An Application to Portfolio Allocation

There are several possible ways to evaluate our argument empirically. One is to examine our underlying assumptions about supporter behavior: Do supporters (especially voters) employ heuristics that establish “yardsticks” by which party performance is judged, and does the use of such heuristics vary with political sophistication? Does party failure to meet the expectations implied by the heuristics result in a loss of support? Another way is to examine our assumptions about elite behavior: is there evidence that party leaders involved in coalition negotiations are concerned about how supporters will perceive bargaining outcomes, and do these concerns shape what leaders are willing to agree to? Finally, an empirical evaluation could investigate the implications of these behavioral assumptions for the actual bargains that are struck by party elites: Do these bargains reflect the trade-offs across more and less observable goods that are predicted by the argument?

Given the limited scope of the current study, we focus on this last avenue of investigation here, leaving the first two for future work. We recognize, of course, that examining these behavioral implications provides only partial evidence. At the same time, it is important to stress that (as we show below) these implications discriminate between our argument and the most prominent alternative explanations of bargaining in the coalition literature, thus increasing confidence that the results we report offer initial support for the theory.

We focus on a salient aspect of coalition bargaining: the allocation of ministerial portfolios. Coalition research, almost since its inception, has confronted a tension between a well-known empirical regularity and the predictions of most theoretical models of coalition bargaining. The regularity is referred to as “Gamson’s Law”—the robust finding, confirming the intuition of Gamson (Reference Gamson1961), that coalition members tend to receive a share of cabinet ministries that is roughly proportional to the share of legislative seats they contribute to the government (Browne and Franklin Reference Browne1973; Warwick and Druckman Reference Warwick and Druckman2006). From a theoretical perspective, Gamson’s Law poses a puzzle because the vast majority of bargaining models do not predict it. Instead, most models lead to the expectation that the distribution of spoils should correspond to the bargaining power of the parties, which often does not coincide with their seat shares (see, e.g., Montero and Vidal-Puga Reference Montero and Vidal-Puga2011; Morelli Reference Morelli1999).Footnote ¹⁰

Our argument provides an avenue for reconciling this tension. The relative size of parties provides a readily available yardstick that supporters may use to assess their party’s performance for outcomes that are easily measured against it. The numerical distribution of cabinet portfolios is an obvious example. It is salient, widely reported, easily understood, and readily assessed using a “relative party size” heuristic. If less sophisticated supporters make use of this heuristic, and party elites are concerned about their perceptions, one would expect these elites to construct bargains that ensure that the numerical allocation of portfolios follows relative size closely.Footnote ¹¹

Of course, the Gamsonian regularity is also consistent with an alternative explanation: a simple commitment by party elites to proportionality; that is, it is possible that party leaders do not worry about supporter perceptions but simply act in accordance with a fairness norm that focuses on party size rather than bargaining power. Indeed, Verzichelli (Reference Verzichelli, Strøm, Müller and Bergman2008, 239) argues that such a proportionality norm underlies coalition bargaining since it “means that each coalition party is rewarded in proportion to its contribution to the parliamentary strength of the government, which corresponds to many people’s ideas of fairness.” Another alternative explanation holds that party leaders might simply use proportionality as a focal solution in structuring the coalition bargain irrespective of voter perceptions (Bäck, Meier, and Persson Reference Bäck, Meier and Persson2009).

The key aspect that distinguishes our argument from these alternatives is that it implies that the distribution of goods is conditional on their observability. If the proportional allocation of spoils is a consequence of an intrinsic commitment to “fair” distribution in accordance with relative size or its focal qualities, then the observability of goods by less sophisticated supporters is irrelevant for their distribution. In contrast, under our argument, it is supporter perceptions that are critical. Easily observable goods are allocated proportionally because those party leaders whose bargaining power is below their seat share—wary of losing the backing of unsophisticated supporters by failing to secure a proportional share of an easily observable good—are trading away less observable goods to parties in a stronger bargaining position. This leads directly to an implication that discriminates between our argument and the alternative accounts. Specifically, our argument implies that—by accepting a Gamsonian distribution of portfolios—parties whose bargaining power exceeds their seat share agree to be “undercompensated” (relative to their bargaining power) in terms of the easily observable share of portfolios, but they are remunerated by side payments in the form of complex goods. Moreover, these side payments increase the more significant the party’s undercompensation. Neither of the alternative accounts leads to this expectation.

There are numerous complex goods that can potentially serve as side payments. Most obviously, parties might demand policy concessions (which, especially for more complex issues, are less likely to be observable for less sophisticated supporters). One attractive and important “complex” good in this context concerns the policy implications that flow from the particular assignment of specific ministries to specific parties. Ministries, headed by a particular cabinet member, play a key role in translating abstract policy agreements into concrete proposals. Legislative initiatives are drafted under the direction of a minister, presented by that minister to the cabinet, and then introduced in parliament. While the precise degree of ministerial influence, and the conditions under which ministers can exercise discretion, continues to be a rich subject of academic debate (Laver and Shepsle Reference Laver and Shepsle1996; Martin and Vanberg Reference Martin and Vanberg2011), there exists a general consensus that ministers can exercise policy influence unless coalition partners expend resources to monitor and constrain ministerial discretion. This implies that a key consideration for party leaders in evaluating a portfolio allocation is the extent to which it grants power to ministers who pose a potential threat.

The extent of this “policy risk” inherent in a portfolio distribution depends on two features. First, given the issues under the jurisdiction of a given ministry, how significant are the policy differences between a coalition partner and the party controlling the portfolio? If the parties have few policy differences, ministerial discretion poses little danger. Second, how important are the issues under the jurisdiction of the ministry to the party and its constituents? If the issues relevant for a particular ministry are not salient for the party, then ministerial discretion is of less concern, even if there are policy differences between the parties. In short, the “policy risk payoffs” of a portfolio allocation for a coalition party depend on the extent to which it confers influence to partners with significantly different preferences on issues the party cares about.

Assessing policy risk is a challenging task for unsophisticated supporters. Doing so requires an understanding of the jurisdictions and responsibilities of ministries as well as of the specific policy goals of the parties and the extent to which these goals are in conflict. In other words, the policy risk aspect of the portfolio allocation is a complex good—and thus provides one way to deliver side payments to parties that are undercompensated with respect to the numerical allocation of portfolios. Moreover, this side payment may be particularly attractive because it is “paid” directly alongside the Gamsonian distribution of portfolios, reducing concerns about enforcement of the compensation agreement. Thus, the theoretical implications outlined earlier lead to the following hypotheses:

Analysis and Findings

To test these hypotheses, we examine original data on the allocation of 5,910 ministerial posts in 308 coalition bargaining situations in 16 European democracies. In SI Appendix B, we provide more detail on the set of governments included, and we also list the archival sources used to create the data on ministry assignments. Across the 308 governments, our sample consists of 910 observations on government parties (the units of analysis).

Hypothesis 1 predicts that parties will receive a share of ministries that corresponds closely to their seat contributions to the government. Numerous studies have examined this relationship, typically via a simple linear regression of the portfolio shares of parties on their relative seat shares. We conduct a similar analysis for our sample, with results shown in Table 1.Footnote ¹²

Table 1.

Effect of Relative Legislative Size on Share of Ministerial Posts

Note: Displayed are ordinary least squares coefficient estimates, with standard errors in parentheses. Dependent variable: Numerical share of cabinet ministries. N: 602 coalition parties (after 308 parties, one per bargaining situation, are randomly dropped from the analysis—see footnote 12). R ²=0.89. *p < 0.10, **p < 0.05, ***p < 0.01.

Table 2.

Portfolio Allocation in Rutte II Government

The parameter estimates are close to those produced by previous studies (see, e.g., Warwick and Druckman Reference Warwick and Druckman2006, 648). The share of seats a party contributes to the government has a strong positive effect on the share of ministries it receives. Notably, the match is not perfectly proportional. The coefficient on Seat Contribution is statistically below unity and the intercept is statistically above zero, meaning that smaller parties receive more than their “fair share” of portfolios, a finding that is in line with previous results (Browne and Franklin Reference Browne1973). This tendency can be seen in Figure 3, which plots the fitted prediction from Table 1 (with 95% confidence bounds) against a 45° line. Most small parties lie above the “perfect proportionality” line, while most large parties lie below it. Nonetheless, on average, for every 5% increase in a party’s seat share, its expected share of ministries increases by approximately 4%.

Of course, as noted earlier, such proportionality is also consistent with some models of coalition bargaining (e.g., Bassi Reference Bassi2013) or a general commitment by party elites to proportionality as a “fair” or “focal” bargaining solution. The distinguishing feature of our argument is that the proportional allocation of portfolios is induced by party elites’ concerns for the perceptions of their supporters, leading us to the expectation (captured in Hypothesis 2) that elites will reconcile the proportional allocation of the number of portfolios with the underlying bargaining strength of the parties by using less observable aspects of the bargain as side payments.

As we discussed, there are numerous complex goods that can serve as side payments. We focus on a complex good that derives from the initial allocation of ministries. Since cabinet ministers are responsible for supervising the formulation and implementation of policies in their domains, gaining control of a ministry can give a coalition party a significant advantage in shaping policy outcomes in its purview. By the same token, a party can incur significant costs from handing over control of a ministry to one of its partners, especially if the ministry deals with salient, divisive issues. Again, we refer to these costs as the “policy risk payoffs” of the allocation. To measure them, we must perform two tasks. First, we have to measure the positions of parties on a set of policy dimensions, and assess the importance they place on those dimensions. Second, we must associate the dimensions with specific ministerial posts.

To measure party positions and issue salience, we must first define the nature of the policy space. This is an inherently subjective task. As Benoit and Laver (Reference Benoit and Laver2006, 69) point out, “there is no such thing as the “true” underlying dimensionality of any policy space,” which means ultimately that, in terms of both the number and the substance of policy dimensions, “[the] type of space we want to measure depends upon the type of model we want to analyze.” In defining the policy space for our purposes, we attempt to satisfy two criteria. First, we want the number of dimensions to be sufficiently high as to adequately capture both the major areas in which ministers make decisions and the areas in which their parties compete. Second, in terms of substance, we want our dimensions to correspond roughly to the set of dimensions other researchers have highlighted as important in party competition and to produce similar patterns of party contestation.

Based on these criteria, we define a policy space of six conceptually distinct dimensions: economic regulation and state services, traditional morality, environmental protection, internationalism and European integration, social cohesion and national identity, and conflict and military power. Using the MARPOR quasi-sentence coding categories, and the scaling approach developed by Lowe et al. (Reference Lowe, Benoit, Mikhaylov and Laver2011), we calculate issue saliency scores and policy positions for parties on these dimensions, described in SI Appendix D. We also present there a confirmatory factor analysis to demonstrate that the chosen MARPOR categories load on the predefined policy dimensions in reasonable ways, and we investigate the face validity of our measures by plotting their averages for party families in postwar Western Europe. While we do not expect our definition of the policy space to be the “last word” on the matter, we are encouraged by these results and believe that future efforts in this area should entail similar construct and face validity checks.

Matching these policy dimensions to specific ministries obviously requires knowledge of ministerial competences. We have assessed these based on information from a variety of sources, including historical descriptions provided by the ministries, formal legislative documents detailing the policy competences of each department, and the types of government bills and directives introduced by ministers. Most of the time, multiple ministers share responsibility within a given policy area; furthermore, any individual minister typically deals with issues in several policy areas. Accordingly, in our coding scheme, ministries that have significant policymaking responsibility in different policy areas are assigned to multiple policy dimensions.Footnote ¹³

To test Hypothesis 2, the key concept we need to capture with the dependent variable is the level of policy risk a government party is exposed to as a result of the agreed-upon distribution of ministries. Is it the case, as we have argued, that a party that “gives away” a number of ministries it “deserves” (based on its bargaining strength) to achieve numerical proportionality for all coalition parties does so in such a way as to lower its policy risk? That is, does an “undercompensated” party generally avoid ceding ministries to ideologically distant partners on dimensions the party cares a lot about? On the other side of the bargain, is an “overcompensated” party generally willing to tolerate greater policy risk to attain numerical proportionality? That is, in exchange for receiving a greater number of ministries than warranted by its bargaining strength, is such a party inclined to allow its undercompensated partner to control ministries on important dimensions on which the preferences of the two parties significantly diverge?

As these questions suggest, and as noted earlier, a party’s policy risk payoff from an allocation depends on two factors: (1) how distant, on the relevant issues, the party is ideologically from the coalition partners that gain control of the ministries and (2) how important those issues are to the party. As these factors increase, the negative policy consequences experienced by the party (and its supporters) will become more severe, and/or the party has to devote greater resources towards constraining its partner’s ministers in an effort to avoid such consequences. We incorporate these factors into our dependent variable, which is constructed in the following steps:

1. (1) We calculate a dimension-specific policy risk payoff for a coalition party $$ c $$ as the average saliency-weighted distance between $$ c $$ and all the ministries (including those controlled by $$ c $$ itself) that are responsible for that dimension (the ministries $$ {M}_d $$),

   $$ \frac{1}{M_d}{\sum}_{m\in {M}_d}{y}_{cd}\left|\right.{i}_{cd} - {i}_{md}\left|\right., $$
   where $$ {y}_{cd} $$ is the relative saliency that party $$ c $$ attaches to dimension $$ d $$, $$ {i}_{cd} $$ is $$ c $$’s policy position on the dimension, and $$ {i}_{md} $$ is the policy position of minister $$ m $$ on the dimension;Footnote ¹⁴

2. (2) we sum the dimension-specific payoffs across the set of (six) policy dimensions $$ \mathrm{D} $$ to create a multidimensional policy risk payoff,

   $$ {\sum}_{d\in D}\frac{1}{M_d}{\sum}_{m\in {M}_d}{y}_{cd}\left|\right.{i}_{cd} - {i}_{md}\left|\right.; $$

3. (3) finally, since we are interested in measuring the policy risk for a party given the number of ministries ceded to other parties (i.e., we are interested in the level of policy risk a party has accepted given the numerical constraints it has agreed to), we divide by the number of portfolios controlled by the party’s partners, $$ {p}_{\mathrm{oc}} $$, thus obtaining the average multidimensional policy risk payoff per partner-controlled ministry:

   $$ \frac{1}{p_{oc}}{\sum}_{d\in D}\frac{1}{M_d}{\sum}_{m\in {M}_d}{y}_{cd}\left|\right.{i}_{cd} - {i}_{md}\left|\right.. $$

In short, the dependent variable, which we refer to as a party’s Portfolio-Adjusted Policy Risk Payoff, taps into a feature of portfolio allocation that parties presumably value highly—the distribution of policy-making authority—but whose details most supporters will be largely unaware of because understanding it requires knowledge that goes well beyond knowing how many ministries a party controls. Even knowing which ministries a party receives would be insufficient. To evaluate the policy risks their party faces from a particular allocation, supporters need to know which ministries are responsible for which policy areas, how the ministries not received by their party are distributed among its coalition partners, and the policy preferences of each of these parties in the areas under their control.

As stated in Hypothesis 2, we expect that parties that are, relative to their bargaining power, undercompensated in the numerical share of portfolios receive side payments in the form of a more favorable policy risk payoff. Thus, we require a measure of bargaining power. There are, of course, multiple approaches to assessing bargaining power. One alternative derives from the literature on power indices. These measures focus on the size of parties and assess each party’s “pivotality” across the set of winning coalitions. The downside of this approach is that it pays no attention to the policy preferences of actors or other factors that potentially strengthen a party’s position in coalition negotiations. This may be sensible in a purely distributive setting, but is problematic in our context since coalition negotiations clearly depend on a host of factors beyond size, including the ideological positions of the potential coalition partners. Therefore, we take a different tack. An important source of leverage for parties in the process of government formation is their ability to join an alternative government to the one they are currently negotiating: parties that have credible “exit options” from the current negotiations are more likely to be influential. Thus, an alternative approach to assessing bargaining power is to focus on parties’ ability to be part of (alternative) governments.Footnote ¹⁵

There are multiple empirical models of coalition formation that would allow us to estimate the probability that a party can get into an alternative government. We rely on a recent model by Kayser, Orlowski, and Rehmert (Reference Kayser, Orlowski and Rehmertn.d.)—hereafter, KOR—that builds on work by Martin and Stevenson (Reference Martin and Stevenson2001; Reference Martin and Stevenson2010). Like these earlier studies, KOR use a conditional logit model to calculate the probability that a particular government is chosen (out of the set of all potential governments), where the probability in their model is a function of nine size-related, policy-related, and contextual factors. Some notable advantages of the KOR specification (versus the Martin and Stevenson specifications) are that it is more parsimonious, excludes certain variables that are difficult to measure reliably, and has a better predictive performance.Footnote ¹⁶

We construct our measure of bargaining power in the following steps:

1. (1) Using the KOR specification, we predict the probability of formation for each potential government in each of our 308 bargaining situations, including the coalition that was actually chosen;

2. (2) for each government party in the chosen coalition, we calculate the party’s probability of being included in an alternative government by summing the formation probabilities of all non-chosen potential governments in which that party is a member (we refer to this quantity as a government party’s exit probability);

3. (3) we normalize each government party’s exit probability by dividing it by the sum of exit probabilities across all government parties.

Thus, our measure of party bargaining power is simply its relative exit probability. Footnote ¹⁷ Using this measure, we create our central covariate, Undercompensation in Portfolio Share, by subtracting the portfolio share received by a party from its bargaining power.

Again, we expect that any party that would receive—on the basis of its bargaining power—a share of portfolios that is less than its perceived “fair share” (i.e., its relative seat contribution) would be willing to trade off complex goods to its partners to achieve that fair-share allocation. What would such a trade-off look like in practice? Before we move to a full analysis, it is useful to consider a single episode of coalition bargaining. We focus here on the last Dutch government in our sample: the second cabinet led by Mark Rutte (see Table 2), which lasted from 2012 to 2017, and consisted of two parties, the VVD and the PvdA.

These parties were roughly equal in size. Given 14 ministerial posts in the cabinet, each party’s “fair share” (as implied by the relative size heuristic) was seven posts. Each party received exactly that number. However, while the parties were roughly equal in size, they were not equal in bargaining power. Given the composition of the legislature (which tilted towards parties of the right), the probability of VVD becoming part of an alternative coalition was almost twice as high as the exit probability for the PvdA. Thus, the Gamsonian allocation of portfolios left the PvdA, relative to its bargaining power, overcompensated with respect to the number of portfolios, while the VVD was undercompensated. Our argument implies that this numerical “bonus” received by the PvdA came at a price. Namely, since the most important issue for both parties in 2012 was how to improve the economy, a matter over which they were quite divided, we would expect the VVD to have taken most of the ministries with jurisdiction on economic matters, thereby lowering the risk that the PvdA could move policy too far away from the VVD’s preferred position.Footnote ¹⁸

With two parties and 14 ministries, there are 3,432 possible ways to allocate portfolios such that each party receives seven ministries. In Figure 4, we display the policy risk payoffs for each of these potential allocations for both parties. One important point to note is that the relationship between payoffs for the two parties is negative, meaning that switching ministries would typically (though not always) produce a loss for one party and a gain for the other. A more crucial point is that the actual bargain that was struck—in line with the expectations of our theory—was much more favorable for the VVD than for the PvdA. Specifically, the VVD’s policy risk payoff was in the top 20% of its possible payoffs. Meanwhile, the payoff for the PvdA was in the bottom 15% of its possible payoffs; that is, there were approximately 3,000 alternative ways to distribute the ministries (holding fixed the number of ministries received by each party at seven) that would have been more favorable for the PvdA. Only three of its ministries dealt directly with economic policymaking (Finance, Education and Culture, and Social Affairs), compared with five possessed by the VVD (Prime Minister, Economic Affairs, Infrastructure and Environment, Health and Welfare, and Housing).Footnote ¹⁹

Is this case emblematic of a more general pattern? Before turning to our analysis, we need to consider a potential confounder—namely, the ideological extremity of a party in relation to its coalition partners. Naturally, a party that finds itself in a government in which it diverges widely from each of its partners on multiple issues should expect to face a higher baseline level of policy risk. If the party’s position vis-à-vis its partners also affects its degree of undercompensation (through its impact on either or both of the covariate’s components, bargaining power or portfolio share), then not taking this factor into account could introduce significant bias into the analysis.

To confront this problem, we include a party-specific distance variable in our model, which we term Distance from the Coalition Center of Gravity, and define as follows:

$$ {\sum}_{d\in D}{y}_{cd}\left|\right.{i}_{cd}-{\sum}_{c\in C}{s}_c{i}_{cd}\left|\right.. $$

The final term represents the coalition’s ideological “center of gravity” on an issue dimension $$ d $$, defined as the weighted-average position of all parties in coalition $$ C $$ on that dimension, with $$ {i}_{cd} $$ representing the ideal point of each party and the weight $$ {s}_c $$ corresponding to the proportion of seats each party contributes to the total controlled by the coalition. On each dimension, we measure the absolute distance between party $$ c $$’s policy position and the coalition center of gravity and then weight that distance by the relative importance of the dimension to the party, $$ {y}_{cd} $$. Finally, we sum these dimension-specific distances across our six policy dimensions to create the final party-specific measure.

In Table 3, we present results from a multilevel random coefficients model. This type of model has several appealing features. First, it accounts for the fact that our party-level observations are nested (clustered) within bargaining situations. Policy risks may vary across bargaining situations for reasons not accounted for by the party-level covariates, inducing positively correlated errors between government parties. Failing to account for such correlations could yield estimates of the standard errors that are too small.Footnote ²⁰ Second, the model allows us to deal with possible heterogeneity in the effects of our covariates by providing estimates of both the overall (or average/fixed) intercept and slope coefficients of interest (shown in the top part of the table) as well as the extent to which the intercepts and slope coefficients at the level of the cluster deviate from the average estimates. Using the estimated variance components of the model (shown in the bottom part of the table), we can calculate best linear unbiased predictions of the random deviations and then use those to generate predicted random intercepts and random slope coefficients for individual bargaining situations.Footnote ²¹ Finally, the model allows us to assess how much unexplained variation in policy risk payoffs is due to differences between parties within bargaining situations versus differences between bargaining situations.

Table 3.

Effect of Portfolio Undercompensation on Policy Risk

Note: Displayed are maximum likelihood coefficient estimates and standard errors from a multilevel random coefficients model (with no constraints imposed on the covariance matrix). Dependent variable: Portfolio-adjusted policy risk payoffs. N: 910 coalition parties (nested in 308 bargaining situations). A likelihood ratio test against a linear model with fixed coefficients leads us to reject the null hypothesis of no difference between the models (p ˂ 0.001). Undercompensation in Portfolio Share: ups; Distance from Coalition Center of Gravity: dccg. *p < 0.10, **p < 0.05, ***p < 0.01.

Based on the results, we can conclude confidently (via a likelihood ratio test) that the multilevel random coefficients specification leads to a significant improvement in fit over a one-level model with fixed coefficients and a fixed intercept ($$ p<0.001 $$).Footnote ²² We can also conclude that most of the unexplained variability in policy risk can be attributed to random (or unmeasured) differences between bargaining situations rather than differences between parties. Specifically, the level-two variance partitioning coefficient (VPC) is approximately 0.65, meaning that only 35% of the total residual variance is attributable to unmeasured differences between parties.Footnote ²³

But how large is this residual variance; that is, how well does our model fit the data? In Figure 5, left panel, we present a quantile-normal plot in which the standardized residuals for our 910 coalition parties are compared against the values one would expect for a standard normal distribution (as represented by the solid 45° line). Most residuals stick fairly close to the line, indicating no serious departures from normality, and there are very few outliers (approximately 2% of cases in each tail). In the right panel, we plot our predicted policy risk payoffs against the actual policy risk payoffs for the 910 parties. In this panel, the dotted line is a 45° line and the solid line is the linear fit from a regression of predicted policy risk payoffs on actual policy risk payoffs. Any deviation between the two lines suggests a systematic pattern in actual policy risk that is not being adequately explained by the model. Fortunately, the figure shows only a small difference between the lines: the solid line has an intercept of approximately 1.31 and a slope of 0.84, meaning that the model slightly overpredicts policy risk payoffs for parties with low risk and slightly underpredicts payoffs for parties with high risk. Nonetheless, the model appears to fit the data quite well overall, perhaps surprisingly so, since it includes only two covariates and their associated random effects.Footnote ²⁴

Figure 5.

Residuals and Predictive Fit for Policy Risk Model

Of course, the most important results from Table 3 concern the estimated effects of these covariates. Beginning with our control variable, we observe that, as expected, the more distant a government party is from the coalition’s ideological center of gravity, the higher its level of policy risk. That is, as a party becomes relatively more extreme, it is more likely to see the ministries not under its control fall into the hands of ideologically hostile partners.

More importantly, the results provide strong support for our central argument. Coalition parties that are undercompensated in their numerical share of portfolios (relative to their bargaining power) tend to receive significantly more favorable policy risk payoffs than parties that are overcompensated. In other words, the set of ministries that are not given to undercompensated parties, but rather to their partners, (a) tend to be those with jurisdiction on policy issues that are relatively less important to the undercompensated parties and/or (b) tend to be assigned to partners with relatively similar policy positions. This supports our central claim that policy risk payoffs appear to be used as “side payments” to parties that are numerically undercompensated.

Of course, a critical question concerns the substantive significance of these side payments. To illustrate this, we calculate the predicted policy risk payoffs for two types of coalition parties in two different bargaining situations. In both situations, the parties are assumed to control approximately half the portfolios in their respective governments. But for one (the undercompensated party), this numerical payoff is considerably less than what is implied by its bargaining power, while for the other (the overcompensated party), the payoff is considerably more than its bargaining power.Footnote ²⁵ For each type of party, we predicted its policy risk payoff using the overall intercept and slope estimates from Table 3 (holding distance from the coalition center of gravity at its sample mean), and we then generated a distribution of 1,000 simulated predictions (drawn from a normal distribution with a mean equal to the predicted effect and a standard deviation equal to the standard error of that effect).

We present the two distributions in Figure 6. Clearly, there is a stark difference in expected payoffs between the parties. Specifically, for the party that is less powerful, but nonetheless receives half the portfolios, the figure shows that the median predicted policy risk payoff is over 50% higher (i.e., less favorable) than for the more powerful party. That is, the party that is undercompensated with respect to the raw number of portfolios can expect to translate its greater bargaining power into a significantly more attractive, albeit less directly observable, set of policy benefits.

Figure 6.

Average Effect of Portfolio Undercompensation on Policy Risk

These findings provide clear support for our argument. But there are two final issues we need to consider. The first concerns the presence of heterogeneity in the effects of the covariates, as captured by the estimated variance components. As noted earlier, we can use these estimates to predict random effects, which tell us how much the estimated intercepts and slopes in individual bargaining situations deviate from the overall (or average) estimates. In SI Appendix H, we present the estimated effect of the undercompensation variable (i.e., the estimated average effect across bargaining situations plus the situation-specific estimated random effect) for each of the 308 governments in the sample.Footnote ²⁶ We do find variation across bargaining situations in the effect of undercompensation on policy risk. There are a few cases in which the effect is in the opposite direction and a few in which undercompensation has no significant effect. But in the overwhelming majority (approximately 90% of bargaining situations), the effect is statistically significant ($$ p<0.05 $$) and in the expected direction. Notably, in approximately 25% of cases, the estimated effect is over 35% larger in magnitude than the average effect shown in Table 3.

The second issue relates to the fact that one component of our undercompensation variable—bargaining power—is itself an estimate produced by another model. As such, it is measured with error, which could lead to biased parameter estimates. Given the predictive success of the KOR model, we do not expect the amount of estimation error to be so great as to alter our conclusions. However, this is ultimately an empirical question. Accordingly, we have chosen to conduct a sensitivity analysis using 100 alternative datasets based on the coefficient estimates and standard errors from the KOR model (see SI Appendix I for a more detailed description of our data-generation procedure). Each new dataset contains a new estimate of party bargaining power and thus a new measure of undercompensation. For each dataset, we reestimate our model and compare the results to those from Table 3.

In Figure 7, the dashed horizontal line represents the original coefficient estimate from Table 3, with the shaded region corresponding to its 95% confidence interval. The 100 dots in the figure are the coefficient estimates for the undercompensation covariate (along with 95% confidence bounds) from the alternative datasets. For easier visualization, we have ordered the estimates according to the amount of measurement error in the datasets (calculated as the sum of squared deviations between the original and new estimates of bargaining power). Also displayed are two Lowess smoothing curves, one for estimates above the original coefficient, and one for estimates below. As one might expect, the curves move away from the original coefficient as measurement error increases. However, even for values of measurement error in the tails, the new coefficient estimates fall within the original estimate’s confidence region, and in all cases we can reject the null of no effect with a very high level of confidence.Footnote ²⁷

Figure 7.

Impact of Measurement Error

Note: The dashed horizontal line is the estimated coefficient for Undercompensation in Portfolio Share from the model shown in Table 3, encompassed by a 95% confidence region. Circles represent the coefficient estimates from 100 simulated data sets (with 95% confidence bounds), arranged in ascending order of estimation error in bargaining power. The solid lines are Lowess smoothing curves for simulated estimates greater than and less than the original coefficent estimate.

Supplementary Materials

To view supplementary material for this article, please visit http://dx.doi.org/10.1017/S0003055420000337.

Replication materials can be found on Dataverse at: https://doi.org/10.7910/DVN/2CITBJ.