Economics, ecologics, and mechanics the dynamics of responding under conditions of varying motivation

1995, 64, 405–431 JO URNAL O F TH E EXPERIMENTAL ANALYSIS O F BEH AVIO R NUMBER ( NO VEMBER) ECONOMICS, ECOLOGICS, AND MECH ANICS: T H E DYNAMICS OF RESPONDING UNDER CONDIT IONS OF VAR YING MOT IVAT ION P ET ER R KILLEEN ARIZO N A STAT E U N IVERSIT Y Th e mech an ics of beh avior developed by Killeen ( 1994) is exten ded to deal with deprivation an d satiation an d with recover y of arousal at th e begin n in g of session s Th e exten ded th eor y is validated again st satiation cur ves an d with in -session ch an ges in respon se rates An omalies, such as ( a) th e positive correlation between magn itude of an in cen tive an d respon se rates in some texts an d a n egative correlation in oth er texts an d ( b) th e greater promin en ce of in cen tive effects wh en magn itude is varied with in th e session rath er th an between session s, are explain ed in terms of th e basic in terplay of drive an d in cen tive motivation Th e models are applied to data from closed econ omies in wh ich ch an ges of satiation levels play a key role in determin in g th e ch an ges in beh avior Relaxation of various assumption s leads to closed-form models for respon se rates an d deman d fun ction s in th ese texts, on es th at sh ow reason able accord with th e data an d rein force argumen ts for un it price as a trollin g variable Th e cen tral role of deprivation level in th is treatmen t distin guish es it from econ omic models It is argued th at tradition al experimen ts sh ould be redesign ed to reveal basic prin ciples, th at ecologic experimen ts sh ould be redesign ed to test th e applicability of th ose prin ciples in more n atural texts, an d th at beh avioral econ omics sh ould sist of th e application s of th ese prin ciples to econ omic texts, n ot th e adoption of econ omic models as altern atives to beh avioral an alysis Key words: econ omics, ecologics, mech an ics, deprivation , satiation , motivation , arousal, deman d fun ction s, drive, in cen tive, models, prin ciples Th is paper compares th ree approach es to th e prediction of beh avior th at is un der th e trol of in cen tives an d supported by motivation al states of var yin g in ten sity Behavioral economics frames beh avior as an exch an ge of goods, an d motivation as th e optimization of th e trade-offs required by th e strain ts of time an d experimen tal text in order to obtain th e best immediate or delay-discoun ted package of goods Ecologics respects th e n atural ecology of th e subject an d rejects th e logic of th e marketplace an d th eoretician for th at of an organ ism adapted by evolution ar y fo r ces to co m p lex n atu r al en vir o n m en ts Ecologics frames beh avior as n ested sets of systems or action pattern s, an d motivation as regulation —th e defen se of setpoin ts with in th ose system states Both of th ese approach es are teleon omic or fun ction al, focusin g on fin al causes, on outcomes: Th e econ omic organ ism beh aves so as to optimize packages of Th is research was supported by NSF Gran ts IBN9408022 an d BNS 9021562 It ben efited greatly from th e reviewers’ commen ts, alth ough it is un likely th ey would en dorse all of th e claims of th is version Address correspon den ce to Peter R Killeen , Departmen t of Psych ology, Box 871104, Arizon a State Un iversity, Tempe, Arizon a 85287-1104 ( E-mail: KILLEEN@ ASU.EDU) goods, an d th e ecologic organ ism beh aves to imize deviation s from optimal setpoin ts in its parameter space Mechanics focuses on th e efficien t rath er th an th e fin al causes of beh avior, an d provides a set of formal causes—a set of math ematical models—th at expan ds simple assertion s of causal agen cy in to more precise fun ction al relation s between variables Th e mech an ical organ ism is n ot beh avin g to o p tim ize an yth in g; in citem en t makes it active, satiation decreases its excitability, an d co-occurren ce of particular respon ses with in cen tives in creases th e probability of th ose respon ses Th e primar y goal of th is paper is to develop th e mech an ics to th e poin t at wh ich it is applicable to th e experimen tal texts th at are favored by econ omic an d ecologic th eorists MECH ANICS A recen t mon ograph ( Killeen , 1994) proposed a mech an ics of beh avior based on th ree prin ciples cern in g th e n ature of arousal, temporal strain t, an d couplin g between respon din g an d in cen tives Th e first prin ciple was th at in cen tives excite respon din g, so th at arousal level ( A) is proportion al 405 406 PETER R KILLEEN to rate of in citemen t ( R; a will be defin ed below) : A ϭ aR ( 1) But th ere are strain ts Th ere is on ly so much time available in wh ich to respon d ( Killeen ’s secon d prin ciple) , an d for a particular target respon se to be differen tially excited by an in cen tive, it must be paired with th at in cen tive; th ey must coreside in th e an imal’s sh ort-term memor y ( th e th ird prin ciple) It is on ly wh en effective tin gen cies couple an in cen tive with a respon se th at th e in cen tive becomes a rein forcer Th ese th ree prin ciples provided th e bases for models of th e beh avior gen erated by various sch edules of rein forcemen t For in stan ce, th e th eor y predicts respon se rates on in ter val sch edules to be Bϭ kR R Ϫ , R ϩ 1/ a ␭ ␭, a Ͼ 0, ( 2) wh ere k is proportion al to th e maximal attain able respon se rate, R is th e rate of rein forcemen t, a is a key parameter wh ose mean in g will be developed below, an d lambda ( ␭) is th e rate of decay of memor y for a respon se Note th at with out th e subtrah en d, th is is essen tially H errn stein ’s h yperbola, wh ich h as been demon strated to predict respon se rate over a wide ran ge of dition s ( see, e.g., de Villiers & H errn stein , 1976) Th e subtrah en d comes in to play on ly at ver y h igh rates of rein forcemen t ( R Ͼ per ute) , wh ere an in creasin g fraction of th e in cen tive bears on th e prior summator y respon se, stren gth en in g it rath er th an th e in strumen tal respon se Because th e subtrah en d is importan t on ly un der ver y h igh rates of rein forcemen t, it will be set to zero for th e rest of th is paper, because th is simplifies an alysis an d in curs on ly a small decrease in goodn ess of fit T he Specific Activation of In cen tives Th e parameter a, wh ich I h ave called th e specific activation , is of greatest cern in th is paper In H errn stein ’s ( 1974) formulation , RO ϭ 1/ a was treated as th e rate of rein fo r cem en t availab le fr o m so u r ces o th er th an th ose sch eduled by th e experimen ter Th is in terpretation h as n ot been supported by subsequen t research ( e.g., Bradsh aw, Szabadi, Ruddle, & Pears, 1983; Dougan & McSween ey, 1985; McSween ey, 1978) Ac- cordin gly, some in vestigators ( e.g., Bradsh aw, Ruddle, & Szabadi, 1981) h ave more agn ostically called th e parameter th e half-life constant, because respon se rate attain s h alf its maximal value wh en R equals RO In earlier work on in cen tive motivation , Killeen , H an so n , an d O sb o r n e ( 1978) sh owed th at each in cen tive delivered un der stan t dition s will gen erate a total of a secon ds of beh avior It follows th at R in cen tives will gen erate th e poten tial for aR secon ds of respon din g, an d th ey called aR th e organ ism’s level of arousal Th e particular form of respon din g gen erated by th at arousal depen ds on th e tin gen cies th at determin e just wh at particular respon se will occur before th e deliver y of th e in cen tive It is th is couplin g of respon ses to in cen tives th at stitutes rein forcemen t Wh en th e couplin g approach es its maximum ( 1.0) , as it does on sh ort ratio sch edules, most of th e beh avior of th e organ ism is cen trated on th e target respon se Wh en th e couplin g is ver y weak, as in sch edules of beh avior-in depen den t rein forcemen t, beh avior is diffuse an d drifts toward adjun ctive forms But in all cases, th e total amoun t of time spen t respon din g is a fun ction of th e arousal level of th e organ ism, wh ich is a product of th e specific activation of th e in cen tives ( a) an d th e rate of th eir deliver y ( R) It is th ese sideration s th at gave rise to Equation We may simplify Equation by droppin g its subtrah en d, an d we may multiply its n umerator an d den omin ator by a to reveal more clearly th e multiplicative in teraction between in cen tive factors summarized by a, an d rate of in citemen t, R: Bϭ kaR aR ϩ ( 3) Equation is h yperbolic in aR because of th e n on lin earities in troduced by ceilin gs on respon se rate Wh en we are operatin g well below th ose ceilin gs, it reduces to th e simple proportion al model, th e first prin ciple of th e mech an ics Wh ereas Equation emph asizes th e relation of th is model to H errn stein ’s h yperbola, Equation remin ds us of th e multiplicative relation between a an d R as th ey join tly determin e arousal level an d respon se rate Terminology It is worth an aside to clarify MECHANICS th e termin ology used th rough out th is paper Th e above equation s were proposed as equilibrium solution s for wh en th e beh avior un der study h as come to a steady state In ph ysics th e study of systems at equilibrium is called statics; an alogously, th e above equation s are part of a statics of beh avior Much of th e recen t research in beh avior an alysis cern s such asymptotic beh avior It derives from a tradition of descriptive beh aviorism; wh en ever a cumulative record is displayed or a regression is fit th rough a scatter of data, th e goal is description Th is is a first step toward a more gen eral scien ce: ‘‘Galileo was cern ed n ot with th e causes of motion but in stead with its description Th e bran ch of mech an ics h e reared is kn own as kinematics; it is a math ematically descriptive accoun t of m o tio n with o u t co n cer n fo r its cau ses’’ ( Frautsch i, O len ick, Apostol, & Goodstein , 1986, p 114) It follows in th e Pyth agorean tradition th at ‘‘approach ed ph en omen a in terms of order an d was satisfied to discover an exact math ematical description ’’ ( Westfall, 1971, p 1) Th ere are man y examples of such a tradition in psych ology today, in cludin g descriptive statistics, th e laws of psych oph ysics, an d th e origin al match in g law Th e study of forces th at cause objects to move is called dynamics; dyn amics stitutes ‘‘a th eor y of th e causes of motion ’’ ( Frautsch i et al., 1986, p 114) Beh avior is th e motion of organ isms, an d th e study of ch an ges in beh avior as a fun ction of motivation , learn in g, an d oth er causal factors stitutes a dyn amics of beh avior Examples in th e beh avioral literature are provided by H iga, Wyn n e, an d Staddon ( 1991) , Staddon ( 1988) , an d Myerson an d Miezin ( 1980) ; Marr ( 1992) provides an over view A framework th at embraces all of th e above special cases is called a mechanics Th is term does n ot n owadays refer to h ypoth etical in tern al mech an ical lin kages; such mach in er y is th e vestige of th e Cartesian tradition in wh ich Newton labored wh en h e began to establish th e modern scien ce of mech an ics Th at mech an ical tradition sough t to provide causal explan ation s of ph en omen a, alth ough such causes were often n arrowly strued as material causes in volvin g th e motion s of particles or aggregation s of matter un derlyin g th e ph en omen a It was on e of Newton ’s ch ief disappoin tmen ts th at h e was n ever able to provide such a ‘‘mech an ical’’ 407 substrate for forces such as gravity, an d h e fin ally repudiated kn owledge of such h ypoth etical causes in h is famous ‘‘h ypoth eses n on fin go,’’ offerin g in stead a precise math ematical description of th e effects of th ose forces H is dyn amical th eor y recon ciled ‘‘th e tradition of math ematical description , represen ted by Galileo, with th e tradition of mech an ical ph ilosoph y, represen ted by Descartes’’ ( Westfall, 1971, p 159) As is th e case in ph ysics, in beh avior an alysis th e term mechanics is someth in g of an atavism; but in both cases, it may be in terpreted as an emph asis on th e an alysis of complex resultan ts in to th eir stituen t forces, as a focus on causal rath er th an statistical explan ation s, an d on math ematical rath er th an mech an ical lin kages between cause an d effect It is in th ose sen ses, on es common to th e beh avior-an alytic tradition , th at it is used h ere It embraces molecular models such as melioration , but n ot teleological models such as th ose predicated upon optimization It in volves th e th eoretical structs of value an d drive Th eoretical structs are as n ecessar y for a scien ce of beh avior as th ey are for an y oth er scien ce ( Williams, 1986) ; th is was recogn ized by Skin n er th rough out h is career, begin n in g with h is argumen t for th e gen eric n ature of th e cepts stimulus an d respon se ( Skin n er, 1935) , th rough h is defen se of drive as a struct th at can make a th eor y of beh avior more parsimon ious overall ( Skin n er, 1938) , to h is fin al writin gs Th e issue, as Skin n er an d oth ers ( Feigl, 1950; Meeh l, 1995) h ave stated, is n ot wh eth er such structs are h ypoth etical, but wh eth er th ey pay th eir way in th e cost-ben efit ratio of structs to prediction s Th is article requires a loan of th e reader’s patien ce as th ese structs are developed an d deployed, in th e h ope th at th e th eor y will in th e en d be judged a worth wh ile tribution to th e experimen tal an d th eoretical an alysis of beh avior Open Versu s Closed Econ omies O n e of th e key dition s th at is assumed to be stan t in Killeen ’s ( 1994) mech an ics, but th at varies substan tially in th e real world, is th e value of th e in cen tive to th e organ ism Th is value depen ds both on th e in trin sic qualities of th e in cen tive—wh at H ull an d h is studen ts den oted by K an d called incentive-motivation—an d th e h un ger, th irst, or ‘‘drive’’ of 408 PETER R KILLEEN th e organ ism, wh ich th ey den oted by D ( e.g., H ull, 1950; Spen ce, 1956) Much of th e early research on th ese factors was an essen tially qualitative an alysis of th e differen tial role th ey played in motivation Th e presen t cern is th e developmen t of a quan titative an alysis, on e th at proceeds by expan din g th e sin gle parameter a ( th e specific activation of an in cen tive) in to compon en ts akin to K an d D H ere th ese structs are developed out of th e already-establish ed statics ( Equation s th rough 3) an d provide th e motivation al ‘‘causes’’ th at tran sform it in to a dyn amics All of th e data an alyzed un der th e origin al formulation of th e mech an ics were derived from an imals at h igh levels of deprivation , wh ich often requires supplemen tar y feedin g in th e h ome cages But beh avioral econ omists h ave argued th at such dition s provide a restricted, perh aps even an omalous, perspective on beh avior, an d th at our an alysis will h ave more ecological validity to th e exten t th at we permit our subjects to earn th eir complete daily ration un der th e strain ts of th e sch edule we study, in th e process often permittin g th em to approach ad libitum repletion by th e en d of th e ( exten ded) daily session Th e tradition al procedure h as been called an open economy because th e subject is main tain ed by food an d water extrin sic to th e sch edule tin gen cies; th e latter arran gemen t h as been called a closed economy Collier, Joh n son , H ill, an d Kaufman ( 1986) ch risten ed th e tradition al open -econ omy procedure th e refinement paradigm, ‘‘developed in classic ph ysics, first en un ciated for an imals by Th orn dike ( 1911, pp 25–29) an d per fected by Skin n er ( 1938) , H ull ( 1943) , th eir studen ts, an d th eir temporaries’’ ( Collier et al., p 113) Because postsession feedin g is on e of th e least importan t distin ction s between open an d closed econ omies, because description of th e procedure as an econ omy stitutes a commitmen t to a particular explan ator y framework, an d because th e refin emen t paradigm is th e ideal text in wh ich to refin e basic prin ciples, th eir term is utilized th rough out th is paper A n umber of research ers h ave adopted th e econ omic an alysis of sch edule effects, with th eir design s often in volvin g n ovel sch edules of rein forcemen t H ursh ( e.g., H ursh , 1984) h as sh own th at th e ver y type of fun ction s an alyzed by Killeen ( 1994) look quite differen t Fig A revision of th e figure drawn by H ursh ( 1980) , sh owin g th e differen ces in pattern s of respon se rates of mon keys un der open an d closed econ omies, as a fun ction of th e in terrein forcemen t in ter val on variablein ter val sch edules Th e cur ves are drawn by Equation 8Ј See H ursh ( 1978) for procedural details an d origin al data u n d er a clo sed eco n o m y Fo r in stan ce H ursh ’s ( 1980) Figure sh owed respon se rate decreasin g sligh tly as th e sch eduled rate of rein forcemen t decreased in an open econ omy, just as we would expect from Equation s an d 3, but increasing markedly in a closed econ omy Figure sh ows th ose data ( derived from H ursh , 1978) Th is stitutes a serious th reat to beh avioral mech an ics an d to all oth er th eories th at en tail th e H errn stein h yperbola H ursh argued th at ‘‘It is th e econ omic system wh ich produced th e differen t results’’ ( 1980, p 223) But just wh at was it about th e differen t systems th at made th e differen ce? H ursh ’s explan ation is in terms of elasticity of demand ‘‘In th e closed econ omy with n o substitutable food outside th e session , deman d was inelastic; in th e open econ omy with stan t food in take arran ged by th e experimen ter, deman d was elastic’’ ( H ursh , 1980, p 233) Elastic goods are th ose such as luxuries for wh ich in creases in price causes decreases in willin gn ess to work for th em or in th e amoun t th at will be paid for th em ( demand) ; in elastic goods are th ose such as basic n eeds for wh ich moderate in creases in cost h ave little margin al effect on deman d; customers will pay wh at th ey h ave to to main tain sumption ( Kooros, 1965; Lea, 1978) Elasticity is measured as th e proportion al ch an ge in de- MECHANICS m an d th at r esu lts fr o m a p r o p o r tio n al ch an ge in price For th e closed econ omy, as th e rein forcemen t rate decreases ( movin g to th e righ t on th e x axis of Figure 1) , price in creases ( an imals get less food per respon se) an d th ere is a comitan t in crease in respon se rates Th e flat fun ction s for th e open econ omy suggest an elasticity n ear un ity, as sh ould be th e case: If you can get it after th e session for free, you sh ouldn ’t work h arder for it wh en prices go up ( Th e proper x axis for th e econ omic an alysis is un it price—respon ses per un it of rein forcer—wh ich is h igh ly correlated with mean time between rein fo r cer s at m o st r esp o n se r ates At lo w respon se rates on in ter val sch edules, h owever, price is positively correlated with respon se rate Strictly speakin g, th is latter depen den cy makes econ omic an alyses in appropriate for in ter val sch edules, because ‘‘In order to deduce th e sh ape of th e deman d for a sumer good, th e first assumption on e sh ould make is [ th at] n o in dividual buyer h as an y appreciable in fluen ce on th e market price; n amely, th e price is fixed’’ Kooros, 1965, pp 51–52.) Beh avioral econ omics provides an in terestin g perspective in a field in wh ich th e data are rich an d complicated an d th e poten tial for bridgin g to an oth er disciplin e is so clear But is it th e righ t perspective? Does respon din g stitute a cost—do an imals meter key pecks th e way h uman s pen n ies? Do th ey an ticipate en d-of-session feedin gs? Just wh y sh ould th e rates un der th e closed econ omy gen erally be lower th an th ose un der th e open econ omy, if in th e latter case an imals can ban k on a postsession feedin g? Wh y sh ould rates fall to n ear zero for th e variable-in ter val ( VI) 20-s sch edule in th e closed econ omy in trast with th e open econ omy? H ow are th ese effects predicted from econ omic th eor y? Elasticity migh t describe, but can n ot explain , th ese differen ces; n or h ave econ omists explain ed wh y elasticity itself sh ould var y tin uously with price, as is usually th e case for beh avioral data A simpler h ypoth esis can explain th e differen ces in th e data un der th ese two experimen tal paradigms: In th e closed econ omy th e subjects are closer to satiation more of th e time, especially at small VI values; subjects from th e open econ omy, bein g h un grier, respon d at a h igh er rate To formalize th is treatmen t requires an expan sion of th e 409 mech an ics to h an dle deprivation an d in cen tive motivation H U NGER Wh ere does deprivation level en ter th e basic prin ciples of rein forcemen t? Th e primar y effect will be on th e specific activation associated with an in cen tive: Th e value of a in Equation will decrease with satiation Th e level of in citemen t th at a small ban an a pellet will provide to a satiated mon key will be less th an th at provided to a h un gr y on e.1 Th e closer an an imal is to its n atural rate of in take un der ad libitum feedin g, th e smaller a sh ould be Similarly, th e in citemen t from a small ban an a pellet will be less th an th at from a large ban an a pellet Th erefore, th e parameter a must be expan ded from a sin gle free parameter to a product of th e organ ism’s h un ger an d th e value of th e in cen tive in alleviatin g h un ger To be crete, let us th in k of th e h un ger drive in th e simplest terms: Con sider th e metabolic system to be a vessel th at stores a fin ite amoun t of food an d utilizes it at a stan t metabolic rate M Th e text permits th e organ ism to acquire n ew food of average magn itude m at th e rate of R ( see th e Appen dix for a review of th e stan ts an d th eir dimen sion s) Depen din g on th e recen t h istor y of depletion an d repletion , th ere will be more or less food in store To be precise, we would n eed to deal with a cascade of storage devices ( i.e., th e mouth , th e stomach , th e bloodstream, th e adipose tissue) , each with th eir own release rates; differen t types of food will affect th ese differen tly Bulky food may fill th e mouth an d stomach but little to alleviate deep h un ger, wh ereas sugars may immediately release stored glucose in to th e bloodstream wh ile leavin g th e stomach relatively empty We will n ot fron t th ose details h ere: Th in k in terms of th e stomach ( or crop) an d some stan dard food such as th ose typically used as rein forcers In th is simplest in stan tiation , th e d eficit is th e em p tin ess o f th e sto m ach Secon dar y motivation al effects on all th e parameters are likely For in stan ce, a weakly motivated organ ism migh t take lon ger to complete a respon se, lowerin g th e ceilin gs on respon se rate ( see, e.g., McDowell & Wood, 1984, an d Equation 3Ј below) But th is paper focuses on th e primar y motivation al effects, wh ose locus of action is on th e parameter a 410 PETER R KILLEEN Ch an ges in th e deficit will depen d on th e balan ce between th e rates of emptyin g th e stomach ( depletion ) an d of fillin g it ( repletion ) over time In th e case in wh ich both th e in put rate ( mR) an d th e output rate ( M) are stan t over th e in ter val t, th e deficit at time t, dt, is dt ϭ d ϩ ( M Ϫ mR) t, ( 4) wh ere d0 den otes th e in itial deprivation level Bou n dary Con dition s It is worth a crete discussion h ere of two of th e variables ( d0 an d M) in Equation 4, because th ey recur th rough out th e paper an d will often be set to fixed values In an open econ omy, th e experimen ter migh t deprive th e organ ism for several days, but n o matter h ow deprived, an imals can eat on ly un til th eir stomach s are full In th ese cases th e in itial deficit d0 takes th e value of th e maximum capacity of th e stomach For rats, th e typical maximum meal size is about g ( see, e.g., Joh n son & Collier, 1989, 1991) For an imals such as pigeon s with a crop or mon keys with ch eek pouch es, a meal can be much more substan tial Th is is also th e case for rats wh en th eir en viron men t permits th em to h o ar d T Reese an d H o gen so n ( 1962) sh owed th at for deprivation times over 24 h r, pigeon s will sume approximately 10% of th eir free-feedin g weigh ts Zeigler, Green , an d Leh rer ( 1971) foun d th at in th e course of an h our, 10 Wh ite Carn eaux th at h ad been deprived to 80% of th eir ad libitum weigh ts sumed 40 g of mixed grain on th e average; th is is sisten t with Reese an d H ogen son ’s estimate of d0 In closed econ omies in wh ich in itial deprivation times are imal, d0 will be small an d may usually be set to zero Un der th ese dition s deprivation will grow with time sin ce th e last meal ( t) accordin g to Equation un til h un ger motivation exceeds th e th resh old, at wh ich poin t an oth er meal will be in itiated Pigeon s of typical size require between 0.5 an d g/ h r to main tain th eir weigh ts between 80% an d 100% of ad libitum, an d th e requiremen ts for rats also fall with in th at ran ge Th ese values for M are sufficien tly smaller th an th e rates of repletion in typical ( open econ omy) experimen ts th at on e may set M ϭ 0, as is don e in all of th e subsequen t an alyses in th is paper Drive Versu s Deficit Wh at is th e relation between th e h un ger drive ht an d deficit dt? Th e simplest model makes h un ger proportion al to deficit, ht ϭ ␥dt, so th at from Equation ht ϭ ␥[ d ϩ ( M Ϫ mR) t] ( 5) Altern ate models of th is basic process are possible Equation is similar to a regulator y model proposed by Ettin ger an d Staddon ( 1983) Town sen d ( 1992) explored a dyn amic motivation al system th at, in place of Equation 5, h ad motivation grow as a fun ction of th e deviation between th e curren t motivation al level an d th e ideal, with a th resh old th at motivation must exceed before respon din g will be in itiated Solution of such a model leads to motivation th at grows expon en tially with time, rath er th an lin early: ht ϭ e␥[ d 0ϩ( MϪmR ) t] Ϫ ␪ ( 6) With th e th resh old equal to 1.0, motivation will be zero wh en deprivation level is zero In th e case of ␪ Ͼ 1, it requires more th an th e imal amoun t of deprivation for th e subject to begin respon din g In th e case of ␪ Ͻ 1, th e subject will tin ue respon din g even wh en satiated ( Morgan , 1974) , eith er because dition in g h as created some beh avioral momen tum or because th e drive is also main tain ed by oth er deprivation s ( e.g., dilute sucrose solution s will assuage both h un ger an d th irst) In th e lin ear model, th resh old effects are absorbed in to th e deficit parameters Th e expon en tial model h as some face validity, in th at in trospection suggests th at th e exigen cy of h un ger seems to grow more steeply th an lin ear with deprivation time It is sisten t with trol-systems an alyses of motivation al systems ( e.g., McFarlan d, 1971; Toates, 1980) Serious studen ts of th ese issues will fin d an excellen t review of th e curren t state of research on appetite an d its n eural an d beh avioral bases in Legg an d Booth ( 1994) Yet an oth er model of h un ger would h ave it grow sigmoidally with deprivation , approach in g a ceilin g at th e h igh est levels of deprivation Such a model is outlin ed in th e Appen dix; its application did n ot improve an y of th e an alyses, an d so it is n ot pursued h ere Equation s an d sh ow th at wh en an an imal becomes satiated ( wh en th e in itial deficit MECHANICS is replaced an d depletion is just balan ced by repletion ) , ht falls below th resh old, drivin g motivation to zero an d carr yin g respon se rate alon g with it Food-motivated beh avior ceases, preven tin g overin dulgen ce th at would drive h un ger levels to a n egative value Con tin gen cies of rein forcemen t th at require sumption for access to oth er in cen tives, h owever, could drive ht to a n egative value In th is case, respon se rates are depressed below free base rates ( Allison , 1981, 1993) , requirin g extern al force or th e passage of time to overcome th at in h ibition Aggregatin g Over a Session For th e lin ear model, th e average drive level over th e course of a session of durationtsess is given by Equation 5, with t ϭ tsess/ ( see th e Appen dix) Un der th e expon en tial drive model, th e situation is more complicated If session duration is stan t, th e average drive level is given by Equation 6, with t ϭ tЈ, some un defin ed fraction of tsess In employin g th e expon en tial model, on e may set tЈ to some arbitrar y value ( e.g., tsess/ 2) an d let th e remain in g parameters adjust th emselves to th at strain t Econ omic Tran slation In econ omic parlan ce, d0 is th e debt, mR is th e wage, an d M is th e cost of doin g busin ess O n ratio sch edules th e rate of rein forcemen t R is an in verse fun ction of th e ratio size ( n) , or price, an d n/ m is th e un it price M, th e rate of utilization of food by a free-feedin g organ ism, is th e coordin ate of th e ideal, or bliss poin t, alon g th e food sumption axis It could be separated in to fixed cost or overh ead ( basal metabolic rate) , an d production cost ( respon se effort) Basal metabolic rate stitutes th e major cost of foragin g an d th us stitutes a sign ifican t ‘‘sun k cost’’ to an y en deavor: O n ce stan din g, it doesn ’t require much more en ergy to an yth in g ( Th is distin ction implies flat optima for models of foragin g th at maximize calories gain ed per calories of effort expen ded; more precise feedback is provided by optimizin g calories gain ed over time expen ded.) Th e parameter ␥ represen ts th e cost of deviation s from th e ideal, an d e␥ provides on e in dex of th e elasticity of deman d If ␥ is large ( an d th us e␥ Ͼ 1) , th e an imal is ver y sen sitive to deviation s from th e ideal rate of repletion , 411 an d deman d is said to be in elastic If ␥ is small ( an d th us e␥ ഠ 1) , th en ch an ges in price elicit on ly imal beh avioral adjustmen ts; deman d for th e commodity approach es un it elasticity If ␥ is n egative ( an d th us e␥ Ͻ 1) , an imals will work less for a commodity as its price in creases, an d deman d is said to be elastic Th is occurs in th e presen ce of substitutes, as wh en food is available for respon din g on oth er levers ( Joh n son & Collier, 1987) Th is in terpretation of elasticity differs from th at of th e econ omists, because th eirs refers to deman d as a fun ction of price but does n ot take deprivation levels in to accoun t Econ omic models are design ed to map population effects, n ot biological on es Saturation of th e market is treated with differen t models th an elasticity ‘‘Decr easin g m ar gin al u tility o f goods’’ captures some of th e idea of satiation , but is usually strued with out referen ce to th e curren t deficit Th e presen t approach predicts th at th e econ omists’ measure of elasticity will ch an ge with price, because on ratio sch edules th e rate of rein forcemen t, R, wh ich appears in th e righ t sides of Equation s an d 6, equals m/ n, th e reciprocal of un it price Motivation varies with price because th at affects th e rate of repletion In deed, H ursh , Raslear, Bauman , an d Black ( 1989) foun d elasticity to var y as a lin ear fun ction of un it price But th is is n ot because ␥ h as ch an ged; our measure of elasticity, e␥, may stay stan t over ch an ges in motivation because we h ave moved th e trollin g variables in to our in depen den t variables ( Equation s an d 6) , an d th erefore n ot n eed to let our th eoretical stan ts var y with our in depen den t variables Ecologic Tran slation M is th e setpoin t repletion rate th at an imals will defen d Equation provides a measure of deviation from th at setpoin t Defen se of th e setpoin t is equivalen t to an imals’ attemptin g to imize th at deviation , th at is, set th e derivative to zero Th e force of th is equilibration is given by ␥ In trol-systems parlan ce, ␥ represen ts th e regulator y gain , or restorin g force Man y differen t arran gemen ts of tin gen cies will gen erate man y differen t stellation s of beh avior, all of wh ich h ave on ly on e th in g in common an d predictable; th e absolute value of Equation will be imized Th is approach th erefore is like th e 412 PETER R KILLEEN H am ilto n ian ap p r o ach to m ech an ics, in wh ich all of th e laws of mech an ics may be derived from imization of a sin gle differen tial equation called th e action It is th e core assumption of regulator y approach es to beh avioral econ omics such as Allison ’s ( 1983) Th e curren t approach also recogn izes th e boun dar y dition s to th is imization : Th e ch an ges in motivation will n ot be revealed in beh avior un til th ey cross a th resh old for action , an d th ey will n ot tin ue on ce th e capacity of th e organ ism is saturated An Application of the Basic Model to Satiation Cu rves H ow does drive level in teract with magn itude or quality of th e in cen tive? Th e simplest assumption is multiplicative: Absen t eith er drive or a viable in cen tive, th e specific activation a must be zero We may call th e in cen tive variable v Th en at ϭ vht Th e value of an in cen tive will n ot gen erally be proportion al to its magn itude, alth ough a lin ear relation may be an adequate approximation if th e ran ge of variation is small In accord with th e above an alysis, for th e lin ear drive model we expan d th e specific activation to at ϭ vht ϭ v ␥[ d ϩ ( M Ϫ mR) t] , ( 7) wh ere ( M Ϫ mR) is th e balan ce between depletion an d repletion , an d its multiplication by t gives th e cumulative effects of th at balan ce Th is equation h as replaced a as a sin gle free parameter with a th ree-parameter model: value v, th e in itial deficit d0, an d th e depletion rate M ( For th e lin ear model th e deviation -cost parameter ␥ is redun dan t with th e value parameter v an d may be absorbed in to it or simply set to 1.0.) Equation may th en be in serted in to Equation to predict r esp o n se r ates o f an im als u n d er in ter val sch edules wh en deprivation levels var y Fisch er an d Fan tin o ( 1968) provided th e data aroun d wh ich th e lin ear model was developed Th ey deprived pigeon s to 80% of th eir ad libitum weigh ts, an d train ed th em to respon d on ch ain ed VI 45 VI 45 sch edules, exten d in g th e sessio n s u n til r esp o n d in g ceased Th e rein forcer sisted of access to a h opper of mixed grain for 2, 6, 10, or 14 s Figure sh ows th e resultin g satiation cur ves in th e termin al lin ks of th e ch ain an d in th e in itial lin ks Alth ough th e data th emselves Fig Respon se rates un der ch ain ed sch edules for pigeon s receivin g differen t duration s of access to th e h opper durin g exten ded session s ( Fisch er & Fan tin o, 1968) Th e data represen ted by filled symbols come from th e termin al lin k, an d th ose by open symbols from th e in itial lin k Th e cur ves are drawn by Equation s an d 7, an d represen t per forman ce for 2-s ( in verted trian gles) , 6-s ( trian gles) , 10-s ( squares) , an d 14-s ( circles) access to food sh ow rath er un excitin g mon oton ic decreases with n umber of feedin gs, th e model provides a ration al fit to th em Th e first step was to estimate th e amoun t of food obtain ed un der th e differen t dition s, because amoun t sumed is n ot proportion al to h opper duration Fortun ately, Epstein ( 1981) publish ed a useful graph givin g th e amoun t sumed from a h opper of th e design used in th is study For th ese h opper duration s th e regression gave th e amoun ts as 0.13, 0.28, 0.35, an d 0.36 g of mixed grain I used th ose n umbers as estimates of m Th e pigeon s’ weigh ts were reduced to 80% of th eir free-feedin g weigh ts To optimize th e goodn ess of fit, I set th e parameter k in Equation to 200 respon ses per ute for th e termin al lin k an d 64 respon ses per ute for th e in itial lin k Th e in itial deprivation d0 took a value of 57 g Th e value parameter v was 1.5 s per rein forcemen t Th e expon en tial drive model provides a comparable fit to th ese data Given th e n ecessar y approximation s, th e fit of th e model to th e data is per2 For Leh igh Valley feeders th e n umber of grams eaten approximates a lin ear fun ction of h opper duration , with a slope of 0.06 g/ s an d an in tercept of 0.2 g ( Epstein , 1985) Pigeon s feedin g ad libitum are less efficien t, with typical eatin g episodes lastin g s, durin g wh ich 0.33 g are sumed ( H en derson , Fort, Rash otte, & H en derson , 1992) MECHANICS h aps acceptable, alth ough respon din g in th e in itial lin ks decreased at a faster rate th an predicted, especially for th e 14-s h opper d itio n ( Len d en m an n , Myer s, & Fan tin o , 1982, foun d a similar h ypersen sitivity in th e in itial lin ks in respon se to variation s in duration of rein forcemen t, as did Nevin , Man dell, & Yaren sky, 1981, in respon se to satiation ) It may be th at in all cases decreased motivation h as its primar y effects on pausin g, an d on ce an an imal h as begun to respon d, it tin ues un til rein forcemen t If th is is th e case, th en pausin g will occur primarily in th e initial links, with animals responding throughout the terminal links Segmenting responding will thus put the greatest leverage of motivation on the earliest segments (See Williams, Ploog, & Bell, 1995, for further analyses of these chain-schedule effects.) We can write th e above models in a more den sed form Set th e metabolic rate M to 0, th e magn itude of th e in cen tive m to 1, an d let th e gain parameter ␥ be absorbed in to v; th en write Equation as Bϭ kR R ϩ 1/ [ v( d Ϫ Rt) ] ( 8) Th is equation reiterates th e above description s, but also provides quan titative prediction s: Because of satiation effects, respon se rate is a quadratic fun ction of rein forcemen t rate Un der dition s of large in itial deficit ( d0) relative to repletion ( Rt) , th e paren th etical expression is essen tially stan t an d can be absorbed by v, wh ich return s to us our simple Equation ( or 3Ј, below) Th e H errn stein h yperbola is th us valid primarily for session s of sh ort duration or low rate of rein forcemen t, wh ere th e in itial deficit outweigh s th e cumulative repletion But satiation effects grow with t, an d become domin an t later in a session If on e is in terested in estimatin g th e parameters in H errn stein ’s h yperbola, th en it is better to use data from early in a session in wh ich repletion ( Rt) is low relative to in itial deficit ( d0) , or from sh ort session s, so th at th e den omin ator is relatively stan t Better yet, use Equation at th e cost of on e addition al parameter ( d0) an d predict th e complete fun ction Note th at th e adden d 1/ [ v( d0 Ϫ Rt) ] in th e den omin ator was in terpreted by H errn stein 413 Fig With in -session satiation effects sh own for gen eral activity as measured by a stabilimeter, an d for lever pressin g Th e data are averaged over two session s in wh ich rats were given two 45-mg pellets for th e first respon se 30 s after th e previous rein forcemen t ( FI 30) Th e cur ves are drawn by Equation 8Ј as RO , th e value of rein forcemen t for oth er ( n on target) respon ses H e an d Lovelan d predicted th at wh en an imals were n ot deprived of th e primar y rein forcer, th ese oth er implicit rein forcers sh ould seem to grow in relative value, th us in creasin g th e value of RO ( H errn stein & Lovelan d, 1972) Th eir data sh owed th is to be th e case; h owever, our in terpretation is more straigh tfor ward: Wh en an imals are n ot greatly deprived, d0 will by defin ition be small, an d th us 1/ [v( d0 Ϫ Rt) ] ( th eir RO ) will be correspon din gly large Th e expon en tial-drive model is n ecessar y for some of th e data on satiation In th at case, Equation may be rewritten as Bϭ kR , R ϩ 1/ ( vht) ( 8Ј) with drive level ht an expon en tial fun ction of deficit ( Equation 6) rath er th an a lin ear fun ction ( Equation 5) In an un publish ed experimen t, Lewis Bizo an d I delivered two 45-mg pellets to rats immediately after a lever press on a fixed-in ter val ( FI) 30-s sch edule Gen eral activity was curren tly measured with a stabilimeter Figure sh ows th e declin e in gen eral activity an d lever pressin g as a fun ction of th e n umber of trials Equation 8Ј drew both cur ves Th e motivation al parameters ( ␥ ϭ 0.3 gϪ1 an d d0 ϭ g) were th e same for both respon ses, wh ereas th e remain in g parameters were un dercon strain ed by th e data 414 PETER R KILLEEN Th e lever-press data are flatter because ceilin gs on respon se rate compress th e top en d of th e fun ction Th e key poin t is th at Equation 8, wh ich predicts a lin ear or cavedown decrease in respon din g, could n ot h ave fit th e cave-up time course of satiation as measured by gen eral activity Equation 8Ј also drew th e cur ves th rough th e data in Figure In both econ omies d0 took th e value of 140 rein forcers an d k was 5,500 respon ses per h our; for th e open econ omy, ␥ ϭ 0.10, an d for th e closed econ omy ␥ ϭ 0.07 Th e key differen ce between th e cur ves is th e degree of repletion permitted with in th e session For th e closed econ omy th e session duration was 6,000 s, so th at tsess/ is 3,000 s, an d th e average session deficit ( th e coefficien t of ␥ in Equation 6) is 140 Ϫ R ϫ 3,000 Th e fixed duration of th e closed econ omy permitted differen tial satiation as a fun ction of rate of rein forcemen t ( R) For th e open econ omy th e session en ded after 180 rein forcemen ts, so tsess/ is 90/ R s, an d th e average session deficit is 140 Ϫ R ϫ 90/ R; th at is, a stan t 50 g Termin atin g session s after a fixed n umber of rein forcers, or in gen eral keepin g session duration proportion al to in terrein forcemen t in ter val ( 1/ R) , fers a stan t average level of motivation Th is is th e key differen ce between th e experimen tal paradigms; it is ‘‘th e econ omic system wh ich produced th e differen t results’’ sh own in Figure It did so by lettin g th e an imals differen tially satiate in on e case but n ot in th e oth er Th e amoun t of food sumed in th ese an d th e Fisch er an d Fan tin o ( 1968) session s was two to five times th e amoun t sumed in a typical session Is th ere eviden ce for th e decrease in respon din g durin g operan t session s of more typical duration ? Th an ks to McSween ey an d h er colleagues, th ere is n ow ample eviden ce of with in -session satiation effects ( see McSween ey & Roll, 1993, for a review) But h er data also sh ow with in -session warm-up effects, so we must digress to a model of th ose WARM-U P Some of th e first eviden ce for with in -session effects from McSween ey’s laborator y came from a study ducted to test th e effects of postsession feedin g on rats th at were Fig Data from McSween ey et al ( 1990) , sh owin g with in -session warm-up an d satiation effects in rats Th e cur ve is drawn by Equation 3, with Equation represen tin g th e satiation effects an d Equation th e warm-up effects required to press a lever for Noyes pellets or, in a differen t dition , to press a key for sweeten ed den sed milk ( McSween ey, H atfield, & Allen , 1990) Alth ough n o effects of postsession feedin g were foun d, a remarkable pattern of rate ch an ges with in th e session was discovered ( see Figure 4) Respon se rates in creased th rough th e first 20 of th e session an d decreased th ereafter, an d th e patter n was vir tu ally id en tical fo r th e two respon ses an d rein forcers Th e decrease in rates may be attributed to satiation of th e kin d seen in th e previous figures To wh at we attribute th e in crease in rates? Killeen an d h is colleagues ( Killeen , in press; Killeen et al., 1978) h ave described similar in creases in rates wh en an imals are first in troduced to a sch edule of periodic rein forcemen t, an d attributed th em to th e cumulation of arousal Such warm-up plays a large role in beh avior main tain ed by aversive stimuli an d a lesser but still measurable role in beh avior main tain ed by relief from h un ger In troduction to th e ch amber itself becomes a dition ed rein forcer an d th erefore a dition ed exciter If th ere were n o loss of th is arousal between session s, even tually each session would begin with rates at th eir asymptotic level But th e an imals calm down between session s For th e presen t purposes, assume th is between -session s loss is complete ( see Killeen , in press, for a more MECHANICS subjects All may be assimilated in mech an istic models of beh avior Th e fin al issue th at must be addressed before mech an ics can begin to stan d as an altern ative to econ omic an d ecologic an alyses is th e relation between th e amoun t of an in cen tive an d its value MAGNIT U DE O F INCENT IVES Wh ereas an imals typically ch oose larger amoun ts of food over smaller amoun ts ( see, e.g., Bon em & Crossman , 1988; Collier, Joh n son , & Morgan , 1992; Killeen , Cate, & Trun g, 1993) , respon se rates often ch an ge little or n ot at all as a fun ction of th e magn itude of th e in cen tive Wh y sh ould th is be? In part, th e an swer depen ds on th e fact th at th e rein forcin g value of an in cen tive is n ot proportion al to its size In th e case in wh ich magn itude is man ipulated by var yin g duration of th e in cen tive, th e reason s for th is are obvious: Th e secon d, th ird, an d nth in stan ts of sumption are n ot tiguous with th e respon se th at brough t th em about; th ey are separated from it by n Ϫ prior in stan ts of sumption ( Killeen , 1985) th at block th eir effectiven ess Th e last in stan ts of a lon g-duration reward stitute a delayed reward Th ose later in stan ts of sumption in creasin gly rein force n ot th e prior operan t respon ses but rath er th e immediately prior summator y respon ses Assume th at each of th e in stan ts of summator y activity in terpolated between a respon se an d th e last in stan t of summator y activity will block th at latter’s effectiven ess by a stan t proportion , ␯ Th en it follows th at th e effectiven ess of an in cen tive sh ould in crease as an expon en tial in tegral fun ction of its duration : v m ϭ v ϱ( Ϫ eϪ␯m) , ( 10) wh ere v m expan ds th e value of an in cen tive from a stan t v to a fun ction of its duration or magn itude ( m) ; v ϱ is th e value of an arbitrarily lon g duration of th at in cen tive, an d ␯ is th e rate of discoun tin g th e in cen tive as a fun ction of its duration Value ( v m) refers to th e psych ological/ beh avioral magn itude of an in cen tive wh ose ph ysical magn itude ( m) may be measured in grams, secon ds, or milligrams per kilogram Incentive motivation refers to th e evaluative or in stigatin g effectiven ess of th e in cen tive th at depen ds on its value 417 in th e text, as represen ted by equation s such as Equation Equation 10 embodies th e maxim of ‘‘margin ally decreasin g utility’’ of in cen tives ( as a fun ction of th eir duration , n ot, as often used in econ omic parlan ce, as a fun ction of n umber of rein forcers) If ␯ is small, th e relation is approximately proportion al; if ␯ is large, in creasin g duration adds ver y little value Killeen ( 1985) foun d th at Equation 10 with ␯ between 0.25 an d 0.75 sϪ1 fit man y of th e ch oice data h e reviewed For th e represen tative value of ␯ ϭ 1/ 2, th e value of s of h opper access h as attain ed 78% of th e maximum possible ( v ϱ) Studies th at man ipulate lon ger duration s are operatin g with in a ver y restricted ran ge Th is model of th e ch an ge in value with ch an ges in th e duration of an in cen tive may be combin ed with Equation s an d to predict per forman ce wh en th e duration of an in cen tive is varied Wh en th e value of an in cen tive is man ipulated by ch an gin g its quality rath er th an by ch an gin g its duration , some utility fun ction oth er th an Equation 10 ( e.g., a power fun ction or a logarith mic fun ction ) may be more appropriate Wh en , for in stan ce, a drug level or sucrose cen tration is man ipulated, a plausible model is v m ϭ m␯, an d th en at ϭ m ␯␥[ d 0ϩ ( M Ϫ mR) t] ( 11) Wh ereas larger in cen tives are margin ally stron ger rein forcers, th ey also decrease th e motivation to work by satiatin g an imals more quickly Th ese effects will ten d to can cel, depen din g on th e ran ge of duration s studied an d th e value of th e deficit th e an imal is attemptin g to satisfy If in itial deficit d0 is large or repletion time t is sh ort or th e rate of repletion mR is small, th e satiation effects will be buffered by d0 an d n et in cen tive effects ( in creasin g respon se rates with in creasin g magn itude) will be foun d Con versely, if d0 is small an d repletion is moderate or large, as is typical of closed econ omies, th e satiation effect will domin ate, an d respon se rates will decrease as a fun ction of magn itude Th e depen den ce of th e sign of th e correlation between magn itude an d respon se rate—positive in th e realm of small in cen tives, n egative in th e realm in wh ich satiation effects domin ate—is sh own in a study by Collier an d Myers ( 1961) , wh o foun d positive covariation of 418 PETER R KILLEEN respon se rates with volume for dilute an d in frequen t sucrose cen tration s an d n egative covariation for frequen t h igh cen tration s Th e auth ors spoke in terms of momen tar y satiation , wh ich is exactly h ow we h ave been speakin g about repletion h ere More particularly, we can take th e derivative of Equation 11 with respect to m an d set it to zero to fin d th e magn itude of m at wh ich th e correlation will go from positive to n egative Th e turn over poin t is m* ϭ ΂1 ϩ ␯΃ ␯ d 0/ t ϩ M R ( 12) O f th e variables un der experimen tal trol, in creases in d0 will exten d th e ran ge of m over wh ich a positive correlation —an in cen tive effect—is foun d; in creases in session duration an d rate of rein forcemen t ( t an d R) will move th e turn over poin t to th e left, leavin g more of th e ran ge to sh ow a n egative correlation —a satiation effect O f course, large values for d0 an d relatively small values for session duration are typical of tradition al experimen tal design s, in wh ich in cen tive effects sh ould th us be th e rule; small values for d0 an d relatively large values for session duration are typical of closed econ omies, in wh ich satiation effects sh ould th us be th e rule Within -Session Effects Versu s Between -Session Effects Ch oice beh avior sh ows greater trol by magn itude of rein forcemen t th an does sin gle-operan t respon din g Th e presen t framework explain s th is result th e followin g way: Th e satiation effects are sh ared by both operan ts in a ch oice situation , leavin g th e in cen tive effects to act differen tially, un buffered by satiation Th e same is true for respon se rates in multiple sch edules, in wh ich satiation effects sh ould gen eralize wh en compon en t duration s are n ot too lon g, leavin g in cen tive effects th e opportun ity for differen tial effectiven ess—an effect kn own as contrast ( Nevin , 1994) It remain s to be seen just h ow much of th e complex literature on beh avioral trast can be un derstood in th ese terms To th e exten t th at th is mech an ics applies, trast sh ould be greatest wh en th ere is least bufferin g by d0; th at is, toward th e en d of session s, in lon ger session s, an d in closed econ omies It sh ould be greater for an imals th at take lon ger to satiate because th ey h ave crops or oth er cach es ( e.g., pigeon s) , compared to th ose th at don ’t ( e.g., rats) Con trast sh ould be greater for in cen tives for wh ich th ere is little satiation ( e.g., electrical stimulation of th e brain , n on n utritive sweeten ers) an d lower for bulky but low-valued in cen tives An alogous prediction s h old for postrein forcemen t pausin g ( see Peron e & Courtn ey, 1992) ( a) Un sign aled with in -session man ipulation s sh ould reflect primarily satiation effects ( lon ger pauses after larger rein forcers) , because th e differen tial magn itudes provide differen tial momen tar y satiation effects immediately after th eir deliver y, wh ereas th e forth comin g in cen tive value is averaged over all duration s of in cen tives ( b) For between session s ch an ges, th e two compon en t effects will ten d to can cel ( c) Sign aled with in -session ch an ges sh ould reflect primarily in cen tive effects, because th e forth comin g in cen tive is particular to per forman ce un der its stimulus trol, wh ereas th e satiation effects will ten d to be averaged across magn itudes Un like respon se rates, th ere is n o ceilin g effect on pause len gth s, wh ich may make th em more sen sitive to ch an gin g motivation al levels th an rates; most of th e effects predicted by th e presen t th eor y may reflect differen ces in th e amoun t of time spen t pausin g or en gagin g in oth er respon ses, rath er th an tin uous ch an ges in respon se rates over a substan tial ran ge In an y case, th e presen t th eor y predicts th at all of th ese effects sh ould be stron gly affected by deprivation level, explain s wh y, an d stipulates th e texts in wh ich satiation versus in cen tive effects will be foun d ECO NO MICS Th e cen tral cern of tradition al econ omics is th e exch an ge of goods for oth er goods, in cludin g labor, an d th at is also th e cern of beh avioral econ omics Experimen tal subjects exch an ge beh avior for goods, or strike balan ces between several goods in return for th eir beh avior With out th is requiremen t for exch an ge of tan gible items, th ere would be un dercon strain t in th eories an d ch aos in th e marketplace: If all th at mattered to h un gr y subjects were maximization of rein forcemen t, all an imals would always respon d at th eir maximum rates un der most tin gen cies MECHANICS Econ omic beh avioral th eor y was in troduced in part because its framework of sacrificin g on e th in g to get an oth er provides a ‘‘ration al’’ basis for th e modulation of respon se rates we see on man y sch edules of rein forcemen t Wh en return rates are ver y low, an imals sh ould respon d with little en th usiasm because doin g so is n ot worth th eir wh ile compared to oth er th in gs th ey could purch ase with th eir labor; wh en th e return rates are ver y h igh , th ey sh ould respon d with little en th usiasm because th ey are close to satiation Th e greatest stren gth of econ omic an alyses lies in th e developmen t of models th at frame th e trade-offs between differen t rein forcers, clarify wh at stitutes a ‘‘bun dle’’ of goods, an d explain th e in teraction s between similar rein forcers th at permit on e to be substituted for an oth er Th e application of econ omic models to beh avior trolled by a sin gle source of rein forcemen t is more problematic, because th ese models are forced to in troduce oth er h ypoth etical goods in volved in th e trade-offs, in a way n ot dissimilar to H errn stein ’s in troduction of RO as a source of competin g rein forcemen t Rach lin an d associates ( Rach lin , 1989; Rach lin , Battalio, Kagel, & Green , 1981; Rach lin & Burkh ard, 1978; Rach lin , Kagel, & Battalio, 1980) treat leisure as a good, so th at depen din g on th e experimen ter’s strain ts, th e an imals must make trade-offs between th e leisure given up by respon din g an d th e material rein forcers th at respon din g provides Th ose trade-offs are motivated by th e subject’s preferen ce for an optimal package of goods un der strain ts of time an d sch edule Staddon ( 1979) assumes th at optimal rates exist for all activities, an d th at an imals are motivated to approach th at locus in beh avioral space th at imizes a weigh ted sum of squares of th e deviation s of each from its optimal rate ( or th at imizes some oth er cost fun ction ) given th e strain ts of time an d sch edule Experimen tal tin gen cies usually require operan t respon din g at a h igh er-th an -optimal rate, so th at such respon din g fun ction s as a cost, much as it does for Rach lin an d associates In Staddon ’s multidimen sion al beh avior space, th e coordin ates of th e ideals of all relevan t dimen sion s defin e a bliss poin t, an d because ever y oth er poin t is in some way in ferior, variation s in an organ ism’s beh avior th at carr y it 419 Fig Data from Kelsey an d Allison ( 1976) plotted by H an son an d Timberlake ( 1983) , alon g with th e cur ves resultin g from th eir model an d from Staddon ’s ( 1979) Reprin ted with permission Superimposed is th e parabola drawn by Equation 16 away from th is global imum are selected again st H an son an d Timberlake ( 1983) focus on regulation , provide a math ematical model of th e equilibrium approach of Timberlake an d Allison ( 1974) , an d derive as special cases Staddon ’s ( 1979) an d Allison ’s ( 1976, 1981, 1993) optimality accoun ts At th e h eart of th e model are th e coupled differen tial equation s kn own as th e Lotka-Volterra system As an exam p le o f its ap p licatio n , th e asym m etr ic cur ve is drawn th rough th e data from Kelsey an d Allison ( 1976) , sh own in Figure Th e dash ed lin e is given by Staddon ’s ( 1979) imum distan ce model In fittin g th eir five-parameter model, H an son an d Timberlake n oted th at th ese fun ction s ‘‘quickly exh aust th e degrees of freedom in h eren t in , for example, six or seven data poin ts’’ ( p 272) Th us, th e most we can h ope for in comparin g th eor y to data is a sisten cy ch eck, a h urdle th at is n ecessar y for th e th eories to clear, but wh ose clearan ce is n ot sufficien t groun ds for us to accept th em Wh eth er or n ot we accept th ese th eories seems to depen d on wh eth er we fin d th eir assumption s gen ial to our in tuition s about beh avior, an d wh eth er th ey make n ovel prediction s Th ere h ave been few n ovel prediction s th at I am aware of H owever, th ey provide n ew structs an d in - 420 PETER R KILLEEN Fig Deman d fun ction s collected an d graph ed by Lea ( 1978) Reprin ted with permission Superimposed is th e model deman d fun ction drawn by Equation 17 dices, such as elasticity of deman d, th at provide altern ative perspectives on beh avior Elasticity is an in dex, ‘‘a n umber derived from a formula, used to ch aracterize a set of data’’ ( American Heritage Dictionary, 1992) In dices are useful because a sin gle n umber can often ch aracterize some crucial aspect of a ph en omen on ( e.g., th e in dex of refraction of optical materials, th e sumer price in dex, etc.) Lea ( 1978) drew deman d cur ves as th e amoun t of an item purch ased as a fun ction of th e price of th e item Wh en th e axes are logarith mic, th e slope of th ese cur ves equals th eir coefficien ts of elasticity ( see, e.g., Koo- ros, 1965) In h is Figures an d 4, Lea drew idealized deman d fun ction s as straigh t lin es of differen t slopes, with items such as coffee an d bread sh owin g th e least decrease in sumption as price is in creased ( deman d for th em is in elastic, as we would expect) , an d items such as h errin g an d cakes sh owin g th e greatest decrease H ere a sin gle n umber—th e coefficien t of elasticity—effectively ch aracterizes a set of data H owever, in h is Figures an d 5, as is th e gen eral case, real data from closed econ omies are cave: Elasticity in creases tin uously with th e price of th e commodity ( see Figure 8) Th is result is MECHANICS about as satisfyin g as would be th e discover y of an ‘‘in verse square law’’ for force as a fun ction of distan ce, but in a world in wh ich th e expon en t varies tin uously with distan ce an d takes th e value of Ϫ2 on ly at on e particular distan ce Elasticity sh ould n ot itself be so elastic! Th e deman d cur ve was design ed for an alysis of decision s by population s, wh ere in creasin g proportion s of th e population may be in fluen ced to purch ase a commodity, perh aps just on ce, as its price decreases It was n ot design ed to an alyze th e repeated purch ases by in dividuals, because such data will be greatly affected by decreasin g margin al utility as magn itude in creases, an d by satiation as rate of sumption in creases As n oted by Staddon ( 1982) , rein forcemen t rate appears on both axes ( R vs n/ R) of th e deman d cur ve, so th at in depen den t an d depen den t variables are in trin sically correlated Such fun ction s provide good stimulus trol of visual an alysis on ly wh en th ey are lin ear an d differen ces in slope may be directly compared Lookin g for secon d-order effects such as differen ces in degree of cur vature is made un n ecessarily difficult by th e tactical ch oice of th ose coordin ates Beh avioral econ omics h as useful th in gs to tell us about substitutability an d complemen tarity ( see, e.g., Green & Freed, 1993; Lea & Roper, 1977) , issues n ot addressed in th is article But wh en applied to sin gle respon se– rein forcer paradigms, th at approach is less useful ( see, e.g., th e commen taries on Rach lin et al., 1981) Th ere are too man y free variables to be tied down ; motivation al ch an ges affect th e parameters wh ile th ey are bein g collected, an d th e core n otion th at an imals prefer n ot to respon d above a relatively low bliss-poin t rate is false, as sh own by Staddon an d Simmelh ag ( 1971) for pigeon s an d by n umerous oth er in vestigators for n umerous oth er organ isms wh ose un econ omical adjun ctive beh avior often over wh elms th eir tin gen t beh avior Th e paired baselin e distributio n s o f r esp o n d in g u sed in r egu latio n models h ave been sh own n ot to predict bliss poin ts, an d th e ratio of in strumen tal to tin gen t respon din g is n ot th e trollin g variable it h as been purported to be ( Tiern ey, Smith , & Gan n on , 1987) Th e econ omic approach does n ot respect m olecu lar tin gen cies of rein forcem en t 421 ( Allison , Buxton , & Moore, 1987) , an d th erefore is prima facie un able to predict th e h uge differen ces in respon din g th at can be obtain ed with brief delays of rein forcemen t, an d is un able even to predict th e profoun d differen ces th at depen d on th e order of exch an ge of goods—th at is, th e differen ces in for ward versus backward dition in g Beh avioral econ omics th erefore does n ot stitute a gen eral th eor y of beh avior It offers some tools for th e comparison of differen t in cen tives an d th eir effects on beh avior wh en satiation an d rein forcemen t tin gen cies are trolled It open s th e door to a beh avio r al an alysis o f co n su m er ch o ice, ab o u t wh ich a mature beh avioral econ omics will h ave much to say ECO LO GICS Collier an d Joh n son an d associates ( Collier et al., 1986, 1992; Joh n son & Collier, 1989, 1991) h ave required rats to work for food un der a variety of dition s, usually on es th at respect th e an imal’s n ormal feedin g routin e, lettin g th e an imals complete meals un in terrupted, an d often exten din g th e session s to permit an imals to acquire most of th eir food with in th e experimen tal text ( i.e., closed econ omies) Th is exten ds th e an alysis of beh avior to a larger time scale But, alth ough perh aps more n atural, it makes it more difficult for th e th eorist to an alyze th e beh avior th at is obtain ed from th ese texts Th e reason for th is is th at un der th ese dition s, rates of rein forcemen t are closely tied to th e pattern s an d rates of th e an imal’s beh avior— rate of rein forcemen t, a key trollin g variable, is n o lon ger an in depen den t variable To un derstan d th is, we must digress to examin e h ow an an imal’s beh avior affects its rate of rein forcemen t Schedu le Feedback Fu n ction s Killeen ( 1994) derived a sch edule feedback fun ction ( SFF) th at predicts th e rate of rein forcemen t on stan t probability VI sch edules, given a stan t rate of respon din g of B respon ses per ute, as R ϭ B( Ϫ eϪR Ј/ B ) , B Ͼ 0, wh ere RЈ is th e programmed rate of rein forcemen t O ver most of its ran ge, th is may be approximated by its Taylor expan sion : 422 PETER R KILLEEN Rϭ BRЈ B ϩ RЈ ( 13) Th is is also th e form of th e SFF suggested by Staddon ( 1977) an d Staddon an d Moth eral ( 1978) It is also th e equation derived if on e assumes th at rein forcers are set up an d respon ses are emitted ran domly an d in sequen ce with rate stan ts of RЈ an d B ( i.e., it is th e mean of series-laten cy devices such as two-step gen eralized gamma distribution s) Wh en respon se rates are h igh , rein forcemen t rate approximately equals th e sch eduled rate RЈ ( divide n umerator an d den omin ator by B an d th en let B go to in fin ity) ; wh en th ey are ver y low, rein forcemen t rate approximately equals th e respon se rate B Equation 13 is accurate on ly in th e ideal case of tin uous en gagemen t of organ ism an d sch edule If an organ ism takes exten ded timeouts from respon din g, obtain ed rates of rein forcemen t are lower ( Baum, 1992; Nevin & Baum, 1980) Th e SFF for ratio sch edules is simply R ϭ B/ n, wh ere n is th e ratio requiremen t Such SFFs are n ot of in terest because we believe th at an imals are sen sitive to h ow th e margin al rates of rein forcemen t are affected by respon din g un der differen t SFFs ( Th is fun damen tal assumption of all molar optimality models h as been effectively discredited by Ettin ger, Reid, & Staddon , 1987.) Rath er, SFFs are importan t because th ey determin e th e rate of rein forcemen t ( a key trollin g variable in Equation s th rough 3) in th e text of an in teractive organ ism Closed systems such as th ose employed by Collier an d associates are closed-loop systems, with th e feedback from respon se rates on rein forcemen t rates closin g th e loop th rough th e SFF To predict beh avior un der such dition s, we in sert th e appropriate feedback fun ction in to th e motivation equation s, an d in sert th ese in to Equation For ratio sch edules, th e solution gen erates th e basic equation of prediction ( Killeen , 1994, Equation 8) For in ter val sch edules, it yields equation s proportion al to Equation 3, but with a sligh tly lower asymptote: Bϭ ( k Ϫ 1/ a) aR Ј , aR Ј ϩ a Ն 1/ k ( 3Ј) No problem: Still th e same old h yperbola! Equation 3Ј sh ows on e of th e reason s th at a h yperbolic model is so robust: Wh en specific activation ( a) is large, Equation 3Ј is equivalen t to Equation But even at low activation wh en obtain ed rein forcemen t rate falls substan tially below its sch eduled value, per form an ce r em ain s a h yp er b o lic fu n ctio n o f sch eduled rein forcemen t rates, merely fin din g a lower asymptote ( k Ϫ 1/ a) Un fortun ately, th e complete equation s of motion for organ isms tain a double feedback loop Not on ly does rate of respon din g affect rate of rein forcemen t ( th at Equation 3Ј compen sates for) , but rate of rein forcemen t determin es th e satiation of th e organ ism, wh ich affects th e value of specific activation a Th e obtain ed rate of rein forcemen t appears in Equation 7, wh ich is an expan sion of a If we in sert Equation 13 in to th at an d attempt to solve it, we get stuck Th e result is a quadratic equation with n o simple solution s ( Equation is quadratic in th e rate of rein forcemen t, but because th at is an in depen den t variable, it caused n o misch ief H ere th e equation s are quadratic in th e depen den t variable, respon se rate.) Quadratic equation s are, of course, n on lin ear; th e n on lin earity is in troduced by h avin g beh avior be a fun ction of a variable ( motivation ) th at itself is a fun ction of beh avior ( wh ich reduces motivation by repletin g th e an imals) Now it becomes impossible to write equation s with all th e kn own s on on e side an d th e un kn own s on th e oth er Th ere is n o simple, complete solution to th is impasse Copin g with Non lin earity Wh en fron ted with a difficult n on lin earity such as th is, we h ave several option s: Experimentally opening the loop We may reduce th e n on lin earity by makin g th e stan t terms large relative to th e var yin g terms Th is mean s large in itial deficits ( d0) relative to repletion rates ( mR) ; Equation s an d 8Ј sh ow th at th is is ach ieved with some combin ation of h igh ly deprived organ isms, small an d in frequen t meals, an d sh ort session s: All of th e beˆ tes n oires th at Collier an d oth er econ omic th eorists h ave repeatedly excoriated It is h ard to dispute th eir poin t th at th ese co n d itio n s o f th e r efin em en t exp er im en t ( i.e., th e stan dard procedures) are n on represen tative extrema un der wh ich th e an imals can display little of th e ran ge of th e n atural repertoire of th eir n ormal in strumen tal an d summator y pattern s O bjects fallin g in a MECHANICS vacuum display little of th e ran ge of th e n atural repertoire of leaves fallin g in an autumn win d It is th rough refin emen t experimen ts th at p h ysicists, ch em ists, an d b eh avio r ists h ave come to un derstan d th e variables of wh ich th eir subject is a fun ction We can h ave simple laws, such as Equation 3, or we can h ave more precise but complicated on es, such as th ose obtain ed by in sertin g Equation 11 in to it; to th e degree th at we wan t precision , we must forgo its complemen t, simplicity ( Killeen , 1993) By opening the loop between controlled and controlling variables, the refinement experiment permits us to explore alternate ways of formulating models to cover the phenomena of interest, to estimate the values of the models’ basic parameters, and to evaluate the adequacy of one model against alternate models (e.g., the linear vs exponential drive models) Surgically opening the loop An oth er way of trollin g th e feedback loop is to open th e esoph agus so th at th e sumed food does n ot fill th e gut Th is is sh am feedin g, a kin d of tin uous bin ge an d purge It provided Pavlov ( 1955) an d Miller ( 1971) with an experimen tal preparation th at effectively addressed certain question s about th e locus of satiety sign als But, because it in sults th e in tegrity of th e organ ism–en viron men t match in a differen t way, it is less useful in addressin g th e question s we pursue cern in g th e beh avior of a wh ole organ ism Postdictions Wh en basic refin emen t experimen ts are completed, we would like a way of th en applyin g th e results to more complex experimen tal arran gemen ts th at are n ot so th eoretically felicitous A mean s to accomplish th is is to give up sch eduled rein forcemen t rate as an in depen den t variable, an d in use the measured rates of reinforcement in our equations of prediction The measured rates of instrumental and contingent behavior are the variables compared by economic theorists such as Staddon (1979) and Rachlin et al (1981) This is a useful tactic in that it demonstrates consistency of the models with data, and in many cases is the best that can be achieved But settling for correlations between dependent variables is less than an optimal solution to the problem; in giving the prime instrument of experimental analysis—control—to the subject by making the paradigm more ‘‘ecologically valid,’’ we are consequently forced to abandon the 423 prime goal of experimental analysis, giving up prediction to settle for postdiction Numerical solutions An oth er option is to fall back on iterative n umerical solution s of th e equation s, wh ich is possible even with th e un kn own on both sides Th is option will be useful in some situation s, but is n ot furth er explored h ere Simplifications Th ere are differen t aspects of th e complete equation s th at we can ign ore for th e sake of a closed-form solution to th e laws of beh avior For in stan ce, in movin g from Equation to Equation 3, we sacrificed th e correction for blockin g of rein forcemen t by previous rein forcemen ts, in currin g some in accuracy at rein forcemen t rates above two per ute Let us n ext table Killeen ’s ( 1994) secon d prin ciple of rein forcemen t by ign orin g th e temporal strain ts on respon din g, an d fall back on h is simplest first prin ciple of arousal, Equation Th en Equation simplifies to an expan sion of th at first an d most basic prin ciple: B ϭ aR ϭ v ␥[ d ϩ ( M Ϫ mR) t] R ( 14) Th is equation is a parabola It describes respon din g at time t in a session as a fun ction of rate of rein forcemen t It also describes th e average respon din g in a session wh en t is set equal to h alf th e session duration ( tsess/ 2; see th e Appen dix) Because we h ave ign ored ceilin gs on respon se rate, we expect th e actual data to be sligh tly less peaked th an a parabola, bein g squash ed in to more of an ellipsoid form Equation 14 provides a good fit to th e data an alyzed by Staddon ( 1979) usin g h is imum distan ce model H owever, some of th ose data were collected in open econ omies, an d th eir down turn at low ratio values is probably due more to th e impoverish ed couplin g of rein forcers to respon ses, wh ich I h ave an alyzed at len gth ( Killeen , 1994) O n ratio sch edules requirin g n respon ses per rein forcemen t, we may substitute th e ratio sch edule feedback fun ction B/ n for R At last, we may write an equation th at can be solved for B! Its solution is Bϭ ΂ ΃ n n MϪ , m vЈ m, v Ј, Ͼ 0, ( 15) wh ere M is th e average depletion , M ϭ d0/ t ϩ M, m is th e magn itude of th e in cen tive, an d vЈ is proportion al to th e in cen tive value of th e 424 PETER R KILLEEN rein forcer, v ( see Equation A5 in th e Appen dix) Equation 15 is a parabola th at in creases to a maximum at n ϭ vЈM / an d decreases toward zero both as n approach es zero ( satiation effects) an d as n becomes ver y large ( strain in g th e ratio, wh ich occurs as n → vЈM , exactly twice th e poin t at wh ich th e maximum occurs) Equation 15 provides a good fit to data such as th ose sh own in Figure 10 of Collier et al ( 1986) It may be preferable to Equation 14, because it predicts respon din g in terms of an in depen den t variable, th e size of th e ratio sch edule n, rath er th an in terms of a depen den t variable, rate of rein forcemen t To calculate th e total n umber of respon ses ( b) in a session of duration tsess, multiply th rough by tsess: bϭ ΂ ΃ n n MϪ t m vЈ sess m, vЈ Ͼ ( 16) Equation 16 provides a reason able fit to th e data in Figure with m an d tsess fixed at 1, vЈ set to 1.2 ϫ 10Ϫ3, an d M ϭ 5,450 licks per session For th e expon en tial drive model ( Equation A6 in th e Appen dix) , th e parabola is skewed to th e righ t an d looks ver y much like H an son an d Timberlake’s ( 1993) cur ve It is a sh ort step to write th e equation for th e deman d fun ction , th e n umber of rein forcers earn ed ( r) as a fun ction of ratio requiremen t, by dividin g Equation 16 by th e n umber of respon ses required per rein forcemen t ( n) If we take th e session as th e un it of time, so th at we can set tsess equal to 1, th en rϭ ΂΃ M n Ϫ m vЈ m m, v Ј Ͼ ( 17) Th is is a model deman d fun ction : Con sumption r is a lin ear fun ction of un it price n/ m, with a slope of Ϫ1/ vЈ an d an in tercept of M / m It is drawn as th e bold lin e in th e logarith mic coordin ates of Figure with m ϭ 1, M ϭ 200, an d vЈ ϭ It h as approximately th e same sh ape as man y of th ose empirical deman d cur ves; it is simple, an d does n ot make th e obviously erron eous econ omic assertion th at th ere is a th in g such as elasticity th at can be assign ed to a good an d th at is in depen den t of its price ( i.e., it does n ot assert th at th e data fall on straigh t lin es in double-log coordin ates) Th e expon en tial drive model provides more flexible deman d cur ves, wh ich are n ecessar y to fit some of th ese data DeGran dpre, Bickel, H ugh es, Layn g, an d Badger ( 1993) h ave systematically reviewed data such as th ose sh own in Figures an d 8, man y in volvin g drug rein forcers Th ey argued for th e use of un it price ( n/ m) as th e proper metric of th e x axis ( as did Timberlake & Peden , 1987, an d H ursh , 1980) Un it price plays a key role in Equation s 15 th rough 17 as well Th e slope of th e deman d cur ve predicted by Equation 17 depen ds n ot on th e variables n an d m, but on ly on th eir ratio.3 Th ere is an importan t differen ce between th e an alysis of DeGran dpre et al ( 1993) an d th e presen t on e DeGran dpre et al plotted th eir data on logarith mic coordin ates A parabola in logarith mic coordin ates is n ot parabolic in lin ear coordin ates, but is skewed to th e righ t Con versely, Equation s 15 an d 16 are skewed to th e left wh en plotted on a logarith mic x axis Th e expon en tial drive model is less skewed th an th e lin ear drive model Wh eth er th e presen t models can provide as good a fit to th e ran ge of available data as h ave th ose of H ursh et al ( 1989) an d DeGran dpre et al ( 1993) remain s to be seen CO NCLU SIO N Mech an istic explan ation s h ave fallen in to disrepute, in part because good on es are h ard to come by, an d in part because th ey elicit images of gears an d pulleys—poor models for th e processes th at beh aviorists seek to un derstan d Goal seekin g, regulation , optimization , or, in gen eral, teleological ( Rach lin , 1992) an d teleon omic ( H Reese, 1994) approach es seem more modern Econ omics, th e scien ce of fin al causes ( Rach lin , 1994) , studies th e goals aroun d wh ich beh avior is organ ized As Rach lin h as n oted in h is sch olarly an d in sigh tful an alyses, we must h ave some sen se of th e purposes of beh avior before we can un derstan d wh at an act is about All four of Aristotle’s causes are n ecessar y for a complete accoun t of beh avior: th e fun ction al goals an d rein forcers ( fin al causes) , effective stimuli ( efficien t causes) , un derlyin g ph ysiology ( ma3 For ver y small values of m, v will covar y with m; for simplicity in th ese an alyses I h ave assumed th at v h as topped out, or at least th at m is n ot experimen tally varied over th e lower en d of its ran ge MECHANICS terial causes) , an d precise metaph ors an d models ( formal causes) In sofar as we ceive of operan t beh avior as bein g un der th e trol of its sequen ces, un derstan din g th e fin al causes of th at beh avior—both th e more proximate causes ( on togen etic, h istories of rein forcemen t) an d th e ultimate causes ( ph ylogen etic, selection pressures) —takes first priority But th at doesn ’t mean th at it must take all our efforts; iden tification of fin al causes is largely a qualitative en deavor, an d may proceed quickly ( we may discover th at on e of th e causes of birds’ sin gin g is defen se of th eir territor y) but workin g out th e mach in er y th at permits th e attain men t of such goals remain s a substan tial project of an alysis Th ere is much to be said for a mech an ics, a scien ce of formal causes, as th e secon d an d most detailed part of th e scien tific en deavor, to guide us in th at an alysis Th e developmen t of simple models based on n aturalistic obser vation s an d laborator y experimen ts leads us to a clearer un derstan din g of th e variables of wh ich beh avior is a fun ction ; th at is, to a clearer un derstan din g of its causes Th e ‘‘essen tial feature of th e Newton ian style is to start out with a set of assumed ph ysical en tities an d ph ysical dition s th at are simpler th an th ose of n ature, an d wh ich can be tran sferred from th e world of ph ysical n ature to th e domain of math ematics Th e rules or proportion s derived math ematically may be compared an d trasted with th e data of experimen t an d obser vation ’’ ( Coh en , 1990, pp 37–38) ; th at is, refin emen t experimen ts Th is leads to modification s of th e model system an d, in turn , of th e experimen tal design , an d aroun d again , with th ese cycles ‘‘leadin g to systems of greater an d greater complexity an d to an in creased vraisemblan ce of n ature’’ ( Coh en , 1990, p 38) ; th at is, ecological validity Math ematics was Newton ’s tool for th e discover y of veræ causæ, true causes: ‘‘Specification of th ose causes was n ot a precon dition for th e struction of model systems, but rath er a product of it’’ ( Coh en , 1990, p 29) An d math ematics, even th e relatively trivial math ematics in th is paper, provides an in valuable formal structure for our metaph orical models: ‘‘It was th e exten sion of Newton ’s in tellectual powers by math ematics an d n ot merely some kin d of ph ysical or ph ilosoph ical in sigh t th at en abled h im to fin d th e mean in g 425 of each of Kepler’s laws’’ ( Coh en , 1990, p 31) Math ematics puts a fin e poin t on th e dull pen cil of metaph or Th e presen t mech an ics provides a relatively parsimon ious quan titative accoun t of man y of th e data It also in troduces th e struct of satiation , a cept th at is in accord with our un derstan din g of n ature an d is overdue for formal recogn ition in our an alyses Mech an ics gen erates a bridge to ecologic an d econ omic an alyses th rough th e explicit utilization of th e cepts of ideal rate of repletion or rein forcemen t ( M, wh ich provides on e coordin ate of th e multidimen sion al ideal, th e bliss poin t) , th e cost of deviation s from it ( ␥) , th e decreasin g margin al utility of rein forcers ( Equation s 10 th rough 12) , an d a role for un it price as an in depen den t variable ( Equation s 15 th rough 17) It is also sisten t with th e ch an ges in respon se rate th at are foun d with in a sin gle session ( Equation 8; see, e.g., Killeen , 1991; McSween ey, 1992) Futh ermore, it leads to a biologically based treatmen t of h un ger th at provides a dyn amic approach to th e steady state assumed by econ omic models Un like th e ecologic an d regulator y approach es, mech an ics does n ot in voke defen se of a setpoin t as a fun damen tal force, but in troduces th at defen se implicitly in equation s th at make deprivation a key factor in motivation ( Equation s th rough 7) It is n ot so much th at an imals defen d a setpoin t, as th at deviation from a setpoin t in creases th e rein forcin g value of even ts th at, as n ature usually h as it, reduces th at deviation Fin ally, in Figures an d it provides altern atives to econ omic an alyses th at are parsimon ious of parameters, derive from simple version s of th e basic prin ciples of rein forcemen t, an d provide in terpretable parameters an d testab le p r ed ictio n s ( Eq u atio n s 15 th rough 17 an d A5 th rough A7) Ecologics calls our atten tion to th e rich in teractive en viron men ts in wh ich an imals h ave evolved an d th at h ave sh aped th eir respon ses to metabolic ch allen ge Its experimen tal results may be ch arted with accuracy, but because it is a dyn amic, path -depen den t, n on lin ear en terprise, th ose results can seldom be predicted from prin ciples Like th e mean ders of a river th at are sisten t with simple an d precise models, th e path s of un ch an n eled beh avior may come to be seen as bein g sisten t with models such as th ose presen ted 426 PETER R KILLEEN h ere, even wh ile th e particular courses of river an d beast may n ever be predictable from th eir prin ciples Prediction an d trol are en gin eerin g ideals, n ot scien tific on es It is th e purpose of refin emen t experimen ts to establish prin ciples; in more ecologically valid experimen ts our goal is to un derstan d, an d un derstan din g is n oth in g oth er th an recogn ition of sisten cy with establish ed prin ciples Like ecologics, econ omics provides in spiration to search for th e en ds aroun d wh ich beh avior is organ ized—its fin al causes—an d th is is wise It provides an approach to un derstan din g th e trade-offs an imals make between altern ate packages of goods, an importan t an d un derrepresen ted area of research But it also seduces us in to usin g th e an alytic framework of econ omists, an d th is is folly Econ omics is n ot on ly th e scien ce of fin al causes; it is also ‘‘th e dismal scien ce.’’ Its complexities an d routin e failures to predict beh avio r fr o m eco n o m ic p r in cip les ar e legen dar y An econ omic beh aviorism th at borrows its structs, rath er th an its goals, 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in beh avior an alysis Behaviorism, 14, 111–124 Williams, B A., Ploog, B O , & Bell, M C ( 1995) Stimulus devaluation an d extin ction of ch ain sch edule per forman ce Animal Learning & Behavior, 23, 104– 114 Zeigler, H P., Green , H L., & Leh rer, R ( 1971) Pattern s of feedin g beh avior in th e pigeon Journal of Comparative and Physiological Psychology, 76, 468–477 Received December 31, 1994 Final acceptance July 14, 1995 MECHANICS 429 APPENDIX Con stan ts an d Dimen sion s Table lists th e symbols an d th eir in terpretation s Lower-case letters are used for all variables except rate variables, wh ich are written in capitals Greek letters are used for stan ts an d parameters Th e secon d column lists th e co n stitu en t d im en sio n s, n o t th e un its For example, A is th e n umber of secon ds of respon din g per secon d; th ese can cel to make it a ‘‘dimen sion less’’ variable b an d r refer to n umber of respon ses an d rein forcers; because coun tin g in volves an absolute scale, th e un its for both are ‘‘coun ts’’; but because th ey are coun tin g differen t th in gs, th ey h ave differen t dimen sion s Session Averages To calculate average respon se rates durin g a session , on e sh ould write th e complete model predictin g respon se rates an d in tegrate it over th e session duration Th is is because with fin ite ceilin gs on respon se rate, even th e most extreme deprivation can on ly elevate respon se rates sligh tly closer to th eir ceilin g It is for th is reason th at th e lin ear an d expon en tial drive models provide equally good fits to man y of th e operan t dition in g data: Th e differen ces between th e drive level predicted by th ose models are greatest at h igh deprivation s, but th at is wh ere respon se rates are n ear th eir ceilin gs an d th us least respon sive to ch an ges in drive levels ( It is also for th is reason th at sigmoidal fun ction s between deprivation an d drive n ot provide a measurable improvemen t in fit to th e data.) Un fortun ately, in tegration of th e complete models yields un gain ly or in soluble forms It is th erefore a worth wh ile simplification to compute th e average drive an d arousal levels over th e course of a session , an d use th ese to predict average respon se rates The linear model For th e lin ear model h un ger level is given by Equation in th e text Th e average h un ger over a session of durationtsess is th e in tegral of th at fun ction with respect to time divided by tsess: h¯ sess ϭ ␥[ d ϩ ( M Ϫ mR) t sess / 2] ( A1) For sh ort session s ( tsess small) , h un ger is determin ed by th e in itial deficit d0, but as sessio n d u r atio n in cr eases, h u n ger ch an ges lin early with it For exten ded session s in Table Sym- Dimen bol sion s A R B r/ s b/ s M g/ s RЈ a r/ s s/ r k b/ s d g h m g/ r t v s s/ r n b/ r b r ␭ b r r/ b ␥ 1/ g ␪ ␣ ␯ 1/ s r/ g ␦ s/ b Mean in g Arousal level; th e amoun t of respon din g elicited by a sch edule of in cen tives in th e absen ce of competition from oth er respon ses Rate of rein forcemen t ( obtain ed) Rate of respon din g; arousal level corrected for respon se duration an d ceilin gs on respon se rate Metabolic rate; assumed stan t an d often set to zero Rate of rein forcemen t ( sch eduled) Specific activation : th e n umber of secon ds of respon din g th at are elicited by a sin gle in cen tive, wh ich depen ds on drive an d in cen tive factors Asymptotic respon se rate on in ter val sch edules Deficit resultin g from a depletion / repletion imbalan ce over time H un ger, a lin ear or expon en tial fun ction of deficit Magn itude of an in cen tive, h ere measured in grams per rein forcer Time Value of an in cen tive, wh ich depen ds on its n ature an d magn itude Number of respon ses required to complete a ratio sch edule Number of respon ses Number of rein forcers Lambda, th e rate of decay of sh ortterm memor y; does n ot play an importan t role in th e presen t developmen t Gamma, th e gain or restorin g force th at tran slates deficit in to drive Th eta, th e th resh old level of motivation for respon din g Alph a, th e rate of warm-up Nu, th e rate of discoun tin g an in cen tive as a fun ction of its magn itude; its dimen sion s depen d on th e in depen den t variable an d th e particular discoun t model ( Equation s 10 or 11) Delta, th e m in im u m in ter resp on se time wh ich tsess is large, h un ger is determin ed primarily by th e balan ce between on goin g metabolic depletion an d repletion , MϪmR The exponential model Calculatin g th e average h un ger durin g a session of durationtsess 430 PETER R KILLEEN fo r th e exp o n en tial m o d el ( Eq u atio n 6) yields a more complicated expression th an is th e case for th e lin ear model: h¯ sess ϭ Ϫe␥d ( Ϫ e␥ ( MϪmR ) t sess) Ϫ ␪ ␥( M Ϫ mR) t sess But th e in tegral may be simplified usin g a power-series expan sion If we retain on ly th e first two terms of th at expan sion , it yields a prediction of h un ger level th at depen ds on ly on th e in itial dition s an d th e stan t of in tegration : h¯ sess ഠ e␥d Ϫ ␪ Because session durationtsess h as disappeared, h un ger depen ds on ly on in itial deprivation level Th is is th e implicit assumption of most tradition al open -econ omy research , wh ich is un cern ed about ch an ges in h un ger durin g th e course of a session If we in clude th e first th ree terms of th e expan sion , we get: h¯ sess ഠ [ ϩ ␥( M Ϫ mR) t sess / 2] e␥d Ϫ ␪ Because ␥ an d d0 may be treated as free parameters, th is is equivalen t to th e lin ear model, Equation A1 Th erefore, th e lin ear model is a special case of th is expon en tial model Th is approximation is best wh en ␥ is ver y small; th at is, in th e case of a un it elastic deman d Addin g a fourth term rein troduces th e n on lin earity as [ ␥( M Ϫ mR) tsess) 2/ 3! It is on ly at th is poin t th at th e models become substan tively differen t; un fortun ately, it is also at th is poin t th at th e approximation becomes as cumbersome as th e exact form As an altern ate tactic to ach ieve a simpler average we may in voke th e mean value th eorem: Wh en we in tegrate a fun ction between two poin ts on th e x axis, th ere is some un specified value of x between th ose poin ts at wh ich th e fun ction will equal th e average over th at ran ge In th e presen t case, for some tЈ between an d tsess, h¯ sess ϭ e␥[ d 0ϩ( MϪmR ) t Ј] Ϫ ␪ ( A2) Th is can fin ally be simplified to: h¯ sess ϭ e␥Ј(MϪmR ) Ϫ ␪ ( A3) wh ere M is a measure of th e average depletion over th e course of a session of duration tsess, M ϭ d0/ tЈ ϩ M, an d ␥Ј is proportion al to th e cost of deviation s ( ␥Ј ϭ ␥tЈ) Th is is th e simplest statemen t of th e basic expon en tial model for average drive level durin g a session Equation A3 may be directly evaluated as lon g as session duration ( wh ich would affect th e implicit tЈ) is n ot varied Average arousal level We may calculate th e average arousal level th rough out a session of duration tsess It is th e in tegral of Equation divided by tsess: [ ] ( Ϫ eϪ␣t sess) A¯ sess ϭ aR Ϫ , ␣( t sess / 2) ␣, t sess Ͼ If session duration is constant, the parenthetical factor can be ignored because it is constant and can be absorbed into a In like manner, if there is little loss of arousal between sessions or session durations are long, as in closed economies, then ( 1/ ␣tsess) is small and the correction is negligible Only in the case of ver y brief sessions (tsess Ͻ 3/ ␣; typically, that is, less than 20 min) will warm-up affect session-average data In other words, in most cases little is usually lost by ignoring the parenthetical factor and setting B ϭ A ϭ aR T he Complete Model for Closed Econ omies The linear model In texts in wh ich ceilin g effects on respon se rate can be ign ored, we may solve th e gen eral model for ratio sch edules From th e first prin ciple ( Equation 1) : B ϭ aR ϭ vhR/ ␦ ( ␦ Ͼ 0) , wh ere v a measure of th e quality of th e in cen tive, h is th e drive level, an d R is th e rate of rein forcemen t ␦ is th e imum in terrespon se time; it appears h ere to vert th e measure of respon se stren gth ( respon se-secon ds per secon d, as given by A in Equation 1) to a measure of discrete respon din g ( B, respon ses per secon d) Th is is a level of explicitn ess n ot n ecessar y for th e body of th is text, but is presen ted h ere for completen ess O n ratio sch edules th e rate of rein forcemen t is per fectly correlated with th e rate of respon din g Th e sch edule feedback fun ction for ratio sch edules is simply R ϭ B/ n, wh ere n is th e ratio requiremen t Substitutin g an d rearran gin g, th is becomes: vh ϭ ␦n ( A4) Th is is a fun damen tal equation of motion for MECHANICS 431 beh avior O n th e left is th e force of an in cen tive—its value times th e drive level of th e organ ism—an d on th e righ t is th e n umber of respon se-secon ds it is required to sustain ( Th e complete equation is ␨vh ϭ ␦n, wh ere ␨ is a measure of th e couplin g between in cen tives an d beh avior, as determin ed by th e tin gen cies o f r ein fo r cem en t; see Killeen , 1994 In th e presen t treatmen t, ␨ is assumed to be stan t at 1.0.) Un der th e lin ear drive assumption ( Equation s or A1) , ed, v will ch an ge with it, over at least part of its ran ge The exponential model In th e case of an expon en tial relation between deprivation an d h un ger, Equation s A3 an d A4 develop in to v ␥[ d ϩ ( M Ϫ mR) t] ϭ ␦n with th e average depletion : M ϭ d0/ tЈ ϩ M, an d m, ␥Ј, v, tЈ Ͼ Th is is th e basic equation of prediction for session averages un der th e expon en tial assumption Th e parameter ␥’ is th e product of th e restorin g force an d tЈ Th e cur ves it gen erates are skewed parabolas, wh ich fit man y of th e data better th an th e lin ear model Th e sideration s of th e previous section on session duration an d magn itude man ipulation s apply h ere also The general drive model Un der extreme deprivation , drive n o lon ger in creases expon en tially with furth er deprivation , but approach es some maximum ( i.e., is sigmoidal) an d may even decrease due to in an ition ( or, in th e case of drugs, due to with drawal) For such extreme deprivation dition s, oth er fun ction s ( e.g., th e Weibull distribution s) migh t be a more appropriate model of th e relation between drive an d deprivation Let us write th e appropriate fun ction of deprivation as h ϭ f [ d] , an d its in verse as d ϭ f Ϫ1[ h] ; th en Equation A4 becomes: We again use th e ratio SFF ( R ϭ B/ n) to elimin ate R, an d rearran ge to get Bϭ ΂ ΃ n n MϪ , m vЈ ( A5) wh ere M ϭd0/ t ϩ M, an d vЈ ϭ v␥t/ ␦, with m, t, ␦ Ͼ This is Equation 15 in the text We may derive the session-average rates by replacing t with tsess/ in the above equations For long sessions, d0/ t becomes negligible and may be omitted, especially in the case of closed economies; conversely, for short sessions and open economies, M may be omitted In general, M may be treated as a free parameter representing average depletion over the course of a session ( part or all of which may be offset by the average repletion during the session, mR) In experimen ts th at termin ate after a fixed n umber of rein forcers, th e value of t ϭ tsess will ten d to covar y with n so th at th e paren th etical term will n ot ch an ge greatly with ch an ges in th e sch edule requiremen t ( n) or un it price ( n/ m) Th is is especially true in closed econ omies in wh ich th e in itial deficit d0/ t is small In th at case, respon se rate will be a mon oton ic fun ction of n/ m In experimen ts th at termin ate after a fixed amoun t of time, respon se rate will be a quadratic fun ction of n, as sh own by Equation A5 If th e magn itude of th e in cen tive, m, is man ipulat- v[ e␥Ј(M Ϫ mR ) Ϫ ␪] ϭ ␦n ; again substitute th e ratio sch edule feedback fun ction an d rearran ge to get Bϭ [ ] ΂ ΃ n ␦n M Ϫ log ϩ␪ , m ␥Ј v ( A6) v f [ d ϩ ( M Ϫ mR) tЈ] ϭ ␦n, wh ose solution is Bϭ [ ] ΂ ΃, n ␦n M Ϫ f Ϫ1 m tЈ v ( A7) with , as before, M ϭd0/ tЈ ϩ M, an d m, v, tЈ Ͼ ... animals responding throughout the terminal links Segmenting responding will thus put the greatest leverage of motivation on the earliest segments (See Williams, Ploog, & Bell, 1995, for further... values of the models’ basic parameters, and to evaluate the adequacy of one model against alternate models (e.g., the linear vs exponential drive models) Surgically opening the loop An oth er way of. .. variable, an d in use the measured rates of reinforcement in our equations of prediction The measured rates of instrumental and contingent behavior are the variables compared by economic theorists such