A. Open Rule and Opinion Writer Models

One of the earliest, and most common, models of collegial court decision-making (largely imported from the legislative setting) is the “open-rule model.” The model assumes that judges on collegial courts decide by majority rule, using an open amendment procedure. One judge proposes a rule, which can then be costlessly amended through alternative proposals by the other judges. An “opinion” emerges when there is no alternative that can secure a majority against the current proposal. Assuming that the case space is one-dimensional and judges’ rule preferences are single peaked, this setting is a straightforward application of the median voter theorem (Black Reference Black1948), leading to the conclusion that the court’s opinion will be written at the ideal rule of the median judge (Epstein et al. Reference Epstein, Hoe, King and Segal2005; Martin, Quinn, and Epstein Reference Martin, Quinn and Epstein2005).

Opinion writer models extend the open-rule model by noting that the opinion assignment process of the court provides a particular judge—the opinion writer—with the opportunity to make a first proposal. Other judges are free to offer counterproposals, but those proposals are costly. As a result, the opinion writer has some agenda-setting power that allows shading the court’s opinion away from the median judge’s ideal rule toward the writer’s own preferred position. But the ability to do so is limited: the opinion writer’s proposal must be sufficiently close to the median judge’s preferred rule to forestall a counterproposal by another judge. This requires that for all other judges, the cost of a counterproposal κ outweighs the potential utility gain from proposing a rule that can beat the opinion writer’s proposal. As a result, the greater the cost of counterproposals, the more leeway the opinion writer has to pull the opinion away from the median judge’s ideal rule.Footnote ¹³ In short, the general conclusion of the open-rule and opinion writer models is that the court’s median judge is in a powerful position, although one that may leave some room for (constrained) influence by opinion writers (Maltzman, Spriggs, and Wahlbeck Reference Maltzman, Spriggs II and Wahlbeck2000; Hammond, Bonneau, and Sheehan Reference Hammond, Bonneau and Sheehan2005; Bonneau et al. Reference Bonneau, Hammond, Forrest and Wahlbeck2007; Lax and Cameron Reference Lax and Cameron2007).

In these models, the asymmetric nature of derived rule preferences is (largely) irrelevant. In the open-rule model, the key feature that drives the court’s opinion to the rule preferred by the median judge is the logic of the median voter theorem, which holds for all single-peaked utility functions, whether asymmetric or not. In the opinion writer model, the asymmetric nature of derived utility functions does matter—but only at the margin. In this model, the extent to which an opinion writer can shade the rule adopted in the court’s opinion away from the median depends on the costs of writing a counterproposal. For any cost $\kappa >0$ of counterproposals, there exists a convex set around the ideal rule of the median judge within which the opinion writer’s proposal must fall in order not to elicit a counterproposal. Denote this set by [r _l , r _r ]. The defining feature of this set is that the median judge must be indifferent between r _l and r _r , that is, $U_M\left(r_l\right)=U_M\left(r_r\right)$ . Given asymmetric rule preferences, this immediately implies that this interval is not symmetric around the median judge’s ideal rule: the edge of the interval will be closer to the median’s ideal rule on the side with greater case density. Substantively, this means that with asymmetric rule preferences, opinion writers to the side of the median with greater case density are more constrained by the median’s preferences than opinion writers on the more sparsely populated side of the median. Put differently, when the legal differences among the opinion writer and the median judge are likely to be consequential in a large number of potential cases, opinion writers are forced to hew more closely to the preferences of the median. Nevertheless, the central conclusion of both models does not change with asymmetric preferences: legal rules will be driven largely by the preferences of the median judge, subject to some influence by the opinion writer.

While the conclusions of these models are largely unaffected by the asymmetry of judicial preferences, note that these models strip away distinctive features of judicial decision-making and are very specific in their scope: they make predictions about the location of the court’s majority opinion (which falls at, or close to, the position of the median judge), but they do not address broader features of judicial behavior. For example, these models do not speak to whether judges will choose to join an opinion or write separately, whether a court is able to issue a majority opinion or will fail to do so, and what the composition of the majority coalition is (if it exists). In other words, these models are not designed to illuminate aspects of judicial behavior that are of obvious interest and importance. In part as a reaction, scholars have recently begun to develop models of judicial decision-making that are designed to take account of the particular institutional features of the judiciary and to speak to precisely these broader aspects. Significantly, in these concurrence models (Carrubba et al. Reference Carrubba, Friedman, Martin and Vanberg2012; Cameron and Kornhauser Reference Cameron and Kornhauser2016), incorporating the distribution of future cases and the resulting asymmetry of rule preferences has significant implications for judicial decision-making.

B. Concurrence Models

Concurrence models focus on two features that distinguish judicial decision-making from ordinary policy making. The first is that judicial decisions involve two separate aspects. Judges must decide on the disposition of the case before them (which party wins), and they must articulate a rule that justifies this outcome. The second is that judges typically cannot be forced to “go along” with an opinion that adopts a legal rule with which the judge disagrees. A judge is always free to write a separate opinion. In particular, a judge who agrees with the majority’s disposition of a case can write a concurrence that supports the disposition but articulates a separate legal rationale (rule) for that decision. By incorporating both of these features, concurrence models not only generate implications for the location of majority opinions but are also able to speak to the conditions under which majority opinions are possible, when plurality opinions result, which judges will join an opinion, and who will write separately. These implications are affected in significant ways by explicit introduction of the distribution of future cases and changes in that distribution.

To show this, we generalize the concurrence model in Carrubba et al. (Reference Carrubba, Friedman, Martin and Vanberg2012). Before we do so, we note a methodological point. The model we sketch is (intentionally) not a full-fledged bargaining model that generates specific equilibrium predictions for the existence and location of majority opinions. As Laver, de Marchi, and Mutlu (Reference Laver, de Marchi and Mutlu2011) have argued in the context of coalition bargaining models, one challenge for full-fledged bargaining models is that equilibrium results typically depend critically on the specific bargaining structure that is assumed (including which players can make offers and in what order). When these bargaining structures reflect “real-world” constraints, this is obviously not problematic. But when the bargaining environment under consideration is essentially unstructured, relying on specific bargaining protocols that drive outcomes is undesirable. Rather, it is advantageous to focus on general characterizations that do not depend on a specific bargaining structure.

This is the approach we take here. The concurrence model we sketch is designed to illuminate—in a very general way—two questions: For a given configuration of judicial preferences on a collegial court, is a majority opinion possible, or can the judges not reach sufficient consensus to issue a majority opinion? Second, if a majority opinion is possible, what is the range of opinions that can secure a sufficient number of votes? Put differently, the goal of the analysis is to characterize the bargaining range within which majority opinions can be located, given the preferences of the judges on the court. Where a majority opinion will emerge within that range depends on the bargaining process among the judges, which is left unspecified.Footnote ¹⁴

Let N denote the set of judges, with |N| odd. Judges must resolve cases, which requires both a (binary) disposition and—in order to issue a majority opinion that creates binding precedent—articulation of a legal rule that a majority of judges support. Following Carrubba et al. (Reference Carrubba, Friedman, Martin and Vanberg2012), we assume that judges care about two facets of the legal rules that they announce in their decisions. One aspect concerns the policy implications of a decision. If a majority opinion is issued, this opinion has precedential value and will guide the decisions of lower courts going forward. Because judges care about how cases are resolved, they receive payoffs from the legal rule that is announced in a majority decision, whether or not the judge signed on to the opinion. Second, we assume that judges care not only about the rule that is enacted; they also care about which rule they express support for (i.e., with which rule they are publicly identified). For judges who sign the majority opinion, these two rules coincide. But for judges who write separately (either in a concurrence or a dissent), these two rules are not the same: they derive policy payoffs from the rule enacted in the majority opinion but receive expressive payoffs from the rule they sign on to in their separate opinion.

The rule preferences that judges have (and that feed into their policy and expressive payoffs) are derived—in keeping with the first part of this article—from the manner in which rules dispose of future cases. Recall that these preferences are single peaked, although typically asymmetric about a judge’s ideal rule (result 1c). Denote the “policy utility” for judge i of rule r by U _i (r) and the “expressive preference” for writing or signing an opinion at $\tilde{r}$ by $E_i\left(\tilde{r}\right)=\alpha U_i\left(\tilde{r}\right)$ , where $\alpha \ge 0$ denotes the weight placed on expressive benefits. Without loss of generality, assume that $U\left(x_i\right)=E_i\left(x_i\right)=0$ . Finally, let $\kappa >0$ denote the cost of writing a separate opinion. Note that if a judge chooses to bear the cost of writing separately, it is a dominant strategy to write at the judge’s ideal point. Given this setup, the payoffs to the alternatives confronting a judge in a particular case are outlined in table 1.

Table 1.Possible Actions and Associated Payoffs under a Concurrence Model of Bargaining

The key feature of concurrence models is that the setup we have just articulated implies that judicial behavior is characterized by a “join region.” Each judge is willing to sign opinions that articulate rules that are sufficiently close to his or her ideal point. But the judge will not sign opinions establishing rules outside of this join region, preferring instead to write separately.Footnote ¹⁵ The intuition is simple: because judges prefer not to be associated with opinions that stray too far from their preferred rule (their expressive preference), they are willing to bear the cost of writing separately if a proposed rule is sufficiently far from their ideal point. For rules that are sufficiently close, they are willing to sign rather than incur the costs of concurring. Formally (focusing on the case in which the judge is pivotal to a proposed majority opinion; other cases are analogous), the judge’s join region is defined as

(6) $\begin{array}{}\text{(6)}& {\mathrm{JR}}_i=\left\{r\in ℝ|\kappa \ge -U_i(r)-E_i(r)\right\}.\end{array}$

This join region is a convex set about the judge’s ideal point (since κ is a constant and the right-hand side increases in the distance of r from the judge’s ideal rule). The set of rules contained in the join region for a judge with ideal rule x_i can therefore be denoted by upper and lower thresholds ${\overline{x}}_i$ and ${\underset{\_}{x}}_i$ , where $x_i\in \left[{\underset{\_}{x}}_i,{\overline{x}}_i\right]$ . This leads to a simple characterization of judicial behavior: judge i is willing to sign majority opinions at $r\in \left[{\underset{\_}{x}}_i,{\overline{x}}_i\right]$ but will prefer to write separately if a majority opinion falls outside of this interval.

Note that given this characterization of judicial behavior, majority opinions can emerge only if there is overlap in the join regions of a majority of judges. This overlapping region is referred to as the majority opinion region. Significantly, this immediately implies that concurrence models can explain plurality opinions: if no overlap exists between the join regions of a majority of judges, no majority opinion can emerge—an outcome that is not possible within open-rule and opinion writer models. Second, unlike the open-rule and opinion writer models that highlight the power of the court’s median judge, concurrence models imply that the legal rule in majority opinions need not be located at, or even near, the ideal rule of the median judge. If the ideal rule of the median judge is not contained in the majority opinion region, that judge cannot hold out for a majority opinion at his or her ideal rule. At best, the judge can bargain for the rule at the boundary of the majority opinion region closest to his or her ideal point.Footnote ¹⁶ Figure 3 provides an illustration for a three-judge court. The figure plots ideal points (x_i ’s) for three judges, as well as a join region around each judge’s ideal point (to make the figure more legible, we separate the judges and their join regions vertically). We also plot the potential case distribution in the lower part of the figure (which is uniform in this figure). In figure 3a , there is overlap between the join regions of judges 2 and 3, allowing for a majority opinion in the overlap region. Importantly, any such majority opinion is constrained to be quite distant from the median judge’s ideal point—precisely because of the need to secure judge 3’s vote. Judge 1, in turn, writes a separate opinion (which, depending on whether he or she agrees or disagrees with the disposition implied by the majority opinion, will either be a concurrence or a dissent). In figure 3b , the median judge has shifted to the left, leading to a disappearance of the overlap in join regions. Now, no two judges can agree on an opinion they are both willing to sign, and no majority opinion is possible.

Figure 3.

Majority and plurality opinions with symmetric join regions. Each panel assumes a uniform case distribution and κ = 0.045. Holding x ₁ = −2.5 and x ₃ = 2 constant, the ideal rule of judge 2 shifts from x ₂ = 0.5 (a) to x ₂ = −0.25 (b). While the join regions for judges 1 (JI₁ = [−3.5, −1.5]) and 3 (JI₃ = [1, 3]) remain unchanged, the join region for judge 2 shifts from JI₂ = [−0.5, 1.5] to JI₂ = [−1.2, 0.7], eliminating the coalition of judges 2 and 3.

C. Incorporating the Distribution of Future Cases

We are now in a position to consider how the introduction of a nonuniform case distribution and the resulting asymmetry of preferences over legal rules affects the conclusions of concurrence models. The central intuition behind the results we present is simple. The key behavioral feature that drives collective decision-making in concurrence models is the join region of individual judges. Overlap between the join regions of judges determines whether a majority opinion can emerge and which judges join or write separately. Because overlap is central, the width and shape of join regions is of primary importance. If judges have standard symmetric rule preferences, join regions are symmetric about the ideal points of judges, and their width is determined by the cost of writing separately and the intensity of judicial preferences.

Relaxing the assumption of a uniform case distribution (and the symmetric rule preferences that result) and introducing nonuniform case distributions (and the asymmetric preferences over legal rules this implies) has significant implications for join regions and therefore decision-making by collegial courts. In particular, there are two features of join regions that are affected. The first is that unless the case density is symmetric around a judge’s ideal rule, the judge’s join region will be asymmetric—that is, the judge is more willing to sign on to opinions that deviate in one direction from the judge’s preferred rule than in the other direction. Second, the greater the density of future cases around a judge’s ideal rule, the more narrow a judge’s join region will be. Put differently, judges are more reluctant to be accommodating if moving away from their ideal rule has implications for a larger number of cases.Footnote ¹⁷ Lemma 1 summarizes these properties when the distribution of future cases is single peaked (proof provided in the appendix).

The collective decisions that judges on a court reach—most importantly, majority opinions—depend on the overlap between the join regions of judges. Significantly, the asymmetry of join regions can have profound consequences for these decisions compared to decisions reached under symmetric join regions (derived from symmetric preferences, such as are implied by a uniform case distribution). Consider two brief examples. In figure 4, we compare join regions and their overlap for a three-judge panel, holding fixed the ideal rule of each judge. In figure 4a , the underlying case distribution is uniform (thus implying symmetric rule preferences), while the underlying case distribution is normal in figure 4b . Note that with a uniform distribution, the majority opinion region is located symmetrically between the ideal rules of the coalition members. With a normal distribution of cases (fig. 4b ), join regions are no longer symmetric. Moreover, the join region of judge 2, whose ideal point is located close to the mean of the case distribution and in the dense region of the case distribution, has shrunk and become much more restrictive. The majority opinion region now significantly advantages this judge.

Figure 4.

Symmetric and asymmetric join regions. Holding constant the ideal rules of the three judges (x ₁ = −3, x ₂ = 0.5, and x ₃ = 2.5) and assuming κ = 0.075, each panel compares join regions derived from a uniform (a) and normal (b) case distribution. Under the uniform distribution, JI₁ = [−4, −1.3], JI₂ = [−1.1, 2.2], and JI₃ = [0.8, 4]. Under a normal case distribution, f(c) = N(−0.5, .75), JI₁ = [−4, −1.6], JI₂ = [0.3, 1.1], and JI₃ = [0.8, 4.0].

Figure 5 presents another example of the potential impact of asymmetric preferences and the underlying case distribution. In this case, figures 5a and 5b show a three-judge panel with rule preferences derived from a normal case distribution, holding fixed the ideal rules of the judges. Note that in figure 5a , a majority coalition between judges 2 and 3 emerges. Now suppose that as a result of some exogenous change—perhaps technological developments—the distribution of future cases has shifted to the right so that the mean of the case distribution has moved toward judges 2 and 3 as depicted in figure 5b . As summarized in lemma 1, the result of this shift is that the join regions of judges 2 and 3 shrink in such a way as to eliminate the overlap between them—and to create overlap between judges 1 and 2 instead. In other words, holding fixed the ideal rules of the judges, an exogenous shift in the case distribution has led to the breakdown of one majority coalition and the emergence of a new one.

Figure 5.

Holding constant the ideal rules of the three judges (x ₁ = −1.5, x ₂ = 0.5, and x ₃ = 2) and assuming κ = 0.25, shifting the mean of the case distribution to the right causes the coalition between judges 2 and 3 to breakdown and a new coalition between judges 1 and 2 to emerge. a, Join regions derived from a case distribution f(c) = N(0, 1), JI₁ = [−4, −0.5], JI₂ = [−0.1, 1.5], and JI₃ = [0.7, 4]; b, join regions drawn from a case distribution f(c) = N(1, 1), JI₁ = [−4, 0.3], JI₂ = [−0.5, 1.1], and JI₃ = [1.3, 4].

These examples illustrate that if judges’ rule preferences are derived from the manner in which legal rules dispose of the stream of cases to which they apply, the distribution of future cases and changes in this distribution have potentially significant impacts because they affect—through the asymmetry of judges’ rule preferences—the potential for majority opinions. But how general are these lessons? Can we identify systematic impacts of the asymmetric rule preferences that are implied by nonuniform case distributions? In the next section, we turn to this issue.

D. Coalition Composition

In this section, we examine how incorporating the underlying distribution of cases (and changes to it) affects the existence and size of majority opinion coalitions in concurrence models. Before turning to those results, an additional lemma is useful to establish the relationship between feasible majority opinion coalitions and changes in the distribution of future cases. Consider two judges with ideal rules $x_i<x_{i+1}$ . Now suppose that there is a change in the distribution of future cases that increases the case density between the two judges in a “smooth” way (i.e., along the entire interval).Footnote ¹⁸ We then have the following lemma:

While we reserve the formal proof for the appendix, the intuition behind this lemma is straightforward. Lemma 1 establishes that as the density of cases surrounding a judge’s ideal rules increases, the join region of the judge narrows. This implies that a density increase between the ideal rules of any two judges with $x_i<x_{i+1}$ will cause the upper boundary of the join region for x _i to decrease ( ${\overline{x}}_i\to x_i$ ) and the lower boundary of the join region for $x_{i+1}$ to increase ( ${\underset{\_}{x}}_{i+1}\to x_{i+1}$ ). As a result, if the join regions overlapped initially, they may no longer do so. Consider the implications.

1. Size of Majority Coalitions

The size of majority coalitions has long been a focus of scholars and commentators on judicial decision-making. One reason for this is a belief that decisions supported by larger majorities (especially unanimous decisions) are more likely to find acceptance among lower courts and citizens and to be long-lasting (Hansford and Spriggs Reference Hansford and Spriggs2006; Benjamin and Desmarais Reference Benjamin and Bruce2012). While the open-rule and opinion writer models do not speak to the size of majority coalitions, concurrence models imply that the size of majority coalitions is determined by the overlap in judges’ join regions. As a result, the size of majority coalitions also depends on the underlying case distribution.

Result 2. An increase in the density of cases within the Pareto set of the judges on a collegial court weakly decreases the size of majority coalitions, provided that the cost of writing separately is sufficiently low.Footnote ¹⁹

Result 2 states that as the case density increases between the judges on a given panel, majority coalitions (weakly) decrease in size. Figure 6 provides an illustration. Consider figure 6a first. All judges have ideal rules to the right of the mode of the case distribution, implying that there is general agreement among the judges on how the majority of cases should be treated. Because there are relatively few cases that divide the judges, join regions are wide, and majority coalitions are likely to be large. Now consider what happens if—holding ideal rules constant—the distribution of potential cases shifts to the right (depicted in fig. 6b ). The result of such a shift is that majority coalitions are now smaller than they were before, despite the fact that the ideal rules judges prefer remain unchanged. What is the intuition behind this result? Substantively, an increase in case density between the judges implies that even if judges’ intrinsic legal preferences have not changed, existing disagreements among them have become more relevant for a greater number of real-world cases, thereby destroying possibilities for compromise that existed previously. There are a number of avenues that might produce such increasing case density. For example, as a result of changes in social norms or technology, certain types of behavior—by citizens or law enforcement officers—might become more prevalent over time, increasing the potential number of cases that will be affected by differences among the legal rules that judges prefer.

Figure 6.

Holding constant the ideal rules (x ₁ = −0.325, x ₂ = 0.5, and x ₃ = 1.75), the coalition between judges 1, 2, and 3 breaks down and a new coalition between judges 2 and 3 emerges when the density of the case distribution increases between the ideal rules of judges 1, 2, and 3. a, Join regions are derived given a distribution of cases f(c) = N(−2, 1), JI₁ = [−0.6, 0.8], JI₂ = [−0.3, 4.0], and JI₃ = [−0.3, 4.0]; b, join regions are derived given a distribution of cases f(c) = N(−1, 1), JI₁ = [−0.4, −0.2], JI₂ = [0.3, 1.0], and JI₃ = [0.7, 4.0].

2. Plurality Decisions

A key consideration for the justices of the US Supreme Court is that the Court’s decision only establishes binding precedent for lower courts if it secures the agreement of a majority of justices. Plurality decisions dispose of the case at hand but have no binding authority. This implies that the conditions under which a majority of justices are able to agree on an opinion, and the conditions under which they will fail to do so, are of obvious importance. Ironically, neither the open-rule nor the opinion writer models speak to this question: these models predict majority opinions close to the ideal rule of the median judge but provide no account of plurality opinions. The concurrence model, in contrast, provides a direct account of majority and plurality opinions: if there is no overlap between the join regions of a majority of judges, no majority opinion can be written. Intuitively, this can happen if judges’ preferences diverge sufficiently: if the ideal rules that judges favor are sufficiently dispersed across the case space, there will be no overlap between the join regions of a majority of judges. More interestingly, changes in the distribution of future cases also affect whether such bargaining failure occurs:

Result 3. For a sufficiently large increase of the density of the case distribution between the ideal rule of the median judge and the judges adjacent to the median, existing majority coalitions will break down, and no majority opinion can emerge, provided that the cost of writing separately is sufficiently low.

A less technical, and more colloquial, way of stating this result is to say that majority opinions are less likely to emerge when the ideal rule of the median judge is located in a region of the case space with a high density of cases. The intuition behind this result is not difficult to see. Increased density between the median judge and the adjacent judges reduces the size of the median’s join region (lemma 1) and may eliminate overlap between the join regions of the median and those of other judges (lemma 2). Since any majority opinion coalition must include the median judge, this implies that majority coalitions may become impossible as case density increases around the ideal rule of the median, leading to the emergence of a plurality opinion.

Figure 7 provides an example. Figure 7a illustrates a majority coalition comprising judges 1 and 2. In figure 7b , we hold the ideal rules of all judges constant but increase the density of the case distribution around the median judge as shown by the new case distribution displayed at the bottom of the figure. As figure 7b shows, the consequence is that the overlap between the join regions of the median and judge 1 disappears, and no majority opinion is possible.

Figure 7.

Holding constant the ideal rules (x ₁ = −2.5, x ₂ = −1, and x ₃ = 2.75) and mean of the underlying distribution (μ = −0.75), increasing case density around the median judge destroys the majority coalition and leads to plurality opinion. a, Join regions are derived given a distribution of cases f(c) = N(−0.75, 1.75), JI₁ = [−4.0, 1.6], JI₂ = [−1.8, −0.3], and JI₃ = [0.8, 4.0]; b, join regions are derived given a distribution of cases f(c) = N(−0.75, 1.05), JI₁ = [−4,0, −1.6], JI₂ = [−1.5, −0.6], and JI₃ = [0.3, 4.0].