({{s}_j},{{R}_k}).$$

that is, party k's expected policy utility is an average of its policy utilities for each party as a prospective MPP, weighted by the probabilities that each party is in fact the MPP. In turn, each party k chooses that strategy s_k that maximizes its expected policy utility, U_k. We note that in the special case where r_f = 0 for all c_f ∈C, indicating no correlation between the unmeasured components of parties’ valence images, Equation 2 simplifies to the familiar logit equation

$${{P}_k}\, = \,\frac{{{{{\exp }}^{\left[ {ag({{s}_k},m)\, + \,{{V}_k}} \right]}} }}{{\mathop{\sum}\limits_{j \in K}^{} {{{{\exp }}^{\left[ {ag({{s}_j},m)\, + \,{{V}_j}} \right]}} } }}.$$

We label this special case the basic model.

The Peripheral Governing Party Moderation Theorem

Suppose that party K is a member of a coalition c_f , that K's optimal position lies strictly between its sincere policy preference R_K and the median voter position m, that this equilibrium position is to the left (right) of all competing parties, and that K's equilibrium probability P(K) of being the MPP is smaller than the MPP probabilities of the other members of the coalition c_f . Then if the collective responsibility coefficient r_f increases, K is motivated to unilaterally moderate its position relative to the median voter position m, i.e. to converge towards the positions of its coalition partners.

In words, the theorem states that when a member of a governing coalition is a peripheral party, i.e. when it promises the most radical right-wing or left-wing policies of all the parties contesting the election, and when moreover this peripheral party's prospects of becoming the MPP are weaker than those of its coalition partner(s), then, as the degree of collective responsibility shared by the coalition members increases, this party experiences strategic pressure to moderate its policies. This result is relevant to the computations reported in Table 1, where the left-most party A has a lower equilibrium probability of being the MPP than does its coalition partner, Party B, and where Party A's equilibrium position becomes increasingly moderate as the collective responsibility coefficient r_AB increases.

We note that the peripheral governing party moderation result is relevant to many real world scenarios where small, non-centrist, parties – such as the German Greens and the French Communists – have entered into governing coalitions with large, mainstream, coalition partners (such as the German Social Democrats and the French Socialists). In these scenarios the peripheral governing party moderation theorem implies that the smaller, more radical, coalition member will experience pressure to moderate its policies by converging towards the position of its more centrist coalition partner.

The fourth pattern in Table 1 relates to how collective responsibility affects parties’ probabilities of being the median parliamentary party (MPP). Note that as voters ascribe more collective responsibility to the governing parties A and B, these parties’ equilibrium probabilities of being the MPP decline (see the second column from the right in Table 1). For instance, for the first scenario in Table 1, we see that when voters ascribe no collective responsibility to A and B, i.e. when r_AB = 0, then these parties’ equilibrium probabilities of being MPP are $$)(--><$>{{P}_A}^{\ast}\, = \,0.224{{,}_\ {}}{{P}_B}^{\ast}\, = \,0.276<$><!--$$ (scenario 1A). However for r_AB = 0.5 these probabilities decline to $$)(--><$>{{P}_A}^{\ast}\, = \,0.185{{,}_{}}\ {{P}_B}^{\ast}\, = \,0.248<$><!--$$ (scenario 1C), and for r_AB = 0.9 these equilibrium probabilities decline even more, to $$)(--><$>{{P}_A}^{\ast}\, = \,0.142{{,}_{}}\ {{P}_B}^{\ast}\, = \,0.244<$><!--$$ (scenario 1E).Footnote 42 The additional scenarios 2A–2E, 3A–3E, and 4A–4E display similar patterns. And, these reductions in the leftist governing parties’ prospects of being MPP, as a function of collective responsibility, imply an increased probability that one of the right-wing opposition parties (C or D) becomes the MPP. The effects of these probability changes are displayed in the RHS column of Table 1, which reports the weighted policy outcome of the election, defined as the mean of the parties’ equilibrium positions weighted by their equilibrium probabilities of being the MPP. We see that as voters ascribe more collective responsibility to the leftist governing parties A and B (i.e. as r_AB increases), this weighted policy mean shifts to the right, away from the governing parties’ preferred positions. These results thereby suggest that collective responsibility depresses the responsible parties’ prospects of being the MPP, which in turn damages their policy expectations. We label this the coalition penalty effect.

Why does collective responsibility depress coalition partners’ probabilities of becoming the MPP? The intuition is that when voters ascribe high degrees of collective responsibility to a coalition of parties, then, substantively, the voters view these parties as being interchangeable, in the sense that the coalition members no longer represent distinct alternatives to the voters.Footnote 43 Thus from the voters’ perspectives, high degrees of collective responsibility effectively collapse the distinct alternatives that coalition members would otherwise represent into a single alternative – an effect that depresses the likelihood that the median voter will prefer some member of the coalition to all the parties outside the coalition.

Additional Illustrative Examples

Scenarios 2A–2E, 3A–3E, and 4A–4E in Table 1 illustrate alternative scenarios in which we vary the parties’ long-term valence images, along with the degree of collective responsibility. In scenarios 2A–2E, for instance, the governing parties A and B are specified as having stronger long-term valence images than the opposition parties C and D, i.e. V_A = V_B = 1, V_C = V_D = 0.Footnote 44 Consistent with the centripetal valence effects (CVE) result that Adams and Merrill prove for the basic model,Footnote 45 we see that the governing parties’ improved long-term valence images motivate them to shift to more radical positions, compared to their positions for scenario 1, in which all parties have equal long-term valence images. The results for scenarios 3A–3E, in which the governing parties A and B are specified as having weaker valence images than the opposition parties (i.e. V_A = V_B = 0, V_C = V_D = 1), again support the CVE result: now the governing parties A and B moderate their positions in order to recoup their electoral prospects. Finally, scenarios 4A–4E illustrate the situation where the centre parties, B and C, have stronger long-term valence images than the peripheral parties A and D, i.e. V_A = V_D = 0, V_B = V_C = 2.Footnote 46 This corresponds to the common real world scenario where a large, mainstream, party such as the Dutch Labour Party, the French Socialists or the German Social Democrats, governs in coalition with a smaller and more radical party (or parties). Compared to scenario 1, in which all parties have equal long-term valence, we see that in scenario 4 the centrist parties shift to (slightly) more radical policies, which again supports the CVE result. And, note that for all of the scenarios in Table 1, increases in the collective responsibility term r_AB are associated with more moderate equilibrium positions for the peripheral coalition member Party A, a pattern that supports the peripheral governing party moderation theorem. By contrast the equilibrium position of the moderate coalition member, Party B, is not very responsive to the value of r_AB .

Extensions to Alternative Scenarios: Opposition ‘Proto-Coalitions’ and Centrist Governing Coalitions

Table 2 reports computations for scenarios that illustrate additional strategic implications of the collective responsibility model. Table 2A presents computations for situations where voters assign collective responsibility to the governing parties A and B and to the ‘proto-coalition’ of opposition parties C and D. For both sets of scenarios illustrated in Table 2A – one where all parties have equal long-term valence images (scenarios 5A–5C) and one where the centrist parties B and C have stronger long-term valence images (scenarios 6A–6C) – we see that the coalition convergence effect operates for both the governing coalition and the opposition proto-coalition: namely, as collective responsibility increases, the peripheral parties A and D converge towards the positions of their coalition partners, as the peripheral governing party moderation theorem predicts. Indeed, when the collective responsibility coefficient is set to the high value r_AB = 0.9, there is minimal policy differentiation within each coalition (i.e. the coalition partners are separated by less than 0.2 units along the 1–7 Left–Right scale) but substantial policy variation between the coalitions, i.e. the policy distance between the coalitions is nearly 1.5 units on the 1–7 scale. These computations thereby illustrate how the strategic incentives associated with collective responsibility can transform a multi-party system into what is effectively a two-party system, in terms of the meaningful policy alternatives offered to the electorate.

Table 2

Equilibrium Positions for Alternative Scenarios

Notes: Party preferences were set at R_A = 1, R_B = 3, R_C = 5, R_D = 7. For these computations parties and voters were assumed to have quadratic policy losses, the median voter's position was m=4, and the policy salience parameter was a = 0.25. The parties’ equilibrium positions are those that maximize their expected policy utilities, which are given by equation 3 in the text; their equilibrium probabilities of being the MPP are calculated using equation 2 in the text.

Table 2B reports computations for scenarios where the governing coalition now comprises the two centrist parties B and C. Again, we see that as the degree of collective responsibility increases, the coalition partners converge towards each other (the coalition convergence effect), and that collective responsibility again depresses the coalition partners’ probabilities of being the MPP (the coalition penalty effect).

In toto, the computations reported in Tables 1–2 suggest that the two central results that Adams and Merrill identified for a basic policy-seeking model – i.e. a model that does not incorporate collective responsibility – extend to the collective responsibility model:Footnote 47namely, an equilibrium result, that an equilibrium configuration in parties’ policy-seeking strategies will exist, and a centripetal valence effects (CVE) result, that parties will moderate (radicalize) their policy strategies when their long-term valence images deteriorate (improve). Our computations and theoretical results also identify three additional effects relating to collective responsibility: a coalition convergence effect, that as coalition partners share more collective responsibility they converge towards each other's policy positions; a peripheral governing party moderation effect, that coalition convergence is often driven by smaller, more radical coalition partners moderating their policies as collective responsibility increases; and a coalition penalty effect, that collective responsibility depresses coalition partners’ probabilities of being the MPP, which in turn diminishes their policy expectations. All the conclusions of this paragraph are further substantiated by our computer simulation analysis in the next section.

Simulation Analysis

To substantiate the conclusions suggested by our illustrative examples and the peripheral governing party moderation theorem, we simulated 1,000 four-party elections in which parameters were chosen randomly from a parameter space, with the parties’ probabilities of being the MPP given by Equation 2 above. Parties A and B were assumed to be the governing parties, and the value of the collective responsibility coefficient r_AB was chosen from a uniform distribution on the interval [0.0, 0.9], while voters were assumed to ascribe no collective responsibility to the opposition parties C and D (i.e. r_CD = 0). Parties’ long-term valence images were chosen independently from a uniform distribution on the interval from 0.0 to 2.0, and the preferred policy positions for parties A, B, C and D were chosen from uniform distributions on the intervals [1.0, 2.5], [2.5, 4.0], [4.0, 5.5] and [5.5, 7.0], respectively. The median voter position was set at m = 4, the policy salience coefficient was set to a = 0.25, and quadratic policy losses were used for voters and parties.

For these simulations we located an equilibrium configuration for all but three of the 1,000 scenarios we generated,Footnote 48 which again supports the hypothesis that the Adams–Merrill equilibrium existence result – which applies to the basic model – extends generally to the collective responsibility model that we analyse here. To assess uniqueness of the equilibrium in each scenario, we generated for each of the 1,000 scenarios two sets of starting values, with each starting value for each party randomly generated on the continuous interval from 1 to 7. In all but five of the 1,000 scenarios the two sets of starting values led to the same set of equilibrium configurations, strongly supporting the conclusion that the equilibria are in nearly all cases unique.Footnote 49 Finally, the evidence is strong that the equilibria are generally global (not just local) because each party's equilibrium position was evaluated as optimal by calculating its utility while varying its location in steps of 0.001 over the entire scale from 1 to 7. Thus, overall, we have strong evidence that, for the collective responsibility model, unique Nash (global) equilibrium configurations virtually always exist for an extensive parameter space.

The parties’ preferred positions and their optimal strategies were then normalized by taking the absolute distance from the median voter's position. Each party j's optimal strategies were regressed on seven independent variables: the party's preferred position R_j ; the square of this position $$)(--><$>R_{j}^{2} <$><!--$$ , the party's long-term valence image V_j ; the degree of collective responsibility that voters assign to the governing parties, r_AB ; and the interaction terms (R_j × V_j ), (R_j × r_AB ), and (V_j × r_AB ), resulting in equations for the optimal strategy of party j of the form:

$${{{s}_j}\, = \,{{\beta }_0}\, + \,{{\beta }_1}{{R}_j}\, + \,{{\beta }_2}R_{j}^{2} \, + \,{{\beta }_3}{{V}_j}\, + \,{{\beta }_4}{{r}_{AB}}\, + \,{{\beta }_5}({{R}_j}\,\times \,{{V}_j})\, + \,{{\beta }_6}({{R}_j}\,\times \,{{r}_{AB}})\, \cr+ \,{{\beta }_7}({{V}_j}\,\times \,{{r}_{AB}}).$$

Regression statistics are presented in Table 3 for each of the four parties. As expected – and as indicated by the positive coefficients on the preferred positions (R_j ) – all four parties move to significantly more extreme positions as their preferred positions become more extreme, but at a diminishing rate (as indicated by the negative coefficients on $$)(--><$>R_{j}^{2} <$><!--$$ ). The moderating effect of greater collective responsibility on the optimal strategy of party A is greater as party A's preferred position becomes more extreme, as indicated by the highly significant negative sign of the coefficient of the interaction term (R_j × r_AB ). In addition, the fact that this coefficient is significant (and negative) for Party A but not for B implies that the policy distance between these parties’ equilibrium positions diminishes as collective responsibility increases – an observation that supports the coalition convergence effect we identified earlier. Furthermore, the highly significant positive coefficient of (V_j × r_AB ) shows that the optimal position of peripheral governing party A becomes more moderate as its valence decreases along with enhanced collective responsibility with its coalition partner B. This effect supports the peripheral governing party moderation effect, that smaller, more radical coalition partners moderate their policies as collective responsibility increases. Finally, we note that the optimal positions of the two parties not in the government coalition (C and D) are not significantly affected by the variables involving the collective responsibility coefficient r_AB of the governing parties, except for the dependence of party C on (R_j × r_AB ) $$)(--><$>({{R}_j}\times {{r}_{AB}})<$><!--$$ .

Table 3

Regression Results for Simulated Data

Notes: In these analyses the dependent variable was the absolute distance between the focal party's equilibrium position and the median voter position. Regression results are based on the 993 (out of 1,000) simulated four-party elections that yielded unique convergent equilibrium configurations (see the text of the article for a description of these simulations). Regression coefficients significantly different from zero at the 0.05 level are indicated by (*); those significant at the 0.01 level, by (**). Values in parentheses are standard errors. (Regression coefficients using all 1000 simulated scenarios and one of the equilibrium sets for each of the seven aberrant cases differ only minutely from those reported above.) Parties A and B were assumed to be the governing parties. The collective responsibility coefficient r_AB was chosen from a uniform distribution on the interval [0.0, 0.9], parties’ long-term valence images V_A ,V_B ,V_C ,V_D were chosen independently from a uniform distribution on the interval from 0.0 to 2.0, while their preferred policy positions R_A , R_B , R_C , R_D were chosen from uniform distributions on the intervals [1.0, 2.5], [2.5, 4.0], [4.0, 5.5], and [5.5, 7.0], respectively. The median voter position was set at m = 4.

The parties’ mean equilibrium probabilities of being the MPP over the simulation runs (see the bottom row of Table 3) indicate that the coalition partners A and B have diminished prospects of becoming the MPP, compared to their counterparts from outside the coalition: specifically, the left-most coalition member, Party A, has a lower mean MPP probability than the right-most party from outside the coalition, party D, while the moderate coalition party, Party B, has a lower mean MPP probability than the moderate party outside the coalition, Party C. As a result the mean weighted policy outcome over the simulations was 4.13, significantly to the right of the median voter position, m = 4. This supports the proposition that collective responsibility diminishes the coalition parties’ policy expectations – the coalition penalty effect that we identified earlier.

Finally, we see that parties’ optimal strategies are more moderate than their preferred positions (see the comparison between the parties’ mean preferred positions and their mean optimal strategies, presented near the bottom of Table 3), most strikingly for the two extreme parties A and D, and especially for party A which is subject to the peripheral governing party moderation effect. This effect results in an equilibrium configuration in which the two parties on the left (and the two parties on the right) are much closer to each other than they are to the locations of the other coalition members. In fact, the gap between the mean optimal strategy for parties B and C is about twice the size of the gap between parties A and B or between parties C and D. Hence, the parties naturally coalesce into centre-left and centre-right coalitions, even though their preferred positions were drawn from equally-spaced intervals.