OSIPP Discussion Paper: DP-2020-E-001

Procrastination and Learning about Self-Control
March 4, 2020
Else Gry Bro Christensen*
RBB Economics, Düsseldorf

Takeshi Murooka†
Osaka School of International Public Policy (OSIPP), Osaka University

Abstract:

We study a model of task completion with the opportunity to learn about own

self-control problems over time. While the agent is initially uncertain about her future self-control, ineach period she can choose to learn about it by paying a non-negative learning cost and spending one
period. If the agent has time-consistent preferences, she always chooses to learn whenever thelearning is beneficial. If the agent has time-inconsistent preferences, however, she may procrastinate
such a learning opportunity. Further, if her time preferences exhibit inter-temporal conflicts between
future selves (e.g., hyperbolic discounting), the procrastination of learning can occur even when thelearning cost is zero. Such procrastination also leads to non-completion of the task. Our results help
explain why people pursue implausible dreams and never start any task instead of taking better
alternatives. When the agent has multiple initially uncertain attributes (e.g., own future self-control
and own ability for the task), the agent's endogenous learning decisions may be misdirected – shechooses to learn what she should not learn from her initial perspective, and she chooses not to learn
what she should.

JEL Codes: C70, D83, D90, D91
Keywords: procrastination, self-control, naivete, hyperbolic discounting, misdirected learning*

Email: e.g.b.christensen@gmail.com†

Email: murooka@osipp.osaka-u.ac.jp

Procrastination and Learning about Self-Control∗
Else Gry Bro Christensen†

Takeshi Murooka‡

March 4, 2020

Abstract
We study a model of task completion with the opportunity to learn about own selfcontrol problems over time. While the agent is initially uncertain about her futureself-control, in each period she can choose to learn about it by paying a non-negative
learning cost and spending one period. If the agent has time-consistent preferences,
she always chooses to learn whenever the learning is beneficial. If the agent has timeinconsistent preferences, however, she may procrastinate such a learning opportunity.
Further, if her time preferences exhibit inter-temporal conflicts between future selves(e.g., hyperbolic discounting), the procrastination of learning can occur even when
the learning cost is zero. Such procrastination also leads to non-completion of thetask. Our results help explain why people pursue implausible dreams and never start
any task instead of taking better alternatives. When the agent has multiple initiallyuncertain attributes (e.g., own future self-control and own ability for the task), theagent’s endogenous learning decisions may be misdirected — she chooses to learn what
she should not learn from her initial perspective, and she chooses not to learn what
she should.
JEL Codes: C70, D83, D90, D91
Keywords: procrastination, self-control, naivete, hyperbolic discounting, misdirected
learning∗

We thank Florian Englmaier, Matthias Fahn, Dominik Fischer, Fabian Herweg, Daisuke Hirata, Alexander Koch, Botond Kőszegi, David Laibson, Yves Le Yaouanq, Johannes Maier, Matthew Rabin, Klaus
Schmidt, Peter Schwardmann, Marco Schwarz, Adam Szeidl, and seminar and conference audiences for helpful comments. Financial support from the Deutsche Forschungsgemeinschaft through CRC/TRR 190 (B01)
and JSPS KAKENHI (JP16K21740) are gratefully acknowledged.
†
RBB Economics, Kasernenstr. 1, 40213 Duesseldorf, Germany (Email: e.g.b.christensen@gmail.com).
‡
Corresponding author. Osaka School of International Public Policy, Osaka University, Machikaneyama
1-31, Toyonaka, Osaka, 560-0043, Japan (Email: murooka@osipp.osaka-u.ac.jp).

11

Introduction

People procrastinate (i.e., make a delay that was not fully anticipated ex-ante) in a range oftasks, including filing tax returns, doing reports, setting up financial portfolios, or making
educational decisions.1 In some situations, people keep procrastinating the same or similartasks again and again. Such “naivete” about own future self-control appears prevalent inmany economic environments, and its implications have been investigated in the literature.
Most of the literature, however, assumes (either explicitly or implicitly) that people do notlearn about their self-control problems over time. This opens up a natural question: underwhat conditions do people fail to learn about their self-control problems?
Building upon the work on naivete about self-control by O’Donoghue and Rabin (1999a,
2001), we investigate models in which an agent faces a task to complete and potentiallyhas time-inconsistent preferences.2 Section 2 introduces our model and discusses its key
assumptions. In our basic model, an agent is initially not sure whether her future selves will
be time-consistent or time-inconsistent, and she may initially underestimate her probabilityof being time-inconsistent. Her time-preference type is persistent over time. In each period,
she voluntarily chooses whether or not to update her beliefs about own self-control problem(which may affect her subsequent actions on task completion) by incurring a non-negative
cost of learning. We highlight the case in which the cost of learning is zero and the agent can(in principle) perfectly learn about her self-control. In contrast to Ali (2011), in our model
the agent cannot commit to any future actions.
Section 3 analyzes an illustrative model of task completion. Our key mechanism is thatthe agent may procrastinate the opportunity to learn about her future self-control type. The
intuition consists of two steps. First, because learning itself changes own future actions, atime-inconsistent agent may indirectly incur a cost from learning if changing future actions
1
For empirical and experimental evidence, see, for example, Ariely and Wertenbroch (2002), DellaVigna
and Malmendier (2006), Augenblick and Rabin (2019), and Le Yaouanq and Schwardmann (2019). Following
O’Donoghue and Rabin (2001) and DellaVigna (2009), we say that an agent “procrastinates” if she plans to
do a task with some probability but actually does it with lower probability, whereas an agent “delays” if sheplans not to do a task with some probability and actually does not do it with the same probability.
2
For papers analyzing task completion and procrastination, see, for example, O’Donoghue and Rabin(1999a, 1999b, 2001, 2008), Ariely and Wertenbroch (2002), Herweg and Müller (2011), and Fischer (2018).

2

involves both future benefit (e.g., returns from setting up a financial portfolio) and future cost(e.g., effort cost to set up a financial portfolio). We highlight that this indirect cost occurs
when an agent’s time preferences exhibit intra-personal conflicts between future selves (e.g.,
hyperbolic discounting), whereas there is no such indirect cost when she has intra-personal
conflicts only between her current self and future selves (e.g., quasi-hyperbolic discounting).
Second, because the time-consistent type of agent initially overestimates the probability
that she will be time-consistent, she (erroneously) believes that she will choose to learn
and complete a task with higher probability than in reality. Because of this overconfidence
about her own future behavior, she may prefer to postpone the learning opportunity inorder to complete the task later rather than sooner.3 As a result, the agent may not use
the opportunity to learn about her own self-control problem, even when the direct cost oflearning is zero. We also derive conditions in which procrastination of learning and noncompletion of a task is a unique equilibrium outcome. Our results help explain why people
procrastinate even when they face similar situations repeatedly.
This non-learning result is in contrast to the one found by Ali (2011), who shows thattime-inconsistent people will learn perfectly about their self-control under flexible commitment devices. We discuss assumptions leading to our non-learning result: absence of commitment devices, voluntary learning, and conflict between future selves. Our result could
help bridge the gap between Ali’s (2011) theoretical results and the empirical relevance of
naivete about self-control. Perhaps surprisingly, our result does not depend on anticipatory
utility, ego utility, private image, or social image (e.g., Kőszegi 2006, Gottlieb 2014, 2016).
Furthermore, even when people are inherently aware of their own self-control problems, whenfacing a new task, they may have an incentive to neglect this awareness and take action as anignorant self. In this sense, we provide a rationale of O’Donoghue and Rabin’s (2001, page127) assumption: “it is central to our analysis that a person not fully learn over time her
3

Note that the second logic looks akin to that in O’Donoghue and Rabin (1999a, 2001). However, we
identify assumptions and conditions for when and how procrastination can occur in the domain of learning
about own self-control problems, whereas O’Donoghue and Rabin (1999a, 2001) assume that an agent cannot
learn about this over time. We further show that procrastination of learning can occur even when the direct
cost of learning is zero under general time-inconsistent preferences, whereas it does not occur under quasihyperbolic discounting as employed by O’Donoghue and Rabin (1999a, 2001).

3

true self-control problem, or, if she does come to recognize her general self-control problem,
she still continues to underestimate it on a case-by-case basis.”
Section 4 analyzes a model of task choice in which an agent first chooses a task to complete
and then engages in the task. To describe the situation, suppose that an agent chooses toeither pursue a university degree, do an apprenticeship, or postpone the decision. If she
knew she were time-consistent, she would go to university now. If she knew she were timeinconsistent, the high upfront cost of the university degree would be too much for her, so shewould be better off choosing the apprenticeship now. But if she is time-inconsistent and does
not know her future self-control type, she may believe that her future selves will be timeconsistent with a high probability. Given this belief and her time-inconsistent preferences,
she may prefer to go to university next year and not now. This is why she does not start theapprenticeship now, and once next year comes she may postpone going to university again;in this way, she may never complete any further education. In addition, she may also notchoose to learn about her own naivete, because she may have an incentive to procrastinate
doing so, as illustrated above. We derive conditions under which a time-inconsistent typeof agent neither chooses a task nor learns about her own future self-control, which leads tonon-completion of any task. Interestingly, lower learning costs can make time-inconsistentagents more likely to procrastinate on the equilibrium path.
As an important extension, Section 5 investigates a case in which an agent initially has
multiple uncertain attributes and can choose to learn about either (or both) of them. For
example, the agent may be initially uncertain about both her own future self-control andher own ability for the task. As analyzed by Carrillo and Mariotti (2000) and Bénabou
and Tirole (2002), under some conditions the agent strictly prefers to avoid learning about
her own ability to motivate her future self to complete the task. This strategic ignorance
can be beneficial from the agent’s initial perspective. Under other conditions, however, weshow that the agent’s endogenous learning decisions are misdirected — she chooses to learn
what, from the initial perspective, she should not learn (in our example, about her own
self-control), and she chooses not to learn what she should learn from the initial perspective

4

(in our example, about her own ability).
Section 6 briefly discusses other extensions. Section 7 concludes. Proofs and additional
analysis are provided in the Appendix.
Related literature. There are several strands of literature that investigate why people do
not learn over time. The most closely related literature to our paper is on strategic ignorance:
Carrillo and Mariotti (2000) and Bénabou and Tirole (2002) show how, under the presence ofself-control problems, an agent may strategically abstain from learning about her own ability
for the task, to keep motivating her own future selves to work harder. In these papers, thestrategic ignorance is beneficial from the agent’s initial perspective. Relatedly, Bénabou andTirole (2004) study how an agent may commit to a personal rule to maintain self-reputation.
The implications of our results are different from those of the previous studies mentioned
here, however: in our model, non-learning is harmful from the agent’s initial perspective
because it leads to non-completion of a task.4 Furthermore, we demonstrate that if theagent is uncertain about both her own self-control and her own ability, then under someconditions the agent’s learning decisions will be misdirected and harmful.
Within the literature on self-esteem, Kőszegi (2006), Gottlieb (2014), and Gottlieb (2016)
investigate models in which an agent may avoid learning because of the presence of ego utility,
private image, or anticipatory utility. In our model, such psychological costs are captured asa direct cost of learning; we show that non-learning can occur even when there is no direct(physical or psychological) cost of learning.
The literature on selective attention, such as Schwartzstein (2014) and Gagnon-Bartsch,
Rabin and Schwartzstein (2018), analyzes situations in which an agent systematically does
not encode a certain type of signal. In contrast to these studies, we focus on the situationin which the cost of learning can be zero and an agent can perfectly learn about her own
self-control problems but may voluntarily choose not to learn about it.5
4

There is also a difference regarding the mechanisms: an agent in the cited papers chooses not to learn
as a means of internal commitment (and to improve her own future payoffs), whereas an agent in our paper
does so because she overestimates the probability that she will behave time-consistently.
5
In a similar way, our results also differ from the literature on non-Bayesian updating rules, in which anagent cannot update her own beliefs according to Bayes’s rule. Further, most of the mechanisms proposed5

Finally, our results on multiple uncertain attributes are related to the recent literatureon learning under misspecified models (Fudenberg, Romanyuk and Strack 2017, Heidhues,Kőszegi and Strack 2018, Hestermann and Le Yaouanq 2020). Building upon and extending
this literature, our mechanism based on procrastination explains why an agent’s learning
decisions can be misdirected even when she can choose to perfectly learn about all initiallyuncertain attributes.

2

Model

This section introduces our basic model. Section 2.1 sets up the model. Section 2.2 discusses
our key assumptions.

2.1

Setup

A risk-neutral agent can choose a task from a menu of tasks in periods t = 1, 2, · · · , T , whereT ≥ 2 is either finite or infinite. Once she chooses a task, she can complete the task fromthe next period on.6 Let X = {x1 , . . . , xN } be a menu of tasks. Each task is represented by
xn = (−cn , bn ): completing the task in period t gives a cost cn ≥ 0 in period t and bringsa benefit bn ≥ 0 in period t + 1. There is no (physical or psychological) cost of choosing atask. The agent can choose (and complete) at most one task during the game.
Let ut denote the agent’s period-t instantaneous utility. There are two types of agents:
time-consistent and time-inconsistent ones. The type of each agent is persistent throughout
the game. For the time-consistent type, her total utility in period t isP∞δ ∈ (0, 1). For the time-inconsistent type, her total utility in period t isP∞

τ =t δ

τ −tuτ withτ =t D(τ − t)uτ ,where D(τ − t) ∈ (0, 1] represents her time discounting in τ − t periods.7 We assume
in previous studies (e.g., conservatisms) would predict that learning is merely slowed down but would occur
eventually (see Benjamin, Rabin and Raymond (2016) for a notable exception).
6
Motivated by real-world examples such as making educational decisions (and then studying) or setting
up a financial portfolio (and then using it to accumulate savings), we focus on situations in which an agent
takes multiple steps to complete a task in Section 4.
7
See, for example, Echenique, Imai and Saito (2020) for the analysis under general time preferences.
Relatedly, Schweighofer-Kodritsch (2018) analyzes Rubinstein’s alternative bargaining problem with general6that D(0) = 1, D(1) < δ, D(t) is strictly decreasing in t with limt→∞ D(t) = 0, andD(1)/D(0) 6= D(2)/D(1).
Both types of agent are initially uncertain about their own future self-control. We assume
that all types share the same initial belief about their own future time consistency, denotedby D̂(t), and that it is D̂(t) = D(t) with probability 1 − q and D̂(t) = δ t with probability q.8
In line with the literature on limited cognition and beliefs as assets (Bénabou 2015, Bénabou
and Tirole 2016), we assume that updating the initial belief about own self-control requires
the agent to voluntarily choose to learn; formally, in each period the agent can acquire asignal about her own self-control by incurring a cost m ≥ 0, while she sticks to her initialbeliefs if she has not acquired the signal. To focus on our main mechanism, we assume thatlearning is perfect and perpetual: the signal is perfectly informative and the agent becomescompletely sophisticated about their own self-control problems for the rest of the game. It
is worth emphasizing that learning is made by the agent’s voluntary choice, as we discuss indetail in Section 2.2.
Let k ∈ {C, I} be the agent’s true discounting type, where C represents a time-consistent
type and I represents a time-inconsistent type. Let UtC (xn ; τ − t) = −δ τ −t cn + δ τ −t+1 bn and
UtI (xn ; τ − t) = −D(τ − t)cn + D(τ − t + 1)bn respectively denote type-C’s and type-I’s total
utility evaluated in period t (not taking into account the learning cost m) when the agent
completes task xn in period τ . Let st ∈ {0, 1} denote the agent’s decision on acquiring asignal about own self-control (i.e., her learning decision) in period t. If the agent has alreadyacquired the signal (i.e., st0 = 1 for some t0 < t), the cost of learning m ≥ 0 has been
paid, and hence the agent’s utility only depends on when she will complete the task. Note
that, at the beginning of period τ , the agent has not learned about her self-control if andonly if st = 0 for all t < τ . When the agent acquires a signal about her type in periodt0 ≥ 0, each type of agent’s subjective expectation of total utility evaluated in period t is
time preferences.
8
In some applications, it could make more sense to assume that a time-consistent type knows own futureself-control, whereas a time-inconsistent type is initially (stochastically) overconfident about own futureself-control. As we will discuss later, none of our results would qualitatively change with this alternative
specification.

7

Ûtk (st0 = 1) = qUtk (x(C); τ (C) − t) + (1 − q)Utk (x(I); τ (I) − t), where x(k) and τ (k) indicate
the chosen tasks and completion periods for an agent of type k after learning her type inperiod t0 . We abbreviate subscript t whenever it is clear. Note that the utility is evaluatedaccording to each type’s own preferences: the only reason why each type of agent cares about
own future preferences is to predict future outcomes, and each period-t self maximizes theagent’s own period-t utility given her (possibly erroneous) expectations.
The timing of the game is as follows. If the agent has not chosen a task, in each period
she takes one of the following actions: learning about her own self-control problem (if she has
not yet done so), choosing a task, or not doing anything. If the agent has chosen a task, ineach period the agent takes one of the following actions: learning about her own self-control
problem (if she has not yet done so), completing the task, or not doing anything.
As an equilibrium concept, we adopt a natural extension of the perception-perfect equilibria (O’Donoghue and Rabin 2001) in which (i) in each continuation game an agent chooses
her best response given her current belief and preference and (ii) an agent keeps using theinitial prior about her own self-control problem if she has not acquired a signal about her own
self-control.9 We focus on pure strategies. While all of our theoretical results are descriptive,
whenever we discuss potential welfare implications, we evaluate each type of agent’s welfare
based on the initial-self’s preference.

2.2

Discussion of Key Assumptions

Voluntary learning. A crucial assumption of our model is that the agent has to voluntarily choose whether to learn about her own self-control problem: whenever updating her
beliefs, she has to “encode” a signal (e.g., she needs to introspect from her own past experience).10 Hence, our model is different from a classical one in which Bayesian updating is
automatically applied. It is also different from a self-confirming equilibrium, as the agent in
9

The second part is a natural extension of O’Donoghue and Rabin (2001), because a (partially) naive
present-biased agent in their original model does not update beliefs about own future self-control over time.
10
In period t = 1, we interpret the learning opportunity as constituting the option for the agent to lookback to similar past experiences to infer her self-control problem. In period t ≥ 2, the agent can try to learn
about her own self-control problem from her own past actions in the game.

8

our model updates her beliefs only when she chooses to do so.11 In our model, learning isthus analyzed through the lens of the agent’s cost-benefit analysis.
Unless the agent chooses to learn, she does not revise her beliefs, and keeps using herprior belief. Hence, unlike Bénabou and Tirole (2004), we rule out self-signaling over time.
We also assume away the possibility that the agent becomes aware of her own self-control
by inferring from her own current preferences and beliefs (i.e., we assume away inferences
like “because my preferences are time-inconsistent today, I must be a time-inconsistent typein the future”).12 These assumptions are consistent with our assumption that the agent
has to voluntarily choose to update her belief, and are also in line with the assumptions in
O’Donoghue and Rabin (1999a, 2001). In contrast to these two papers, however, our model
does not assume that the possibility of learning is excluded. Indeed, our results complement
O’Donoghue and Rabin (1999a, 2001) in the sense that, given the conditions provided in thispaper, their original insight and results are robust to introducing the opportunity to learn
about own self-control.
(At most) one action per period. Another assumption is that the agent cannot choose
learning and other actions at the same time (i.e., an agent has to spend one period to update
her beliefs). While this assumption is restrictive, it captures the notion that we need to spend
time thinking about ourselves whenever changing our self-perception; this “time cost” is key
to generating an intra-personal conflict between future selves (and hence procrastination onlearning). In other words, an agent in our model can be Bayesian, but the Bayesian inference
is not free: the agent needs to pay the cost of learning m ≥ 0 and take some time to think.13
11

Esponda and Pouzo (2016) provide a framework to analyze agents with misspecified models. Note that
our model is also different from theirs, because in our model the agent voluntarily chooses whether or not
to update her own initial belief.
12
Akin to the discussion in Footnote 8 of O’Donoghue and Rabin (2001), a time-inconsistent type in ourmodel does not make such an inference. That is, a time-inconsistent type cannot revise her beliefs unlessshe encodes a signal. By contrast, all of our results are qualitatively robust (and often become simpler) if
a time-consistent type becomes aware of her own future self-control by inferring from the fact that she iscurrently time-consistent.
13
Note that this assumption may not be necessary if future selves can misuse the acquired information fromthe perspective of today’s self. To both simplify the analysis and highlight the procrastination mechanism,however, we shut down such a possibility in our model.

9Although the direct cost of learning can be zero, because of the delay introduced byvoluntary learning (as described above), the agent may also incur a cost from non-learning.
One of the contributions of this paper is to highlight how an agent may not learn about her
own future self-control even when the direct learning cost is zero (i.e., m = 0) and how thiscan be a unique equilibrium outcome. Our model also allows a positive learning cost (i.e.,m > 0), which represents, for example, a physical recollection cost or a psychological cost ofdeteriorating self-esteem (Kőszegi 2006; Gottlieb 2014, 2016). Incorporating such a learning
cost is plausible in many real-world situations, and comparative statics with respect to mhence generate additional implications for such settings.
No commitment. In the model, the agent cannot commit to any particular future behavior, including future learning decisions. In Section 6, we discuss the case in which an agent
can take up a commitment device. If the agent can commit to specific future actions, underzero learning cost, non-learning will not occur in our model. In this respect, our non-learning
result complements the study by Ali (2011), who shows that time-inconsistent people willlearn perfectly about their self-control if they can take up a flexible commitment device anddo Bayesian updating automatically.

3

Task Completion

This section analyzes an illustrative model of task completion; we focus on the case in which
the agent has already selected a task x = (c, b).14 We can then restrict attention to theagent’s learning and task completion choices. To shed light on our main mechanism in thesimplest manner, we further assume throughout this section that the agent can work on thetask only after acquiring a signal about her own self-control.15 In Section 4, we analyze afull model without these assumptions.
14

Alternatively, this can be a situation in which the agent is facing a menu of only one task.
As we formally investigate in Section 4, this assumption simplifies a situation in which working on atask without learning is costly (e.g., the agent cannot choose an optimal task without knowing her owntype), and hence the agent does not want to engage in any task before learning about her own self-control.
15

10

3.1

Task Completion: Setup

The timing of the game in this section is as follows. If the agent has not acquired a signal
st ∈ {0, 1} about her self-control problem, in each period she chooses to either acquire it(st = 1) or not (st = 0). If the agent has acquired it, in each period she chooses whether tocomplete the task or not.
In what follows, we focus on the most interesting case, in which each type of agent prefers
to complete the task as soon as possible rather than never:Assumption 1 (Task is Worthwhile). (i) U I (x; 0) > 0, (ii) min{U C (x; 1), U I (x; 1)} > m.
Because U C (x; 0) ≥ U I (x; 0) by assumption D(1) < δ, Assumption 1 (i) means that eachtype of agent prefers to complete the task right now as opposed to never .16 Assumption 1(ii) means that each type of agent prefers to acquire information right now and to complete
the task in the next period rather than never.17 Note that under Assumption 1, both typesof agent agree that the task should be completed. Hence, uncertainty and naivete about
own future type is the only reason for not acquiring the signal and for non-completion of the
task.

3.2

Illustrative Example in a Finite Horizon ModelIn this subsection, we illustrate a simple numerical example with a finite time horizon (i.e.,
4 ≤ T < ∞) to highlight how procrastination on learning can occur in our model. Becauseof the finite time horizon, equilibrium can be pinned down by backward induction and isgenerically unique. See Appendix B for the full analysis of the finite horizon model.
1
Suppose that m = 0, c = 1, b = 32 + for a sufficiently small  > 0, δ = 1, and D(t) = 1+rt

with r = 12 , that is, a time-inconsistent type’s preference exhibits hyperbolic discounting. In
this case, it can be shown that in every continuation game the time-consistent type always
If U I (x; 0) < 0, a time-inconsistent type never completes the task. In this case, if U C (x; 0) < 0 or if
the time-inconsistent type would not want the task to be completed in any future period, then obviously
the time-inconsistent type would never encode the signal. Otherwise, the time-inconsistent type may havean incentive to acquire a signal, but procrastination on acquiring the signal can still occur in this case.
17
If U C (x; 1) < m, then a time-consistent type never encodes a signal.
16

11

acquires information about her own type (if she has not acquired it yet) and always works
on the task (if she has acquired information). For the time-consistent type, U I (x; τ ) > 0
for all τ , and completing the task in two periods is the most preferred for her, that is,maxτ U I (x; τ ) = U I (x; 2). As the agent has to learn her type before completing the task,
the earliest point in time the task can be completed is t = 2 (with acquisition of informationin t = 1).
Next, we analyze the behavior of the time-inconsistent type by backward induction. First,
we characterize the behavior of the time-inconsistent type in t = T . If she has acquired
information, then she completes the task, because U I (x; 0) = −c + D(1)b = 23  > 0. If she
has not acquired information, then she cannot complete the task, because t = T is the lastperiod, and hence, her payoff is zero.
Second, we characterize the behavior of the time-inconsistent type in t = T − 1. If she
has already acquired information, then she does not complete the task, because U I (x; 1) =1
−D(1)c + D(2)b = 12
+ 12  > U I (x; 0). Intuitively, akin to O’Donoghue and Rabin (1999a,

2001) as well as the subsequent literature on behavior of sophisticated time-inconsistentagents, a time-inconsistent type may prefer to delay the completion of the task by one
period rather than to complete it now. Note that this result itself is not procrastination, as
the agent has correct beliefs about when the task will be completed and her actual decisions
follow the beliefs. If she has not yet acquired information, then she acquires it because bothtypes will complete the task in the next period, and hence Û I (sT −1 = 1) = U I (x; 1) > 0 =Û I (sT −1 = 0).
Third, consider a continuation game in t = T − 2 in which the time-inconsistent type
has not acquired information. If she chooses to acquire it now, she anticipates that she willcomplete the task in t = T − 1 with probability q and do so in t = T with probability 1 − q,and hence Û I (sT −2 = 1) = qU I (x; 1) + (1 − q)U I (x; 2). If she does not acquire information
now, she anticipates that she will complete the task in t = T with probability one, and hence
1+ 25  > U I (x; 1), she
Û I (sT −2 = 0) = U I (x; 2). Because U I (x; 2) = −D(2)c + D(3)b = 10

does not acquire information in t = T − 2. Intuitively, a time-inconsistent type prefers to12delay acquiring information in order to delay the completion of the task.
Finally, we show the condition under which the time-inconsistent type procrastinates acquiring information. Consider continuation games in t ≤ T −3, in which the time-inconsistent
type has not acquired information. If she acquires it now, she anticipates that she will complete the task in the next period with probability q and do so in some future period withprobability 1 − q.18 Because maxτ U I (x; τ ) = U I (x; 2) in this case, her anticipated expected utility when she acquires information is at most qU I (x; 1) + (1 − q) maxτ U I (x; τ ) =qU I (x; 1) + (1 − q)U I (x; 2). If she does not acquire information now, she anticipates that
she will do so in the next period and will complete the task in the next two periods withprobability q. Hence, her anticipated expected utility when she does not acquire information
is at least qU I (x; 2). This implies that if qU I (x; 2) > qU I (x; 1) + (1 − q)U I (x; 2), or if q > 67
when  approaches 0, the time-inconsistent type will choose not to acquire information in anyt ≤ T − 3.19 Hence, if q > 76 , the time-inconsistent type procrastinates acquiring information
until t = T − 2 on the equilibrium path.
Intuitively, because a time-inconsistent type is (probabilistically) overconfident about own
future self-control, she erroneously thinks that she will acquire information with probability
q in the next period, even when she does not do so now. Hence, as in O’Donoghue andRabin (1999a, 2001), the time-inconsistent type may procrastinate information acquisition— she thinks that she will acquire information with probability q in the next period, but she
actually will not do so.20 However, in contrast to O’Donoghue and Rabin (1999a, 2001), theagent in our model is never surprised by a zero-probability event on the equilibrium path,even when she keeps procrastinating.
If the time-inconsistent type is sufficiently naive (i.e., q is sufficiently close to 1), she(erroneously) believes that time inconsistency is of little relevance to her. Hence, she be18

Here, for simplicity of exposition, we do not describe exact equilibrium strategies and beliefs of thetime-inconsistent type. See Appendix B for the full characterization.
7
19
For a sufficiently small but positive  > 0, the condition becomes q + 24
(2q − 1) > 67 .
20
Note that, for continuation games in t ≤ T − 3 in which the time-inconsistent type has not acquired
information, the beliefs for her future learning decisions are wrong: in reality, if she does not learn about her
own type in period t = T − 3, she will not do so in period t = T − 2 either (because her type is persistentover time). However, she is naive about it exactly because she has not learned about it yet (and hence keeps
using her initial prior belief).

13

lieves that if she acquires information now, she is almost certainly going to complete thetask tomorrow, and if she does not acquire it now, she believes that she will most likely
acquire it in the next period and then complete the task in the next two periods. The naive
time-consistent type (erroneously) believes that this would be the main trade-off she is facing: acquiring information now and completing the task tomorrow, or acquiring information
tomorrow and completing the task in two periods. This leads to procrastination if the latter
option outweighs the former one.
Perhaps surprisingly, and beyond the original logic of O’Donoghue and Rabin (1999a,
2001), procrastination in our model can occur even when m = 0. This is because the conflict
between future selves exists if U I (x; 2) > U I (x; 1), which never arises under quasi-hyperbolic
discounting but can occur under hyperbolic discounting (as well as other functional forms ofdeclining impatience over time).

3.3

Analysis in an Infinite Horizon ModelIn this subsection, we analyze a model with T = ∞ and show under which conditions a timeinconsistent agent will procrastinate learning indefinitely, which implies never completing thetask on the equilibrium path. Specifically, we look for an equilibrium in which: I) once theagent learns about her own self-control, both types will complete the task in the next period;II) the agent always acquires the signal when she is time-consistent; and III) the agent will
never acquire information when she is time-inconsistent.
We first specify continuation games of type I: the agent’s behavior in continuation games
in which the agent has already learned her type. If the agent has learned that she is timeconsistent, she chooses to complete the task in any period, because Assumption 1 (i) implies
U C (x; 0) = −c + δb > 0. If the agent has learned that she is time-inconsistent, Assumption 1(i) ensures that there exists an equilibrium in continuation games in which the agent always
completes the task in the next period after learning, as shown in the proof. When showing
the existence of a non-learning equilibrium, we focus on such an equilibrium in continuation
games. We then derive the condition in which non-learning for time-inconsistent agents is a14

unique equilibrium outcome.
We next investigate the agent’s behavior regarding learning about her own self-control.
Consider continuation games of type II, where the agent is time-consistent. In this case, Assumption 1 (ii) ensures that she chooses to learn about her own self-control in our candidate
equilibrium.
Now suppose continuation games of type III, where the agent is time-inconsistent. Notethat her beliefs about her own future actions depend on her beliefs about her own future selfcontrol. On the one hand, the agent (erroneously) believes that she will be time-consistentand hence will acquire information in the next period, with probability q. On the other hand,she (correctly) anticipates that she will not acquire information in any future period, withprobability 1 − q, in our candidate equilibrium. To show the existence of this equilibrium,
we first focus on an equilibrium in continuation games in which the time-inconsistent type
always completes the task in period t + 1 if she learns about own self-control in periodt. Given these beliefs, if the time-inconsistent type chooses not to learn, her anticipatedexpected utility is:hiÛtI (st = 0) = q −D(1)m + U I (x; 2) + (1 − q) · 0.
In contrast, if the time-inconsistent type chooses to learn, her anticipated expected utility
is:ÛtI (st = 1) = −m + U I (x; 1).
Hence, there exists an equilibrium in which the time-inconsistent type prefers not to learn
in each period if:
ÛtI (st = 0) > ÛtI (st = 1)hi⇐⇒ q U I (x; 2) − U I (x; 1) + m [1 − qD(1)] > (1 − q)U I (x; 1).

(1)

Next, we derive the condition under which a time-inconsistent type never chooses to learn
about her self-control problems — and hence never completes the task — in any equilibriumoutcome. In the proof, we show that in any continuation game, a time-consistent type will15

always choose to learn and complete the task immediately if(q − δ)U C (x; 1) + m(1 − δ) ≥ 0.

(2)

Suppose continuation games where a time-inconsistent type has not learned. Denotethe maximum and minimum utility of a time-inconsistent type from completing a task byII

U (x) = maxτ U I (x; τ ) and U I (x) = minτ U I (x; τ ), respectively. Note that U (x) > 0 ≥U I (x) by Assumption 1 and that limτ →∞ U I (x; τ ) = 0. Then, the lower bound of anticipatedexpected utility when the time-inconsistent type chooses not to learn is:
h

i

h

i

U It (st = 0) = q −D(1)m + U I (x; 2) + (1 − q) −D(1)m + U I (x) .
To see the intuition here, note that a time-inconsistent type knows how a time-consistent
type will behave. With probability q, she believes that she will be time-consistent in the
future, acquire the signal, and complete the task in two periods. In contrast, however, withprobability 1 − q she believes that she will be time-inconsistent in the future. In this case,the lowest possible payoff is to pay the information acquisition cost as soon as possible but
to complete the task in the least desired period. Similarly, an upper bound on anticipatedexpected utility when the time-inconsistent type chooses to learn is:II

U t (st = 1) = −m + qU I (x; 1) + (1 − q)U (x).
Hence, in any equilibrium outcome, the agent prefers not to learn if:
I

U It (st = 0) > U t (st = 1)
h

h I

i

i

⇐⇒ q U I (x; 2) − U I (x; 1) + m [1 − D(1)] > (1 − q) U (x) − U I (x) .

(3)

The result is summarized as follows.
Proposition 1. Suppose T = ∞ and Assumption 1 holds.
(i) If Inequality (1) holds, there exists an equilibrium in which a time-inconsistent typenever learns about her own future self-control (and hence never completes the task).
(ii) If Inequalities (2) and (3) hold, a time-inconsistent type never learns about her own
future self-control (and hence never completes the task) in any equilibrium outcome.
16

The intuition of the result is two-fold. First, if the agent is time-inconsistent, she may
prefer to complete a task later rather than sooner. Specifically, if the time-inconsistent typeprefers to complete the task in two periods rather than in one period (i.e., if U I (x; 2) >U I (x; 1)), then she has an incentive to delay acquiring the signal in order to delay taskcompletion. Second, because the time-inconsistent type is (probabilistically) overconfident
about her own future self-control, she overestimates the likelihood that she will acquire thesignal in the future. Specifically, if the degree of naivete q is sufficiently large, the timeinconsistent type (erroneously) believes that she will most likely acquire the signal in thenext period; hence, she prefers not acquiring it now, to delay task completion. Because thetime-inconsistent type fails to infer from her own actions that she cannot be time-consistentin the future unless acquiring a signal about her own type, she may procrastinate indefinitely.
It is worth mentioning that procrastination on learning about own self-control can occur even when the time-inconsistent type prefers to complete the task in the next period
rather than never (i.e., U I (x; 1) > m). Intuitively, because the time-inconsistent type isoverconfident about her own future self-control, she underestimates the possibility that she
will procrastinate on learning. As a result, she may prefer to postpone a valuable learning
opportunity.
Furthermore, if the time-inconsistent type is sufficiently naive and prefers a delay, procrastination of learning is a unique equilibrium outcome. It extends the result in Section 3.2
in which non-learning can occur even when m = 0. Further, even when m = 0, non-learning
can be a unique equilibrium outcome:Corollary 1. Suppose T = ∞, Assumption 1 holds, and U I (x; 2) > U I (x; 1). Then, there
exists q̄ ∈ (0, 1) such that for any q > q̄ a time-inconsistent type never learns about own
future self-control (and hence never completes the task) in any equilibrium outcome.

17

4

Task Choice

In Section 3, we assumed that the agent already faces the need to complete a certain task.
This section analyzes situations in which an agent first chooses a task to complete and thenengages in the task; for example, one could choose to either pursue a university degree, doan apprenticeship, or postpone the decision. Formally, the agent in this section has multiple
tasks, but she can choose only one task throughout the game, and the optimal choice depends
on her type. In this section, we allow the agent to complete a task without acquiring a signal
about her own type. As an illustration, we first focus on a case where the agent is facing
two possible tasks to choose from, and then provide a more general analysis.

4.1

Analysis with a Two-Task Menu

Suppose T = ∞ and the agent can choose one of the following two tasks: x = (−c, b) and
x0 = (0, b0 ) with −c + δb > δb0 > 0 > −c + D(1)b. We also assume that minτ ≥0 U I (x, τ ) =U I (x, 0) = −c + D(1)b. As described in Section 2.1, we assume that the task cannot becompleted in the same period it is chosen. We derive the conditions for an equilibrium inwhich the time-inconsistent type never learns about her own self-control problems or chooses
a task, whereas the time-consistent type chooses to learn about her own self-control problems
and then complete task x on the equilibrium path.
First, we characterize the agent’s task-completion behavior after learning her own type.
It is straightforward to show that if the agent has chosen task x0 , she will complete thattask in every period irrespective of her type. Also, if the agent has chosen task x, she willcomplete that task in every period if she is time-consistent and will never complete the task
if she is time-inconsistent.
Second, suppose the agent has learned her own type but has not yet chosen a task. In
this case, it is straightforward to show that the agent chooses x in any period if she istime-consistent and chooses x0 in any period if she is time-inconsistent.
Third, suppose the agent has chosen a task but has not learned her own type. If theagent has chosen task x0 , she will complete the task in every period irrespective of her type.
18

Also, if the agent has chosen task x, she will complete the task in every period if she istime-consistent and will never complete the task if she is time-inconsistent.
Fourth, suppose that the agent has neither learned about her own type nor chosen a task
and that the agent is time-consistent. Given the behavior shown in the above continuation
games, the perceived expected utility of the time-consistent type for each action in ourcandidate equilibrium is as follows:• Choosing x without learning own type: qU C (x; 1) + (1 − q) · 0.
• Choosing x0 without learning own type: U C (x0 ; 1).
• Learning own type: −m + qU C (x; 2) + (1 − q)U C (x0 ; 2).
• Not doing anything: q(−mδ + U C (x; 3)) + (1 − q) · 0.
Note that for the time-consistent type, not doing anything is dominated by choosing x
without learning own type. Hence, if the agent is time-consistent, she will choose to learn if
and only if:−m + qU C (x; 2) + (1 − q)U C (x0 ; 2) > max{qU C (x; 1), U C (x0 ; 1)}orm
1 − (1 − q)δ C 0m(1 − q)δ C 0
U (x ; 0) −> qU C (x; 0) >U (x ; 0) + 2
1−δ
δ(1 − δ)δδ

(4)Note that Inequality (4) always holds for m = 0 and δ close to one. In what follows , wefocus on the case in which Inequality (4) holds.
Finally, suppose that the agent has neither learned about her own type nor chosen a task
and that she is time-inconsistent. If she chooses to learn, she expects that she will choosetask x in the next period and then complete it in two periods with probability q, whereas she
will choose task x0 in the next period and then complete it in two periods with probability
1 − q. Hence, the perceived expected utility of the time-inconsistent type for each action in
our candidate equilibrium is as follows:19• Choosing x without learning own type: qU I (x; 1) + (1 − q) · 0.
• Choosing x0 without learning own type: U I (x0 ; 1).
• Learning own type: −m + qU I (x; 2) + (1 − q)U I (x0 ; 2).
• Not doing anything: −q(mD(1) + U I (x; 3)) + (1 − q) · 0.
Thus, if the agent is time-inconsistent, she will choose not to do anything if(

1
1−q I 0m
U (x; 3) − mD(1) > max U (x; 1), U I (x0 ; 1), U I (x; 2) +U (x ; 2) −
q
q
q
I

I

)

(5)

and Inequality (4) holds.
The intuition is as follows. First, the time-inconsistent type thinks that she will acquirea signal about her own self-control if she is time-consistent. Second, because she is timeinconsistent, she may prefer to complete a task later rather than sooner. For example, if
m = 0 and b0 is sufficiently close to zero, then the time-inconsistent type has an incentive topostpone acquiring the signal to delay task completion if U I (x; 3) > max{U I (x; 1), U I (x; 2)}.
Third, because the agent is (probabilistically) overconfident about her own future self-control,
she underestimates the likelihood that she will not acquire the signal in the future. The next
proposition summarizes the result:
Proposition 2. Suppose T = ∞ and the agent faces a menu of x = (−c, b) and x0 = (0, b0 ).
If Inequalities (4) and (5) hold, there exists an equilibrium in which a time-inconsistent typenever learns about her own future self-control or chooses any task on the equilibrium path.
Proposition 2 highlights a perverse welfare effect of non-learning in our model: If thetime-inconsistent type were to know that she is time-inconsistent, then she would choosetask x0 , which would give her strictly positive utility. But because she (erroneously) willthat she is time-consistent with a high probability, she believes that she would choose task x
in the future with a high probability. Given this (erroneous) belief, she prefers to do nothingin order to delay completing task x. Because her type is time-inconsistent and is persistentover time, however, she forever keeps doing nothing on the equilibrium path.
20

Proposition 2 may help explain why people pursue implausible dreams and never actually
start anything instead of taking better alternatives . As a real-world example, suppose thatan agent can choose whether to pursue a university degree or do an apprenticeship. If she
knew she were time-consistent, she would go to the university this year. If she knew she
were time-inconsistent, she would realize that the high upfront cost of the university degree
is too much for her, so she would choose the apprenticeship this year. But if she is timeinconsistent and does not know her future type, she may believe that her future selves would
be time-consistent with a high probability, and hence, may plan to go to university next
year. This is why she does not start the apprenticeship this year, and as a result, ends up
never completing any further education.
1
Example 4.1. Consider the case in which D(t) = 1+rt
, r = 21 , and δ ' 1. Then, theassumption −c + δb > 0 > −c + D(1)b holds if and only if cb ∈ (1, 32 ).
(i) If m = 0, Inequality (4) always holds and Inequality (5) is equivalent to the following
condition:
0

25 bc
15 b0
q > max,.
0
1 + 20 bc 4 c()

0Hence, for bc < 51 there exists a q̄ < 1 such that for all q > q̄ the time-consistent type acquires
information and completes task x, while a time-inconsistent type never does anything on theequilibrium path.
011(ii) If m = 50c and bc = 10
, Inequality (4) and (5) become:

(

)

15
4
1
> q > max,.
b
550 c − 55 44
Hence, for cb ∈ ( 98 , 23 ), there is a range for q such that the time-consistent type acquires
information and completes task x, while a time-inconsistent type never does anything on theequilibrium path.
Example 4.1 (i) shows that a sufficiently naive time-inconsistent type (i.e., q is close toone) procrastinates information acquisition and task choice indefinitely if b0 is small. The
intuition is as follows: if q is large, the time-inconsistent type believes that x is likely to be21

the optimal task; therefore, choosing task x0 becomes unattractive. Also, because choosing
task x now will lead to the completion of the task in the next period with probability q,
the time-inconsistent type might prefer to postpone choosing the task. Therefore, not doing
anything is the only way to commit to postponing the completion of task x — which leads
to procrastination.
Note that if b0 approaches zero in Example 4.1 (i), the degree of naivete q ensures procrastination goes to zero. This means that when the value of knowing that she will betime-inconsistent is sufficiently small, the agent becomes more likely to procrastinate andstick to suboptimal decisions from her initial perspective. Similarly, if c increases while cb isfixed, the time-inconsistent type is more likely to procrastinate.
Example 4.1 (ii) illustrates a case in which the time-consistent type may choose a task
without having acquired information under a positive m, posing additional restrictions on
the degree of naivete q. Intuitively, if the time-consistent type believes that she will bea particular type with sufficiently high probability (q is close to either zero or one), theinformation has low perceived value and she will choose a task without learning her own
type. This consideration imposes an upper bound of q for the existence of our candidate
equilibrium. Interestingly, the range of q in the candidate equilibrium can be smaller in
Example 4.1 (ii) than in Example 4.1 (i); that is, lower learning costs can make timeinconsistent agents more likely to procrastinate on the equilibrium path.

4.2

Analysis with a General Menu

This subsection analyzes a generalized version of the above infinite-horizon model, including
more than two tasks in a menu. We look for an equilibrium in which the time-inconsistent
type never learns about own type nor chooses a task on the equilibrium path.
Let X = {x1 , . . . , xN } be a menu of tasks of the form xn = (−cn , bn ) where cn , bn ≥ 0
for each n. Without loss of generality, we assume that U C (xn ; 0) > 0 for all i.21 Given this,in any continuation games in which the time-consistent type of agent has already chosen
21

Any task xn that gives the time-consistent type negative utility would never be considered or completed
by either type because of the assumption D(1) < δ.

22

xn , she will always choose to complete the task. We also assume that each type of agent
has strict preferences for any task completed in any period: U k (x; t) 6= U k (x0 ; t0 ) for any
x, x0 6= x, k ∈ {C, I}, and t, t0 ≥ 0. Note that this assumption holds generically.
Let xk0 = argmaxxn U k (xn ; 0) denote the most preferred task by type k when she completes
Ckka task. Note that xC
0 = argmaxxn U (xn ; t) for any t ≥ 0. Let x1 = argmaxxn s.t. U I (xn ;0)>0 U (xn ; 1)

denote the most preferred task by type k that is feasible to complete for both types, giventhat the agent will complete a task in the next period. Because D(1) < δ, U I (xn ; 0) > 0
implies U C (xn ; t) > 0 for any t. Denote the maximum utility level of the time-inconsistentI

type by U = max(xn ,t) U I (xn ; t). To focus on interesting cases, we assume the following:
I
I
I
C
Assumption 2 (Feasibility of Task). (i) U C (xC
0 ; 0) > 0 > U (x0 ; 0), (ii) U (x0 , 1) > 0.

Assumption 2 implies that the time-inconsistent type will never complete the timeconsistent type’s preferred task and that the optimal task depends on her type.22 Note
Cthat Assumption 2 implies xC
0 6= x1 .

In what follows, we derive sufficient conditions for the non-learning outcome, in which
the time-inconsistent type does not do anything while the time-consistent type learns own
type in t = 1, chooses xC
0 in t = 2, and completes it in t = 3. Note again that because of
the assumption U C (xn ; 0) > 0 for all xn ∈ X, in any continuation games in which the timeconsistent type of agent has already chosen xn , she will always choose to complete the task.
We focus on the following candidate equilibrium: for continuation games of type I, in which
the agent has first learned own type and then has chosen a task xn , the time-consistent type
always completes it while the time-inconsistent type completes it immediately after choosing
it if U I (xn , 0) > 0; for continuation games of type II, in which the agent has not chosen atask but has learned own type, the time-consistent type always chooses xC
0 while the timeinconsistent type always chooses task xI1 ; and for continuation games of type III, in which
the agent has not done anything yet, the time-consistent type always learns own type while
If Assumption 2 (i) were not satisfied, the time-consistent agent would always be better of picking xC
0
immediately without learning her type. If Assumption 2 (ii) were not satisfied, it might be the best for theagent to not complete any task.
22

23the time-inconsistent type does not do anything.23
In the proof, we show that there is an equilibrium in which each type of agent takes a best
response in continuation games of types I and II. In what follows, we investigate continuation
games of type III.
The time-consistent type of agent prefers to learn own type immediately rather than tochoose a task immediately without learning (or not doing anything) ifCI− m + qU C (xC
0 ; 2) + (1 − q)U (x0 ; 2)

n

o

C
C
C
C
> max qU C (xC
0 ; 1), qU (x1 ; 1) + (1 − q)U (x1 ; 1) ,(6)

where qU C (xC
0 ; 1) is the upper bound of the agent’s expected payoff when she chooses atask that is not feasible for the time-inconsistent type to complete, and qU C (xC
1 ; 1) + (1 −q)U C (xC
1 ; 1) is the upper bound of the payoff when she chooses a task that is feasible forboth types to complete.
Because the time-inconsistent type of agent believes that once she chooses a task she willcomplete it in the next period with probability q, she prefers to do nothing rather than tochoose a task immediately without learning ifI

II
q(−mD(1) + U I (xC
0 ; 3)) > qU (x1 ; 1) + (1 − q)U .

(7)

Similarly, because the time-inconsistent type of agent believes that once she learns her own
type she will choose xC
0 in the next period with probability q, she prefers to do nothing rather
than to acquire information ifI

ICq(−mD(1) + U I (xC
0 ; 3)) > −m + qU (x0 ; 2) + (1 − q)U .

(8)

The result is summarized as follows:
Proposition 3. Suppose T = ∞ and Assumption 2 holds. If Inequalities (6), (7), and(8) also hold, there exists an equilibrium in which the time-consistent type acquires information immediately and completes her preferred task, whereas the time-inconsistent typeprocrastinates indefinitely.
I

23

As we derive only sufficient conditions for the non-learning outcome using the upper bound U , we do
not need to explicitly specify continuation games in which the time-inconsistent type of agent has chosen atask xn but has not learned own type yet.

24For illustration, suppose that m = 0 and δ is close to 1. In this case, for a sufficiently
large q, Inequality (6) holds: the time-consistent type always acquires information. Also, form = 0 and sufficiently large q, Inequalities (7) and (8) holds if:IIICU I (xC
0 ; 3) > max{U (x1 ; 1), U (x0 ; 2)}.

Intuitively, the time-inconsistent type thinks that she will almost surely acquire informationin the next period, choose xC
0 in two periods, and complete it in three periods. If her
perceived utility of completing xC
0 in three periods is higher than completing any task within
two periods due to her time inconsistency, then she prefers to do nothing — leading to
non-completion of any task.

5

Procrastination and Misdirected LearningThis section investigates an extension to our basic model by including learning about anadditional payoff-relevant attribute, which we call ability. Suppose that there are two initiallyuncertain attributes: self-control and ability. Though the agent is initially uncertain aboutboth of them, she can acquire perfectly informative signals about each attribute.
As we now have two attributes, we introduce further notation. Let qd ∈ (0, 1) be theprobability with which the agent believes her discounting is exponential, that is, the probability that she is time-consistent, and let (1 − qd ) be the probability the agent attaches tohaving a discount function D(t). Additionally, the agent now has beliefs about her ability. We assume that there are only two ability types: high ability aH > 0, and low ability
aL < aH . The agent has beliefs about her ability: she believes with probability qa ∈ (0, 1)that she is of high ability and with probability (1 − qa ) that she is of low ability.
To highlight our key mechanism and results in a simple manner, we examine a modelin which the agent has an opportunity to costlessly and perfectly learn about her own
ability in t = 0 and then plays the game described in Section 3 with m = 0. That is, theagent faces a task x with payoff (−c, bi ) where bi = ai b. The agent’s expected ability isâ = qa aH + (1 − qa )aL , and the expected benefit of the task is b̂ = âb. Slightly abusing25

the notation, let Utk (bi ; τ − t) = −D(τ − t)c + D(τ − t + 1)bi denote a type-k agent’s total
utility evaluated in period t when she completes a task in period τ . We abbreviate subscriptt whenever it is clear.
To focus on interesting cases, throughout this section we consider the case in which
both types would acquire information about their own self-control and work on the task
as soon as possible if they do not learn about their own ability. Formally, we assume thatmaxτ U I (b̂; τ ) = U I (b̂; 1) > 0 and U I (b̂; 0) > 0. The latter condition implies U C (b̂; 0) =−c + δ b̂ > 0, and hence both types would acquire information about own self-control in t = 1
and work on the task in t = 2 if they would not learn about their own ability in t = 0. It
also implies that if each type would learn about own ability in t = 0 and realize that own
ability would be bH , then both types would acquire information about their own self-control
and work on the task as soon as possible.
We investigate two cases. First, consider the case in which U C (bL ; 0) > 0 and U I (bL ; 2) >0 > U I (bL ; 0). In this case, a time-inconsistent agent never works on the task once she learns
that her ability is aL . However, from the t = 0 perspective, both types strictly prefer tocomplete the task in t = 2. Also, because by the assumption, maxτ U I (b̂; τ ) = U I (b̂; 1) > 0,
both types would immediately learn about their own self-control and then complete the task
if they do not learn about their own ability. That is, the agent would strictly prefer to avoid
learning about her own ability to motivate her future self to complete the task, as analyzed
by Carrillo and Mariotti (2000) and Bénabou and Tirole (2002).
Second, consider the case in which U C (bL ; 0) < 0, U I (bL ; 2) < 0, and U I (bH ; 2) >U I (bH ; 1). The first condition implies that U I (bL ; 0) < 0. In this case, an agent of either type never works on the task once she learns that her ability is aL . We show that if(1−qd ) k
U k (bL ; 2) < − qa1−q
U (bH ; 2) < 0 holds for each type k ∈ {C, I}, then both time-consistentaand time-inconsistent types strictly prefer to acquire information about their own ability
in t = 0. As shown in Section 3, if U I (bH ; 2) > U I (bH ; 1), a time-inconsistent type may
procrastinate learning about own self-control. That is, learning about own ability can leadto non-completion of the task even when ability is bH .

26

The following proposition summarizes the results:
Proposition 4. Suppose T = ∞, m = 0, maxτ U I (b̂; τ ) = U I (b̂; 1) > 0, and U I (b̂; 0) > 0.
(i) Assume U C (bL ; 0) > 0 and U I (bL ; 2) > 0 > U I (bL ; 0). There exists an equilibrium inwhich a time-inconsistent type of agent strictly prefers to not learn about own ability andboth types of agent complete the task in t = 2.
(1−qd ) k(ii) Assume U C (bL ; 0) < 0, U I (bH ; 2) > U I (bH ; 1), and U k (bL ; 2) < − qa1−q
U (bH ; 2)afor type k ∈ {C, I}. There exists an equilibrium in which a time-inconsistent type of agent
strictly prefers to learn about own ability but never learns about own future self-control orcompletes the task.
On the one hand, the result in Proposition 4 (i) is in line with the literature on strategic
ignorance (Carrillo and Mariotti 2000, Bénabou and Tirole 2002). In this literature, an agent
chooses not to learn about her own ability as a means of internal commitment to improve
her own future payoffs. In our model, the agent chooses not to learn about own ability tocommit to completing the task in t = 2.
On the other hand, Proposition 4 (ii) highlights that a time-inconsistent type may notlearn about own future self-control. Non-learning about own self-control in this case isharmful from the period-0 perspective, because it leads to a non-completion of a task. Thisis different from the literature on strategic ignorance, as it focuses on learning about ability
when the agent perfectly knows her own future self-control problems.
Interestingly, in contrast to Carrillo and Mariotti (2000) and Bénabou and Tirole (2002),
the time-inconsistent type of agent endogenously chooses to learn about own ability, which isharmful from the initial perspective. Without learning about her own ability, an agent of thistype will complete the task in t = 2, and her expected utility from her initial perspective
will be higher than in the case when she procrastinates and hence never completes thetask. Because an agent of the time-inconsistent type is initially overconfident about her own
future self-control, however, she takes the same action as the time-consistent type does, whichleads to procrastination in the end. In particular, the result and its welfare consequences inBénabou and Tirole (2002) — that sophisticated time-inconsistent agents optimally choose27not to learn about their own ability — can be reversed when we investigate partially naive
agents.
Most importantly, Proposition 4 (ii) highlights how the agent’s endogenous learning decisions can be misdirected under naivete about her own future self-control — she chooses
to learn what she should not learn from her initial perspective (i.e., about her ability) andchooses not to learn what she should (her self-control). Complementing recent papers suchas Heidhues et al. (2018) and Hestermann and Le Yaouanq (2020), our mechanism based on
procrastination explains why an agent’s learning decisions can be self-defeating even when
she can voluntarily choose to learn about all uncertain attributes.

6

Further Extensions and Discussion

Leisure goods. So far, we have only considered situations in which the agent faces investment goods: tasks or goods with an immediate cost and a delayed benefit (DellaVigna andMalmendier 2004). A natural extension is to analyze the agent’s learning incentives if they
face leisure goods: a task or good with an immediate benefit and a delayed cost. To describe
the implications of such a case in a simple manner, consider an infinite horizon model in
Section 3 with m = 0 and a modification under which the agent now receives a benefit b > 0
immediately and a delayed cost c > 0 in the next period upon the completion of a task.
In this case, if U I (x; 1) < 0 < U I (x; 0), an agent of the time-inconsistent type may
not learn about her own future self-control to prevent herself from completing the task.
Intuitively, she chooses not to learn about her own future self-control as a means of internal
commitment, to avoid temptation. This result is akin to Proposition 4 (i) as well as theliterature on strategic ignorance (Carrillo and Mariotti 2000, Bénabou and Tirole 2002),
while the agent here would not learn about her own future self-control rather than her own
ability.
Commitment to future actions. We discuss the predictions of our model when we allow
for commitment technologies. First, in the infinite horizon model in Section 3, if the agent
28

were given the opportunity to commit to acquiring information in any future period, thenboth types of agent would commit to doing so. Intuitively, as a time-inconsistent type would
prefer to complete a task in some future period rather than never by Assumption 1, she would
commit to acquiring the information in the future if such a commitment were possible. By
a similar logic, if the agent were given the opportunity to commit to acquiring informationin any future period, the non-learning results in Section 4 would not occur. Therefore, theassumption of non-commitment is crucial to derive our non-learning result.
Negative learning cost. While we have highlighted how an agent may not learn about
her own future self-control even when the learning cost is zero (i.e., m = 0), this result implies
that non-learning can occur for some small m < 0. In other words, time-inconsistent people
may be willing to pay a positive cost to stay ignorant. Notably, as our underlying mechanism
is procrastination regarding learning opportunities, this willingness to pay to stay ignorantcan occur even in the absence of any type of self-esteem concern (Kőszegi 2006; Gottlieb
2014, 2016).
Partial naivete and sophistication. In the literature, partial naivete is often modeled
as a point belief about own future self-control, specifically as β̂ ∈ (β, 1) for quasi-hyperbolic
discounting. To allow for the possibility of learning, we assume that an agent’s belief about
her own future self-control is a bi-modal distribution where the true value is in its support.
First, we discuss how we can extend our results to general non-degenerate distributions
for own future self-control. They can be extended straightforwardly, as in Section 3, because
the agent faces only one task and the only reason an agent cares about her own future selfcontrol is to predict how likely it is the task will be completed and when. Specifically, for agiven general non-degenerate distribution on own future self-control, an agent of either type
can compute the probability of completing the task in t periods for each t ≥ 1 when she
does not learn now. Given these probabilities, each type of agent maximizes own perceived
utility. By contrast, the analysis in Section 4 under general non-degenerate distributions on
own future self-control can be much more complicated, as different types can choose different
29tasks at different points in time. The analysis in such cases is left for future research.
Second, there are other ways of modeling partial naivete even within a bi-modal distribution. For example, instead of an agent’s type being constant over time, agents canhave stochastic preferences in each period. Specifically, the true probability of being timeconsistent in each period is qL < 1 for the time-inconsistent type and qH = 1 for thetime-consistent type. The realization is i.i.d. across time, but the time-inconsistent typeis uncertain about the probability of being time-consistent in a given period. An agent of
this type believes with probability q ∈ (0, 1) that she will be time-consistent with probability 1, and with probability 1 − q that she will be time-consistent with probability qL . Ina given period, she believes she will be time-consistent in future periods with probabilityq̂ ≡ q · qH + (1 − q)qL = qL + (1 − qL )q > qL . Defining partial naivete in this way is akinto the approach in models of cue-based consumption, where agents are facing stochastic
temptation and are overestimating how well they will withstand it. In the limit case whereq = 0, the two types are identical to the ones in our model, in both actions and beliefs. If
qL > 0, that is, both types of agent are time-consistent in any given period with a positive
probability, the beliefs are distinct from those in our setting. Our main results would not be
qualitatively changed by this alternative specification, however.
Last but not least, it could make more sense in many situations to assume that a timeconsistent type is certain about own future self-control from the beginning of the game,whereas a time-inconsistent type is initially (stochastically) overconfident about own futureself-control. Given the assumption that a time-inconsistent type cannot infer own type
without acquiring information, our results would qualitatively be the same under this specification, because the main force of our results is that an agent of the time-inconsistent typewants to procrastinate on her learning opportunity, to delay completing the task.

7

Concluding Remarks

This paper provides a new mechanism explaining why people do not learn about their selfcontrol problems over time. We find that individuals may procrastinate on a free learning30

opportunity indefinitely, even though learning about her own self-control problems would
make the agent better off. Notably, our results do not rely on agents having image concerns
or on the use of overconfidence as commitment. In our model, naivete about own self-control
problem is often harmful.
One potential future direction of research is applying our mechanism to situations thatinvolve strategic interactions between players. For example, when firms offer contracts which
specify a base contract and a set of options, they often have an incentive to make consumers
procrastinate on cancelling or switching the option when a fraction of consumers are naive(DellaVigna and Malmendier 2004, Heidhues and Kőszegi 2010, Murooka and Schwarz 2018).
Our results imply that firms may have an incentive to let consumers endogenously procrastinate on learning about their own naivete by offering such “exploitative” contracts. How
procrastination on learning about own self-control problems may interact with strategic concerns of other parties is left for future research.

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33

A

Appendix: Proofs

A.1

Proof of Proposition 1.

(i) We derive conditions for an equilibrium in which the agent always acquires information
about her own type when she is time-consistent, never when she is time-inconsistent, andagents of both types will complete the task in the next period once they know their own
type.
First, we investigate continuation games, in which the agent has already learned her type.
If the agent has learned that she is time-consistent, she always chooses to complete the task,
because of Assumption 1 (i), which implies U C (x; 0) = −c + δb > 0. Suppose, in contrast,that the agent has learned that she is time-inconsistent. Because D(t) is decreasing in t,−cD(t) + bD(t + 1) approaches zero as t goes to +∞, Assumption 1 (i) ensures that there
exists a τ ∈ N such that U I (x; 0) = −c + D(1)b ≥ −cD(τ ) + D(τ + 1)b = U I (x; τ ). Let
denote τ = min{τ |U I (x; 0) ≥ U I (x; τ )}. Consider an equilibrium candidate in continuation
games in which, if the agent has learned in period τ that her type is time-inconsistent, shewill complete the task in periods t = τ + 1 + nτ , where n ∈ N0 . Then, akin to O’Donoghue
and Rabin (1999a, 2001), by the definition of τ , the agent strictly prefers not to complete
the task in t 6= τ + 1 + nτ and weakly prefers to complete it in t = τ + 1 + nτ . Hence,
there exists an equilibrium in continuation games in which the time-inconsistent type always
completes the task in the next period after learning.
Second, we investigate continuation games in which the agent is of the time-consistent
type and in which she has not acquired the signal. In this case, the expected payoff of her
acquiring information immediately in our candidate equilibrium is:
Û C (s0 = 1) = −m + U C (x; 1).
If she does not do so, in our candidate equilibrium she believes that she will acquire information in the next period if she is a time-consistent type and that she will never do so if she
is a time-inconsistent type. Hence, her expected payoff is:
Û C (s0 = 0) = q(−mδ + U C (x; 2)) + (1 − q) · 0 = qδ(−m + U C (x; 1)).
34

Under Assumption 1 (ii), such an agent always prefers to acquire information immediately
in our candidate equilibrium.
Continuation games in which the agent is a time-inconsistent type and in which she has
not acquired the signal are characterized in the main text.
(ii) Note that in any continuation game in which the agent is a time-consistent typeand in which she has acquired the signal, she always chooses to complete the task, under
Assumption 1 (i).
We here prove that in continuation games in which an agent of a time-consistent type has
not learned her type, she always chooses to learn. Given the above behavior, note that the
expected payoff of her acquiring information immediately is at least −m + qU C (x; 1) + (1 −q)U I (x; τ ) > −m + qU C (x; 1), whereas the expected payoff of her not acquiring information
is at most −δm + U C (x; 2) = −mδ + δU C (x; 1). Hence, in any such continuation games, thetime-consistent type always chooses to learn if (q − δ)U C (x; 1) + m(1 − δ) ≥ 0.
Continuation games in which the agent is a time-inconsistent type and in which she has
not acquired the signal are characterized in the main text.

A.2

Proof of Proposition 2.

Provided in the main text.

A.3

Proof of Proposition 3.

We derive conditions for the equilibrium outcome described in the main text. Specifically, weinvestigate the following candidate equilibrium: for continuation games of type I, in which
the agent has first learned her own type and then has chosen a task xn , the time-consistent
type always completes the task while the time-inconsistent type completes it in the next
period after choosing the task if U I (xn , 0) > 0 and never completes it if U I (xn , 0) ≤ 0; forcontinuation games of type II, in which the agent has not chosen a task but has learned her
own type, the time-consistent type always chooses xC
0 while the time-inconsistent type always

35

chooses task xI1 ; and for continuation games of type III, in which the agent has not done
anything yet, the time-consistent type always learns own type while the time-inconsistent
type does not do anything.
It is straightforward to check that the time-consistent type takes the best response in eachcontinuation game of types I and II. Suppose a continuation game of type I with the timeinconsistent type of agent. When U I (xn , 0) ≤ 0, there is an equilibrium in the continuation
game in which the agent never completes the task. When U I (xn , 0) > 0, by the same
logic as O’Donoghue and Rabin (1999a, 2001) and the above analysis, she will complete thetask in every τ n periods where τ n = min{τ |U I (xn ; 0) ≥ U I (xn ; τ )}. Hence, there exists anequilibrium in continuation games in which the time-inconsistent type always completes thetask in the next period after choosing a task if she has already learned her own type. Given
this, if the time-inconsistent type has learned her own type but has not chosen a task, herbest response is to choose xI1 .
In continuation games in which the time-consistent type of agent has chosen task xn but
has not learned her own type, by U C (xn , 0) > 0 for all xn ∈ X and she can complete thetask without learning, she always chooses to complete it. In continuation games in whichan agent of the time-inconsistent type has chosen task xn but has not learned her own type,although equilibrium behavior depends on parameters, by construction, her payoff is at mostI

U in any equilibrium and in any period.
The conditions for continuation games of type III are characterized in the main text.

A.4

Proof of Proposition 4.

(i) As in the main text, the assumption U I (bL ; 0) < 0 ensures that an agent of a timeinconsistent type never works on the task once she learns that her ability is aL . However,by the assumption U I (bL ; 2) > 0, from the t = 0 perspective, agents of both types prefer
to complete the task in two periods rather than never. Also, the assumption U I (b̂; 0) =−c + D(1)b̂ > 0 implies that −c + δ b̂ > 0 and maxτ U C (b̂; τ ) = U C (b̂; 0). Hence, if the agent
does not learn about her own ability in t = 0, then agents of both types would immediately

36

learn about their own self-control in t = 1 and complete the task in t = 2. Therefore, suchan equilibrium exists.
(ii) We look for an equilibrium in which an agent of a time-consistent type acquires
information about own ability and then works on the task if and only if she learns that her
ability is aH while an agent of a time-inconsistent type acquires information about her own
ability and never works on the task on the equilibrium path, and both types will work on
the task as soon as possible if they do not acquire information about their own ability in
t = 0.
Note first that agents of both types never work on the task once they learn that theirability is aL . We here show that an agent of a time-consistent type chooses to learn about
her own ability in t = 0 and then completes the task if and only if her ability is aH . Given
the above strategies, her anticipated expected utility in t = 0 when she chooses to learn
about her own ability is:hiqa qd · U C (bH ; 2) + (1 − qd ) · 0 + (1 − qa ) · 0 = qa qd U C (bH ; 2) > 0.
Because of the assumptions, agents of both types complete the task in t = 2 if they choose
not to learn about their own ability in t = 0. Their anticipated expected utility in t = 0
when they choose not to learn about their own ability is:qa · U C (bH ; 2) + (1 − qa ) · U C (bL ; 2).
Hence, an agent of the time-consistent type strictly prefers to learn about her own ability in(1−qd ) C
t = 0 if U C (bL ; 2) < − qa1−q
U (bH ; 2).
a

We now show that the time-inconsistent type strictly prefers to acquire information about
own ability in t = 0. In t = 0, given the above strategies, an agent of this type’s anticipatedexpected utility when she chooses to learn about her own ability is:hiqa qd · U I (bH ; 2) + (1 − qd ) · 0 + (1 − qa ) · 0 = qa qd · U I (bH ; 2).
Because of the assumption maxτ U I (b̂; τ ) = U I (b̂; 1) > 0, her anticipated expected utility in
37t = 0 when she chooses not to learn about her own ability is:qa · U I (bH ; 2) + (1 − qa ) · U I (bL ; 2).
Hence, the time-inconsistent type strictly prefers to learn about her own ability in t = 0 if
(1−qd ) I
U (bH ; 2).
U I (bL ; 2) < − qa1−qaFrom t = 1 on, notice that if Inequality (1) with b = bH and m = 0 holds, or equivalently
if U I (bH ; 2) > U I (bH ; 1) > 0, then the equilibrium behavior for each type in t ≥ 1 is exactly
the same as in Section 3.

B

Task Completion: Analysis in a Finite Horizon Model

This section analyzes when procrastination occurs in a task completion setting under a finite
time horizon (i.e., T < ∞). It corresponds to environments in which there is a deadline fortask completion (e.g., filing a tax return).
Decisions after learning. By Assumption 1, the task is worthwhile for both typesof agents.

Because of the properties of finite horizon decision problems, if the agents

have learned their type, they will complete the task at latest in the last period.

Letτ = min{τ |U I (x; 0) ≥ U I (x; τ )} denote the earliest period in which the agent prefers tocomplete the task immediately rather than in τ periods; by Assumption 1, both types of
agents then complete the task in t = T if they have not yet done so. When the task will becompleted is summarized in the following lemma:
Lemma B.1. Suppose T < ∞ and Assumption 1 holds. Consider a continuation game inperiod τ ≥ 1 in which the agent has already learned her type. Then,(i) If the agent is time-consistent, she completes the task immediately.
(ii) If the agent is time-inconsistent, she completes the task in the first period t ≥ τ thatsatisfies t = T − nτ , where n ∈ N0 .

38

Proof. For (i), note that U C (x; k) = δ k U C (x; 0) for the time-consistent type. Hence, in any
period t, the agent will always complete the task immediately if she is time-consistent andthe task has not yet been completed.
We now prove (ii). In t = T , the payoff from completing the task for a time-inconsistentagent is U I (x; 0). If she does not complete the task in t = T , she receives zero payoffs. By
Assumption 1, the agent prefers to complete the task.
In t = T − 1, the agent knows that the task will be completed in t = T if it is notcompleted in the current period. Hence, she completes the task if U I (x; 0) ≥ U I (x; 1). By
induction applied in each t ≤ T − 2, the agent will always complete the task immediately
when U I (x; 0) ≥ U I (x; 1).
If U I (x; 0) < U I (x; 1), the sophisticated time-inconsistent type would not complete thetask in t = T − 1, instead postponing task completion until t = T . If this is the case, thenin t = T − 2 the agent knows that not completing the task now means that the task will
be completed two periods from now; hence, the agent will complete the task in t = T − 2
if and only if U I (x; 0) ≤ U I (x; 2). If this inequality does not hold, then the agent keeps
postponing task completion until period t = T − τ that satisfies U I (x; 0) < U I (x; τ ). If there
does not exist such a τ ≤ T − 1, then no matter when she has learned about own self-control,
she postpones completing the task until t = T . If such a τ ≤ T − 1 does exist, then she
postpones at most τ periods.24 Because τ = min{τ |U I (x; 0) ≥ U I (x; τ )}, the agent knowsthat she will complete the task in t = T and in t = T − τ if τ < T .
In t = T − τ − 1, the agent knows that if she postpones the task, it will be completed
in the next period, and she will therefore postpone to complete the task now if and only if
τ > 1. As described above, she keeps postponing for τ periods. By induction, the agent
will complete the task in period t ≥ 1 if and only if t ∈ {t ∈ N|t = T − nτ , n ∈ N0 }. This
completes Lemma B.1.
Lemma B.1 (ii) implies that a time-inconsistent type completes the task in every τ periods. The intuition underlying it is as follows. Because the agent is perfectly sophisticated24

From Assumption 1, we know that if T is large enough, there exists τ < T such that the agent completes
the task immediately rather than in τ periods.

39

about her future self-control after learning, she knows she will complete the task in the lastperiod and therefore not complete it in the second-last period if U I (x; 0) < U I (x; 1). In thethird-last period, she will complete the task if she prefers completing it immediately overcompleting it in two periods (i.e., U I (x; 0) ≥ U I (x; 2)); otherwise, she will not. By induction,
the agent completes the task in any period t = T − nτ , and there is a cycle of length τ .
Note that this result does not constitute procrastination, as the agent has a correct belief
about when the task will be completed, and her actual decisions coincide with the beliefs.
This result is akin to that of O’Donoghue and Rabin (1999a, 2001) as well as subsequent
literature on the behavior of sophisticated time-inconsistent agents .
Learning decisions. Next, we investigate conditions under which each type of agent acquires the signal. We focus on deriving the conditions for our equilibrium of interest: one inwhich the time-inconsistent type procrastinates on information acquisition while the timeconsistent type always acquires information.
We first analyze the actions of the time-consistent type. Note that in this case she isunderconfident about her own future self-control: she thinks she will be time-inconsistent
with probability 1 − q. Nevertheless, if U I (x; 1) ≤ U I (x; 0), she will complete the task in thenext period with probability one. Otherwise, she takes into account the delay in case her
future self will be time-inconsistent, and she will complete the task only in period t = T −nτ .
In this case, the most stringent condition on acquiring information is that the agent expects adelay of τ periods if her future self is time-inconsistent. To be precise, if U I (x; 1) > U I (x; 0),
the time-consistent type acquires information if the following inequality holds:q(1 − δ)U C (x; 1) ≥ (1 − δq − (1 − q)δ τ −1 )m.

(9)

The result is summarized in the following lemma:
Lemma B.2. Suppose T < ∞ and Assumption 1 holds. If an agent is time-consistent, shewill acquire information in any continuation game if either of the following holds:(i) U I (x; 1) ≤ U I (x; 0),40

(ii) U I (x; 1) > U I (x; 0) and Inequality (9) holds.
Proof. (i) If U I (x; 1) ≤ U I (x; 0), both time-inconsistent and time-consistent types will complete the task immediately upon learning. Therefore, the payoff from acquiring information
for the time-consistent agent is ÛtC (st = 1) = −m + U C (x; 1) > 0, where the last inequality
is from Assumption 1.
We show that, if the time-consistent agent postpones information acquisition, she will geta lower payoff no matter what her beliefs are about her future information acquisition. If she
believes a time-consistent type will acquire information in k periods and a time-inconsistent
type in h periods, her payoff will be:
ÛtC (st = 0) = q(−δ k m + U C (x; k + 1)) + (1 − q)(−δ h m + U C (x; h + 1))= (qδ k + (1 − q)δ h )(−m + U C (x; 1)) < −m + U C (x; 1).
Therefore, if U I (x; 1) ≤ U I (x; 0), then the time-consistent agent always completes the task
immediately.
(ii) If U I (x; 1) > U I (x; 0), the time-inconsistent type may postpone completing the task.
From Lemma B.1, the time-inconsistent type will postpone the task for cycles of τ periods.
In t = T −1, the agent knows the task will be completed in the next period. In t = T −2, thetime-inconsistent type believes that the task will be completed in t = T − 1 with probability
q. Therefore, she will acquire information if:
ÛTC−2 (sT −2 = 1) = −m + qU C (x; 1) + (1 − q)U C (x; 2) ≥ −δm + U C (x; 2) = ÛTC−2 (sT −2 = 0)m
⇔ U C (x; 1) ≥ .
q
By the same logic, in t = T − τ where τ ≤ τ , the agent knows that the time-inconsistent
type will postpone until t = T to complete the task and until t = T −1 to acquire information.
Hence, the time-consistent type prefers acquiring information immediately if− m + qU C (x; 1) + (1 − q)U C (x; τ ) ≥ q(−δm + U C (x; 2)) + (1 − q)(−δ τ −1 m + U C (x; τ ))⇔ U C (x; 1) ≥[1 − qδ − (1 − q)δ τ −1 ]m.
(1 − δ)q(10)
41

The right-hand side of Inequality (10) is increasing in τ . Because τ is the longest period inwhich the time-inconsistent agent will delay completing the task upon learning her type, thetime-consistent agent will always complete the task immediately if:[1 − qδ − (1 − q)δ τ −1 ]m.
U (x; 1) ≥(1 − δ)qC

This completes Lemma B.2.
Condition (i) in Lemma B.2 ensures that both types will complete the task immediately
upon learning their type, and hence that the time-consistent agent will acquire information
immediately to maximize her expected payoff. Condition (ii) in Lemma B.2 covers the casewhere the time-inconsistent agent postpones task completion in some periods. In this case,the time-consistent agent is facing a trade-off: acquiring information immediately leads toincurring cost m with certainty, but to completing the task in the next period only withprobability q. If Condition (ii) in Lemma B.2 is satisfied, however, the time-consistent agent
will not postpone acquiring information in any period. Note that Lemma B.2 holds when mis close to zero and/or q is close to one.
We next analyze the actions of the time-inconsistent type. First, we illustrate conditions
in which the time-inconsistent type delays acquiring information in t = T − 2. Because
the last period for an agent of this type to acquire information in order to complete thetask is in t = T − 1, Assumption 1 (ii) and Lemma B.1 ensure that the agent will acquire
information in t = T − 1. Given Lemma B.2, acquiring information in t = T − 2 will leadthe time-consistent type to complete the task in t = T − 1, but the time-inconsistent typewill delay the task until t = T if U I (x; 1) > U I (x; 0). Hence, for the time-inconsistent agent
in t = T − 2, the subjective expected payoff from acquiring information immediately is
ÛTI −2 (sT −2 = 1) = −m + qU I (x; 1) + (1 − q)U I (x; 2).
If the time-inconsistent agent does not acquire information, she (correctly) anticipatesthat she will acquire it in the next period and complete the task with certainty in two periods.
Hence, the subjective expected payoff from not acquiring information in t = T − 2 isÛTI −2 (sT −2 = 0) = −mD(1) + U I (x; 2).
42

Therefore, the time-inconsistent agent will prefer not to acquire information in t = T − 2 if
− mD(1) + U I (x; 2) > −m + qU I (x; 1) + (1 − q)U I (x; 2)⇔ q(U I (x; 2) − U I (x; 1)) + m(1 − D(1)) > 0.

(11)

Note that she will always prefer to delay if U I (x; 2) > U I (x; 1). Intuitively, a time-inconsistent
type indirectly incurs a cost from learning in this case, because learning itself changes the
behavior or future selves.
We now show the condition under which the time-inconsistent type procrastinates onacquiring information in t = T − 3. In period t = T − 3, she wrongly believes that she
will acquire information in t = T − 2 with positive probability. Suppose that U I (x; 2) >U I (x; 1) > U I (x; 0) and that Inequality (9) holds. If so, upon acquiring information int = T − 3, the agent believes that she will complete the task in the next period if and only
if her future type will be time-consistent. Hence,ÛTI −3 (sT −3 = 1) = −m + qU I (x; 1) + (1 − q)U I (x; 3).
If she does not acquire information in t = T − 3, she (erroneously) expects that she will
acquire information in the next period with probability q. Hence,ÛTI −3 (sT −3 = 0) = q(−mD(1) + U I (x; 2)) + (1 − q)(−mD(2) + U I (x; 3)).
Note that the beliefs for this agent’s future learning decisions are wrong: in reality, if she
does not learn her own type in t = T − 3, she will not do so in t = T − 2, because her typeis persistent over time. However, she is naive about this fact precisely because she has notacquired information about her own type yet, and hence, keeps using her initial prior belief.
Hence, the agent will not acquire information in t = T − 3, while believing that she will
acquire information in t = T − 2 with probability q if
ÛTI −3 (sT −3 = 0) > ÛTI −3 (sT −3 = 1)
⇔ q(U I (x; 2) − U I (x; 1)) + m(1 − qD(1)) > m(1 − q)D(2).

43

(12)

Note that the logic of procrastination here — a time-inconsistent type (in our model, probabilistically) overestimates own future self-control and hence takes no action now — is akinto that one in O’Donoghue and Rabin (1999a, 2001).
The logic and derivations are the same in periods t < T −3, although there are additional
conditions depending on τ . The result for procrastination of learning decisions for the timeinconsistent type of agent until t = T − 1 is summarized as follows:
Proposition B.1. Suppose T < ∞ and Assumption 1 holds. If U I (x; 2) > U I (x; 1) >U I (x; 0), Inequality (9), andq(U I (x; 2) − U I (x; 1)) + m(1 − qD(1)) > max (1 − q)[mD(nτ ) − U I (x; nτ + k) + U I (x; k)]n,kwhere n ∈ N0 , k ∈ N, k < τ , and nτ + k ≤ T also hold, then there exists a uniqueequilibrium outcome in which a time-inconsistent type will procrastinate learning about
own future self-control until t = T − 1.
Proof. If U I (x; 1) > U I (x; 0), the time-inconsistent type of agent will complete the task inperiods t = T − nτ where n ∈ N0 . Under Assumption 1, the time-inconsistent type will
acquire information in t = T − 1 and complete the task in t = T . In t = T − 2, if an agent of
the time-inconsistent type acquires information, she believes that the task will be completedwith probability q in t = T − 1 and with 1 − q in t = T . If she does not acquire informationin t = T − 2, she will acquire information with certainty in t = T − 1 and complete the taskwith certainty in t = T . Therefore, the time-inconsistent agent will postpone informationacquisition if:− m + qU I (x; 1) + (1 − q)U I (x; 2) < −mD(1) + U I (x; 2)⇔ q(U I (x; 2) − U I (x; 1)) + m(1 − D(1)) > 0.

(13)

Given this inequality and τ > 2, then in t = T − 3, acquiring information immediately means
the task will be completed in t = T − 2 with probability q and in t = T with probability
1 − q. If the time-inconsistent agent postpones now, she believes that with probability q theinformation will be acquired in t = T − 2 and the task will be completed in t = T − 1, but
44

with probability 1 − q the information will be acquired in period t = T − 1 and the task
will be completed in t = T . The time-inconsistent type of agent will therefore postpone
information acquisition if:− m + qU I (x; 1) + (1 − q)U I (x; 3) < q(−mD(1) + U I (x; 2)) + (1 − q)(−mD(2) + U I (x; 3))⇔ q(U I (x; 2) − U I (x; 1)) + m(1 − qD(1)) > +m(1 − q)D(3).
In general, when t = T − τ where 2 ≤ τ ≤ τ , the time-inconsistent type of agent will
postpone acquiring information ifq(U I (x; 2) − U I (x; 1)) + m(1 − qD(1)) > m(1 − q)D(τ − 1).
The right-hand side of this inequality is decreasing in τ , so the inequality becomes themost stringent when τ = 2. Hence, if q(U I (x; 2) − U I (x; 1)) + m(1 − qD(1)) > m(1 − q)D(1)or equivalently if q(U I (x; 2) − U I (x; 1)) + m(1 − D(1)) > 0, the time-inconsistent agent
will postpone information acquisition for at least τ periods. A sufficient condition for thetime-inconsistent agent to procrastinate for τ periods is U I (x; 2) > U I (x; 1).
In t = T − τ − 1, the time-inconsistent agent knows that if she acquires information, thetask will be completed with certainty in the next period, and hence her expected payoff isÛT −τ −1 (sT −τ −1 = 1) = −m + U I (x; 1).
If she chooses not to acquire information, she knows that the time-consistent type will acquire
information in the next period, but the time-inconsistent type will only acquire informationin t = T − 1. Hence, her expected payoff isÛT −τ −1 (sT −τ −1 = 0) = q(−mD(1) + U I (x; 2)) + (1 − q)(−mD(τ ) + U I (x; τ + 1)).
Thus, she will prefer not acquiring the information in t = T − τ − 1 ifq(U I (x; 2) − U I (x; 1)) + m(1 − qD(1)) > (1 − q)(mD(τ ) − U I (x; τ + 1) + U I (x; 1)).
When q approaches 1, this condition reduces to Inequality (13). Whether this condition is
generally stronger or weaker than Inequality (13) depends on parameters . If this condition45

holds, however, the agent will procrastinate for an additional τ periods. This follows fromthe same argument as above.
By induction, for each t = T − nτ such that t ≥ 1 and n ∈ N0 , we need to check whether
the agent will prefer to acquire information or not. The agent is always comparing theoutcomes of acquiring information now and completing the task with certainty in the next
period, on the one hand, and the case where the time-inconsistent type procrastinates until
the second-last period and the time-consistent type acquires information in the next periodon the other. Therefore, if
ÛT −nτ −k (sT −nτ −k = 1) = −m + qU I (x; 1) + (1 − q)U I (x; k)< ÛT −nτ −k (sT −nτ −k = 0) = q(−mD(1) + U I (x; 2)) + (1 − q)(−mD(nτ ) + U I (x; nτ + k)), which is rewritten asq(U I (x; 2) − U I (x; 1)) + m(1 − qD(1)) > (1 − q)(mD(nτ ) − U I (x; nτ + k) + U I (x; k)), for any n ∈ N0 and k ∈ N where k < τ and nτ + k ≤ T , then the time-inconsistentagent will procrastinate information acquisition for any t < T − 1. Note also that, given thiscondition, we have derived the time-inconsistent type’s learning decision in all continuation
games and hence have uniquely pinned down the equilibrium outcome.
To see the underlying intuition here, suppose that a time-consistent agent always chooses
to acquire information and completes the task in every period. Intuitively, because an agentof the time-inconsistent type is (probabilistically) overconfident about her own future selfcontrol, she erroneously thinks that she will acquire information with probability q in thenext period. Note that the time-inconsistent agent evaluates her anticipated future outcomesin terms of her current (i.e., time-inconsistent) preferences. Hence, akin to O’Donoghue andRabin (1999a, 2001), the time-inconsistent agent may procrastinate information acquisition— in any period t < T − 1, she thinks that she will acquire information with probability q
in the next period, but in reality she will not do so, with probability one.
Perhaps surprisingly, and beyond the original logic of O’Donoghue and Rabin (1999a,
2001), procrastination in our model can occur even when m = 0. This is because the time46

inconsistent type indirectly incurs a cost from learning if U I (x; 2) > U I (x; 1), which neveroccurs under quasi-hyperbolic discounting. Corollary B.1 highlights the result:Corollary B.1. Suppose T < ∞ and U I (x; 2) > U I (x; 1) > U I (x; 0). Then, for anym ≥ 0, there exists a q̄ < 1 such that for any q ∈ [q̄, 1) there is a unique equilibriumoutcome in which the time-consistent type will acquire information immediately while thetime-inconsistent type will procrastinate on acquiring information until t = T − 1.

47