A Dynamic Model of Doctrinal Choice

Abstract

1. INTRODUCTION

The US federal court system divides functions among courts at different levels of the judicial hierarchy. The Supreme Court articulates legal doctrine that will guide decisionmaking by the lower federal courts. The lower courts are allocated the task of law application—they implement the guidance given by the Supreme Court in the form of legal doctrine by applying it to the myriad factual situations presented in actual cases. As a result, while the Supreme Court has the power to articulate broad principles, it must rely on the lower federal courts to determine outcomes across the mass of cases in accordance with its directions.

When deciding cases, the Supreme Court not only decides the outcome—which party wins—it also must choose how to articulate the doctrine that will govern similar, subsequent cases. That decision has been variously characterized as a choice between vagueness and specificity, narrowness and breadth, or, most famously, standards and rules. That choice in turn determines how much discretion lower courts will have when applying the precedent, thereby influencing the outcomes they reach.

Although the choice between rules and standards has been thoroughly explored by jurisprudence and law and economics scholars (Kennedy 1976; Diver 1983; Schlag 1985; Sullivan 1992; Kaplow 1992; Sunstein 1995; Posner 2010, 747–752), little attention has been paid to how doctrine operates in the context of the judicial hierarchy. This article formally studies the choice of doctrinal form using a dynamic model of upper and lower court interactions. The aim is to understand how the Supreme Court's efforts to influence case outcomes in the lower courts will affect their choice of doctrinal form. In addition to considering the factors that influence an initial choice on the spectrum between rules and standards, we also explore the dynamics of changes in doctrinal form. Why might the Court shift from a rule-like command to a more open-ended standard; or transform a standard into a more rigid rule-like command? Although our model is motivated by the Supreme Court's interactions with lower courts,² it is applicable whenever the functions of law declaration and law application are separated.

The conventional framework for the discussion of rules and standards is well established. Rules tend to be hard edged and determinate; they constrain subsequent decisionmakers and limit in advance the relevant factors for future decisions. Standards are softer and more open ended; they afford greater discretion to later courts to determine what is relevant to the decision in a case.

For law and economics scholars, the impact of the legal form on the behavior of private parties has been the foremost concern. Thus, Kaplow (1992) suggests that rules provide low-cost information to private actors, while Cooter & Ulen (1988) and Fischel (1985) note that rules facilitate private ordering: the clearer the entitlement, the easier it is to trade (cf. Johnston 1995). Others have argued that a bad actor can more easily evade rules, suggesting that standards may be preferable to ensure compliance with public norms (Kennedy 1976, 1773; Rose 1988, 600).

Although theoretical arguments might favor one form or the other, in practice, legal doctrine often shifts in form from a rule to a standard, or standard to a rule, sometimes cycling back again to an earlier form (Schauer 2003, 2005; Posner 2010, 749; Holmes 1881, 110).

In Miranda v. Arizona,³ for example, the Supreme Court replaced a “totality of the circumstances” test for the voluntariness of a confession with the hard-edged requirement that an explicit warning about the right to remain silent must be given in order for statements by a suspect to be admissible. Conversely, in Planned Parenthood v. Casey,⁴ the Court replaced Roe v. Wade's⁵ trimester approach, which set sharp boundaries for determining when a women has a right to terminate a pregnancy, with an undue burden standard that permits consideration of a wide variety of factors. Johnston (1995) and Rose (1988) document similar shifts between rules and standards in private law doctrines.

Some explain these shifts as the result of changing court personnel and a resulting shift in policy preferences (Sullivan 1992). Others argue that the disadvantages of rules in shaping private behavior become apparent when a rule is used, pushing courts to switch to standards. Likewise, the disadvantages of standards become apparent when they are used, pushing courts to switch to rules (Rose 1988; Johnston 1991).

Up to now, little attention has been paid to how interactions between upper and lower courts influence doctrinal form. This neglect is surprising in some ways, because lower courts play a crucial role in implementing whatever rules or standards are laid down by the Supreme Court. For the vast majority of litigants, it is the decisions of the lower courts—how they apply established doctrines—that give meaning and force to the pronouncements of the Supreme Court. Nevertheless, the rules and standards literature pays little attention to the vertical relationship between courts. At the same time, judicial politics scholars, who do pay attention to institutional structure and inter-court dynamics, traditionally ignored the function and form of doctrine. For a long time, the only concern was judicial votes: doctrine and legal reasoning were viewed as merely cover for judges' policy preferences, the true drivers of decisionmaking.

As a result of this gap, only a handful of prior studies have explored formally the role of legal doctrine in the judicial hierarchy.⁶McNollgast (1995) argue that doctrine serves as a form of communication, informing lower courts of the range of acceptable outcomes. Their model assumes that lower court judges are politically motivated actors, seeking to achieve their preferred outcomes through their decisionmaking. They do not seek to explain the particular form that doctrine takes, but rather describe doctrine as a signal by the Supreme Court to lower courts, informing them how to avoid reversal. Likewise, Bueno de Mesquita & Stephenson (2002) depict doctrine as a form of inter-court communication, emphasizing tradeoffs in achieving accurate communication when appellate judges choose between following or breaking with precedent. Trial court judges, in their model, are not strategic actors, but are assumed to seek to follow appellate court doctrine faithfully.

Cross et al. (2012), Lax (2012, forthcoming), and Jacobi and Tiller (2007) focus more directly on the choice of doctrinal form, albeit in one period models. These scholars posit that lower courts largely follow doctrine, but that they nevertheless may be able to exercise discretion depending upon the doctrinal form. Because lower courts may have differing policy preferences, higher courts choose a doctrinal form to optimally control lower court decisionmaking. The doctrinal form decision is characterized as a dichotomous one: the higher court must choose between a fully determinate rule and an open-ended standard. In these models, factors such as the characteristics of the judges, ideological conflict, and the mix of cases presented drive doctrinal choice.

Consistent with the case space model,⁷ we view legal rules as guiding which cases (bundles of facts) should result in what outcomes, and follow the common assumption that judges have preferences that they seek to advance through their decisionmaking.⁸ We also utilize the conventions of the rules and standards literature: what differentiates a rule from a standard is the ease with which new factors can be incorporated in future decisions (Kaplow 1992; Sunstein 1995). At the same time, we seek to explain the dynamic evolution of doctrine observed in practice. Although changes in judicial preferences may produce shifts in doctrinal form,⁹ we develop an endogenous account, focusing on how repeated interactions between an upper and lower court might drive doctrinal change. Note that judicial preferences remain important in our model—specifically, the degree of preference divergence between upper and lower courts influences doctrinal form. We do not, however, rely on a simple ideological shift—e.g. a more conservative Supreme Court—to explain changing doctrinal form.

Our model thus entails a number of innovations over prior work. First, we abandon the polarized views that characterize lower court judges either as unconcerned with legal principles, or as faithfully following Supreme Court doctrine without any regard for their own preferences. The model assumes that lower court judges have preferences over case outcomes that vary from Supreme Court preferences. At the same time, the form of the doctrine the Supreme Court articulates plays a critical role in their decisionmaking because doctrinal form makes deviations from Supreme Court precedent by the lower court more or less costly. Simply put, it is costlier for a lower court to distinguish precedent when phrased as a rule than when phrased as a standard.¹⁰

Second, the Supreme Court does not face a dichotomous choice between a rule or a standard, but rather selects from a continuum of doctrinal possibilities that are more or less determinate, more or less constraining in subsequent cases. So viewed, the Supreme Court faces a tradeoff. More standard-like doctrine will make it easier for a lower court to take into account unanticipated, but relevant information, thereby avoiding problems of over- and under-inclusiveness. However, less determinate doctrine raises the risk that the lower court will use its discretion to take account of new factors the Supreme Court considers irrelevant.

Third, the model does not rely on fear of reversal to explain lower court behavior (e.g. Songer et al. 1994; McNollgast 1995). Accounts that rely principally on the Supreme Court's reversal power present a puzzle: why do lower court judges, who are assumed to be motivated by political, not legal factors, largely appear to follow precedent, even when the risk of reversal by the Supreme Court in any particular case is vanishingly small? A number of mechanisms, such as Supreme Court “auditing” or litigant “signaling” have been suggested to explain the phenomenon (e.g. Cameron et al. 2000; Songer et al. 1994). The fact remains, however, that “reversal is a particularly unimpressive sanction … where the likelihood of reversal by the Supreme Court in any individual case is so small as to render it essentially meaningless as a sanction” (Haire et al. 2003, 146).

In our framework, the Supreme Court exercises power by establishing the doctrine that frames the lower court's subsequent choices. Changes in the doctrine impact the cost structure shaping lower courts' decisionmaking by altering the amount of discretion they have. Of course, changing doctrine will often entail reversal of a lower court's decision, but it is not necessary to our model that lower court judges particularly fear reversal; it is enough that they wish to avoid changes in doctrine that will constrain their discretion in future cases.

We use a repeated game model to capture the dynamic interaction between upper and lower courts, moving beyond the static games in the literature.¹¹ The repeated game model predicts that doctrine will evolve endogenously as the Supreme Court learns that its prior doctrine is not producing the “correct” results in enough cases.

More concretely, in an equilibrium of the repeated game, when the Supreme Court issues a standard, the lower court attempts to cooperate by only incorporating new information when the Supreme Court would want the information considered. Unfortunately, the lower court is occasionally mistaken about the Supreme Court's preferences. The Supreme Court responds to mistaken applications of precedent by shifting the doctrine to more rule-like commands. Such commands cabin or constrain the lower court's discretion. Too much constraint, however, is costly to the Supreme Court, because in some cases new information that it would find relevant is excluded from consideration. Hence the possibility of cycling back to a standard.

The equilibrium is inspired by the work on repeated games with imperfect public information, first articulated by Green & Porter (1984) and subsequently developed by Abreu, Pearce, & Stacchetti (1990) and Fudenberg, Levine, & Maskin (1994). As in this earlier work, a public signal—the Supreme Court's loss from the outcomes of cases in the lower court—is imperfectly correlated with the lower court's behavior because those outcomes could be the result of the the lower court either (1) not cooperating or (2) making a mistake in assessing what the Supreme Court wanted.

The model also relates to the economics literature on rule development. For example, Chassang (2010) considers how two players learn to cooperate when they do not share knowledge of which actions are productive. Ellison & Holden (2008) model how rules are chosen and revised within a firm given constraints on the principal's ability to communicate precisely with the agent in the face of uncertainty about future states of the world. We examine similar questions in the particular context of the judicial hierarchy, but focus on the type of instructions (doctrinal form) apart from its substantive content.

The article proceeds as follows. Section 2 informally develops the intuitions that underlie our model. Section 3 sets out the one-period model. It shows how the Supreme Court's choice of doctrinal form responds to: (i) the weight the Supreme Court places on avoiding considerations of factors it deems irrelevant; (ii) the weight the Supreme Court places on a failure of the law to properly reflect changing circumstances; and (iii) the degree of preference divergence between the courts. This one-period model formally captures many of the intuitions of the rules and standards literature. Section 4 presents the repeated game model and derives the main result: the endogenous ocsillation between rules and standards over time. Section 5 discusses actual doctrinal evolution consistent with the model and considers the model's implications and limitations. Section 6 concludes. Proofs of all the propositions are in Appendix A.

2. INFORMAL DESCRIPTION OF DOCTRINAL CHOICE

Rules and standards do not exist in their pure form. Any given rule, no matter how absolute it appears, is inevitably ambiguous around the edges (Hart 1961; Schauer 2005). A subsequent court may avoid the rule by declaring that the rule does not apply to the new situation, or that the pending case falls within an exception. Conversely, when applying a standard, courts do not have unlimited freedom to decide a case. At a minimum, the facts and outcome of the case in which a standard is articulated constrain how it is applied in the future. If, for example, the Supreme Court determines that a constitutional violation exists after balancing a particular set of facts, a later court has little discretion to reach a different outcome in a subsequent case presenting identical facts, even when the precedent is phrased as an open-ended standard. Rules and standards, then, do not represent a rigid dichotomy, but rather the endpoints on a continuum (Sullivan 1992).

Where the Court's directive falls on the continuum between rules and standards matters because lower court judges care about following legal norms. Kim (2007). The proposition that judges follow legal precedent established by a higher court, as well as pursuing their preferred policy, is increasingly accepted as a theoretical matter (Cross et al. 2012; Lax 2012. forthcoming; Friedman 2006), and consistent with empirical evidence (e.g. Gruhl 1980; Songer & Sheehan 1990; Westerland et al. 2010; Benesh & Reddick 2002).

Because lower court judges follow legal precedent, the choice of doctrinal form affects how costly it is to incorporate new information.¹² Rules specify in advance the factors relevant to determining the outcome of a case (Shavell 2007; Sunstein 1995; Kaplow 1992), thereby narrowing the range of permissible considerations and constraining the exercise of discretion (Sullivan (1992, 64)). A lower court judge who wishes to take account of a new factor will have to expend effort devising a plausible justification for avoiding the rule or creating an exception. In addition, by doing so a judge may violate her internal norms of role performance and risk incurring reputational costs in the form of professional or even public criticism, or undermining the legitimacy of the judiciary as a whole.

Standards, in contrast, allow the decision-maker to take account of all relevant circumstances, even those not identified ex ante, affording greater freedom in determining the outcome ex post (Sullivan 1992, 59; Kaplow 1992; Sunstein 1995). The judge who considers additional factors when applying a standard is unlikely to be viewed as violating legal norms and, therefore, faces much lower costs of doing so.¹³

From the Supreme Court's perspective, the more rule-like the doctrine, the more likely it is that the lower courts will follow the directive, considering only the facts identified by the Supreme Court as relevant. However, the Supreme Court has limited knowledge about the great variety of factual situations in which the rule might become applicable and knows that it cannot anticipate all of these circumstances. In some future cases, a new factor may arise that the Court would prefer to be taken into account. If the Court has articulated a rule-like command, the cost of incorporating new information is greater, and therefore the lower court is less likely to take it into account. The result is the oft-noted problem of the under- and over-inclusiveness of rules. On the other hand, if the Supreme Court issues a more standard-like command, the lower court will have greater leeway to incorporate not only new information that the Supreme Court would agree is relevant, but also factors that the Court would not want considered.

Because the Supreme Court cannot know in advance what new factors will arise in subsequent cases, it faces a tradeoff in choosing between rules and standards.¹⁴ A more rule-like command prevents the incorporation of new information by the lower court when the Supreme Court would want that information to affect the outcome. But rules have an upside too: they prevent the incorporation of information in those cases where the Supreme Court would deem the new information irrelevant.

A concrete example illustrates these points. Suppose that the Supreme Court must decide under what conditions a confession obtained from a criminal suspect is involuntary, such that it would be inadmissible in a subsequent criminal trial. The Court has a range of options for articulating how the admissibility of confessions should be determined in future cases:

• Announce a “pure” standard: state that the admissibility of a confession depends upon its “voluntariness” in light of all the surrounding circumstances.

• Announce a balancing test: state that factors suggesting coercion (such as the length of the interrogation and the age of the suspect) should be balanced against other factors indicating voluntariness (such as the lack of physical coercion and the provision of information regarding constitutional rights). How these factors balance out will determine whether an incriminating statement should be admitted.

• Announce a presumption: state that if a confession was obtained while the suspect was in police custody, it is presumptively inadmissible unless the prosecution can establish circumstances showing that the statement was voluntarily given.

• Announce a “pure” rule: state that a confession obtained through interrogation while the suspect was in police custody is not admissible unless the police gave a specific warning informing the suspect of his constitutional rights before questioning began.

These options are points on a continuum. Depending upon how detailed and determinate the instructions given by the Supreme Court, the established doctrine could fall at one of these points or elsewhere along the spectrum between a rule and a standard.

Now suppose that the lower court must decide a subsequent case in light of the doctrine laid out by the Supreme Court, and that it includes a circumstance not considered in the precedential case. For example, suppose the precedential case involved interrogation of a suspect while in isolation at a police station. The new case involves a confession elicited by police questioning of a suspect by the side of a road after his vehicle was stopped for a traffic violation. The lower court believes that the new circumstance—the fact that the questioning occurred in a public place—should be taken into account in deciding the voluntariness of the confession.

The lower court's willingness to do so, however, will depend upon the form of the legal doctrine set out in the relevant precedent. More specifically, if the Supreme Court has articulated an open-ended standard, the lower court can take into account the new setting quite easily, at very low cost. If, on the other hand, the Supreme Court has articulated a rule, the lower court will feel more constrained. It may still take the new setting into account in reaching its conclusion, but will find it more costly to do so. Avoiding the outcome suggested by the precedential case will require it to make a reasoned argument that questioning a suspect by the side of a public road is not “police custody,” that the concerns regarding coercion that motivated the rule are not present in that context and that, therefore, no explicit warning is required to render the confession voluntary.

Of course, the Supreme Court did not consider the admissibility of confessions obtained outside the police station in the earlier case. Once the new issue is raised, the Court may agree with the lower court that the public setting should make a difference in the outcome. Knowing that such situations may arise, the Court may want to insure that such an unanticipated, relevant factor is taken into account by formulating the doctrine as a standard. Yet, by doing so, the Court also makes it easier for the lower court to take into account another unanticipated factor, one that it would regard as irrelevant.

For example, a lower court may wish to admit statements made by a suspect while in custody, but not in response to specific questioning by the police. The Supreme Court, however, may view the fact that no specific questions were posed as irrelevant to determining the voluntariness of the confession. If the relevant doctrine is framed as a standard, the lower court will face little cost in incorporating this new factor to avoid the requirement of a prior warning.

Thus, the Supreme Court's choice of doctrine—how rule-like to make the legal directive—depends upon how it trades off losses when the lower court fails to incorporate relevant information against losses when the lower court incorporates irrelevant information (from the Supreme Court's perspective).

Having this informal description in hand, we now build the one-period model. In the formal model that follows, we necessarily make several simplifying assumptions. First we put aside the complications of collegial decisionmaking and treat the Supreme Court as a unitary actor. In our model, the Justices do not have independent preferences over doctrinal form; rather, they choose doctrine to produce the most “correct” outcomes in the future. We assume that any intra-court debate over the best outcomes can be captured with a single parameter—e.g. the preference of the median Justice—as is commonly assumed in studies of the judicial hierarchy (e.g. McNollgast 1995; Westerland et al. 2010; Bueno de Mesquita 2002).

Similarly, in order to explore the possibility of strategic interactions over time, we simplify the multi-tiered structure of the judicial hierarchy by focusing on a generic lower court tasked with applying Supreme Court precedent. For ease of exposition, we build the model assuming a single lower court, and consider later the complications created by multiple lower courts. Following the description of the one-period model, we turn to the repeated model.

3. DOCTRINAL CHOICE: THE ONE-PERIOD MODEL

When deciding a precedential case, the Supreme Court does two things—it determines the outcome (which party prevails) and it chooses the form of the doctrine. Focusing on the second aspect, the Court's choice can be formalized as the selection of a doctrinal form, α, between 0 and 1, which indicates the looseness of the legal directive. If α = 0, the Supreme Court articulates a pure rule. If α = 1 , the Supreme Court sets a pure standard. As noted above, doctrinal form exists on a continuum, and so the Court may select α between 0 and 1.

After the Supreme Court announces the doctrine, subsequent cases arise that involve a new, unanticipated factor that the lower court believes should alter the outcome. The new unanticipated factor will fall into one of two categories. With probability (1 − p), the Supreme Court agrees that the new factor is relevant to the decision and should change the outcome. Stated differently, both courts would agree that excluding the new information by strict application of the precedent would produce an incorrect outcome due to the precedent's over- or under-inclusiveness. With probability p, the courts disagree: the Supreme Court believes the new factor is not relevant, whereas the lower court believes it should be incorporated in order to reach the correct outcome.¹⁵

These probabilities thus capture the extent of the preference conflict between the courts. With probability p , the courts have divergent preferences. With probability 1 − p, the courts have shared preferences with regard to the new factor.

3.1. The Lower Court

When the lower court encounters cases involving a new factor it deems relevant and not anticipated by existing precedent, it must decide whether to reach a different outcome in those cases than called for by strict application of the precedent. The greater the proportion of cases in which it takes account of the new information and reaches a different outcome, the greater the benefit to the lower court. At the same time, the lower court's cost of incorporating new information will depend on the doctrinal form. More standard-like precedent makes it easier to incorporate new information; more rule-like directives make it more costly to do so.

In addition, the lower court could spend effort deciphering what the Supreme Court itself would prefer done, if it were aware of the new factor. Notably, even if the lower court tries to do this, it might make a mistake, erroneously believing that the Supreme Court would consider the new factor relevant when it does not.

In the one-period model, the lower court's effort choice is simple. Because we assume that the Supreme Court cannot respond to the lower court's decision by reversing the case or, since it is a one period model, revising the precedent, the lower court has no incentive to attempt to discern the Supreme Court's preferences. Instead, the extent to which it incorporates new information will turn solely on the doctrinal form chosen by the Supreme Court. As we will explore in the next section, in the repeated game the Supreme Court can respond to the lower court's choices, and so the lower court's decision to expend effort or not, and the possibility of mistake, become important.

Because we focus on the situation where the lower court believes the new factor is relevant, in the one-period model it decides the proportion of the cases (r) in which the new factor affects the outcome. It makes this choice to maximize its payoff, defined as

The benefit to the lower court increases as the proportion of cases in which the new factor is incorporated increases (B_r > 0), but at a decreasing rate (B_rr < 0). The cost increases as the lower court incorporates information in more cases (C_r > 0) and does so at an increasing rate (C_rr > 0).¹⁶ In addition, as the looseness of the legal command increases, the marginal cost of incorporating new information decreases (C_rα < 0). In the event the Supreme Court issues a pure standard, there is no cost to incorporating new information (C(r ; 1) = 0)) and the lower court will take the new factor into account in every case, even where the factor is not highly compelling.

At the interior solution, maximization by the lower court results in the following first order condition

Solving yields an optimal fraction of cases (r*(α)) in which the lower court will consider the new factor. Focusing on the interior solution, 0 < r*(α) < 1, an intuitive comparative static can be derived, namely

In words, as the legal command becomes looser—less rule-like—the lower court incorporates the new factor in a larger proportion of the cases.

3.2. The Supreme Court

In deciding the form of the legal command, the Supreme Court anticipates the reaction of the lower court. When deciding subsequent cases, the lower court could deviate from the Supreme Court's preferred outcome in two directions. First, the lower court and Supreme Court (if it had considered the issue) might agree that an unanticipated new factor is relevant (a situation that arises with probability 1 − p). Yet, despite this agreement the lower court fails to allow the new factor to play a role in the resolution of some proportion of the cases. The lower court does so because the form of the legal directive constrains its decision. In other words, strict application of precedent is over- or under-inclusive. Denote the weight on this type of loss by L.

On the other hand, the lower court might have a differing view of the relevance of the new factor (which occurs with probability p) and might incorporate in some cases when the Supreme Court would deem it irrelevant. Taking the new factor into account makes the law less predictable without improving the outcome, leading to the wrong result in the eyes of the Supreme Court. Denote the weight on this type of loss by E.

The Supreme Court understands that the proportion of cases where the presence of the new factor will alter the outcome reached by the lower court depends on the doctrinal form. Given this information, the Supreme Court sets α to maximize its expected payoff. Formally, the Supreme Court maximizes

The terms in the payoff function are squared because the Supreme Court's loss from either type of error increases at an increasing rate. In other words, the first few cases where the lower court's decision diverges from the Supreme Court's preferences are less costly to the Court than subsequent cases. This functional form can be justified by assuming that the Supreme Court's payoff decreases more sharply the more the overall pattern of outcomes differs from what it believes to be correct.

Given this payoff function, the Supreme Court faces a tradeoff in selecting doctrinal form. If it makes the command more standard-like (α → 1), the lower court responds by incorporating new information in a larger proportion of cases. If, ex post, the Supreme Court does not believe the new factor should be relevant, it suffers a loss from the undesirable incorporation in these cases.

On the other hand, if the Supreme Court makes the legal command more rule-like (α → 0), the doctrine constrains, meaning the lower court incorporates new information in a smaller fraction of cases. As a result, the lower court declines to incorporate new information even when, ex post, the Supreme Court thinks the factor should be relevant to the outcome. Accordingly, a rule-like doctrine can also lead to losses for the Supreme Court.

In the one-period problem, the Supreme Court picks the form of the doctrine to balance the two competing effects. The first proposition establishes the relationship between the optimal form of doctrine (α*) and the parameters of the model.  

Proposition (1) captures the core insights of the rules and standards literature. In areas of law where lower courts have preferences that diverge widely from the Supreme Court's (i.e. p is large), the Court will likely resort to more rule-like commands. In areas of law where the situation on the ground is rapidly changing, unanticipated information is more likely to be relevant, and therefore the weight placed on losses from an over- or under-inclusive doctrine (L) is likely to be large. As a result, more standard-like commands should be attractive to the Supreme Court. And in areas where the Supreme Court considers predictability to be particularly important, it will view the incorporation of a new, but irrelevant factor as particularly costly (i.e. E is large). Thus, it will resort to more rule-like doctrines.

What remains unexamined is doctrinal evolution: movements between rule-like and standard-like commands over time. Legal doctrine often displays these sorts of movements. These changes could be explained by external shocks, such as a change in the composition and hence, the preferences, of the Supreme Court. We are interested, however, in how such movements might arise endogenously, which requires a dynamic model. Before considering the repeated game, we first consider the payoffs in the one-period model and the possibility for inter-court cooperation.

3.3. Payoffs in the One-Period Model and The Possibility of Cooperation

Given the optimal choice of doctrinal form, the Supreme Court's expected payoff in the one-period model is

The lower court's expected payoff is

Now imagine the courts anticipate the following “deal”. The Supreme Court issues a pure standard (α = 1), granting the lower court maximum discretion to decide whether or not to incorporate a new factor. The lower court agrees to use that discretion as the Supreme Court would prefer: to incorporate the new factor only in cases the Supreme Court would agree that the factor is relevant. The Supreme Court is better off under such an arrangement because it minimizes its losses due to the over- or under-inclusiveness of a rule, while also avoiding losses from the incorporation of irrelevant information. And because the lower court sometimes agrees with the Supreme Court that the new information is relevant, it may be better off cooperating, rather than having its decisions tightly constrained by doctrine.

In order to cooperate in this way, the lower court must discern the Supreme Court's preferences regarding the new factor. For example, it could examine the text of judicial opinions, including non-binding concurrences and dissents, or consider other writings by the Justices and attempt to effectuate the broad policy goals expressed by the Court.

Even if the lower court tries to cooperate, however, it might not accurately predict the Supreme Court's preferences. It is hardly a simple matter to discern the preferences of another actor regarding situations that actor has never previously encountered, and so the lower court faces a probability of β that it will make a mistake. In this model, a mistake occurs when the lower court believes the Supreme Court would consider the new factor relevant when, in fact, it does not.¹⁷

The lower court cooperates by incorporating the new factor in all cases where it perceives the Supreme Court would want the factor considered. The lower court perceives agreement when it correctly recognizes that the Supreme Court agrees and when it mistakenly believes so, even though the two courts actually disagree. The lower court's expected payoff from cooperation is

Given the risk of mistake (β) in circumstances in which the Supreme Court prefers no incorporation (p), the Supreme Court's expected payoff from this deal is

Of course, in the one-period model such a deal is impossible. If given doctrinal flexibility, the lower court will incorporate new information in all the cases it wants to, including ones where it perceives the Supreme Court will disagree. And knowing this, the Supreme Court would not grant complete discretion.

Repeated interactions, however, create the possibility that the payoffs from cooperation might be realized. In particular, assume that π_lc^coop > π_lc* and Π_sc^coop > Π_sc*: the courts prefer to cooperate if possible.

4. REPEATED GAME MODEL

Suppose now that the one-period game repeats an infinite number of times. When deciding a precedential case, the Supreme Court decides the outcome and selects the doctrinal form in which to express the legal principle. The lower court applies the precedent in a number of subsequent cases involving a new, unanticipated factor and decides whether to change the outcome based on the new factor. Both courts observe the outcomes across the group of cases and the payoff to the Supreme Court. The Supreme Court then decides whether to keep the form of the doctrine or to change it. In the next period, the lower court confronts another new factor and applies the established or revised precedent. Outcomes and payoffs are observed. The Supreme Court again decides whether to respond, and so on. The courts share a common discount factor δ.

Notably, in this model, the Supreme Court does not review and reverse the outcome in any individual case. Instead, its power resides in its ability to change the doctrinal form, making it more or less constraining going forward. Of course, in reality, the Court does so by hearing and deciding a specific case. It might, however, change the form of a doctrine—making it more rule-like or more standard-like—without necessarily reversing the decision below. Because we are primarily interested in the Court's power over doctrine rather than its reversal power, we ignore for now the process of selecting individual cases for review.

In one equilibrium of the repeated game, the Supreme Court issues a pure standard in the initial phase—the trust phase—anticipating the benefits of cooperation.¹⁸ As seen above, if the lower court incorporates in all cases where it perceives agreement, its expected payoff is

Alternatively, the lower court might act non-cooperatively. The lower court acts non-cooperatively when it flips the outcome in all cases where it believes the new factor is relevant to the outcome, no matter whether it believes the Supreme Court would agree or not. Non-cooperation does not necessarily mean that a lower court judge is acting in a deliberately ideological or defiant manner. It might also occur when a judge makes little effort to overcome quick, intuitive judgments (Guthrie et al. 2007). The lower court's expected payoff from acting non-cooperatively is

After the lower court chooses the proportion of cases to flip the outcome, the Supreme Court observes the case outcomes in that period. Both courts also learn whether the Supreme Court agreed or disagreed about the relevance of the factor; that is whether the Supreme Court suffered a loss from the lower court's resolution of the cases is a public signal. If the Supreme Court agrees with the lower court's conclusions about the new factor, it suffers no loss. If their preferences differ, but the lower court fully cooperated and made no mistakes, its decisions would again coincide with what the Supreme Court prefers. It might turn out, however, that the lower court has incorporated a new factor that the Supreme Court deems irrelevant.

At this point, the Supreme Court suffers loss because of the erroneously decided cases, but it cannot readily determine the cause. One possibility is that the lower court cooperated—that is, it tried to discern the Supreme Court's preferred outcome—but made a mistake. Another possibility is that the lower court did not even try to anticipate the Supreme Court's preferences, but simply incorporated the new factor in all the cases where it thought it relevant. Because the Supreme Court cannot directly observe the payoff to the lower court, it faces a non-trivial signal extraction problem—it cannot know whether the lower court failed to cooperate or made a mistake in assessing the Supreme Court's preferences.¹⁹

Unable to distinguish these two situations, the Supreme Court attempts to deter non-cooperation by issuing more constraining, rule-like doctrine following any loss. This doctrinal directive corresponds to the optimal legal form in the one-period equilibrium, α*. In this second phase, the constraint phase, the lower court chooses the proportion of cases incorporating new information based on α*, r(α*). The restrictive doctrinal form limits the Supreme Court's losses from non-cooperative behavior by the lower court. Yet, the restrictive doctrinal form also dampens lower court incorporation when the Supreme Court would agree the new factor is relevant. Thus, the Court suffers losses under the more constraining standard, and after a period of time, t, reverts back to a standard.

More formally, this interaction can be captured as follows: the Supreme Court enunciates a standard and the lower court chooses whether to incorporate or not when it perceives disagreement. Let V⁺ be the expected value of the stream of payoffs to the lower court when it cooperates (i.e. only incorporates in cases when it perceives inter-court agreement).

which reduces to

(1)

This equation captures the immediate payoff to the lower court from cooperating plus the probabilities that the trust phase will continue in the future or be terminated by the imposition of a constraint phase following the mistaken application of precedent.

In this equation, V⁻ is the expected value of the stream of payoffs to the lower court in the constraint phase, assuming that that phase lasts for t periods:

Plugging V⁻ into the definition of V⁺ yields

In the trust phase, the lower court might choose not to cooperate. If it does so, it gains the immediate benefit of non-cooperation in the initial period (π_lc^no-coop) and continued payoffs from cooperation (V ⁺) if the Supreme Court happens to agree with it (a situation that occurs with probability 1 − p). At the same time, the lower court risks a lower stream of payouts if the Supreme Court disagrees and imposes a constraining doctrine (a situation that occurs with probability p). In order for the strategies described above to constitute a sub-game perfect equilibrium, the lower court must be better off cooperating, even with the risk of mistake than not cooperating in the trust phase. This condition can be stated formally as

On the left hand side of the equation above, plug in for V⁺ from Equation (1). Then, after some manipulation, observe that it is in the lower court's interest to cooperate whenever

or

(2)

Equation (2) expresses the basic tradeoff in the repeated model for the lower court. Not cooperating in the trust phase gives the lower court a gain of B(1) if the courts disagree and the lower court would not have made a mistake in inferring the Supreme Court's intent. Cooperating, by contrast, means that the lower court preserves the boost in the future stream of payoffs associated with avoiding imposition of the constraint phase.²⁰ The inequality ensures that the one-time, short-term gain from not cooperating is not worth the long-term cost when the Supreme Court responds by imposing greater constraints.

Finally, in equilibrium, the lower court's discounted stream of payoffs in the repeated game is

Lemma (1) in Appendix A shows that this payoff is strictly decreasing in the length of the doctrinal constraint phase, t.

Turning to the Supreme Court, let W⁺ be its discounted stream of payoffs in the trust phase if the lower court cooperates and W ⁻ be its discounted stream in the constraint phase. We have

and

Solving for W⁺ results in

As with the lower court, the payoff to the Supreme Court is strictly decreasing in the number of periods of more constraining doctrine.²¹

The next proposition establishes an equilibrium of the repeated game.  

This proposition immediately leads to the question about the “optimal” length of time the Supreme Court should stick with more rule-like constraining doctrine.²² Since the payoffs to both courts decrease in ⁠, the constraint phase that maximizes W⁺ and V ⁺ is the shortest possible constraint phase where Equation (2) just holds.²³Equation (2) depends on the probability of preference difference between the courts, as stated in the next proposition.  

5. LEGAL DOCTRINE IN THE JUDICIAL HIERARCHY

5.1. An Example of Doctrinal Evolution

The model suggests that doctrine may sometimes oscillate between standards and more rule-like doctrine back to standards again. This type of oscillation in doctrinal form is illustrated by the Supreme Court's recent cases interpreting the Sixth Amendment's Confrontation Clause. These cases raise the question whether the testimony of a witness who is not available for cross-examination at trial may constitutionally be used by the prosecution in a criminal trial. In 1980, the Supreme Court in Ohio v. Roberts²⁴ summed up the general approach it had been taking in these cases: when a witness is not available for cross-examination at trial, his statement is admissible “only if it bears adequate ‘indicia of reliability’ ”.²⁵ This interpretation of the Confrontation Clause was clearly framed as a standard. The test afforded judges a great deal of discretion in determining the reliability of proffered testimony, and left them free to consider any of a number of factors that they might consider relevant to that inquiry.

In 2004, in Crawford v. Washington,²⁶ the Court repudiated its earlier approach and replaced it with something much closer to a rule. Specifically, the Court held that when it comes to “testimonial evidence,” prior statements cannot be admitted at a criminal trial unless the witness is unavailable and the defendant had a prior opportunity to cross-examine the witness. As the Court put it, “the only indicium of reliability sufficient to satisfy constitutional demands is … confrontation”.²⁷ Thus, at least as to testimonial evidence, the Court's new test removed the discretion of judges to inquire into the reliability of out-of-court statements.

The model suggests that the Supreme Court moved from the standard articulated in Roberts to the far more constraining Crawford rule, because it observed the lower courts applying the standard in ways it disagreed with, incorporating factors it believed irrelevant. Although the Supreme Court stated that it had “no doubt that the courts below were acting in utmost good faith” when assessing reliability, it explained that it was changing the doctrine because the prior standard left “too much discretion in judicial hands”.²⁸ In the view of the Court, the standard was “amorphous”, “unpredictable”, and “manipulable”,²⁹ allowing “countless factors” to bear on the question of whether a statement was reliable. The Court reviewed dozens of lower court decisions to demonstrate how different courts treated the same factor in opposite ways, or relied on erroneous factors in deciding whether a statement was reliable.³⁰

The rule enunciated in Crawford was clearly intended to constrain lower courts, restricting their discretion to admit prior testimony that had not been tested by cross-examination. However, the model suggests that imposition of a rule will be costly because it will constrain lower courts from considering new information, even when the Supreme Court would agree that it is relevant.

Thus, seven years after Crawford, the Supreme Court in Michigan v. Bryant ³¹ decided another Confrontation Clause case that significantly moved the relevant test back in the direction of a more open-ended standard. While nominally accepting the framework established in Crawford, the Court in Bryant greatly expanded the discretion of judges to determine whether the prior statement of an unavailable witness could be admitted at a criminal trial. It did so by creating an open-ended standard for determining whether a statement was “testimonial evidence” subject to the requirement of cross-examination or “non-testimonial” and therefore outside the concerns of the Confrontation Clause. According to the majority, this inquiry turned on the “primary purpose” for which a statement had been taken, which courts should determine by “look[ing] to all of the relevant circumstances”.³² The dissenters in Bryant criticized the majority for moving back toward “open-ended” and “amorphous” inquiries into the “totality of the circumstances bearing upon reliability”.³³ Rejecting their criticisms that the new standard was unworkable, the majority asserted that “we … are unwilling to sacrifice accuracy for simplicity”.³⁴

The recent evolution of doctrine in the Confrontation Clause cases thus appears to fit the model well. Of course, there were significant changes in the membership of the Court between 2004 and 2011, but changing preferences alone do not offer a persuasive explanation for the doctrinal shifts. The Crawford court's move to overturn Roberts, did not follow any significant ideological shift, but came after the Court's membership had been stable for ten years. On the other hand, although Bryant's move back toward a standard occurred after significant changes in the Court's membership, the Court's overall ideological composition did not shift dramatically.³⁵ In any case, with Scalia and Ginsberg in dissent, and Roberts and Alito joining Sotomayor's majority opinion, the votes in Bryant did not fall along traditional ideological lines. Interestingly, the two centrist judges—Kennedy and Breyer—joined the majority in both Crawford and Bryant. Thus, a traditional policy preference story cannot easily explain the shift from standard to rule back to standard in the Confrontation Clause cases.

5.2. Implications and Extensions

As discussed above, the one-period model formally captures many of the insights of the rules and standards literature—particularly, the tradeoffs entailed in the choice between rules and standards and the effect of preference differences between the courts. The repeated game model offers additional insights into how doctrinal choice is shaped by upper and lower court interactions. It suggests that when courts interact repeatedly over time, lower courts will be motivated to follow Supreme Court precedent in order to preserve their discretion in the future. Notice that the lower court complies in this model not out of fear of reversal, but rather because it values doctrinal flexibility—the ability to easily incorporate new information about unanticipated factors in future cases. The lower court sometimes makes mistakes, however, in applying the standard. Mistakes lower the payoff for the Supreme Court and induce it to move to a more constraining rule. Rules are costly as well, because of their over- and under-inclusiveness and so the Supreme Court may decide to give greater discretion again to the lower court.

Note that our model is not inconsistent with the explanation that changed judicial preferences may sometimes produce doctrinal shifts. But such exogenous preference changes are not necessary to explain doctrinal evolution—shifts between rules and standards can result endogenously from the interaction between upper and lower courts with unchanging preferences.

Judicial preferences do play an important role in our model, albeit a different one from most models of the judicial hierarchy. A greater divergence between the upper and lower court's views increases the risk of mistake and the incentive for non-cooperation—i.e. to fail to even attempt to figure out how the Supreme Court would want the doctrine applied to new circumstances. As a result, greater preference difference increases the length of time doctrinal constraints must be imposed following a mistake in order to provide greater deterrence.

Significantly, our model suggests that with repeated interactions, upper and lower courts will have an incentive to cooperate, even when their preferences diverge. Thus, standards, or at least more standard-like doctrine, may be observed even when the Supreme Court and lower court do not share values. Because it seeks to preserve its discretion in future cases, a lower court is motivated to try to discern what the Supreme Court would want done even when it has discretion (in the short run) to decide cases as it likes. Moreover, different legal issues may present different tradeoffs for the Supreme Court, depending upon the relative risks of loss from over- and under-inclusiveness versus lack of predictability in each issue area. Thus, the model suggests that the same Supreme Court facing the same lower court (i.e. given a constant p) might choose more rule-like commands in one substantive area and more standard-like doctrine in another.

The model developed here necessarily entailed some simplifying assumptions. For example, it focused on an equilibrium involving a pure standard and cooperation between the courts; however, other equilibria may be possible. The equilibrium discussed above entails a trust phase allowing the lower court complete discretion, alternating with a constraint phase that is sufficiently long to insure that the lower court has an incentive to try to discern the Supreme Court's preferences with respect to unanticipated factors when it is given discretion. Depending upon the parameters of the model, however, the required length of the constraint phase might be very long, with each extra period of constraining doctrine hurting the Supreme Court. As a result, the equilibrium studied might not be the one that maximizes the Supreme Court's long-run payoff.

One could imagine an equilibrium where the Supreme Court announces something more constraining than a pure standard in the trust phase. In such a situation, the lower court might still have considerable discretion to exercise; however, its payoff from not cooperating would fall because the more constraining doctrine itself would make incorporating new factors in a large number of cases more costly. Since the benefit from non-cooperation is smaller, the Supreme Court would need to impose constraining doctrine for fewer periods to maintain the lower court's incentives to cooperate. Thus, alternative equilibria might exist involving oscillation between standard-like (but somewhat constraining) doctrine, and more rule-like constraints, but for shorter periods of time.

The model also assumes that the Supreme Court interacts with a single lower court, although in fact there are multiple lower courts. In our model, the Supreme Court observes deviations from its preferred pattern of outcomes and can respond by imposing more constraining doctrine, thereby giving the lower court an incentive to cooperate. If there are multiple lower courts, the interaction becomes more complicated. Imagine, for example, that there are ten lower courts instead of one, all applying the Supreme Court's precedent to an array of factual situations. The Supreme Court begins by setting a pure standard, which the lower courts then apply. In the next period, the Court might observe one of the lower courts to have deviated from its preferred pattern of outcomes. It could choose to impose a more constraining doctrine to incentivize cooperation; however, in doing so, it would not only be “punishing” the non-compliant lower court, but also imposing constraints on the other nine, thereby incurring unncessary losses in future cases due to the rigidity of the rule.

In such a situation, the Supreme Court might decide to tolerate the deviation rather than imposing constraints on all ten lower courts by revising the doctrine. Knowing this, another lower court may be tempted not to cooperate. If enough of the lower courts fail to cooperate, however, the cost of allowing the lower courts to exercise broad discretion will outweigh the costs of constraining doctrine and the Supreme Court will once again find it worthwhile to impose a more rule-like doctrine. The strategic lower court, then, must anticipate what other lower courts will do and when its own non-cooperation will shift the costs for the Supreme Court enough that it will impose constraints on all of the lower courts. The existence of multiple lower courts does not necessarily eliminate the possibility of cooperation between upper and lower courts, although it significantly complicates formal modeling of their interaction.

Although we have focused here on the Supreme Court's power to set precedent, the model might usefully be extended by also taking into account the Court's reversal power. That is, the Court might be thought of as having two tools to influence outcomes in the lower courts—its power to set precedent that binds all lower courts and its power to correct error by reversing a particular decision. When the Court uses its reversal power solely for purposes of error correction, the form of the doctrine does not change significantly—the Court simply signals to one deviating lower court that it has erred in applying the doctrine.

Using the two tools of doctrinal choice and error correction in tandem might give the Supreme Court more leverage over lower court decision-making when confronted by multiple lower courts. So, for example, if a single lower court deviates from the Supreme Court's preferences, the possibility of an error-correcting reversal may allow the Court to respond when changing the doctrine in a way that constrains all lower courts would be too costly. These complications go beyond the formal model developed here, but suggest further avenues for study.

6. CONCLUSION

Judicial politics scholars have focused attention on the importance of inter-court interactions in shaping judicial decisionmaking, but have largely neglected or discounted the role of legal doctrine. Other scholars have emphasized the choice between different doctrinal forms without taking account of the dynamic interaction between upper and lower courts. Our model demonstrates the advantages of taking account of both institutional structure and legal doctrine, and of doing so using a dynamic model of decisionmaking. It takes seriously the role of doctrine in shaping the decisions of lower courts, while simultaneously considering how repeated interactions between courts with divergent preferences will influence their behavior.

The one-period model captures the basic intuitions of the rules and standards literature. Rules are more determinate, restraining discretion at the point of application, but also risk producing errors of over- and under-inclusiveness. Standards allow consideration of unanticipated factors that should affect outcomes, but also fail to constrain undesirable exercises of discretion. Examined in the context of the judicial hierarchy, these characeristics of rules and standards highlight the tradeoff facing the Supreme Court: it cannot simultaneously minimize error costs by lower courts and maximize its control over their decisionmaking. Thus, how determinate or not it makes a legal command will depend upon how it weighs the relative costs and the degree of preference divergence between it and the lower courts.

The dynamic model suggests additional insights. First, it explains how changes in doctrinal form emerge endogenously from the interaction between upper and lower courts. Even when there is no change in the courts' relative policy preferences, doctrine may move from a standard to a more rule-like form, simply because of the difficulty confronting the Supreme Court in communicating its policies and the risk that lower courts will make mistakes in applying the doctrine. Conversely, insisting on an inflexible command imposes error costs because lower courts are constrained from considering relevant, new information, and so the Supreme Court may swing back to a more open-ended standard after a period of time. The degree of preference divergence between the upper and lower court remains relevant, as it affects the duration of time constraining rules are likely to be imposed.

Significantly, the model suggests that with repeated interactions over time, a Supreme Court and lower court with differing preferences may nevertheless have an incentive to cooperate. The Court may grant discretion to the lower court as it shapes doctrine because it knows that it cannot anticipate all the factual circumstances that will arise in the future. And the lower court has an incentive to use that discretion in accordance with the Supreme Court's preferences despite the fact that the risk of reversal in any given case is exceedingly low. Rather than resulting from out-sized fear of reversal, the willingness of lower courts to follow precedent can be understood as a form of cooperation with the Supreme Court. Compliance increases the likelihood of a payoff for lower courts in the form of future doctrines that afford them greater discretion. Non-compliance risks provoking the Supreme Court to impose constraining rules.

References

Appendix A

A.1. PROOF OF PROPOSITION (1)

r*(α) is the best response of the lower court as a function of the form of the legal command. This best response is upward sloping in α; that is, ⁠. The Supreme Court's payoff, Π_sc, is only a function of α through its effect on the best response. As a result, the Supreme Court, in effect, selects r, corresponding to a value of α in order to maximize Π_sc. And so, this reduces to a simple decision problem, where we can apply the monotone comparative static results (Milgrom & Shannon 1994; Ashworth & Bueno de Mesquita 2006). That result states that the choice variable is weakly increasing in the parameter if the objective function satisfies the single crossing property. A sufficient condition for this property to hold is that the cross partial of the objective function with respect to the choice variable and the parameter is positive (Ashworth & Bueno de Mesquita 2006, 219). 

Proof of (A)

Following the technique outlined in Ashworth & Bueno de Mesquita (2006, 221), define the parameter ρ = −p. From above, a sufficient condition for α* to be a weakly increasing function of ρ is that ⁠. Given the best response of the lower court, we know that

So

and

which is positive. And so, α* is a weakly increasing function of ρ, which implies that α* is a weakly decreasing function of p.

Proof of (B)

We know that

So

which is positive. And so, α* is a weakly increasing function of L.

Proof of (C)

Define the parameter χ = −E. A sufficient condition for α* to be a weakly increasing function of χ is that ⁠. We know that

which is positive. α* being a weakly increasing function of χ implies that it is a weakly decreasing function of E.  ▪

A.2 PROOF OF PROPOSITION (2)

We claim that the following set of strategies constitutes a subgame perfect equilibrium of the repeated game.

A.2.1 The Supreme Court's strategy

In the trust phase, the Supreme Court sets α = 1 if, in the previous period, it realized a payoff of 0. The constraint phase starts if the Supreme Court observes a loss in the prior period.

In the constraint phase, the Supreme Court sets α_t = α* forperiods if, in the previous period, it realized a payoff less than 0. Defineas the smallest integer such that Equation (2) holds.

At date 1 of the constraint phase, the Supreme Court sets α_t = α*.

At dates {2 … t − 1} of the constraint phase, the Supreme Court sets α_t = α* if, in prior period, it realized a payoff of (i) −r(α*)²E or (ii) −(1 − r(α*))²L. If, in the prior period, the Supreme Court observed any other payoff, it starts theperiod constraint phase again.

A.2.2 The Lower Court's strategy

In the trust phase when α_t = 1, the lower court sets r = 1 if it believes that the Supreme Court would view the new factor as relevant. The lower court sets r = 0 if it believes that the Supreme Court would view the new factor as irrelevant.

In the trust phase if α_t ≠ 1, the lower court plays its single period best response, setting r = r(α_t).

In the constraint phase (which follows the public signal that the Supreme Court suffered a loss after issuing a pure standard), the lower court always plays its single period best response, r(α), no matter the form of the legal doctrine.

To prove these strategies form a subgame perfect equilibrium of the repeated game requires that these strategies form a Nash equilibrium in each subgame.

Equation (2)ensures that the lower court does not deviate in the trust phase, assuming theperiod of constraining doctrine is in the offing.

Suppose that the lower court deviates in the constraint phase. According to the Supreme Court's strategy, following any realized payoff other than −r(α*)²E or −(1 − r(α*))²L, it restarts the constraint phase. To avoid this prospect, the lower court's best response is to set r = r(α*), guaranteeing the Supreme Court one of these two payoffs. Given that the lower court responds with its single-period best response, the Supreme Court cannot gain by setting α_t ≠ α*.

To conclude the proof, suppose that the Supreme Court deviates in the trust phase, setting α≠1. The lower court's strategy calls for it to play its one-period best response to any doctrinal form other than α = 1 . This best response will ensure that the Supreme Court experiences a loss that period, which, in turn, triggers the constraint period. This deviation cannot be profitable since setting the directive at α_t = 1, preserves the Supreme Court’s chance at a 0 payoff that period plus a continuation of the trust phase.    ▪  

Proof.

Notice that V⁺(t) > V ⁺(t + 1) whenever

which can be written as

or

which holds if

or

which is assumed to be true.    ▪

A.3 PROOF OF PROPOSITION (3)

The proof proceeds in two steps. First, we show that the LHS of (2) is increasing in⁠, the number of periods of constraining legal doctrine. The RHS is clearly independent of⁠. Second, we pick the smallest value ofsuch thatEquation (2)holds. Finally, allow p to vary. If the LHS decreases in p it follows, all else equal, that a (weakly) higher value ofis needed to maintain incentive compatibility for the lower court.

Step One:

RewriteEquation (2)as

B(1)?>

Plugging in for V⁺results in

Multiplying both sides by 1 − δ and rewriting the denominator yields

The bracketed term on the LHS can be written as

Collecting terms gives

(3)

By assumption, π_lc^coop > π_lc*. It follows that the LHS ofEquation (3)is increasing in t if

which reduces

or

The inequality must hold since δ^t+1 < δ^t.

Step Two:

Return toEquation (3). Take the derivative of the LHS with respect to p, resulting in

Note thatand π_lc* < π_lc^coopby assumption. As a result, the LHS ofEquation (3)decreases in p. The RHS is constant, meaning a weakly higher value of t is needed to maintain incentive compatibility.   ▪