2.1 Timing

In the first stage, justices collectively decide whether to hear the case. To capture minority quotas such as the Rule of Four, hearings require consent from m = n − 1/2 justices. Hearings impose costs c ≥ 0 on each justice.Footnote ³

If the case is not heard, the game ends with x _sq persisting.

Otherwise, there is a dispositional vote decided by majority rule. Each justice votes whether to move policy left or right from x _sq.Footnote ⁴ We denote rightward votes d = 1 and leftward votes as d = 0. After the dispositional vote, justices in the dispositional majority determine the new policy location, which we refer to as the final ruling.Footnote ⁵

2.2 Preferences

Policy preferences are represented by quadratic loss utility.Footnote ⁶ Thus, each justice i has ideal point $\hat {x}_i \in {\opf R}$. Without loss of generality, we order justices $\hat {x}_1 < \hat {x}_2 < \cdots < \hat {x}_n$.

Along with the final ruling's location, justices care about dispositional votes.Footnote ⁷ One might wonder why justices ever vote against the majority; joining allows them to participate in bargaining and possibly influence the final outcome. In practice, however, dissenting may signal future behavior (Ginsburg, Reference Ginsburg2010) or increase a case's media coverage (Bryan and Ringsmuth, Reference Bryan and Ringsmuth2016). Moreover, justices may care about reputation, which could be damaged by joining dispositions they oppose.

Dispositional motivations are represented by $I\cdot 0 + ( 1 - I) \cdot l( x_{{\rm sq}}- \hat {x}_i)$; where I = 1 if justice i joins her preferred disposition and I = 0 otherwise, and l() is a concave loss function single peaked at 0. Voting against dispositional motivation is costlier for justices farther from the status quo.Footnote ⁸

Justice i's payoff from not hearing the case is $-( x_{{\rm sq}} - \hat {x}_i) ^2$. Because each justice i incurs costs c from a hearing, i's payoff from hearing a case resulting in final ruling $x^\ast$ is $U_i( x^\ast ) - c$, where

$$U_i( x^\ast ) = \left\{\matrix{-( x^\ast - \hat{x}_i) ^2 \hfill & {\rm if\, vote\, for\, preferred\, disposition} \hfill \cr - ( x^\ast - \hat{x}_i) ^2 + l( x_{{\rm sq}}- \hat{x}_i) \hfill & {\rm if\, vote\, against\, preferred\, disposition.} \hfill }\right.$$

2.3 Final rulings

In practice, final rulings likely result from bargaining among majority justices. We do not explicitly model such bargaining. Instead, we assume final rulings satisfy four properties. Let dispositional majority M have associated final ruling $x^\ast _M$.

Properties of final rulings:

1. (1) If justice i is in dispositional majority M, then $0 \leq {\partial x^\ast _M}/{\partial \hat {x}_i} \leq 1$.

2. (2) If justice j satisfies $\hat {x}_j \geq \; x^\ast _{M_1}$ for dispositional majority M ₁ and $M_2 = M_1 \cup j$, then $x^\ast _{M_2} \geq \; x^\ast _{M_1}$. Symmetrically, $\hat {x}_j \leq \; x^\ast _{M_1}$ implies $x^\ast _{M_2} \leq \; x^\ast _{M_1}$.

3. (3) Let $\overline {\hat {x}}_M$ denote the rightmost justice in dispositional majority M and $\underline {\hat {x}}_M$ denote the leftmost. Then $x^\ast _M \in [ \underline {\hat {x}}_M,\; \overline {\hat {x}}_M]$.

4. (4) Let justice i be in dispositional majorities M ₁ and M ₂, where M ₁ ⊂ M ₂. Then $( {\partial x^\ast _{M_1}/}{\partial \hat {x}_i}) \geq ( {\partial x^\ast _{M_2}}/{\partial \hat {x}_i})$.

Broadly, the properties say that all justices in the dispositional majority can (weakly) affect final rulings. By Property 1, shifting a dispositional majority justice moves final rulings weakly in that same direction. Property 1 also limits the size of that shift. Property 2 implies that adding a justice to a dispositional majority shifts the final ruling weakly toward that justice's ideal point. Property 3 assures that final rulings cannot be improved for every majority member. Property 4 states that justices have weakly more influence on the final ruling's location in smaller majorities than in larger majorities.Footnote ⁹

These reduced-form properties for judicial bargaining align with microfounded settings studied elsewhere, including Lax and Cameron (Reference Lax and Cameron2007), Carrubba and Clark (Reference Carrubba and Clark2012), and Parameswaran et al. (Reference Parameswaran, Cameron and Kornhauser2019). Our approach allows us to take the bargaining equilibria as given and focus on the implications for, and consequences of, case selection.

3.1 Dispositional vote

We study Subgame Perfect Equilibria in weakly undominated strategies (hereafter just equilibria). In particular, we analyze equilibria featuring ‘monotonic’ dispositional majorities: either the dispositional vote is unanimous, or there is a justice i such that each justice j votes to move the threshold rightward, d _j = 1, if and only if j ≥ i. In Lemma B.1 in the appendix, we show that sufficiently strong dispositional motivations ensure that such equilibria exist and henceforth we focus on that case.

In equilibrium, each justice i has a unique cutpoint, $\check {x}_i$, fully characterizing her dispositional voting strategy. If $x_{{\rm sq}} < \check {x}_i$, then i votes to shift precedent rightward (d _i = 1) and, similarly, $x_{{\rm sq}} \geq \check {x}_i$ implies she votes to shift precedent leftward (d _i = 0). Additionally, the cutpoints have the same order as ideal points, which implies monotonic dispositional majorities.

We can use the dispositional cutpoints to easily order final opinions for heard cases as a function of x _sq. Let $x^\ast _i$ denote the final opinion of the dispositional majority if d _j = 0 for all j ≤ i and d _j = 1 for all j > i. For example, in a five-member court hearing a case with $x_{{\rm sq}} \in ( \check {x}_2,\; \check {x}_3)$, the final opinion $x_2^\ast$ is written by justices 3, 4, and 5. Properties 2 and 4 imply that final opinions are ordered $x^\ast _{{n + 1}/{2}} \leq x^\ast _{{n + 1}/{2} + 1} \leq \cdots x^\ast _n = x^\ast _0 \leq x^\ast _1 \leq \cdots \leq x^\ast _{{n-1}/{2}}$. Intuitively, the left-leaning bare majority writes the leftmost ruling, $x^\ast _{{n + 1}/{2}}$, and the right-leaning bare majority writes the rightmost ruling, $x^\ast _{{n-1}/{2}}$.Footnote ¹⁰

3.2 Case hearings

We first establish a necessary condition for a justice to vote to hear a case: she must be in the dispositional majority. In heard cases, the final ruling is worse than x _sq for minority justices. Intuitively, they never want to bear costs of hearing cases resulting in worse outcomes.

For case selection with quota m = n − 1/2, it suffices to focus on two justices who are the decisive pivots, P _L = n − 1/2 and P _R = n + 3/2. We focus on P _R, as P _L is analogous. Optimally hearing a case requires the benefit from setting new precedent to exceed the cost of hearing and deciding the case. Formally, if $x_{{\rm sq}} \in ( \check {x}_i,\; \check {x}_{i + 1}]$ for i < n + 1/2, then the case is heard if and only if

(1)$$U_{P_R}( x^\ast _i) - c\geq -( x_{{\rm sq}} - \hat{x}_{P_R}) ^2.$$

We can characterize whether the court hears $x_{{\rm sq}} \leq \check {x}_{{n + 1}/{2}}$ using conditions similar to (1), and $x_{{\rm sq}}> \check {x}_{{n + 1}/{2}}$ using analogous conditions for P _L.

Lemma 2 shows existence of an interval of cases around $\hat {x}_{{n + 1}/{2}}$ that are not heard. In the main text, we focus analysis on this interval to emphasize our key points. Yet, there can exist other intervals of non-heard cases. In the appendix, we fully characterize which cases are heard and show that the main analysis illustrates the primary takeaways.

In $[ \underline {x},\; \overline {x}]$, the left endpoint is the most moderate status quo that the right hearing pivot votes to hear. Similarly, the right endpoint is the most moderate status quo that the left hearing pivot votes to hear. All cases between the endpoints are not heard and their final ruling is x _sq. See Figure 1 for an illustration.

Figure 1. Effects of more extreme hearing pivots on case selection and final rulings.

Figure (a)–(c) each display a five member court. To ease illustration, each justice $i$ has arbitrarily strong dispositional motivation, so $\hat {x}_i \approx \check {x}_i$. Curly braces indicate the sets of status quo, $x_{{\rm sq}}$, mapping to final outcomes, which are not depicted spatially. In (a), final rulings satisfy $x_4^\ast \leq x_5^\ast = x_0^\ast \leq x_1^\ast$. In (b), justice $4$ shifts rightward to $\hat {x}^\prime _4 \geq \hat {x}_4$. The court hears more left-leaning cases, $\underline {x}^\prime > \underline {x}$, but the effect on right-leaning cases is ambiguous. A left-leaning bare majority is now possible, if $x_{{\rm sq}} \in ( \overline {x}^\prime ,\; \check {x}^\prime _4)$, so the leftmost ruling decreases from (a): $x^{\ast \prime }_3 < x^\ast _i$ for $i = 0,\; 1,\; 4,\; 5$. For each majority also possible in (a), rulings shift right: $x^{\ast \prime }_i > x^\ast _i$ for $i = 0,\; 1,\; 4,\; 5$. In (c), $\hat {x}^{\prime \prime }_4 \geq \hat {x}^{\prime }_4$. The court hears the same right-leaning cases, $\overline {x}^{\prime \prime } = \overline {x}^\prime$, and more left-leaning cases, $\underline {x}^{\prime \prime } > \underline {x}^\prime$. The leftmost ruling, $x^{\ast \prime }_3$, is unchanged and rulings shift rightward for each majority also possible in (b). Right-leaning bare majorities are now possible, if $x_{{\rm sq}} \in ( \check {x}_2,\; \underline {x}^{\prime \prime }$), so the rightmost ruling increases from (b): $x^{\ast \prime \prime }_2 > x^{\ast \prime }_i$ for $i = 0,\; 1,\; 3,\; 4,\; 5$.

We are interested in how changing justices’ ideal points alters which cases are heard and final rulings. First, we show that more extreme hearing pivots are willing to hear more moderate cases.

If a hearing pivot becomes more extreme, two main forces produce Proposition 1. First, status quo on the opposite side of the spectrum become less favorable to the shifting pivot, which encourages granting certiorari. Second, the final ruling can move, which may help or harm the shifting pivot. If it helps her, then this second force complements the first. But if the final ruling becomes less favorable, then the second force discourages granting certiorari and counteracts the first force. Yet, Property 1 implies that the first force always dominates. Thus, the overall effect encourages hearing the case. For example, the overall effect of a more conservative hearing pivot increases x, so the court hears cases with more moderate liberal precedents.

Although Proposition 1 conveys key forces, it only applies to the central interval of non-heard cases. Proposition 2 extends the result. The key forces are similar.

3.3 Change in dispositional majorities

We have shown that more extreme hearing pivots are willing to hear more moderate cases. Next, we study the effects on (i) the composition of dispositional majorities and (ii) final rulings. In general, more extreme hearing pivots make observed decisions more divisive and final rulings more extreme.

Lemma 2 implies that x is the most moderate left-leaning status quo that P _R hears. Let $\hat {x}_{P_R}^\prime \geq \hat {x}_{P_R}$, with corresponding x^′ ≥ x, by Proposition 1. Thus, cases x _sq ∈ (x, x^′) are not heard if $\hat {x}_{P_R}$ is the right hearing pivot, but are heard if she is replaced by $\hat {x}_{P_R}^\prime$. For these cases, either the dispositional majority is identical to that if x _sq = x, or it has fewer justices. Thus, more extreme hearing pivots lead to smaller dispositional majorities for cases with precedent on the opposite side of the spectrum.

More extreme hearing pivots also affect dispositional majorities for cases on their own side. For example, there are cases for which $\hat {x}_{P_R}$ joins the dispositional majority but $\hat {x}_{P_R}^\prime$ joins the minority. This more extreme hearing pivot shrinks the dispositional majority from n + 1/2 + 1 to n + 1/2.

Fixing the court, smaller dispositional majorities produce more extreme final rulings. Thus, changes in dispositional majorities can affect final rulings. For example, some cases will have one fewer conservative in their dispositional majorities and, in turn, more liberal final rulings.

3.4 Changes in other justices

We now discuss how non-pivotal justices affect case selection. Even though non-pivotal justices are not decisive in case selection, their effect on final rulings can alter $[ \underline {x},\; \overline {x}]$. If final rulings shift toward a hearing pivot, then the court hears more cases with precedents opposite that pivot. Symmetrically, if final rulings shift away from a hearing pivot, then the court hears fewer cases with precedents opposite that pivot.

For example, a sufficiently extreme justice may pull final rulings farther away from a hearing pivot and expand the interval of non-heard cases. Consider a dispositional majority containing a justice $\hat {x}_j \geq \hat {x}_{P_R}$ and producing final ruling $x^\ast \geq \hat {x}_{P_R}$. If j becomes more extreme, the distance between $\hat {x}_{P_R}$ and $x^\ast$ increases by Property 1. Appointing very extreme justices therefore presents a trade-off: heard cases have more extreme final rulings, but the court may hear fewer cases with moderate precedents.Moving the court median without changing the overall ordering is roughly equivalent to shifting any other non-hearing-pivot justice. However, the interval characterized in Lemma 2 can shift through changes in dispositional majorities, as well as through changes in final rulings. And if $\underline {x} = \hat {x}_{{n + 1}/{2}}$, then increasing $\hat {x}_{{n + 1}/{2}}$ can cause such changes even if the final ruling is unchanged. See Figure 2 for an example.

Figure 2. Effects of other justices on case selection and final rulings.

Figure 2 depicts a seven-member court. To ease illustration, each justice $i$ has arbitrarily strong dispositional motivation, so $\hat {x}_i \approx \check {x}_i$. If legislator $6$ shifts right from $\hat {x}_6$ to $\hat {x}^\prime _6$, as depicted in (b), then the court hears more cases and the final ruling increases for all heard cases, i.e. $x^{\ast \prime }_i > x^\ast _i$ for all $i = 0,\; 1,\; 6,\; 7$. If legislator $7$ shifts right from $\hat {x}_7$ to $\hat {x}^\prime _7$, as depicted in (c), then the court hears fewer left-leaning cases and the same right-leaning cases. Final rulings weakly increase for all heard cases.

If the ordering changes such that a former hearing pivot is now the median justice and a new justice is a hearing pivot, then there are multiple effects. Beyond the already noted effects of moving the hearing pivot, changing the court median switches some cases from one dispositional majority to the other. A new median justice may actually be more important than previously understood because, as far as we are aware, the combined impact on the dispositional majority and case selection has not been noted.