ISSN (Print) 0473-453X

Discussion Paper No. 1092

ISSN (Online) 2435-0982

LEARNING
WHILE SETTING PRECEDENTS
Ying Chen
Hülya Eraslan

June 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Learning While Setting PrecedentsHülya Eraslan∗

Ying Chen

June 28, 2020
Abstract
A decision maker makes a ruling on a random case in each period. She is uncertain about the correct ruling until conducting a costly investigation. A rulingestablishes a precedent, which cannot be violated under binding precedent. We
compare the information acquisition incentives, the evolution of standards andthe social welfare under nonbinding and binding precedents. Compared to nonbinding precedent, under binding precedent, information acquisition incentives
are stronger in earlier periods, but become weaker as more precedents are established. Although erroneous rulings may be perpetuated under binding precedent,
welfare can be higher because of the more intensive investigation early on.
JEL Classification: D02, D23, D83, K4.
Keywords: precedent; binding precedent; information acquisition; transparency.

∗

Chen: Department of Economics, Johns Hopkins University, ying.chen@jhu.edu. Eraslan: Department of Economics, Rice University and Institute of Social and Economic Research, Osaka University,
eraslan@rice.edu. We thank discussants Tom Clark, Ming Li and Jennifer Reinganum; we also thankScott Baker, Xinyu Hua, Lewis Kornhauser, Aniol Llorente-Saguer, Kathryn Spier, Bruno Strulovici,
Eric Talley, Richard Van Weelden, Rakesh Vohra, Jingyi Xue and seminar and conference participants
at Chicago, George Mason, HKUST, Johns Hopkins, Kyoto, LSE, Nagoya, Northwestern, NUS, NYU,
Osaka, Princeton, Rice, Singapore Management University, Texas A&M, Warwick, Law and Economic Theory Conference 2015, Texas Theory Camp 2015, NSF Decentralization Conference 2016,
Cornell Political Economy Conference 2016, Econometric Society Asia Meeting 2016, Stony Brook
Law and Economics Workshop 2016, Game Theory World Congress 2016, Microeconomics Workshop
at SHUFE 2017, Game Theory Workshop at Zhejiang University 2017, 5th Annual Formal Theory andComparative Politics Conference at Emory, HKUST Workshop on Law and Economics 2017, Econometric Society Asia Meeting 2018, University of Queensland and University of Technology Sydneyfor helpful comments and suggestions. Eraslan thanks the International Joint Research Promotion
Program (Osaka University).

1

In our progress towards political happiness my station is new; and, if I may use theexpression, I walk on untrodden ground. There is scarcely any part of my conduct
wch. [sic] may not hereafter be drawn into precedent. Under such a view of the duties
inherent to my arduous office, I could not but feel a diffidence in myself on the one
hand; and an anxiety for the Community that every new arrangement should be madein the best possible manner on the other.
George Washington, 9 January 1790

1

Introduction

Decision makers within organizations typically face and must address a series ofproblems under uncertainty over time. For example, an organization’s executive could
be tasked with deciding whether to accept or reject a series of project proposals, andgovernment agencies such as the FDA evaluate proposals for approval or rejection. For
reasons of consistency, fairness, predictability and public confidence, a cost is often
associated with treating similar cases differently. Thus, a yes-or-no decision (which
we call a ruling) regarding a current case may affect future decisions by creating a
precedent. Acquiring the information needed to make the correct ruling is costly, butthese decisions may have far-reaching consequences through two channels. First, theinformation acquired may reveal some general principle applicable not just to the caseat hand but to future cases as well. Second, as the ruling on the current case becomesa precedent, the decision maker may incur some cost in making a different ruling on afuture case that has similar characteristics, and this cost associated with overturninga precedent may prevent the decision maker from making certain rulings, even when
it becomes clear that an erroneous ruling was made on a previous case. Our primary
goal in this article is to understand how the dynamic consequences of precedents affectthe decision maker’s incentive to acquire information and in turn the overall qualitiesof the decisions.
This kind of dynamic problem solving has a broad range of applications, including
judge-made law, which provides a natural application for our study because of thecentral role of precedents in common law systems. For example, the U.S. Bankruptcy
Code allows firms to restructure their debt under Chapter 11 by confirming a reorganization plan. This can be achieved if all classes of creditors and interest holders voteto accept the plan. Without the consent of all classes, confirmation is possible only if2the reorganization plan is judged to be “fair and equitable” to each class that voted
against it. However, the definition of “fair and equitable” within the Bankruptcy Codeis not entirely clear and the standard has therefore been set gradually over time by the
courts on a case-by-case basis.
Another reason that judge-made law provides a good application for our study isvariation within the legal system in the effects that precedents can have on futurecases that are similar. United States law, for example, has two kinds of precedents:
persuasive and binding. Under a persuasive precedent, a judge is not required to follow
previous rulings but can use the information acquired in the ruling of a previous case.
Under binding precedent, in contrast, the principle of stare decisis requires that a judge
must follow previous rulings when they apply. We note that both kinds of precedents
exist in the legal system. In the U.S. legal system, for example, a lower court is boundby a precedent set by a higher court in its region, but cases decided by lower courts orby peer or higher courts from other jurisdictions are only persuasive and not binding.1
Given our objective, the institutions of common law system provide a fitting framework
to anchor our analysis.
We analyze the DM’s incentives an infinite-horizon model. We find that in early
periods, when few precedents have been established, the DM’s incentives to acquire
information are stronger if precedents are binding. But as more precedents are established over time, the incentive to acquire information becomes weaker under binding
precedents. Our results suggest that decision makers spend more time and resources
deliberating on cases when an organization is in an early stage of development and lesswhen its rules and standards have matured. Furthermore, this contrast is especiallypronounced when precedents are binding.
To see why, note that the cost of making a wrong decision is higher under binding precedent because the DM has to follow precedents in the future when they arebinding even if the previous rulings that established these precedents turn out to be
erroneous. Because of the long-term repercussions of an early erroneous ruling whenprecedents are binding, a DM who faces few precedents is more inclined to acquire
information to avoid making mistakes. (This point is illustrated by the “diffidence”
and “anxiety” expressed by the first U.S. President George Washington in a letter tohistorian Catharine Macaulay.) As more precedents are established over time, however,1Modeling the legal hierarchy is beyond the scope of our study. For models of judicial learning andprecedent that incorporate the legal hierarchy, see, for example, Bueno De Mesquita and Stephenson[2002] and Callander and Clark [2017].

3

the value of information becomes lower under binding precedent because the DM, now
bound by previous rulings, may not be able to use the newly acquired information,and is therefore discouraged from acquiring information under binding precedent.2 Aninteresting parallel is the “two-phase” process discussed in the seminal work on organizations by March and Simon [1958]: when a new organizational unit is created todevelop a new program, there is “a spurt of innovative, program-developing activity”
which automatically diminishes as the program is elaborated and the unit is “boundand hampered by traditions and precedents.”
The appropriate social welfare criterion for our model is not obvious because theDM is the only agent explicitly modeled. Provided the rest of society does not bear
the cost of information acquisition but cares about the rulings, the payoffs derived
from the ruling decisions may constitute a reasonable measure of the social welfare.3
For example, judicial decision-making can be viewed as a principal-agent problem inwhich society delegates important decisions to courts. Because the court’s rulings havebroad implications that affect the society at large, it would be misleading to use theDM’s payoff alone to measure the social welfare. When using payoffs from rulings
as our welfare measure, we find that the social welfare can be higher under binding
precedent than under nonbinding precedent.4 This is because the social benefit comingfrom the more intensive investigation conducted by the DM early on can outweighlosses attributable to the persistent mistakes in ruling potentially arising under binding
precedent. Our result shows that increasing the cost of violating precedent can provide
an effective way to strengthen a DM’s incentive to gather information and improve thequality of rulings.
Although our measure of social welfare excludes the cost of investigation, this cost
nonetheless indirectly affects it through the effects on incentives. When the cost ofinvestigation is high relative to the importance of the rulings, the gain from more
intensive early investigation under binding precedent does not compensate for the losses
resulting from perpetuating erroneous rulings. As a result, the social welfare is lower
under binding precedent. But when the investigation cost is relatively low, the gain
2

Even though the lack of information acquisition seems consistent with the current U.S. President’s
behavior, we do not believe that our model provides a good explanation for it.
3
The seminal article by Tullock [1971] discusses the tension between the private cost of informationacquisition and the public benefit of correct decisions, which results in under-investment in informationacquisition, but it does not provide any solutions. For a synthesis of solutions to this problem, see
Stephenson [2010].
4
Our result continues to hold when we allow for weights on the DM’s utility relative to society as
long as this weight is small.

4

from more intensive early investigation under binding precedent dominates, resulting
in higher social welfare.5,6 As a result, our finding is that binding precedent tends
to induce higher social welfare when decisions are more important, pointing out anadvantage of a system that gives more important rulings more binding power such as
the hierarchy in the U.S. legal system.
Apart from institutions for which costs of violating precedent are heterogeneous,such as those in the legal system, other interpretations make our analysis applicable.
Specifically, the cost of violating precedent may correspond to the degree of transparency in the decision making process. DMs whose actions are hidden face no penalty
in treating similar problems differently. However, a DM whose actions are visible tothe public can be held accountable for inconsistency or unfairness when overturning a
precedent. In light of this, our findings demonstrate a social benefit associated withtransparency in decision making that has not being pointed out in the literature. Bymaking a violation of precedent public and thus punishable, transparency encourages
more deliberate and careful initial decision making.7 Indeed, some organizations choose
to publish their decisions and refer to these decisions as precedents. For example, theU.S. Department of Justice Executive Office for Immigration Review publishes precedent decisions, and some Australian universities publish their transfer credit decisions
in an online database.8
Related literature
The study most closely related to ours is Baker and Mezzetti [2012], which analyzes the process by which a long-lived court makes rulings under uncertainty whenprevious rulings become precedent. Unlike our study, they do not explicitly distinguishbetween nonbinding precedent and binding precedent and, in effect, analyze only nonbinding precedent. The focus of Baker and Mezzetti [2012] is to show how analogicalreasoning arises endogenously as optimal behavior of a DM when resources are limited.
5This result is established analytically in a discrete framework in section 5. We also show it
numerically in the continuous model because closed-form solutions are not obtainable. (Computercode and results are available upon request.)
6
As mentioned earlier, there may be other advantages of binding precedent such as consistency andpredictability. We deliberately do not consider them because our focus is to explore the implicationsof binding precedent for information acquisition.
7
Increasing the transparency of an agent’s action, or of the consequence of an agent’s action, canhave subtle effects. See, for example, Prat [2005], Fox [2007], Levy [2007], Fox and Van Weelden[2012].
8
See https://www.uscis.gov/laws/precedent-decisions and http://www.uq.edu.au/uqabroad/uqcredit-precedent-database.

5In contrast, the focus of our study is the comparison of a DM’s incentives to acquire
information under nonbinding versus binding precedent.
Like Baker and Mezzetti [2012], several other articles provide reasons for why
stare decisis arises endogenously in equilibrium (Kornhauser [1989], Rasmusen [1994],
Daughety and Reinganum [1999], Talley [1999], Anderlini, Felli, and Riboni [2014],
and Cameron, Kornhauser, and Parameswaran [2017]). Levy [2005] looks at a modelin which judges are driven by career concerns and finds that binding precedent increases
the probability that a ruling contradicting precedent will be subject to reversal, whichis costly for the judge and therefore causes her to behave inefficiently. Another perspective on precedent is provided by Gennaioli and Shleifer [2007a]: they analyze a modelin which the law evolves when judges distinguish cases from precedents by adding newmaterial dimensions while at the same time endorsing the existing precedent.
A primary insight obtained through our study is that although mistakes may perpetuate when overruling a precedent is costly, it is precisely this ex post inefficiency
that motivates the decision maker to acquire more information so as to avoid making
the mistakes in the first place. Our article is therefore connected to a strand of literature showing how ex post inefficient rules provide incentives for agents to gather
costly information, leading to better decision-making overall. Li [2001] shows that it
can be optimal for a group to commit to a “conservative” decision rule that biases
against the alternative favored by the group’s common prior because it alleviates the
problem of free-riding in information acquisition among the group’s members. Szalay[2005] investigates how the set of feasible actions affects an agent’s decision regarding
how much information to acquire. He finds that it may be optimal for the principal to
give only extreme options to the agent to motivate him to collect more information.9
The remainder of this article is organized as follows. In the next section, we present
our model. In section 3, we discuss a three-period example to illustrate some of theintuition before analyzing the infinite-horizon model in section 4. Then, in section 5,
we compare the social welfare under nonbinding and binding precedents. In section
6, we discuss various extensions of our model and conclude. Appendix A contains theproofs and Appendix B presents our formal results on partial learning.

9

In contrast, in Aghion and Tirole [1997], Baker, Gibbons, and Murphy [1999] and Armstrong
and Vickers [2010], to motivate the agent to exert more effort to acquire information, the principal
allows a larger set of options, including some that are undesirable for the principal.

6

2

Model

A decision maker (DM) in an organization regulates a set of activities by permitting
or banning them. In each period, a new case arises which must be decided by the DM.
The DM prefers to permit activities that she regards as beneficial and ban activities
which she regards as harmful.
As in Baker and Mezzetti [2012], we model a case by x ∈ [0, 1]. For example,
the DM could be regulating the duration of non-competition agreements allowed in
employment contracts, and x might correspond to the duration of the non-competeclause in a given contract. Alternatively, the DM could be regulating copyright, andx may correspond to the distance between the characteristics of a new product andan established intellectual property. The DM has a threshold value θ that determines
whether x is socially beneficial or not. The preference parameter θ is unknown initially,
and we assume that θ is distributed according to a continuous cumulative distributionfunction F with support [θ, θ] where 0 ≤ θ < θ ≤ 1. In the context of non-competition
agreements, we can think of θ as the socially optimal duration of non-compete clause
for a given industry, and the DM would like to permit x if and only if x ≤ θ. In thecontext of copyright, we can think of θ as the distance beyond which property rightis not violated, and the DM would like to permit x if and only if x ≥ θ. As such,our model can be applied to either vertically differentiated cases (as in the example
of non-competition agreement) or horizontally differentiated cases (as in the example
of copyright). For expositional simplicity, for the remainder of the article, we assume
that the DM regards x ≤ θ as socially beneficial and would like to permit it if and only
if x ≤ θ.
Denote the case at time t by xt . We assume that the cases are independent acrossperiods and each has a continuous cumulative distribution function G. In practice, onerole that precedent plays is to provide guidance to various agents in society so thatthey can take proper precautions or regulate their activities. As such, precedents may
affect the distribution of cases. Having endogenous arrival of cases is interesting but isbeyond the scope of our article. 10
The precedent at time t is captured by two numbers Lt and Rt where Lt is thehighest case that was ever permitted and Rt is the lowest case that was ever banned
10

For a model of judicial learning with endogenous arrival of cases, see Parameswaran [2018]. For
other articles that incorporate endogenous arrival of cases, see Che and Yi [1993] and Fox and Vanberg[2014].

7

by time t. Assume that L1 = 0 and R1 = 1, that is, the initial precedent is consistent
with the DM’s preferences and does not impose any mistake in ruling.
In period t, after case xt is realized, the DM chooses whether to conduct an investigation or not before deciding whether to permit or ban the case. An investigation
allows the DM to learn the value of θ at a fixed cost z > 0. We assume that the
DM learns about her preference parameter θ through investigation. Alternatively, we
can assume that the DM learns about her preferences in terms of the consequences ofcases, but does not know the consequence of a particular case unless she investigates.
To illustrate, let c(x) denote the consequence of a case x and assume that c(x) = x + γ,
where γ is the same across cases and initially unknown. The DM would like to permit
case x if c(x) is below some threshold c̄ and would like to ban it otherwise. Supposethat the DM knows c̄ and observe x, an an investigation reveals γ. This alternativemodel is equivalent to ours.
Because we focus on the incentives in case-by-case decision making, our model
assumes that the DM decides whether to investigate after a case is realized. An alternative timing is for the DM to decide whether to learn θ before the realization of a case.
In this alternative setup, we can show that generically, the equilibrium outcomes are
the same under nonbinding and binding precedents. In the knife-edge case in which the
DM randomizes between learning and not learning under nonbinding precedent, shewould choose to learn with probability 1 in the first period under binding precedent,
so the result still holds that the DM has a stronger incentive to learn early on underbinding precedent.
To give a concrete interpretation of our model here, consider a Director of Graduate
Studies regulating the funding for conference presentations by students. A case x
corresponds to an application by a student to present at a certain conference. Any casehas its own characteristics (the selectiveness of the conference, the cost of funding, thestudent’s general performance, etc) that carry different benefits and opportunity costs.
As a modeling device, we assume that these potentially multi-dimension characteristics
can be reduced to a single dimension (see Baker and Mezzetti [2012] for a discussionof this reduction). A full investigation requires using time and resources to study thefacts of each case to uncover how the benefits and the opportunity costs are associatedwith the characteristics of the cases.
The stark form of the learning process assumed here is for tractability: it simplifies
the analysis significantly because either the DM is fully informed or her belief is the8same as the prior. If learning is partial, then we have to keep track of the DM’s
belief in addition to the precedent as a state variable, which complicates the analysis
significantly. We revisit this issue and extend our analysis to partial learning in section
6, showing that our main results are robust.
Let s = ((L, R), x). For expositional convenience, we refer to s as the state eventhough it does not include the information about θ. Let S = [0, 1]3 denote the set of
possible states.
Denote the ruling at time t by rt ∈ {0, 1}, where rt = 0 if the case is banned andrt = 1 if the case is permitted. After the DM makes her ruling, the precedent changes
to Lt+1 and Rt+1 . If xt was permitted, then Lt+1 = max{Lt , xt } and Rt+1 = Rt ; if xt
was banned, then Lt+1 = Lt and Rt+1 = min{Rt , xt }. Formally, the transition of theprecedent is captured by the function π : S × {0, 1} → [0, 1]2 where(π(st , rt ) =(Lt , min{Rt , xt }) if rt = 0(max{Lt , xt }, Rt ) if rt = 1.

(1)

We consider two institutions: nonbinding precedent and binding precedent. Under
nonbinding precedent, the DM is free to make any ruling; under binding precedent, theDM must permit xt in period t if xt ≤ min{Lt , Rt } and must ban xt if xt ≥ max{Lt , Rt }.
To understand this assumption, note that if xt ≤ min{Lt , Rt }, then there must be somecase higher than xt that was permitted in the past and there is no case lower than xt
that was banned in the past, so the only ruling that is consistent with precedent is topermit xt . Similarly, if xt ≥ max{Lt , Rt }, then there must be some case lower than xt
that was banned in the past and there is no case higher than xt that was permitted in
the past, so the only ruling that is consistent with precedent is to ban xt . Note that
under binding precedent, after the DM learns the value of θ, she may be bound to makecertain rulings even if she knows them to be erroneous. If we allow the DM to overturnprecedents once they are found to be erroneous, then the precedents effectively become
nonbinding.11
More generally, we say that a ruling regarding x violates precedent (L, R) if x ≤min{L, R} and the DM bans x or if x ≥ max{L, R} and the DM permits x. We canthink of the cost of violating precedent to be infinite when it is binding and zero when
11

Indeed, legal scholars recognize that it is “an essential feature of any coherent doctrine of stare
decisis” that any overruling should not be solely based on the belief that a prior ruling is erroneous(Nelson [2001]).

9

it is nonbinding. We focus on these two extremes to highlight the difference in the
incentives that the DM faces.12
Note that, Lt < Rt on the equilibrium path under binding precedent. But it ispossible that Rt < Lt off the equilibrium path; in this case, for xt ∈ (Rt , Lt ), the DM
can either permit or ban xt even under binding precedent. This is because if the DM
permits xt , the ruling is supported by precedent because there is a higher case that has
been permitted before, and if the DM bans xt , the ruling is still supported by precedentbecause there is a lower case that has been banned before.
The payoff of the DM from the ruling rt on case xt in period t is given by(u(rt ; xt , θ) =0
if xt ≤ θ and rt = 1, or xt ≥ θ and rt = 0,−`(xt , θ) otherwise,

where `(xt , θ) > 0 for xt 6= θ is the cost of making a mistake, that is, permitting acase when it is above θ or banning a case when it is below θ. Assume that `(x, θ) is
continuous in x and θ for x 6= θ, strictly increasing in x and strictly decreasing in θ if
x > θ and strictly decreasing in x and strictly increasing in θ if x < θ. (We allow there
to be a discontinuity at x = θ to reflect a fixed cost of making a mistake in ruling.)
For example, if `(x, θ) = f (|x − θ|) where f (y) : R+ → R+ is continuous for y > 0,strictly increasing, and f (0) = 0, then these assumptions are satisfied.
The dynamic payoff of the DM is the sum of her discounted payoffs from the rulings
made in each period net of the cost of violating a precedent and net of the discounted
investigation cost if the DM carries out one. The discount factor is δ ∈ (0, 1).

3

An example

To illustrate the intuition, we present a three-period example. We characterize theoptimal investigation and ruling policies for each period using backward induction.
We then compare the information acquisition incentives under the two institutions. As
shown by Proposition 1, three periods is the shortest time horizon to illustrate thedifference in the DM’s dynamic incentives under the two institutions.

12

If we allow the DM to choose in each period whether to make her ruling binding, the analysis
would be the same as the case of nonbinding precedent because the DM would not choose to makeher ruling a binding precedent if she did not conduct an investigation.

10

Nonbinding precedent
In any period, if the DM has investigated in a previous period, then θ is known andshe permits or bans case x according to θ. If the DM has not investigated in a previous
period, then her belief about θ is the same as the prior. Continuity of the DM’s payoff
function implies that there exists x̂ ∈ (θ, θ̄) such that
Z θ̄

Z θ̄
u(1; x̂, θ)dF (θ) =θu(0; x̂, θ)dF (θ),(2)θ

that is, x̂ is the case such that the uninformed DM is indifferent between permitting
and banning. Lemma 1 shows that an uninformed DM permits any case below x̂ andbans any case above x̂. For expositional simplicity, we assume that the DM permits
the case when indifferent between permitting and banning.
Lemma 1. Under nonbinding precedent, in any period t, if the DM is uninformed, shepermits xt if xt ≤ x̂ and bans xt if xt > x̂.
This characterization of the ruling decision of an uninformed DM under nonbinding
precedent holds not just for the three-period example but for any time horizon.
Now we analyze the DM’s investigation decisions. We say that case x triggers aninvestigation if the DM conducts an investigation when the case is x. If DM decides
to investigate in period t, her payoff is −z in period t and 0 in all future periods. The
following lemma says that when the investigation cost is sufficiently low, the uninformed
DM investigates with positive probability in each period (that is, the cases that she
investigates has positive measure); the cases that trigger an investigation in period t
forms an interval; and the interval of investigation is larger in an earlier period. (Forexpositional simplicity, we assume that he conducts an investigation when indifferent
between investigating and not investigating.) Intuitively, for the cases that fall in the
middle, it is less clear to the DM whether she should permit or ban and the expected
cost of making a mistake is higher. Hence, the value of investigation for these cases ishigher. Thus if case x triggers an investigation in period t and case x0 > x triggers aninvestigation in period t, then any case in [x, x0 ] triggers an investigation in period t.
Moreover, because the DM can use the information she acquires in an earlier period
for later periods, the value of investigation is higher in an earlier period, resulting in
more investigation in an earlier period.

11Lemma 2. In the three-period example under nonbinding precedent, the set of casesthat trigger an investigation in period t, denoted by XtN , is an interval (possibly empty).
Moreover, X3N ⊆ X2N ⊆ X1N .

Binding precedent
We first show that in each period t, the cases that trigger an investigation form a(possibly degenerate) interval under binding precedent as well. Moreover, if the DM’s
hands are tied regarding case xt , that is, if xt ≤ Lt or if xt ≥ Rt , then xt does not
trigger an investigation.
Lemma 3. Under binding precedent, for any precedent (Lt , Rt ), the set of cases thattrigger an investigation in period t, denoted by XtB (Lt , Rt ), is an interval (possibly
empty). Moreover, XtB (Lt , Rt ) ⊆ (Lt , Rt ).
This characterization of the investigation decision of an uninformed DM underbinding precedent holds not just for the three-period example but for any time horizon.
In the next proposition, we compare the investigation decisions under binding precedent and nonbinding precedent.
Proposition 1. In the three-period example, the uninformed DM investigates more under binding precedent than under nonbinding precedent in period 1, and investigates lessunder binding precedent than under nonbinding precedent in periods 2 and 3. Specifically, for any (Lt , Rt ), t ∈ {1, 2, 3},(i) X1N ⊆ X1B (L1 , R1 ).
(ii) If [θ, θ̄] ⊆ [L2 , R2 ], then X2N = X2B (L2 , R2 ); otherwise X2N ⊆ X2B (L2 , R2 ).
(iii) X3B (L3 , R3 ) = (L3 , R3 ) ∩ X3N .
To understand this result, first consider the last period. The reason for the DM
to investigate less in period 3 under binding precedent is that an investigation has novalue if x3 ≤ L3 or if x3 ≥ R3 because the DM must permit any x3 ≤ L3 and mustban any x3 ≥ R3 no matter what the investigation outcome is; moreover, because
period 3 is the last period, the information about θ has no value for the future either.
For x3 ∈ (L3 , R3 ), the DM faces the same incentives under binding and nonbinding
precedent and therefore the same set of cases trigger an investigation.
12

If the precedent in period 2 satisfies [θ, θ̄] ⊆ [L2 , R2 ], then investigation avoids
mistakes in ruling in the current period as well as the future period even under binding
precedent. In this case, the DM faces the same incentives under binding and nonbinding
precedent and therefore the same set of cases trigger an investigation. However, if theprecedent in period 2 does not satisfy [θ, θ̄] ⊆ [L2 , R2 ], then even if x2 ∈ (L2 , R2 ) andthe DM investigates, mistakes in ruling can still happen in period 3 under binding
precedent if θ ∈/ [L2 , R2 ] because the DM is bound to follow the precedent. In thiscase, the value of investigation is lower under binding precedent than under nonbinding
precedent and therefore the DM investigates less under binding precedent.
Because the precedent in period 1 satisfies [θ, θ̄] ⊆ [L1 , R1 ], investigation avoids
mistakes in ruling in the current period as well as in future periods even under binding
precedent. However, for x1 ∈ (θ, θ̄), if the DM makes a ruling without an investigation
when x1 is realized, then she changes the precedent in a way such that [θ, θ̄] 6⊆ [L2 , R2 ].
As discussed in the previous paragraph, the binding precedent arising from this rulingpotentially results in mistakes in the future and diminishes the DM’s incentive toinvestigate future periods, which in turn lowers the DM’s dynamic payoff. Hence, theDM’s payoff from not investigating in period 1 is lower under binding precedent which
gives her a stronger incentive to investigate early on.
Proposition 1 makes two points: first, the DM investigates more under binding
precedent than under nonbinding precedent early on; second, the DM investigates lessunder binding precedent in later periods. We use backward induction in establishing this result. Because backward induction does not apply when the time horizon
is infinite, the analysis of the infinite-horizon model requires a different methodology.
Moreover, formalizing the second point in the infinite-horizon model is somewhat tricky
because what cases trigger investigation under binding precedent in later periods depend on the realized path of the cases. We address these complications in the next
section.

4

Optimal policies

We first define the optimal policies under the two institutions. Note that undereither institution, the DM’s problem is a Markov decision problem. In both cases, we
restrict attention to stationary Markov policies, which is without loss of generality (see,for example, Theorem 6.2.9 in Puterman [2005]).

13

Definition of optimal policies under nonbinding precedent
With nonbinding precedent, the payoff-relevant state in any period is the realizedcase x ∈ [0, 1] and the information about θ.
If θ is known at the time when the relevant decisions are made, then it is optimal
not to investigate; to permit x if x ≤ θ; and the DM’s payoff is 0.
If θ is unknown at the time when the relevant decisions are made, a policy for the
DM is a pair of functions σN = (µN , ρN ), where µN : [0, 1] → {0, 1} is an investigation
policy and ρN : [0, 1] → {0, 1} is an uninformed ruling policy, with µN (x) = 1 if andonly if an investigation is made when the case is x and ρN (x) = 1 if and only if case x
is permitted.
For each policy σN = (µN , ρN ), let VN (·; σN ) be the associated value function, thatis, VN (x; σN ) represents the dynamic payoff of the DM when she is uninformed, facescase x in the current period, and follows the policy σN . In what follows, we suppress
the dependence of the dynamic payoffs on σN for notational convenience. Given any
R1
dynamic payoff VN , let EVN∗ denote its expected value, that is, EVN∗ = 0 VN (x0 )dG(x0 ).
∗∗and the associated value function VN∗= (µ∗N , ρ∗N ) is optimal if σN
The policy σNsatisfy the following conditions:(N1) The uninformed ruling policy satisfies ρ∗N (x) = 1 if and only if for any case x
Z θ̄

Z θ̄
u(1; x, θ)dF (θ) ≥θu(0; x, θ)dF (θ).
θ

(N2) Given VN∗ and the uninformed ruling policy ρ∗N , the investigation policy for theuninformed DM satisfies µ∗N (x) = 1 if and only if for any case x−z ≥ ρ∗N (x)Z θ̄u(1; x, θ)dF (θ) + (1 − ρ∗N (x))

θ

Z θ̄

u(0; x, θ)dF (θ) + δEVN∗ .

θ∗(N3) Given σN, for any case x, the dynamic payoff satisfies(
VN∗ (x) = max −z,Z θ̄u(1; x, θ)dF (θ) + δEVN∗ ,

θ

Z θ̄

)
u(0; x, θ)dF (θ) + δEVN∗.

θ

Condition (N1) says that the DM chooses the ruling that maximizes the expected
payoff in the current period. The ruling decision depends only on the current period

14

payoff because the ruling does not affect the DM’s continuation payoff under nonbinding
precedent. Condition (N2) says that when uninformed, a case triggers an investigation
if and only if the DM’s dynamic payoff from investigating is higher than her expected
dynamic payoff from not investigating. If a case triggers an investigation, then θ
becomes known and no mistake in ruling will be made in the current period as wellas in the future. In this case, the dynamic payoff of the DM is negative of the cost ofinvestigation. If a case does not trigger an investigation, then the DM’s dynamic payoff
is the sum of the expected payoff in the current period and the continuation payoff. By
the Principle of Optimality, this payoff is the value function given in condition (N3).
Note that under non-binding precedent, the uninformed DM’s optimal policies are
history independent. (Of course the optimal policies depend on whether the DM isinformed or not, which in turn depends on history.) When we consider partial learningin Appendix B, however, informational states become richer, and because previous
cases determine what information was learned, the DM’s optimal policies depend on
history in more interesting ways.
Definition of optimal policies under binding precedent
With binding precedent, the payoff-relevant state in any period is the precedentpair (L, R), the realized case x, and the information about θ.13
If θ is known at the time when the relevant decisions are made, then it is optimal
not to investigate. Moreover, the optimal ruling policy is as follows. If θ ∈ (L, R), thenit is optimal to permit x iff x ≤ θ. If θ ≤ L, then it is optimal to permit x iff x ≤ L.
If θ ≥ R, then it is optimal to permit x iff x < R.
Let C(L, R) denote the expected dynamic payoff of the DM when the precedentis (L, R), conditional on θ being known when decisions regarding the cases are madewhere the expectation is taken over θ before it is revealed and over all future cases x.
Note that C(L, R) has two components, one is when future cases fall below L and theother is when future cases fall above R. We denote the first component by c(L) andthe second by c(R). Formally,

13

For expositional simplicity, we consider precedents with L < R in our analysis of binding precedent. Under binding precedent, this must happen on the equilibrium path and the analysis is without
loss of generality.

15

c(L) =

0

if L ≤ θ,RLRL 1u(1; x, θ)dG(x)dF (θ)
1−δ θ
θ

0
c(R) =
 1 R θ̄ R θ u(0; x, θ)dG(x)dF (θ)
1−δ

R

R

if L > θ,
if R ≥ θ̄,
if R < θ̄,andC(L, R) = c(L) + c(R).

To see how we derive c(L) and c(R), note that if θ < L and x ∈ (θ, L], then theDM’s payoff is u(1; x, θ) because she has to permit x, but if θ ≥ L, then the DM’s
payoff is zero if the case falls below L; similarly, if θ > R and x ∈ [R, θ), then the DM’s
payoff is u(0; x, θ) because she has to ban x, but if θ ≤ R, then the DM’s payoff is zero
if the case falls above R. Note that c(L) is decreasing in L and c(R) is increasing in
R, and therefore C(L, R) is decreasing in L and increasing in R. Intuitively, C(L, R)
is the negative of the expected cost in ruling mistakes due to the binding power of theprecedent L and R, and therefore, it is lower (that is, the expected cost is higher) when
the precedent is tighter.
If θ is unknown at the time when the decisions regarding the cases are made, apolicy for the DM is a pair of functions σB = (µB , ρB ), where µB : S → {0, 1} isan investigation policy and ρB : S → {0, 1} is an uninformed ruling policy, whereµB (s) = 1 if and only if an investigation is made when the state is s, and ρB (s) = 1 if
and only if case x is permitted when the state is s.
Let A(s) denote the DM’s dynamic payoff if she investigates in state s = ((L, R), x),
not including the investigation cost. Formally, let L be the (possibly degenerate)interval [θ, max{L, θ}] and R be the (possibly degenerate) interval [min{R, θ̄}, θ̄], wehaveZ xA(s) = 1L (x)Z θ̄u(1; x, θ)dF (θ) + 1R (x)θu(0; x, θ)dF (θ) + δC(L, R).
x

For each policy σB = (µB , ρB ), let VB (·; σB ) denote the associated value function,that is, VB (s; σB ) represents the dynamic payoff of the DM when the state is s, θ is
unknown, and she follows the policy σB . In what follows, we suppress the dependence
16

VB on σB for notational convenience. For notational convenience, let EVB (L, R) =R1
V (L, R, x0 )dG(x0 ).
0 B
Recall that the transition of precedent is captured by the function π, defined in (1).
The policy σB∗ = (µ∗B , ρ∗B ) is optimal if σB∗ and the associated value function VB∗ satisfy
the following conditions:(B1) Given VB∗ , for any state s, the uninformed ruling policy satisfies ρ∗B (s) = 1 ifeither x ≤ L or x ∈ (L, R) andZ θ̄u(1; x, θ)dF (θ) + δEVB∗ (π(s, 1)) ≥Z θ̄u(0; x, θ)dF (θ) + δEVB∗ (π(s, 0));θθand ρ∗B (s) = 0 if either x ≥ R or x ∈ (L, R) andZ θ̄u(1; x, θ)dF (θ) + δEVB∗ (π(s, 1)) <

θ

Z θ̄

u(0; x, θ)dF (θ) + δEVB∗ (πB (s, 0));θ(B2) Given VB∗ and the uninformed ruling policy ρ∗B , for any state s, the investigation
policy for the uninformed DM satisfies µ∗B (s) = 1 if and only if−z + A(s) ≥ ρ∗B (s)Z θ̄u(1; x, θ)dF (θ)θ+(1 − ρ∗B (s))

Z θ̄u(0; x, θ)dF (θ) + δEVB∗ (π(s, ρ∗B (s))).

θ(B3) Given σ ∗ , for any state s, the dynamic payoff satisfies(
VB∗ (s) = maxZ θ̄−z + A(s),u(1; x, θ)dF (θ) + δEVB∗ (π(s, 1)),

θ

Z θ̄

)
u(0; x, θ)dF (θ) + δEVB∗ (π(s, 0)) .

θUnder binding precedent, the ruling decision may change the precedent, which
in turn may affect the continuation payoff. As such, condition (B1) says the rulingdecision depends on both the current period payoff and the continuation payoff. In
particular, the DM chooses the ruling that maximizes the sum of the current period
payoff and the continuation payoff, taking into consideration how her ruling affects theprecedent in the next period. Condition (B2) says that the DM chooses to investigate17

a case if and only if her dynamic payoff from investigating is higher than her expected
dynamic payoff from not investigating.
If x triggers investigation, then θ becomes known. When the precedents are binding, however, mistakes in ruling can still happen if θ < L or if θ > R. In this case,the dynamic payoff VB∗ (s) is A(s) minus the cost of investigation. If x does not trigger
investigation, then the DM’s dynamic payoff is the sum of the expected payoff in thecurrent period and the continuation payoff. Using the Principle of Optimality, we formalize this in condition (B3).
Existence and uniquenessWe next show that the DM’s value functions and optimal policies as defined in(N1-P3) and (B1-B3) exist and are unique.
Proposition 2. Under either nonbinding precedent or binding precedent, the DM’s
optimal policy exists and is unique.
To prove this, we first apply the Contraction Mapping Theorem to show that thevalue functions VN∗ and VB∗ exist and are unique. The optimality conditions (N1-N2)∗
and (B1-B2) then uniquely determine the optimal policies σNand σB∗ respectively. We
next turn to the characterization of the value function and optimal policy.

Analysis of nonbinding precedent
If the DM already investigated in a previous period, then she knows the value of θ
and would permit or ban a case according to θ. The following result is analogous toLemma 2 in the example.
Lemma 4. Under nonbinding precedent, the set of cases that trigger an investigationin any period is an interval.
Let XN = {x : µ∗N (x) = 1}, that is, XN is the set of cases that trigger an investigation under nonbinding precedent. Recall x̂ is the case such that the uninformed DM
is indifferent between permitting and banning. Let
Z x̂
`(x, θ)dF (θ)

ẑ =

(3)

θ

denote the expected cost from making uninformed ruling on x̂. If XN 6= ∅, let aN =inf{x : µ∗N (x) = 1} and bN = sup{x : µ∗N (x) = 1}.
18

We next show that if the DM faces a case such that there is no uncertainty about
what the correct ruling is for that case (that is, if x ≤ θ or if x ≥ θ̄), then there is no
investigation. Intuitively, because of discounting, it is optimal to delay investigation
until it is useful immediately even though the information from investigation is valuable
for future rulings. In this case, the optimal ruling decision is to permit x if and only ifx ≤ x̂ as shown in Lemma 1.
Let λ denote the expected payoff if the DM makes a ruling without an investigation,that is,Z x̂ Z θ̄Z θ Z θ̄u(1; x, θ)dF (θ)dG(x) +

λ=
θ

θ

u(0; x, θ)dF (θ)dG(x).
x̂

θ

δλ
Let z ∗ = ẑ − 1−δ
. We also show that the uninformed DM investigates with positive probability if the investigation cost is below the threshold z ∗ . In that case, theuninformed DM permits any case below aN and bans any case above bN .

Proposition 3. Under nonbinding precedent, any case x ∈/ (θ, θ̄) does not trigger aninvestigation. If z > z ∗ , then XN = ∅, and the uninformed DM permits x if x ≤ x̂and bans x otherwise. If z ≤ z ∗ , then XN = [aN , bN ] 6= ∅ and the uninformed DM
permits x if x < aN and bans x if x > bN .
R1
Suppose XN 6= ∅. Recall that EVN∗ = 0 VN∗ (x0 )dG(x0 ). Proposition 3 says thatthe optimal policies are characterized by aN and bN . To find aN and bN , note that



δEVN∗




R θ̄ u(1; x, θ)dF (θ) + δEV ∗NθVN∗ (x) =


−z




R θ̄ u(0; x, θ)dF (θ) + δEV ∗N
θ

if x ≤ θ, or if x ≥ θ̄,
if θ < x < aN ,(4)

if x ∈ [aN , bN ],if bN < x < θ̄.

To see how we derive this, note that if x ≤ θ or if x ≥ θ̄, then the DM does notinvestigate and makes no mistake in her ruling in the current period. In this case, hercurrent-period payoff is 0 and her continuation payoff is δEVN∗ . If θ < x < aN or if
bN < x < θ̄, the DM does not investigate in the current period and incurs some cost of
making a mistake in expectation. Because θ remains unknown, her continuation payoffis δEVN∗ . If x ∈ [aN , bN ], then the DM investigates. Because she makes no mistakein her ruling both in the current period and in all future periods, her current period
payoff is −z and her continuation payoff is 0.
19

From (4), we have
EVN∗ = −z[G(bN ) − G(aN )] + δEVN∗ [G(aN ) + 1 − G(bN )]Z aN Z θ̄
Z θ̄ Z θ̄+u(1; x, θ)dF (θ)dG(x) +u(0; x, θ)dF (θ)dG(x).
θ

bN

θ

θ

For any a, b such that θ < a ≤ b < θ̄, let
Z a Z θ̄h(a, b) =Z θ̄ Z θ̄u(1; x, θ)dF (θ)dG(x) +θu(0; x, θ)dF (θ)dG(x).
b

θ

(5)

θ

That is, h(a, b) is the expected payoff of the DM before the realization of the case when
the investigation interval is [a, b]. Then
EVN∗ =

h(aN , bN ) − z[G(bN ) − G(aN )].
1 − δ[G(aN ) + 1 − G(bN )](6)

Because the DM is indifferent between investigating and not investigating whenx = aN or when x = bN , we have
Z θ̄
−z =

u(1; aN , θ)dF (θ) + δEVN∗ =

θ

Z θ̄

u(0; bN , θ)dF (θ) + δEVN∗ .

(7)θ

We can solve for EVN∗ , aN , bN from equations (6) and (7) which gives us thecharacterization of the optimal policies. Plugging these in (4), we can solve for thevalue function VN∗ (x).

Analysis of binding precedent
We now consider binding precedent. We first establish that the value function VB∗
is decreasing in L and increasing in R; and the optimal investigation policy µ∗ is also
decreasing in L and increasing in R. This result says that as the precedent gets tighter,
the DM investigates less and her payoff also becomes lower.
In what follows, let S e denote the set of possible precedents that can arise on theequilibrium path under binding precedent, that is, S e = {(L, R) ∈ [0, 1]2 : L < R}.
Proposition 4. Suppose the precedent (L̂, R̂) is tighter than (L, R), that is, L ≤ L̂ <R̂ ≤ R. Under binding precedent, for any case x ∈ [0, 1], if it triggers an investigation
under precedent (L̂, R̂), then it also triggers investigation under precedent (L, R), that20

is, µ∗B (L, R, x) is decreasing in L and increasing in R. Moreover, the value function
VB∗ (L, R, x) is decreasing in L and increasing in R, and EVB∗ (L, R) is continuous in L
and R for any (L, R) ∈ S e .
We next show that as in the example, the set of cases that the DM investigates is
an interval, and moreover, if the DM’s hands are tied regarding a case, then that casedoes not trigger an investigation.
Proposition 5. Under binding precedent, for any precedent (L, R) ∈ S e (i) the set ofcases that trigger an investigation is an interval; (ii) if x ≤ L or if x ≥ R, then x does
not trigger an investigation.
Let XB (L, R) = {x : µ∗B (L, R, x) = 1} denote the investigation interval under(L, R). For any (L, R) ∈ S e such that XB (L, R) 6= ∅, let a(L, R) = XB (L, R) andb(L, R) = sup XB (L, R).
We next show that, analogous to Proposition 1 in the three-period example, theDM investigates more under binding precedent than under nonbinding precedent earlyon but investigates less under binding precedent in later periods. We formalize the firstpart of this statement by comparing the investigation intervals in the first period. To
formalize the second part, we characterize a limit investigation interval under binding
precedent, which allows us to show that eventually there is less investigation underbinding precedent.
Before we give a formal definition, we discuss the idea. Suppose that given thethe initial precedent, the set of cases that trigger an investigation is nonempty (ifit is empty, then no investigation will be carried out in any period). For notational
simplicity, let a1 = a(L1 , R1 ) and b1 = b(L1 , R1 ). Recall that the the initial precedent
is consistent with the DM’s preference, that is, L1 < θ and R1 > θ̄. Hence, we have
L1 < a1 ≤ b1 < R1 , and x1 triggers an investigation if and only if x1 ∈ [a1 , b1 ].14
If x1 ∈ [a1 , b1 ], then it triggers an investigation immediately. If x1 ∈/ [a1 , b1 ], then the
DM makes a ruling without any investigation and changes the precedent to (L2 , R2 ) =(x1 , R1 ) if she permits the case and to (L2 , R2 ) = (L1 , x1 ) if she bans the case. Notethat the resulting new precedent satisfies L2 < a1 and b1 < R2 . Monotonicity ofµ∗B in L and R as established in Proposition 4 implies that the investigation intervalin period 2, if nonempty, satisfies a(L2 , R2 ) ≥ a1 and b(L2 , R2 ) ≤ b1 . Therefore we
14

In Lemma A.2, we show that if L < a(L, R) ≤ b(L, R) < R, then the investigation interval underprecedent (L, R) is closed.

21

have L2 < a(L2 , R2 ) ≤ b(L2 , R2 ) < R2 and the DM investigates x2 if and only if
x ∈ [a(L2 , R2 ), b(L2 , R2 )]. An iteration of this argument shows that on any realized
equilibrium path, given the precedent (L, R), the investigation interval (if nonempty)
satisfies L < a(L, R) ≤ b(L, R) < R. The investigation intervals either converge to∅ or to some nonempty set [â, b̂] such that if the precedent is (L, R) = (â, b̂), thena(L, R) = â and b(L, R) = b̂.
We now define a limit investigation interval under binding precedent, denoted by∞
XB . Construct a sequence {an , bn , Ln , Rn } as follows. For any n, given Ln and Rn ,if XB (Ln , Rn ) 6= ∅, then let an = a(Ln , Rn ), bn = b(Ln , Rn ) and pick Ln+1 and Rn+1
such that Ln < Ln+1 < an and bn < Rn+1 < Rn ; if XB (Ln , Rn ) = ∅, then let
n, an+1 = an , bn+1 = bn , Ln+1 = Ln , Rn+1 = Rn . Note that an is
an = bn = Ln +R
2
increasing and bn is decreasing. Because a monotone and bounded sequence converges,
lim an and lim bn are well defined. Now let XB∞ = (lim an , lim bn ). Note that a limit
investigation interval under binding precedent may depend on the particular sequence{Ln , Rn } we pick. In particular, if XB (Ln , Rn ) = ∅ for any n, then XB∞ = ∅.
Recall that XN is the set of cases that trigger an investigation under nonbinding
precedent.
Proposition 6. The DM investigates more under binding precedent than under nonbinding precedent early on but investigates less under binding precedent in later periods.
Specifically, for any limit investigation interval XB∞ under binding precedent, we have
XB∞ ⊆ XN ⊆ XB (L1 , R1 ).

5

Social welfare

Because binding precedent places constraints on what the DM can do in terms ofher rulings, her payoff is clearly higher under nonbinding precedent than under binding
precedent. (If the DM is given the choice between the two institutions, then she would
prefer the nonbinding precedent. A third alternative is for the DM to be given the rightto decide on a case-by-case basis whether to make her ruling binding. Because the DM
would not make the ruling on any case binding without learning θ first, the equilibriumoutcome is the same as that under the institution of non-binding precedent.) However,
because the decisions may affect the society at large, as in the case of court rulings, theDM’s payoff is not a good measure of social welfare in the presence of this externality.
If the rest of the society does not bear the cost of information acquisition but cares
22about the rulings, then a reasonable measure of social welfare may be simply the payoffs
coming from the ruling decisions. A standard social welfare function is the utilitariansocial welfare function which is the sum of the DM’s payoff and the payoffs of themembers of the society from the DM’s rulings. We can think of our social welfare
function as a limit case of this social welfare function. Specifically, for simplicity, we
focus on the social welfare function that puts zero weight on the DM’s utility, but given
the continuity of the social welfare function, our results continues to hold as long as
the weight on the DM’s utility is small. This is plausible when the society consistsof many agents whose welfare are affected by the DM’s rulings. Formally, we define asocial welfare function VNS (x) under nonbinding precedent and a social welfare function
VBS (x, L, R) under binding precedent as follows.
Under nonbinding precedent, the optimal policy that the DM chooses is given by(µ∗N , ρ∗p ). If µ∗N (x) = 1, then the current ruling as well as all future rulings are correct, and therefore VNS (x) = 0. If µ∗N (x) = 0, then the social welfare consists of the
expected social cost from the potential mistake in ruling today as well the discounted
continuation payoff EVpS . In this case,

VNS (x) = max

(Z
θ̄

)

Z θ̄
u(1; x, θ)dF (θ),

u(0; x, θ)dF (θ)+ δEVNS .

θθ

Similarly, under binding precedent, if µ∗B (x) = 1, then VBS (s) = A(s), and if µ∗B (x) =0, then
VBS (s) = max(Zθ̄u(1; x, θ)dF (θ) + δEVBS (π(s, 1)),

θ

Z θ̄

)
u(0; x, θ)dF (θ) + δEVBS (π(s, 0)) .

θ

R1
Let the expected social welfare under nonbinding precedent be EVNS = 0 VNS (x0 )dG(x0 )R1
and the expected social welfare under binding precedent be EVBS = 0 VBS (L1 , R1 , x0 )dG(x0 ).
We next compare the social welfare under the two institutions.

Welfare comparison
We first discuss the special cases of δ = 1 and δ = 0 because welfare comparison isstraightforward in these cases.
If δ = 1, then the DM investigates all x ∈ [θ, θ] in the first period and the equilibrium
23

policies are the same under binding and nonbinding precedent. To see this, first notethat it is straightforward to show that the investigation interval under nonbinding
precedent is [θ, θ] as δ goes to 1. Also note that the investigation interval under binding
precedent contains the investigation interval under nonbinding precedent in the first
period for any δ by Proposition 6. Because no case outside [θ, θ] is ever investigated,
it follows that the investigation interval under binding precedent in the first period is
also [θ, θ]. It follows that the social welfare coincides under the two institutions.
If δ = 0, then the equilibrium investigation interval is the same whether precedentis binding or not because the DM cares about only the current decision. Given thatthe social welfare function puts zero weight on future decisions when δ = 0, social
welfare is the same under the two institutions as well. The more interesting case iswhen the social welfare function is evaluated at a different discount factor than theDM’s. This is plausible because one interpretation of δ = 0 is that it represents asituation in which a new DM is appointed in each period and each DM cares about
only her own decision, but the society at large cares about the welfare implicationsof the decisions made in different periods and thus has a non-zero discount factor. In
this case, even though the equilibrium investigation decisions are the same under thetwo institutions, the rulings may be different after the DM learns θ. Under nonbinding
precedent, she makes the correct ruling in all future periods, but does not necessarily
do so under binding precedent if θ ∈/ (L, R). Hence, social welfare is higher under
nonbinding precedent when the DM has zero discount factor but the social welfare
function has a positive discount factor. Intuitively, with short-sighted DMs, making
the precedent binding does not provide stronger incentive for information acquisitionand only results in distortions in rulings.
We have considered a special case here in which the society is more far-sighted
than the DM, whose discount factor is 0. In general, when society gives more weightto the future than the DM does and the DM also has a positive discount factor, it ishard to say whether the society would prefer binding or non-binding precedent. On
the one hand, the DM faces more constraints in terms of what rulings she can makeunder binding precedent; on the other hand, because the DM has a positive discountfactor, the more intensive early learning that arises from binding precedent has positive
spillover effects.
In the special cases where δ = 0 or δ = 1 discussed above, the stronger incentive
for the DM to acquire information early on is absent, resulting in higher social welfare

24under nonbinding precedent. But the opposite comparison is possible too, as illustrated
in the following example.
Example 1. Suppose that θ is uniformly distributed on [0.2, 0.8], x is uniformly distributed on [0, 1], δ = 0.95, z = 0.1 and `(x, θ) = |x − θ|.
In this example, more cases trigger investigation in the first period under binding
precedent than under nonbinding precedent, but as the precedents are established overtime, the set of cases that trigger an investigation eventually becomes empty underbinding precedent.
To provide a sharper characterization of which institution is more conducive tohigher social welfare, we next consider a simple and tractable discrete framework.

Discrete framework
In this subsection, suppose xt is uniformly distributed on {xL , xH } and θ is uniformly distributed on {θL , θM , θH } with θL < xL < θM < xH < θH . Assume that under
the prior, it is optimal to permit xL and ban xH . Let cL = `(xL , θL ) and cH = `(xH , θH )and assume, without loss of generality, that cL ≤ cH . As the decisions become more
important, cL and cH become higher because the cost of making an erroneous rulingis higher for more important decisions. As before, the initial precedent L1 and R1 areconsistent with the DM’s preferences, that is, L1 ≤ θL and R1 ≥ θH .
The following lemma is an analog of Proposition 3, which characterizes the optimal
policies under nonbinding precedent for the continuous framework. It also says that
more cases trigger an investigation when the cost of investigation is lower.
Lemma 5. There exist z N and z̄N , both linearly increasing in cL and cH , such that
under nonbinding precedent, if z ≤ z N , both xL and xH trigger investigation; if z N <z ≤ z̄N , only xH triggers investigation; if z > z̄N , there is no investigation.
A similar result holds for the binding precedent, in the sense that, in the first period,
when the cost of investigation is low enough, both cases trigger investigation; when thecost of investigation is sufficiently high, there is no investigation; and in between, only
xH triggers investigation. The next result formalizes this observation, and establishes
a comparison of the cost cutoffs relative to nonbinding precedent.
Lemma 6. There exist z B and z̄B , both linearly increasing in cL and cH , such that in
the first period under binding precedent, if z ≤ z B , both xL and xH trigger investigation;25

if z B < z ≤ z̄B , only xH triggers investigation; if z > z̄B , there is no investigation.
Furthermore, z̄N = z̄B and z N < z B .
Note that when z ≤ z N , both cases trigger investigation immediately under either
institution, and when z > z̄B = z̄N , there is no investigation under either institution.
In these cases, the social welfare is the same under both institutions. By contrast, social
welfare is different under the two institutions when z ∈ (z N , z B ], or when z ∈ (z B , z̄N ].
When z ∈ (z N , z B ], both cases trigger investigation immediately under binding
precedent but only xH triggers investigation under nonbinding precedent. As a result,the social welfare under binding precedent is higher than the social welfare under
nonbinding precedent due to more information acquisition.
When z ∈ (z B , z̄N ], only xH triggers investigation in the first period under both
institutions. Consequently, if xH is drawn in the first period, then the payoff is thesame under both institutions. If xL is drawn in the first period, however, the DM
permits it without investigation, and the precedent becomes L2 = xL , R2 = R1 . Under
nonbinding precedent, she continues to permit xL without investigation until xH isdrawn. At that point, she investigates and decides all cases correctly from then on.
By contrast, under binding precedent, when xH is drawn, depending on the parametervalues, it may or it may not trigger investigation. If it does not trigger investigation,social welfare is lower under binding precedent due to less information acquisition. If it
triggers investigation, social welfare is still lower under binding precedent because theDM cannot make full use of the information: if the investigation reveals that θ = θL ,it is socially optimal to ban xL but the DM has to follow the precedent and permit
xL . To summarize, we have (i) If z ≤ z N or z > z̄N = z̄B , then welfare is the sameunder nonbinding precedent and binding precedent. (ii) If z N < z ≤ z B , then welfare
is higher under binding precedent. (iii) If z B < z ≤ z̄N = z̄B , then welfare is higher
under nonbinding precedent. Because z N , z B , z̄N , z̄B are all linearly increasing in
cL and cH , they all shift to the right as the decisions become more important. The
following proposition summarizes these observations.
Proposition 7. There exists a threshold, increasing in the importance of decisions,such that if the cost of information acquisition is lower than the threshold, then welfare is higher under binding precedent; otherwise welfare is higher under nonbinding
precedent.
Proposition 7 implies that the institution of binding precedent tends to induce
higher social welfare when decisions are more important. This is consistent with the26observation that rulings from higher courts, which are typically more important, havemore binding power.

6

Conclusion

We have analyzed a simple model designed to capture an important consequenceof the authority of the precedent, a common institution which can make violating
previously established rulings costly. The main insight obtained is that this cost canserve to motivate decision makers to acquire information more intensively early on andtherefore can be socially beneficial. There are many ways to enrich and extend themodel; as concluding remarks, we discuss some promising directions.
Partial learning. The information structure we assumed so far is stark: an investigation reveals θ perfectly. We relax this assumption by considering partial learning.
As in Baker and Mezzetti [2012], we now assume that an investigation reveals only
whether it is optimal to permit or ban the case at hand, which is informative about
the value of θ, but may not perfectly reveal it. With partial learning, the informational
state is much richer than before,15 thereby complicating the analysis significantly. For
tractability, we continue to use the discrete framework introduced in the preceding
section. In this setting, if the DM investigates when the current case is xL and findsthat it should be banned, she infers that θ = θL , but if she finds that it should bepermitted, she infers that θ ∈ {θM , θH }. Similarly, if the DM investigates when thecurrent case is xH and finds that it should be permitted, she infers that θ = θH , but
if she finds that it should be permitted, she infers that θ ∈ {θL , θM }. In Appendix B,
we show that the analogs of Lemma 5, Lemma 6 and Proposition 7 hold in this partial
learning setting, demonstrating the robustness of our results.
Varying degrees of bindingness. We have compared two opposite mechanisms:
either it is costless to violate a precedent (the nonbinding mechanism) or it is infinitely
costly to do so (the binding mechanism). In practice, there may be intermediatemechanisms. One example is a mechanism that commits to a kind of “partially binding”
precedent in the sense that a ruling establishes the precedent if and only if the distance
between L and R is greater than a pre-specified length. Under this mechanism, the“nonbinding range” no long shrinks once the two precedents are sufficiently close to
15

Recall that under perfect learning, the DM either knows θ perfectly or her belief about θ is thesame as her prior.

27

each other. Another possibility is to have binding precedent only for the cases banned
but not for the cases permitted, which is observed in certain areas of law, for example,
in antitrust practice.16 A third mechanism to consider is to have some cost to violate
a precedent. This cost may not be so large as to completely deter a DM from goingagainst a previous ruling when new information makes it clear that doing so is sociallybeneficial.17 The third mechanism can be analyzed by extending our model to allow aricher cost structure associated with violating precedent, reflecting the varying degreesof bindingness that exist in different institutions. These richer structures permits the
exploration of such institutional design questions as the optimal degree of bindingness.
This is especially important if there are different decision makers in different periods,
an extension that we discuss next.
Different decision makers in different periods. We have considered a single longlived DM and the only externality is the social implications of the rulings. If different
DMs make decisions in different periods, then the model becomes a stochastic game,and new complications arise. The smallest departure from our model would be a settingin which the DMs have the same preferences regarding the rulings (a common θ) and theinformation learned about θ from one DM’s investigation is passed down to future DMs(for example, through written opinions), but the information acquisition cost is still
private.18 Because information acquisition is costly, the DMs have incentives to freeride. But because this free-riding incentive is present under both nonbinding precedentand binding precedent, our main point that the current DM has a stronger incentive
to acquire information under binding precedent than under nonbinding precedent inearlier periods but a weaker incentive to do so in later periods still holds. Even though
this framework is a game rather than a single-agent decision problem, we can still
formulate it in a way that allows us to apply contraction mapping theorem to establish
existence of a unique equilibrium. Our numerical analysis shows that the main results
are robust in this extension.19
A more significant departure is a setting in which the DMs have different preferences
regarding outcomes, perhaps because of different ideological leanings. Then, under16

We thank an anonymous referee for suggesting these two mechanisms.
The conflict between stare decisis and the cost of judicial error has been recognized and studied
by legal scholars. See, for example, Lash [2014].
18
In section 5, we discussed the case of δ = 0, which can be interpreted as modeling short-lived
DMs who care only about their own decisions. Here we consider short-lived DMs who also care about
future decisions made by other DMs.
19
We omit the details, but the uniqueness result and numerical analysis are available upon request.
17

28

binding precedent, a DM may distort her decision in the direction that she favors so
as to tie the hands of future DMs having different preferences than hers. This kind of
inefficiency does not arise if precedents have only nonbinding influence.20
Precedents as organizational memory In their seminal 1963 book, Cyert andMarch point out that “organizations have memories in the form of precedents, andindividuals in the coalition are strongly motivated to accept the precedents as binding.”
And as observed by Levitt and March [1988], the details of the future path of anorganization “depend significantly on the processes by which the memory is maintained
and consulted.” As such, how precedents shape the direction of an organization depends
on the processes through which they are conserved and retrieved. This raises a numberof interesting questions. For example, how does the technology of conservation andretrieval of precedents affect the performance of the organization? Is it optimal tohave atrophy of organizational memory? Specifically, should more recent precedents
have more authority?
Evolving preferences. We have assumed that θ is fixed over time, reflecting stablepreferences. However, for certain matters, views on gay marriage for instance, social
preferences may change over time. The rigidity of binding precedent may delay anorganization’s response to such changes in preferences. Our model does not capture
the kind of inefficiency that arises in this case under binding precedent.
Dynamic delegation. Implicitly in our model, the DM is delegated to make decisions
regarding cases that arise over time. Our analysis of binding precedents shows that
imposing history-dependent constraints on the set of feasible actions for the DM can
improve social welfare. A broader question is what would be the optimal delegation
mechanism in settings like ours. This opens up a new line of research because the literature on optimal delegation has mostly focused on static settings (see, for example, theseminal article by Holmström [1984] and the recent works by Alonso and Matouschek[2008], Armstrong and Vickers [2010] and Amador and Bagwell [2013]). Recently, newmodels have been developed to study delegation in dynamic settings (see, for example,
Li, Matouschek, and Powell [2017] and Lipnowski and Ramos [2017]). However, thesearticles consider a conflict of interest between the principal and the agent in terms ofthe difference in preferred actions but not in terms of costly information acquisition.
20

On the other hand, as pointed out in Gennaioli and Shleifer [2007b], when judges have ideologicalbiases, the ability to overrule previous decisions creates instability in the law, which may result inother inefficiencies.

29How to use the dynamic link across decisions to design the optimal delegation mechanism to motivate information acquisition is an exciting question. We plan to explore
this and the other interesting questions discussed above in future research.

30

Appendix A: Proofs
Proof of Lemma 1: If x < θ, then the DM strictly prefers to permit the case because
x < θ regardless of what the realization of θ is. Likewise, if x > θ, then the DM strictlyprefers to ban the case. For any x ∈ [θ, θ̄], the difference in expected payoff between
banning the case and permitting the case is given by
Z x

Z θEθ [u(0; x, θ) − u(1; x, θ)] =−`(x, θ)dF (θ) −x−`(x, θ)dF (θ).
θ

Given the assumptions on `(x, θ), it follows that Eθ [u(0; x, θ) − u(1; x, θ)] is continuous
and increasing in x. Because Eθ [u(0; x, θ) − u(1; x, θ)] < 0 if x = θ and Eθ [u(0; x, θ) −u(1; x, θ)] > 0 if x = θ̄, there exists x̂ ∈ (θ, θ̄) satisfying equation (2). Note thatEθ [u(0; x, θ) − u(1; x, θ)] < 0 for x < x̂ and Eθ [u(0; x, θ) − u(1; x, θ)] > 0 for x > x̂.
Lemma 1 follows immediately.
Proof of Lemma 2: Consider period 3 first. If x3 < θ, then the DM permits x3
without an investigation; if x3 > θ̄, then the DM bans x3 without an investigation. Forx3 ∈ [θ, x̂), if the DM does not investigate, then she permits the case by Lemma 1 andRθ
her expected payoff is θ u(1; x3 , θ)dF (θ), which is decreasing in x3 . For x3 ∈ (x̂, θ̄],if the DM does not investigate, then she bans the case by Lemma 1 and her expectedRθ
payoff is θ u(0; x3 , θ)dF (θ), which is increasing in x3 .
RθLet z ∗ = θ u(1; x̂, θ)dF (θ) > 0. If z > z ∗ , then X3N = ∅. If z ≤ z ∗ , then X3N =RθRθ[x3 , x̄3 ] where x3 < x̂ and x̄3 > x̂ satisfy θ u(0; x3 , θ)dF (θ) = θ u(1; x3 , θ)dF (θ) = z.
Let EVtN be the expected continuation payoff of the uninformed DM in period t.
We have
EV3N =

Z x3 Z θ

Z θZ θ
u(0; x, θ)dF (θ)dG(x) − [G(x̄3 ) − G(x3 )]z > −z.

u(1; x, θ)dF (θ)dG(x) +
θ

θ

x̄3

θ

Now consider period 2. For any x2 ∈ [0, 1], the continuation payoff for the uninformed DM if she investigates is −z. For x2 ∈/ [θ, θ], because the DM’s expected payoffis δEV3N > −δz if she does not investigate and −z if she investigates, it is optimal for
her not to investigate.
Consider x2 ∈ [θ, x̂) and suppose the DM does not investigate. Because she permitsRθsuch a case, her expected payoff in period 2 is θ u(1; x2 , θ)dF (θ) Similarly, for x2 ∈(x̂, θ], if the DM does not investigate, she bans x2 and her expected payoff in period 2
31

is

Rθ

u(0; x2 , θ)dF (θ).
Now consider the DM’s optimal investigation policy in period 2. For x2 ∈ [θ, x̂), it
is optimal for the DM to investigate in period 2 iff
θ

Z θ
−z ≥

u(1; x2 , θ)dF (θ) + δEV3N .

θ

Similarly, for x2 ∈ (x̂, θ], it is optimal for the DM to investigate in period 2 iff
Z θ
−z ≥

u(0; x2 , θ)dF (θ) + δEV3N .

θHence, if z > z ∗ − δEV3N , then X2N = ∅. If z < z ∗ − δEV3N , then X2N = [x2 , x̄2 ]where x2 and x̄2 satisfy
Z θ
−z =

u(1; x2 , θ)dF (θ) + δEV3N =θZ θu(0; x̄2 , θ)dF (θ) + δEV3N .

θ

Because EV3N < 0, we have x2 ∈ (θ, x3 ) and x̄2 ∈ (x̄3 , θ̄), which implies that X3N ⊆ X2N .
It also follows that
#
#
Z "Z
Z "Z
x2

EV2N =

θ

θ

θ

θ

θ

u(1; x, θ)dF (θ) + δEV3N dG(x) +u(0; x, θ)dF (θ) + δEV3N dG(x)x̄2θ− [G(x̄2 ) − G(x2 )]z.
Note that
EV3N =Z aZ θ
maxa,b∈[θ,θ̄],b>aθ

Z θZ θ
u(1; x, θ)dF (θ)dG(x)+u(0; x, θ)dF (θ)dG(x)−[G(b)−G(a)]z,

θ

b

θ

and EV3N < 0. It follows that EV2N < EV3N < 0.
Now consider period 1. By a similar argument as in period 2, if z > z ∗ − δEV2N ,then X1N = ∅; if z < z ∗ − δEV2N , then X1N = [x1 , x̄1 ] where x1 and x̄1 satisfy
Z θ
−z =

u(1; x1 , θ)dF (θ) + δEV2N =θZ θu(0; x̄1 , θ)dF (θ) + δEV2N .

θ

Because EV2N < EV3N , it follows that x1 ∈ (θ, x2 ) and x̄1 ∈ (x̄2 , θ̄) and therefore
32

X2N ⊆ X1N .
Proof of Lemma 3: Consider period 3 first. Recall that under nonbinding precedent,
the investigation interval is X3N . Because under binding precedent, investigation has
no value if x3 ≤ L3 or if x3 ≥ R3 , it follows that x3 triggers an investigation if and only
if x3 ∈ X3N ∩ (L3 , R3 ). Hence, X3B (L3 , R3 ) is an interval and X3B (L3 , R3 ) ⊆ (L3 , R3 ).
Let k(L, R) denote the DM’s expected payoff in period t under binding precedentwhen the precedents are (L, R) conditional on θ being known where the expectation istaken over θ before it is revealed and over all possible cases x. FormallyZ Z LZ Z θu(1; x, θ)dG(x)dF (θ) +k(L, R) =
L

R

θ


u(0; x, θ)dG(x)dF (θ)

R

where L is the (possibly degenerate) interval [θ, max{L, θ}] and R is the (possibly
degenerate) interval [min{R, θ̄}, θ̄]. That is, k(L, R) = (1 − δ)C(L, R) where C(L, R)
is defined on page 21. Note that k(L, R) is decreasing in L and increasing in R;
k(L, R) ≤ 0 and k(L, R) < 0 if L > θ or if R < θ̄.
To prove Lemma 3 for periods 1 and 2, we first establish Claim 1 below. Let
EVtB (L, R) denote the uninformed DM’s expected equilibrium continuation payoff inperiod t under binding precedent given that the precedent in period t is (L, R).
Claim 1. If EVtB (L, R) is decreasing in L and increasing in R, then the set of casesthat trigger an investigation in period t − 1 is an interval for any precedent in periodt − 1.
Proof: Suppose that EVtB (L, R) is decreasing in L and increasing in R. Fix precedentBin period t − 1 by (Lt−1 , Rt−1 ). Suppose that x0 and x00 > x0 are in Xt−1
(Lt−1 , Rt−1 ).
000
BWe next show that any x̃ ∈ [x , x ] is also in Xt−1 (Lt−1 , Rt−1 ).
Let g p (L, R, x) be the DM’s current-period payoff if she permits the case without
investigation in state s = (L, R, x) and g b (s) be her current-period payoff if she bans thecase without investigation in state s. Note that for any (L, R), g p (L, R, x) is decreasing
in x and g b (L, R, x) is increasing in x.
Suppose x̃ ∈ (Lt−1 , Rt−1 ). If the DM investigates, then her continuation payoff
is −z + δk(Lt−1 , Rt−1 ). Suppose, without loss of generality, that x̃ < x̂ so that it isB
optimal for the uninformed DM to permit x̃. Because x0 ∈ Xt−1
(Lt−1 , Rt−1 ), we havep0B0−z + δk(Lt−1 , Rt−1 ) ≥ g (Lt−1 , Rt−1 , x ) + δEVt (max{x , Lt−1 }, Rt−1 ). Because g p is33

decreasing in x, it follows that g p (Lt−1 , Rt−1 , x0 ) > g p (Lt−1 , Rt−1 , x̃). Moreover, becausex̂ > max{x0 , Lt−1 } and EVtB is decreasing in L, we have EVtb (max{x0 , Lt−1 }, Rt−1 ) >EVtB (x̂, Rt−1 ). Hence, we have −z+δk(Lt−1 , Rt−1 ) ≥ g p (Lt−1 , Rt−1 , x̂)+δEVtB (x̂, Rt−1 ),which implies that it is optimal for the DM to investigate when the case is x̃.
Suppose x̃ ≤ Lt−1 . Then the DM has to permit x̃ regardless of whether she
investigates or not. Hence, x̃ triggers an investigation iff −z + δk(Lt−1 , Rt−1 ) ≥B
δEVtB (Lt−1 , Rt−1 ). Because x0 ∈ Xt−1
(Lt−1 , Rt−1 ) and x0 < x̃, we have −z+δk(Lt−1 , Rt−1 ) ≥B(Lt−1 , Rt−1 ). A similar argument shows that
δEVtB (Lt−1 , Rt−1 ), implying that x̃ ∈ Xt−1
B
B
(Lt−1 , Rt−1 ) is an interval for any(Lt−1 , Rt−1 ) if x̃ ≥ Rt−1 as well. Hence, Xt−1x̃ ∈ Xt−1(Lt−1 , Rt−1 ).
We next show that EV3B (L, R) is decreasing in L and increasing in R. Consider
precedents (L3 , R3 ) and (L̂3 , R̂3 ) such that L̂3 ≤ L3 and R̂3 ≥ R3 . As shown before,
under the precedent (L3 , R3 ), it is optimal for the DM to investigate iff x3 ∈ (L3 , R3 ) ∩X3N . By following the same policy under precedent (L̂3 , R̂3 ), the DM receives thesame payoff as under precedent (L3 , R3 ). Hence, EV3B (L3 , R3 ) ≤ EV3B (L̂3 , R̂3 ), andit follows from Claim 1 that X2B (L2 , R2 ) is an interval. Consider x2 ∈/ (L2 , R2 ). It
triggers an investigation iff −z + δk(L2 , R2 ) ≥ δEV3B (L2 , R2 ). Because EV3B (L2 , R2 ) ≥−z + k(L2 , R2 ), it follows that δEV3B (L2 , R2 ) > −z + δk(L2 , R2 ) and therefore x2 does
not trigger an investigation. Hence, X2B (L2 , R2 ) ⊆ (L2 , R2 ).
We next show that EV2B (L, R) is decreasing in L and increasing in R. Consider
precedents (L2 , R2 ) and (L̂2 , R̂2 ) such that L̂2 ≤ L2 and R̂2 ≥ R2 . We show that
if the DM follows the same policy under precedent (L̂2 , R̂2 ) as the optimal policy
under (L2 , R2 ), then her continuation payoff is higher under precedent (L̂2 , R̂2 ) thanunder (L2 , R2 ). Consider first x2 ∈ X2B (L2 , R2 ). Then x2 ∈ (L2 , R2 ), and in thiscase, the DM’s continuation payoff is −z under either (L2 , R2 ) or (L̂2 , R̂2 ). Next consider x2 ∈/ X2B (L2 , R2 ). Without loss of generality, suppose x2 ≤ x̂, which implies
that the uniformed DM permits x2 . In this case, the precedent in period 3 becomes(max{x2 , L2 }, R2 ). If the DM follows the same policy under precedent (L̂2 , R̂2 ), thenthe precedent in period 3 becomes (max{x2 , L̂2 }, R̂2 ). Because EV3B (L, R) is decreasing
in L and increasing in R, we have EV3B (max{x2 , L2 }, R2 ) ≤ EV3B (max{x2 , L̂2 }, R̂2 ).
Because the DM’s payoff in period 2 is the same under either (L2 , R2 ) or (L̂2 , R̂2 ),it follows that her continuation payoff in period 2 is higher under precedent (L̂2 , R̂2 )than under (L2 , R2 ). A similar argument shows that the result holds for x2 > x̂.
Hence, EV2B (L, R) is decreasing in L and increasing in R. Claim 1 then implies
34

that X1B (L1 , R1 ) is an interval. Because L1 = 0, R1 = 1 by assumption, we have
X1B (L1 , R1 ) ⊆ (L1 , R1 ).
Proof of Proposition 1: Consider period 3 first. As shown in the proof of Lemma
3, under binding precedent, x3 triggers an investigation iff x3 ∈ X3N ∩ (L3 , R3 ).
Now consider period 2. First suppose [θ, θ̄] ⊆ [L2 , R2 ]. Then the incentive of the
DM in period 2 is the same under binding precedent as under nonbinding precedent.
In this case, X2B (L2 , R2 ) = X2N . Next suppose [θ, θ̄] 6⊆ [L2 , R2 ]. Recall that if XtN 6= ∅,then XtN = [xt , x̄t ]. We show below that under binding precedent, x2 does not trigger
an investigation.
Recall that the DM is indifferent between investigating and not investigating whenx2 = x2 in period 2 under nonbinding precedent. That is, we haveZ θ̄u(1; x2 , θ)dF (θ) + δk(x3 , x̄3 ) − δz[G(x̄3 ) − G(x3 )].

−z =

(A1)

θ

Consider binding precedent. If x2 ∈/ (L2 , R2 ), then x2 does not trigger an investigation, as shown in Lemma 3. Next consider x2 ∈ (L2 , R2 ). Without loss of generality,
suppose that x2 ≤ x̂, which implies that the uninformed DM would permit x2 , resulting in L3 = x2 and R3 = R2 . Because X3B (L3 , R3 ) = (L3 , R3 ) ∩ [x3 , x̄3 ], the differencein the DM’s continuation payoff between investigating and not investigating when thecase is x2 is− z + δk(L2 , R2 )
#
"Z
θ̄
−
u(1; x2 , θ)dF (θ) + δk(max{x2 , x3 }, min{R2 , x̄3 }) − δz[G(min{R2 , x̄3 }) − G(max{x2 , x3 })] .
θBecause max{x2 , x3 } = x3 , this is equal to
Z θ̄
−z −

u(1; x2 , θ)dF (θ) + δ[k(L2 , R2 ) − k(x3 , min{R2 , x̄3 })] + δz[G(min{R2 , x̄3 }) − G(x3 )].
θ

Substituting for −z from (A1), the difference in the DM’s continuation payoff between
investigating and not investigating when x2 = x2 isδ[k(x3 , x̄3 ) + k(L2 , R2 ) − k(x3 , min{R2 , x̄3 })] − δz[G(x̄3 ) − G(min{R2 , x̄3 })].

35

If x̄3 ≤ R2 , then k(x3 , x̄3 ) + k(L2 , R2 ) − k(x3 , min{R2 , x̄3 }) = k(L2 , R2 ) < 0; ifx̄3 > R2 , then k(x3 , x̄3 ) + k(L2 , R2 ) − k(x3 , min{R2 , x̄3 }) = k(L2 , x̄3 ) < 0. Note alsothat G(x̄3 )−G(min{R2 , x̄3 }) ≥ 0. Hence, the DM’s payoff is higher by not investigatingwhen x2 = x2 and therefore x2 does not trigger an investigation under binding precedent(L2 , R2 ).
A similar argument establishes that under binding precedent, x̄2 does not trigger
an investigation in period 2. Because both X2N and X2B (L2 , R2 ) are intervals, it follows
that X2B (L2 , R2 ) ⊆ X2N .
Now consider period 1. Recall that if X1N 6= ∅, then X1N = [x1 , x̄1 ]. We show
below that x1 triggers an investigation under binding precedent. Recall that the DM isindifferent between investigating and not investigating when x1 = x1 under nonbinding
precedent. That is,
Z θ̄
−z =

u(1; x1 , θ)dF (θ) + δEV2N

θ

Z θ̄
=

u(1; (x1 , θ)dF (θ) + δk(x2 , x̄2 ) + δ 2 [1 − G(x̄2 ) + G(x2 )]k(x3 , x̄3 )θ− δz[G(x̄2 ) − G(x2 ) + δ(G(x̄3 ) − G(x3 ))].
Under binding precedent, if the DM investigates in period 1, her continuation payoff
R θ̄
is −z; if the DM does not investigate, her continuation payoff is θ u(1; x1 , θ)dF (θ) +δEV2B (x1 , R1 ). Note that EV2B (x1 , R1 ) < V2N because the DM can follow the same
policy under nonbinding precedent as the optimal policy under binding precedent andreceive a higher payoff. Hence, x1 triggers an investigation in period 1 under binding
precedent.
A similar argument establishes that under binding precedent, x̄1 triggers an investigation in period 1. Because both X1N and X1B (L1 , R1 ) are intervals, it follows that
X1N ⊆ X1B (L1 , R1 ).
Proof of Proposition 2: Let F denote the set of bounded measurable functions
on S taking values in R. For f ∈ F, let ||f || = sup{|f (s)| : s ∈ S}. An operator
Q : F → F satisfies the contraction property for || · || if there is a β ∈ (0, 1) such thatfor f 1 , f 2 ∈ F, we have ||Q(f 1 ) − Q(f 2 )|| ≤ β||f 1 − f 2 ||. For any operator Q thatsatisfies the contraction property, there is a unique f ∈ V such that Q(f ) = f .
We prove the proposition for binding precedent. A similar and less involved argu-

36

ment shows uniqueness under nonbinding precedent as well.
Let g p (s) be the DM’s current-period payoff if she permits the case without investigation in state s and g b (s) be her current period payoff if she bans the case without
investigation in state s. Formally,g p (s) =g b (s) =R θ̄ u(1; x, θ)dF (θ)θ

if x < R,−∞

if x ≥ R,

R θ̄ u(0; x, θ)dF (θ)

if x > L,θ−∞

if x ≤ L.
R1
For any V ∈ F and (L, R) ∈ S e , let EV (L, R) = 0 V (L, R, x0 )dG(x0 ).
For V ∈ F and any s ∈ S, define
T V (s) = max{−z + A(s), g p (s) + δEV (max{x, L}, R), g b (s) + δEV (L, min{x, R})}.
(A2)
∗
∗
∗
Note that VB as defined in (B3) satisfies VB = T VB .
Suppose that V 1 , V 2 ∈ F and consider any s ∈ S e ×[0, 1]. Without loss of generality,
suppose that T V 1 (s) ≥ T V 2 (s). There are three cases to consider.
(i) Suppose that T V 1 (s) = −z + A(s). Because T V 1 (s) ≥ T V 2 (s), we have
T V 2 (s) = −z + A(s). It follows that T V 1 (s) − T V 2 (s) = 0.
(ii) Suppose that T V 1 (s) = g p (s) + δEV 1 (max{L, x}, R). We have
|T V 1 (s) − T V 2 (s)| ≤ g p (s) + δEV 1 (max{L, x}, R) − g p (s) − δEV 2 (max{L, x}, R)
Z 1
 1

V (max{L, x}, R, x0 ) − V 2 (max{L, x}, R, x0 ) dG(x0 )
≤δ
Z0 1

 1
≤δ
|V (max{L, x}, R, x0 ) − V 2 (max{L, x}, R, x0 )| dG(x0 )]0≤ δ||V 1 − V 2 ||.
(iii) Suppose that T V 1 (s) = g b (s) + δEV 1 (L, min{x, R}). Then a similar argument
as in case (ii) shows that |T V 1 (s) − T V 2 (s)| ≤ δ||V 1 − V 2 ||.
Because either |T V 1 (s) − T V 2 (s)| = 0 or |T V 1 (s) − T V 2 (s)| ≤ δ||V 1 − V 2 || for anys ∈ S in all three cases, we have ||T V 1 − T V 2 || ≤ δ||V 1 − V 2 || and therefore T is acontraction. Because T is a contraction, it has a unique fixed point. Because VB∗ as37

defined in (B3) satisfies T VB∗ = VB∗ , there is a unique VB∗ . Once we solve for VB∗ , (B1)and (B2) determine the optimal policies ρ∗B and µ∗B uniquely.
Proof of Lemma 4: Suppose x0 and x00 trigger an investigation. Then, by (N1) and(N2), we have(Zθ̄−z ≥ min)Z θ̄u(1; x, θ)dF (θ),

u(0; x, θ)dF (θ)+ δEVN∗

θ

θ

R θ̄
R θ̄
for x ∈ {x0 , x00 }. Consider x̃ ∈ [x0 , x00 ]. Because − θ u(1; x, θ)dF (θ) and − θ u(0; x, θ)dF (θ)
are monotone and therefore quasioncave in x, it follows that
( Z
)
Z θ̄
θ̄
min −
u(1; x, θ)dF (θ), −u(0; x, θ)dF (θ)θθ

is quasiconcave in x. Hence
( Z
)
Z θ̄
θ̄
z ≤ min −u(1; x̃, θ)dF (θ), −u(0; x̃, θ)dF (θ) − δEVN∗θθ

and therefore x̃ triggers an investigation.
Proof of Proposition 3: We first show that if z > z ∗ , then ρ∗N (x) = 1 iff x ≤ x̂,λµ∗N (x) = 0 for any x ∈ [0, 1], and EVN∗ = 1−δ
. Condition (N 1) is clearly satisfied
given the definition of x̂. Condition (N 2) is satisfied because for any x, the payofffrom not investigating is at least as high as −ẑ + δEVN∗ , which is greater than −z if
z > ẑ − δEVN∗ = z ∗ . And (N 3) is satisfied by plugging in ρ∗N , µ∗N and EVN∗ .
Note that if z < z ∗ , then µ∗N (x) = 0 for any x ∈ [0, 1] would violate (N 2). Hence,
XN = ∅ if and only if z > z ∗ .
We next show that XN ⊂ (θ, θ̄). If XN = ∅, then it is clearly true. We next showby contradiction that if XN 6= ∅, then θ < aN < bN < θ̄. Suppose aN ≤ θ. Consider
x = aN . Because aN ≤ θ, the DM’s dynamic payoff equals −z if she investigates, andequals δEVN∗ if she does not investigate. Because aN = inf{x : µ∗ (x) = 1}, it follows
that −z ≥ δEVN∗ . Note that for any x > θ̄, the DM’s dynamic payoff is −z if she
investigates, and δEVN∗ if she does not investigate. Hence, it must be the case thatany case x > θ̄ triggers an investigation. It follows that bN = 1. Moreover, because38

aN ≤ θ, the DM makes the correct decision for any case x ≤ aN . It follows that
EVN∗ =Z aN

δEVN∗ dG(x) − z(1 − G(aN )) = δG(aN )EVN∗ − z(1 − G(aN )).

0

Because −z ≥ δEVN∗ , we have EVN∗ > δEVN∗ , but this is impossible because EVN∗ < 0.
Hence, we have aN > θ. A similar argument shows that bN < θ̄.
We next prove that the uninformed DM permits x < aN and bans x > bN . Note thatthe left-hand-side of (N 2) achieves its minimum at x̂, which implies that if XN 6= ∅,then x̂ ∈ XN , that is, aN ≤ x̂ ≤ bN . It follows from Lemma 1 that the uninformed DM
permits x < aN and bans x > bN .
Proof of Proposition 4: Recall from equation (A2) that
T V (s) = max{−z + A(s), g p (s) + δEV (max{x, L}, R), g b (s) + δEV (L, min{x, R})}.
Let KV (s) = 1 if T V (s) = −z + A(s) and KV (s) = 0 otherwise. Recall that F
denote the set of bounded measurable functions on S taking values in R. To prove theproposition, we first establish the following lemma.
Lemma A.1. For (L, R) ∈ S e , if V ∈ F satisfies the following properties: (i) V isdecreasing in L and increasing in R, (ii) EV (L, R) − EV (L̂, R̂) ≤ C(L, R) − C(L̂, R̂),and (iii) KV is decreasing in L and increasing in R, then T V also satisfies theseproperties, that is, (i) T V is decreasing in L and increasing in R, (ii) ET V (L, R) −ET V (L̂, R̂) ≤ C(L, R) − C(L̂, R̂), and (iii) KT V is decreasing in L and increasing in
R.
Proof: We first show that if V ∈ F is decreasing in L and increasing in R, then T V
is also increasing in L and decreasing in R. Fix x ∈ [0, 1]. If V is decreasing in L
and increasing in R, then EV (max{x, L}, R) and EV (L, min{x, R}) are decreasing in
L and increasing in R. Note that A(s) is decreasing in L and increasing in R, g p (s) isconstant in L and increasing in R, g b (s) is constant in R and decreasing in L. Hence,
T V (s) is decreasing in L and increasing in R.
Let ŝ = (L̂, R̂, x). We next show that if V ∈ F satisfies properties (i), (ii), and (iii),
then ET V (L, R) − ET V (L̂, R̂) ≤ C(L, R) − C(L̂, R̂). Consider the following cases.
(a) Suppose T V (ŝ) = −z + A(ŝ). Then KV (ŝ) = 1. Because KV is decreasing in
L and increasing in R, we have KV (s) = 1, which implies that T V (s) = −z + A(s).
Hence T V (s) − T V (ŝ) = A(s) − A(ŝ).
39

(b) Suppose T V (ŝ) > −z + A(ŝ). Without loss of generality, suppose that T V (ŝ) =g (ŝ) + δEV (max{x, L̂}, R̂). Note that KV (ŝ) = 0 and x < R̂. Suppose KV (s) = 1.
Then T V (s) = −z + A(s) and T V (s) − T V (ŝ) < A(s) − A(ŝ). Suppose KV (s) = 0
and T V (s) = g p (s) + δEV (max{x, L}, R). Thenphi
T V (s) − T V (ŝ) = δ EV (max{x, L}, R) − EV (max{x, L̂}, R̂)≤ δ[EV (L, R) − EV (L̂, R̂)] ≤ δ[C(L, R) − C(L̂, R̂)].
Suppose T V (s) = g b (s) + δEV (L, min{x, R}). There are two cases to consider, either
x > L̂ or x ≤ L̂. First suppose x > L̂. Then g b (ŝ) = g b (s). Because T V (ŝ) ≥g b (ŝ) + δEV (L̂, min{x, R̂}), it follows that T V (s) − T V (ŝ) ≤ δEV (L, min{x, R}) −δEV (L̂, min{x, R̂}) ≤ δ[C(L, R) − C(L̂, R̂)]. Next suppose x ≤ L̂. Note that T V (ŝ) =g p (ŝ) + δEV (max{x, L̂}, R̂) = g p (ŝ) + δEV (L̂, R̂) and A(ŝ) = g p (s) + δC(L̂, R̂). Hence,A(ŝ) − T V (ŝ) = δC(L̂, R̂) − δEV (L̂, R̂). Note also that T V (s) = g b (s) + δEV (L, x) ≤g b (s) + δEV (L, R) and A(s) ≥ g b (s) + δC(L, R). Hence A(s) − T V (s) > δC(L, R) −
δEV (L, R). It follows that A(s) − T V (s) − A(ŝ) + T V (ŝ) > δC(L, R) − δEV (L, R) −
δC(L̂, R̂) + δEV (L̂, R̂) ≥ 0. Therefore T V (s) − T V (ŝ) ≤ A(s) − A(ŝ). It follows thatfor all x ∈ [0, 1], we have T V (s) − T V (ŝ) ≤ A(s) − A(ŝ), and therefore ET V (L, R) −ET V (L̂, R̂) ≤ E[A(s) − A(ŝ)] = C(L, R) − C(L̂, R̂).
Lastly we show that if V ∈ F satisfies properties (i), (ii), and (iii), then KT V isdecreasing in L and increasing in R. Because KT V (s) ∈ {0, 1} for any s ∈ S, it issufficient to show that if KT V (ŝ) = 1, then KT V (s) = 1.
Suppose KT V (ŝ) = 1. Consider x ∈ (L̂, R̂) first. Then we have
− z + A(ŝ) ≥ max{g p (ŝ) + δET V (x, R̂), g b (ŝ) + δET V (L̂, x)}.

(A3)

Note that in this case, A(ŝ) = δC(L̂, R̂), A(s) = δC(L, R), g p (ŝ) = g p (s), g b (ŝ) = g b (s).
As established earlier, if V ∈ F satisfies properties (i), (ii), and (iii), then T V is decreasing in L and increasing in R and C(L, R) − C(L̂, R̂) ≥ ET V (L, R) − ET V (L̂, R̂).
Because L < L̂ < x < R̂ < R, we have max{L, x} = max{L̂, x} = x and min{x, R} =min{x, R̂} = x. It follows that ET V (max{L, x}, R)−ET V (max{L̂, x}, R̂) = ET V (x, R)−
ET V (x, R̂) and ET V (L, min{x, R})−ET V (L̂, min{x, R̂}) = ET V (L, x)−ET V (L̂, x).
Because C(x, R) − C(x, R̂) ≥ ET V (x, R) − ET V (x, R̂) and C(L, R) − C(L̂, R̂) ≥C(x, R)−C(x, R̂), it follows that C(L, R)−C(L̂, R̂) ≥ ET V (max{L, x}, R)−ET V (max{L̂, x}, R̂).
Similarly, because C(L, x)−C(L̂, x) ≥ ET V (L, x)−ET V (L̂, x) and C(L, R)−C(L̂, R̂) >40

C(L, x) − C(L̂, x), it follows that and C(L, R) − C(L̂, R̂) ≥ ET V (L, min{x, R}) −ET V (L̂, min{x, R̂}). It then follows from (A3) that−z + A(s) ≥ max{g p (s) + δET V (max{L, x}, R), g b (s) + δET V (L, min{x, R})}and therefore KT V (s) = 1.
Next consider x ∈/ (L̂, R̂), and without loss of generality, suppose that x ≤ L̂.
In this case, A(ŝ) − δC(L̂, R̂) = g p (ŝ) and A(s) − δC(L, R) ≥ g p (s). Specifically,A(s) − δC(L, R) = g p (s) if x ≤ L and A(s) − δC(L, R) = 0 if L < x ≤ L̂. Hence,− z + A(s) − g p (s) − δET V (max{L, x}, R) − [−z + A(ŝ) − g p (ŝ) − δET V (max{L̂, x}, R̂)]≥ δ[C(L, R) − C(L̂, R̂)] − δ[ET V (x, R) − ET V (x, R̂)] ≥ 0.
It follows that −z + A(s) ≥ g p (s) + δET V (max{L, x}, R). We next show that −z +A(s) ≥ g b (s) + δET V (L, min{x, R}). If x ≤ L, then clearly −z + A(s) ≥ g b (s) +δET V (L, min{x, R}). Suppose L < x ≤ L̂. Note that −z+A(ŝ) ≥ g p (ŝ)+δET V (L̂, R̂)
implies that −z + δC(L̂, R̂) ≥ δET V (L̂, R̂). Because A(s) = δC(L, R) ≥ g b (s) +δC(L, R) and C(L, R) − C(L̂, R̂) ≥ ET V (L, R) − ET V (L̂, R̂), it follows that −z +A(s) ≥ g b (s) + δET V (L, R) ≥ g b (s) + ET V (L, min{x, R}). Hence KT V (s) = 1.
Because VB∗ = T VB∗ and Lemma A.1 shows that the contraction mapping T preservesproperties (i), (ii) and (iii), it follows that VB∗ is decreasing in L and increasing in R.
It also follows that EVB∗ (L, R) − EVB∗ (L̂, R̂) ≤ C(L, R) − C(L̂, R̂). Because C(L, R)
is continuous in L and R, we have EVB∗ (L, R) is continuous in L and R. Because theoptimal policy satisfies µ∗B (s) = KVB∗ (s), it also follows from Lemma A.1 that µ∗B isdecreasing in L and increasing in R.
Proof of Proposition 5: Fix (L, R) ∈ S e . We first prove part (i). Note that g p (s)is decreasing in x and g b (s) is increasing in x. Because VB∗ is decreasing in L by
Proposition 4, VB∗ (π(s, P ), x0 ) = VB∗ ((max{L, x}, R), x0 ) is decreasing in x. Similarly,
because VB∗ is increasing in R, VB∗ (π(s, B), x0 ) = VB∗ ((L, min{R, x}), x0 ) is increasing inx.
Suppose xi (i = 1, 2) triggers an investigation. Then we have−z + A(L, R, xi ) ≥ g p (L, R, xi ) + δEVB∗ (max{L, xi }, R)(A4)−z + A(L, R, xi ) ≥ g b (L, R, xi ) + δEVB∗ (L, min{R, xi }).

(A5)

41

Recall that L = [θ, max{L, θ}] and R = [min{R, θ̄}, θ̄]. Suppose x̃ ∈ [x1 , x2 ]. There
are three cases to consider.
(a) Suppose 1L (x̃) = 1R (x̃) = 0. Then A(L, R, x̃) ≥ A(L, R, xi ) for i = 1, 2.
Because x̃ > x1 , g p (s) is decreasing in x and EVB∗ (max{L, x}, R) is decreasing in x, it
follows from (A4) that−z + A(L, R, x̃) ≥ g p (L, R, x̃) + δEVB∗ (max{L, x̃}, R).
Because x̃ < x2 , g b (s) is increasing in x and EVB∗ (L, min{R, x}) is increasing in x, it
follows from (A5) that−z + A(L, R, x̃) ≥ g p (L, R, x̃) + δEVB∗ (L, min{R, x̃}).
Hence, it is optimal for the DM to investigate when x = x̃.
(b) Suppose 1L (x̃) = 1. Then 1R (x̃) = 0 and 1R (x1 ) = 0. Moreover, we haveA(L, R, x̃) = g p (L, R, x̃)+δC(L, R), A(L, R, x1 ) ≥ g p (L, R, x1 )+δC(L, R), g b (L, R, x̃) =−∞, and g b (L, R, x1 ) = −∞. From (A4), we have−z + δC(L, R) ≥ δEVB∗ (max{L, x1 }, R).
Because EVB∗ (max{L, x}, R) is decreasing in x and x̃ > x1 , it follows that−z + δC(L, R) ≥ δEVB∗ (max{L, x̃}, R)
and therefore−z + A(L, R, x̃) ≥ g p (L, R, x̃) + δEVB∗ (max{L, x̃}, R).
Because g b (L, R, x̃) = −∞, we also have−z + A(L, R, x̃) > g b (L, R, x̃) + δEVB∗ (L, min{R, x̃}).
Hence, it is optimal for the DM to investigate when x = x̃.
(c) Suppose 1R (x̃) = 1. Then 1L (x̃) = 1L (x2 ) = 0. Moreover, we have A(L, R, x̃) =bg (L, R, x̃) + δC(L, R), A(L, R, x2 ) ≥ g b (L, R, x2 ) + δC(L, R), g p (L, R, x̃) = −∞, andg p (L, R, x2 ) = −∞. A similar argument as in case (ii) shows that it is optimal for the42

DM to investigate when x = x̃.
We next prove part (ii) by contradiction. Consider a case x ∈/ (L, R). Because
the DM has to follow the precedent in the current period regarding x, the differencein her current period payoff between investigating and not investigating is −z. If she
investigates, her continuation payoff is δC(L, R); if she does not investigate, she does
not change the precedent because x ∈/ (L, R) and therefore her continuation payoff isδEVB∗ (L, R).
Suppose x triggers an investigation. Then we have −z + δC(L, R) ≥ δEVB∗ (L, R).
Note that neither side of the inequality depends on x, which implies that any x ∈/(L, R) triggers an investigation given precedent (L, R). It follows from part (i) of the
proposition that any x ∈ [0, 1] triggers an investigation given precedent (L, R). Hence,
EV because∗B (L, R) = −z + C(L, R), which contradicts −z + δC(L, R) ≥ δEVB∗ (L, R)
because −z < 0 and δ < 1. Hence, x ∈/ (L, R) does not trigger an investigation.
Lemma A.2. If L < a(L, R) ≤ b(L, R) < R, then XB (L, R) is closed, and the DM
is indifferent between investigating and not investigating under precedent (L, R) if x =a(L, R) or if x = b(L, R).
Proof: Suppose L < a(L, R) ≤ b(L, R) < R and consider x ∈ (L, R). Becausex ∈ (L, R), the DM does not have to follow any precedent in her ruling of x. It follows that her expected current-period payoff in making a ruling regarding x without
an investigation is continuous in x for x ∈ (L, R) because `(x, θ) is continuous in x
for any x 6= θ. Because EVB∗ is also continuous by Proposition 4, the DM’s dynamic
payoff if she makes a ruling regarding x without an investigation is continuous in x forx ∈ (L, R). Recall that the DM’s dynamic payoff if she investigates when the case is
x is −z + A((L, R), x). Because x ∈ (L, R), we have A((L, R), x) = δC(L, R), which isconstant in x. Hence, the DM’s dynamic payoff if she investigates when the case is x
is also continuous in x. It follows the DM is indifferent between investigating and notinvestigating when x = a(L, R) and when x = b(L, R). Recall that we assume thatwhen the DM is indifferent between investigating and not investigating, she investigates. Hence, if L < a(L, R) ≤ b(L, R) < R, then the set of cases that the uninformed
DM investigates is the closed interval [a(L, R), b(L, R)].
Proof of Proposition 6: Recall that a1 = a(L1 , R1 ) and b1 = b(L1 , R1 ). We firstshow that XN ⊆ XB (L1 , R1 ). Recall that XN = [aN , bN ] when XN 6= ∅. We next showthat a1 ≤ aN . A similar argument shows that b1 ≤ bN .
43

Recall from (5) that for any a, b such that θ < a ≤ b < θ̄,Z θ̄ Z θ̄

Z a Z θ̄h(a, b) =u(1; x, θ)dF (θ)dG(x) +θθu(0; x, θ)dF (θ)dG(x).
bθNote that for any L, R, we have
EVB∗ (L, R) ≤ h(a(L, R), b(L, R)) + [G(b(L, R)) − G(a(L, R))] [−z + δC(L, R)]+ δEVB∗ (L, R) [G(a(L, R)) + 1 − G(b(L, R))] ,where the inequality comes from the property that EVB∗ (L, R) is decreasing in L andincreasing in R and that when the DM makes a decision without an investigation, theprecedent either stays the same or gets tighter.
It follows that
EVB∗ (L, R) ≤h(a(L, R), b(L, R)) + [G(b(L, R)) − G(a(L, R))] [−z + δC(L, R)].
1 − δ [G(a(L, R)) + 1 − G(b(L, R))]

Because C(a(L, R), b(L, R)) ≤ 0, we have EVB∗ (L, R) ≤ h(a(L,R),b(L,R))−z[G(b(L,R))−G(a(L,R))].
1−δ[G(a(L,R))+1−G(b(L,R))]
Consider any case x ∈ (L1 , a1 ). Because x < a1 , it does not trigger an investigation
R θ̄
given precedent (L1 , R1 ) and we have θ u(1; x, θ)dF (θ)+δEVB∗ (x, R1 ) > −z. It follows
that
Z θ̄

u(1; x, θ)dF (θ) > −z − δEVB∗ (x, R1 ).

θ

Recall that under nonbinding precedent, aN satisfiesZ θ̄u(1; aN , θ)dF (θ) = −z − δEVN∗ .

(A6)

θ

R θ̄
R θ̄
Because EVB∗ (x, R1 ) ≤ EVN∗ , it follows that θ u(1; x, θ)dF (θ) > θ u(1; aN , θ)dF (θ)
and therefore x < aN . Because this is true for any x < a1 we have a1 ≤ aN . A similarargument shows that bN ≤ b1 .
Suppose that XB∞ 6= ∅. Let XB∞ = (â, b̂). We next show that aN < â < b̂ < bN .
Note that if the precedent satisfies L = â and R = b̂, then x triggers an investigation44

if and only if x ∈ (â, b̂). Hence, we have

h
i
h
i
EVB∗ (â, b̂) = h(â, b̂) + −z + δC(â, b̂) G(b̂) − G(â) + δEVB∗ (â, b̂) G(â) + 1 − G(b̂) ,which implies that
EVB∗ (â, b̂) =h(â, b̂)
1 − δ[G(â) + 1 − G(b̂)]

+ −z + δC(â, b̂)G(b̂) − G(â)
1 − δ[G(â) + 1 − G(b̂)]. (A7)

Moreover, by Lemma A.2, we haveZ θ̄−z + δC(â, b̂) =u(1; â, θ)dF (θ) + δEVB∗ (â, b̂) =

θ

Z θ̄

u(0; b̂, θ)dF (θ) + δEVB∗ (â, b̂).

θ

From the indifference condition above and (A7), we haveZ θ̄u(1; â, θ)dF (θ) = −z + δC(â, b̂) − δEVB∗ (â, b̂)θ=−(1 − δ)z + (1 − δ)δC(â, b̂) − δh(â, b̂)
1 − δ[G(â) + 1 − G(b̂)].

Because (1 − δ)C(â, b̂) = h(â, b̂), it follows thatZ θ̄u(1; â, θ)dF (θ) =θ−(1 − δ)z
1 − δ[G(â) + 1 − G(b̂)].

Similarly,Z θ̄u(0; b̂, θ)dF (θ) =θ−(1 − δ)z
1 − δ[G(â) + 1 − G(b̂)].

From (6) and (A6), we have that under nonbinding precedent, aN and bN satisfy
Z θ̄

Z θ̄
u(1; aN , θ)dF (θ) =θu(0; bN , θ)dF (θ) =θ−(1 − δ)zδh(aN , bN )−.
1 − δ[G(aN ) + 1 − G(bN )] 1 − δ[G(aN ) + 1 − G(bN )]

R θ̄
R θ̄
For a ∈ [θ, θ̄], let β(a) be defined by θ u(1; a, θ)dF (θ) = θ u(0; β(a), θ)dF (θ).
Note that bN = β(aN ), b̂ = β(â) and β(a) is decreasing in a. Also, let A equal theδh(aN ,bN )> 0.
constant − 1−δ[G(aN )+1−G(bN )]45

Note that aN is the solution toZ θ̄u(1; a, θ)dF (θ) =θ−(1 − δ)z+A
1 − δ[G(a) + 1 − G(β(a))]

and â is the solution toZ θ̄u(1; a, θ)dF (θ) =θ−(1 − δ)z.
1 − δ[G(a) + 1 − G(β(a))]

R θ̄
If a = θ, then θ u(1; a, θ)dF (θ) = 0. Moreover, because −(1 − δ)z − δh(aN , bN ) < 0−(1−δ)z−(1−δ)z
N ,bN )+ A < −(1−δ)z−δh(a, it follows that 1−δ[G(a)+1−G(β(a))]+A < 0and 1−δ[G(θ)+1−G(θ̄)]
1−δ[G(θ)+1−G(θ̄)]
for a = θ. Hence, for any a ∈ [θ, aN ), we haveZ θ̄u(1; a, θ)dF (θ) >θ−(1 − δ)z+ A.
1 − δ[G(a) + 1 − G(β(a))]

Because A > 0, this implies that for any a ∈ [θ, aN ), we haveZ θ̄u(1; a, θ)dF (θ) >θ−(1 − δ)z.
1 − δ[G(a) + 1 − G(β(a))]

It follows that â > aN . Because β(a) is decreasing in a, it follows that b̂ < bN .
cL
Proof of Lemma 5: We first show that if z ≤ 3(1−δ), then both xL and xH trigger aninvestigation. Note that if both cases trigger an investigation, then EVN∗ = −z. Thisis optimal if −z ≥ − 13 cL + δEVN∗ and −z ≥ − 13 cH + δEVN∗ , which are satisfied whencLz ≤ 3(1−δ).
cL
H +δcLWe next show that if 3(1−δ)< z ≤ (2−δ)c, then only xH triggers an investi6(1−δ)
gation. Note that if xH but not xL triggers an investigation, then EVN∗ = 12 (−z) +z+ 31 cL1(− 13 cL + δEVN∗ ), implying that EVN∗ = − 2−δ
. Hence, it is optimal that only xH
2
1
triggers an investigation if −z ≥ − 3 cH + δEVN∗ and xL if −z < − 31 cL + δEVN∗ , whichcL
H +δcLare satisfied when 3(1−δ)< z ≤ (2−δ)c.
6(1−δ)(2−δ)cH +δcL
Finally, we show that if z >, then neither xH nor xL triggers an inves6(1−δ)
L +cH
tigation. Note that if neither xH or xL triggers an investigation, then EVN∗ = − c6(1−δ).
1∗Hence, it is optimal that neither cases trigger an investigation if −z < − 3 cH + δEVN
H +δcLand −z < − 13 cL + δEVN∗ , which are satisfied when z > (2−δ)c.
6(1−δ)(2−δ)cH +δcL
cL
Lemma 5 follows by letting z N = 3(1−δ) and z̄N = 6(1−δ) .

46

Proof of Lemma 6: Consider x1 = xL . If the DM investigates, her payoff is −z, andif she does not, her payoff is − 13 cL + δEVB∗ (xL , 1). Hence, xL triggers an investigation
under the initial precedent if −z ≥ − 31 cL + δEVB∗ (xL , 1).
To find EVB∗ (xL , 1), note that under precedent (xL , 1), xL does not trigger investiδδ
cL ≥ − 13 cH − 6(1−δ)
cL −gation, and xH triggers investigation if and only if −z − 6(1−δ)δ
2−δc , that is, z ≤ 6(1−δ) cH .
6(1−δ) H
2−δHence, if z ≤ 6(1−δ)
cH , then EVB∗ (xL , 1) = 12 [− 31 cL + δEVB∗ (xL , 1)] + 21 [−z −
δ
z
1
2−δ
c ], which implies that EVB∗ (xL , 1) = − 2−δ
− 6(1−δ)cL . And if z > 6(1−δ)
cH ,6(1−δ) L
then EVB∗ (xL , 1) = 12 [− 31 cL + δEVB∗ (xL , 1)] + 21 [− 13 cH + δEVB∗ (xL , xH )]. Because there
11cL − 6(1−δ)
cH , which
is no investigation under (xL , xH ), we have EVB∗ (xL , xH ) = − 6(1−δ)
11implies that EVB∗ (xL , 1) = − 6(1−δ)
cL − 6(1−δ)
cH .
∗Substituting EVB (xL , 1) in the condition −z ≥ − 13 cL + δEVB∗ (xL , 1), we find that2−δ
if z ≤ 6(1−δ)
cH , then under the initial precedent, xL triggers investigation if −z ≥
2

(2−δ)
z
1
2−δ
− 31 cL + δ[− 2−δ
− 6(1−δ)cL ], that is, if z ≤ 12(1−δ)
2 cL . And if z > 6(1−δ) cH , then under
1
1
the initial precedent, xL triggers investigation iff −z ≥ − 13 cL + δ[− 6(1−δ)
cL − 6(1−δ)
cH ],
δ
2−δ
cL + 6(1−δ)
cH .
that is, if z ≤ 6(1−δ)
2

(2−δ)
cH
2−δ
2−δ
2−δ
2−δ
Note that 6(1−δ)
cH > 12(1−δ)> 2−2δ
and 6(1−δ)
cH > 6(1−δ)
cL +2 cL if and only if c
L
cH
2−δ
δ
c if and only if cL > 2−2δ .
6(1−δ) H
2

(2−δ)
2−δ
2−δ
2−δ
2−δ
δ
Suppose ccHL > 2−2δ. Then 6(1−δ)
cH > 12(1−δ)
2 cL and 6(1−δ) cH > 6(1−δ) cL + 6(1−δ) cH .
2(2−δ)
It follows that xL triggers investigation under the initial precedent iff z ≤ 12(1−δ)
2 cL .
2

(2−δ)
2−δ
2−δ
2−δ
2−δ
δ
Suppose ccHL ≤ 2−2δ. Then 6(1−δ)
cH ≤ 12(1−δ)
2 cL and 6(1−δ) cH ≤ 6(1−δ) cL + 6(1−δ) cH .
2−δ
cH or if
It follows that xL triggers investigation under the initial precedent if z ≤ 6(1−δ)
2−δ
2−δ
δ
2−δ
δ
c < z ≤ 6(1−δ) cL + 6(1−δ) cH , that is, if z ≤ 6(1−δ) cL + 6(1−δ) cH .
6(1−δ) H
Next consider x1 = xH . An investigation takes place if and only if −z ≥ − 13 cH +
δEVB∗ (0, xH ). Straightforward calculation similar to the above shows that if z ≤2−δz1
c , then EVB∗ (0, xH ) = − 2−δ
− 6(1−δ)cH , and in this case, xH triggers an investi6(1−δ) L
2

(2−δ)
2−δ
1
1
∗
gation if z ≤ 12(1−δ)
2 cH . And if z > 6(1−δ) cL , then EVB (0, xH ) = − 6(1−δ) cL − 6(1−δ) cH ,2−δδand in this case, xH triggers an investigation if z ≤ 6(1−δ)
cH + 6(1−δ)
cL . Because
2

(2−δ)
2−δ
2−δ
δ
≤ 12(1−δ)
2 cL and 6(1−δ) cL ≤ 6(1−δ) cH + 6(1−δ) cL , it follows that xH triggers2−δδ
an investigation if z ≤ 6(1−δ)
cH + 6(1−δ)
cL .
2−δ
c
6(1−δ) L

2

(2−δ)
cH
2−δ
δ
2−δ
Let z̄B = 6(1−δ)
cH + 6(1−δ)
cL . And let z B = 12(1−δ)> 2−2δ
and z B =2 cL if c
L
cH
2−δ
δ
2−δ
c + 6(1−δ) cH if cL ≤ 2−2δ . Note that z B < z̄B . It follows that under the initial
6(1−δ) L
precedent, if z ≤ z B , then both xL and xH trigger an investigation; If z B < z ≤ z̄B ,then only xH triggers an investigation; if z > z̄B , then neither triggers an investigation.

47

cL
H +δcL
Recall that z N = 3(1−δ)
and z̄N = (2−δ)c. Hence, z N < z B and z̄N = z̄B .
6(1−δ)

Appendix B: Partial learning
Let θLM denote the informational state such that the DM believes that θ = θLand θM with equal probability. Define θM H and θLM H similarly. In what follows, we
assume that `(xL , θL ) < `(xL , θM ) and `(xH , θH ) < `(xH , θM ), which imply that under
nonbinding precedent, the DM finds it optimal to permit xL in state θLM and to ban
xH in state θM H .
Consider the initial informational state θLM H . As the next two lemmas show, thecharacterization and the comparison of the optimal investigation policies with partial
learning are qualitatively the same as those in Lemma 5 and 6.
0such that under nonbinding precedent in state
Lemma B.1. There exist z 0N and z̄N00, only xH
θLM H , if z ≤ z N , both xL and xH trigger an investigation; if z 0N < z ≤ z̄N0, there is no investigation.
triggers an investigation; if z > z̄N

Proof: For any informational state sθ , let EVN∗ (sθ ) denote the DM’s equilibrium payoffin state sθ . Consider informational state θLM . Note that xH does not trigger aninvestigation, and if xL triggers an investigation, then EVN∗ (θLM ) = 12 (0+δEVN∗ (θLM ))−z
1z, which implies that EVN∗ (θLM ) = − 2−δ
. Because it is optimal to investigate in state21
L (2−δ)θLM if −z ≥ − 2 cL +EVN∗ (θLM ), it follows that xL triggers an investigation if z ≤ c4(1−δ).
cL (2−δ)cL (2−δ)cLz∗Hence, EVN (θLM ) = − 2−δ if z ≤ 4(1−δ) and EVN (θLM ) = − 4(1−δ) if z ≥ 4(1−δ) .
H (2−δ)Similarly, in informational state θM H , xH triggers an investigation iff z ≤ c4(1−δ).
cH (2−δ)cH (2−δ)cHz∗∗Hence, EVN (θM H ) = − 2−δ if z ≤ 4(1−δ) and EVN (θM H ) = − 4(1−δ) if z > 4(1−δ) . In
H (2−δ)what follows, we focus on the case in which z ≤ c4(1−δ)
for simplicity. (Similar resultscH (2−δ)can be derived when z > 4(1−δ) .)2−δConsider state θLM H . We first show that if z ≤ (1−δ)(6−δ)
cL , then both xL andxH trigger an investigation. Note that if both xL and xH trigger an investigation,
6−δthen EVN∗ (θLM H ) = −z + 3δ (EVN∗ (θM H ) + EVN∗ (θLM )) = − 3(2−δ)
z. Because xL triggers
an investigation iff −z + 23 δEVN∗ (θM H ) ≥ − 13 cL + δEVN∗ (θLM H ) and xH triggers aninvestigation iff −z + 32 δEVN∗ (θLM ) ≥ − 31 cH + δEVN∗ (θLM H ), it follows that both trigger2−δ
an investigation iff z ≤ (1−δ)(6−δ)
cL .
2−δ
H +δcL ]We next show that if (1−δ)(6−δ) cL < z ≤ (2−δ)[(2−δ)c, then only xH triggers2(1−δ)(6−δ)
an investigation. Note that if only xH triggers an investigation, then EVN∗ (θLM H ) =48

1[− 13 cL + δEVN∗ (θLM H )] + 21 [−z + 32 δEVN∗ (θLM )], which implies that EVN∗ (θLM H ) =
2
2δz
1
1
[− 13 cL − z + 23 δEVN∗ (θLM )] = 2−δ
[− 13 cL − z + 3(2−δ)]. Because xH triggers an
2−δ
2
1
∗
∗
investigation iff −z + 3 δEVN (θLM ) ≥ − 3 cH + δEVN (θLM H ) and xL does not trigger an investigation iff −z + 32 δEVN∗ (θM H ) < − 13 cL + δEVN∗ (θLM H ), it follows that if2−δ
H +δcL ]c < z ≤ (2−δ)[(2−δ)c, then only xH triggers an investigation.
(1−δ)(6−δ) L
2(1−δ)(6−δ)
L +cH
If neither xL or xH triggers an investigation, then EVN∗ (θLM H ) = − c6(1−δ). Becauseneither case triggers an investigation if −z + 23 δEVN∗ (θLM ) < − 31 cH + δEVN∗ (θLM H ) andH +δcL ]if −z + 32 δEVN∗ (θM H ) < − 31 cL + δEVN∗ (θLM H ), it follows that if z > (2−δ)[(2−δ)c,2(1−δ)(6−δ)

then no investigation takes place in state θLM H .
2−δ
H +δcL ]0cL and z̄N.
= (2−δ)[(2−δ)c
The lemma follows by letting z 0N = (1−δ)(6−δ)
2(1−δ)(6−δ)
Lemma B.2. There exists z 0B and z̄B0 such that in the first period under binding precedent, if z ≤ z 0B , both xL and xH trigger investigation; if z 0B < z ≤ z̄B0 , only xH triggers
0
investigation; if z > z̄B0 , there is no investigation. Furthermore, z̄N= z̄B0 and z 0N < z 0B .
Proof: Under binding precedent and with partial learning, a state s consists of theprecedent (L, R) and the informational state sθ . Let EVB∗ (L, R, sθ ) denote the DM’s
equilibrium payoff in state s. Consider the state in the first period, (L1 , R1 , θLM H ). The
case xL triggers an investigation iff −z+ 32 δEVB∗ (xL , R1 , θM H ) ≥ − 13 cL +δEVB∗ (xL , R1 , θLM H ).
In state (xL , R1 , θLM H ), the DM’s problem is the same as in perfect learning. To
see this, note that given the precedent (xL , R1 ), the DM must permit xL . Therefore xL does not trigger an investigation and the value of information from perfectlearning or partial learning when xH triggers an investigation is the same. As shown2−δcH , then xH triggers an investigation andin the proof of Lemma 6, if z ≤ 6(1−δ)
z12−δEVB∗ (xL , R1 , θLM H ) = − 2−δ − 6(1−δ) cL ; if z > 6(1−δ)
cH , then xH does not trigger an
11∗
investigation and EVB (xL , R1 , θLM H ) = − 6(1−δ) cH − 6(1−δ)
cL .
In state (xL , R1 , θM H ), the DM’s problem is the same as in nonbinding precedentbecause in the informational state θM H , it is optimal to permit xL . It follows thatz
EVB∗ (xL , R1 , θM H ) = − 2−δ
.
∗Substituting for EVB (xL , R1 , θM H ) and EVB∗ (xL , R1 , θLM H ), we find that the con(2−δ)2dition for xL to trigger an investigation in state (L1 , R1 , θLM H ) is z ≤ 4(1−δ)(3−2δ)cL for2(2−δ)
δ(2−δ)2−δ
2−δ
z ≤ 6(1−δ)
cH and is z ≤ 2(1−δ)(6−δ)
cL + 2(1−δ)(6−δ)
cH for z > 6(1−δ)
cH .
2

2

(2−δ)
(2−δ)
δ(2−δ)2−δ
2−δ
Note that 6(1−δ)
cH ≥ 4(1−δ)(3−2δ)cL and 6(1−δ)
cH ≥ 2(1−δ)(6−δ)
cL + 2(1−δ)(6−δ)
cH areboth equivalent to ccHL ≥ 6−3δ. Hence, if ccHL ≥ 6−3δ, then xL triggers an investigation in
6−4δ
6−4δ

49

2

(2−δ)
cL ; if ccHL < 6−3δ, then xL triggers an investigation
state (L1 , R1 , θLM H ) iff z ≤ 4(1−δ)(3−2δ)
6−4δ
2

(2−δ)
δ(2−δ)
in state (L1 , R1 , θLM H ) iff z ≤ 2(1−δ)(6−δ)
cL + 2(1−δ)(6−δ)
cH .
We can derive the conditions for xH to trigger an investigation in a similar way. Be(2−δ)2cause ccHL < 6−3δ, xH triggers an investigation in state (L1 , R1 , θLM H ) iff z ≤ 2(1−δ)(6−δ)
cH +
6−4δ
δ(2−δ)c .
2(1−δ)(6−δ) L
2

2

(2−δ)
(2−δ)
δ(2−δ), then 4(1−δ)(3−2δ)cL < 2(1−δ)(6−δ)
cH + 2(1−δ)(6−δ)
cL . If ccHL ≥Note that if ccHL ≥ 6−3δ
6−4δ
δ(2−δ)(2−δ)2(2−δ)26−3δ
, let z 0B = 4(1−δ)(3−2δ)
cL ; if ccHL < 6−3δ, let z 0B = 2(1−δ)(6−δ)
cL + 2(1−δ)(6−δ)
cH . Also,
6−4δ
6−4δ
2
δ(2−δ)(2−δ)
0
let z̄B0 = 2(1−δ)(6−δ)
cH + 2(1−δ)(6−δ)
cL . Note that z̄N= z̄B0 and z 0N < z 0B . Lemma B.2

follows.
Hence, if z ≤ z 0N , then both cases trigger an investigation in the first period under0= z̄B0 , then neither case trigger an investigation in the firsteither institution; if z > z̄Nperiod under either institution; if z 0N < z ≤ z 0B , then under binding precedent, bothcases trigger an investigation in the first period whereas under nonbinding precedent,
0
only xH triggers an investigation in the first period; if z < z 0B < z̄N, then only xH
triggers an investigation in the first period under either institution. Accordingly, we
have the following comparison of social welfare analogous to Proposition 7.
Proposition B.1. If the cost of information acquisition is sufficiently low or sufficiently high, then welfare is the same under two institutions with partial learning,0
otherwise, it is different. Formally, (i) If z ≤ z 0N or z > z̄N= z̄B0 , then welfare is thesame under nonbinding precedent and binding precedent. (ii) If z 0N < z ≤ z 0B , then0
welfare is higher under binding precedent. (iii) If z 0B < z ≤ z̄N= z̄B0 , then welfare ishigher under nonbinding precedent.

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permitted

bannedFigure 1: Evolution of precedents.

nonbinding

Figure 2: Comparison of the DM’s dynamic payoffs

53

nonbinding

Figure 3: Comparison of social welfare54