Optimal Stopping under Present-Biased Preferences

Transcription

1 Opimal Sopping under Presen-Biased Preferences Qing Li, Javad Nasiry School of Business and Managemen, Hong Kong Universiy of Science and Technology, Clear Waer Bay, Kowloon, Hong Kong SAR., imqli, Peiwen Yu School of Managemen, Fudan Universiy, Shanghai, PR China, We sudy he opimal sopping problem in which decision makers (agens) have ime-inconsisen, presenbiased preferences. The agens may be naive and unaware of he bias or sophisicaed and aware, and he sopping problem may involve immediae rewards or coss. We invesigae wheher agens follow a hreshold sopping policy as well as differences in agen behavior. Prior work shows ha an agen wih sandard imeconsisen preferences follows a hreshold sopping policy if he marginal payoff is monoone in sae. We show ha, under presen-biased preferences, similar condiions are sufficien for naifs and when rewards are immediae for sophisicaes o follow a hreshold sopping policy. Ye if coss are immediae, hen sophisicaes migh no follow a hreshold policy because of preempive sopping o avoid fuure self-conrol problems. We also compare he ses of saes under which differen agens sop. Sophisicaes are always more likely o sop han naifs. When rewards are immediae, naifs are more likely o sop han ime-consisen agens; however, he converse holds when coss are immediae. These findings exend he lieraure o seings where he saes of agens are imporan. We discuss implicaions of our resuls in wo sopping-problem examples, one in projec managemen and he oher in healh care. Key words : opimal sopping, presen-biased preferences, quasi-hyperbolic discouning, dynamic programming 1. Inroducion In an opimal sopping problem, a decision maker (agen) maximizes her expeced uiliy by making a binary choice, in each ime period, o sop or o coninue. The decision-making environmen evolves sochasically, and he problem erminaes when he decision maker sops (Ross 1983). Opimal sopping problems appear frequenly in operaions, finance, markeing, and economics (for a review, see Oh 2010) and are ofen sudied under exponenial discouning, which assumes ha he discoun rae beween any wo ime periods is he same irrespecive of when he rae is evaluaed. However, his assumpion is no psychologically or normaively plausible (Frederick e al. 2002). Decision makers are biased oward immediae graificaion, and hey prefer o enjoy rewards now and o pospone coss. Wriing a review, for example, involves immediae coss and possible fuure 1

2 2 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences benefis; in conras, smoking involves immediae rewards and delayed consequences. Hence he reviewer is discouraged from wriing he review and he smoker is encouraged o indulge. Such preferences are known as presen-biased preferences (O Donoghue and Rabin 1999, 2001). Our goal in his paper is o sudy opimal sopping problems under such preferences when eiher rewards or coss are immediae. Exponenial ime discouning implies ha he agen exhibis ime-consisen behavior; in oher words, he choice beween wo alernaives is independen of when he comparison is made. Ye presen-biased preferences lead o ime inconsisency because preferences may change over ime. For insance, suppose ha spending an hour on some unpleasan ask a monh from now is preferable o spending wo hours on he same ask wo monhs from now. Afer a monh has elapsed, however, mos people pospone he ask insead of doing i as originally planned. Thus, agens exhibi selfconrol problems. The decision maker may be naive or sophisicaed ha is, she may no foresee her fuure self-conrol problems and plan jus as if she were ime consisen, or she may be aware of he bias and plan accordingly. Following he exan lieraure, we model presen-biased preferences by assuming ha he decision maker has quasi-hyperbolic ime preferences; his is ofen referred o as he (β, δ) model (see, e.g., Laibson 1997). In his framework, he one-period discoun facor for he immediae fuure is βδ, where a lower β corresponds o a sronger bias for immediae payoffs. However, he one-period discoun facor for all fuure payoffs is δ. Because naifs are unaware of heir fuure self-conrol problems, when making decisions in he curren period hey misakenly suppose ha heir long-run uiliy is he same as ha of a ime-consisen (TC) agen. We can herefore define heir decisionmaking problem by way of a dynamic programming recursion. Unlike naifs, sophisicaes are fully aware of heir fuure self-conrol problems and can be viewed as a collecion of disinc selves, each making decisions based on her own preferences prevailing a he ime. To idenify he choices made by each self, we solve he muli-player game by backward inducion and find he subgame-perfec Nash equilibrium. In order o esablish he problem s srucural properies, we focus on he marginal payoff, or he difference in payoff beween sopping now and sopping one period laer. Prior work shows ha if he marginal payoff is monoone in sae, hen a TC agen follows a hreshold sopping policy (see, e.g., Oh 2010). We find ha similar condiions are sufficien for naifs o follow a hreshold sopping policy under boh immediae coss and immediae rewards. For sophisicaes, however, hese condiions guaranee a hreshold srucure only when rewards are immediae. When coss are immediae, sophisicaes migh no generally follow a hreshold policy because hey migh preempively sop o avoid fuure self-conrol problems.

3 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences 3 We compare he ses of saes under which differen decision makers sop. When rewards are immediae, he se of sopping saes for TC agens (TCs, hereafer) is a subse of ha for naifs, which iself is a subse of ha for sophisicaes. This means ha sophisicaes are more likely o sop and collec he immediae rewards han are naive or ime-consisen agens. The reason is ha naifs, because hey are unaware of heir fuure self-conrol problems, value he opion o coninue more han do sophisicaes. In addiion, if we assume ha ime-consisen behavior is ideal, hen our resuls show ha sophisicaion induces premaure sopping decisions and hence does no benefi he agen. When coss are immediae, he se of sopping saes for naifs is a subse of ha for imeconsisen agens and a subse of ha for sophisicaes. In oher words, naifs are he leas likely o sop (he mos likely o pospone). However, he comparison beween ime-consisen agens and sophisicaes is in general indeerminae owing o he opposing effecs of presen-biased preferences and sophisicaion. In his case, we provide sufficien condiions for he presen-bias effec o dominae he sophisicaion effec; under hese condiions, sophisicaes value he opion o coninue more han do TCs and hence are less likely han he laer o sop. Thus, sophisicaion miigaes procrasinaion and benefis he agen. Our resuls are general in he sense ha hey do no depend on wheher he sopping decision follows a hreshold srucure. Our work conribues o he operaions managemen and also he economics lieraure. The operaions managemen lieraure on presen-biased preferences is sparse. Su (2009) develops a model of consumer ineria in which consumers delay purchases even when i is opimal o purchase immediaely. He argues ha such inerial behavior is consisen wih hyperbolic ime preferences of naive agens in a purchase decision seup ha is characerized as an immediae cos problem (he paymen precedes he consumpion). Plambeck and Wang (2013) sudy he pricing and scheduling of a service wih immediae coss when cusomers have quasi-hyperbolic preferences. They show ha charging for subscripion is opimal for he service provider, especially when cusomers are naive. In a projec managemen conex characerized by cosly immediae effors, Wu e al. (forhcoming) sudy he opimal conrac design and eam composiion for achieving projec goals when he workers have hyperbolic ime preferences. Gao e al. (2014) sudy he dynamic pricing problem of a monopolis selling o sraegic consumers wih quasi-hyperbolic preferences and show ha, from he seller s perspecive, a policy of cream skimming (reducing prices) is generally opimal. We conribue o his lieraure by sudying he implicaions of quasi-hyperbolic ime preferences in a general opimal sopping problem ha may involve immediae coss or immediae rewards. Presen-biased preferences have been sudied exensively in economics under various seups; see Frederick e al. (2002) for a review. In he paricular area of sopping problems, O Donoghue and Rabin (1999) are closes o our research. In heir model, here is no uncerainy and he payoffs

4 4 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences are no dependen on saes. Hence, srucural properies are irrelevan, and he comparaive resuls focus on he ime a which differen agens sop no on he sopping regions ha we are ineresed in. There are alernaive mehods in economics for modeling self-conrol problems and heir consequences in special cases of opimal sopping ime problems. Miao (2008) uses he empaion proneness model of Gul and Pesendorfer (2001) o show ha empaion can resul in eiher procrasinaion or preproperaion (excessive hase) in an opimal opion exercise problem. Fudenberg and Levine (2006) obain similar resuls wih a dual-self model in which he decisionmaking process is modeled as a game beween a shor-run impulsive self and a long-run paien self. We apply he quasi-hyperbolic discouning framework originally developed by Phelps and Pollak (1968) in a more general sopping problem. The res of he paper is organized as follows. Secion 2 oulines he basic componens of our model for differen agens. In Secion 3, we esablish he srucural properies of he opimal sopping problem under presen-biased preferences. Secion 4 compares he sopping behavior of differen ypes of agens. In Secion 5, we show how our model applies in operaional seings characerized by eiher immediae rewards or immediae coss. Secion 6 concludes. 2. Model We assume ha ime is discree and use s o denoe as he sae in period. In each period 1, an agen mus decide wheher o sop (decision 1) or o coninue (decision 0). If she sops, he process ends and she receives a payoff r 1 (s ) immediaely as well as a payoff r f (s ) in he fuure. If she coninues, she receives a payoff r 0 (s ) immediaely and he sae will become S 1 (s ) in he nex period 1. Here S 1 (s ) is a random variable dependen on s. We assume ha if he agen does no sop in all periods 1 hen she mus sop a period 0. All payoffs can be eiher posiive or negaive. In his paper, we adop he following simple framework o model presen-biased preferences; i was developed by Phelps and Pollak (1968) and has since been exensively used in he lieraure. Definiion 1. Le c i be an agen s insananeous uiliy in period i. She has (β, δ)-preferences if her ineremporal uiliy in period can be represened by 1 U (c, c 1,..., c 1 ) = c + β δ i c i, where 0 < β, δ 1. Wihou loss of generaliy, we le δ = 1 hroughou he paper. When β = 1, he preferences are ime consisen. Le v (s ) denoe he maximum oal expeced payoff from period o he end of he horizon when he sae is s. For TCs, he dynamic programming recursion can be wrien as v (s ) = max{r 1 (s ) + r f (s ), r 0 (s ) + Ev 1 (S 1 (s ))}, i=1

5 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences 5 where v 0 (s 0 ) = r0(s 1 0 ) + r f 0 (s 0 ). Agens may have differen beliefs abou heir presen-biased preferences. A naive agen is unaware ha she is ime inconsisen; she believes ha her fuure behavior is he same as ha of a imeconsisen agen. Therefore, naifs will sop if and only if r 1 (s ) + βr f (s ) > r 0 (s ) + βev 1 (S 1 (s )). In conras, a sophisicaed agen can foresee her self-conrol problems and correcly anicipaes her fuure behavior. We le u 1 (s 1 ) denoe a sophisicaed agen s expeced payoff from period 1 o he end of he horizon. Then a sophisicae will sop if and only if r 1 (s ) + βr f (s ) > r 0 (s ) + βeu 1 (S 1 (s )). Here u 1 (s 1 ) is he perceived value funcion from period- self s perspecive, in which all fuure payoffs are weighed equally. This funcion reflecs he long-run uiliy of sophisicaes (O Donoghue and Rabin 1999). In general, u (s ) is no equal o he maximum of he wo erms r 1 (s ) + βr f (s ) and r 0 (s )+βeu 1 (S 1 (s )). Insead, a sophisicaed agen compares hese wo erms o deermine her opimal acion a period and hen based on he opimal acion updaes u (s ) according o he following recursive relaionship: u (s ) = We also assume ha u 0 (s 0 ) = r 1 0(s 0 ) + r f 0 (s 0 ). { r 0 (s ) + Eu 1 (S 1 (s )) if i is opimal o coninue a s ; r 1 (s ) + r f (s ) if i is opimal o sop a s. We consider wo cases. If r 1 (s ) r 0 (s ) hen here is an immediae benefi from sopping relaive o coninuing; hus he agen receives an immediae reward from sopping. In conras, if r 1 (s ) < r 0 (s ) hen here is an immediae cos from sopping relaive o coninuing; so in his case, he agen incurs an immediae cos from sopping. The disincion is immaerial for TCs bu is criical for naifs and sophisicaes. 3. Threshold Sopping Srucure A quesion ofen asked in sopping problems is wheher and under wha condiions a hreshold policy is opimal. Under a hreshold policy, sopping is he opimal acion if and only if he sae is greaer (or smaller) han a hreshold. Threshold srucure reveals complemenariy (or subsiuabiliy) beween saes and acions; i also enables easy compuaion and implemenaion of he opimal policy.

6 6 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences 3.1. Time-consisen Agens The condiions under which a hreshold policy is opimal for TCs are well esablished in he lieraure (see, e.g., Oh 2010). We summarize he resuls here. Firs we define he marginal payoff of coninuing for one more period: M T C (s ) = r 0 (s ) + E[r 1 1(S 1 (s )) + r f 1(S 1 (s ))] r 1 (s ) r f (s ). The marginal payoff represens he difference in payoff beween (a) coninuing for one more period and hen sopping and (b) sopping immediaely in he curren period. If he marginal payoff is decreasing (resp., increasing) in sae and he sae ransiion is sochasically increasing, hen i is opimal o sop if and only if he sae is above (resp., below) a cerain hreshold. The concep of sochasic increasing is defined nex. Definiion 2. A se of random variables {X(θ) θ R} is sochasically increasing in θ if Eg(X(θ)) is increasing in θ for all increasing funcions g. Throughou we shall make he following assumpion on sae ransiions. Assumpion 1. The sae ransiion S 1 (s ) is sochasically increasing in s. According o his assumpion, a larger sae s in he curren period leads o a sochasically larger sae S 1 in he nex period. For insance, Assumpion 1 is me when he sae ransiion akes he following muliplicaive and addiive form: S 1 (s ) = D,1 s + D,2, where D,1 and D,2 are random variables and D,1 is posiive. The implicaions of his assumpion are discussed (wihin differen problem conexs) in Secion 5. The following resul on hreshold policies is adoped from Oh (2010). Lemma 1. The following saemens hold in boh he immediae reward and he immediae cos cases: (i) if M T C (s ) is decreasing in s, hen here exiss a hreshold s T C such ha TCs sop if and only if s s T C ; (ii) if M T C (s ) is increasing in s, hen here exiss a hreshold s T C only if s s T C Naive Agens such ha TCs sop if and Much as we did for ime-consisen agens, for naifs we define he marginal payoff of coninuing for one more period: M n (s ) = r 0 (s ) + βe[r 1 1(S 1 (s )) + r f 1(S 1 (s ))] r 1 (s ) βr f (s ). This definiion reflecs he presen-bias effec. Unlike he ime-consisen case, here he monooniciy of M n is no sufficien o guaranee a hreshold sopping rule. A naive agen s period- self

7 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences 7 regards M n (s ) as he marginal payoff of coninuing for one more period and hen sopping a period 1. However, she is unaware of her fuure self-conrol problems and believes ha her fuure selves would behave like TCs. She will herefore likewise view Mi T C (s i ) as her marginal payoffs of coninuaion in fuure periods (i 1). A hreshold policy is no guaraneed unless boh M T C and M n are monoone. The resuls are presened formally in he following heorem. Theorem 1. The following saemens hold in boh he immediae reward and he immediae cos cases: (i) if boh M T C (s ) and M n (s ) are decreasing in s, hen here exiss a hreshold s n such ha naifs sop if and only if s s n ; (ii) if boh M T C (s ) and M n (s ) are increasing in s, hen here exiss a hreshold s n such ha naifs sop if and only if s s n ; Under some condiions, he monooniciy of M n (s ) implies he monooniciy of M T C (s ); under oher condiions, he opposie is rue. The relevan condiions are given in he following lemma. In applicaions, if hese condiions are saisfied hen we need only check he monooniciy of eiher M T C (s ) or M n (s ); we do no need o check boh. Lemma 2. (i) Suppose r 1 (s ) r 0 (s ) is decreasing in s. Then M T C (s ) is decreasing in s if M n (s ) is decreasing in s. (ii) Suppose r 1 (s ) r 0 (s ) is increasing in s. Then M n (s ) is decreasing in s decreasing in s Sophisicaed Agens if M T C (s ) is Unlike naifs, sophisicaes correcly anicipae heir fuure behavior. Alhough M n (s ) capures he presen-bias effec of he period- self, i does no capure fuure selves presen-bias effec, of which he period- self is fully aware and accouns for when pondering wheher o sop or coninue. As a resul, he sophisicae s sopping behavior may no follow a hreshold policy even when boh M n (s ) and M T C (s ) are monoone. Consider he following example wih a wo-period planning horizon and immediae coss. There are wo possible saes in each period: s {0, 1}. Assume he sae remains unchanged over ime; ha is, le S 1 (s ) = s. Then we clearly have a sochasically increasing sae ransiion. Le β = 0.5, and suppose r 0 (s ) = r 0 0(s 0 ) = r 1 0(s 0 ) = r f 0 (s 0 ) = 0. The reward and cos schedules are r 1 2(s 2 ) = 2, r f 2 (s 2 ) = 6 + 4s 2 ; r 1 1(s 1 ) = 5, r f 1 (s 1 ) = 9 + 3s 1.

8 8 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences By definiion of M n (s ), i is easy o see ha M n 1 (0) = 0.5, M n 1 (1) = 1; M n 2 (0) = 1, M n 2 (1) = 0.5. Thus M n (s ) is decreasing in s and M T C (s ) is also decreasing in s by Lemma 2. The acions aken by differen agens and he payoffs from hose acions can be compued using backward inducion. Table 1 summarizes he payoffs from differen acions for all ypes of agens. Period 2 Period 1 Sae 0 Sae 1 Sae 0 Sae 1 Sop Coninue Sop Coninue Sop Coninue Sop Coninue TCs Naifs Sophisicaes Table 1 Payoffs of differen acions for differen ypes of agens Observe from Table 1 ha TCs always sop irrespecive of he saes. Furher, naifs only sop a sae 1 in period 1. Boh are hreshold policies in line wih Lemma 1 and Theorem 1. However, sophisicaes sop a sae 0 bu no a sae 1 in period 2 ye sop a sae 1 bu no a sae 0 in period 1. Such behavior is obviously no reflecive of a hreshold policy. When he sae is 1, for sophisicaes he period-2 self s discouned payoff is /2 = 3 when sopping. If he sophisicae coninues, she correcly anicipaes ha she will end up sopping in period 1 and so receive a discouned payoff of ( )/2 = 3.5, which is greaer han 3. In his case, he period-2 self coninues. When he sae is 0, if he period-2 self sops hen her discouned payoff is 2 + 6/2 = 1. If he sophisicae coninues, she also correcly anicipaes ha her period-1 self will likewise coninue and hence will end up wih a discouned payoff of 0. So in his case, he period-2 self chooses o sop. Jus like TCs, sophisicaes weigh all fuure payoffs equally, which fuure selves disagree. In his example, if he sae is 0 hen, despie he period-2 self s wishes o coninue and le he period-1 self sop in period 1, she knows ha he period-1 self would no sop. The period-2 self is hus beer off by sopping immediaely. The non-hreshold sopping behavior occurs when he coss are immediae. When rewards are immediae, however, he monooniciy of M n (s ) and M T C (s ) guaranees a hreshold policy for sophisicaed agens. In his case, if he presen self plans o coninue for one period and hen sop, he nex-period self would never objec because he immediae reward from sopping is hen even more irresisible. This poin will become clearer in Secion 4, where we compare he behavior of TCs, naifs, and sophisicaes. Resuls concerning hreshold policies are presened in our nex heorem.

9 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences 9 Theorem 2. In he immediae reward case, he following saemens hold: (i) if boh M T C (s ) and M n (s ) are decreasing in s, hen here exiss a hreshold s s such ha sophisicaes sop if and only if s s s ; (ii) if boh M T C (s ) and M n (s ) are increasing in s, hen here exiss a hreshold s s such ha sophisicaes sop if and only if s s s. 4. Comparaive Saics We now compare he sopping behavior of differen ypes of agens. The noaion R i represens he sopping region for a ype-i agen a period for i {T C, n, s}. Tha is, for ype-i agens, sopping is beer han coninuing if and only if s R i. In he following heorem, we compare hese regions and describe how hey change as a funcion of β. In he examples given by O Donoghue and Rabin (1999) in which here is no uncerainy and saes are irrelevan he sopping regions are eiher empy or infinie. Theorem 3. (i) If rewards are immediae, hen R T C R n R s and boh R n and R s are decreasing in β. (ii) If coss are immediae, hen R n R s, R n R T C and R n is increasing in β. By his heorem, for a given sae, if naifs sop hen sophisicaes also sop regardless of wheher i is he coss or he rewards ha are immediae. This is because naive agens, being unaware of heir self-conrol problem, assign a higher value o he opion of coninuing han do sophisicaed agens. When rewards are immediae, naifs are more likely o sop han TCs because hey believe hey will behave like TCs in he fuure bu are emped by immediae rewards now. When β increases, he sopping regions for naifs and sophisicaes are moving closer o ha of TCs. When coss are immediae, naifs are less likely o sop han TCs because naifs weigh more heavily he immediae cos from sopping. When β increases, he sopping region for naifs increases and hence is closer o ha of TCs. Somewha surprisingly, hese resuls are very robus and do no depend on wheher sopping follows a hreshold policy; neiher do hey depend on wheher sopping is opimal when he sae is low or when i is high. Suppose ha a hreshold policy is opimal for TCs, naifs, and sophisicaes and ha heir respecive hresholds are denoed s T C, s n, and s s. In he immediae reward case, for example, i follows from Theorem 3 ha if sopping is opimal if and only if he sae is higher han he hresholds, hen s s s n s T C ; if sopping is opimal if and only if he sae is lower han he hresholds, hen s s s n s T C. Theorem 3 allows us o beer undersand why sophisicaes may follow a non-hreshold policy when coss are immediae. Given ha naifs follow a hreshold policy, any non-hreshold behavior

10 10 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences mus be driven by sophisicaion. In oher words, sophisicaes may sop preempively o avoid fuure self-conrol problems whereas naifs unaware of he self-conrol problem may coninue. Take, for example, he case where boh M n and M T C are decreasing. To simplify he argumens, suppose saes remain unchanged over ime and here are only wo saes: low and high. Suppose furher ha here is a hreshold policy for periods 1, 2,..., 1 (his is rue a leas for period 1). Now assume ha, a he low sae, he presen self would wan he period-( 1) self o sop bu knows ha her period-( 1) self would acually coninue. As a resul, she sops preempively. When we swich o he high sae, he period-( 1) self migh coninue bu migh also sop. In he laer even, he self-conrol problem becomes moo and here is no longer any need for preempive sopping a period. The consequence would be a non-hreshold policy: sopping a he low sae bu no a he high sae. When rewards are immediae, however, self-conrol problems have differen effecs. In he low sae, we suppose similarly ha for he presen self s perspecive he bes ime o sop is a some period < 1. Sophisicaes foresee ha heir period-( 1) selves will no be able o resis he empaion and so will sop a period 1, which for he presen self is no as good as sopping a period. Therefore, he presen self sops preempively. When we change o he high sae, he period-( 1) self will also sop because of he hreshold policy. Because M n is decreasing, if, in he low sae, sopping now is beer han coninuing for exacly one period followed by sopping, hen he same holds in he high sae. The comparison beween R T C and R s is in general indeerminae when coss are immediae. As noed by O Donoghue and Rabin (1999), in heir examples he behavior of sophisicaes can be explained by wo effecs: he presen-bias effec leads o procrasinaion while he sophisicaion effec leads o preproperaion. These wo effecs work in opposie direcions. The sophisicaion effec alleviaes procrasinaion. However, i is possible ha sophisicaes sop even sooner han TCs; ha is, he former migh exercise preempive overconrol. In our seing, he sophisicaion effec makes he sopping region for sophisicaes even larger han ha for TCs. The nex heorem idenifies condiions under which such preempive overconrol would no occur. Theorem 4. When coss are immediae, R s R T C holds for all saes s and periods : (i) r f (s ) Er f 1(S 1 (s )) Er 0 1(S 1 (s )) 0; (ii) r 0 (s ) r 1 (s ) Er 0 1(S 1 (s )) Er 1 1(S 1 (s )); (iii) M T C (s ) 0. is rue if one of he following condiions Under hese sufficien condiions, i pays o be sophisicaed. Tha is, sophisicaion under hese condiions can lead agens who suffer from presen-bias effec o make choices ha are closer o

11 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences 11 hose ha TCs would make which long-run selves would appreciae. Condiion (iii) in Theorem 4 is obvious since TCs always sop in his case. Condiion (ii) means ha he ne payoffs of sopping are decreasing in ime. Condiion (i) is a necessary condiion for r f (s ) u 1 (s ) r 1(s f ) u 2 (s ), which means ha he ne fuure payoffs are decreasing in ime. Under hese condiions, he fuure self-conrol problems are no ha severe, of which sophisicaes are fully aware. Hence he phenomenon of preempive overconrol does no arise. Anoher way o undersand (i) and (ii) is ha, under hese condiions, he long-run uiliy of sophisicaes (u ) is no much lower han ha of TCs (v ). In paricular, we can show under condiion (i) ha u (s ) 1 β [v (s ) (1 β)(r 0 (s ) + r f (s )] and under condiion (ii) ha u (s ) 1 β [βv (s ) (1 β)(r 0 (s ) r 1 (s )]. 5. Examples Here we provide wo examples of sopping problems. In one, rewards are immediae; in he oher, coss are immediae. We discuss he condiions and resuls derived in previous secions in hese specific conexs Projec Managemen Suppose a manager is considering when o sop a produc developmen projec. The sae s measures he produc s performance, and a higher sae corresponds o a beer performance. According o Assumpion 1, he beer he performance in he curren period, he sochasically beer he performance in he nex period. If he manager coninues he process, she incurs a developmen cos r 0 (s ) < 0 in he curren period; if she sops he projec, she incurs no cos in he curren period and so r 1 (s ) = 0. The reward from he projec r f (s ) will come in he fuure. This is an immediae reward case even hough he curren-period payoffs are negaive. Suppose ha r 0 (s ) and Er 1(S f 1 (s )) r f (s ) are boh decreasing in s. The former means ha i is more cosly o improve an already high performance; he laer means ha he marginal benefi of coninuing he projec is lower when he performance is higher. So in his scenario, TCs, naifs and sophisicaes all sop he process if and only if produc performance exceeds cerain hresholds. From he ime-consisen perspecive, Theorem 3 implies ha naifs may be sopping when he performance is no high enough and ha sophisicaes may be sopping when he performance is even less sufficien. Suppose r 0 (s ) and Er 1(S f 1 (s )) r f (s ) are boh increasing in s. This could happen if, for example, r 0 is a fixed cos (i.e., is independen of s ), S 1 (s ) = s + 1, and r f (s ) = b(s a) + for

12 12 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences all. The funcional form of r f means ha if performance is below a hen here is no reward; his kind of payoff has been much discussed in he projec managemen lieraure (e.g., Huchzermeier and Loch 2001; Saniago and Vakili 2005). In his case, TCs, naifs, and sophisicaes all sop he process if and only if he performance is below cerain hresholds; in oher words, hey abandon he projec if progress seems hopelessly slow. Here TCs view naifs and sophisicaes as quiers and naifs hink he same of sophisicaes Underaking Medical Examinaions A person needs o decide wheher and when o ake a medical examinaion o deec for poenial cancer. Examinaions are cosly and generae unpleasan feelings. The sae s represens he individual s healh saus, where a higher sae corresponds o being less healhy. Assumpion 1 means ha he more unhealhy a person is now, he more unhealhy she will be (sochasically) in he fuure. This is a common assumpion in he healhcare lieraure. Under hese circumsances, i is reasonable o assume ha r 0 (s ) = 0 and r 1 (s ) < 0 and ha r 1 (s ) is independen of he sae. The fuure reward r f (s ) represens he healh benefi afer underaking he medical examinaion. This is an immediae cos case. If Er 1(S f 1 (s )) r f (s ) is decreasing in s, hen boh M T C (s ) and M n (s ) are decreasing. This condiion means ha he marginal healh benefi from delaying he examinaion is lower (or ha he marginal cos is higher) when he subjec is more unhealhy. By Lemma 1 and Theorem 1, boh TCs and naifs choose o underake he examinaion if and only if he healh saus is higher (worse) han cerain hresholds. Sophisicaes may or may no use a hreshold sopping rule. According o Theorem 3(ii), here are siuaions where naifs pospone aking he examinaion bu boh TCs and sophisicaes would no pospone if hey were in he same siuaion. For a given healh saus, do sophisicaes ever undergo he examinaion when TCs do no? If delaying medical examinaions brings no healh benefi ha is, if r f (s ) Er f 1(S 1 (s )), which is probably rue in mos cases hen condiion (i) in Theorem 4 is saisfied. And since he cos of aking he examinaion is mos likely fixed and independen of ime, i follows ha condiion (ii) in Theorem 4 is also saisfied. Therefore, sophisicaes would no exercise overconrol. In his conex, sophisicaion can alleviae he negaive effecs creaed by presen-bias effec, which is appreciaed by he long-run selves. These resuls provide an alernaive behavioral perspecive from which we can beer undersand why (besides reasons of regre, worry or perceived risk) individuals migh decline o engage in such prevenive medical inervenions as vaccinaion (Chapman and Coups 2006; Connolly and Reb 2003). If presen-biased preferences are a driver of individual healh decisions hen he naure of remedies o improve hose decisions will be affeced.

13 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences Conclusion In his paper, we sudy opimal sopping problems under general assumpions abou he payoff srucure and sae ransiions when decision makers have quasi-hyperbolic ime preferences. We show ha marginal payoff s monooniciy in sae suffices for a hreshold sopping policy o be opimal for all ypes of agens when he rewards are immediae. However, when coss are immediae, hese condiions ensure he opimaliy of hreshold policy only for ime-consisen agens and naive agens. In his case, sophisicaes migh no follow a hreshold policy because hey are inclined o aking preempive acion ha will preclude fuure self-conrol problems. We also compare he behavior of differen agens and show ha, when rewards are immediae, sophisicaed agens are he mos likely o sop premaurely. When coss are immediae, naive agens are he ones mos likely o procrasinae. Our model applies o a wide array of binary sop coninue decisions in projec managemen, financial decisions, healh-care decisions, and markeing. For example, unpleasan and cosly prevenive medical inervenions such as vaccinaion may be posponed by presen-biased paiens, exacerbaing heir healh saus. Fuure research could focus on he implicaions of our findings in a variey of operaional problems and on heir possible remedies. References Chapman, G. B., E. J. Coups Emoions and prevenive healh behavior: Worry, regre, and influenza vaccinaion. Healh Psychology 25(1) Frederick, S., G. Loewensein, T. O Donoghue Time discouning and ime preference: A criical review. Journal of Economic Lieraure 40(2) Fudenberg, D., D. K. Levine A dual-self model of impulse conrol. American Economic Review 96(5) Gao, X., X. Chen, Y-J. Chen Dynamic pricing wih ime-inconsisen consumers. Working paper. Gul, F., Pesendorfer, W Tempaion and self-conrol. Economerica 69(6) Huchzermeier, A., C. H. Loch Projec managemen under risk: Using he real opions approach o evaluae flexibiliy in R&D. Managemen Science 47(1) Laibson, D Golden eggs and hyperbolic discouning. Quarerly Journal of Economics 112(2) Miao, J Opion exercise wih empaion. Economic Theory 34(3) O Donoghue, T., M. Rabin Doing i now or laer. American Economic Review 89(1) O Donoghue, T., M. Rabin Choice and procrasinaion. Quarerly Journal of Economics 116(1) Oh, S Opimal Sopping Problems in Operaions Managemen. PhD disseraion, Sanford Universiy.

14 14 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences Phelps, E. S., R.A. Pollak On second-bes naional saving and game-equilibrium growh. Rev. Econom. Sud. 35(2) Plambeck, E., Q. Wang Implicaions of hyperbolic discouning for opimal pricing and scheduling of unpleasan services ha generae fuure benefis. Managemen Science 59(8) Ross, S. M Inroducion o sochasic dynamic programming. Academic Press. Saniago, L. P., P. Vakili On he value of flexibiliy in R&D projecs. Managemen Science 51(8) Su, X Model of consumer ineria wih applicaions o dynamic pricing. Producion and Operaions Managemen 18(4) Wu, Y., K. Ramachandran, V. Krishnan. Managing cos salience and procrasinaion in projecs: Compensaion and eam composiion. Producion and Operaions Managemen, forhcoming. Appendix. Proofs We firs inroduce he following noaion for (respecively) TCs, naifs, and sophisicaes. B T C (s ) = r 0 (s ) + Ev 1 (S 1 (s )) r 1 (s ) r f (s ), B n (s ) = r 0 (s ) + βev 1 (S 1 (s )) r 1 (s ) βr f (s ), B s (s ) = r 0 (s ) + βeu 1 (S 1 (s )) r 1 (s ) βr f (s ). Recall from Secion 2 ha a ype i agen sops a sae s in period if and only if B i (s ) 0. Proof of Lemma 1. We prove only par (i) of he lemma; par (ii) can be proved similarly. Firs, we can show ha he following relaionship beween B T C B T C (s ) = r 0 (s ) + Ev 1 (S 1 (s )) r 1 (s ) r f (s ) (s ) and M T C (s ) holds: = M T C (s ) + E(v 1 (S 1 (s )) r 1 1(S 1 (s )) r f 1(S 1 (s ))) = M T C (s ) + E max{0, r 0 1(S 1 (s )) + Ev 2 (S 2 (S 1 (s ))) r 1 1(S 1 (s )) r f 1(S 1 (s ))} = M T C (s ) + E max{0, B 1(S T C 1 (s ))}. We shall idenify he hreshold policy by using inducion o show ha B T C (s ) is decreasing in s. By assumpion, B T C 1 (s 1 ) = M T C 1 (s 1 ) is decreasing in s 1. Suppose B T C 1(s 1 ) is decreasing in s 1 for some 1 1, hen he funcion max{0, B T C 1(s 1 )} is decreasing s 1. Since he sae variable S 1 (s ) is sochasically increasing in s, i follows ha he funcion Emax{0, B T C 1(S 1 (s ))} is decreasing in s. By assumpion, M T C (s ) is decreasing and so B T C (s ) is decreasing in s, which complees he inducion. Proof of Theorem 1. The proof is similar o he one for Lemma 1. Noe ha we have he following relaionship beween B n (s ) and M n (s ): B n (s ) = M n (s ) + βemax{0, B T C 1(S 1 (s ))}.

15 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences 15 Here, he monooniciy of M T C (s ) is needed o ensure he monooniciy of B T C (s ). Then, given our addiional assumpion on he monooniciy of M n (s ), we conclude ha hreshold sopping rules are opimal for naifs. Proof of Lemma 2. The resuls are an immediae consequence of he following wo equaliies: M T C (s ) = 1 β M n (s ) + ( 1 β 1)(r1 (s ) r 0 (s )); and M n (s ) = βm T C (s ) (1 β)(r 1 (s ) r 0 (s )). Proof of Theorem 2. We prove only par (i), since par (ii) can be shown similarly. For sophisicaes, B s (s ) = r 0 (s ) + βeu 1 (S 1 (s )) r 1 (s ) βr f (s ) = M n (s ) + βe[u 1 (S 1 (s )) r 1 1(S 1 (s )) r f 1(S 1 (s ))]. We use inducion on u (s ) r 1 (s ) r f (s ) o show ha B s (s ) is decreasing. I is obvious ha u 0 (s 0 ) r 1 0(s 0 ) r f 0(s 0 ) = 0 is decreasing. Suppose ha u 1 (s 1 ) r 1 1(s 1 ) r f 1(s 1 ) is decreasing for some 1 0. Then B s (s ) is decreasing, which resuls in a hreshold policy. To complee he inducion, we mus show ha u (s ) r 1 (s ) r f (s ) is also a decreasing funcion. From he discussion in Secion 2, we have { r 0 (s u (s ) = ) + Eu 1 (S 1 (s )) if s < s s ; r 1 (s ) + r f (s ) if s s s. Hence u (s ) r 1 (s ) r f (s ) = { r 0 (s ) + Eu 1 (S 1 (s )) r 1 (s ) r f (s ) if s < s s ; 0 if s s s. We know ha r 0 (s ) + Eu 1 (S 1 (s )) r 1 (s ) r f (s ) is decreasing because r 0 (s ) + Eu 1 (S 1 (s )) r 1 (s ) r f (s ) = M T C (s ) + E[u 1 (S 1 (s )) r 1 1(S 1 (s )) r f 1(S 1 (s ))] is decreasing. In addiion, we know ha r 0 (s ) + Eu 1 (S 1 (s )) r 1 (s ) r f (s ) r 0 (s ) r 1 (s ) + 1 β (r1 (s ) r 0 (s )) 0 for s s s. Therefore, u (s ) r 1 (s ) r f (s ) is decreasing in s. Proof of Theorem 3. (i) We know ha u (s ) v (s ) for any sae s because v (s ) is he maximum oal discouned reward under a consisen discoun rae. Therefore, B s (s ) B n (s ) and we have R n R s. (ii) In wha follows, we rewrie B i (s ) as B i (s, β) in order o sress is dependence on β. For sophisicaes, a key componen in defining B s (s, β) is he funcion u (s ). We also rewrie u (s ) as u (s, β) because of is dependence on β. Similarly, we also rewrie R i as R i (β) for i = {n, s}. Suppose β 1 β 2, hen we need o show ha R i (β 2 ) R i (β 1 ) for i = n, s. We firs look a he behavior of naifs. If s R n (β 2 ), hen B n (s, β 2 ) 0 and B n (s, β 1 ) = r 0 (s ) + β 1 Ev 1 (S 1 (s )) r 1 (s ) β 1 r f (s ) = β 1 B n (s, β 2 ) + (r 0 (s ) r 1 (s ))(1 β 1 ) β 2 β 2 0.

16 16 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences Therefore, s R n (β 1 ) and he proof for naifs is complee. To prove he resul for sophisicaes, we mus perform an exra inducion on he funcion u (s, β). More specifically, we shall prove ha u (s, β) is increasing in β. This is rue when = 0 because u 0 (s 0, β) = r 1 0(s 0 ) + r f 0(s 0 ) is independen of β. Suppose u 1 (s 1, β) is increasing in β. When s R s (β 2 ), we have B s (s, β 2 ) 0 and B s (s, β 1 ) = r 0 (s ) + β 1 Eu 1 (S 1 (s ), β 1 ) r 1 (s ) β 1 r f (s ) r 0 (s ) + β 1 Eu 1 (S 1 (s ), β 2 ) r 1 (s ) β 1 r f (s ) = β 1 B s (s, β 2 ) + (r 0 (s ) r 1 (s ))(1 β 1 ) β 2 β 2 0. Therefore, s R s (β 1 ) and we have R s (β 2 ) R s (β 1 ). To complee he inducion, we sill need o show ha u (s, β 1 ) u (s, β 2 ). For s R s (β 2 ), he resul holds since u (s, β 1 ) = u (s, β 2 ) = r 1 (s ) + r f (s ). For s R s (β 1 ) bu s / R s (β 2 ), The inequaliy holds because u (s, β 2 ) = r 0 (s ) + Eu 1 (S 1 (s ), β 2 ) r 1 (s ) + r f (s ) = u (s, β 1 ). r 0 (s ) + Eu 1 (S 1 (s ), β 2 ) r 1 (s ) r f (s ) r 0 (s ) r 1 (s ) + 1 β 2 (r 1 (s ) r 0 (s )) 0 for s / R s (β 2 ). Finally, for s / R s (β 1 ) we also have This complees he inducion. u (s, β 2 ) = r 0 (s ) + Eu 1 (S 1 (s ), β 2 ) r 0 (s ) + Eu 1 (S 1 (s ), β 1 ) = u (s, β 1 ). (iii) Suppose β 1 β 2, hen we need o show ha R n (β 1 ) R n (β 2 ). When s R n (β 1 ), we have B n (s, β 1 ) 0 and Therefore s R n (β 2 ), compleing he proof. Proof of Theorem 4. (i) For sophisicaes, B n (s, β 2 ) = r 0 (s ) + β 2 Ev 1 (S 1 (s )) r 1 (s ) β 2 r f (s ) = β 2 B n (s, β 1 ) + (r 0 (s ) r 1 (s ))(1 β 2 ) β 1 β 1 0. B s (s ) = βeu 1 (S 1 (s )) + r 0 (s ) r 1 (s ) βr f (s ) = B T C (s ) + E[βu 1 (S 1 (s )) v 1 (S 1 (s )) + (1 β)(r 0 1(S 1 (s )) + r f 1(S 1 (s )))] +(1 β)[r f (s ) Er f 1(S 1 (s )) Er 0 1(S 1 (s ))].

17 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences 17 The resul will follow from our demonsraion ha if B T C (s ) 0 hen B s (s ) 0. For his i suffices o show ha βu (s ) v (s ) + (1 β)(r 0 (s ) + r f (s )) 0 for any period and sae s. The proof is by inducion. When = 0, we have βu (s ) v (s ) + (1 β)(r 0 (s ) + r f (s )) = (1 β)(r 0 (s ) r 1 (s )) 0. Suppose βu 1 (s 1 ) v 1 (s 1 ) + (1 β)(r 1(s 0 1 ) + r 1(s f 1 )) 0, hen R s R T C. We consider hree cases as follows. Case 1: s R s. Then boh TCs and sophisicaes sop. We have βu (s ) v (s ) + (1 β)(r 0 (s ) + r f (s )) = (1 β)(r 0 (s ) r 1 (s )) > 0. Case 2: s R T C bu s / R s. Then TCs sop bu sophisicaes coninue. We have βu (s ) v (s ) + (1 β)(r 0 (s ) + r f (s )) = β[eu 1 (S 1 (s )) r f (s )] + r 0 (s ) r 1 (s ) = B s (s ) 0. Case 3: s / R T C. Then boh TCs and sophisicaes coninue. We have βu (s ) v (s ) + (1 β)(r 0 (s ) + r f (s )) = βeu 1 (S 1 (s )) Ev 1 (S 1 (s )) + (1 β)r f (s ) (1 β)[r f (s ) Er f 1(S 1 (s )) Er 0 1(S 1 (s ))] The firs inequaliy holds because of he inducion hypohesis. Hence for any sae s, we have which complees he inducion. (ii) We can decompose B s (s ) as 0. βu (s ) v (s ) + (1 β)(r 0 (s ) + r f (s )) 0, B s (s ) = βeu 1 (S 1 (s )) + r 0 (s ) r 1 (s ) βr f (s ) = βb T C (s ) + E[βu 1 (S 1 (s )) βv 1 (S 1 (s )) + (1 β)(r 0 1(S 1 (s )) r 1 1(S 1 (s )))] +(1 β)[r 0 (s ) r 1 (s ) (Er 0 1(S 1 (s )) Er 1 1(S 1 (s )))]. The resul follows from showing ha if B T C (s ) 0 hen B s (s ) 0. I suffices o show ha, for any period and sae s, we have βu (s ) βv (s ) + (1 β)(r 0 (s ) r 1 (s )) 0. The proof is by inducion. When = 0, we have βu (s ) βv (s ) + (1 β)(r 0 (s ) r 1 (s )) = (1 β)(r 0 (s ) r 1 (s )) 0. Suppose βu 1 (s 1 ) βv 1 (s 1 ) + (1 β)(r 1(s 0 1 ) r 1(s 1 1 )) 0, hen R s R T C. Again we consider hree cases.

18 18 Li, Nasiry, and Yu: Opimal Sopping under Presen-Biased Preferences Case 1: s R s. Then boh TCs and sophisicaes sop. We have βu (s ) βv (s ) + (1 β)(r 0 (s ) r 1 (s )) = (1 β)(r 0 (s ) r 1 (s )) > 0. Case 2: s R T C bu s / R s. Then TCs sop bu sophisicaes coninue. We have βu (s ) βv (s ) + (1 β)(r 0 (s ) r 1 (s )) = β[eu 1 (S 1 (s )) r f (s )] + r 0 (s ) r 1 (s ) = B s (s ) 0. Case 3: s / R T C. Then boh TCs and sophisicaes coninue. We have β[u (s ) v (s )] + (1 β)[r 0 (s ) r 1 (s )] = βe[u 1 (S 1 (s )) v 1 (S 1 (s ))] + (1 β)[r 0 (s ) r 1 (s )] (1 β)[r 0 (s ) r 1 (s ) (Er 0 1(S 1 (s )) Er 1 1(S 1 (s )))] 0. The firs inequaliy holds because of he inducion hypohesis. Thus he inducion is complee. (iii) If M T C (s ) 0 for all periods and saes s, hen i follows from B T C (s ) = M T C (s ) + Emax{0, B 1(S T C 1 (s ))} ha he inequaliy B T C (s ) 0 always holds. Hence TCs always sop, and he resul follows.