Temporal horizons in decision making

Temporal Horizons in Decision MakingMarc Wittmann and Martin P.PaulusUniversity of California,San Diego,and Veterans Affairs San Diego Healthcare System The perception of duration is crucial when we make choices between immediate and delayed rewards.Immediate rewards are valued more than the same rewards if they are delayed.Preferences for earlier rewards become even stronger when the reward can be received within a subjectively proximate time frame.A “rational”decision to wait for a delayed reward may be overruled by an “impulsive”choice when the option for the closer reward falls within this proximate time range.Based on findings on circadian and circannual physiological rhythms,we suggest that there are 2time units that are both biologically and culturally determined and have an impact on human experience and behavior:the day and the year.We highlight results of a neuroimaging study showing that rewards with delays up to 1year are discounted differently than reward delays longer than 1year.This duration-dependent discounting is associated with specific brain activation in the striatum.We present various conceptualizations of subjective time incorporated in parametric models of intertemporal decisions that may lead to a better understanding of human choice behavior.Keywords:time perception,biological rhythms,delay discounting,intertemporal choice,impul-sitivityEvery day we make choices that have imme-diate as well as delayed consequences;that is,individuals frequently make decisions based on predictions of rewards on different time scales.We may choose to save money for retirement,thereby opting for a momentary loss of money to gain future benefits.Someone may have to make a decision on whether to spend inherited money on a luxury vehicle or to invest that money for gaining a higher profit in the long run.The time until the beneficial higher outcome of the investment can be received (in a few years)is viewed as a cost and is weighed against thebenefits of the possible outcomes (the joy of pos-sessing the automobile now or greater financial power in a few years).In economic terms,the expected utility of a choice is viewed as a function of the value of the reward,the probability it will be received,and the magnitude of the delay before the reward is received (Read,2001).The process of deciding whether to opt for an immediate reward or for a delayed but higher reward is strongly related to scholarly and pro-fessional success in life (Mischel,Shoda,&Rodriguez,1989).To function effectively,one has to voluntarily postpone impulsive urges for immediate gratification and persist in goal-directed behavior to achieve positive outcomes that are delayed in future (Zimbardo &Boyd,1999).However,rewards that are received sooner are often preferred over future rewards as the subjective value of a reinforcer is dis-counted as a function of the delay (Ainslie,1975).Thus,the anticipated time span between the choice and the reception of the reward is an important factor that determines our decisions.Following this reasoning,time is an essential variable in understanding the mechanisms of choice behavior.However,the underlying psy-chological and neural processes that account for the perception of time,and that lead an individ-ual to select between choices with differentMarc Wittmann,Department of Psychiatry,University of California San Diego,and Veterans Affairs San Diego Healthcare System,San Diego,CA;Martin P.Paulus,De-partment of Psychiatry and Laboratory of Biological Dy-namics and Theoretical Medicine,University of California,San Diego;and Veterans Affairs San Diego Healthcare System,San Diego,CA.This research was funded by Grant 1R03DA020687-01A1from the National Institute on Drug Abuse and by a grant from the Kavli Institute for Brain and Mind.We acknowledge Gal Zauberman for discussions on the topic.We also thank two anonymous reviewers whose comments led us in the right direction in developing our concepts.Correspondence concerning this article should be ad-dressed to Marc Wittmann,Department of Psychiatry,Uni-versity of California,San Diego,La Jolla,CA 92093.E-mail:wittmann@Journal of Neuroscience,Psychology,and Economics ©2009American Psychological Association 2009,Vol.2,No.1,1–111937-321X/09/$12.00DOI:10.1037/a00154601delays,are only beginning to be explored(Ta-kahashi,2005;Wittmann,Leland,&Paulus, 2007;Zauberman,Kyu Kim,Malkoc,&Bett-man,in press).Because individuals choose re-wards as a function of perceived duration,it is conceivable that a person’s estimate of duration will influence her intertemporal decisions.We have recently proposed that impulsive intertem-poral decisions as seen in many impulsive indi-viduals could be due to the fact that time inter-vals are perceived subjectively as lasting too long and,thus,that a given time interval is associated with too high of a cost,which in turn leads to the selection of the alternative with a more immediate outcome(Wittmann&Paulus,2008). Intertemporal decisions in everyday life as well as in laboratory decision-making tasks dif-fer considerably in their temporal characteris-tics.Delays of reward options ranging from seconds to minutes and years to decades will most probably be processed by different mech-anisms(Wittmann,1999).Here,we point out two important time intervals that have shown to be of relevance for temporally structuring hu-man life and the decisions we make:the dura-tion of the day(24hr)and the time span of the year(365days).In this article,wefirst discuss findings of experimental delay discounting tasks as they relate to the experience of time. Several very recent conceptualizations have in-corporated the notion that the subjective expe-rience of time is an integrative part in models explaining intertemporal choice behavior.Fur-thermore,we present aspects of research on time perception(duration processing)in which we show that different processing mechanisms of the brain are responsible for different time scales.Finally,in(a)combining the notion of subjective time perception as being logarithmic, and(b)using a parametric model that separates between a near future and a far future,we provide means to analyze subjects’choices in intertemporal decisions.In this context,we dis-cuss how the perception/anticipation of a year might influence decision making.Decision Making,Impulsivity,and Time PerceptionDelay discounting behavior in human par-ticipants is experimentally assessed by pre-senting individuals with hypothetical or real options.For example,subjects have to decide whether they would prefer$10right now or $30in a week.As the duration of the delay to the second option increases(e.g.,increasing to2 or3weeks),preference for this option de-creases,with this temporal discounting pattern being described by a hyperbolic function(Ain-slie,1975;Lane,Cherek,Pietras,&Tcher-emissine,2003;Madden,Begotka,Raiff,& Kastern,2003)or a quasi-hyperbolic function (Laibson,1997).The discounting function is steeper at shorter intervals(the rate of discount-ing is higher)and becomesflatter as the delay of the reward increases(the rate of discounting is lower)(Takahashi,in press).Analyses of con-sumer behavior show how money and time are traded against each other and lead to the hyper-bolic discounting function(Hantula&Bryant, 2005).Delay discounting paradigms are one ap-proach to quantify impulsive behavior and self-control(Arce&Santisteban,2006;Prelec& Loewenstein,1997).In these tasks,impulsive individuals show an even steeper discounting function;that is,they prefer the smaller but earlier reward over the larger but later reward more often than more self-controlled subjects. The ability to delay gratification is generally associated with the notion of willpower(Met-calfe&Mischel,1999;Bechara,2005).Individ-uals,in contrast,who choose immediate grati-fication at the expense of important long-term interest are judged to be impulsive(Ainslie, 1975;Critchfield&Madden,2006).If one con-ceptualizes human behavior as resulting from the competition between multiple“agents”within a person,then our more impulsive self competes with our more controlled and rational self(Laibson,1997;Takahashi,in press).De-pending on the temporal distance of rewards and costs and the subjective perception of these distances,either the impulsive self or the self-controlled self will be more likely to dominate choice behavior.Impulsive behavior as quanti-fied in delay discounting tasks is seen in a wide range of patient populations,such as drug users (Kirby,Petry,&Bickel,1999)or children with attention-deficit/hyperactivity disorder(Bark-ley,Edwards,Laneri,Fletcher,&Metevia, 2001).High levels of impulsivity have also been associated with maladaptive decision mak-ing in daily life(e.g.,risky health-related be-havior,the frequency of speeding tickets and2WITTMANN AND PAULUScollisions,and chronic hostility;Bogg&Rob-erts,2004).It is important to note,however,that an overly strong emphasis on the future can also be dysfunctional,because it deprives one of impor-tant hedonistic qualities in life that can only be experienced at the present moment.An exces-sive farsightedness can lead to long-lasting re-grets of having missed out on pleasures of life. These regrets of“overcontrol”are even longer lasting than regrets of giving in to short-term temptations(Keinan&Kivetz,in press).Thus, it appears that adaptive functioning depends on a balance between the future-oriented reflective system and the impulsive system,which is ori-ented more on the present moment.This bal-ance enables us to successfully choose between commodities that have different values and magnitudes of delay.Eventually,the ability to create positive and memorable moments is the basis for the subjective experience that our life periods lasted satisfyingly long(Wittmann& Lehnhoff,2005).Also,the impulsive choice is not necessarily an irrational choice,as one has to deal with contingencies that may make it impos-sible to receive the delayed reward(Takahashi, in press).Recently,we have proposed that impulsive intertemporal decision making may be causally related to the experience of time(Wittmann& Paulus,2008).Impulsive individuals would ex-perience time differently because they overesti-mate the duration of time intervals,compared with more self-controlled individuals.As a con-sequence,the value of delayed rewards is dis-counted more strongly.Empirical evidence,for example,shows that sleep-deprived subjects relative to well-rested controls are more impul-sive,they discount delayed rewards more strongly,and they overestimate time intervals in the multiple-seconds range(Reynolds&Schiff-bauer,2004).Impulsive patients with borderline personality disorders as well as patients with orbitofrontal cortex lesions,who were more im-pulsive during behavioral tests,considerably overestimated time intervals between10and 90s(Berlin&Rolls,2004;Berlin,Rolls,& Kischka,2004).Patients with cocaine and methamphetamine dependence participating in an inpatient drug treatment program(stabilized and without acute craving symptoms)overesti-mated the duration of a53-s interval,which was mediated by higher self-reported impulsivity (Wittmann,Leland,et al.,2007).Thus,findings from clinical populations clearly demonstrate that the subjective experience of time(specifi-cally an overestimation of duration)is associ-ated with higher levels of both self-rated and behaviorally tested impulsivity.These individ-uals are more likely to experience a slowing down of subjective time and an overestimation of duration in situations when they are not able to act on their impulsive urges;that is,when they have to wait for a delayed reward and thus are confronted with the passage of time.Our explanation relating increased attention to time to an overestimation of duration and,as a consequence,to greater discounting of de-layed rewards in impulsive individuals is par-tially supported by experiments showing that when subjects increase their attention to time,as manipulated in experimental settings,they change choice preferences—less discounting in the near future but stronger discounting in the far future(Ebert&Prelec,2007).Moreover,it is a consistentfinding that people accept more readily a longer delay for a reward when time is described by a calendar date than when it is described in terms of the actual delay(LeBoeuf, 2006;Read,Frederick,Orsel,&Rahman, 2005).An explanation for thisfinding is that descriptions of the actual reward delay(as op-posed to indicating a date)direct attention more strongly to the duration of the time interval,thus resulting in a greater awareness of duration, which in turn leads more frequently to a rejec-tion of the longer delay.It is well known that attending more strongly to time leads to larger time estimates(Zakay&Block,1997).In this context,it is important to note that recent models,complemented by empirical ev-idence,explain how the subjective experience of time leads to hyperbolic discounting across time intervals.This specific discounting pattern might be due to the fact that people’s estimates of anticipated duration do not correspond to objective time in a linear way but that objective time is perceived logarithmically(Takahashi, 2005;Zauberman et al.,in press).Following the Weber–Fechner law,which states that human perception can be described as a logarithmic function of the physical stimulus,subjective time estimates also do not accurately(not lin-early,but logarithmically)match changes in objective time(interpretable as a lack of sensi-tivity in the time domain;Takahashi,Oono,&3TEMPORAL HORIZONS IN DECISION MAKINGRadford,2008).This lack of sensitivity over time can explain hyperbolic discounting if in-tertemporal decisions are a function of objective time.However,if discount rates are displayed not as a function of objective time but of sub-jective time,they appear to be constant over time(a hyperbolic pattern is no longer visible; Zauberman et al.,in press).Although more re-search is needed to determine the causal rela-tionship between decision making and time per-ception,the mentioned studies indicate that the subjective experience of time is a decisive fac-tor in decision making.Effects of Anticipated Durationon Choice BehaviorThe phenomenon of preference reversals (i.e.,a previously preferred option is no longer chosen when another option with a subjectively higher value is available sooner;Ainslie,1975; Green,Fristoe,&Myerson,1994)emphasizes how the subjective perception of time affects the evaluation of reward delays.For example, when having to decide whether to formally ac-cept an invitation to a party in a week or to decline it and to state that one has to study for an exam,preference for making arrangements to study on that evening(7days ahead)might be prevailing.However,when faced with the deci-sion to go to a party tonight or to study for the exam,the temptation to go to the party may be stronger than the intention to study.In this case, a reversal of preferences has occurred.The strength of preferences for an earlier reward relative to a later reward increases as both rewards approach in time and the earlier reward(the party tonight)can be consummated very soon(Kirby&Marakovic,1996).Decision making,thus,is influenced by the subjective estimate of duration,which depends on tempo-ral processes partitioning time intervals into dis-crete units.Options within a certain time range (e.g.,today)are treated differently(being more proximate)as the same options that are received only beyond that time range(e.g.,tomorrow). For example,when the critical event—whether to attend a party or to study—is still several days away,the student may prefer the option associated with the delayed gratification;that is, he will prefer to study.However,as soon as that evening becomes today’s evening,the student switches preference and accepts the invitation to the party.The choices“entered”a subjective time zone that demarcates events as being per-sonally close or more distant.In other words, the temporal attribute of the event changed be-cause of the different time horizon.With this change,the experienced value of the earlier reward increased.At the same time,the antici-pated cost of working tonight also increased considerably.One reason why people change their behavior as a function of time is that they tend to have more abstract notions of how they would decide in situations when they are psychologically more distant;that is,when they have to decide on hy-pothetical scenarios or,relevant to our discussion, when the events are temporally distant(Malkoc, Zauberman,&Ulu,2005;Trope&Liberman, 2003).Thus,the concreteness of mental represen-tations(“tonight I have to either study or go to the party”)may account for the present-biased pref-erence(Malkoc&Zauberman,2006).Moreover, because of the temporal proximity and the con-creteness of the situation,interoceptive states and visceral factors will have a greater impact and influence decisions so that we are more likely to prefer immediate gratification(Loewenstein, 1996).We propose that options lying within a subjective time frame that defines temporal prox-imity will be interpreted differently from options that lie beyond the time frame.Decisions on op-tions vary considerably,depending on the timing of consumption;that is,whether a specific out-come will be received within or after a certain time period.Ebert and Prelec(2007)have recently developed a parametric model of discounting(the constant-sensitivity discounting model)that incor-porates this idea of a time frame that separates a near future from a far future.In the stretched exponential discount function,a parameter of im-patience,a,marks the boundary between a sub-jective near future and a subjective far future.A time-sensitivity parameter,b,affects the discount-ing rates for the near and far future,thus,shaping the discounting function in a way that a time horizon becomes more or less pronounced.Exper-imental manipulations that aim at increasing sen-sitivity to time lead to a decrease in discounting in the near-future period and increase discounting in the far future,thus increasing the categorical dis-tinction between a personally close time zone and more distant time.4WITTMANN AND PAULUSTime Frames for Cognition and Behavior Neuropsychological and psychophysical evi-dence suggests that temporal information pro-cessing is critical for many aspects of cognitive functioning and results in specific temporal pat-terns of human perception and behavior(von Steinbu¨chel and Po¨ppel,1993;Wittmann, 1999).Basic timing processes are an integrative part of the cognitive system.Nevertheless,the cognitive and neural systems of human time processing are incompletely understood.How-ever,investigations based on neuropsychologi-cal approaches have yielded some progress in understanding mechanisms and principles that underlie perception of time in the milliseconds-to-seconds range,which has resulted in an ex-tensive literature on the psychology and under-lying neural systems of temporal processing (Harrington et al.,2004;Kagerer,Wittmann, Szelag,&von Steinbu¨chel,2002;Zakay& Block,1997).Several neural models of subjec-tive time perception are based on the notion that an internal clock,along with attentional and working memory processes,contribute to dif-ferent temporal processing mechanisms across different time scales(Lewis&Miall,2003; Po¨ppel,1997;Trevarthen,1999).Specifically, studies show that the integrity of dopamine neu-rotransmission within the frontostriatal circuitry of the brain is an essential component for the processing of time in the seconds range(Buhusi &Meck,2005;Harrington et al.,2004). There is accumulating evidence of different time perception mechanisms operating at differ-ent time scales.Specifically,a timing system responsible for the milliseconds range can be discerned from other timing systems for longer durations(the seconds range;Lewis&Miall, 2003).Furthermore,it has also been shown that the processing of intervals up to2to3s is different from that of intervals exceeding this time range(Fraisse,1984;Po¨ppel,1997;Witt-mann et al.,2007).The fact that individuals are more accurate when information is presented in temporal intervals that do not exceed2–3s supports the idea that events are integrated into a unit or perceptual gestalt(Ulbrich,Churan, Fink,&Wittmann,2007).Human beings orga-nize behavior also on longer time scales and temporal units,and research has identified fur-ther temporal processes associated with differ-ent longer time ranges that are relevant in ev-eryday decision making.For example,cyclic properties that are appar-ent in the recurrence of days and the repetition of the seasons are another aspect of time.Nature has adapted to these properties by establishing endogenous biological clocks in organisms.The circadian(about1day)clock and a circannual (about1year)clock regulate fundamental as-pects of physiology and behavior and are en-trained(synchronized)by the periodicity of light.Specifically,daily rhythms persist with a period of approximately24hr that has been shown for many biological functions in humans, ranging from many aspects of physiology to the sleep/wake cycle and to cognitive functioning (Roenneberg,Daan,&Merrow,2003;Witt-mann,Dinich,Merrow,&Roenneberg,2006). In addition,circannual physiological rhythms, likewise entrained by light,appear to have an influence on the rhythmic appearance of emo-tion and behavior(Roenneberg&Aschoff, 1990;Wirz-Justice,Kra¨uchi,&Graw,2001). The seasons and the accompanying physiologi-cal adaptations in mammals shaped human hunting,farming,and reproduction behavior, temporally ordered within a year’s cycle(Bron-son,2004).Even today,social,political,and economic planning is packed into1-year time frames.The astronomical constraints of the day and night cycle,as well as the yearly periodicity of the seasons,shape biological processes on Earth and likewise affect human cognition and behav-ior.It is,therefore,conceivable that these ef-fects have also an impact on the subjective experience of duration.Events that lie beyond such a period of time(tomorrow or next year, respectively)are treated categorically differ-ently compared with events that lie within that period(today,this year).A temporally proxi-mate reward(within the time range)will more likely be chosen than a temporally more distant reward(beyond the time range).A choice be-tween options that can be received today versus tomorrow will be treated differently compared with a choice between options delayed by to-morrow and by two days.Despite the fact that both pairs of options differ in reward delay by1 day,relative preference for today’s earlier re-ward will be much higher than that of tomor-row’s earlier reward.Since ancient times,the day and the year have formed natural time units5TEMPORAL HORIZONS IN DECISION MAKINGfor measuring time intervals in calendars and synchronizing individual schedules(Waugh, 1999).We have been socialized to measure duration and to plan ahead by using these tem-poral intervals.Moreover,these two time units correspond to periodicfluctuations in physio-logical functions.Thus,both culturally and bi-ologically,the day and the year have a signifi-cant impact on human experience and behavior. The duration of what is considered a subjec-tively proximate time interval will definitely vary.In decisions that rely on bodily sensations (e.g.,pain,thirst,hunger,or sexual drive),the experienced moment may lie in the seconds-to-minutes range.If an option for fulfilling these basic drives and needs is,for example,1hr ahead,that would feel enormously distant. When feeling very hungry,the option to just eat what is left in the fridge now could dominate over the possibility to drive to a restaurant with excellent food and the opportunity to meet friends(when one would have to wait for an-other60min before one could actually consume food in a sociably pleasant surrounding).That is,physiological constraints and visceral re-sponses will determine the degree to which proximate or distant options will be perceived as such.A direct link between the perception of time and physiological processes has recently been proposed by Craig(2008),who claims that time and temporal processing relate to emo-tional and visceral processes because they share a common underlying neural system,the insular cortex and the interoceptive system.Decision-making processes have also been linked to emo-tion processing and were found to be related to insular cortex activation(Paulus,Rogalsky, Simmons,Feinstein,&Stein,2003).Besides the frontostriatal areas of the brain,the insular cortex has repeatedly been shown to be in-volved in time perception tasks using functional neuroimaging(Ferrandez et al.,2003;Lewis& Miall,2003).Moreover,we recently discovered specific accumulator-type neural activity in the posterior insula which is involved in the encod-ing of multiple-second intervals(Wittmann, Simmons,Aron,&Paulus,2008).Time units of1day and of1year will be more relevant in decisions that are not so tightly related to imminent bodily needs but are more related to long-term planning,such as economic forecasting.Support for our hypothesis comes from an analysis of past studies on temporal discounting of monetary rewards(Frederick, Loewenstein,&O’Donoghue,2002):Discount rates were plotted against the time horizons of the individual studies and were shown to be consistently high for delays up to1year.Vir-tually no decline of discounting rates was visi-ble with longer delays;that is,a plateau of small discounting rates was reached for delays be-tween1year and over10years.This meta-analysis thus reveals that subjects perceive the 1-year unit as a discrete entity.The Influence of the1-Year Time Frameon Choice BehaviorWhat we suggest here is that one defining factor of what is subjectively experienced as being proximate or temporally distant is shaped by cognitive mechanisms that partition time based on culturally learned and physiologically determined time units.Studies that integrate research on the subjective experience of dura-tion and intertemporal decision making are scarce.We have observed some support for the existence of different subjective time scales in a recent study in which subjects had to decide between an immediate and a varied delayed hypothetical monetary reward during event-related functional magnetic resonance imaging (Wittmann,Leland,&Paulus,2007).Subjects performed with different discounting slopes for shorter(Ͻ1year)than for longer(Ն1year) delays(see Figure1),when time on the x axis is plotted logarithmically,thus reflecting more closely the subjective experience of duration (Zauberman et al.,in press).Moreover,differ-ential activation in the striatum accounted for the two delay discounting slopes(the delay equivalent value represents the discounted value of the delayed and greater reward).Neural activation associated with the more immediate rewards(Ͻ1year)relative to the more delayed rewards(Ն1year)was associated with in-creased activation in the head of the left caudate nucleus and putamen(see Figure1).Ourfind-ings of the duration-dependent involvement of the striatum corroborates a model that assumes that different pathways of the corticobasal gan-glia circuit may be specialized for reward pre-diction and action selection on different time scales(Tanaka et al.,2004).Our results are consistent with this model because they demon-strate that the striatum can influence whether6WITTMANN AND PAULUSone chooses a delayed reward that will be re-ceived within 1year versus one that will be received after 1year.In addition,the differ-ences between shorter versus longer delays in the caudate and ventral striatum supports the idea that these structures encode value as a function of time.The specific differences in discounting slopes between options Ͻ1year and Ն1year may well be due to cognitive anchoring;that is,subjects process future re-wards relative to an anchor point defined by a biologically relevant period of time.Rewards delayed by intervals longer than 1year would be moved out of an annual time horizon and thus would appear to be qualitatively much fur-ther away leading to the effect of stronger dis-counting of delayed rewards.To furthermore validate our conclusion that our data represents behavior following the per-ceptual categorization of time intervals into a more proximate time zone and more distant time intervals,we combined two recent concep-tualizations.Because subjective time is a loga-rithmic function of objective time (Takahashi,2005;Zauberman et al.,in press),we used in-dividuals’discounting behavior of our study with delays on the x axis transformed logarith-mically.Similar to a recent comparison of re-lated discount models (Takahashi et al.,2008),we then fit the individual discounting perfor-mance using the constant-sensitivity discount-ing model (Ebert &Prelec,2007)applying (a)apresent-future dichotomy function,(b)an expo-nential,and a (c)hyperbolic ing the statistical computing software R (current version:2.7.0),we estimated goodness-of-fit measures (ad-justed R 2)according to these functions.According to the model by Ebert and Prelec (2007),the constant-sensitivity discount function is:f ͑t ͒ϭexp ͑͑Ϫat ͒b ͒(1)where a marks the boundary between the near and far future,b is a parameter affecting dis-counting for near and far future outcome,and f(t )is the discount factor at different time points in time t .A derivate of Equation 1enables a linear fit of the constant-sensitivity function (Equation 2)Ϫlog ͑Ϫlog f ͑t ͒͒ϭϪb log a Ϫb log t(2)where b is the slope and Ϫb log a is the inter-cept.That way,we can recover a:a ϭexp ͑Ϫintercept/slope ͒.(3)Following this notation,the exponential function is:Ϫlog f ͑t ͒ϭat(4)and the hyperbolic function is:Ϫlog f ͑t ͒ϭϪb log (a )Ϫb log (a )(5)Because the R 2values across the subjects were skewed,we used nonparametric statistics to test for differences in the goodness-of-fit measure across the three probed functions.The Friedman test for multiple comparisons reveals significant differences over the functions,␹2(2)ϭ10.3,p Ͻ.006.Wilcoxon’s tests show that the best fit of the present–future dichotomy function (aver-age R 2ϭ0.689;see Table 1)is significantly different from those of the exponential function (average R 2ϭ0.615;z ϭ1.992,p Ͻ.046)and the hyperbolic function (average R 2ϭ0.534;z ϭϪ2.691,p Ͻ.007).Taken together,these data provide further evidence that a time-horizon approach as modeled with a stretched exponential function as suggested by EbertandFigure 1.Different discounting slopes for delays Ն1year and for delays Ͻ1year (Wittmann,Leland,&Paulus,2007).7TEMPORAL HORIZONS IN DECISION MAKING。