Uncertainty in Judicial Decision Making

Much of the focus on judicial uncertainty in existing theoretical work has been on the uncertainty faced by an upper court when deciding whether to review a lower court’s decision (see Beim, Hirsch, and Kastellec Reference Beim, Hirsch and Kastellec2014; Cameron, Segal, and Songer Reference Cameron, Segal and Songer2000; Spitzer and Talley Reference Spitzer and Talley2000). These approaches center case facts as the source of uncertainty for an upper court. An inability to directly observe the facts of a case translates into uncertainty over whether a lower court decision is compliant.

This uncertainty, however, differs from what courts face when deciding new legal issues. In these cases, courts observe the facts of the case, but face uncertainty into how these facts translate into relevant outcomes. To better understand judicial decision making under uncertainty, I conducted a handful of exploratory interviews with former clerks from the U.S. Circuit Courts of Appeals.Footnote ⁴ In these semistructured interviews, I asked the clerks about the process of deciding cases involving new legal issues. While uncommon relative to the overall caseload of a court, cases sometimes arose where no precedent applied. The clerks noted that uncertainty would often arise following some sort of shock to an existing legal rule. This shock could take the form of a new law, a recent Supreme Court case, or facts that required a new understanding of old law. The uncertainty faced by the clerks in these cases was not about what the facts of the case were but instead about how these cases related to existing preferences in light of the shock.

The clerks noted that when these circumstances occurred, they were not completely in the dark about what to do. Instead, they would look to what other courts had written as guideposts to understand the new legal landscape. This did not mean that the clerks’ judges always followed their peers; instead, they looked to the actions of other circuits, updated their beliefs, and then took the action they thought was best.

Model Illustration

As an example, consider a court reviewing the constitutionality of a search. The court observes the facts of the case—what was searched, the scope of the search, how the search was conducted—and must decide whether to admit $ (d=con) $ or exclude $ (d=lib) $ the evidence. The court’s preferences over dispositions can be characterized in terms of how intrusive a search was. Figure 1 visualizes these preferences for two different courts. The courts’ cut points divide levels of intrusiveness into two regions: levels of intrusiveness that justify exclusion and those that do not. $ {C}_2 $ is more liberal than $ {C}_1 $ . As their cut point is further to the left, a lower level of intrusiveness is required before they prefer to exclude evidence from a search; whereas, $ {C}_1 $ , the more conservative court, requires a higher level of intrusiveness before they would exclude the evidence from a search.

Figure 1. Preferences over Dispositions

Note: The top panel displays the preferences for a conservative court, with a cut point further to the right. The intrusiveness of a search must be relatively high before they are willing to exclude the evidence. In contrast, $ {C}_2 $ , who has a cut point further to the left, is more liberal as they will exclude evidence from searches with low levels of intrusiveness.

Uncertainty manifests in the mapping from case facts to intrusiveness. Consider the use of a new technology, such as a thermal camera, to conduct a search of a house.Footnote ⁸ Whether the court wants to exclude the evidence depends on how intrusive the search was; however, the court does not directly observe the intrusiveness of the search. Instead, the court sees that a thermal camera was used to detect the amount of heat emanating from the house. How these observable case facts translate into intrusiveness may be unclear. The thermal camera can “see” through walls, which may increase the intrusiveness, but what it is seeing is heat as opposed to images, which may decrease the intrusiveness.

If the court faces uncertainty over how intrusive the search is, this can lead them to be unsure of which disposition is correct for the case. Figure 2 captures this understanding of uncertainty. The same observed case facts result in two different levels of intrusiveness depending on the size of the shock. For case facts in the shaded region, the court’s preferred disposition depends on whether the shock is $ {\omega}_1 $ or $ {\omega}_2 $ . Courts then have an interest in learning about $ \omega $ in order to choose the “correct” disposition.

Figure 2. Case Space Conceptualization of Judicial Uncertainty

Note: In the shaded region, the court’s preferred disposition differs depending on the realization of $ \omega $ .

Model Discussion

Before proceeding to the analysis, three features of the model warrant discussion. First, the distribution that generates a court’s signal depends on the preferences of the court. When deciding new legal issues, the information courts have available comes primarily from litigants. Litigants are not concerned with providing judges with information for the sake of learning about the law but instead are trying to convince a judge (or judges) why they should rule for the litigant’s side. Litigants will target information at judges whose votes they seek to secure (Hazelton and Hinkle Reference Hazelton and Hinkle2022). Consequently, the information available to judges is often specific to what that judge should do for the case at hand and not about the world more generally. By conditioning the distribution of signals on the preferences of the court hearing the case, I account for this strategic incentive of litigants.

Second, in line with previous work on social learning, I assume that actors receive private signals before making their decision (see Bikhchandani, Hirshleifer, and Welch Reference Bikhchandani, Hirshleifer and Welch1992). When courts decide cases, their decisions are often accompanied by an opinion outlining the reasons for their decision. This opinion may provide information about the court’s signal. Talley (Reference Talley1999) examines this setting under the additional assumption that courts have the same preferences. Opinions, however, are written after the court has decided which disposition to choose. The opinion that can be written is conditional on which decision the court chooses (Parameswaran, Cameron, and Kornhauser Reference Parameswaran, Cameron and Kornhauser2021). Consequently, a court’s opinion may not always be a perfect accounting of its private information.

Finally, I assume that a court’s utility is independent of the actions of other actors. Under the model, courts have no inherent preference for uniformity and face no cost for being part of a circuit split. This assumption transforms the court’s choice into a decision-theoretic one. Importantly, the courts’ decisions are not completely independent. The second court can still be influenced by the actions of the first court; however, each court’s utility depends only on their own actions.

Canonical models of informational cascades also assume that actors’ payoffs are independent of each other’s actions (see Bikhchandani, Hirshleifer, and Welch Reference Bikhchandani, Hirshleifer and Welch1992). Under my framework, this assumption allows me to easily extend the model and look at the behavior of multiple courts as opposed to just two. Additionally, while courts observe each other’s actions, they are not obligated to follow one another. Any cost of a circuit split is informal.Footnote ⁹ While some judges may internalize a cost to generating a split, this is not universal (Klein Reference Klein2002). Finally, in the Courts of Appeals, courts face uncertainty over whether a future circuit will even decide the same question and, if so, what the preferences of this court might be. Utilizing a decision-theoretic framework allows me to abstract away from this additional uncertainty and instead focus on what information courts can learn from each other’s actions.Footnote ¹⁰

First Court

I begin by analyzing the behavior of the first court. I assume that courts are unconcerned with how their decisions will be perceived by future courts. Consequently, the first court will simply choose the disposition that has a higher probability of being correct.Footnote ¹¹ The first court has two sources of information available about the state of the world: their signal and their prior beliefs over $ \omega $ . While the signal is influential on the court, it is not deterministic. The shock to case facts is limited in value. If the observed case facts are extreme relative to the court’s cut point, no shock is large enough to change which disposition they prefer ex ante. For $ \widehat{x}\in (0,{y}_i-\Omega )\cup ({y}_i+\Omega, 1\hskip-1pt ) $ , a court’s choice is the same irrespective of signal.

Even if a shock is theoretically large enough to change which disposition a court prefers, they will not necessarily follow their signal. Signals are sometimes wrong. The accuracy of the signal affects its influence on a court’s decision. If $ \widehat{x}<{y}_1-\Omega (2p-1) $ , the case facts are so conservative that the court “ignores” the signal and always chooses $ d=con $ . If $ \widehat{x}\ge {y}_1+\Omega (2p-1) $ , the case facts are so liberal that they will always choose $ d=lib $ .

Whether the signal will affect which disposition is chosen depends on the observable case facts, $ \widehat{x} $ . Figure 3 divides the case-space into three regions. For observed case facts in regions I and $ III $ , which disposition the court chooses does not depend on their signal. Only in region $ II, $ does their disposition depend on the signal. If case facts are in this region, then the court will choose the disposition that matches their signal. The size of region $ II $ depends on p. When signals are very accurate $ (p\sim 1 $ ), courts will follow their signal for even relatively extreme case facts. In contrast, for inaccurate signals $ (p\sim \frac{1}{2}) $ , the court’s choice of disposition almost never depends on their signal.

Figure 3. Disposition Choice of the First Court

Note: For observed case facts in Region I, the court always chooses $ d=con $ . In region $ III $ , the court always chooses $ d=lib $ . In Region $ II $ , the court will choose the disposition that matches their signal.

Second Court

Whether the second court learns anything from the first court’s actions depends on the relationship between $ {y}_1 $ and $ \widehat{x} $ . The second court does not directly observe the first’s signal. Instead, the second court must rely on both the observable case facts and the first court’s disposition to update their beliefs about $ \omega $ . When case facts are extreme relative to the first court’s cut point, a disposition provides no information about $ \omega $ because the first court chooses the same disposition regardless of signal. As the first court’s signal is private, the second court learns nothing about $ \omega $ . In contrast, if the case facts are more moderate (relative to the court), the first court’s disposition is identical to their signal. As dispositions are public, the second court then learns the first court’s signal.

From the perspective of a second court, there are two distinct types of decisions which are as follows.

As the second court observes $ \widehat{x} $ and $ {y}_1 $ , they can always derive whether the first court’s decision is informative or uninformative. If the first court’s decision was uninformative, the second court proceeds as if they were the first decision maker.Footnote ¹² They observe case facts $ \widehat{x} $ and a signal $ {s}_2 $ about their preferred disposition for the case. The second court then chooses $ {d}_2 $ . Instead, if the first court’s decision was informative, they learn that $ {s}_1={d}_1 $ .

Preference heterogeneity leads to differences in the influence of the first court’s decision on the second. The first court’s signal is dependent on their preferences. If the first and second courts have different preferences, this affects what the second court can learn from the first’s actions. As an example, consider $ {y}_2<{y}_1 $ and $ {d}_1=lib $ , that is, the first court is more conservative than the second and chose the liberal disposition. The top panel of Figure 4 displays this preference arrangement. Assume this decision is informative. As $ {d}_1=lib $ , the second court learns that $ {s}_1=\widehat{lib} $ . A signal of $ {s}_1=\widehat{lib} $ means that, with probability p, $ {y}_1\le x $ . As the first court is more conservative than the second $ ({y}_2<{y}_1) $ , this implies that $ {y}_2<{y}_1\le x $ . The second court’s cut point is also to the left of the case facts.

Figure 4. Potential Location of $ \widehat{x}+\omega $ Based on a Signal of $ {s}_1=\widehat{lib} $

Note: In A, the two courts prefer the same disposition. In B, the courts prefer different dispositions.

In any case, where a more conservative court prefers the liberal disposition, the second court does as well. This concept is analogous to the “Nixon goes to China” principle discussed in the judicial auditing literature. In models of judicial auditing, an upper court does not review a liberal disposition chosen by a more conservative lower court, as they would always agree with the decision (Cameron, Segal, and Songer Reference Cameron, Segal and Songer2000). In my model, when a conservative court chooses the liberal disposition, future liberal courts can infer that they would have made the same choice.

When the first court is more liberal than the second $ ({y}_1<{y}_2) $ , as in the bottom panel of Figure 4, this relationship changes. Again assume that $ {d}_1=lib $ . Unlike before, the first court prefers $ d=lib $ for a larger set of case facts than the second court. While the two courts might agree on many cases, there is an interval of case facts between the first and second courts’ cut points for which the first court prefers $ {d}_1=lib $ and the second court prefers $ {d}_2=con $ .

The second court receives their own signal as well. Using both signals, the second court updates their beliefs about $ \omega $ . Choosing a disposition is then simple. The court compares the probability that the case facts are to the left of their cut point with the probability that they are to the right and chooses the disposition corresponding to whichever probability is higher.

What Contributes to Splits?

A split occurs if the two courts choose different dispositions. I analyze the factors that make a split more likely to occur. Under the model, courts have no innate preference for uniformity in their decisions. They are only concerned with choosing a “correct” disposition according to their ideal rule. The court’s beliefs about which disposition is correct are central to their decision. As the novel feature of my model is preference heterogeneity, I focus on how the ideological distance between the courts affects the probability of a split.

Learning from Previous Decisions

First, I analyze the effect of ideological distance on the second court’s beliefs over $ \omega $ . Assume the first court chose $ {d}_1=lib $ . An increase in the second court’s belief that the liberal disposition is correct increases the probability that they will choose the same disposition—the second court becomes more likely to always choose $ {d}_2=lib $ , regardless of signal, and less likely to always choose $ {d}_2=con $ . Together, these effects make it more likely that the second court will choose the same disposition as the first court.

If the first court’s decision is uninformative, the second court learns nothing from the disposition. Assume the first case is informative such that the second court learns $ {s}_1 $ . Consider $ {s}_1={s}_2=\widehat{lib} $ and a second court more conservative than the first $ ({y}_1<{y}_2) $ . How likely are these signals given $ \omega $ ?

Figure 5 divides the case space into three regions. In Region A, $ \widehat{x}+\omega $ is to the left of both courts. Within this region, both courts prefer a disposition of $ d=con $ ; however, both courts received signals of $ s=\widehat{lib} $ . For $ \omega $ to be in Region A, both of these signals would have to be incorrect. Moving to the right, Region B is the interval between the courts’ cut points. For values of $ \omega $ in this interval, the first court prefers $ d=lib $ while the second court prefers $ d=con $ . If $ \omega $ were in this region, the second court’s signal would still be incorrect; however, the first court’s signal is now correct. For both signals to be correct, $ \omega $ would need to be in Region C ( $ \omega >{y}_2-\widehat{x} $ ), where both courts prefer a disposition of $ d=lib $ . When updating their beliefs about $ \omega $ , the second court weights their prior beliefs by the probability that neither, one, or both of the signals are correct.

Figure 5. Accuracy of $ {s}_1={s}_2=\widehat{lib} $ Based on the Value of $ \omega $

Note: The figure demonstrates what would have to be true of the signals received by the courts for $ \omega $ to be in various regions of the space.

The second court is able to learn something from an informative case regardless of the disposition chosen or the relative position of the previous court. How much the court can learn, however, depends on these factors. Importantly, “learning” does not immediately translate to more accurate beliefs. Even if the court is able to perfectly intuit the signals of the other courts, these signals are not perfectly accurate. When signals are imperfect, there is always a chance that incorrect information persists (Bikhchandani, Hirshleifer, and Welch Reference Bikhchandani, Hirshleifer and Welch1992).

Proposition 1. Assume the first court’s decision is informative. As the distance between the courts’ cutpoints increases, the second court is less likely to believe that $ {d}_1 $ is their preferred disposition.

Intuitively, as the ideological distance between the two courts increases, the set of cases for which the two courts prefer different dispositions increases. Consequently, the relative weight the second court places on the first’s signal decreases. As the first court’s signal becomes less influential on the second court’s beliefs, the second court is less likely to take the same action. A corollary of this proposition is that a court learns most from a perfect ally. Heterogeneous cut points reduce the amount of information a court can learn from a single case.

Learning from Private Information

The two courts might choose the same disposition because the second court observed the decision of the first, updated their beliefs, and chose the same disposition as a result. This process is our common understanding of why uniformity occurs. However, the courts may choose the same disposition even if the second learns nothing from the actions of the first (Volden, Ting, and Carpenter Reference Volden, Ting and Carpenter2008). Outside of previous dispositions, a court has two sources of information available to them when deciding a case: the location of the case facts and a private signal. If both courts are faced with extreme case facts, prior beliefs over $ \omega $ will outweigh any of the influence signals have on the courts’ decisions. Similarly, both courts might have received the same signal and chosen the same disposition as a result. Any combination of these three processes could result in uniformity; however, only the first is the result of learning from another court’s actions.

Consider an example from policy diffusion literature—why two cities enact a ban on smoking in restaurants (see Shipan and Volden Reference Shipan and Volden2006; Reference Shipan and Volden2008). It may be that the second city learned from the first’s policy. After the first city enacted a ban, the second observed the effects, updated its beliefs about the utility of a ban, and, as a result of this updating, decided to ban smoking. This explanation fits squarely in our understanding of policy diffusion. However, both cities could adopt a ban in the absence of learning; that is, both could have received private information about the negative consequences of secondhand smoke and enacted a ban as a result. Either of these processes could lead both cities to ban smoking, but we would refer only to the former as policy diffusion.

What looks like courts learning from one another may actually be courts simply taking actions consistent with their private information about the case. If the ideological distance between two courts is related to the private information they receive, it could lead us to incorrectly attribute uniformity to learning from previous decisions (Volden, Ting, and Carpenter Reference Volden, Ting and Carpenter2008). If we could directly observe the information available to the courts, we would be able to account for the influence of private information on uniformity. However, we are limited in our ability to observe the case facts and the courts’ signals.Footnote ¹³ Consequently, I model how courts’ private information relates to the relationship between ideological distance and the probability of a circuit split. Combining these results with Proposition 1 then gives the expected effect of ideological distance.

The probability that the courts receive the same signal and the probability that case facts are extreme vary as the ideological distance between the courts increases. To isolate the effect of a court’s private information, I calculate the effect of ideological distance assuming that the courts do not observe each other’s actions. Formally, I assume that regardless of the actions of the first court, the second court always has a posterior belief equal to their prior, $ G(\omega |{d}_1=d,{y}_1)=U[-\Omega, \Omega ] $ . The assumption that dispositions are private is unrealistic; however, it allows me to identify whether there is an effect of ideological distance that is not related to the second court learning from the first.

To characterize the effect of ideological distance, we need to distinguish between two types of orderings.

Definition 3 (Bias-compatible). A disposition-court pair is bias-compatible if the first court is more liberal than the second and the first court chose the liberal disposition. Similarly, a pair is bias-compatible if the first court is more conservative than the second and chose the conservative disposition.

Definition 4 (Bias-incompatible). A disposition-court pair is bias-incompatible if the first court is more liberal than the second and chose the conservative disposition, or vice versa.

Whether a decision is bias-compatible or bias-incompatible is relative to the cut point of the second court. The same decision by the same first court could be bias-compatible or bias-incompatible, depending on whether the first court is more conservative or more liberal than the second court.

Ideological distance relates to the probability of a split through two mechanisms: changes in the probability of expected case facts and changes in probability that $ {s}_2=\widehat{lib} $ . For building intuition, I focus on changes in the probability of expected case facts. Propositions are with respect to the combined effect of both mechanisms.

As a comparison point, I first assume the courts have the same cut point. The top panel of Figure 6 plots the potential locations of the observable case facts, $ \widehat{x} $ , conditional on the first court choosing $ {d}_1=lib $ . The space can be broken down into two regions. If $ \widehat{x}\in A $ , the second court will choose $ {d}_2=lib $ if and only if she receives a signal of $ {s}_2=\widehat{lib} $ . Instead, if $ \widehat{x}\in B $ , the second court will always choose $ {d}_2=lib $ .

Figure 6. Possible Locations of $ \widehat{x} $ for Different Preference Arrangements

Note: In Region A, the second court chooses $ {d}_2=lib $ if and only if $ {s}_2=\widehat{lib} $ . In Region B, the second court always chooses $ {d}_2=lib $ . In Region C, the second court never chooses $ {d}_2=lib $ . If $ {y}_1<{y}_2 $ , increasing the ideological distance between the two courts increases the size of region C. If $ {y}_2<{y}_1 $ , increasing ideological distance reduces the size of Region A. The probability that $ \widehat{x} $ is in Region D is $ 0 $ . C $ (y)=y+{G}^{-1}(1-p) $ . $ \overline{C}(y)=y+{G}^{-1}(p) $ .

Moving down a panel, I consider the bias-compatible case, where the first court is more liberal than the second. I note first that Regions A and B remain the same as before; however, there is a third region: C. If $ \widehat{x}\in C $ , the second court will choose $ {d}_2=con $ regardless of which signal they received. As the distance between $ {y}_1 $ and $ {y}_2 $ increases, the size of Region C increases. Consequently, the probability that the two courts choose different dispositions increases. This leads to the following proposition.

Proposition 2. Assume dispositions are private. As the ideological distance between the courts increases, the probability of a circuit split increases for bias-compatible decisions.

The effect of ideological distance on the probability of a circuit split is in the opposite direction for bias-incompatible decisions. Returning to the third panel of Figure 6, the second court is now more conservative than the first. As with the special case, where the two courts shared a cut point, there are only two regions: A and B. The size of Region A, however, is smaller than before. While the absolute size of Region B does not increase, its relative size does. The greater the distance between the first and second court, the greater the probability that $ \widehat{x}\in B $ . When $ \widehat{x}\in B $ , the second court always chooses $ {d}_2=lib $ , so legal uniformity is assured. Consequently, increasing the distance between the two courts decreases the probability of a circuit split for bias-incompatible cases. This leads to the following proposition.

Proposition 3. Assume dispositions are private. As the distance between the courts increases, the probability of a circuit split weakly decreases for bias-incompatible decisions.

The Effects of Ideological Distance

As a reminder, increasing ideological distance reduces the influence of the first court’s decision on the second court’s actions. Combining this effect with the results concerning private information allows me to characterize the effect of ideological distance on uniformity. For bias-compatible decisions, increased ideological distance results in a decreased probability that the courts choose the same disposition.

Proposition 4. For bias-compatible decisions, as ideological distance between the two courts increases, the probability of a circuit split increases.

This result yields a clear empirical prediction for bias-compatible cases: increased ideological distance should yield more circuit splits. The clear empirical prediction, however, prevents one from determining what type of learning is occurring. Both effects are in the same direction, so a positive effect could result from learning from previous decisions or from private information (Figure 7).

Figure 7. Expected Effect of Ideological Distance on the Probability of a Circuit Split

Bias-incompatible cases provide an opportunity to identify an independent effect of ideology on the probability of a disagreement resulting from courts learning from their peers. For these cases, the direction of the effect of ideological distance depends on what type of information the court is learning from. Focusing only on learning from peers, ideological distance has a positive effect on the probability of a circuit split. In contrast, ideological distance has a negative effect on the probability of a circuit split if we look only at a court’s private information. Taken together, this results in unclear expectations on the effect of ideological distance on the probability of a circuit split; however, a positive effect can only occur if courts are learning from each other’s decisions.

Proposition 5. For bias-incompatible decisions, a positive effect of ideological distance on the probability of a circuit split is evidence of courts learning from each other’s decisions.

Prevalence of Circuit Splits

Unlike previous work, I capture both circuit splits and legal uniformity among the courts. How common were circuit splits? Of the 239 legal questions, there were 83 circuit splits (about 35% of questions). Despite ideological variation across panels and circuits, there is substantial uniformity in the Courts of Appeals. On almost two-thirds of issues, circuits were in agreement with one another. For more recent cases, the circuits still have an opportunity to split on a question of first impression. As I include legal issues that started as late as 2021, a split may not yet have occurred. Subsetting to issues, where the last decision was in or before 2015 (171 issues) yields a split in about 35% of questions, the same as all issues.

Circuit splits occur in a minority of cases of first impression. While the presence of a split distinguishes these cases, so do other factors of the case. Figure 9 (panel a) examines the number of circuits that weighed in on a legal question. Legal questions that divided the circuits were heard by more courts on average than questions, where the circuits agreed. The median number of courts that ruled on a decision that divided the circuits is six; whereas, the median number of courts that rule on a question where the courts agreed is five.

Figure 9. Number of Decisions on Issues of First Impression

Note: Panel a compares splits and cases with uniformity. Panel b compares all splits and cases where the split occurred in the second period.

This relationship is not caused by the weakly increasing probability of a circuit split in the number of courts that decide an issue. Subsetting to cases, where the split occurred in the second period (the earliest it can happen), the relationship persists. Figure 9 (panel b) compares these splits to all splits.

My data allow me to calculate how likely a circuit split is, conditional on the number of courts that decide an issue. Figure 10 plots the circuit split hazard rate. A circuit split is most likely to happen immediately. In the second period, a split happens in around one out of every five issues—by far the most likely time for a split to occur. After this point, the probability of a split appears to weakly decrease. If an issue makes it to a seventh court, the probability of a split drops below 5%. After period 9, the observed hazard rate drops to zero.Footnote ²²

Figure 10. Split Hazard Rate

Note: Hazard rate is calculated as the proportion of previously uniform cases in period $ t-1 $ that result in a split in period t.

Testing Predictions

The model generates predictions on when legal uniformity is more likely to occur. While the model only focuses on the first two courts that decide an issue, the main theoretical predictions apply to multiple courts as well. When multiple courts have already decided an issue, the next court in the sequence is able to potentially learn from all of the previous decisions. What it learns from these decisions, however, is governed by the same structure as the two-court model. The current court learns previous courts’ signals from informative decisions, and decisions by more ideologically distant courts are less influential. Similarly, the relationship between two courts’ private information and ideological distance does not depend on the other courts that decide an issue. Consequently, it is unaffected by the inclusion of more decisions.Footnote ²³

To capture an environment with multiple actors deciding the same issue, I rely on an empirical approach commonly used to study policy diffusion (Volden Reference Volden2006). Studies of policy diffusion seek to capture how the actions of multiple other actors can influence current decision-makers. This approach corresponds well to the process of legal development and has been used to study the transmission of precedent across state (Hinkle and Nelson Reference Hinkle and Nelson2016). Within each issue, I create a dyad between each pair of courts that decided the question. For example, if three courts decide an issue, there are three dyads: $ ({C}_1,{C}_2),({C}_2,{C}_3) $ , and $ ({C}_1,{C}_3) $ . I create a Disagreement variable that is equal to $ 1 $ if the two courts chose different dispositions and $ 0 $ otherwise. To measure ideological distance, I utilize Judicial Common Spaces Scores (Boyd Reference Boyd2015; Epstein et al. Reference Epstein, Martin, Segal and Westerland2007; Giles, Hettinger, and Peppers Reference Giles, Hettinger and Peppers2001).Footnote ²⁴ Divergence is measured as the distance between the panel medians.Footnote ²⁵

I begin by examining the relationship between ideological distance and disagreement, pooling together bias-compatible and bias-incompatible cases. Columns 1 and 2 in Table 1 present results from this simple regression. In both models, the effect of ideological distance on disagreement is not significant and close to zero. An interpretation of these results that stopped there would suggest that ideological distance does not contribute to the occurrence of circuit splits.

Table 1. OLS Estimates of the Effect of Ideological Distance on Disagreement between Circuits

Note: Robust standard errors are clustered at the issue of first impression. * $ p<0.05 $ .

The formal model, however, suggests that this apparent null result is masking important heterogeneity in the relationship between ideological distance and agreement. To capture this effect, I include a variable for whether the first court’s decision is bias-compatible. I code the variable such that it is equal to $ 1 $ if the previous court is more liberal than the second and chose the liberal disposition or if they are more conservative than the second and chose the conservative disposition, and $ 0 $ otherwise. I interact this variable with the ideological distance between the two courts. I estimate models including the aforementioned variables and fixed effects for each issue of first impression.

Columns 3 and 4 in Table 1 display results from this analysis.Footnote ²⁶ I begin with the direct effect of a case being bias-compatible. When ideological distance is equal to $ 0 $ , there is no difference in the probability of a circuit split between bias-compatible and bias-incompatible cases. The coefficient for bias-compatible is insignificant and near zero. Turning next to ideological distance, we need to distinguish between bias-compatible and bias-incompatible orderings. The main effect of ideological distance captures the effect of distance on the probability of a split for bias-incompatible cases. When a previous decision is bias-incompatible, increasing the distance between the two courts reduces the probability of a circuit split. As the two courts move further apart, the current court is more likely to choose the same disposition as the previous court. When the previous court is more liberal and chose the conservative disposition, or vice versa, ideological distance increases agreement among the circuits.

The relationship between ideological distance and the probability of a split is opposite for bias-compatible cases. The effect of distance on disagreement among bias-compatible cases is captured by the sum of the coefficients for ideological distance and the interaction between distance and bias-compatible. Consistent with the predictions from the formal model, I find that, for bias-compatible decisions, as ideological distance increases, the current court is less likely to choose the same disposition as the previous court. In other words, a circuit split is more likely to occur when the previous court is ideologically distant and the first court renders a bias-compatible decision.

The results from the analysis suggest the importance of distinguishing between the two types of cases. Looking only at the effect of ideological distance on agreement suggests that ideology has no effect. This apparent null effect is instead the result of significant but opposite effects of ideological distance based on the ideological ordering of the courts.

Figure 11 (panel a) plots the predicted probability of a circuit split as ideological distance increases, distinguishing between bias-compatible and bias-incompatible cases. Figure 11 (panel b) plots the difference in probability of a circuit split between bias-compatible and bias-incompatible cases along with the 95% confidence intervals of this difference. When the courts have the same cut point, there is no significant difference in the probability of a circuit split between bias-compatible and bias-incompatible cases. As ideological distance increases, however, different patterns emerge. The probability of a split decreases for bias-incompatible cases, while the probability increases for bias-compatible cases. What is the substantive significance of this difference? The mean liberal panel $ (y<0) $ and the mean conservative panel $ (y>0) $ have an ideological distance of $ 0.64 $ . This results in a $ 10\% $ difference in the probability of a circuit split depending on whether the first court’s decision was bias-compatible or bias-incompatible. As the average probability of a split is around only $ 35\% $ , this difference is notable.

Figure 11. Effect of Ideological Distance on the Probability of a Circuit Split

Note: Panel a displays the predicted probability of a circuit split as ideological distance increases. The darker, purple line is the probability for bias-incompatible cases while the lighter, orange line is the effect for bias-compatible cases using estimates from model 4. Panel b plots the difference in predicted probability between bias-compatible and bias-incompatible cases. Issue of first impression fixed effects is included. Shaded regions show $ 95\% $ confidence intervals. Robust standard errors clustered at the issue of first impression.

How do the above findings relate to previous work on judicial decision making? Ubiquitous in the judicial auditing literature is the idea that supervising courts rarely review decisions that run counter to a court’s known bias. A liberal decision made by a more conservative lower court or a conservative decision made by a more liberal lower court is unlikely to be reviewed (Beim, Hirsch, and Kastellec Reference Beim, Hirsch and Kastellec2014; Cameron, Segal, and Songer Reference Cameron, Segal and Songer2000). The results from the model and subsequent analysis demonstrate that a similar principle exists with respect to the probability of a circuit split. A conservative decision made by a more liberal court indicates that conflict is unlikely. These results can be characterized in terms of liberal and conservative dispositions. If the first court chose a liberal disposition, a circuit split is less likely as the relative conservativeness of the first court increases. If the first court chose a conservative disposition, a circuit split is more likely as the relative conservativeness of the first court increases.

Distinguishing Mechanisms

The above findings provide evidence that ideological distance between courts affects the probability of agreement. Importantly, the effect of ideological distance for bias-compatible matches the prediction of the model. Increasing the distance between two courts increases the probability that a circuit split occurs. For bias-incompatible cases, results are consistent with the model, as the model allows for a positive or a negative effect; however, a negative effect does not allow us to say anything about whether the findings result from courts learning from one another.

If courts were making decisions independently, we should observe no relationship between when in the sequence of decisions a court acts and either the probability of subsequent agreement (Proposition 6) or the probability that a split occurs in that period (Proposition 7). First, I test whether sequence order is related to the probability of subsequent agreement. Figure 12 captures this relationship. In contrast to the flat relationship that a model with independent decision making predicts, we observe a positive relationship between the period in which a court makes a decision and subsequent agreement with that decision.

Figure 12. Relationship between the Probability of Subsequent Agreement and the Period in Which a Court Acted

Note: The dashed line is the empirical average probability of agreement between two courts. Dots indicate average probabilities of agreement with a court deciding in period t. The solid line is a loess estimate with shaded region depicting $ 95\% $ confidence interval.

A similar result arises with respect to when the minority decision in a circuit split is made. If the circuits were making decisions independently, we would expect the distribution of when minority decisions are chosen to look identical to the distribution of cases overall.Footnote ²⁷ Instead, Figure 13 demonstrates that if all but one court chooses the same disposition, the minority decision is more likely to occur in earlier periods. Both of these findings provide evidence that judicial behavior is not consistent with courts making actions independently based solely on their private information.Footnote ²⁸

Figure 13. Period in Which the Minority Decision Occurred

Note: Solid, gray bars plot proportions of period where a court chose $ lib\hskip0.3em (con) $ conditional on all other courts choosing $ con\hskip0.3em (lib) $ . Black, striped bars plot the proportions of all decisions occurring in the period. Figure subsets to issues involving at least three decisions in order to code a minority decision.