THE UNCOVERED SET, PARTIES AND LEGISLATIVE OUTCOMES

The essence of formal modeling is to develop abstract representations of real-world situations and to use these models to predict real-world behavior and outcomes. Such models facilitate critical tests between competing theories. In the debate over party influence in the U.S. Congress, the technique of choice is spatial modeling, where preferences and outcomes are specified in terms of points in space.

How do preferences translate into outcomes in spatial voting games? In a one-dimension spatial model, the Median Voter Theorem (MVT) states that the expected outcome of majority-rule voting, given an open agenda and single peaked preferences, is the ideal point of the median voter (the core or Condorcet winner). Thus, if we know the median voter's ideal point, we can predict that it would be the outcome of any majority-rule voting in these games. Put another way, the median voter's ideal point is the only enactable outcome in the game. An outcome is enactable if there exists some admissible agenda such that (1) the outcome could be the ultimate result when sophisticated legislators vote through the agenda using majority rule and (2) the agenda itself could receive majority support when brought to the floor.

It is well known that the MVT does not generalize to cases where multiple dimensions are needed to describe preferences and outcomes, implying that outcomes in these games are sensitive to agendas, voting rules, and other constraints (Shepsle 1979, 1986). The so-called Chaos Theorems state that majority-rule decision making, unchecked by institutional constraints, can go “from anywhere to anywhere,” rendering the ultimate outcome indeterminate. However, further work showed that if voters or legislators in these settings consider the ultimate consequences of their actions, rather than choose myopically between alternatives presented at each decision point, majority-rule voting will yield an outcome in a relatively small area, the uncovered set (McKelvey 1986, Miller 1980).

Formally, let N be the set of n voters or legislators. Assume that n is odd and for any agent, i ∈ N, preferences are Euclidian and defined by an ideal point ρ_i. Let x,y be elements of the set X of all possible outcomes. A point x beats another point y by majority rule if it is closer than y to more than half of the ideal points. A point x is covered by y if y beats x and any point that beats y beats x. The uncovered set includes all points not covered by other points. In essence, the uncovered set generalizes the MVT to multidimensional spatial models. In a one-dimensional spatial model, the uncovered set is a single point—the median voter's ideal point. In a multidimensional game, the uncovered set is a relatively compact region within the space, unless ideal points satisfy stringent conditions in which a single core point exists and the uncovered is this core point.

The significance of the uncovered set lies in the potential to specify the set of possible majority-rule outcomes in these games and in the real-world situations these games intend to capture. If y covers x, y dominates x as an outcome of a majority-rule voting game (McKelvey, 1986–8): if y covers x, any outcome that ties y defeats or ties x and any outcome that defeats y also defeats x. Therefore, strategic legislators should eliminate covered points from the voting agenda. Instead of promoting outcomes that are bound to be defeated later in the game, sophisticated legislators should promote points in the uncovered set that may survive the voting process (Cox 1987, 419).¹

Although much effort has focused on characterizing the size, shape, and location of the uncovered set in spatial games, a general result has eluded scholars up to now.²

The grid-search technique also allows us to determine whether the uncovered set's theoretic attractiveness is matched by an ability to predict real-world outcomes. Predictive power is crucial to our analysis: if the uncovered set does not capture actual outcomes, its size or location provides no insight into party influence or its observability. Bianco et al. (2004b) reanalyze data from canonical majority-rule experiments, showing that the uncovered set is a very good predictor of experimental outcomes—depending on the experiment, over 90% (often 100%) of outcomes are in the uncovered set. Experiments using 5-player and 35-player groups (Bianco et al. 2004c) provide additional evidence of the uncovered set's predictive power. Bianco, Jeliaskov, and Sened (2004a) also show that, more often than not, the theoretically derived uncovered set is consistent with actual outcomes in the contemporary U.S. Congress.

The Uncovered Set and the Observability of Party Influence

The theory of conditional party government (Aldrich 1995; Aldrich and Rohde, 1998, 2001) states that majority-party influence stems from its leaders' control of the agenda, debate, and institutions. When parties are polarized (similar policy goals within each caucus, disagreement across caucuses), “the majority party acts as a structuring coalition, stacking the deck in its own favor—both on the floor and in committee—to create a kind of “legislative cartel” that dominates the legislative agenda (Cox and McCubbins, 1993, 270; see also 2003).” As a result, “…the greater the degree of satisfaction of the condition in conditional party government, the farther policy outcomes should be skewed from the center of the whole Congress toward the center of opinion in the majority party” (Aldrich and Rohde 10–11).

Some evidence supports conditional party government. Aldrich and Rohde (1998) find polarization in legislators' ideal points in the contemporary House. Aldrich and Battista (2000) explain this polarization in terms of electoral forces, finding a positive relationship between the effective number of parties in a state and various measures of legislative polarization. However, the key prediction of conditional party government—that outcomes will favor the majority party—has never been tested.

Krehbiel (1999, 2000) argues that outcomes favoring the majority party are a natural consequence of polarization, thus imply nothing about majority-party power: “Parties are said to be strong exactly when, viewed through a simple spatial model, they are superfluous (Krehbiel 1999, 35).” That is, when legislative parties are polarized, outcomes will lie closer to majority party ideal points because the party contains a majority of legislators, not because of anything party leaders do. This claim is derived from a unidimensional spatial model with two parties, where all legislators are party members. Thus, the median floor legislator is necessarily a majority party member. The MVT predicts that the outcome of this voting game will be the median voter's ideal point. Therefore, as the party medians diverge toward opposite ends of the dimension, the median voter—and expected outcomes—moves away from the center of the distribution and toward the majority party. However, this effect is the result of polarization coupled with majority rule; it does not require any actions by party leaders.

Figure 1 contains a nine-voter example of Krehbiel's argument. The figure contains two 1-dimensional spatial games where ideal points for the majority party are denoted by diamonds and for the minority party by squares. The top example depicts a situation where ideal points are not polarized. As the figure indicates, the expected outcome is the median voter's ideal point, located near the center of the plot. In the bottom example, ideal points are polarized with majority-party legislators on the left-hand side of the plot. The expected outcome is the median legislator's ideal point. However, with the change in the distribution of ideal points, the median legislator's ideal point, and, thus, the expected outcome, has shifted leftward. This shift, however, is driven by legislators' preferences, without any additional effect resulting from actions taken by party leaders.

Measuring Party Influence: The Problem of Observability in One Dimension

The concern about the observability of party influence is not a technical point. Conditional party government and Krehbiel's (1999) counterhypothesis are based on very different conceptions of the limits on majority-party action. The implicit assumption under conditional party government is that there is a wide range of enactable outcomes. The task for party leaders is to decide which of these outcomes are most acceptable to their caucus and to select procedures that generate this outcome. In contrast, in Krehbiel's model, there is only one enactable outcome—the median voter's ideal point. Thus, the majority party is fundamentally constrained by floor preferences. If majority-party leaders propose a nonmedian outcome or procedures that would lead to a nonmedian outcome, their proposal or agenda will not gain majority support.³

This debate provides an entry point for our analysis. Krehbiel's (1999) findings are based on a one-dimensional spatial model. The question is, does it generalize to more realistic two-dimensional situations? Our analysis begins with the premise that information on the size and location of the uncovered set in real-world legislatures offer a resolution to concerns about the observability of party influence—and a new scenario for the observable exercise of party influence. In particular, although the uncovered set is a single point in a one-dimensional game, it may well be larger in a more realistic multidimensional game. If so, the size of the uncovered set may create opportunities for agenda setting in multiple dimensions, even if it is offset toward the majority party as in a one-dimensional game. This scenario is shown in Figure 2.

Party Influence in Two Dimensions

Figure 2 gives a hypothetical configuration of ideal points and uncovered set in two dimensions for a nine-voter, polarized party legislature. The figure labels the majority and minority centroids (the average ideal point for each party), draws a dashed line between centroids, and labels the midpoint of this line. The dark region is a hypothetical uncovered set, one that is typical of the real-world data presented later. Note that the uncovered set is relatively large and offset toward the majority party. In fact, it is completely on the majority party side of the line linking the party centroids. This location implies that actual outcomes, located inside the uncovered set, will always favor majority party legislators to some extent regardless of what their leaders do—just as in Krehbiel's (1999) one-dimensional model. However, the larger size of the uncovered set creates a new opportunity for the exercise of majority-party power. Rather than allowing outcomes to be determined by unconstrained floor action, majority-party leaders can pick a point inside the uncovered set, presumably one that is close to ideal points in their caucus, and use procedural strategies that yield this outcome on the floor. One such outcome is labeled as “agenda point” in Figure 2—note that all majority-party legislators prefer this outcome to everything else in the uncovered set. Suppose that party leaders structured the legislative process to yield this outcome. If an observer considered these outcomes and knew where the uncovered set was located, he or she would conclude that the distribution of preferences conveyed a natural advantage to majority-party legislators—but that majority-party power was being exercised at the margin of this advantage.

In sum, in a one-dimensional spatial model of legislative action, it is hard to see how the majority party could influence outcomes at all. However, this result may not hold in more realistic spatial models or in the real-world legislatures they depict. Our aim here is to derive uncovered sets for a variety of real-world legislatures using two-dimensional preference data and to assess their size and location relative to the majority and minority parties, allowing a direct resolution of the party influence debate.

More specifically, our focus is on two measurements. The first is a comparison of the average distance between outcomes in the uncovered set and the ideal points of legislators in the majority and minority parties. For some legislature, let M be the set of legislators in the majority party (m in number), N be the set of minority party legislators (n in number) and U the set of outcomes in the uncovered set (u in number). Let D_ij be the distance between legislator i's ideal point, ρ_i, and an outcome j in the uncovered set. Let D_m (D_n) denote the average distance between the ideal points of legislators in the majority (minority) party and outcomes in the uncovered set:

We express the difference in these measures as a percentage: 100·(D_n-D_m)/D_n. This measure varies from −100 to 100. If the uncovered set is equidistant from majority- and minority-party legislators, it will equal zero. If the uncovered set is closer to the majority party on average, as per Krehbiel (1999), the measure will be positive. For example, if the average distance between majority party legislators and uncovered set outcomes is half as large as the distance between these outcomes and minority party legislators, the measure will equal 50. Negative values imply that the uncovered set is closer to the minority party than to the majority party. Difference-of-means tests will be used to assess the statistical significance of the difference in average distance between the majority and minority parties.

Our second measurement focuses on the potential gains from agenda setting within the uncovered set. That is, to what extent can efforts to produce one particular uncovered set outcome improve on the situation where leaders are inactive and all uncovered set outcomes are equally possible? In general, it is not obvious which outcome will be the focus of party leaders' agenda-setting efforts. In real-world legislatures, this calculation is a complex process involving the preferences of caucus members; factional splits in the caucus; and the availability of side payments, threats, and strategic behavior on behalf of the relevant legislators. Our analysis approximates this calculation as follows. Rather than considering legislators' utilities or payoffs, we focus on distance: to what extent can agenda setting within the uncovered set bring outcomes closer to the ideal points of majority party legislators compared to the expected distance given no leadership action?⁴

We assess the potential for majority party agenda setting as follows. For a given legislature, let ρ_m=[sum ]_i∈M ρ_i/m denote the center of gravity for the majority party ideal points, where ρ_i = (x_i, y_i) is a two-dimensional vector denoting legislator's i's ideal point. We interpret it as an approximation of the outcome that majority-party leaders and caucus members would like to enact if possible.⁵

The next step is to determine how much party leaders can improve on this baseline. Let x ∈ U be the uncovered set outcome that is closest to ρ_m and denote the distance between x and ρ_m by C_x. Outcome x is the best that party leaders can do in terms of using agenda setting to satisfy their caucus—anything closer to ρ_m is outside the uncovered set and, therefore, unenactable.

Our analysis of the potential for agenda setting in a given legislature focuses on the difference between C and C_x. We scale both distances by expressing them as a percentage of the range of legislator ideal points on the x-axis to normalize the results across different legislatures and data sources. In addition, scaling provides insight into the substantive significance of the two distances and the differences between them.⁶

UNCOVERED SETS AND CONDITIONAL PARTY GOVERNMENT

This section assesses the potential for conditional party government in contemporary legislatures by analyzing the size and location of uncovered sets for various sessions of the U.S. House of Representatives and state legislatures. Regarding our data, there is an ongoing and unresolved debate over the appropriate technique for recovering legislators' ideal points from observed behavior (votes).⁷

• Constant-space ideal points calculated using NOMINATE (Poole and Rosenthal, 1997) for the 81st, 86th, 91st, 96th, 101st, and 106th U.S. House of Representatives.
• Ideal points derived from a linear model for the same House sessions (Groseclose and Snyder 2000).
• Ideal points for the 106th House calculated with a Markov Chain Monte Carlo technique (Jackman, Clinton, and Rivers 2004).
• Ideal points for ten state legislatures from the late 1990's (Aldrich and Battista, 2002) calculated using NOMINATE.

Our analysis strategy is as follows. First, we present uncovered sets for the 106th U.S. House calculated from three different datasets. Although the distribution of ideal points and the size and location of the uncovered set vary across these plots, all show the same pattern: a substantial uncovered set that is closer to the majority-party legislators compared to those in the minority party. Our next step examines Krehbiel's (1999) conjecture about polarization and outcomes by comparing the average distance of uncovered sets to majority- and minority-party legislators. Then, we analyze the potential for agenda setting inside these uncovered sets.

The Observability of Majority-Party Power

Our analysis provides strong support for the idea that when legislators are polarized by party, the uncovered set is closer to the majority party than to the minority party); thus the measurement of party influence is confounded by the majority party's natural advantage. Figure 3 gives three examples for the 106th U.S. House. The top plot in Figure 3 gives ideal points, and the uncovered set for the 106th House calculated using Poole–Rosenthal ideal points; the middle plot uses Groseclose–Snyder estimates; the bottom uses Jackman–Rivers scores. In all three plots, idea points for minority-party Democrats are denoted as diamonds and located on the left-hand side, whereas ideal points for majority-party Republicans are squares on the right-hand side. The shaded region is the uncovered set estimated using our grid-search procedure.

Uncovered Sets for 106th U.S. House

The plots show that the different ideal point estimation techniques yield different results, both in terms of the preferences ascribed to each legislator and in terms of the uncovered sets calculated from these ideal points. Such variation may be due to differences in the estimation techniques, or auxiliary assumptions such as the salience of each dimension or the exclusion of certain votes (Jackman, personal communication). Even so, the uncovered sets are similar in two important respects. First, they are not located in the center of the distributions of ideal points. Rather, they are shifted toward the cluster of majority-party ideal points. In addition, the plots reveal that the uncovered sets are fairly substantial in size, occupying about 10% of the Pareto Set. In other words, the intuition of the MVT, that only one outcome is enactable, does not hold in more realistic multidimensional spatial games. This finding is consistent with our hypothesis about the potential gains from agenda setting.

Figure 4 expands our analysis of the location of uncovered sets in real-world legislatures by reporting the relative position of the uncovered set in eleven U.S. state legislatures and six sessions of the U.S. House. As noted earlier, we have two sets of ideal points for five House sessions and three sets for one session. We omit the single data point from Jackman, Clinton, and Rivers, 2004 in this and subsequent figures; however, these data are consistent with those presented here.

The Majority Party's Built-in Advantage

The number reported for each legislature is the ratio of the average distance between majority-party legislators and uncovered set outcomes,D_m=[sum ]_{i ∈ M,j∈ U}D_ij/(m·u), and the average distance between minority-party legislators and uncovered set outcomes, D_n=[sum ]_{i ∈ N,j∈ U}D_ij/(n·u), expressed as a percentage 100·(D_n-D_m)/D_n. For clarity, we present plots for different venues (state legislatures vs. the U.S. House) and different methods of calculating ideal points (Heckman–Snyder vs. Poole–Rosenthal). Percentage difference bars that are statistically significant at .05 or better are in dark color; empty bars reflect lower significance levels.

As the figure shows, with one exception, all of the legislatures in our analysis have uncovered sets that are closer on average to majority-party legislators than to legislators in the minority party.⁹

Figure 5 extends the analysis, showing that the position of the uncovered set relative to the majority and minority parties is influenced by polarization: as the average distance between majority- and minority-party legislators increases, the uncovered set moves relatively closer to the majority party.

Expected Outcomes and Polarization

These findings about the location of uncovered sets in real-world legislatures suggest that concerns over the observability of party influence are well founded. Regardless of whether preferences are specified using one dimension or two, when legislators are polarized by party, the set of feasible outcomes favors is closer to the majority party. Thus, an observer who considered actual outcomes relative to legislators' ideal points without examining the uncovered set would conclude that the location of these outcomes indicated a majority-party cartel at work, when in fact party leaders might be inactive or powerless. In this sense, our analysis provides a partial confirmation of Krehbiel's (1999) conjecture concerning the observability of party influence.

Agenda-Setting and Majority-Party Influence

The fact that polarization gives the majority party a built-in advantage in terms of outcomes does not imply that conditional party government is impossible. As noted earlier, when the uncovered set contains many outcomes, majority party leaders can pick one and formulate an agenda that yields this outcome as the result of majority-rule voting. Earlier, we described distance measures that characterized this potential. Figure 6 reports these measures for the three datasets in our analysis—again, we separate results by estimation method and venue.

The Potential for Majority-Party Influence via Movement within the Uncovered Set

The results highlight the potential for agenda-setting: in all cases, the uncovered set outcome that is closest to the majority-party consensus is noticeably closer than the average uncovered set outcome. For example, in the case of the 106th House using the Groseclose–Snyder data, the average distance of uncovered set outcomes to the caucus center of gravity point, ρ_m, is about 25% of the x-axis range (note the uncovered set for this session is shown in Figure 3). However, by using an agenda strategy that yields the closest possible uncovered set outcome, majority party leaders can cut this distance in half: the distance between the majority center of gravity and this outcome is only 13% of the x-axis range. Additional analysis—omitted here, but available on request—shows that the difference in the two distances is higher given higher levels of polarization.

In substantive terms, these results show that in real-world legislatures, by choosing an appropriate agenda, party leaders can move legislative outcomes considerably closer to the preferences of their caucus compared to the expected results given leader inaction. Because these efforts involving movement within the uncovered set—that is, selecting one enactable outcome and devising an agenda that yields it—this potential for majority-party influence exists at the margin of whatever inherent advantages are conveyed to the majority by the location of the uncovered set.

The charts also show that the potential gains from agenda setting vary across legislatures. For example, among the state legislatures, some states (e.g., Maine) have uncovered sets that are close to the majority caucus to begin with (low C), whereas in others, the uncovered set lies farther away (e.g., Connecticut, where C is high). Similarly, in some states, agenda setting provides relatively substantial gains over the situation where leaders are inactive (e.g., Vermont; note the difference between C and C_x), whereas in other states the gains are more modest (e.g., South Carolina, where C and C_x are similar).

These results suggest that the value of conditional party government is not the same for all legislatures. Polarization may be a necessary condition for conditional party government to operate, but it is not sufficient. Depending on the distribution of legislators' preferences, both the level of disagreement between the parties and the level of agreement within the majority party, legislators in the majority caucus may decide that the range of outcomes that are possible given an inactive leadership are sufficiently good that the cost of empowering leaders outweighs the gains.

More generally, the data in Figure 6 only describe the potential gains from agenda setting. There is no assurance that the majority caucus will agree on which outcome to implement or that party leaders will respond to a caucus mandate by devising appropriate agenda strategies. What these results establish is that in a realistic model of the legislative process, the potential exists for the majority party to use its control over legislative procedures to make real improvements in legislative outcomes—improvements that occur at the margin of whatever advantages are conveyed to the majority party by polarization.