J Risk Uncertain (2009) 38:117–141 DOI 10.1007/s11166-009-9061-5
Pre-commitment and flexibility in a time decision experiment Marco Casari
Published online: 11 February 2009 # Springer Science + Business Media, LLC 2009
Abstract An agent with dynamically inconsistent preferences may deviate from her plan of action as the future draws near. An exponential discounter may do exactly the same when facing an uncertain future. Through an experiment we compare preference-based vs. uncertainty-based explanations for choice reversal over time by eliciting choices for pre-commitment and flexibility. Evidence of widespread commitment favors a preference-based explanation. Keywords Experiments . Intertemporal choices . Dynamic consistency . Commitment . Flexibility . Uncertainty JEL Classification C91 . D90 . D81
This study was initiated when visiting the Autònoma University of Barcelona, Spain. Discussions with Howard Rachlin have been particularly valuable. Earlier versions of this manuscript have also benefited from the comments of Alberto Bisin, Jordi Brandts, Gabriele Camera, Colin Camerer, Gary Charness, Pierre Courtois, Guillaume Frechette, David Laibson, Dan Levin, Rosella Nicolini, Charlie Plott, Arno Riedl, Kip Viscusi, and of participants at seminars at Kyoto Sangyo University, University of Maastricht, University of Venezia, New York University, Capua Workshop in Behavioral Economics, the Autònoma University of Barcelona, Purdue University, and at the ESA meeting in Montreal. All remaining errors are mine. Henrik Nordin and Ricardo Flores provided valuable research assistance. Many thanks to Aurora Gallego and Nikos Georgantis for letting me use the LEE (Laboratori d’Economia Experimental) of the Universidad Jaume I of Castellon, Spain, and to Juan Gómez Pérez for the technical help. The financial support from an EU Marie Curie Fellowship and the Russell Sage Foundation (grant #: 98-04-05) is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations in this material are those of the author and do not necessarily reflect the views of the European Commission or of the Russel Sage Foundation. M. Casari (*) Department of Economics, University of Bologna, Piazza Scaravilli 2, 40126 Bologna, Italy e-mail:
[email protected]
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In a medical study, pregnant women were asked 1 month before delivery whether they preferred to have anesthesia during labor. The anesthesia would offer immediate pain relief but would also expose the newborn to a small risk of long-term side effects. Most women preferred to avoid anesthesia. Despite their earlier statements, during active labor many did choose anesthesia over pain (Christensen-Szalanski 1984). Had they had a choice 1 month prior to delivery, would they have signed a statement committing to withhold anesthesia? Or, would they have been willing to pay to have the opportunity during labor to change their previous choice? This paper provides experimental evidence regarding similar types of decisions, but in situations involving money. Intertemporal decisions have recently received considerable attention from behavioral economists (for a review, see Frederick et al. 2002). We provide and discuss new empirical evidence regarding preferences for pre-commitment and flexibility, which is important to discriminate between competing models of time discounting. Typically, empirical studies focus on subjects’ discount rate for short and long time horizons: (a) Would you prefer $100 in 2 days or $110 in 2 months? (b) Would you prefer $100 in 1 year and 2 days or $110 in 1 year and 2 months? Most participants in experiments prefer $100 in 2 days in (a), and then reverse their choice by preferring $110 in 2 months in (b). In line with this finding, in our study about 65% of participants reversed their choices over time. Under the canonical exponential discounting model, such choices are interpreted as time inconsistent behavior (Thaler 1981; Benzion et al. 1989; Loewenstein 1987; Chapman 1996).1 Scholars have attributed this choice reversal to discount rates declining for longer time horizons and put forward alternative discounting functions (Kirby 1997; Ainslie 1992; Ainslie and Haendel 1983). The present study does not test for the best functional form of discounting but investigates what type of motive causes choice reversal.2 To this end, we elicit preferences for pre-commitment: (c) Would you prefer $109 in 1 year and 2 months, or being able to choose 1 year from today, between $100 in 2 days or $110 in 2 months? If a subject that previously reversed her choice now prefers $109 in 1 year and 2 months, she reveals a strict preference for commitment, i.e. to restrict her future choice set.3 Understanding whether decision makers are willing to pay to restrict their choice set is relevant for several reasons. First, if time-inconsistent agents will improve their welfare through commitment, then the practical impact of time inconsistency may be
1
Harrison et al. (2002) and Rubinstein (2003) and others have questioned this finding. Pender (1996) explains it through a seasonality effect. Notice that when choices are defined over sets of outcomes and not over single outcomes, choice reversal can originate from dynamically consistent preferences (Gul and Pesendorfer 2001).
2
For a comparison among alternative discounting functions see Benhabib et al. (2006) and Tanaka et al. (2005). Strotz (1955) introduced the expression “strategy for pre-commitment.” In this paper we will use precommitment and commitment interchangeably.
3
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quite limited. Given voluntary commitment, policy makers’ roles would simply be to ensure that time-inconsistent agents have access to cheap commitment options. On the contrary, if agents are unaware of their inconsistency (naïve), they will not voluntarily commit (Strotz 1955; O’Donoghue and Rabin 1999). In this case, optimal policies may call for strict constraints on choice sets.4 Although the implications of naïveté or sophistication are profound, the behavioral evidence is still quite limited (Ashraf et al. 2006; Ariely and Wertenbroch 2002; DellaVigna and Malmendier 2006; Benartzi and Thaler 2004). The first contribution of this paper is to present empirical evidence on the degree to which the preference for commitment is common. Second, while choice reversal over time may originate from inconsistent time preferences, it may also be the result of uncertainty about future outcomes. Halevy (2008) is a recent contribution in this direction. In the example about pregnant women, both issues may be at work. On one hand, choice reversal may be due to the temptation to ask for anesthesia that arises while experiencing pain. Given such a preference-based explanation, commitment could be an appealing option for the decision-maker. An alternative explanation for choice reversal may be ignorance about the future level of pain. Only during labor does a woman realize the level of pain, and only then can she make an informed choice about whether or not to take anesthesia.5 Under this uncertainty-based explanation, commitment is never a good option. Far from being optimal, committing can prevent a decision-maker from revising a decision in light of new, relevant information. Actually, in the presence of uncertainty, there may be a strict preference for flexibility, i.e. for a larger choice set. We also elicited a person’s preference for flexibility: (d) Would you prefer $111 in 1 year and 2 months, or being able to choose 1 year from today, between $100 in 2 days or $110 in 2 months? A novel aspect of this study is that subjects faced both commitment and flexibility choices. Hence, we were able to classify subjects into those that preferred commitment, those that preferred flexibility, and those that were “in-between.” We report that about 60% of those who reversed their choice did commit. Moreover, the lower the cost of commitment the more subjects chose to commit. Interestingly, certain smoking habits were correlated with reversing choices and selecting commitment. This evidence provides direct support for a preference-based explanation of choice reversal over time. Among those who reversed their choices, 14% preferred flexibility and 26% were “in-between.” The minority of subjects who preferred flexibility suggests two qualifications for the earlier conclusion in favor of a preference-based explanation. On one hand, this evidence rules out that the presence of uncertainty was the only motive for choice reversal over time. On the other hand, it shows that uncertainty 4
However, the welfare implications of commitment are not clear-cut. For one, by committing, an agent forgoes the value of flexibility. Moreover, it is sometimes unclear how to use the agent’s preferences for welfare comparisons. In Gul and Pesendorfer (2001) the welfare criterion is clear: the agent “is unambiguously better off when ex ante undesirable temptations are no longer available.”
5
Evidence is mixed on this point (Christensen-Szalanski 1984). In a follow-up interview 1month postpartum, many women regretted their choice and expressed their preference to avoid using anesthesia (support for the preference-based explanation). Women at their first childbirth experience were more likely to reverse their choice than the average (support for the uncertainty-based explanation).
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about future states was a relevant motive for some people. Finally, the subjects characterized as “in-between” did not clearly select either commitment or flexibility. Under a preference-based explanation, these subjects are considered “naïve” because they fell into temptation but were unaware of it. Being in the “in-between” category was also correlated with having lower intellectual abilities. In conclusion, according to our estimates, most subjects were “sophisticated” and not “naïve.” The paper is structured as follows. Section 1 outlines the experimental design employed, while Section 2 puts forward some theoretical predictions of uncertaintybased and preference-based models. Section 3 presents the results. Section 4 is a literature survey. Conclusions follow in Section 5.
1 Experimental design and procedures We studied the time preferences of 120 subjects recruited from the undergraduate population of Jaume I University of Castellon, Spain.6 Each subject faced a series of choices between a smaller-sooner payment (SS) and a larger-later (LL) payment with delays between 2 days and 22 months. Choices were divided into three parts: a first part to measure impatience, a second part to detect choice reversal, and a third part to assess preferences for commitment and flexibility. The procedures for parts one and two share similarities with the titration procedure used in Mazur (1987) and Kirby and Herrnstein (1995). In all parts special attention was given to equalize the implicit costs among alternative options. For instance, no option involved an immediate payment, so subjects had to return to the lab to cash rewards no matter what the choice was. In part one, each decision had four parameters: the amount of the two possible payments and their delays. Both payment amounts were held constant throughout the procedure at 100€ and 110€. The goal of part one was to elicit an approximate measure of impatience D* by fixing the delay of the smaller-sooner payment at 2 days, SS= (100€, 2 days), and varying the delay D of the larger-later payment, LL=(110€, D), up and down until the subject was approaching indifference between the two payments. The main reason for eliciting D* was not to accurately estimate impatience but to be able to individually calibrate choices in parts two and three. An example will clarify the procedure of part one. Consider a subject who selects LL=(110€, 10 days) over SS=(100€, 2 days), but chooses SS over LL=(110€, 17 days). In this example the point of the switch from LL to SS narrows down the subject’s indifference point between the two options to a delay in between 10 and 17 days. We take D*=17 as our measure of impatience, i.e. the waiting time in days for LL in that decision where the subject switches from LL to SS.7 To minimize trembling hand mistakes, following the switch, one or two additional decisions were 6
Six sessions were run between April 26 and April 28, 2004. The actual last payment took place on May 30, 2005. Recruitment was done through announcements in classes. Half of the subjects were women and a large majority was comprised of economics or business majors. Basic statistics on the sample composition are in Table 6 in the “Appendix”.
7
Unless a subject has a discount factor of one, hence is perfectly patient, there always exists a long enough delay to induce the switch in favor of the sooner-smaller reward.
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121 SS: (100€, FED+2 days)
Fig. 1 Willingness to wait in parts one and two of the experiment FED = Front-end delay
LL: (110€, FED+D days) Decision during the session
prompted with longer delays D, and answers had to confirm the switch for part one to stop. Moreover, a list of all part-one decisions was presented and each one of them could be changed before moving to part two. In part two, subjects faced a series of decisions with a front-end delay (FED) added to both rewards SS and LL in order to detect possible choice reversals (Figs. 1 and 2). The smaller-sooner reward maintained a consistent delay of FED+2 days and the larger-later reward maintained a delay of FED+D* days, which ensured throughout part two that the waiting time difference between SS and LL was constant. The remaining parameters of the decisions were unchanged. This design increased the chances to detect the presence of choice reversal patterns. To continue with the example, part two would open with a decision with a FED of 7 days between SS=(100€, 9 days) versus LL=(110€, 24 days). Assume that in this opening decision the subject chose SS. Then the following choice would be (100€, 16 days) versus (110€, 31 days). If the subject kept choosing SS, the FED would become progressively longer. If, instead, at this point a subject chose LL, such choice was the hint of a choice reversal. Following this switch from SS to LL, one or two additional decisions were prompted with longer FED delays, and the subject had to confirm the switch in order for part two to stop. In case the choice reversal was confirmed, we define FED*, which in the example is 14 days, as the shortest frontend delay that induced the subject to reverse her choice from SS to LL. The procedure stopped in any case when the front-end delay FED exceeded 395 days.8 Finally, a list of all part two decisions was presented, and each one of them could be changed before moving to part three. Throughout the experiment, particular care was taken to avoid calendar effects, i.e. the inability or unwillingness to cash the reward on a particular day because of events such as birthdays, examinations, or traveling out of town.9
8
Although the number of decisions in parts one and two could vary across participants, there was no advantage for someone to finish earlier because that would not have changed the time the person could leave the room. In part one, the elicited maximum wait D* was 276 days. In part two, the elicited frontend delay was between 7 and 395 days. The elicited furthest possible time of payment in the experiment was 611 days.
9
If payment fell on Saturday, Sunday, or an official holiday the delay was automatically adjusted either backward or forward to make it easy for subjects to cash the reward. The summer period (July 25– September 5) was excluded, as well as Christmas vacations (December 22–January 6) and several other national or local festive days (for example October 9 and 12, November 1, December 6 and 8, February 27, March 19, May 1, June 29). Classes ran up to May 28, and then there were exams up to June 30. The university was closed in August. About 90% of the subjects’ families lived in the university town or in the region (Valencia province). More details on the procedure can be found in Casari (2006).
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Fig. 2 Soft commitment in decision 1 (Table 1)
Decision during the session
SS
(100€, FED* +2)
LL
(110€, FED*+D*)
SS’
(98€, FED* +2)
LL’
(110€, FED* +D*)
Later decision by email
Part three included eleven decisions aimed at measuring preferences for commitment and flexibility (Table 1). These choices had two peculiarities. First, instead of only two alternative payments, each decision could involve either three or four. Second, each decision comprised two choices that took place on two separate days: one on the day of the experimental session and the other through email at a later date, sometimes months later. Part three gives interesting insights especially for those subjects who reversed their choices. In part three, a subject could either commit immediately to the larger-later reward {LL} or postpone the choice between a larger-later reward and a smaller-sooner reward, {LL, SS} (Table 1, decision 5). The commitment strictly eliminated the possibility of later choosing another option: in particular, once made, it irreversibly ruled out the possibility at a future date to opt for SS. On the contrary, postponing the choice allowed a subject to wait exactly FED* days and send an email stating a preference for either LL or SS.10 Amount and delays of payments were structured in a way to resemble decisions already made in parts one and two. In particular, in the lab a subject faced SS=(100€, FED*+2) versus LL=(110€, FED*+D*). Remember that the subject faced the same decision in part two, and those who reversed their choices chose LL. The subject could commit to LL and would need to send an email after FED* days. As an alternative, she could postpone the choice and, after having waited FED* days, she could send an email choosing between SS=(100€, 2) versus LL=(110€, D*). Remember that the subject faced this same decision in part one and chose SS. A commitment for LL at the time of the lab session reveals an awareness of the future temptation to choose SS. Part three included several variants of the choice above where a cost was added either to the restricted set {LL} or to the larger set {SS, LL} in terms of a lower payment amount or a longer delay. In one case, strict commitment was available for free (decision 5), and in the other two cases it was available for the cost in terms of a modified restricted set {LL′}. In decision 7, LL′ offered a lower amount than LL while in decision 10 LL′ exhibited a longer delay than LL. We also studied soft commitment using choices with four alternative rewards SS, SS′, LL, LL′. In the lab a subject selected one of the two sets, {SS, LL} or {SS′, LL′}, and in a later email she chose an option within the set already selected (Fig. 3). In three cases LL′ = LL and SS′ offered either a lower amount (decisions 1 and 2) or a longer 10
It was explained that only replies arriving within two days before or after the specified day would be considered valid. Almost all of the responses were received on the specified day. The response rate was 64.1%.
100 100 100 100
100 100 100
100 100−2 100 100
FED*+2 FED*+2 FED*+2 FED*+2
FED*+2 FED*+2 FED*+2
FED*+2 FED*+2 FED*+2 FED*+2+5 (c)
110 110 110
110 110 110 110
FED*+2 FED*+2 FED*+2 FED*+2+5 (a)
Amount, € Delay, t
FED*+D* 110−2 FED*+D* 110−2 FED*+D*+2 (b) 110 FED*+D*+5 (d) 110
FED*+D* FED*+D* FED*+D*
FED*+D* FED*+D* FED*+D* FED*+D*
Amount, € Delay, t 100−2 100−6 100−6 100
110 110 110−2 110
Amount, €
FED*+D* FED*+D* FED*+D* FED*+D*
110 110 110 110
FED*+D* 110 FED*+D* 110−2 FED*+D*+2 (b) 110
FED*+D* FED*+D* FED*+D* FED*+D*
Amount, € Delay, t
Larger-later reward, LL′
87.2 96.1 82.0 89.7
61.5 17.9 23.1
12.8 7.7 10.2 17.9
% choices for option Ba
Notes: In part three of the experiment, subjects faced eleven choices. Each line in the table describes a choice between option A and option B. Some options include a follow-up decision between a sooner-smaller reward (SS) and a larger-later reward (LL). D* Delay in days that was elicited in part one, FED* individual front-end delay in days elicited in part two. For “calendar” reasons the actual delay of rewards may have been different for some subjects. Delays in days may have been (a) 5 or 7; (b) 2, 3, 4; 46 for one subject; (c) between 1 and 7; 0 for two subjects; (d) 5, 6, 7, and for five subjects it was either 46 or 50. In no case was the later reward at an earlier date than the earlier reward. Two additional decisions were made after the eleven listed above; they are not reported because they are not relevant for the present analyses, 12: SS=(FED*+2, 100−5), LL= (FED*+D*, 110), SS′=(FED*+2, 110−6) and 13: SS=(FED*+2+2, 100), LL=(FED*+D*+2, 110), LL′=(FED*+D*+2, 110) a Based only on subjects who did reverse their choices (N=78)
Soft commitment 1. Monetary cost, low 2. Monetary cost, high 3. Monetary cost, high 4. Time cost, low Strict commitment 5. No cost 7. Monetary cost, low 10. Time cost, low Flexibility 6. Monetary gain, low 8. Monetary gain, low 9. Time gain, low 11. Time gain, low
Delay, t
Sooner-smaller reward, SS′
Sooner-smaller reward, SS
Larger-later reward, LL
Option B
Option A
Table 1 Decisions regarding commitment and flexibility in the experiment
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Fig. 3 Strict commitment in decision 7 (Table 1)
t0
Decision during the session
SS
(100€, FED* +2)
LL
(110€, FED*+D*)
LL’
(108€, FED* +D*+k)
t1
Later decision by email
delay (decision 4). In another case SS′ offered a substantially lower amount than SS and LL′ offered a slightly lower amount than LL (decision 3). Finally, there were four choices regarding flexibility. These decisions were formally similar to those pertaining to strict commitment (Fig. 3), but instead of paying a cost to restrict the choice set, a subject had to pay a cost to make it wider. More precisely, option LL involved a monetary cost (decisions 6 and 8) or a delay cost (decisions 9 and 11) with respect to option LL′. The payments related to choices over time were never immediate. Subjects had to come back at a future date at least 2 days after the session. Payments were made in cash, hence no trip to a bank was necessary.11 For choices in part one, only one participant per session received a large, delayed payment. There were 20 participants in each session. A volunteer randomly drew a number out of a bag immediately at the end of the session to select one of the lucky persons. The selected person received a signed letter on university letterhead promising a later payment of 100€ or 110€. The exact date and amount of the actual payment reflected one of the subject’s choices in part one of the experiment. The choice paid out was selected through a second random draw.12 For choices in parts two and three, the computer randomly selected one other person per session for a large payment between 94€ and 110€ and randomly selected one choice in part two or part three. His or her name and the selected decision were not immediately revealed to the participants. Instead, along with all participants’ contact information, the name was placed in an envelope that was then sealed in front of them and signed by two subjects.13 Only the subjects who emailed their follow-up choices were eligible to receive this second large payment. In particular, a choice for commitment did not save the cost of sending an email at the pre-specified
11 A payment of 5.61€ ($6.68) per person was given right after the session. It included a 3€ show-up fee (2€ in the first two sessions) and a performance-based fee for some correct answers in the questionnaire, as explained in the result session. Another component is discussed in footnote 21. The exchange rate at the time of the experiment was 1€ = $1.19. 12
All post-session payments were carried out by the personnel affiliated with the Jaume I University Laboratory of Experimental Economics (LEE). These personnel teach in the department of economics and conducted experiments and payments in previous experiments, so they were familiar to the subjects. Only the persons selected for the large payments had the burden of returning to collect money.
13
The envelope was stored in the laboratory, and any participant could ask to have it opened when all decisions and payments had been completed, which was several months later. This procedure gave credibility to the promise of a later payment while maintaining an incentive for all subjects to send the follow-up email. When the date of payment approached, the selected person was privately contacted to arrange an appointment for the payment day.
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date, because a lack of a confirmation message would have precluded participation in the draw for the payment. Subjects were seated at computer terminals and separated by partitions. Instructions were distributed and read aloud. First, instructions for a risk attitude task were read and the corresponding decisions taken with pen and paper.14 Then, time choice instructions were read and the corresponding decisions were made via computers. Finally, a questionnaire was distributed. No communication among subjects was allowed. Including instruction and reading time, a session lasted between 2 and 2.5 h. Overall, the average payment per subject was 16.06€ ($19.11).
2 Predictions This section outlines the predictions of alternative models of time discounting and puts forward five statements that provide guidelines to interpret the experimental results. The key question is what explains choice reversal. According to a preference-based explanation, an agent reverses her choice over time because she has present-biased preferences, instead of exponential time preferences. According to an uncertainty-based explanation, an agent reverses her choice over time because future promises carry an implicit risk of default that is characterized by a declining hazard rate; alternatively, there may exist other forms of risk or uncertainty about aspects of the decisional situation, with similar consequences.15 The remaining part of this section will clarify these two explanations, which are not mutually exclusive. With few exceptions, the empirical literature has discussed time preferences within a “certainty” scenario, where there is neither risk nor uncertainty at future dates about (a) actually receiving the reward, (b) the wealth position of the agent, (c) the agent’s preferences, and (d) other aspects of the decisional situation. We now illustrate the predictions on choice reversal, commitment, and flexibility for an exponential and a hyperbolic discounter under a certainty and an uncertainty scenario (Table 2). The “uncertainty” scenario is the most general situation. All statements rely on the following standard assumptions. First, the instantaneous utility ut(ct) of each agent does not change over time, i.e. ut(ct)=u(ct) for every period t=0,…,T; second, the utility function increases in consumption, ∂u/∂ct >0; and, third, it is weakly concave, ∂2u/∂2ct ≤0. In other words, the statements below do not rely on the instantaneous utility being linear in consumption. Consider a “certainty” scenario where the net present utility can be represented as X U0 ðcÞ ¼ t ¼ 0;...;T lðtÞuðct Þ ð1Þ where c=(c0, c1,…,cT) is a consumption profile and l(t) is a discount function, which we normalize as l(0)=1. When an agent discounts exponentially, l(t)=δt, 14
Complete instructions are available upon request to the author. Elicitation of risk attitudes closely followed the procedure of Holt and Laury (2002) and results are not reported here. Random draws and lottery payments were carried out at the end of the session.
15
In Amador et al. (2006) agents expect to receive in the future relevant information regarding their preferences. Fernandez-Villaverde and Mukherji (2000) focus on possible shocks on income as it is common in the macroeconomic literature.
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Table 2 Predictions of different models of time preferences
Certainty scenario a) Exponential discounting b) Hyperbolic discounting Uncertainty scenario c) Exponential discounting d) Hyperbolic discounting
Any possible case of choice reversal? (1)
Possible willingness to pay for commitment? (2)
Possible willingness to pay for flexibility? (3)
No Yes
No Yesa
No No
Yes Yes
No Yesa
Yes Yes
Notes: The table does not explicitly list the quasi-hyperbolic discounting model. If a 2-day delay is still treated as “present” then the quasi-hyperbolic model yields similar predictions in this experiment as the hyperbolic model. If not, predictions are the same as the exponential discounting model. In this paper we assume that a delay of up to 2 days is considered “present” and so the quasi-hyperbolic and hyperbolic discounting models yield similar predictions a Stands for “yes” if the agent is sophisticated and for “no” if the agent is naïve
with a constant discount factor 0≤δ≤1, she will never reverse her choices (Table 2, line a). On the contrary, when the discount factor is not constant but depends on the horizon, an agent with an established plan for a far-away future might not implement it as that future draws near (Strotz 1955). When the discount factor is smaller for shorthorizon rewards than for long-horizon rewards, we refer to the agent as “present-bias.” We take hyperbolic discounting, l(t)=1/(1+k·t), with k>0, as the reference model for such preferences but the predictions apply more generally to many present-biased models. For any hyperbolic discounter, one can construct a larger-later reward (LL) and a smaller-sooner reward (SS) that pay at a fixed, future date where today the agent prefers the LL and then, at some point in the future, the agent switches to the SS (Table 2, line b). When scholars reported empirical evidence of choice reversal over time, it was rationalized as a preference-based phenomenon explained by hyperbolic discounting. This explanation is only one among many possible explanations. Another prediction of hyperbolic discounting is that agents may be willing to commit, i.e. given two sets of alternatives A and B choose the set B over A, where both set contain the same, best element, and where A is a superset of B (Akerlof 1991; O’Donoghue and Rabin 2001; Gul and Pesendorfer 2001). Commitment is attractive because it removes future temptations to deviate from the established plan. The likelihood of commitment choices depends on two factors, which are now briefly discussed: the agent’s assessment of the extent of her future self-control problems and the cost of commitment. An agent is called “sophisticated” if she is aware of her future-self control problems, and “naïve” otherwise. Statement 1. Under a preference-based explanation, a sophisticated agent who reverses her choice over time may be willing to commit, while a naïve agent who reverses her choice over time is never willing to commit. A sophisticated, hyperbolic discounter may commit or not depending on the price at which she can buy commitment. For instance, if the net present value forgone by committing is higher than the maximum willingness to pay to avoid temptation, then a hyperbolic discounter will not commit.
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Statement 2. Under a preference-based explanation, the higher the cost of commitment, the less frequently sophisticated agents subject to temptation will choose to commit. In an uncertainty scenario statements 1 and 2 still hold but the above analysis becomes richer. There exist forms of uncertainty for which choice reversals over time can be a rational path for an exponential discounter. What uncertainty cannot explain is why the exponential discounter chooses to pay for commitment.16 To illustrate the effect of uncertainty, we will focus on the case of implicit default risk of the promise to deliver the reward. Such promise may lack credibility and the reward may fail to materialize entirely or arrive with delay (Benzion et al. 1989; Dasgupta and Maskin 2005). There may also be a chance of the agent being unable to cash the reward because of unforeseen logistical obstacles, such as having to leave on a business trip or simply losing a coupon for a spa massage. Under this uncertainty scenario, time discounting is the product of time preferences and of the likelihood of the consumption in a given time period, s(t). The expected net present utility can be represented as E½U0 ðcÞ ¼
X t ¼ 0;...;T
lðtÞsðtÞuðct Þ
ð2Þ
Uncertainty has three important implications. First, for some probability distributions of s(t), an exponential discounter may rationally reverse her choice over time. Consider the following example adapted from Sozou (1998). Bob has to choose between two promises from Jim: a promise of a cake and of a bottle of wine. Assume that Bob values the wine w more than the cake c, u(w)=3, u(c)=2, and that he has a belief about the likelihood of actually receiving the item from Jim, which models promises with higher delays as less credible, more precisely, s(t)=1/(1+t) where t is the delay in months. If Bob is risk neutral and perfectly patient, λ(t)=1 he will prefer a promise of wine in 3 months over a promise of a cake in 2 months, 1/4 u(w)> 1/3 u(c). Yet, after 2 months have passed he will reverse his choice and prefer the cake to the wine in a month, 1/2 u(w)
