Stare Decisis and Judicial Log-Rolls: A

Gains-from-Trade Model*

Charles M. Camerona, Lewis A. Kornhauserb , and Giri Parameswaranc

aPrinceton University, ccameron@princeton.edu

bNew York University School of Law, lewis.kornhauser@nyu.edu

cHaverford College, gparames@haverford.edu

Abstract

The practice of horizontal stare decisis requires that judges occasionally decide

cases “incorrectly”. What sustains this practice? Given a heterogeneous bench, we

show that the increasing differences in dispositions property of preferences generates

gains when judges trade dispositions over the case space. These gains are fully realized

by implementing a compromise rule – stare decisis. Absent commitment, we provide

conditions that sustain the compromise in a repeated game. When complete com-

promises become unsustainable, partial compromises still avail. Moreover, judges may

prefer to implement partial compromises even when perfect ones are sustainable. Thus,

stare decisis is consistent with a partially-settled-partially-contested legal doctrine.

*We thank participants at workshops at Toulouse School of Economics, New York University Law School,the 2015 Conference on Institutions and Law-Making at Emory University, and the 2015 American Law andEconomics Association Meetings.

1 Introduction

Every legal system has a body of previously decided case law, “precedent.” Legal systems

differ, however, in the role that precedent plays in future decisions. In common law jurisdic-

tions, though not in civil law jurisdictions, precedent constitutes a source of law. Common

law systems have thus developed a set of practices that facilitates the development of law

from precedent. Among the most important of these, or at least among the most discussed,

is stare decisis. Under stare decisis, today’s court, when confronted by a case similar to an

earlier case, follows the principles or rules of decision laid down in the prior case.1 Stare deci-

sis may thus require a court to adhere to a decision that it believes incorrectly decided. Put

differently, stare decisis may require a court to decide a case wrongly. An obvious question

is, why would courts adopt such a practice?

In this paper we investigate the political logic of stare decisis. In the model, the court

consists of a bench of judges that divide into two factions (called ‘L’ and ‘R’ in the model);

each case is decided by a judge drawn randomly from the bench.2 Judges receive utility

from the disposition of cases but judges from different factions disagree about the correct

rule to apply to cases. Critically, the judges in the majority faction have no formal device

for committing the minority faction to the majority’s most-preferred rule. Nor does such a

device oblige it to obey the other faction’s most-preferred rule. We explore the circumstances

under which the entire bench nonetheless can maintain a practice of stare decisis in which

1The term “stare decisis” derives from the Latin phrase “Stare decisis et non quieta movere,” an in-junction ”to stand by decisions and not to disturb the calm.” Within a given jurisdiction, the practice ofvertical stare decisis – how a lower court treats the previously decided cases of a higher court – differs fromthe practice of horizontal stare decisis – how a court treats its own previously decided cases. The practicemay vary across constitutional, statutory, and common law areas of law. In this paper, we provide a modelof horizontal stare decisis. Stare decisis rests on another practice that indicates when the instant case isgoverned by some (and by which) prior case. This practice also underlies the related practice of “distin-guishing.” For a brief discussion of stare decisis see Kornhauser (1998). For more extended discussions seeCross and Harris (1991), Duxbury (2008) and Levi (2013).

2We thus offer a stylized model of intermediate courts of appeal in which a panel of judges is drawnfrom a wider bench. If the judges on the court divided into two factions, each panel (with an odd number ofjudges) would, absent panel effects, decide as the faction of the majority of its members would. The practiceof drawing a panel from a bench is common on intermediate courts of appeal in the United States as well ason appellate courts in other legal systems. Many international courts have a similar practice.

1

all of the judges adhere to precedent and employ a common rule for disposing of cases, one

different from the most-preferred rule of either faction.

Two features of the model are important. First, we assume that each ruling by a judge

of the incumbent faction affects the utility of all judges on the bench, regardless of faction,

because the non-ruling judges view those decisions as jurisprudentially, socially, economically,

ideologically, or morally right or wrong. Thus, a judge (or the faction to which she belongs)

is not a stationary bandit who benefits only when she renders decision (Olson, 1993). Nor

are the judges non-consequentialists who strive only to “do the right thing” for its own sake

while in office irrespective of what others have done before or will do after. Rather, they

care about the actions of the judiciary whether they decide or not. As a result, they are

concerned about the response of other judges to their rulings.

Second, the model is set in case space rather than a generic policy space (see Kornhauser

(1992b), Lax (2011)). The judges are obliged to dispose of concrete cases by applying rules

to the cases. Hence, the actors are indeed judges on a Court rather than legislators in a

parliament. This aspect of the model is important for more than mere verisimilitude; in fact

it is critical for the logic of stare decisis. In case space, the logical structure of legal rules

creates the possibility of gains from trade in cases across factions of jurists. These gains

from trade arise if judicial utility for dispositions exhibits what we call increasing differences

in dispositions -- the utility differential between a correctly and incorrectly decided case in-

creases the “easier” the judge thinks the case. When judges have preferences with increasing

differences in dispositions, each faction can do better by adhering to a compromise rule.

In essence the compromise rule allows each faction to trade relatively low-value incorrectly

decided cases in exchange for receiving relatively high-value correctly decided cases from the

other faction. We identify the conditions under which such a compromise rule is sustainable

in a repeat play setting absent a commitment device.

The main points of the analysis are the following: As just noted, the structures of le-

gal doctrine and dispositional preferences create the opportunity for intra-court gains from

2

trade, a kind of ideological log-roll. But, the requirements to support stare decisis over all

possible cases are demanding. Much less demanding is partial stare decisis, a practice in

which all judges adhere to a common rule for relatively extremal cases but follow their own

preferred rule for more interior cases (we make “extremal” and “interior” precise shortly).

The phenomenon of partial stare decisis emerges naturally from the model but appears to be

novel in the legal literature. Finally, we identify sets of parameter values that can support

complete and partial stare decisis. Roughly speaking, a practice of stare decisis is both more

valuable and more easily sustained the more polarized the court and the greater the degree

of ideological heterogeneity among the court’s judges (e.g., the greater the imbalance in the

sizes of the L and R factions).

The paper is organized as follows. In the remainder of this section, we provide a simple

motivating example that illustrates how stare decisis allows an ideologically diverse bench

deciding cases to realize gains from trade. We also briefly discuss relevant literatures. Sec-

tion 2 lays out the model, including a formal representation of cases and of judicial utility.

Section 3 examines the stage game, establishing that autarky is the unique Nash equilibrium

in a one-shot setting. However, autarky is inefficient. Thus the judges find themselves in

a continuous action prisoner’s dilemma. Section 4 investigates the conditions under which

a practice of stare decisis is sustainable absent commitment but with an infinite stream of

cases. Section 5 considers equilibrium selection, exploring the implications of two plausi-

ble bargaining protocols selecting an equilibrium, the agenda-setter protocol and the Nash

Bargaining solution. The final section discusses the results, distinguishing the log-rolling

mechanism from risk aversion (the critical feature of many models of compromise). This

section also suggests extensions.

1.1 Judicial Logrolling: An Illustration

A simple example provides a great deal of intuition about how doctrine structures judicial

log-rolls. In this example, the two factions, L and R, decide four cases: x1 = 14, x2 = 1

4,

3

x3 = 34, and x4 = 3

4. The two factions alternate in deciding cases beginning with a judge

from the L faction, so the L faction decides cases x1 and x3 while R faction decides cases x2

and x4. We denote a disposition of a case (e.g., liable, not liable) as d(xt). Critically, each

faction has a conception of the correct disposition of a case given its location, and receives

a payoff from a correct disposition and a smaller payoff from an incorrect disposition. More

specifically the dispositional utility functions are:

uL (dt;xt) =

0 if xt≥0 and dt = 1, or xt < 0 and dt = 0 [correct dispositions]

− |xt| if xt ≥ 0 and dt = 0, or xt < 0 and dt = 1 [incorrect dispositions]

uR (dt;xt) =

0 if xt≥1 and dt = 1, or xt < 1 and dt = 0 [correct dispositions]

− |1− xt| if xt ≥ 1 and dt = 0, or xt < 1 and dt = 1 [incorrect dispositions]

These dispositional utility functions are shown in Figure 1. In the figure, the utilities from

“incorrect” dispositions are shown with dashed lines; the utility from “correct” \ dispositions

is shown by the dark horizontal line at zero.

First consider the “autarkic” situation in which each judge holding power disposes of

the instant case using her most-preferred legal rule, so each judge decides her assigned case

”correctly” according to her own dispositional utility function. Because L decides x1 and x3

she receives the dispositional value of a correctly decided case (zero) for these cases; these

values are labeled uL(x1) and uL(x3) in the left-hand panel of Figure 1, and are shown in

the left panel of Table 1. For R, however, L’s disposition of these cases results in wrongly

decided cases, which afford much less utility. These utility values are labeled uR(x1) and

uR(x3) in the left-hand panel of Figure 1. Particularly bad for R is L’s disposition of the

distant case x1. Again these values are shown in Table 1. In a similar fashion, because R

decides cases x2 and x4. R receives the value of a correctly decided case for these cases

4

Figure 1: Dispositional Utility in the Example.

1

4

1

2

3

41

Case Location

-1

4

-3

4

Dispositional Utility

1

4

1

2

3

41

Case Location

-1

4

-3

4

Dispositional Utility

Incorrect Disposition for L Incorrect Disposition for LIncorrect Disposition for R Incorrect Disposition for R

uL(x1), u

R(x2)

uL(x2)

uR(x1)

uL(x3), u

R(x4)

uR(x3)

uL(x4)

uR(x1), u

R(x2)

uL(x1), u

L(x2)

uL(x3), u

L(x4)

uR(x3), u

R(x4)

The dashed lines in the panel show the dispositional utility a faction receives from an “in-correct” disposition. “Correct” dispositions yield a utility of zero to both factions. In theleft-hand panel each faction decides its cases according to its own lights (“autarky”). In theright-hand panel, both factions decide cases using a compromise rule (stare decisis) whichsometimes forces a faction to decide a case incorrectly, according to its most-preferred doc-trine. The utility to the factions of each disposition of each case in the example are labeled,e.g., uL(x1).

Table 1: Stage Payoffs Under Autarky and Stare Decisis.Autarky

Period 1 2 3 4Case 1

414

34

34

Ruling Judge L R L RDisposition 1 0 1 0L-utility 0 -1

40 -3

4

R-utility -34

0 -14

0

Stare DecisisPeriod 1 2 3 4Case 1

414

34

34

Judge L R L RDisposition 0 0 1 1L-utility -1

4-1

40 0

R-utility 0 0 -14

-14

Note that under Autarky, the sum of L’s payoffs is -1, as is the sum of R’s payoffs. Bycontrast, under Stare Decisis, this sum is −1

2for each type of judge.

(labeled uR(x2) and uR(x4) in the left-hand panel of the figure) while L receives the value of

incorrectly decided cases (uR(x2) and uR(x4) in the figure). Using the left panel of Table 1

it will be seen that the sum of L’s utilities across the cases is −1 as is that of R.

Now consider payoffs when both judges obey a controlling precedent, one in which the

legal standard lies in the interval (14, 3

4), for example, 1

2. This means that L can decide x3 = 3

4

correctly by her lights, but must decide x1 = 14

incorrectly. The resulting utility values for

the two factions are shown in the right-hand panel of Figure 1 and compiled in the right

panel of Table 1. Using the table, it will be seen that each judge’s sum of the (undiscounted)

stage utilities is now −12; the payoffs under stare decisis are much better.

5

What is the explanation? First, note in Figure 1 that the utility functions display risk

neutrality in the relevant region of the case-space, so the explanation does not lie in utility

smoothing for risk-averse agents. The explanation lies in the IDID property and the structure

of doctrine. To understand the result, first define the utility differential between a correctly

decided and an incorrectly decided case for Judge i, ∆i(x). (In Figure 1, the utility differential

is simply the distance between the x-axis and the appropriate dashed line.) For Judge

L for cases in [0, 1], ∆L(x) = 0 − (−x) = x. For Judge R for cases in [0, 1], ∆R(x) =

0 − (1 − x) = (1 − x). L’s utility differential increases as x increases from 0, so disposing

distant cases correctly is more important than disposing close cases correctly. Similarly, R’s

utility differential falls as x increase from 0 toward 1 – in other words, it is more important

for R to get distant cases correct than nearby ones. This feature of the utility functions is

the increasing differences in dispositional value property of adjudication.

Now again consider Table 1. Under either autarky or stare decisis, Judge L views two

cases as being decided correctly and two cases as decided incorrectly. However, the cases

differ. Under autarky, the two cases decided correctly are x1 = 14

and x3 = 34, one close

case and one far case. (The two incorrectly decided cases are of course x2 = 14

and x4 = 34,

again a close case and a far case). Under stare decisis, the two cases decided correctly are

x3 = 34

and x4 = 34, two far cases. (The two cases decided incorrectly are x1 = 1

4and x2 = 1

4,

two close cases). In other words, in moving from autarky to stare decisis, Judge L trades a

nearby correctly decided case (located at 14) for a distant incorrectly decided case (located

at 34), which is then decided correctly. Because of the increasing differences property, this

trade leaves Judge L much better off. By symmetry, the trade also leaves Judge R much

better off – she trades a correctly decided near case for an incorrectly decided far case, that

then is decided correctly. Finally, note how the compromise stare decisis doctrine structures

and allows this beneficial trade or log roll to take place.

In this example, we have not constructed an equilibrium; we have just demonstrated

how the stage payoffs work and how a compromise standard coupled with a practice of stare

6

decisis creates an opportunity for a favorable, implicit log-roll between factions of judges.

We analyze the equilibrium of our formal model of the structure of this trade.

1.2 Related Literature

A vast literature on stare decisis exists but almost none of it addresses the practice’s sus-

tainability.3 Two papers, however, consider mechanisms close to that analyzed here.

O’Hara (1993) offers an informal argument that includes many elements of our formal

model. She considers a pair of ideologically motivated judges who engage in what she calls

“non-productive competition” over the resolution of a set of cases that should be governed

by a single rule.4 The judges hear cases alternately. She recognizes that they each might do

better by adhering to a compromise rule. She characterizes their interaction as a prisoner’s

dilemma and then relies on the folk theorem to argue for the existence of a mutually preferred

equilibrium. Because her argument is informal, the nature of the exchange between the

two judges is unclear. Similarly, she cannot provide comparative statics for the degree of

polarization or the relative sizes of the factions (i.e., the relative probability that each judge

will hear a case).5 Nor does she consider partial stare decisis.

Rasmusen (1994) also puts a formal structure on O’Hara’s argument. He develops an

overlapping generations model in which each judge in a sequence of judges hears n+ 1 cases.

3Much of the literature there is normative. Relevant literature includes Goodhart (1930), Levi (2013),Llewellyn (2012), Cross and Harris (1991), MacCormick and Summers (1997) and Duxbury (2008), as wellas Postema (1986) and Schauer (1987) on the desirability of the practice. A sizable literature in PoliticalScience searches for empirical evidence on the extent of stare decisis on the U.S. Supreme Court. See Spaethand Segal (2001), Knight and Epstein (1996), Brisbin (1996), Brenner and Stier (1996), and Songer andLindquist (1996). Several authors investigate reasons why a court might adopt a practice of stare decisis.See for example Kornhauser (1989), Bueno de Mesquita and Stephenson (2002), Blume and Rubinfeld (1982),Heiner (1986) and Gely (1998).

4She notes that, in other instances, the judges might specialize, each generating law in a different areaof law and following the precedent of the other judge on other areas of law.

5Several formal models of courts simply assume a practice of stare decisis; these models treat horizontalstare decisis as an exogenous constraint on judges’ behavior (Jovanovic (1988), Kornhauser (1992a)). Inother words, these models assume a commitment device whereby one generation of judges may bind thehands of its successors. Jovanovic presents a very abstract model in which he shows that ideologicallydiverse judges would do better in terms of both ex ante and ex post efficiency if they were constrained bya rule of stare decisis. Our model identifies conditions under which rational, self-interested judges might infact successfully adopt such a rule.

7

He identifies conditions under which each judge adheres to the decisions of the previous n

judges and each of the n judges who succeed him adheres to his decision in the (n+ 1)th

case. One might understand this structure as exhibiting the gains from trade that drives

our model but the trade here differs in three respects from that which underlies our model.

First, in Rasmusen’s model, trade occurs across rules rather than across cases governed by

a single rule. Second, trade occurs across generations of judges; in our model, the trade

occurs among judges who decide cases contemporaneously or sequentially. In addition, the

utility function in the model is not fully consequential: a judge receives no disutility from

subsequent judges continuing to adhere to the decisions of prior judges that she considers

wrongly decided, but the judge does receive utility from their adhering to her decision.6 The

model has no measure of ideological polarization. Nor does the model address partial stare

decisis.

Stare decisis affords a kind of political compromise between the two factions of judges.

One might expect, therefore, that general analyses of political compromise might apply in this

judicial setting (see for example Kocherlakota (1996), Dixit, Grossman and Gul (2000), and

Acemoglu, Golosov and Tsyvinski (2011)). However, those analyses rely on the risk aversion

of the actors, who perceive a certain compromise policy as more attractive than a lottery

over non-compromise policies. We return to this point in more detail in the Discussion,

but the mechanism here is not risk aversion but gains from trade, induced by the logical

structure of legal rules and judicial preferences over case dispositions. Hence our model of

judicial compromise is distinct from standard analyses of political compromise. The gains-

from-trade mechanism is arguably a signature of the judicial setting.

6Her preferences are thus not consequential. But they don’t seem to be expressive either. She has afterall endorsed the decisions of the prior judges so future adherence to these wrongly decided cases should havesome negative impact.

8

2 The Model

There are two factions of judges, L and R, that form a bench. In each period t = 1, 2, ..., a

single judge hears a case xt ∈ X ⊂ R. Cases are drawn for a common knowledge distribution

F (x). We assume F is continuous, has bounded support, and admits a density f (x). When

a case arrives before the court, a judge is chosen at random to decide it. The chosen judge is

from the L faction with probability p, and from the R faction with probability 1− p.7 The

variable p indicates the probability that the L faction holds power, but it can also be seen as

a measure of the heterogeneity of the bench. More precisely the bench is most heterogeneous

when p = 12

and becomes less heterogeneous (more homogeneous) as p approaches 0 or 1.

2.1 Cases, Dispositions, Rules, and Cutpoints

In the model, a case x connotes an event that has occurred, for example, the level of care

exercised by a manufacturer. A judicial disposition d ∈ D = {0, 1} of the case determines

which party prevails in the dispute between the litigants. Judges dispose of cases by applying

a legal rule. A legal rule r maps the set of possible cases into dispositions, r : X → D. We

focus on an important class of legal rules, cutpoint-based doctrines, which take the form:

r (x; y) =

1 ifx ≥ y

0 otherwise

where y denotes the cutpoint. For example, in the context of negligence, the defendant is

not liable if she exercised at least as much care as the cutpoint y.8 Judges from different

factions differ in their assessment of the ideal rule. Formally, the ideal cutpoint for a judge

7Suppose the bench is composed of n judges, nL of whom belong to the L faction and nR of whom belong

to the R faction. Then one can view p = nL

n and 1− p = nR

n .8Other examples include allowable state restrictions on the provision of abortion services by medical set

providers; state due process requirements for death sentences in capital crimes; the degree of proceduralirregularities allowable during elections; the required degree of compactness in state electoral districts; andthe allowable degree of intrusiveness of police searches. Many other examples of cutpoint rules may suggestthemselves to the reader.

9

from faction j ∈ {L,R} is yj, where yL < yR.

The logical structure of cutpoint rules implies both consensus and conflict between two

judges who enforce, or wish to enforce, two different cutpoints. We are particularly interested

in the conflict region – the [yL, yR] interval where they disagree about the correct disposition.

For a case x < yL, both judges agree that the appropriate resolution of the case is ‘0’.

Similarly, when x > yR, the two judges agree that the appropriate resolution of the case is

‘1’. Only when x ∈[yL, yR

]do the judges disagree: yL < x < yR implies that the L judge

believes the appropriate resolution of x is ‘1’ while the R judge believes the appropriate

resolution is ‘0’.

To simplify the analysis, we assume[yL, yR

]⊂ supp (F ), which implies that F is strictly

increasing over the region of conflict. In footnote 15 we show that this assumption is benign

in that it merely rules out trivial multiplicities of equilibria. Finally, we parameterize the

degree of polarization ρ by the fraction of cases in the region of conflict; ρ = F(yR)−F

(yL).

2.2 Dispositional Utility

The governing judge’s disposition of the case affords utility to all judges, reflecting their

conceptions of the ‘correct’ disposition of the case. Stage utility for judge i in each period t

is determined by the dispositional utility function:

u(d;x, yi) =

h(x; yi) if d = r(x; yi)

g(x; yi) if d 6= r(x; yi)

where yi connotes judge i’s most-preferred standard. In words, judge i receives h(x; yi) if

the judge in power disposes of the case ‘correctly’, that is, if the ruling judge reaches the

same disposition as if she employed a rule incorporating judge i’s most-preferred cutpoint yi.

Conversely, judge i receives g(xt; yi) if the judge in power disposes of the case ‘incorrectly’.

Both the ruling judge and the non-ruling judge receive utility each period.

We make the following assumptions about dispositional utility:

10

1. h(x; yi) ≥ g(x; yi) for all x (it is better that a case be correctly disposed rather than

not);

2. h(x; yi) is (weakly) increasing in the distance∣∣x− yi∣∣ (a correctly decided case far

from the preferred cutpoint yields (weakly) greater utility than a correctly decided

case closer to the preferred standard);

3. g(x; yi) is (weakly) decreasing in the distance∣∣x− yi∣∣ (an incorrectly decided case far

from the preferred cutpoint yields (weakly) less utility than an incorrectly decided case

closer to the preferred cutpoint); and,

4. h(x; yi) − g(x; yi) is strictly increasing in∣∣x− yi∣∣. This is the increasing differences

in dispositions (IDID) property, which requires that the net-benefit from correctly

disposing of a case (rather than not) is increasing in the distance of the case from the

preferred cutpoint.

As an example, the functions h(x; yi) = 1 and g(x; yi) = 0 display properties 1-3 but violate

the IDID property. The functions h(x; yi) = 0 and g(x; yi) = −(x − yi)2 display all four

properties.

For notational convenience, we denote by η (x; yi) = h (x; yi) − g (x; yi), the net benefit

of correctly disposing of case x. Assumption 1 implies that η (x; yi) ≥ 0 for all x, and

assumption 4 implies that η (·; yi) is decreasing in x when x < yi and increasing otherwise.

This latter property amounts to asserting that −η (x; yi) has a single peak at yi. We assume

that the functions h and η are continuous, which ensures they are integrable with respect

to the distribution function F . Beyond these, we make no further assumptions about the

properties of η; in particular, η is not assumed to be convex — judges need not be risk averse

over case dispositions.

11

2.3 Strategies and Equilibrium

A history of the game ht indicates all the prior cases and the dispositions afforded them by the

ruling judges. The set of possible histories at the beginning of time t is thusHt = X t−1×Dt−1.

A behavioral strategy σj for ruling judge j is a mapping from the set of possible histories and

the set of possible current cases into the set of dispositions σj : Ht ×X → D ; a strategy σj

for judge j is a history dependent selection of a legal rule. We focus on symmetric strategies,

so that all judges from the same faction choose the same strategy.

Let(σL, σR

)be a pair of strategies for L and R-type judges, respectively. Let V i (ht)

denote the expected lifetime utility of judge i after history ht, given strategies(σL, σR

).

Lifetime utility satisfies the Bellman equation:

V i (ht) = p

∫ [u(σL (ht, x) ;x, yi

)+ δV i

(ht, x, σ

L (ht, x))]dF (x) +

+ (1− p)∫ [

u(σR (ht, x) ;x, yi

)+ δV i

(ht, x, σ

R (ht, x))]dF (x)(1)

where δ ∈ [0, 1) is the common discount factor. The lifetime utility V i is simply the expected

discounted stream of stage utilities that judge i receives, given the strategies chosen by each

faction.

A pair of strategies(σL, σR

)is a sub-game perfect equilibrium if, for every x ∈ X and

after every history ht ∈ Ht:

(2) σi (ht, x) ∈ arg maxd∈{0,1}

{ui(d;x, yi

)+ δV i (ht, x, d)

}We solve for symmetric sub-game perfect equilibria.

2.4 The Practice of Stare Decisis

The norm of stare decisis is in effect if all judges would dispose of the same case in the same

way. For example, suppose at time t, the strategies direct judges from factions L and R

12

to adjudicate cases according to cutpoints zLt and zRt (respectively), where zLt < zRt . Then,

all judges will identically dispose of cases x < zLt and x > zRt . The law in those regions

is settled ; there is common agreement about how such cases will be decided. By contrast,

different judges will differently dispose of cases in the region[zLt , z

Rt

]. In this region, the

law remains contested. There is an established norm of stare decisis to the extent that cases

arise in the settled region of the case space.

Two special cases are worth noting. First, if zLt = yL and zRt = yR, then the period t

regime is characterized by autarky ; each judge simply applies their ideal legal rule, and there

is no attempt at implementing a consistent, compromise legal rule.9 The contested region

of law comprises the entire zone of conflict between the two factions. Second, if zLt = zRt ,

then the period t regime is characterized by complete stare decisis ; the judges would decide

all cases in the same way. The law is settled over the entire case space.

As we show in the following section, it is natural to expect yL ≤ zLt ≤ zRt ≤ yL. If so,

the cut-points(zLt , z

Rt

)divide the region of conflict

[yL, yR

]into three sub-regions. In region[

yL, zLt], L faction judges are required to ‘compromise’ by returning the disposition favored by

R faction judges. Similarly, in region[zRt , y

R], R faction judges are required to compromise

by returning the disposition favored by L faction judges. The norm of stare decisis is

maintained in these regions. By contrast, in the region(zLt , z

Rt

), the law is contested. Judges

of neither type compromise, and instead each disposes of cases according to their preferred

rule.

In an actual practice of (horizontal) stare decisis, judges adhere to the decisions rendered

by prior courts. The legal literature disagrees about what in the prior decision “binds” the

judge: the disposition, the announced rule, or the articulated reasons.10 Our formalism

seems consistent with judges respecting either a previously announced rule yC or respecting

9Technically, the norm of stare decisis still prevails for all cases arising outside the region of conflict.However, cooperation over this region is trivial, given that all judges agree about the ideal disposition of casesin these regions. In this paper, we focus attention on the ability of judges to sustain norms of cooperationin cases where their ideal dispositions diverge.

10For some discussion see Kornhauser (1998).

13

the dispositions of prior decisions that give rise to a coherent rule as in the model in Baker

and Mezzetti (2012).

3 The Stage Game

3.1 Payoff functions in the Stage Game

In this section we analyze the stage game. We begin with an analysis of the one-period

payoff functions.

The game has the structure of a two-stage lottery, where the first stage lottery determines

which faction will decide the case (and implicitly, which rule will be applied), whilst the

second stage lottery selects the case to be decided. This feature allows us to treat the payoff

function essentially as the judge’s expected utility over legal rules. In the second stage of the

lottery we calculate the judge’s utility function for legal rules or policies. This policy utility

function derives from the judge’s dispositional utility function. We then calculate the judge’s

payoff function as her expected policy utility from the first stage lottery. This perspective

on the payoff functions allows us easily to display the sub-optimality of the equilibrium of

the stage game and, later in the discussion section, to distinguish the mechanism that we

analyze and that depends on IDID in case space from a mechanism that is situated in an

abstract policy space and depends on risk aversion.

The dispositional utility function u (d;x, yi) represents judge i’s preferences over second-

stage outcomes. Note that we can write:

u(d;x, yi

)= h

(x; yi

)− 1

[d 6= r

(x; yi

)]η(x; yi

)Let vi (z) =

∫x∈X u (r (x; z) ;x, yi) dF (x) denote judge i’s expected second stage utility,

given that cutpoint rule z is being applied.11 We refer to this as the judge’s policy utility;

11This is judge i’s interim utility — after the resolution of the first stage lottery, but before the resolutionof the second stage lottery.

14

it captures the judge’s preferences over different rules. Note well that policy preferences

are not primitives in their own right, but rather, are induced by the judge’s dispositional

preferences and the distribution of cases F . Policy preferences satisfy12:

(3) vi (z) = E[h(x; yi

)]−∣∣∣∣∫ z

yiη(x; yi

)dF (x)

∣∣∣∣Let

(zL, zR

)be a pair of judicial rules applied by judges from factions L and R (respec-

tively). Judge i’s (ex ante) expected utility is:

vi(zL, zR

)= pvi

(zL)

+ (1− p) vi(zR)

= E[h(x; yi

)]− p

∣∣∣∣∣∫ zL

yiη(x; yi

)dF (x)

∣∣∣∣∣− (1− p)

∣∣∣∣∣∫ zR

yiη(x; yi

)dF (x)

∣∣∣∣∣(4)

which is the sum of three terms13. The first term is independent of the chosen legal rule, and

establishes the judge’s baseline utility if every case were decided in accordance with her ideal

rule. The second term is the expected loss incurred when a judge from faction L decides

a case differently from judge i’s ideal. Suppose yL < zL < yR. If judge i is herself from

faction L, then this will occur when x ∈[yL, zL

]; when the actual disposition is ‘0’ although

her ideal disposition would be ‘1’. If judge i is from faction R, then this will occur when

x ∈[zL, yR

]; when the actual disposition is ‘1’ although her ideal disposition would be ‘0’.

Similarly, the third term is the expected loss incurred when a judge from faction R decides

a case differently from judge i’s ideal.

Example 1. Suppose h (x; yi) = 0 for all x, and g (x; yi) = − |x− yi|, which implies

η (x; yi) = |x− yi|.14 Further suppose X ∼ U [x, x], where x < yL < yR < x, which

12The absolute value corrects the sign of the integral if it is negatively oriented; i.e. if yi > z.13Without confusion, we use vi (z) to denote judge i’s policy utility if the rule z is applied, and vi

(zL, zR

)to denote the expected policy utility from a first stage lottery generating rule zL with probability p, and rulezR with probability 1− p.

14This very tractable utility function has been deployed by others (e.g., see Fischman (2011)). Further-more, in other work, two of us present other arguments that recommend the choice of h

(x; yi

)= 0 for all x,

more generally (see Cameron and Korhnauser (2017)).

15

ensures that every case in the conflict region arises with positive probability. Denote

ε = 12

(x− x), the distance between the mean and boundary cases. Then (3) yields:

vi (z) = − 14ε

(z − yi)2and (4) yields: vi

(zL, zR

)= − 1

4ε

[p(zL − yi

)2+ (1− p)

(zR − yi

)2].

3.2 Equilibria and Optima

Consider a one-shot version of the game in which each judge chooses a rule to maximize her

ex-ante utility, taking as given the strategy of the other judge.

Lemma 1. In the one shot game, it is a strictly dominant strategy for each judge to imple-

ment her ideal rule.

Lemma 1 makes the straightforward point that, absent any other incentives, the best that

any judge can do is implement her own ideal policy whenever she is chosen to adjudicate

a dispute. Autarky is a Nash equilibrium of the one-shot adjudication game. However, as

the next Lemma shows, Autarky is not Pareto optimal. The judges could do better by

cooperating.

Lemma 2. The set of complete stare decisis regimes defines the Pareto frontier. Formally,

a pair of rules(zL, zR

)is Pareto optimal iff yL ≤ zL = zR ≤ yR.

Lemma 2 highlights the main insight of this paper; that judges can enjoy gains from trade

when judges from both factions resolve cases in the same way. This ‘cooperation’ amounts

to the application of an identical rule to all cases that arise in the conflict zone. Indeed,

the gains from trade typically exhaust only when the judges settle upon a complete stare

decisis regime; i.e. if the law is fully settled.15 To build intuition for the result in Lemma

15The assumption that F is strictly increasing ruled out trivial multiplicities of optima. Relaxing thisassumption, Lemma 2 generalizes to the claim: ‘A rule pair

(zL, zR

)is Pareto optimal iff F

(yL)≤ F

(zL)

=

F(zR)≤ F

(yR).’ To see this, suppose F is weakly increasing and F

(zL)

= F(zR)

with zL 6= zR. Then the

partial stare decisis regime(zL, zR

)is Pareto optimal because it induces the same pattern of case dispositions

as the complete stare decisis regime(zC , zC

), where zL ≤ zC ≤ zR. The strategies are observationally

equivalent. Although the rule-pairs are distinct in principle, they disagree over a measure-zero set of cases,and so their distinction is trivial. Hence, restricting attention to strictly increasing F is without loss ofgenerality. A similar insight applies to Lemma 1 which generalizes to the claim: ‘If F

(zi)

= F(yi), then

implementing rule zi is a dominant strategy for judge i.’

16

Figure 2: An Edgeworth Box Representation of Gains From Trade

The diagram assumes F is uniform over the region of conflict.

2, consider the following ‘Edgeworth Box’-type diagram, which represents the preferences

of both types of judges in policy-space. The square box represents all pairs of legal rules

whose cutpoints lie in the region of conflict. The top-left corner of the box corresponds to

Autarky, since(zL, zR

)=(yL, yR

). The top-right corner of the box corresponds to an R

faction judge’s ideal legal rule, whilst the bottom-left corner of the box corresponds to an

L faction judge’s ideal rule. The 45◦ line is the locus of all complete stare decisis regimes

since zL = zR. Intuitively, we should expect rules in the upper triangle, since in this region,

zL ≤ zR. (Rule-pairs in the lower triangle have the property that each type of judge prefers

the rule implemented by the other type of judge, to their own.)

The downward sloping dashed lines have slope −p1−p and connect all rule-pairs that share

the same expected rule. Indifference curves for R faction judges improve in the north-

east direction, whereas indifference curves for L faction judges improve in the south-west

17

direction. The slopes of indifference curves are given by:16:

∂zR

∂zL|vi(zL,zR) constant = − p

1− p·f(zL)

f (zR)·η(zL; yi

)η (zR; yi)

As the expression makes clear, indifference curves must be downward sloping. This should

be intuitive; an increase in zL is a concession from L to R faction judges that improves

the utility of R faction judges and lowers the utility of L faction judges. To compensate,

the R faction judges must make a concession to L faction judges, which requires zR to fall.

Furthermore, the IDID property implies that, at any given point, the indifference curves

of L faction judges are less steep (than those of R faction judges) in the upper triangle,

and steeper in the lower triangle.17 Hence partial stare decisis regimes cannot be Pareto

optima. By contrast, along the 45◦ line, all indifference curves have slope equal to −p1−p .

These represent the set of Pareto optimal rules.

The shaded area represents the set of legal rule-pairs that constitute Pareto improvements

over Autarky. The logic of gains from trade are made clear by noting that from any rule pair(zL, zR

)with zL 6= zR, there emanates a ‘lens’ of Pareto improvements that widens in the

direction of the 45◦ line. By the IDID property, the net utility gain from correctly deciding

cases close to the ideal cut-point is small relative to the gain from correctly deciding cases far

from the ideal cut-point. This implies that both types of judges value concessions by the other

type to equivalent concessions18 of their own, which is reflected in their indifference curves

16To see this, take some rule pair(zL0 , z

R0

). Since η is continuous and F admits a density, vi

(zL, zR

)is differentiable in the neighborhood of

(zL0 , z

R0

), with: ∂vi

∂zL= −pη

(zL; yi

)f(zL)sgn

(zL − yi

)and ∂vi

∂zR=

− (1− p) η(zR; yi

)f(zR)sgn

(zR − yi

). Let vi0 = vi

(zL0 , z

R0

)and φi

(zL, zR

)= vi

(zL, zR

)−vi0. Notice that

φi(zL0 , z

R0

)= 0 and that ∂φi

∂zR= ∂vi

∂zR6= 0 (except when zR = yi). Then, by the implicit function theorem,

there exists a function zR(zL, vi0

)s.t. vi

(zL, zR

(zL; vi0

))= vi0 in the neighbourhood of

(zL0 , z

R0

). This defines

the indifference curve containing policy(zL0 , z

R0

). Moreover:

∂zR(zL,vi0)∂zL

= − p1−p

f(zL0 )f(zR0 )

η(zL0 ;yR)η(zR0 ;yR)

sgn(zL−yi)sgn(zR−yi) .

17To see this, note that the slopes of the indifference curves differ only in the ratioη(zL;yi)η(zR;yi)

. Suppose

zL < zR. The IDID property implies that η(zL; yL

)< η

(zR; yL

)and η

(zL; yR

)> η

(zR; yR

). Hence, we

haveη(zL;yL)η(zR;yL)

< 1 <η(zL;yR)η(zR;yR)

.18By equivalent concessions, we mean that the expected fraction of cases implicated is the same. Suppose

zL < zL′< zR

′< zR. Then the concessions by L from zL to zL

′, and by R from zR to zR

′are equivalent

18

having different slopes. These differing valuations straight-forwardly produce opportunities

for mutual gains from trade.

We note, importantly, that whilst the Edgeworth Box suggests that the mechanism driv-

ing our results is trading over policies, in fact, what the judges are actually trading is case

dispositions. Straight-forwardly, policies are not ‘things’ that judges possess and can trade

with one another. What they do possess is the ability (when recognized) to decide cases.

Choosing among different policies amounts to the judges making trades about how they

will dispose of various cases. Although the judges’ choices are over options in policy space,

the import of those policy choices is determined by their consequence for outcomes in case

space. This connection between the case space and policy space is crucial to understanding

this paper’s main insights.

Finally, we note that Lemmas 1 and 2 together imply that the incentives for the judges

in the adjudication game are strategically equivalent to a (continuous action) Prisoners’

Dilemma. Although there are potential gains from cooperation, the individual incentives

for each judge cause them to optimally choose a Pareto inferior regime in equilibrium. In

the next section, we investigate the possibility that repeated interaction between the players

may provide incentives for cooperation.

4 Sustainability of Stare Decisis

The gains from trade offered by complete stare decisis make it an attractive regime for judges.

But judges have no mechanism to commit themselves to such a regime. Confronted with a

case she must decide incorrectly under stare decisis, a judge may be sorely tempted to defect

and decide the case correctly according to her own lights. We thus seek to identify conditions

that will sustain a practice of stare decisis as an equilibrium in an infinitely repeated game

between the judges on the bench.

provided that p[F(zL

′)− F

(zL)]

= (1− p)[F(zR)− F

(zR

′)]

.

19

4.1 Sustainable Stare Decisis

It is a well known result that non-Nash outcomes can be sustained in a repeated game if the

players employ strategies that punish deviations from the designated action profile. Whilst

a variety of potential punishment strategies exist, in this section, we study a focal strategy:

grim trigger Nash reversion.19 Let Ht

(zC)

be the set of histories up to time t in which all

previous actions were consistent with judges respecting the complete stare decisis regime(zL, zR

)=(zC , zC

). Consider the pair of strategies

(σL, σR

)defined by:

σi (ht, x) =

r(x; zC

)ifht ∈ Ht

(zC)

r (x; yi) otherwise

Each judge respects the complete stare decisis regime provided that it had been respected in

all previous periods, and reverts to autarky otherwise. This pair of strategies can be sustained

as an equilibrium if, but only if, for any case that a judge is recognized to adjudicate,

adherence to stare decisis yields a larger discounted lifetime utility stream than deviating to

autarky. A critical point to note is that the greatest temptation to defect from stare decisis

occurs when the case confronting the ruling judge is located precisely at the compromise

policy zC , for such a case is the most-distant case the judge is obliged to decide incorrectly

under stare decisis. Hence, if the ruling judge would adhere to stare decisis when x = zC ,

then she will adhere to stare decisis for any other case as well. In this most demanding case,

judge i will adhere to stare decisis provided:

g(zC ; yi

)+

δ

1− δvi(zC , zC

)≥ h

(zC ; yi

)+

δ

1− δvi(yL, yR

)

19Of course, other types of punishments strategies exist, and the analysis that follows can be easilymodified to accommodate different punishments. Importantly, we acknowledge that punishments exist thatare more severe than Nash reversion. However, we also note that such punishments may not necessarily becredible if δ is sufficiently small. By contrast, since Autarky is a Nash equilibrium of the stage game, thethreat of Nash reversion is always credible. We also note that cooperation may be sustained by less severepunishment strategies, especially when δ is large.

20

which implies:

δ ≥η(zC ; yi

)η (zC ; yi) + [vi (zC , zC)− vi (yL, yR)]

Deviation to autarky improves the deliberating judge’s short run utility (since she can

now dispose of cases according to her ideal rule), but causes expected utility to fall in all

future periods, since reversion to autarky represents a Pareto-deterioration. If the judge

values future cases sufficiently much, then the long run losses will outweigh the short-run

gains, and cooperation will be sustainable.

For each i ∈ {L,R}, let

δi(zC)

=η(zC ; yi

)η (zC ; yi) + [vi (zC , zC)− vi (yL, yR)]

and let δ(zC)

= max{δL(zC), δR

(zC)}

. Since to sustain an equilibrium, all judges must

be incentivized to cooperate, we need that δ ≥ δ(zC).

Proposition 1. A complete stare decisis regime at compromise threshold zC can be sustained

as an equilibrium of a repeated game, provided δ ≥ δ(zC). Moreover:

1. there exist z, z ∈(yL, yR

)with z ≤ z, such that δ

(zC)> 1 whenever zC < z or zC > z.

2. there exists a unique δ∗ ∈ (0, 1) and z∗ ∈ (z, z), such that δ∗ = δ (z∗) and δ(zC)≥ δ∗

for all zC. Further δ(zC)

is decreasing for zC < z∗ and increasing for zC > z∗.

3. the correspondence Z (δ) ={zC |δ ≥ δ

(zC)}

is monotone in the sense that δ′ > δ

implies Z (δ) ⊂ Z (δ′).

Several points are worth noting. First, although they are Pareto optimal, not all complete

stare decisis regimes are sustainable. If the compromise legal rule is skewed too far towards

the ideal legal rule of either faction, then cooperation breaks down. The thresholds z and

z bound the set of potentially sustainable compromise policies. These correspond to the

boundaries of the shaded region in Figure 2 along the 45◦ line; they are the subset of Pareto

optimal rules that are Pareto improvements over autarky.

21

Figure 3: Sustainable Stare Decisis.

δL

δR

'Triangle'

z1

2z 1

zC

δ*

1

The horizontal axis represents compromise policies zC , the vertical axis represents the dis-count factor δ. The upward sloping curve is δL, the critical discount factor for L judges.The downward sloping curve is δR. The discount factor cannot be larger than 1. Completestare decisis is sustainable when the discount factor is larger than both critical values. Thefigure assumes the environment in Example 1, with yL = 0, yR = 1, ε = 2

3, and p = 1

2.

Second even among the set of Pareto improving compromises (zC ∈ [z, z]), these policies

are only sustainable if the discount factor is sufficiently large. The function δ(zC)

determines

how large the discount factor must be before a compromise at zC is sustainable. Obviously,

if δ(zC)> 1, then it is impossible to sustain cooperation around the cutoff point zC ,

whereas if δ(zC)

= 0, then such a compromise is guaranteed to be sustainable. As part

(2) of the proposition shows, δ(zC)

is always strictly greater than zero, and so if judges

are sufficiently impatient, then none of the Pareto optimal rule-pairs are implementable in

equilibrium. Moreover, δ∗ is the minimum discount factor needed to sustain cooperation at

some compromise cutpoint.

Third, as the discount factor increases, the set of compromise policies that can be sus-

tained grows. We see this in Figure 3,which plots the set of compromise policies that can be

sustained for each value of δ. The sustainable set is the triangle-like region bounded by the

line δ = 1 and the curves δL(zC)

and δR(zC). The bottom tip of the triangle represents

the pair (z∗, δ∗). Monotonicity is evidenced by the triangle ‘widening’ as δ increases.

The failure to sustain Pareto optimal policies stems from a problem of commitment.

Although both types of judges favor compromise policies ex ante, once chosen, the presiding

judge has a short-run incentive to defect and dispose of cases according to her ideal legal rule.

22

Cooperation in the short run is sustained by the threat of a breakdown of cooperation in the

future. Hence, cooperation requires that the short-run cost of adhering to the agreement is

smaller than the discounted value of gains from trade. If δ is low, then cooperation is not

sustainable as the future gains from trade are insufficient to compensate the presiding judge

for short run losses.

We can interpret the size of the discount factor in two ways. On the one hand, if cases

arrive at a fixed rate, a high δ implies that judges are relatively far-sighted and place sufficient

value on the disposition of future cases. On the other hand, if the case arrival rate is not

fixed, a high δ may imply that the next case will arrive relatively soon into the future, and so

the gains from defection will be relatively short-lived. By contrast, a low δ may imply that

a long period of time will pass before a judge from the other faction has the opportunity to

reverse the current judge’s decision, and so the gains from defection may persist for a long

time. Under this interpretation, our model predicts that compromise policies are more likely

to arise in areas of the law that are frequently litigated, than in areas where controversies

arise rarely.

4.2 Comparative Statics

In the previous sub-section, we characterized the set of compromise policies zC that can be

sustained in equilibrium, given the discount factor δ. In this sub-section, we are interested

in how these sustainable compromises are affected by heterogeneity and polarization. In

analyzing the comparative statics, we will be concerned with how the underlying parameters

affect: (i) the functions δL and δR, (ii) the compromise policy z∗ that is implementable over

the broadest set of discount factors (abusing terminology, the ‘most frequent’ compromise

policy), (iii) the lowest discount rate at which some compromise is feasible δ∗, and (iv) the

highest and lowest compromise policies (z and z, resp.) that can be sustained .

23

4.2.1 Heterogeneity and Sustainability

Recall, heterogeneity is a measure of the distribution of power between L and R judges on

the bench. In our model, this is captured by the parameter p. Heterogeneity is highest

when p = 12, so that both factions are equally powerful. Heterogeneity has the following

implications for equilibrium policy:

Lemma 3. Following an increase in p:

1. L-faction (resp. R-faction) judges become less (resp. more) amenable to compromise.

Formally, δL(zC)

(resp. δR(zC)) is increasing (resp. decreasing) in p.

2. The ‘most frequent’ compromise policy z∗ moves closer to L’s ideal (z∗ is decreasing

in p). The effect on the lowest discount factor δ∗ = δ (z∗) is ambiguous.

3. The highest and lowest sustainable policies both move closer to L’s ideal (z and z are

both decreasing in p), although the effect on the range of compromise policies (z − z)

is ambiguous.

Lemma 3 makes several points. First, it shows that, as the L faction gains more power,

L-faction judges will be less amenable to compromise whilst R-faction judges become more

amenable. In the left panel of Figure 4 below, this manifests as an upward shift in the δL

curve and a downward shift in the δR curve. There is a set of compromise policies that L

faction judges would previously support, but which they no longer will. Similarly, there is

a set of compromise policies that R faction judges would previously not support, but which

they now will. As judges become more politically powerful, they become less amenable

to compromise. The intuition is straight-forward; as p increases, the cost of autarky falls

for L-faction judges and rises for R-faction judges. The implications for the benefits from

compromise are obvious.

Second, the ‘most frequent’ compromise policy z∗ (i.e. the bottom tip of the triangle in

Figure 3) shifts towards the L-faction’s ideal. Similarly, the highest and lowest sustainable

24

Figure 4: The Effect of Heterogeneity on the Set of Sustainable Compromise Policies.

δL

δR

δL'

δR'

z' z z' z* z 1zC

1

z

z*

z

1

4

1

2

3

41

p

0.5

1

The figure assumes the environment in Example 1, with yL = 0, yR = 1, and ε = 23. In

the left panel, the solid lines represent δL and δR when p = 12, whilst the dashed lines

represent the same functions when p = 34. We see that δL shifts up whilst δR shifts down,

and consequently z,z∗ and z all decrease. (Note, in this particular example, z∗ and z′ happento coincide.) The right panel depicts z,z∗ and z as p increases from 0 to 1.

compromise policies (z and z, respectively) both shift towards the L-faction’s ideal. These

results follow immediately from the preceding insight that L-faction judges can be more

demanding, whilst R-faction judges will be less so. By contrast, the effect on the range

of discount factors that can sustain a compromise is ambiguous. If the degree to which L

faction judges have become more demanding exceeds the degree to which R faction judges

have become more amenable, then compromises will be less likely, and vice versa. The

range of compromise policies (z − z) that can be sustained as an equilibrium is similarly

ambiguous.

Figure 4 depicts these comparative statics for our working example. The left panel shows

how the curves δL and δR shift in response to an increase in p, and the implications for the

sorts of compromise policies that can be sustained. The right panel shows how the highest,

most frequent and lowest compromise policies change as p increase from 0 to 1.

4.2.2 Polarization and Sustainability

In our model, polarization ρ is measured as the fraction of cases in the region of conflict,

F(yR)− F

(yL).20 As ρ increases from 0 to 1, so does the fraction of cases over which the

20At first blush, this definition might seem odd. Why not simply define polarization as the absolutedistance between the ideal cutpoints yR − yL? But this ignores that the case-space, as defined so far, is

25

judges will disagree, and hence, the likelihood of conflict.

Let F (x; [a, b]) denote the conditional distribution of X, given that X ∈ [a, b]. To analyze

the effect of a change in polarization, note that judicial preferences over legal rules can be

written:

vi(zL, zR

)= E [h (θ)]−ρ

{p

∣∣∣∣∣∫ zL

yiη(x; yi

)dF(x;[yL, yR

])∣∣∣∣∣+ (1− p)

∣∣∣∣∣∫ zR

yiη(x; yi

)dF(x;[yL, yR

])∣∣∣∣∣}

As before, the relevant component of utility is the second term, which is the product of

the degree of polarization and expressions that only depend on the conditional distribution

of cases within the region of conflict. In performing the comparative static analysis, we

consider changes in the case-distribution that cause the degree of polarization to change,

holding fixed the conditional distribution in the region of conflict. Hence, changing our

measure of polarization changes the likelihood that the judges will be in conflict, but does

not change the relative likelihood of certain controversial cases arising over others.21

Lemma 4. Following an increase in polarization ρ = F(yR)− F

(yL)

(supposing that the

conditional distribution FX∈[yL,yR] is unchanged):

1. Both L and R faction judges become more amenable to compromise. Formally, δL(zC)

and δR(zC)

are both decreasing in ρ, for zC ∈ [z, z].

2. The ‘most frequent’ policy z∗and the highest and lowest sustainable policies (z and z,

resp.) are unchanged.

3. The lowest discount factor at which stare decisis is sustainable δ∗decreases.

Polarization has a remarkable effect in the model: it increases the gains from trade and

endowed neither with a natural 0 case, nor with a standard unit of measurement. Hence, we have two degreesof freedom, and it is without loss of generality to fix yL and yR, and to scale all other cases accordingly. (Inour example, we specify yL = 0 and yR = 1.) Hence, yL and yR will be invariant in our model. However,we can think of these cutpoints as being ‘closer’ or ‘further’ from each other, relative to the set of cases thatwill likely arise, given the case generating process. This motivates our chosen measure for polarization.

21It should be clear that changing the conditional distribution will change the relative value of compromiseto L and R faction judges, independent of the distance between their ideal points.

26

Figure 5: The Effect of Polarization on the Set of Sustainable Compromise Policies.

δL

δR

δL' δ

R'

z z* z 1zC0

0.5

1

The figure assumes the environment in Example 1, with yL = 0, yR = 1, and p = 12. The

solid lines represent δL and δR when ρ = 34

(i.e. ε = 23), whilst the dashed lines represent the

same functions when ρ = 13

(i.e. ε = 32).

hence expands the set of compromise policy/discount-factor pairs(zC , δ

)that are consistent

with stare decisis. The ‘ triangle’ becomes larger. The result is somewhat counter-intuitive;

one might have expected that the more the factions disagree, the less likely they would

be amenable to compromise. But this ignores the gains from trade logic that underpins the

model. As polarization increases, the expected losses from autarky become larger, since more

cases will now be decided ‘incorrectly’ if the presiding judge is from the opposing faction.

This implies that the gains from trade have become larger, making defection costlier, and

cooperation easier to sustain.

Figure 5 depicts the effect of a decrease in polarization in the case of our working example.

Given the uniform distribution, polarization in the working example is given by ρ = 12ε

. An

increase in polarization causes both the δL and δR curves to shift downward. This implies

that more compromise policies become sustainable for any given δ < 1. Moreover, both

curves shift down by the ‘same amount’, so that the policy z∗ at which they intersect (i.e.

the ‘most-frequent’ compromise policy) is unchanged.

4.3 Partial Stare Decisis

The previous subsections focused on the sustainability of Pareto optimal legal regimes as

equilibria of a repeated game. We showed that, any complete stare decisis regime that

27

represented a Pareto improvement over autarky could be sustained if δ were sufficiently

large. We also showed that for δ small enough (in fact, δ < δ∗), that it was impossible

to sustain a complete stare decisis equilibrium. In fact, these insights generalize to partial

stare decisis regimes as well. Let(zL, zR

)be a pair of legal rules that constitute a Pareto

improvement over autarky (i.e. which are contained in the shaded lens in Figure 2). For

each i ∈ {L,R}, let

δi(zL, zR

)=

η (zi; yi)

η (zi; yi) + [vi (zL, zR)− vi (yL, yR)]

and let δ(zL, zR

)= max

{δL(zL, zR

), δR

(zL, zR

)}. Then the partial stare decisis regime(

zL, zR)

is sustainable provided that δ ≥ δ(zL, zR

).

Proposition 2. The following are true:

1. the correspondence Z (δ) ={(zL, zR

)| δ ≥ δ

(zL, zR

)}is monotone in the sense that

δ′ > δ implies Z (δ) ⊂ Z (δ′).

2. autarky is always sustainable (i.e.(yL, yR

)∈ Z (δ) for all δ ∈ [0, 1]).

3. autarky is the only sustainable regime for δ small enough. Formally, there exists

δ∗∗ ∈ [0, δ∗) such that Z (δ) ={(yL, yR

)}for all δ ≤ δ∗∗. Furthermore δ∗∗ = 0 iff

∂η(x;yi)∂x|x=yi = 0 for each i.

4. for intermediate values of δ, partial stare decisis regimes are sustainable, although

complete stare decisis regimes are not. Formally, if δ ∈ (δ∗∗, δ∗), then there exists(zL, zR

)6=(yL, yR

)∈ Z (δ), and if zL = zR, then

(zL, zR

)/∈ Z (δ).

Proposition 2 bears many similarities to Proposition 1, which is to be expected; the same

logic underpins both claims. The second part of Proposition 2 has no analogue in Proposition

1, but it is not interestingly new. It simply reflects a result we have previously stated, that

since autarky is a Nash equilibrium of the stage game, it must be an equilibrium in the

repeated game.

28

The fourth part is substantially different; it states that there are a range of environments

over which partial stare decisis regimes may be sustainable whereas complete stare decisis

regimes cannot. To build intuition for this result, suppose, beginning from autarky, the

judges progressively move their thresholds towards the middle until the thresholds meet.

By Lemma 2, we know that whenever the thresholds do not coincide, there are further

gains from trade to be exploited. However, the lion’s share of the gains from trade are

actually realized at the start. To see this, consider the L faction judges; they lose little in

the original compromise since the disutility from incorrectly deciding cases close to their

threshold is small, whilst the utility gain from having cases far from their threshold (close

to R’s threshold) decided ‘correctly’ is large. As the judges progressively compromise, the

size of the gains fall relative to the losses, and so the net gains from trade, whilst positive,

decrease.

Recall, also, that sustaining cooperation becomes harder as the compromise threshold

moves further away from the judge’s ideal. Hence, as judges compromise more, the marginal

disutility from compromising in the stage game increases, whilst the marginal utility gain in

the continuation game decreases. It may be that, on net, the net gains (adding costs and

benefits in the stage and continuation game) become negative before the thresholds meet.

I.e. it may be more desirable to sustain a partial compromise than the complete one.

Proposition 2 also shows that, if the judges are sufficiently present biased, then no com-

promise may be possible at all, and the only feasible regime is autarky.22

The intuition for these results can be seen in Figure 6. In each panel, the thin outer

curves represent the locus of policies over which the L and R-faction judges (respectively)

are indifferent to autarky. (These curves were represented in Figure 2 as well.) The lens

enclosed by these indifference curves represents the set of Pareto improvements over autarky.

22To understand the condition on the derivative of η, note that since η is increasing, starting from autarky,an ε-compromise by both factions of judges generates a first order gain-from-trade; future utility from thecompromise increases in the first order. If η′ > 0 at autarky, then the immediate utility loss from compromiseis also of the first order. When judges are sufficiently impatient, the latter will dominate the former. Bycontrast, if η′ = 0 at autarky, then the immediate utility loss from compromise is of the second order. Thefuture gains from trade will always dominate, and so a sustainable compromise always exists.

29

These are also the set of sustainable (partial) compromises when δ = 1 (since in this case,

judges do not care about short run losses). The thick inner curves represent the locus of

policies which are just sustainable (for the L and R-faction judges, respectively) given some

discount factor δ < 1. In the left hand panel, δ > δ∗, and so some complete stare decisis

compromises are sustainable. In the right hand panel δ < δ∗, and so none of the complete

stare decisis regimes are feasible, but some partial regimes are. The set of policies contained

within the smaller lens is precisely Z (δ).

As δ decreases the inner curves move ‘towards each other’, causing the set of sustainable

policies to shrink. This is the monotonicity property in part 1 of the proposition. The δ∗

defined in Proposition 1, is the δ for which the two curves to intersect precisely on the 45◦

line. In both panels, at Autarky, the inner curves expand ‘away from one another’. However,

since these curves also move towards one another as δ decreases, there may be some δ small

enough, such that the curves become tangent to one another at Autarky. This defines δ∗∗.

For δ ≤ δ∗∗, the two curves no longer enclose a lens of policies. there is no lens. Autarky

is the unique sustainable equilibrium. (However, if η′(yL; yL

)= 0, then L’s curve must be

horizontal at autarky for any δ, and if η′(yR; yR

)= 0, then R’s curve must be vertical at

autarky for any δ. In either case, it is impossible for the curves to ever be tangent, and so a

lens is enclosed for all δ > 0.)

5 Equilibrium Selection

In the previous section, we described the set of compromise policies that could be sustained

in equilibrium. We now suggest a plausible mechanism to select from among the potential

equilibria. In doing so, we are particularly concerned about determining when or whether

courts will implement complete stare decisis regimes, and when partial stare decisis regimes

will prevail.

Imagine, prior to the start of the repeated game, the factions engage in pre-play bargain-

30

Figure 6: Sustainable Legal Rules and the Discount Factor

Autarky

ICL0

ICR0

Z(�)

Slope =-p/(1-p)

ICR1

yL yRzL

yL

yR

Autarky

ICL0

ICR0

Z(�)

Slope =-p/(1-p)

yL yRzL

yL

yR

z

The figure assumes the environment in Example 1, with p = 13. In the left panel, δ = 0.88,

whilst in the right panel, δ = 0.83.

ing over the compromise policy. Whilst we remain agnostic as to the specific details, we have

in mind a bargaining protocol a la Rubinstein (1982) or Baron and Ferejohn (1989), where

the outside-option is autarky. Let γ denote the common discount factor in the pre-play bar-

gaining, whilst δ remains the discount factor in the repeated game proper. In each period of

pre-play bargaining, some faction is recognized (somehow) to propose a compromise policy

from the sustainable set Z (δ). If the compromise policy is agreed to by the opposing faction,

then pre-play bargaining concludes and the adjudication game begins. If the compromise

policy is rejected, then the process is repeated in a subsequent period of pre-play bargaining

until a compromise is achieved. It is well known that no-delay equilibria of bargaining games

of this sort exist (see Rubinstein (1982), Banks and Duggan (2006)).

Rather than solve the pre-play bargaining model for generic γ, we focus on two special

cases: γ = 0 and γ = 1. A standard result from the bargaining literature is that the

advantage to the proposer is decreasing in the discount rate γ. Hence, our special cases

provide the outer limits of the equilibria that we should except to see arise more generally.

When γ = 0, there is effectively a single around of bargaining, which gives significant power

to the proposer. The faction recognized to propose policy can freely set the agenda, subject

31

to respecting the sustainability constraint. By contrast, as γ → 1, both factions are able

to make counter-proposals arbitrarily quickly, which significantly reduces the agenda setting

power of the initially recognized faction. Binmore, Rubinstein and Wolinsky (1986) show

that this limit case converges to the Nash Bargaining solution.

5.1 Agenda Setter (γ = 0)

We begin by considering the case of an agenda setter.

Proposition 3. Under the agenda setter framework:

� If δ = 1, then the agenda setter will choose a complete stare decisis regime. The

compromise policy will be(zL, zR

)= (z, z) if the L faction sets the agenda, and(

zL, zR)

= (z, z) if the R faction sets the agenda.

� If δ < 1, then the agenda setter will choose a partial stare decisis regime, unless δ < δ∗∗,

in which case the only feasible regime is autarky.

Proposition 3 shows that if the faction choosing the compromise is sufficiently empowered,

it will generically choose a Pareto inferior partial stare decisis regime (unless δ = 1), even

when Pareto optimal regimes are sustainable.23 This seems counter-intuitive – we might

have conjectured that the agenda setter would choose the most favorable sustainable Pareto

optimum. To build intuition for this result, note that judge i’s policy utility from an arbitrary

rule pair(zL, zR

)can be decomposed as the difference of two terms: the policy utility

associated with the complete stare decisis regime implementing the expected policy E [z] =

pzL + (1− p) zR, less a term proportional to the variance of the rule.24 Each judge prefers

a complete stare decisis regime (since the variance term is zero) to any other rule-pair that

23Of course, the chosen policy will be constrained Pareto optimal, given the sustainability constraint. Ifnot, the agenda setter could improve her utility further.

24Formally, we take a second order Taylor approximation of vi(zL, zR

)centered at E [z] = pzL +

(1− p) zR. This gives vi(zL, zR

)≈ vi (E [z] , E [z]) − ψi (E [z])V ar (z), where ψi (z) = d

dz

(η(z; yi

)f (z)

)and V ar (z) = p

(zL − E [z]

)2+ (1− p)

(zR − E [z]

)2.

32

induces the same expected rule, ceteris paribus. But as we showed in Proposition 2, there may

be partial stare decisis regimes that are sustainable, even when the corresponding complete

stare decisis regime is not. Given this constraint, the proposing faction faces a trade-off

between pulling the expected legal rule closer to their ideal (which improves utility along

the first dimension) and increasing the variance of the rule (which decreases utility along the

second dimension). At the optimum, the proposer is willing to introduce some variance into

the rule, in order to bring the implemented rule closer to her ideal, in expectation. Hence,

the proposing faction chooses not to realize all potential gains from trade. Even when a

perfect compromise is available, we may expect to observe legal doctrines that are partially

settled and partially contested.

This is made apparent in the left panel of Figure 6.The figure depicts the case when the

R faction proposes the agenda. As before, the shaded region represents the set of sustainable

policies, which in this case includes some complete stare decisis policies. The proposer will

choose the policy that puts him on the highest achievable indifference curve (IC1R) given the

sustainability constraint. The slopes of the R faction judges’ indifference curves, and the

boundary of Z (δ) make clear that the optimum is incompatible with perfect perfect stare

decisis.

5.2 Nash Bargaining (γ = 1)

We now turn our attention to outcomes under the Nash Bargaining approach, which we

understand as the limiting case of factions who are symmetrically empowered during the

pre-adjudication bargaining stage. In contrast to the results in the previous subsection, we

show that complete stare decisis is chosen over a range of discount factors, suggesting that

symmetrically empowered factions will be more likely to achieve a perfect compromise.

33

Recall, the Nash Bargaining solution solves:

max(zL,zR)

[vL(zL, zR

)− vL

(yL, yR

)] [vR(zL, zR

)− vR

(yL, yR

)]s.t.(zL, zR

)∈ Z (δ)

Proposition 4. Under the Nash bargaining framework, there exists a unique zNB ∈ (z, z)

and δNB ∈ [δ∗, 1) such that the Nash bargaining solution is:

� a complete stare decisis regime(zL, zR

)=(zNB, zNB

)whenever δ ≥ δNB

� a partial stare decisis regime whenever δ < δNB, unless δ < δ∗∗, in which case the only

feasible regime is autarky.

What distinguishes the Nash Bargaining result from the Agenda Setter case, is that sym-

metry in bargaining power requires both factions to enjoy an ‘equal’ share of the bargaining

surplus. (By contrast, the agenda setter was able to extract the lion’s share of the surplus

for herself.) This naturally pushes the factions to settle on a compromise that is roughly in

‘the middle’ of the set of potentially Pareto improving policies. Hence, even as decreases in

δ causes the set of sustainable compromises to narrow, this ideal compromise will continue

to remain sustainable until δ because sufficiently small.

In Propositions 1 and 2, we showed that complete stare decisis equilibria were only sus-

tainable for δ sufficiently large, but that partial stare decisis equilibria remained potentially

sustainable even as δ became small. An immediate implication is that unless judges are

sufficiently patient, the law would be characterized by doctrine that was coherent only in

part. Propositions 3 and 4 extend this insight further by showing that even when a perfectly

consistent doctrine could be sustained, its implementation is not guaranteed, and depends

on the relative bargaining strengths of judges in building doctrine.

34

6 Discussion and Conclusion

In this section, we wish to clarify the mechanism underlying our model and to suggest some

extensions.

6.1 Case Space and IDID vs Policy Space and Risk Aversion

Our exposition and our Edgeworth box diagrams might suggest that our analysis might be

situated in policy space. After all, a model in which judicial preferences over policies were

risk averse would also lead to factions preferring a certain compromise rule over the ideal

rule of each. But this suggestion misunderstands the mechanism underlying our analysis.

It is IDID not risk aversion that permits the factions to sustain a practice of stare decisis.

There are several ways to understand the centrality of case space and IDID to our argument.

First, and most obvious, in each period, the presiding judge renders judgment by an-

nouncing a disposition. The court never announces a policy in our model, it simply applies

a rule to the case before it. The applied rule summarizes a pattern of behavior; it has no inde-

pendent existence. Indeed, the rule it implies may be implicit rather than explicit, emerging

from the common practice of the court. This feature of the model parallels the operation of

common law reasoning in Anglo-American law. In traditional English common law practice,

the court, as in our model, announces only a disposition and never a rule though each judge

on the panel offers reasons for her decision on the disposition. Even in US practice, though

the court commonly does announce a rule, the court still renders judgment case-by case.

Thus compliance with a rule and punishment for deviation from the practice can only be

accomplished by the choice of dispositions.

Second, inspection of our proof of proposition 1 reveals that it rests on an analysis of

the judicial utility of compliance with stare decisis relative to deviation from it in the worst

case. If no judge has an incentive to deviate in the worst case, no judge ever has an incentive

to deviate. Thus the logic of our argument rests squarely in case space.

35

Third, the factions realize gains from trading case dispositions not from trading in policies.

It is here that IDID plays a central role. The IDID assumption provides the factions with

a medium of exchange through which they can realize their gains from trade. Consider, for

example, two judges, each of whom has a constant loss dispositional utility function; i.e.

each receives 0 from a correct disposition and negative utility from an incorrect disposition.

To be precise, let L receive a utility of −.5 from an incorrect disposition while R receives

−1. In autarky, L has an expected utility of −.25 while R has an expected utility of −.5.

There are gains from trade here – social welfare would be higher if the court adopted R’s

preferred rule to decide all cases but R cannot make a side payment to L and consequently

there is no mechanism to sustain a stare decisis equilibrium at R’s ideal point.

When dispositional preferences satisfy IDID the factions can trade dispositions on cases

because, in the conflict zone, a case that is close to faction L’s ideal point is far from faction’s

R’s ideal point. This discrepancy in valuation underlies the motivating example with which

we started and the logic of proposition 1. Without IDID, trades are not possible.

Nevertheless, one might think that IDID simply provides micro-foundations for risk aver-

sion over policies by inducing strictly concave preferences over policies. But the policy utility

function vi (z) need not be strictly concave. Judges are not necessarily risk averse over the

first stage lottery. One way to see this is to note that, if f and η are differentiable, then for

vi to be concave, we need:

(5) sgn

(d2vi (z)

dz2

)= −sgn

(z − yi

) [f (z) η′

(z; yi

)+ f ′ (z) η

(z; yi

)]< 0

Straightforwardly, the condition is satisfied in the case of the uniform distribution (i.e. when

f ′ (z) = 0). However, this is not guaranteed to be true for more general case distributions.

For example, in the region of conflict, concavity of both vL and vR requires that:

η′(z; yR

)η (z; yR)

<f ′ (z)

f (z)<η′(z; yL

)η (z; yL)

36

for each z ∈(yL, yR

), which will not be satisfied if f ′(z)

f(z)is either sufficiently negative or

sufficiently positive.

Perhaps the key difference between assuming IDID on dispositional utility and risk aver-

sion over policy utility, rests on the fact that the derivation of preferences over policies

depends both on preferences over dispositions and on the distribution of cases that court

will decide. As a consequence, one cannot conclude that the dispositional utility underly-

ing a strictly concave policy utility satisfies IDID. It may not.25 Thus, though the factions

would like to share the risk by adhering to a rule of stare decisis, they cannot accomplish

this through trading dispositions over cases.

Adjudication is a paradigmatic example of a task in which the decision maker must

sort cases or events into categories. Preferences over dispositions takes this task seriously

and attributes to the decision maker preferences that reward her for correctly sorting the

case. Many bureaucrats face similar tasks – e.g., insurance agents declining coverage or

immigration officials granting asylum. Understanding the role that various assumptions,

including IDID, on such dispositional preferences is thus apt to have wide application.

6.2 Partial Stare Decisis

We introduced rules of partial stare decisis. These rules have been ignored in both the formal

and legal literatures on judicial behavior. Nevertheless these rules play an important role

in our analysis. Over a range of δ, the only, non-autarkic equilibria are partial stare decisis

rules. Moreover, we saw that, in the agenda setter framework, a partial stare decisis rule

that does not exhaust the potential gains from trade is the equilibrium selected even when

a complete stare decisis rule would be sustainable and would exhaust the gains from trade.

25However, it can be shown that if the policy space is constructed not in terms of the location of cut-points (in case space), but rather the quantile of the case distribution implicated by the cut-point, then theIDID assumption over dispositional preferences does imply concavity of policy utility over this differentlyconstructed space. Formally, for each z, let ζ = F (z) denote the fraction of cases that will for which ajudge applying rule z will assign disposition ‘0’. Let νi (ζ) denote policy preferences over quantiles, and notestraightforwardly that νi (ζ) = vi

(F−1 (ζ)

). The IDID property of dispositional preferences implies νi is

concave in ζ.

37

Recall that under a rule (zL, zR) (with zL 6= zR), the bench decides cases, for x < zR,

as the right faction would. Similarly, it decides cases x > zR as the L faction would. In the

interval (zL, zR), we have autarky. This structure implies that a body of law governed by

a rule of partial stare decisis may look incoherent over the interval (zL, zR). Moreover the

larger this interval, the more incoherent the body of law will appear. With this incoherence

as a marker, we might identify a number of highly criticized doctrines as bodies of law

governed by partial stare decisis. The fact that no clear complete rule has emerged should

not conceal the fact that judges with distinct jurisprudential views on the ideal rule have

managed to reduce the extent of their disagreement and to adhere partially to stare decisis.

Consider, for example, search and seizure law in the United States. The Constitution

protects individuals against “unreasonable searches and seizures” of their persons and prop-

erty. Challenges to warrantless searches and seizures are frequent and the law is generally

considered incoherent.26 There are clear cases, at both extremes, on which the left and the

right are agreed. Between these extremes, the decisions seem chaotic and incoherent. That

incoherence is consistent with partial stare decisis as, between the extremes, decisions will

be rendered by the judge designated to decide the case.

In terms of our model, one might identify the one-dimensional case space as the “degree

of intrusiveness of the search”. At the extreme intrusive end of the spectrum lie searches

incident to unwarranted intrusions into the individual’s home; on the other end of the spec-

trum lie “searches” of, say, cars that uncover drugs in “plain view.” Admittedly, the area

in between the endpoints of the policy may be vast but the courts can be understood as

following a rule of partial stare decisis.

As a second, somewhat stylized, example, consider the rules governing which terms con-

stitute the contract. This issue arises often in on-line transactions when a consumer assents

not only to the core terms of the contract (such as price and physical product description)

as well as to “additional terms” that are more or less easily accessed. In other cases, these

26Wasserstrom and Seidman (1988), for example, cite critics on both the right and the left.

38

additional terms are only revealed after the product is delivered (“buy now, terms later”

agreements). These questions arise frequently both in on-line and physical transactions.

For purposes of the model, consider a one-dimensional case space that one might call “con-

spicuousness”. Conspicuousness refers to prominence of a notification of the existence of

additional post-formation terms and the ease with which they may be accessed. Courts gen-

erally agree that low conspicuousness implies unenforceability of the “hidden” terms while

high conspicuousness implies enforceability.27 For intermediate degrees of conspicuousness,

consumer-oriented judges decide one way – typically that the post-formation terms are not

part of the contract – while business-oriented judges decide the other way.

The use of rules of partial stare decisis thus identify an underlying structural coherence

to bodies of law that superficially appear to be chaotic and confused.28.

6.3 Extensions

We have assumed that the case-generating process does not vary with the prevailing rule of

decision on the court. This assumption is common in models of courts but highly likely to

be counterfactual. Litigants are apt to settle “easy” cases and litigate “hard” ones. Case

difficulty will depend on the prevailing rule of decision. Thus we might expect that, under

autarky, most litigated cases will fall within the conflict zone and very few outside it. Under

a rule of complete stare decisis, we might expect most litigated cases to fall near the cut-off

point zC of the compromise legal rule while under a rule of partial stare decisis, litigated cases

might fall primarily within the interval [zL, zR]. Rule-dependent case-generating processes

will thus alter the size of the gains from trade available to the two factions. Further work

might determine the extent to which this undermines or supports a practice of stare decisis.

We explicitly model a structure similar to that of an intermediate court of appeals in

27Restatement of Consumer Contracts § 2 (Am. Law Inst., Discussion Draft, 2017). We thank FlorenciaMarotta-Wurgler for directing us to this doctrine and source.

28There are other potential explanations. The law might, of course, simply be confused. Or the law mightbe evolving with the judges clarifying the law over time. On this latter account, we should observe the sizeof the interval (zL, zR) shrinking over time.

39

which cases are decided by randomly drawn panels of judges. But we think a similar mecha-

nism might be at play on courts such as the Supreme Court of the United States that sit en

banc. Author-influence models of these courts, (e.g. Lax and Cameron (2007)), suggest that

the author of an opinion has significant power to structure the announced rule. When the

bench divides into two distinct factions, a dynamic similar to the one modeled may emerge.

But extant models of opinion assignment are formulated in policy space rather than case

space so the conclusions of our model do not follow immediately.

A Appendix

Proof of Lemma 1. We prove the more general result discussed in footnote 15. For con-

ceteness, consider a judge from faction L. Let zL be any threshold. Then:

vL(yL, zR

)− vL

(zL, zR

)= p

∣∣∣∣∣∫ zL

yLη(x; yL

)dF (x)

∣∣∣∣∣ ≥ 0

which implies that implementing her ideal cutpoint derives at least as much utility as any

other cutpoint, ceteris paribus. Furthermore, since η(x; yL

)> 0 for all x 6= yL and since

F is weakly increasing, the expression must be strictly positive whenever F(zL)6= F

(yL).

Hence zL is optimal only if F(zL)

= F(yL). Moreover, if yL ∈ int {supp (F )}, then F is

strictly increasing in the neighborhood of yL, and so F(zL)

= F(yL)

implies zL = yL.

Proof of Lemma 2. (Only if ⇒). Suppose F(zL)< F

(yL). Then, since η (x; yi) > 0 for

all x 6= yi and since F is weakly increasing:

vL(yL, zR

)− vL

(zL, zR

)= p

∫ yL

zLη(x; yL

)dF (x) > 0

and:

vR(yL, zR

)− vR

(zL, zR

)= (1− p)

∫ yR

zLη(x; yR

)dF (x) > 0

40

which implies that(yL, zR

)is a Pareto improvement over

(zL, zR

). Hence, if

(zL, zR

)is

Pareto optimal, F(zL)≥ F

(yL). Using similar arguments, we can show that F

(yL)≤

F (zi) ≤ F(yR)

for i ∈ {L,R}.

Now, suppose F(zL)6= F

(zR). WLOG suppose zL < zR and let zC ∈

(zL, zR

)be such

that F(zC)

= pF(zL)+(1− p)F

(zR). By the continuity of F , such a zC must exist. Then:

vL(zC , zC

)− vL

(zL, zR

)= (1− p)

∫ zR

zCη(x; yL

)dF (x)− p

∫ zC

zLη(x; yL

)dF (x)

> (1− p)∫ zR

zCη(zC ; yL

)dF (x)− p

∫ zC

zLη(zC ; yL

)dF (x)

= η(zC ; yL

) [pF(zL)

+ (1− p)F(zR)− F

(zC)]

= 0

where the second line makes use of the fact that η(·; yL

)is strictly increasing on the interval[

zL, zR], that F

(zL)< F

(zC)< F

(zR), and that F is weakly increasing. By a sym-

metric argument, we can show that vR(zC , zC

)> vR

(zL, zR

). Hence

(zC , zC

)is a Pareto

improvement over(zL, zR

). The proof follows by contrapositive.

(If⇐). Let F(yL)≤ F

(zL)

= F(zR)≤ F

(yR), but suppose that

(zL, zR

)is not Pareto

optimal. Then there exists some other pair(zL′, zR

′)which is a Pareto improvement over(

zL, zR). Clearly, if F

(zL′)

= F(zL)

and F(zR′)

= F(zR), then vL

(zL′, zR

′)= vL

(zL, zR

)and vR

(zL′, zR

′)= vR

(zL, zR

), which is not a Pareto improvement. Hence, either F

(zL′) 6=

F(zL)

or F(zR′) 6= F

(zR).

Suppose[F(zL′)− F (zL)] [F (zR′)− F (zR)] ≥ 0. Recall that vL is decreasing in zL

and zR (whenever these are above yL) and vR is increasing in zL and zR (whenever these are

below yR). It follows that either vL(zL′, zR

′)< vL

(zL, zR

)or vR

(zL′, zR

′)< vR

(zL, zR

).

Hence, we must have[F(zL′)− F (zL)] [F (zR′)− F (zR)] < 0. Since F

(zL)

= F(zR),

this implies zL′ 6= zR

′.

WLOG, suppose zL′< zR

′. (A symmetric arugment holds for the reverse case.) Let

41

z ∈(zL′, zR

′)be s.t. F (z) = pF

(zL′)

+ (1− p)F(zR′)

. If F (z) ≥ F(zL)

= F(zR), then

we know that vL (z, z) ≤ vL(zL, zR

). Moreover, by arguments in an earlier part of this proof,

we know that vL (z, z) > vL(zL′, zR

′). Then, since vL

(zL′, zR

′)> vL

(zL, zR

), it must be

that F (z) < F(zL). By a symmetric argument for R, we can conclude that F (z) > F

(zL),

which is a contradiction.

Proof of Proposition 1. The first part of the proposition follows immediately from the

discussion that precedes it. It remains to prove the enumerated points. First, we show that,

as defined, δ(zC)> 0 for all zC . It suffices to show that the denominators of both δL and

δR are always positive. To see this for δL, note that:

η(zC ; yL

)+ vL

(zC , zC

)− vL

(yL, yR

)=η(zC ; yL

)− p

∫ zC

yLη(x; yL

)dF (x) + (1− p)

∫ yR

zCη(x; yL

)dF (x)

≥η(zC ; yL

)− p

∫ zC

yLη(x; yL

)dF (x)

≥η(zC ; yL

)− p

[F(zC)− F (y)

]η(zC ; yL

)>0

where the 4th line makes use of the IDID property. A similar approach verifies the result

for δR.

To show (1), note that δ(zC)> 1 provided that ∆vi

(zC)

= vi(zC , zC

)− vi

(yL, yR

)< 0

for either i ∈ {L,R}. Since η and F are both continuous, so are vi(zC , zC

)are ∆vi

(zC).

Now, ∆vL(zC)

is decreasing in zC . (To see this, take z′ > z. Then ∆vL (z′) = ∆vL (z) −∫ z′zη (x; yi) dF (x) ≤ ∆vL (z), since η ≥ 0.) Furthermore, ∆vL

(yL)> 0 and ∆vL

(yR)< 0.

Then, by the intermediate value theorem, there exists a unique z ∈(yL, yR

)s.t. ∆vL (z) = 0,

and ∆vL (z) > 0 provided that z < z. Similarly, we can show that ∆vR(zC)

is increasing in

zC , and that ∆vR(yL)< 0 and ∆vR

(yR)> 0. Then, there exists a unique z ∈

(yL, yR

)s.t.

∆vR (z) = 0, and ∆vR (z) > 0 provided that z > z.

42

It remains to show that z ≤ z. Suppose not. Then for every zC ∈[yL, yR

], either

∆vL(zC)< 0 or ∆vR

(zC)< 0. But this implies that there are no Pareto improvements

over Autarky, which contradicts Lemma 2. Hence z ≤ z.

To show (2), note that we have now shown that, for i ∈ {L,R}, ∆vi(zC)> 0 for every

zC ∈ (z, z). Since:

δi(zC)

=1

1 + ∆vi(zC)η(zC ;yi)

then δi(zC)

is increasing in zC iff∆vi(zC)η(zC ;yi)

is decreasing in zC . Now, we previously showed

that ∆vL is decreasing and ∆vR is increasing in zC , for zC ∈[yL, yR

]. Moreover, in this

region, η(zC ; yL

)is increasing and η

(zC ; yR

)is decreasing. Hence

∆vL(zC)η(zC ;yL)

is decreasing and

so δL(zC)

is increasing in zC . Likewise,∆vR(zC)η(zC ;yR)

is increasing and so δR(zC)

is decreasing

in zC .

Finally, let D(zC)

= δL(zC)−δR

(zC). We have shown that D

(zC)

is strictly increasing

in zC .By construction, δL (z) = 1 and δR (z) = 1. Since δL (resp. δR) is increasing (resp.

decreasing), it follows that δL (z) < δR (z) = 1 and δR (z) < δL (z) = 1. Hence D (z) < 0

and D (z) > 0. Furthermore, since η and F are continuous, so are δi and D. Then, by the

intermediate value theorem, there exists a unique z∗ ∈ (z, z) s.t. D (z∗) = 0. Now, since

δ(zC)

= max{δL(zC), δR

(zC)}

, the previous discussion implies:

δ(zC)

=

δR(zC)

zC < z∗

δL(zC)

zC > z∗

Then, by the properties of δL and δR, δ(zC)

is decreasing on (z, z∗) and increasing on (z∗, z).

Finally, this implies δ(zC)≥ δ (z∗) = δ∗ for all zC .

Proof of Lemma 3. To see (1), recall from the proof of Proposition 1 that δi(zC)

is

strictly decreasing in∆vi(zC)η(zC ;yi)

. Moreover, note that the denominator of this expression is

43

independent of p, and that:

∆vi(zC)

= sgn(zC − yi

) [∫ yR

zCη(x; yi

)dF (x)− p

∫ yR

yLη(x; yi

)dF (x)

]

Since∫ yRyL

η (x; yi) dF (x) > 0, then ∆vL(zC)

is strictly decreasing in p and ∆vR(zC)

is

strictly increasing in p. Hence, δL(zC)

and δR(zC)

are strictly increasing and strictly

decreasing in p, respectively.

Next, to see (2), recall that z∗ is the solution to D(zC)

= δL(zC)− δR

(zC)

= 0. We

showed in proposition 1 that δL and δR are increasing and decreasing in zC , respectively,

and so D is increasing in zC . Moreover, part (1) of this lemma showed that D is strictly

increasing in p. Now, let p′ > p, and let D (z∗; p) = 0 and D (z′; p′) = 0. Suppose z′ ≥ z∗.

Then 0 = D (z′; p′) ≥ D (z∗; p′) > D (z∗; p) = 0, which is a contradiction. Hence z′ < z∗.

To see (4), note that z and z are defined by: δR (z) = 1 and δL (z) = 1. Let p′ > p. Let

z = z (p) and z′ = z (p′). Similarly, let z = z (p) and z′ = z (p′). Suppose z′ ≥ z. Then, since

δR is increasing in p, 1 = δR (z′, p′) ≥ δR (z; p′) > δR (z; p) = 1, which is a contradiction.

Hence z′ < z. An analogous argument shows that z′ < z.

Proof of Lemma 4. To see (1), recall from the proof of Proposition 1 that δi(zC)

is

strictly decreasing in∆vi(zC)η(zC ;yi)

. Again, note that the denominator of this expression is inde-

pendent of ρ, and that the numerator can be re-written:

ρ

{sgn

(zC − yi

) [(1− p)

∫ yR

zCη(x; yi

)dF(x;[yL, yR

])− p

∫ zC

yLη(x; yi

)dF(x;[yL, yR

])]}

Now, since we are in the region of gains from trade, ∆vi(zC)≥ 0, and so the term in braces

must be non-negative. Further note that the term in braces only depends on the conditional

distribution of F , and so is independent of ρ. Hence ∆vi is increasing in ρ, and so δi is

weakly decreasing in ρ.

Part (3) and the results concerning z and z in part (2) follow straightforwardly, using

44

similar arguments to those in the proof of Lemma 3. To show the result about z∗ in part

(2), note that z∗ solves∆vL(zC)η(zC ;yL)

=∆vR(zC)η(zC ;yR)

. Let ϕi(zC)

denote the term in braces, so that

∆vi(zC)

= ρϕi(zC). Then, z∗ is the solution to:

ϕL(zC)

η (zC ; yL)=

ϕR(zC)

η (zC ; yR)

But since ϕi and η are both independent of ρ, it follows that z∗ must be as well.

Proof of Proposition 2. Recall that(zL, zR

)is sustainable at δ if φi

(zL, zR, δ

)=

−η (zi; yi) + δ1−δ∆v

i(zL, zR

)≥ 0 for i ∈ {L,R}. Since η is continuous and vi are both con-

tinuous, then φi is jointly continuous in(zL, zR, δ

). To show (1), suppose

(zL, zR

)∈ Z (δ).

Then φi(zL, zR, δ

)≥ 0 which implies ∆vi

(zL, zR

)≥ 0 (since η ≥ 0). Then, if δ′ > δ,

φi(zL, zR, δ′

)≥ φi

(zL, zR, δ

)≥ 0 (and the first inequality is strict if ∆vi

(zL, zR

)> 0).

Hence(zL, zR

)∈ Z (δ′), and so Z (δ) ⊂ Z (δ′).

To show (2), note that since η (yi; yi) = 0, φi(yL, yR, δ

)= 0 for every δ ∈ [0, 1]. Hence(

yL, yR)∈ Z (δ) for every δ.

To show (4), first note that by construction, φi (z∗, z∗, δ∗) = 0 for each i. Take some

ε > 0, and let zL′= z∗ − (1− p) ε and zR

′= z∗ + pε. Note that zL

′ 6= zR′. We seek to show

that φi(zL′, zR

′, δ∗)> 0. For concreteness, take i = L. Then:

d

dεφL(zL′, zR

′, δ∗)

= (1− p) η′(zL′; yL)

+δ∗

1− δ∗p (1− p)

[η(zL′; yL)f(zL′)− η

(zR′; yL)f(zR′)]

and so ddεφL(zL′, zR

′, δ∗)|ε=0 = (1− p) η′

(z∗; yL

)> 0. Hence, for ε > 0 small enough,

φL(zL′, zR

′, δ∗)> φL (z∗, z∗, δ∗) = 0. We can similarly show that φR

(zL′, zR

′, δ∗)> 0. Now,

since φi is continuous and increasing in δ, there exists δ′ < δ∗, such that(zL′, zR

′) ∈ Z (δ′).

But by construction, we know that there is no complete stare decisis regime (z′, z′) for which

(z′, z′) ∈ Z (δ′). This proves (4).

Finally, to show (3), let δ∗∗ = sup{δ|Z (δ) =

{(yL, yR

)}}. Since Z (0) =

{(yL, yR

)},

the sup is well defined. Moreover, δ∗∗ ∈ [0, δ∗), since δ∗∗ ≤ δ′ < δ∗. We now show the

45

relationship between δ∗∗ and η′ (yi; yi). First, note that φi(yL, yR, δ

)= 0. Now, define

zL′′

= yL + (1− π) ε′ and zR′′

= yR − πε′, for some π ∈ [0, 1] and ε′ > 0. By choosing π

appropriately,(zL′′, zR

′′)spans every rule pair in an ε′−neighborhood of

(yL, yR

). Again we

focus on φL, and note that the proof for φR is analogous. Then:

d

dε′φL(zL′′, zR

′′, δ)

= (1− π) η′(zL′′; yL)

+δ

1− δ

[(1− π) pη

(zL′′; yL)− π (1− p) η

(zR′′; yL)]

and so ddε′φL(zL′′, zR

′′, δ)|ε′=0 = − (1− π) η′

(yL; yL

)+ δ

1−δπ (1− p) η(yR; yL

). Now, if η′ =

0, then ddε′

> 0, and so(zL′′, zR

′′) ∈ Z (δ) whenever(zL′′, zR

′′)is in an ε′-neighborhood of(

yL, yR). We have δ∗∗ = 0. Suppose instead η′ > 0. Then, for δ small enough d

dε′< 0, and so

for every(zL′′, zR

′′)in the neighborhood of

(yL, yR

),(zL′′, zR

′′)/∈ Z (δ). Hence δ∗∗ > 0.

Proof of Proposition 3. Suppose judge i is the agenda setter. Her problem is:

max(zL,zR) vi(zL, zR

)s.t.

(zL, zR

)∈ Z (δ). By Proposition 2, we know that

(zL, zR

)∈ Z (δ)

provided that φi(zL, zR, δ

)≥ 0 for each i ∈ {L,R}. and zR ≥ zR

(zL;φR = 0

). Suppose

δ > δ∗∗ (otherwise, autarky is the only sustainable equilibrium, and so that is what is cho-

sen.) Intuitively, the agenda setter holds judges of the other faction to indifference. Hence,

the problem becomes: max(zL,zR) vi(zL, zR

)s.t. φ−i

(zL, zR, δ

)≥ 0.

Forming the Lagranian, and solving for the first order conditions implies:viLviR

= ∂φ−i/∂zL

∂φ−i/∂zL,

where vij = ∂vi

∂zj. Recall that:

viLviR

= p(1−p)

η(zL;yi)f(zL)η(zR;yi)f(zR)

. Moreover, if i = R, then

∂φ−i/∂zL

∂φ−i/∂zR=

1−δδη′(zL; yL

)+ pη

(zL; yL

)f(zL)

(1− p) η (zR; yL) f (zR)

and if i = L, then:

∂φ−i/∂zL

∂φ−i/∂zR=

pη(zL; yR

)f(zL)

1−δδη′ (zL; yL) + (1− p) η (zR; yR) f (zR)

. Let(zL, zR

)be a maximizer. If zL = zR (i.e. if the optimum is characterized by a complete

stare decisis regime), thenviLviR

= p1−p . But, by inspection, it is clear that ∂φ−i/∂zL

∂φ−i/∂zR|zL=zR = p

1−p

46

iff δ = 1. Hence, a complete stare decisis regime is chosen only if δ = 1.

Proof of Proposition 4. Let ϕ(zL, zR

)= ∆vL

(zL, zR

)∆vR

(zL, zR

), where

∆vi(zL, zR

)= vi

(zL, zR

)− vi

(yL, yR

). For each δ, the Nash Bargaining outcome is

the solution to: max(zL,zR) ϕ(zL, zR

)s.t.

(zL, zR

)∈ Z (δ).

First, consider the unconstrained optimum. Clearly, the optimum must be Pareto op-

timal, or we can find a rule-pair that generates a larger value for the objective. Hence,

the Nash Bargaining solution maximizes ϕ (z). By Proposition 1 ∆vL (z, z) > 0 for

z < z and ∆vR (z, z) > 0 for z > z, with z < z. Hence ϕ (z) > 0 iff z ∈

(z, z), and so obviously the optimizer must lie in this open interval. Now, ϕ′ (z) =[η(z; yR

)∆vL (z, z)− η

(z; yL

)∆vR (z, z)

]f (z). Solving ϕ′ (z) = 0 implies: ∆vR(z,z)

∆vL(z,z)=

η(z;yR)η(z;yL)

. Notice, the left-hand term is strictly increasing in z and the right-hand term is

strictly decreasing in z. Hence there is a unique zNB that is the unconstrained Nash Bar-

gaining solution. Let δNB = δ(zNB

). Recall that δ∗ ≤ δ (z) < 1 for all z ∈ (z, z). Hence

δNB ∈ [δ∗, 1). Now, since(zNB, zNB

)is the unconstrained optimum, it is also the constrained

optimum whenever it is feasible given the constraint. This will be true whenever δ > δNB.

Finally, we must show that the optimal zL 6= zR when δ ∈[δ∗, δNB

). (It is obvious for

δ < δ∗, since complete stare decisis regimes are unsustainable.) Since δ < δNB, one of the

constraints (φL = 0 or φR = 0) binds. Formulating the Lagranian and taking first order

conditions implies ∂ϕ/∂zL

∂ϕ/∂zR= ∂φ−i/∂zL

∂φ−i/∂zLat the optimum. We easily verify that if zL = zR, then

∂ϕ/∂zL

∂ϕ/∂zR= p

1−p 6=∂φ−i/∂zL

∂φ−i/∂zL.

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