risky theories—the effect of variancs e on foraging … · lumba livid) are given a choice ......

AMER. ZOOL., 36:402-434 (1996)

Risky Theories—The Effects of Variance on Foraging Decisions1

ALEX KACELNIK AND MELISSA BATESON

Department of Zoology, University of Oxford, South Parks Road,Oxford OX1 3PS, United Kingdom

SYNOPSIS This paper concerns the response of foraging animals to vari-ability in rate of gain, or risk. Both the empirical and theoretical literaturesrelevant to this issue are reviewed. The methodology and results fromfifty-nine studies in which animals are required to choose between for-aging options differing in the variances in the rate of gain available aretabulated. We found that when risk is generated by variability in theamount of reward, animals are most frequently risk-averse and sometimesindifferent to risk, although in some studies preference depends on energybudget. In contrast, when variability is in delay to reward, animals areuniversally risk-prone. A range of functional, descriptive and mechanisticaccounts for these findings is described, none of which alone is capableof accommodating all aspects of the data. Risk-sensitive foraging theoryprovides the only currently available explanation for why energy budgetshould affect preference. An information-processing model that incorpo-rates Weber's law provides the only general explanation for why animalsshould be risk-averse with variability in amount and risk-prone with de-lay. A theory based on the mechanisms of associative learning explainsquantitative aspects of risk-proneness for delay; specifically why the delaybetween choice and reward should have a stronger impact on preferencethan delays between the reward and subsequent choice. It also explainswhy animals should appear to commit the "fallacy of the average," max-imising the expected ratio of amount of reward over delay to reward whencomputing rates rather than the ratio of expected amount over expecteddelay. We conclude that only a fusion of functional and mechanistic think-ing will lead to progress in the understanding of animal decision making.

INTRODUCTION 15-S fixed delay to water and pecks on theWe begin with three examples to intro- o t h e r l e a d t o a variable delay to water av-

duce the behavioral phenomenon we are eragmg 15 s, the birds prefer the variableseeking to understand. First, if bumblebees d e l a y k ey ( C a s e et al- 1 9 9 5 ) - A n d thirdly,(Bombus edwardsi) are given a mixed array w h e n yellow-eyed juncos (Junco phaeono-of two types of artificial flowers one of "«) «« § i v e n a c h o i c e o f t w o feeding sta-which provides a constant volume of 0.1 jxl tions, o n e o f w h i c h o f f e r s f o u r m i l l e t s e e d s

of nectar, and the other a variable volume o n every visit and the other a variable num-with 1 |xl on 10% of encounters and nothing b e r that i s either one or seven seeds withon the other 90%, the bees show a strong 5 0 % probability, the birds' preference de-preference for the constant type (Wadding- pends on the ambient temperature: at 1°Cton et al, 1981). In contrast, if pigeons [Co- they prefer the variable option, but at 19°Clumba livid) are given a choice of two keys, they prefer the constant option (Caraco etarranged such that pecks on one lead to a al, 1990).

In all three cases these animals are facedwith two options with apparently equal

'From the symposium on Risk Sensitivity in Behav- f a t e s f ^ b different variances. Thewral Ecology presented at the Annual Meeting of the . &

American Society of Zoologists, 4-8 January 1995, at variation IS experimentally programmed toSt. Louis, Missouri. occur at the level of individual choices such

402

RISKY THEORIES 403

that it is impossible for a forager to knowexactly what it will obtain if it chooses thevariable option; all it can theoretically learnis the probability distribution of the possi-ble outcomes of making such a choice. Inthe foraging literature this type of environ-mental stochasticity is usually referred to asrisk. The word risk is also used sometimesin connection with danger of predation, butin the present context we use it exclusivelyto indicate variance. The above examplesare not isolated cases, but are representativeof a whole area of research by both psy-chologists and behavioural ecologists de-voted to understanding the effects of riskon the foraging decisions made by animals.In general, if an animal is allowed to choosebetween two foraging options that differ inrisk, the chances are that it will show a pref-erence for one option over the other evenif this means choosing the option offeringa somewhat lower average rate of gain. Inother words animals tend to be risk-sensi-tive.

From an evolutionary point of view theseresults are important because they challengethe basic assumptions of classical optimalforaging theory. In Charnov's (1976a, b)two seminal models it is assumed that theforager is adapted to a homogeneous habitatwith stochastic but stable properties inwhich fitness is an increasing function ofthe rate of energy gain per unit of timespent foraging. Both of these models pro-vide deterministic rules that cope with en-vironmental variability by operating withonly the averages of the relevant variablessuch as the average energy content of preyitems, the average time taken to handle preyitems or the average travel time betweenpatches. However, the results describedabove make it clear that considering onlyaverage values of the variables involved inforaging decisions is insufficient, since ifthe foragers only had access to estimates ofthe average nectar volume, the average timedelay to water or the average number ofseeds they should have treated the fixed andvariable options as equivalent.

Within behavioral ecology, the dominantevolutionary framework for explaining an-imals' responses to variance is risk-sensi-tive foraging theory. This body of theory

depends on assuming that there is a non-linear relationship between the rate of gainassociated with each prey item and fitness(for a review see McNamara and Houston,1992; Smallwood, 1996). Given a non-lin-ear function, F(x), Jensen's inequality statesthat for a range of values of x the averageof ¥{x) is not equal to F(x) where x is theaverage of the values of x. This producesrisk-sensitivity as follows. If a forager isfaced with two patches offering equal av-erage rates of gain and if the function re-lating rate of gain to fitness is positivelyaccelerated, then, because deviations abovethe average give greater fitness gains thandeviations below the average, it pays to berisk-prone. Conversely if the function isnegatively accelerated then by the same rea-soning symmetrical deviations from the av-erage cause fitness losses, and it pays to berisk-averse.

A paradigmatic risk-sensitive foragingmodel (Stephens, 1981) considers a smallbird that has to reach a minimum thresholdof reserves by nightfall if it is to surviveuntil morning. This threshold provides thenecessary non-linear relationship betweenrate of gain and fitness. As in the junco ex-ample, it is assumed that the bird is facedwith two foraging options offering the sameexpected number of seeds with one fixedand the other variable. If the fixed optionoffers a rate of gain sufficiently high to takethe bird above the threshold {i.e., it is on apositive energy budget) then it should berisk-averse, whereas if the rate is not suf-ficiently high {i.e., the bird is on a negativeenergy budget) then the bird's only chanceof survival is to be risk-prone and gambleon the variable option giving an above av-erage rate of gain. This has been summar-ised in the daily energy budget rule (Ste-phens, 1981) which states that a forager ona positive budget should be risk-averse buta forager on a negative budget risk-prone.

Our aim in this paper is to review someof the theoretical frameworks proposed toaccount for the phenomenon of risk sensi-tivity. While we show that risk-sensitiveforaging theory has had some success, ourprimary aim is to demonstrate that there area number of alternative frameworks that de-serve equal attention. We bring together

404 A. KACELNIK AND M. BATESON

several lines of thought based on functional,mechanistic and descriptive considerationsthat, while being clearly connected, haveremained isolated from each other. In somecases we show that these theories make dif-ferent predictions about how foragersshould respond to risk. For example, theydiffer in their predictions relating to the ef-fects of the energy budget of the subjects,which dimension is risky (since this can beeither the amount of a commodity or theforaging time associated with obtaining it),and even details of the procedure such ashow stimuli are used to signal the differentoptions. Before launching into the theory,we therefore start by reviewing what is cur-rently known about the responses of realanimals to risky choices.

RISK-SENSITIVE FORAGING: A CRITICALREVIEW OF THE EVIDENCE

Our aim in this section is to identifywhether there are any consistent patterns inthe responses of animals to risk. The reviewencompasses both the psychological andecological literatures, however we have ex-cluded the vast literature on human gam-bling behavior since we are doubtful wheth-er some of the factors known to affect hu-man decision making, such as for examplethe exact words used to frame a question,or the dress code of a casino (for examplessee Kahneman and Tversky, 1984; Wage-naar, 1988), have any direct equivalents inanimals. Specifically, we have chosen to re-view the results of experiments in whichanimals are given a choice between two for-aging options that offer different levels ofvariance in the rate of gain available. Inmost cases the options are arranged suchthat the average rate is the same in bothoptions (by rate we mean the currency ofclassical optimal foraging theory which islong-term rate, defined as the ratio of ex-pected gain divided by expected foragingtime), however in some experiments the av-erages may also differ in order that infor-mation can be obtained on how animalstrade-off average rate of gain against vari-ance in rate of gain. In these latter cases itis often possible to extrapolate whether thesubjects would be risk-averse or risk-prone

were the average rate in the two options infact the same.

As well as recording the preferences ofthe animals, studies were scored accordingto a number of variables that may influencethe extent or direction of risk sensitivity.Table 1 includes the following informationfrom each study: the species; the dimensionthat was variable or risky; the choice pro-cedure employed; and finally, whether thestudy involved manipulation of the ener-getic (or water) budget of the subjects. Weare particularly interested in whether thepredictions of the energy budget rule de-scribed above are born out by experimentalevidence since, as we will argue later, thisprediction is currently unique to risk-sen-sitive foraging theory. There are manyproblems inherent in trying to extract gen-eral patterns from comparative data, and inthis case it is particularly likely that thevarying methodologies employed in the dif-ferent studies affected how the animals be-haved. We therefore begin by discussingtwo aspect of methodology that are likelyto affect animals' responses to risk.

The dimension that is riskyRisk can be generated experimentally in

a number of ways that can be broadly di-vided into varying either the amount offood available in a foraging option or thetime delay associated with obtaining food.While both types of manipulation are ca-pable of generating risk in the rate of gainassociated with an option, it is important forboth theoretical and practical reasons to dif-ferentiate between the two types of risk.From a theoretical point of view the dis-tinction is important because risk-sensitiveforaging theory makes different predictionsfor variable amounts and delays in somecircumstances (see later), and also some ofthe alternative theories we consider differin their treatment of the two sorts of vari-ance.

From a practical point of view the meansused to generate risk can affect the rate ofgain experienced by the subjects. It is im-portant to establish empirically that there isa linear relationship between the amount offood programmed by the experimenter andthat taken by the animal. If we start by con-

RISKY THEORIES 405

sidering variability in amount of reward,Table 1 includes studies in which variabilityis in the number of similarly sized fooditems, the duration of access to a food hop-per, and the volume or concentration of nec-tar delivered. Although many of the exper-iments claim that the mean amounts ob-tained from constant and variable optionswere equal, in the cases where amount iscontrolled by the length of access given toa food hopper this is rarely verified (e.g.,Essock and Rees, 1974; Menlove et al.,1979). In fact, it has been demonstrated thatthe amount of grain consumed by pigeonsfeeding from a hopper is not always a linearfunction of the length of hopper access (Ep-stein, 1981). An exception is the study onstarlings by Reboreda and Kacelnik (1991)in which the necessary linear relationshipbetween duration of access and grams offood consumed is found over the range ofdurations they used in an experiment. Aswith amount, delay to obtaining food canalso be controlled in a number of ways. Inthe language of operant psychology thestudies in Table 1 use either interval, timeor ratio schedules (operant terminology isexplained in the Appendix). However, theprogrammed delay need not necessarilyhave a linear relationship to the delay theanimal experiences. This is especially likelyto be a problem with interval and ratioschedules in which the length of the delayexperienced will depend on the rate atwhich the animal responds.

A second practical problem is that ex-periments that explicitly claim to varyamount almost invariably also vary the timeassociated with acquiring the food, becauselarge rewards will often take longer to de-liver, to handle and to consume than smallones. This source of variability is generallyignored, and can lead to apparent risk sen-sitivity in a long-term rate maximizer ifthere is a non-linear relationship betweenthe size of a food item and the time it takesto handle. For example, Possingham et al.(1990) have shown that a bee maximisingits long-term rate of intake should be risk-prone to variability in volume, given theobserved increasing decelerating relation-ship between the volume of nectar takenfrom a flower and the time spent on the

flower (Hodges and Wolf, 1981). This ef-fect of handling time will be most importantin animals such as bees where it comprisesa substantial portion of each foraging cycle.Effects of handling time are probably min-imised in the experiments in which amountis varied by manipulating the concentrationof a fixed volume of nectar (e.g., Wunderleand O'Brien, 1985; Banschbach and Wad-dington, 1994; Waddington, 1995), al-though even here there may be differencesin the time taken to acquire or process nec-tar of different concentrations. It is notablethat recent experiments on bees that use thistechnique to equalise the mean energy re-ceived from fixed and variable flowers havenot found any risk sensitivity (Banschbachand Waddington, 1994; Waddington, 1995;Perez and Waddington, 1996).

A final remark relates to the pre-1990studies of Caraco and his colleagues (Car-aco et al, 1980; Caraco, 1981; Caraco,1983) and others who have followed theirmethods (e.g., Moore and Simm, 1986). Inan attempt to maintain experimental ani-mals on the stated energy budget, the fol-lowing inter-trial interval was made pro-portional to the size of the food reward justobtained, thus maintaining the intake rate ofthe animal at a set level regardless of thevariance in the size of rewards. However, asecondary effect of this was that both delayto obtaining reward and amount of rewardwere variable in the variable option. Thusthese experiments confound the effects ofvariability in amount and delay. Caraco etal. (1990) subsequently addressed this crit-icism by using ambient temperature ratherthan rate of intake to control the energybudgets of the subjects, thereby placing theanimals on a positive or negative energybudget without changing the amount or tim-ing of feedings.

In summary, it is important to notice howrisk is generated in an experiment, since themeans used can influence both the theoret-ical model that is appropriate to the prob-lem and the long-term rate of intake expe-rienced by the subjects. Some cases of ap-parent risk sensitivity may be explainedsimply as the result of long-term rate max-imising.


The choice procedure and measure ofpreference

A variety of methods have been used topresent animals with choices of two forag-ing options and measure their resultingpreferences. The study by Cartar (1991) isunique in utilising naturally occurring dif-ferences in the variance in profitability oftwo wild flower species; all others rely ongenerating the reward distributions artifi-cially. In the simplest technique, the subjectis allowed to forage freely on an array ofdishes or artificial flowers of two differenttypes, generally identified by colouredmarkers. Preference is measured as the pro-portion of visits made to the flowers ordishes of each type. This procedure is oftenused with bees (Real, 1981; Waddington etal., 1981), and seed-eating and nectarivo-rous birds (Caraco, 1982; Wunderle andO'Brien, 1985; Barkan, 1990; Tuttle et al,1990). It has the advantage of mimickingnatural foraging well, but it allows limitedcontrol of the rate at which the subjects for-age and the reward distributions they ex-perience if they revisit depleted sites. Theseproblems can be solved by using just twofeeding stations, one for each option, thatcan be replenished after each visit {e.g.,Croy and Hughes, 1991; Banschbach andWaddington, 1994). A further extension in-volves scheduling the availability of food atthe two patches according to a discrete tri-als procedure, in which there is an inter-trialinterval {e.g., Caraco et al, 1980). Proce-dures of this type allow precise control ofthe animals' experience at the cost of re-duced resemblance to natural foraging. Amajor advantage is that an animal can beforced to experience both options in trialsin which no choice is given before thosetrials in which choice is measured. The de-velopment of side preferences, in which ananimal repeatedly uses one feeding stationindependently of the reward schedule it of-fers, is however often a problem (Ha et al,1990; Ha, 1991; Reboreda and Kacelnik,1991).

Many choice procedures are implement-ed using operant techniques; birds can betrained to peck keys or hop on perches forfood rewards, and mammals such as rats

can be trained to press levers. Choice ex-periments usually involve two operanda,and the animals are taught that either theposition of an operandum or a particularcue, such as the colour of light projectedonto a pecking key, is associated with a par-ticular distribution of rewards. Differentschedules of reinforcement can be pro-grammed on the operanda. In scheduleswhere the animal makes a choice with asingle response, preference is measured asthe proportion of times each of the optionsis chosen. Sometimes additional measuresthat may correlate with preference are alsotaken, such as the latency to accept eachoption in forced trials or pecking rate dur-ing a delay (Reboreda and Kacelnik, 1991;Bateson and Kacelnik, 1995a). A numberof studies have used titration procedures inwhich either the amount of food or the de-lay in the fixed option is adjusted until apoint is found where the subjects are indif-ferent between the fixed and variable option{e.g., Mazur, 1984; Mazur, 1986; Batesonand Kacelnik, 1996). This technique has theadvantage of providing a quantitative mea-sure of the value subjects attribute to thevariable option. It may also provide a moresensitive measure of preference.

Apart from the discrete trials procedurementioned above in which the subjects mayhave to peck a key or press a lever a singletime to register their choice, there are twoother types of operant choice procedure thatare frequently used: concurrent schedulesand concurrent-chains schedules. In con-current schedules, a separate schedule of re-inforcement is independently programmedon each operandum, and the subject is al-lowed to switch between the two, being re-warded whenever the requirements of theschedule on either of the operanda are met.Preference is measured as the proportion ofthe total time or total responses allocated toeach option. Concurrent schedules can beused to test preference for fixed versus vari-able amounts by making the two schedulesidentical apart from the amount of rein-forcement given {e.g., Hamm and Shettle-worth, 1987). Alternatively, if the delay toreward in one of the schedules is fixed andin the other it is mixed or variable, the tech-nique can be used to measure preference for

RISKY THEORIES 407

variability in delay or number of responsesrequired to obtain reinforcement. For ex-ample, Ha et al., used concurrent variableratio schedules, one of which had a highervariance than the other, to measure prefer-ence for variability in the number of re-sponses required per reward (Ha et al.,1990; Ha, 1991).

The concurrent-chains procedure in-volves an initial link in which two identicalratio or interval schedules are concurrentlyavailable on two operanda. When the re-quirement on one of these schedules is met,the other operandum becomes unavailableand the subject has to complete the terminallink schedule on the current operandum.For risk-sensitivity experiments the termi-nal links are generally a fixed-interval ver-sus a mixed-interval or variable-interval(Herrnstein, 1964; Davison, 1969; Davison,1972), a fixed-ratio versus a mixed- or vari-able-ratio (Fantino, 1967; Morris, 1986;Ahearn and Hineline, 1992), or a fixed-timeversus a mixed- or variable-time (Rider,1983a; Zabludoff et al, 1988). Preferenceis measured as the proportion of initial linkresponses made on each operandum, thusgiving a continuous measure of the strengthof preference for the fixed and variable ter-minal links in each trial.

There is some evidence to suggest thatthe preference obtained from an experimentis likely to be affected by the type of choiceprocedures and index of preference used.For example, Caraco (1982) compared theresults obtained from white crowned spar-rows choosing between fixed and variablesources of seeds in an experiment with dis-crete choice trials programmed at two feed-ing stations with the results from anotherexperiment in which the birds foraged free-ly from an array of dishes covered with dif-ferently coloured paper discs. He found thatin both experiments the birds were risk-averse, but for similar variance to mean ra-tios this preference was more extreme in thediscrete trials paradigm. Hamm and Shet-tleworth (1987) argue on the basis of theirresults that preferences are also likely to bemore extreme when measured with a dis-crete trials procedure than with a concurrentschedule. However this claim is difficult toassess from their data because factors other

than just the choice procedure, such as thevariance of the variable option, werechanged between their two experiments.Rider (1983*) tested preference for thesame fixed-ratio (FR50) versus mixed-ratio(MR1/99) schedule using both a concurrentprocedure and a concurrent-chains proce-dure. He found strong risk aversion in theformer experiment and risk proneness in thelatter. However, the risk aversion in the con-current schedules experiment is easily ex-plained because the rat only has to make asingle response to determine whether themixed-ratio is currently 1 or 99, and if it is99 it can switch to responding solely on thefixed-ratio lever. This experiment demon-strates that the results obtained from con-current schedules need to be interpretedwith caution.

General patterns in risk-sensitive foraging

Having highlighted some of the possiblecaveats in the interpretation of the availableevidence we proceed with the results of thestudies. Fifty-nine papers are summarisedin Table 1, and while this is not an exhaus-tive list we believe it is a fair representationof the literature. The table includes studieson a total of 28 species consisting of 10insects (all bees and wasps), two fish, 12birds and four mammals. Risk-sensitivepreferences are observed in all of these tax-onomic groups. However, in the light of ourearlier comments regarding the studies onbees, it is possible that some of the risk sen-sitivity reported in this group could in factbe explained as an outcome of unequal av-erage rates of gain in the two options. Allstudies involve artificial manipulation of ei-ther the subjects or their food sources; onlyone study uses wild unmanipulated animalsliving in their natural environment (black-capped chickadees, Barkan, 1990), and onlyone study uses a natural unmanipulatedfood source (two species of wild flowers,Cartar, 1991). Thirty-eight studies investi-gate the response of animals to variabilityin amount of food, 18 to variability in delayto food, and three studies compare the twotypes of variability. Studies differ in wheth-er or not they investigate the effects of ma-nipulating energy budgets; we shall start by


TABLE 1. Summary of experiments investigating animals' responses to variability in their food sources. Studiesare divided into those on insects, fish, birds and mammals.

SpeciesQuantity variable or risky,

and nature of reward Choice procedure

InsectsPaper wasp (Vespula

vulgaris).Bumblebee (Bombus

sandersoni)Bumblebee (Bombus

edwardsi)Bumblebee (Bombus

pennsylvanicus)Bumblebee (Bombus

occidentalism

Bumblebee (Bombusmelanopygus, B. mixtusand B. sitkensis)

Honey-bee(Apis mellifera ligustica)

Bumblebee (Bombusfervidus)

FishBitterling (Rhoderus

sericus)

Fifteen-spined stickleback(Spinachia spinachia)

BirdsBlue jay (Cyanocitta

cristata)

Grey jay (Persoreuscanadensis)

Grey jay (Persoreuscanadensis)

White-crowned sparrow(Zonotrichia leucophrys)



White-throated sparrow(Zonotrichia albicollis)

Dark-eyed junco (Juncohyemalis)

Volume of 40% sucrose



Volume of sucrose soln

Volume of sucrose solnNB Unequal means

Net profitability, however, mostvariance originates from nectarvolume

Sucrose concentration

Sucrose concentration

Number of trout pellets

Free foraging on array of artificialflowers




Artificial flowers

Free foraging on unmanipulatednaturally occurring flowers(seablush and dwarf huckleber-ry)

Discrete choices between twopairs of artificial flowers

Discrete choices between twopairs of artificial flowers

Discrete choices of two feedingpatches

Number of prey items (Artemia) Free foraging at two feedingpatches

Number of mealworm halves Discrete choice trials

Ratio to obtain 97-mg cat pellets Concurrent (VR, VR)

Ratio to obtain 97-mg cat pellets Concurrent (VR, VR)

Number of millet seeds



Number of thistle seeds


Discrete choices of two feedingstations

Free foraging on array of fooddishes


Free foraging on array of fooddishes


Dark-eyed junco (Juncohyemalis)

Yellow eyed junco (Juncophaeonotus)

Yellow eyed junco (Juncophaeonotus)

Yellow-rumped warbler(Dendroica coronata)




Number of Tenebrio larvae





RISKY THEORIES 409

TABLE 1. Extended.

Manipulation of energetic statusof subjects Results: response to risk

Reference (numbers are used foridentification in Figures 1 and 2)

No manipulation

No manipulation

No manipulation

No manipulation

Sucrose added to or removedfrom colony reserves

Sucrose added to or removedfrom colony reserves

Honey combs removed from oradded to colony

No manipulation

Risk-averse

Risk-averse

Risk-averse

Risk-averse

Risk-averse when reserves en-hanced, but risk-prone/no pref-erence when depleted

No preference when reserves en-hanced, but risk-prone whendepleted

No preference in either treatment

No preference

Real (1981)—1

Real (1981)—1

Waddington et al. (1981)—2

Real et al. (1982)—3

Cartar and Dill (1990)—4

Cartar (1991)—5

Banschbach and Waddington(1994)—6

Waddington (1995)—7

Prior deprivation and feeding ratemanipulated

Prior deprivation or satiation ma-nipulated

Risk-averse with short deprivation Young et al. (1990)—8but risk-prone with long depri-vation

Risk-averse when sated but risk- Cray and Hughes (1991)—9prone when hungry

No manipulationMaintained at 80% free feeding

weightsMean ratio requirement manipu-

lated under closed economyAs above plus manipulation of

day length and ambient temper-ature

Maintained on positive budgets


Prior deprivation and inter-trial in-terval manipulated




Ambient temperature manipulated


Day length used to manipulatepremigratory state

Risk-averse

Risk-prone independent of treat-ment

Risk-prone independent of treat-ment

Risk-averse

Risk-averse

Risk-averse on positive budgetsand risk-prone on negative bud-gets

Risk-averse

Risk-averse on positive budgets,no preference on balanced bud-gets and risk-prone on negativebudgets

Risk-averse

Risk-averse on positive budgetsand risk-prone/no preference onnegative budgets


Pre-migratory birds risk-averseand postmigratory risk-prone

Clements (1990)—10

Ha et al. (1990)—11

Ha (1991)—12

Caraco (1982)—13

Caraco (1982)—13

Caraco (1983)—14

Tuttle etal. (1990)—15

Caraco (1981)—16

Caraco and Lima (1985)—17

Caraco et al. (1990)—18

Caraco etal. (1980)—19

Moore and Simm (1986)—20

410

TABLE 1. Continued.

A. KACELNIK AND M. BATESON



European starling (SturnusvulgarL)

Length of access to hopper withturkey crumbs (amount)

Discrete choice trials scheduledon two keys (FI, FI)

European starling (Sturnusvulgaris)

Interval to turkey crumbs Discrete choice trials (FI, VI)





Bananaquit (Coerebaflaveola

Bananaquit (Coerebaflaveola)



Great tit (Parus major)

Black-capped chickadee(Parus atricapillus)

Pigeon (Columba livid)












Pigeon (Columba livia)



Number of units of turkey crumbs

Interval to turkey crumbs

Interval to turkey crumbsNB unequal means

Interval to turkey crumbsNB unequal means

Volume of 30% sucrose soln

Concentration of 10 (JLI sucrosesoln

Volume of 0.47 M sucrose soln

Volume of sucrose soln

Number of fly pupae

Number of sunflower seeds

Interval to hopper access

Amount of hopper access


Ratio to obtain mixed grain

Number of pellets

Ratio to obtain reward


Interval to reward


Ratio to obtain hopper access

Number of 75-mg pellets

Number of 20-mg pellets


Interval to scoop of water (0.3 or1.00 ml depending on condi-tion)

Time to 3-s hopper accessNB unequal means

Discrete choice trials scheduledon two keys (FI, FI)

Discrete choice trials scheduledon two keys (FI, VI)

Discrete choice trials scheduledon two keys (FI, MI) with ad-justing FI

Discrete choice trials scheduledon two keys (FI, VI) with ad-justing FI






Wild birds freely foraging on ar-ray of holes in logs

Concurrent chains scheduled ontwo keys (FI, VI)

Concurrent (VI, VI) scheduled ontwo keys

Concurrent (FR, FR) scheduledon two keys

Discrete choice trials (FR, MR)scheduled on two keys

Discrete choice trials scheduledon two keys

Concurrent chains (FR, VR)scheduled on two keys

Concurrent (FI, MI)

Concurrent chains (FI, MI)

Concurrent chains (FI, MI)

Concurrent chains (FR, MR)

Concurrent (VI, VI)

Discrete choice trials

Concurrent chains (FI, FI)

Concurrent chains (FI, VI)

Discrete choice trials (FT, MT),(FT, VT) or (FT, RT)

Adjusting procedure

RISKY THEORIES 411

TABLE 1. Extended. Continued.


Reference (numbers are used foridentification in Figures 1 and 2)

No intended manipulation, howev-er subjects differed in theirability to exploit hopper accesstime

No intended manipulation, howev-er subjects differed in theirability to exploit hopper accesstime

No manipulation

No manipulation

No manipulation

Risk-averse/no, preferenceCorrelation betwen aversion and

amount of food obtained acrosssubjects

Risk-prone. Negative correlationbetween proneness and amountof food across subjects

Risk-averse/no preference

Risk-prone

Risk-prone

Reboreda and Kacelnik (1991)—21

Reboreda and Kacelnik (1991)—21

Bateson and Kacelnik (1995a)—22

Bateson and Kacelnik (1995a)—22

Bateson and Kacelnik (1996)—23

Number of units of turkey crumbsmanipulated

No manipulation

No manipulation

No manipulationBudgets probably negativeNo manipulationBudgets neutralNo manipulationBudgets probably positiveRate of intake manipulated, but

budgets always positiveMaintained at 80% free feeding

weightsMaintained at 80% free feeding


weightsMaintained at 75-80% free feed-

ing weightsMaintained at 80% free feeding






weightsMean length of VI manipulated

Maintained at 80% free feedingweights


Prior state, volume of rewards andlength of sessions all manipu-lated


Risk-prone on both positive andnegative budgets


No preference

Risk-averse

Risk-averse

Risk-averse

Risk-averse

Risk-prone

No preference

Risk-prone/no preference

Risk-prone

Risk-prone

Risk-prone

Risk-prone

Risk-prone

Risk-prone

Risk-prone

No preference in all treatments

Risk-prone

Risk-averse

Risk-prone in all treatments

Risk-prone

Bateson and Kacelnik (in press)—24

Wunderle and O'Brien (1985)—25

Wunderle and O'Brien (1985)—25

Wunderle et al. (1987)—26

Wunderle and Cotto-Navarro(1988)—27

Stephens and Ydenberg (1982)—28

Barkan (1990)—29

Herrnstein (1964)—30

Staddon and Innis (1966)—31

Essock and Rees (1974)—32

Morris (1986)—33

Young (1981)—34

Ahearn et al. (1992)—35

Cicerone (1976)—36

Davison (1969)—37

Davison (1972)—38

Fantino (1967)—39

Hamm and Shettleworth (1987)—40

Mazur (1984, 1986)-^t3, 44

Menlove et al. (1979)—41

Case et al. (1995)—42

Mazur (1984, 1986)-^3, 44


TABLE 1. Continued.





MammalsRat (Rattus norvegicus)

Rat (Rattus norvegicus)


Rat (Rattus norvegicus)Rat (Rattus norvegicus)Rat (Rattus norvegicus)




Amount of hopper accessNB unequal meansAmount of hopper accessNB unequal means

Number of units (0.24 g) of foodor number of drops of water

Time to 45-mg pellets

Time to 45-mg pellets

Number of 45-mg pelletsTime to 45-mg pelletsRatio to obtain 45-mg pellets

Ratio to obtain 45-mg pellets

Number of food pellets

Number of scoops of water

Discrete choice trialsAdjusting procedureDiscrete choice trialsAdjusting procedure

Discrete choices in E-maze

Discrete choices in Y-maze (FT,MT)

Discrete choices in maze (FT,MT)

Discrete choices in mazeConcurrent chains (FT, MT)Concurrent (FR, MR)

Concurrent chains (FR, MR)

Discrete choice trials (FR, FR)


Rat (Rattus norvegicus) Time to 45-mg pellets Concurrent chains (FT, MT)



Rhesus monkey (Macacamulatto)

Common shrew (Sorexaraneus)

Common shrew (Sorexaraneus)

Round-eared elephantshrew (Macroscelidesproboscideus)

Number of 45-mgNB unequal meansNumber 45-mg pelletsNB unequal meansNumber of popcorn kernels

Number of mealworm segments Discrete choice trials

Discrete choice trials on two lev-ers

Discrete choice trials on two lev-ers


Number of mealworm segments

Number of mealworms


Discrete choice trials from twotrays

summarising the results of the studies thatdid not.

Studies that did not manipulate energybudgets

We list 22 studies in which the responseof animals to variability in amount is in-vestigated with no manipulation of energybudgets. A range of different responses to

variability is observed: the majority of thesestudies report risk aversion, while some re-port results close to indifference, and a fewreport risk proneness (Fig. 1). It is possiblethat some of this variation could be attrib-uted to the energetic status of the subjectsin line with the predictions of the energybudget rule. Of the risk-averse animalssome were certainly on positive budgets

RISKY THEORIES 413

TABLE 1. Extended. Continued.


Reference (numbers are used foridentification in Figures I and 2)

Maintained at 80% free feeding Risk-proneweights

Maintained at 80% free feeding No preferenceweights

Mazur (1985)-^t5

Mazur (1989) Experiment 1.—46

Rate of intake manipulated

No manipulation, but deprived be-fore sessions

No manipulation

No manipulationNo manipulationMaintained at 80% free feeding


weightsNumber of forced trials before

choices manipulatedNumber of trials in a session ma-

nipulated

Body weight and mean time delaymanipulated

Number of trials per session ma-nipulated

Number of pellets per reward ma-nipulated

No manipulation

Length of ITI manipulated

No manipulationSubjects probably on negative

budgetsPrior deprivation, ITI and ambient

temperature all manipulated

No preference with low rate of in-take, but risk-prone with high rateRisk-prone

Risk-prone

Risk-averse/no preferenceRisk-proneRisk-averse, but see text for why

this is misleadingRisk-prone

Risk-averse in all treatments

Risk-averse on positive and bal-anced budgets, no preferenceon negative budgets

Initially risk-averse, but subse-quently no preference/risk-prone when 85% free-feedingweight, but risk-prone/no pref-erence when below 85%NB changes in body weightwere confounded with changesin the variance of the variableoption

Increasing risk-aversion with few-er trials

Increasing risk-aversion with larg-er rewards



Risk-prone/no preference

Risk-averse/no preference on pos-itive budgets, and no preferenceon negative budgets

Leventhal et al. (1959)—47

Pubols (1962)—48

Logan (1965)—49

Logan (1965)—49Rider (1983a)—50Rider (1983fc)—51

Rider (19836)—51

Battalio et al. (1985)—52

Kagel et al. (1986ft)—53

Zabludoff et al. (1988)—54

Hastjarjo et al. (1990)—55

Hastjarjo et al. (1990)—55

Behar (1961)—56

Barnard and Brown (1985)—57

Barnard et al. (1985)—58

Lawes and Perrin (1995)—59

{e.g., Caraco, 1982; Turtle et al, 1990; Car-aco and Lima, 1985; Stephens and Yden-berg, 1982), however in one study the sub-jects were probably on negative energybudgets (Wunderle et al, 1987). Of therisk-prone animals, Barnard et a/.'s (1985)shrews were probably on negative energybudgets, and the other three studies (Essock

and Rees, 1974; Young, 1981; Mazur,1985) were on pigeons maintained at 80%of their free-feeding weights that could alsohave been on negative budgets. However,the difficulty of trying to explain the vari-ation seen is exemplified by the studies onpigeons: despite the fact that similar pro-cedures were employed and all the birds


Wg>

'•a"55oCD

E3

•i tZ15

1 0 -

F -

5

n -

AMOUNT

l23101315172225262728414956

72225313246495658

32344558

DELAY

| 51 |

222330333536373839434448495051

Risk preference

FIG. 1. The effect of the variable dimension (amount or delay) on the direction of risk-sensitive preferences.The data are from those studies in Table 1 that did not manipulate the energy budgets of the subjects. In caseswhere a study is ambiguous in its findings (e.g., "risk-averse/no preference" in Table 1) both outcomes areincluded. The numbers within the bars indicate which studies in Table 1 contribute to the bar.

were maintained at 80 or 85% of their free-feeding weights, two experiments foundrisk aversion (Menlove et al., 1979; Hammand Shettleworth, 1987), three indifference(Staddon and Innis, 1966; Essock and Rees,1974; Hamm and Shettleworth, 1987) andthree risk proneness (Essock and Rees,1974; Young, 1981; Mazur, 1985). Thisvariation can not be explained by any ob-vious differences in the schedules used.

We list 15 studies in which the effects ofvariability in delay to obtain reward wereinvestigated with no manipulation of energybudgets (Fig. 1). The results show that withinterval, time and ratio schedules, animalsare almost universally risk-prone (the onlyexception being the concurrent schedulestudy of Rider (1983&) discussed earlier, inwhich a misleading measure of preferencewas used). Given that the majority of thesestudies were carried out by psychologistsusing pigeons maintained at as low as 75%

of their free-feeding body weights, it is pos-sible that the subjects may have been onnegative energy budgets. However, thisseems unlikely because pigeons are rela-tively large birds that can be maintained on75-80% of their free-feeding weights forlong periods of time, and it is not clear howthese levels of deprivation relate to naturalbody weights and requirements in the wild.Also since the majority of the daily rationis generally received in the experiment, therate of intake experienced must be suffi-cient to result in a positive energy budget.

Three studies (Logan, 1965; Reboredaand Kacelnik, 1991; Bateson and Kacelnik,1995a) have compared responses to vari-ability in amount and delay. All have foundthat animals (rats and starlings) are risk-averse when variability is in amount butrisk-prone when variability is in delay, sup-porting the general trend observed betweenstudies (Fig. 1).

RISKY THEORIES 415

V)CD

T3

I»*—O

oE

12

10-

8 -

6 -

4 -

2 -

0

AMOUNT

89141618192057

45

21535559

62940475255

r

Effect of budget manipulation

FIG. 2. The size of the effect of budget manipulations. The data are from those studies in Table 1 that manip-ulated the energy budgets of the subjects. In cases where a study is ambiguous in its findings both outcomesare included. The numbers within the bars indicate which studies in Table 1 contribute to the bar.

Effects of energy budget manipulationsWe list 24 studies that manipulated the

energy budgets of the subjects: 18 of theseinvestigated risk-sensitive preferences whenvariability was in amount, five when vari-ability was in delay and one study whenvariability was in both dimensions (Fig. 2).Of the experiments with variability inamount, 14 found some evidence for a shifttoward risk-proneness when energy budgetswere reduced and towards risk aversionwhen they were increased, although onlyeight of these studies showed a completeswitch in preference between significantrisk proneness and significant risk aversion.Despite the apparent level of support for theenergy budget rule shown by the literature,we suspect that the real number of failuresto obtain the predicted switch in preferencewith budget is actually greater, since studiesthat fail to reject the null hypothesis of lackof effect of budget are bound to be harderto publish and are probably often not sub-mitted at all. Some failures may also be sal-vaged for publication if a convincing posthoc argument can be constructed for why aswitch in preference would not have beenpredicted. A possible example is Barkan's(1990) study involving treatments with twovery different rates of intake, in which hejustifies the continuing risk aversion of the

birds on the basis of careful calculationsshowing that they were always on positivebudgets.

Of the few experiments with variabilityin delay, all have met with failure to dem-onstrate convincing shifts in preference. Haand colleagues (Ha et al, 1990; Ha, 1991)tried without success to induce a switch inpreference in gray jays as did Bateson andKacelnik (in press) in starlings. Similar fail-ure was met by Case et al. (1995) usingwater as a reinforcer for pigeons and ma-nipulating water budgets. A study on ratsby Zabludoff et al. (1988) found some ev-idence for a switch in preference. The ratsbecame risk-prone as their body weightswere reduced from 85% to 75% of theirfree-feeding weights. However, this exper-iment is difficult to interpret because the de-crease in body weight is confounded withan increase in the variance of the more vari-able option, and other studies have shownincreasing risk proneness with increasingvariance in delay to reward. More convinc-ingly, Reboreda and Kacelnik (1991) reporta correlation between decreasing riskproneness and increasing efficiency at ex-tracting food during a period of hopper ac-cess in different individual starlings; how-ever this was not manipulated experimen-tally. Thus, there is little direct evidence


that energy budget affects risk sensitivitywhen variability is in delay to receivingfood. Instead, risk proneness for delayseems universal.

There is some suggestion in the birds andmammals that body weight might explainwhich species respond to budget manipu-lations. Of the studies that did not find ev-idence for the predicted shift most were onlarger bird species such as pigeons (Hammand Shettleworth, 1987; Case et al, 1995),jays (Ha et al, 1990; Ha 1991) and star-lings (Bateson and Kacelnik, in press) orrats (Leventhal et al, 1959; Battalio et al,1985; Kagel et al., 1986*; Hastjarjo et al,1990). It is possible that light species withfew reserves might be more likely to havebeen subject to selection for short-fall min-imisation and thus energy budget-associat-ed switches in risk-sensitivity. In an attemptto control for phylogenetic confounds suchas differences in metabolic rate betweenbirds and mammals, we analysed the rela-tionship between body weight and the ef-fects of budget on risk sensitivity across thebird species only, and found that changesin budget were more likely to result in ap-propriate switches in foraging preferencesin small birds (Fig. 3). However, a closerinspection of the data reveals that all of thestudies with variability in delay have beenconducted on larger species such as pi-geons, jays and starlings. Thus it is notclear at present whether the lack of an effectof budget in these studies is due to the di-mension that was varied or the bodyweights of the subjects since these two vari-ables are confounded.

To summarise our conclusions:

• There are enough well controlled studiesfor us to conclude that risk sensitivity isa real phenomenon. In a few cases (thebees in particular) the animals may haveexperienced unequal average rates. How-ever this does not apply in all studies.

• There is a general difference in animals'response to variability in amount and inthe delay to reward: animals are more of-ten risk-averse when variability is inamount but risk-prone when variability isin delay.

• There is growing support for energy bud-

•3

'(D

o

o

uuu -

100-

10-

14O

16OOO1918O20

cm21 21

cm40 42

24

A11 12

<f *°Size of effect of budget manipulation

FIG. 3. The size of the response of bird species to amanipulation of budget plotted against body weight.'Switch' indicates a switch in preference as predictedby the energy budget rule, and 'Some effect' indicatesa change in the right direction but not a completeswitch in preference, 'No effect' indicates either noeffect of budget or an effect in the opposite directionto that predicted. Body weights were taken from Dun-ning (1993). The numbers refer to the studies in Table1. The empty circles indicate studies that investigatedvariability in amount the filled circles variability indelay.

get having some role in determining thedirection of risk-sensitive preferenceswhen variability is in amount: animals onpositive energy budgets are often morerisk-averse and animals on negative bud-gets more risk-prone.

• There is little evidence for a comparableeffect when variability is in delay. How-ever, experiments with variability in de-lay have not been performed in the same(low body weight) species that show aswitch in preference with variability inamount.

PROBLEMS WITH TESTING RISK-SENSITIVEFORAGING THEORY

It is clear that the energy budget rule isinsufficient to explain the patterns of risk-sensitivity in the literature. However, de-spite the fact that the majority of experi-mental tests have focused on the energybudget rule, this rule is not a universal pre-

RISKY THEORIES 417

diction of all risk-sensitive foraging mod-els. Stephens' original risk-sensitive forag-ing model has spawned a number of vari-ants that explore the effects of modifyingvarious of his original assumptions. It is notour intention to give an exhaustive reviewof the theory here, since this has been doneelsewhere (e.g., see McNamara and Hous-ton, 1992), but rather to give a taste of thecurrent level of sophistication of the theoryand complexity of its predictions.

One of the most important constraints inStephens' model lies in the fact that the for-ager is only allowed to make a single for-aging decision, and is then required to stickwith this for the remainder of the day. Thisled to the criticism that risk pronenesswould be very rare because it would onlyoccur when the forager had a probability ofdying of over 50% that day (Krebs et al.,1983). However, if the forager is allowed tomake sequential decisions that can vary ac-cording its current state, which may be af-fected by the outcome of previous deci-sions, then risk proneness becomes far lessdangerous and therefore more likely. Thisis because a risk-prone forager that has arun of good luck that takes it back onto apositive trajectory can switch to risk aver-sion rather than chancing the possibilitythat it will drift back below the budget line(Houston and McNamara, 1982; McNa-mara, 1983; McNamara, 1984). Whether asingle or sequential choice model is morerealistic for a given foraging situation willdepend on the degree to which an animalcommits itself when it makes a choice. Sin-gle choice models may be more appropriateto large-scale patch choice decisions,whereas sequential choice models will beappropriate to modelling prey choice withina patch.

A second assumption of Stephens' modelis that the only way to die is by failing tomeet the critical level of reserves by night-fall. There is no possibility of starvingwhile foraging, which for small mammalssuch as the shrew is a very real danger. Ifthe model is modified so that a forager isassumed to forage continuously to stayabove the lethal limit of reserves, and if themean net gain is positive, then the optimalpolicy is always to be risk averse (McNa-

mara and Houston, 1992). The introductionof unpredictable interruptions to foragingsuch as bad weather or presence of a pred-ator can further change this prediction. Forexample, Barnard et al. (1985) have shownthat the need to insure against the possibil-ity of interruptions makes risk pronenessoptimal at intermediate levels of reserves.Continuous foraging models are furthermodified if a constraint is placed on themaximum level of reserves a forager canstore (McNamara and Houston, 1990).Houston and McNamara (1985) have com-bined the possibility of death by falling be-low a lethal boundary while foraging andthe need to build up reserves to survive thenight in a single model. In this case the op-timal policy is a compromise been those inthe separate models: to be risk-averse at alllevel of reserves except for a wedge shapedregion in the reserves-time space near todusk when it is optimal for an animal belowthe budget line to be risk-prone in order toget above the critical level of reserves.

A third limitation of Stephens' originalmodel, and in fact all of the others men-tioned so far, is that the optimality criterionhas been restricted to maximization ofprobability of survival. There may be ani-mals, particularly insects, that have the op-tion of using energy acquired for immediatereproduction. McNamara et al. (1991) haveused a model in which reproduction occursabove a certain level of reserves, and resultsin a reduction in reserves, to show that thepolicy that maximises lifetime reproductivesuccess is different from one that minimisesmortality. In the reproduction models risk-proneness can occur at high levels of re-serves since there are conditions underwhich a risk-prone decision could take aforager over a threshold above which it canreproduce (see also Bednekoff, 1996). Giv-en certain parameters, it can be shown thatas reserves increase, the optimal policychanges successively from risk aversion (toescape an immediate lethal boundary) torisk proneness (to have a chance of meetinga daily requirement) back to risk aversion(when the requirement can be reached butthere is no chance of reproduction) and fi-nally back to risk proneness again (whenthis offers the possibility of reproduction).


A final deficiency with the models dis-cussed so far is that they only consider vari-ance in amount of food. Zabludoff et al.(1988) have modelled choice between a for-aging option in which there is a fixed delayto food and one in which the delay has thesame mean but is variable. They show thatif the forager must reach a critical level ofreserves by nightfall, then the energy bud-get rule describes the optimal policy in thissituation. However, McNamara and Hous-ton (1987) have modelled the effect of bothvariability in amount and delay on foragingpreferences. They considered an animal for-aging within a finite time horizon (a day forexample), by the end of which it must haveexceeded a certain critical level of reservesto survive. They concluded that when vari-ability is in amount the animal should berisk-averse if its current level of reserves issufficient for it to be above the criticalthreshold at dusk, and risk-prone if they arebelow. However, for delay the picture ismore complex, and there are typically fourregions in the reserves-time space, two inwhich it is optimal to be risk-averse for de-lay, and two in which it is optimal to berisk-prone. Thus, there are regions in whichit is optimal to be risk-prone in time butrisk-averse in reserves and vice versa. Vari-ability in amount and time have differenteffects because although both affect vari-ance in rate of intake, for an animal with ashort time horizon variable delays eat un-predictably into the foraging time left in theday, whereas variable amounts do not.

The message from this brief overview ofrisk-sensitive foraging theory is that, asHouston and McNamara have stressed(Houston, 1991; McNamara et al, 1991;McNamara and Houston, 1992), there is nosingle model of risk-sensitive foraging andno single prediction. Notably, the energybudget rule is not a universal prediction ofrisk-sensitive foraging theory. This com-plexity has a number of ramifications forhow risk-sensitive foraging theory can betested. As pointed out by Houston and Mc-Namara (Houston, 1991; McNamara andHouston, 1992) failures to support the en-ergy budget rule are inconclusive, since itis always possible to claim that the wrongmodel has been tested. In our view this

damages rather than supports the value ofrisk-sensitive foraging theory as an explan-atory framework since it is often difficult toensure that the correct model is being testeda priori. A related problem is that it is prac-tically impossible to make quantitative pre-dictions from risk-sensitive foraging mod-els. Even if the correct model can be chosenfor a given situation, there will still be toomany unknowns to predict quantitativelyhow an animal should respond to a givenmanipulation. A related problem is that it isdifficult to assess the fitness benefits boughtby a shift in preference of a given magni-tude. For instance, Caraco et al.'s (1990)juncos switched from an average of 60%preference for variability under a cold re-gime to an average of 37% preference forit under thermoneutral conditions, but wehave no idea of the magnitude of the benefitaccrued by this switch.

A further problem with testing risk-sen-sitive foraging theory, and in fact most nor-mative models, is that it is necessary tomake assumptions about the mechanisms ananimal uses to assess important variablessuch as energy budget or rate of intake.While the many different manipulationsthat have been tried are all theoretically ca-pable of modifying energy budgets, thisdoes not mean that a shift in budget willnecessarily be registered by the animal,since this depends on the proximate mech-anism it uses to assess its budget. For ex-ample, in the natural habitat ambient tem-perature may be easy to measure and pro-vide a reliable correlate of energy require-ments. However, an animal that has evolveda rule-of-thumb to change its risk prefer-ence using ambient temperature may fail toregister a change in energy requirements in-duced by another means such as a periodof restricted access to food. At present wedo not know how animals assess their en-ergetic status, meaning that a failure todemonstrate an effect of an energy budgetshift on preference cannot be interpreted asimplying that the energy budget rule wouldnot predict behavior in the wild. A similarargument could be applied to the assess-ment of variance in rate of gain. This prob-lem is only partially solved by doing ex-periments in the field (e.g., Cartar, 1991),

RISKY THEORIES 419

since under more natural conditions it isvery difficult to control and measure the ex-perience of individual foragers, thus intro-ducing a different set of problems.

From this discussion it becomes clearthat it will be exceedingly difficult to testthe ecological validity of risk-sensitive for-aging theory by rejecting predictions ofmodels, since the rejection of a model canalways be attributed to some cause otherthan the general validity of the theory. Twodifferent tacks can be taken: either to testpredictions of risk-sensitive foraging theorythat are common to all of the models, or totest single predictions that while not com-mon to all risk-sensitive foraging modelsare unique to risk-sensitive foraging theory,and therefore if confirmed will suggest thatthe theory is applicable to behavioral deci-sions in at least some circumstances. Thislatter approach suffers from the problemthat other theoretical frameworks may sub-sequently be found that will predict thesame pattern of behaviour.

A candidate for a universal prediction ofrisk-sensitive foraging theory is that allrisk-sensitive foraging models depend onthe environmental variance being risky orunpredictable rather than just variable. Thuswe can predict that if the observed prefer-ences have evolved for the reasons pro-posed by risk-sensitive foraging theory, anoptimal animal should be risk-sensitiveonly in the face of unpredictable varianceand not predictable variance. We have re-cently tested this idea in starlings with theresult that the birds appear not to treat pre-dictable and unpredictable variable delaysdifferently in the manner predicted by risk-sensitive foraging theory (Bateson and Ka-celnik, in press). However, our predictionmay in practice be invalid, because pre-dictable variance is probably uncommon inthe natural world making it possible that an-imals have evolved a rule-of-thumb oftreating all variability as risk whether it ispredictable or unpredictable.

The energy budget rule emerges as thebest candidate for a prediction that is cur-rently unique to risk-sensitive foraging the-ory, and we suggest that a sound empiricaldemonstration of this would be strong evi-dence for relevance of the theory. Caraco

et al.'s (1990) demonstration of the pre-dicted switch in juncos has been heraldedas just such a test (e.g., Houston and Mc-Namara, 1990), however we claim belowthat the test has some weaknesses and de-serves replication and extension.

How much did the juncos know and whendid they know it?

Risk sensitivity predictions assume fullknowledge of the stable environmental sta-tistics (for a new model that does not seeMcNamara, 1996). It is our view that at-tempts ought to be made to demonstrateknowledge independently from preferences,lest the observed choices are due to uncer-tainty about the parameters of the problemand not to risk sensitivity. Here we ask howmany trials a forager has to experience be-fore it can estimate the probabilities of thealternative outcomes of a variable optionwith a given level of certainty. Consider aforaging option with two alternative out-comes, a small prey item programmed withprobability p and a large prey item pro-grammed with probability q = (1 — p). Ex-perimental animals have to estimate theseprobabilities from experience. Let the ani-mal experience n independent trials of thisoption, on a proportion ps of which it getsa small item and qs a large item. The con-fidence interval for the proportion, ps is giv-en by

Ps±

where z is the z-score corresponding to thedesired level of confidence e.g. for 95%confidence, z = 1.96. If the level of accu-racy is defined as ±d, where

d = z

then by rearranging the above expression itis possible to show that the number of trials,n, necessary to say that P(ps — d < p < ps+ d) > 0.95 is

z_d

For given values of z and d, the value of


TABLE 2. Percentage confidence in an estimate of pfrom an observed proportion based on different num-bers of trials for three levels of accuracy.

Number of

<n)

10204080

160320640

0.05

24.834.547.362.979.492.698.9

Accuracy of estimate (d)

0.1

47.363.079.492.698.9

100.0100.0

0.2

79.492.698.9

100.0100.0100.0100.0

n will be at a maximum when the productpsqs is at its maximum, i.e., 0.5-0.5 = 0.25.Assuming this scenario, to achieve 95%confidence that the true probability, p, lieswithin ±0.1 of the proportion experienced,ps, the animal would need to experience,and remember, the outcomes of 96 trials.This applies to most risk-sensitivity exper-iments because the two outcomes of thevariable option are often programmed tooccur an equal number of times in a sessionto ensure the animal gets equal experienceof both. Table 2 shows the percentage con-fidence in estimates of p based on differentnumbers of trials and at different levels ofaccuracy.

Caraco et al. (1990) tested juncos' pref-erences by giving each bird 40 trials a dayfor three days. The first 16 trials each daywere forced trials of which half were of thevariable option, therefore in the extremecase where an animal chooses the fixed op-tion exclusively in the choice trials it willonly have experienced the variable option24 times in the entire experiment. In factCaraco et al. report that many of the birdsexhibited similar behavior on all three daysof the test, suggesting that they had reachedasymptotic performance after only eight ex-periences of the variable option. It can beseen from Table 2 that the animals cannothave had a confident estimate of the param-eters of the variable option given thisamount of experience. McNamara (1996)has shown that uncertainty of the parametercan change the predictions of a risk-sensi-tive foraging model with the general effectof making risk proneness less likely. Twomeasures that can be taken to combat thisdifficulty are first, to provide evidence that

the subjects' preferences are stable, andsecond, to demonstrate that they know thepossible outcomes in the two options on of-fer. In the case of reward size this could bedone by examining whether when given ex-tra seeds the subject pauses after havingcollected the programmed number, and inthe case of delay to reward pecking ratescan be used to demonstrate whether thesubject knows when food is due {e.g., Bate-son and Kacelnik, 1995a).

To summarise our arguments in this sec-tion:

• Risk-sensitive foraging theory is a suiteof models that make different predictionsabout how animals should respond to vari-ability in both amount and delay dependingon the precise biological scenario assumed.The energy budget rule is not a universalprediction of all risk-sensitive foragingmodels.

• It will therefore be difficult to test thegeneral ecological validity of risk-sensitiveforaging theory by rejecting specific mod-els.

• The evidence for switches in prefer-ence with energy budget is the strongestsupporting evidence for risk-sensitive for-aging theory, however studies need to bereplicated since even the most cited haveweaknesses on close inspection.

ALTERNATIVE APPROACHES

Time discounting based on the probabilityof being interrupted

Our purpose in this section is to discussone effect of variability in delay to food.However, to introduce the argument westart by considering a human subject whois given a one-off choice between gaininga reward immediately and another, of thesame magnitude, to be delivered some timelater. In this context a one-off choice is onethat occurs once only in the life of the sub-ject. Note that at this stage we are not in-troducing any programmed variance in thedelays to reward.

One expects an immediate reward to bepreferred over a delayed reward for a va-riety of reasons. First, for suitable timescales and rewards such as money or off-spring in an expanding population, there is

RISKY THEORIES 421

an added value of early gains because themoney could be made to work or the off-spring could reproduce, so that at the endof the longer delay, a reward obtained ear-lier has increased by gaining compound in-terest. Second, there is the probability ofloss by interruption. If a promised rewardhas a chance of being lost during the wait-ing time, the longer the delay the greaterwill be the cumulative probability that thereward does not arrive. A delayed rewardwill have a lower expected value than animmediate one because expected value isthe result of multiplying its worth by theprobability of its being delivered. This ar-gument implies loss of value of the delayedreward, while the first one (interest) in-volves gain in value of the early reward.Other logically distinct reasons can be sug-gested for effects of delay on the value ofsingle rewards. For instance, if reward val-ue depends on the subject's state, and statemay change during the waiting time, thevalue associated with a delayed reward mayhave greater uncertainty because of unpre-dictability in the state of the subject itselfin the future. Finally, the delayed rewardmay be less desirable if the waiting timecannot be freely employed for other pur-poses. This would apply if the choice wasone of a series rather than a one-off event.By choosing an immediate reward, time issaved that can be used for making morechoices in the future. Similarly, when a re-ward is lost because of an interruption, thisreleases time for pursuing further rewards.For present purposes we do not distinguishbetween these various effects of delay, be-cause although they differ in nature, the ef-fects on the relative value of immediate anddelayed rewards may be similar.

The value of a delayed reward V, can bewritten as V = F(A, d) where A is the worthof the reward if obtained immediately, d isthe delay from the point of judgement andF is the discounting function (F(A, 0) = A).There is a sizeable theoretical and empiricalliterature on the shape of the discountingfunction, F (for a recent discussion seeMyerson and Green, 1995). A normativeapproach (Samuelson, 1937), predicts theshape of F given certain additional assump-tions. If the reason underlying discounting

is the probability of loss through interrup-tion, the discounting functions for rewardsof different magnitude are equal, the chanc-es of the reward being lost are constant perunit of waiting time, and the subject is fullyinformed of all the parameters of the prob-lem, then F ought to be a negative expo-nential function, since V depends on the cu-mulative probability that a loss might occurbetween the time of the choice and the timeof reward delivery. This approach has dom-inated foraging theory (e.g., Kagel et al,1986a; Stephens and Krebs, 1986; McNa-mara and Houston, 1987).

There have been some attempts to ex-plore how humans actually discount singledelayed events. In a recent example, Myer-son and Green (1995; see also Green andMyerson, 1996) gave subjects a choice be-tween two notional sums of money, si tobe delivered immediately and s2 to be de-livered after a delay, d. By systematicallyvarying s 1 for each pair of values of s2 andd they found the value of si at which eachsubject switched preference from one op-tion to the other. This gives an estimate ofthe relative value of the immediate and thedelayed rewards from which the shape ofthe time discounting function, F, can be ob-tained. In common with other similar stud-ies this experiment shows that the shape ofthe discounting function is hyperbolic rath-er than exponential, meaning that for a giv-en absolute increase in d the fraction of val-ue lost decreases as d increases (cf. an ex-ponential function where the proportionalreduction in value with an increase in d isindependent of d). Therefore, the empiri-cally determined discounting function is in-consistent with the function predicted underthe assumptions outlined above. Specifical-ly, the function that most closely fits avail-able data is of the form

V = A/(a + kd), (1)

where a and k are constants with a smalland k close to unity.

We now extend the argument to includea one-off choice between a fixed and a vari-able delay option. In the fixed option a re-ward of fixed magnitude A is promised fordelivery after a fixed delay df, while in thevariable option a reward of fixed magnitude


A is promised after a delay of either df — 8or df + 8, with equal probability. The ex-pected delay in the variable option is dv =0.5 [(df - 8) + (df + 8)] = ds. Note thatbecause we are dealing with a one-offchoice, the subject is told the parameters ofthe variable option rather than experiencingthe variance personally. The discounted val-ue of the fixed option is Vf = F(A, df), butthe subject may employ different algo-rithms to assign value to the variable op-tion. For instance, it may use

Vvl = F(A, dv) (2a)

or

Vv2 = 0.5[FG4, df- 8) F(A, df + 8)](2b)

In Equation 2a the subject discounts rewardvalue using the expectation of the delay,while in Equation 2b it averages the dis-counted value of each possible outcome.Thus Vvl = Vf, but for concave-up discount-ing functions, such as the empirically de-rived hyperbola or the predicted negativeexponential, by Jensen's inequality, Vv2 >Vv, = Vf. Thus, to understand how variabil-ity in delay affects preference, both theshape of the discounting function and thealgorithm used for computing expected val-ue (whether subjects use Equation 2a or 2b)come into play.

In the case of a one-off choice betweena fixed and a variable amount option thesituation is different. Consider a fixed op-tion with a reward of fixed magnitude Afpromised for delivery after a fixed delay df,and a variable option with a reward of ei-ther Af + 8 or Af- 8, with equal probability,promised after a fixed delay of d. As abovethe expected magnitude of the variable re-ward is Av = 0.5[(Af - 8,d) + (Af + 8,d)]= Af, and the discounted value of the fixedreward is Vf = F(Af, d). However, as longas F is a linear function of A, as is the casefor most of the putative equations for value,then F(AV, d) = 0.5[F(Af + 8) + F ^ - 8)].Therefore variability in amount per seshould have no effect on preference. If,however, A and d are correlated, as will bethe case if large food items take longer tohandle than small ones, then variability in

amount can affect preference through its ef-fects on delay. We return to this issue later.

We now turn to whether subjects shoulduse Equation 2a or 2b, regardless of theshape of the discounting function. A subjectfacing the one-off choice used in this ex-ample should use Equation 2b, because it isthe expected values of the two possible out-comes that count. However, when a seriesof repeated choices is involved the problemdiffers because the time lost during the de-lay to reward in one choice affects the timeat which the next choice can be made.Thus, when rate of reward per unit time isto be maximized, as in most of the prob-lems addressed by classical optimal forag-ing theory, the use of Equation 2a is to beexpected.

In summary, by combining the empiricalevidence for a concave-up time discountingfunction and the normative argument forusing the average of discounted payoffs(Equation 2b) we predict that in a one-offchoice a reward promised after a variabledelay ought to be preferred to a rewardpromised after a fixed delay equal to thearithmetic mean of the possible delays inthe variable option. In spite of the varietyof closely related tasks in the human liter-ature on self-control, we have not yet foundempirical tests of this prediction.

REPEATED CHOICES AND THE RATEALGORITHM

Hyperbolic discounting without variability

Many experiments on non-human ani-mals have examined the effect of length ofdelay and variability in delay on rewardvalue. These experiments have resulted inhyperbolic time discounting functions (Ma-zur, 1987; Rodriguez and Logue, 1988) andin preference for variable delays (Herrn-stein, 1964; Davison, 1972; Gibbon et al,1988; Reboreda and Kacelnik, 1991; Bate-son and Kacelnik, 1995a). Consequently, itis tempting to generalise the ideas from theprevious section and treat the animal resultsas equivalent to the human work on timediscounting using one-off choices. Thereare however problems in translating be-tween human and animal work becauseone-off choices with variance, as described

RISKY THEORIES 423

above, cannot be offered to animals. Hu-mans can be told the parameters for a one-off choice, but animal subjects must betrained by repeated exposure to the two al-ternatives. These are generally identified bydifferent stimuli followed by programmeddelays and reward amounts, thus choicesare among arbitrary conditioned stimulirather than the rewards themselves. Train-ing gives the subject information about theparameters of the options on offer and theopportunity to learn whether or not trialsare ever interrupted during the delay. Giventhat animals do not treat probabilistic re-wards (partial reinforcement) as they treatcertain rewards (Mazur, 1985), it is clearthat they can adjust to various probabilitiesof interruptions to foraging. It thereforeseems unparsimonious to assume that hy-perbolic discounting in animals that havebeen maintained on deterministic schedulesfor long periods is a response to the prob-ability of interruption. While it could be ar-gued that animals are hardwired to expectsome baseline level of interruptions, we in-vestigate whether there are alternative ex-planations for hyperbolic discounting in an-imals faced with a sequence of choices.

In a sequence of choices, immediate re-wards are more valuable than delayed onesbecause the time saved can be used to pur-sue further rewards. Longer delays imply aloss of foraging opportunity. An appropri-ate measure of value in such situations israte of gain, the ratio of expected gain overexpected time which is the currency of clas-sical optimal foraging theory. Similarly toEquation 1, Value = Rate = F(A, d) = Al(t+ d), where t includes time intervals otherthan d, such as handling time, inter-trial in-terval or travel time. Therefore rate dropshyperbolically with d. There is no need toinvoke any variability to predict hyperbolicdiscounting.

Repeated choices and variability in delayOnce we have a sequence of choices, we

can consider the effects of introducing vari-ability in d. For a stimulus that is pairedwith a variable delay, even in the absenceof uncertainty, the distinction betweenEquations 2a and 2b applies: value can becalculated either by discounting the average

reward size by the average of the delays, orby calculating the average of the discountedoutcomes. In the context of rate maximi-zation, the algorithm resulting from apply-ing Equation 2a is long-term rate, alsonamed the Ratio of the Expectations ofamount and time (RoE), and from Equation2b, short-term rate or the Expectation of theRatios of amount and time (EoR, Batesonand Kacelnik, 1995a; Bateson and Kacel-nik, 1996). These currencies were initiallydiscussed in the foraging literature in thecontext of the so-called "fallacy of the av-erages" (Templeton and Lawlor, 1981; Gil-liam et al., 1982; Turelli et al, 1982).

For one-off choices we claim that Equa-tion 2b is the appropriate currency, but witha sequence of choices the problem is dif-ferent. Here RoE (i.e., Equation 2a) char-acterises the benefit obtained over a forag-ing period and therefore it is this measureof rate that should be maximized. Accord-ing to RoE, two food sources have the samevalue if one gives a reward after a fixeddelay and another gives the same rewardafter a variable delay of the same averageduration. However, experimental resultsshow that animals have a strong preferencefor food sources with variable delays, andthe exact value of this preference is com-patible with assuming that variable foodsources are characterised by somethingclose to Equation 2b, or EoR (Mazur, 1984;Mazur, 1986; Bateson and Kacelnik, 1996).Also relevant here, the literature shows thatbees are risk-averse when there is variabil-ity in the volume of nectar present in flow-ers (Real, 1981; Waddington, 1981), andHarder and Real (1987) have claimed thatthis finding is compatible with bees usingEoR. The correlation between nectar vol-ume, A, and handling time, d, in bees re-sults in A appearing in both the numeratorand denominator of F resulting in a concavedown function relating the volume of nectartaken to intake rate. This relationship doesnot lead to risk sensitivity if bees are max-imising RoE, but if they are maximisingEoR then risk aversion is predicted by Jen-sen's inequality (see Caraco et al., (1992)for a full analysis of the effects of covari-ance between A and d on risk sensitivityunder maximization of EoR). The magni-


tude of this effect of variability in amountwill depend on the proportion of total for-aging time that is spent handling the preyitem. In bees this is large, however in manyof the bird studies the effects of handlingtime are likely to be swamped by longerinter-trial intervals, and variability inamount is unlikely to lead to detectablerisk-aversion even if there is a correlationbetween amount and handling time.

We do not know precisely the cost ofmaximising EoR rather than RoE, and itcould be argued that perhaps in natural en-vironments it may make little differencewhich algorithm an animal uses. If there isno variation in the denominator, t + d,Equations 2a and 2b become identical.However, measurements of the distributionof inter-prey intervals experienced by star-lings foraging on natural pastures suggestthat this argument cannot help us to under-stand why animals appear to maximiseEoR, since the birds would have experi-enced significantly reduced rates of ener-getic intake if foraging decisions were con-trolled by this algorithm (Bateson andWhitehead, 1996).

Differential weight of various timecomponents

From the perspective of rate maximiza-tion, all the time allocated to a feedingevent is unavailable for performing otheractions. Time lost pursuing a prey item andtime spent handling it cause the same lossof foraging opportunity, and this is why thedenominator in the equation for rate is thesum of all the expected times in a foragingcycle, travel, searching, handling etc. Thisapplies whether the subjects use RoE (asexpected for rate maximising) or EoR (asrequired to fit the evidence). Nevertheless,it has been shown (Snyderman, 1987) thattimes intervening between the arbitraryconditioned stimulus and the reward havemuch greater impact on the value of an op-tion than those following the reward. Afood item associated with a short choice-reward delay and a long post-reward delayis preferred over one in which the lengthsof the two delays are reversed such that thechoice-reward delay is long and the post-reward delay short. This strong impact of

choice-reward delays is also found in stud-ies using two sources, one of which hasvariable delays. For example, in an exper-iment with starlings one stimulus was fol-lowed by a fixed delay before food and an-other by either of two delays (Bateson andKacelnik, 1996). There were two additionaltimes in the foraging cycle, the reward time(or handling time) and the inter-trial inter-val. The fixed delay was adjusted by titra-tion until the subjects were indifferent be-tween the two food sources. We calculatedthe predicted fixed delay at which the twostimuli ought to be equally valuable ac-cording to six algorithms resulting from ap-plying either RoE or EoR, each with threecombinations of time components (namelyincluding only choice-reward delay, addingthe handling time and further adding the in-ter-trial interval). The point of indifferencewas very close to that predicted by EoRmaximization computed with only thechoice-reward delay.

Interruptions could be brought into thepicture by claiming that this predominanceof the period preceding the reward is dueto the possibility of natural interruptions,since in nature interruptions may be morelikely during this period than at other times.We do not favour this approach for two rea-sons. First, there is no evidence for this noris it likely that evidence will be forthcom-ing. Relative susceptibility to interruptionsduring various foraging components isbound to be highly situation-specific and in-accessible to research. Second, the resultsare obtained after long-term training with-out interruptions. Since laboratory animalscan respond appropriately when interrup-tions do occur, it is parsimonious to avoidthe involvement of this ad hoc postulate.

Summarising our arguments about timediscounting and rate algorithms:

• Time discounting based on interruptionsis only relevant to one-off choices, and itis not supported by the finding that thediscounting function appears to be hy-perbolic rather than exponential and ap-pears to be insensitive to the lack of in-terruptions.

• For sequences of choices without vari-ability, rate maximization predicts a hy-

RISKY THEORIES 425

perbolic relationship between value anddelay.

• When variability is present, rate maxi-mization favours Equation 2a (RoE); butthe empirical results favour Equation 2b(EoR).

• Maximization of EoR explains the hy-perbolic time discounting function andrisk-proneness when variability is in de-lay. It can also explain risk aversionwhen variability is in amount if there isa correlation between amount and han-dling time and if handling time makes upa significant portion of the foraging cy-cle.

• All of the rate maximising arguments failto account for the predominance of thedelay between choice and the delivery ofreward over other times in the foragingcycle.

Associative learning and delay variabilityIn this section we offer a mechanistic ex-

planation for risk proneness for delay, forthe use of EoR and for the special relevanceof the interval between stimulus and re-ward. We suggest that all of these findingscan be explained by considering the pro-cesses by which animals learn about causalrelationships in the environment (see Mo-ntague et al. (1995) for a related argument).Our idea is that the general principles ofassociative learning have evolved underbroader selective pressures than those act-ing on foraging decisions, and that theylead to deviations from optimality in someforaging tasks.

The link with associative learning comesfrom the fact that the training of subjects inforaging experiments follows a protocolcompatible with that used in standard stud-ies of Pavlovian conditioning. In a condi-tioning experiment the subject is exposed toan originally neutral, conditioned stimulus(CS) and after some time delay it receivesa meaningful, unconditioned, stimulus (US)such as a food reward. Rewards that followshortly after the stimulus onset strengthenthe association with the CS more efficientlythan those that occur after a longer delay.This makes functional sense since such amechanism is more likely to allow the an-imal to respond to causal relationships in

its environment (Dickinson, 1980). Thefunction relating both the rate of learningand its asymptotic strength to the length ofthe CS-US delay is also non-linear, and ap-pears to be hyperbolic. The hyperbolic re-lationship can be explained if, for exampleas has been shown by Gibbon et al. (1977),learning is proportional to the ratio of theinter-trial interval to the CS-US delay. Thisis adaptive, since this ratio indicates howmuch extra information the CS providesabout the timing of an impending reward.Alternatively, a hyperbolic relationshipcould be explained if the strength of theCS-US association depends on the ratio ofthe size of a prey item to the CS-US delay.If learning occurs each time a reward is re-ceived, the value of the association with theCS will approximate Equation 2b, becauseto learn according to Equation 2a, the sub-ject would have to remember the set ofCS-US delays, compute the average onthese intervals and then attribute value tothe CS. It is unlikely that these intervalsshould be stored before the animal knowsof their significance. Given a hyperbolic re-lationship, Jensen's inequality suggests thatan animal should learn more efficientlyabout a CS that is separated from its US bya variable delay than it will about a CS sep-arated from the US by a fixed delay of thesame mean duration. Working on the as-sumption that differences in the efficiencyof learning about different CSs may trans-late into preferences in a choice test be-tween the CSs, this idea could explain an-imals' preferences for options with variabledelays to reward. If the subject is trainedwith two stimuli, one followed by a fixedCS-US delay and another followed by ei-ther of two delays with the same mean asthe fixed one, as in our starling experimentmentioned above, the attribution of valueby the reward that follows a variable delaywill be greater, and the subject will showpreference for that CS.

Following this interpretation, the reasonwhy the interval between stimulus and re-ward is more influential than other periodsin the cycle is that learning occurs at thetime when food is encountered by retro-spective assignment of value to the CS. Thestimulus does not gain any associative val-


ue at the end of the handling time or theinter-trial interval. Therefore, this accountcopes with Snyderman's (1987) finding thatthe CS-US interval is of special importance.Thus, in summary, the learning model ex-plains the features of animals' responses tovariable delays, but it cannot currently ex-plain responses to variability in amount, al-though some modifications could allow this(see Montague et al., 1995). Also, it cannotbe used to explain the results from one-offchoice experiments in humans where nolearning is involved.

Psychophysics and memoryOur final perspective takes variability

into the head of the forager. So far we haveignored the problems of measurement inthat none of the ideas discussed above makeany reference to the accuracy with whichinformation is processed. Here we considerhow errors in perception or memory willaffect animals' responses to fixed and sto-chastic environments and how they couldgenerate sensitivity to variance in eitheramount or delay.

An attempt to analyse the effects of mea-surement error on optimal prey choice isprovided by Yoccoz et al. (1993). Theymodel the effects of perceptual error by as-suming explicit relationships between thereal energy content (G) and handling time(7") of each prey item and what the foragerestimates these to be (X and Y respectively).The random variables X and Y are con-structed from the random variables G andT by adding normally distributed errorswith a mean of zero such that X = G + EGand Y = T + ET. Introducing this type ofperceptual error into a sequential encounterdiet choice model generates several predic-tions concerning the behavioural tactics for-agers should employ. Of interest in thepresent context, foragers are predicted to besensitive to the variance in the expected en-ergy content of food items, preferring eitherenvironments with highly predictable ener-gy content of food items or fairly unpre-dictable energy content of food items evenwhen the resources are characterised by thesame mean values. We shall not expand onthese predictions because we question thevalidity of the assumptions used to derive

them. The model assumes that perceptualerror is independent of the magnitude of thequantity being perceived, and as we shallshow below this is contrary to what isknown to be the case.

Weber's lawPut simply, Weber's law says that the dif-

ference in stimulus magnitude required tosee two stimuli as different is proportionalto the mean value of the stimuli. This min-imum difference is called the Just Notice-able Difference or JND. The law is com-patible with modelling subjective estimatesas increasing in proportion to the logarithmof the stimulus magnitude. We shall not dis-cuss details of deviations of this law andthe problems of any attempt to model sen-sation. It will suffice here to accept that thelaw is successful in a variety of domainsand to go on to examine some of its con-sequences.

Let us start by assuming that an animalforms a representation of the magnitude ofsome stimulus, such as the size or the inter-capture interval for a prey type. Let us fur-ther assume for the moment that there is novariability in the objective experience,namely the prey has a fixed size and it takesalways the same time to find. On each en-counter with that prey type, in accordanceto Weber's law the subject will register avalue within the range of its true value plusor minus one JND. The distribution of val-ues experienced will be close to a normaldistribution centred on the true value of thestimulus, and with width proportional to it(because of the proportionality of the JND).This notion introduces a form of variabilitythat is generated by the individual itself, butthat leads to interesting interactions withenvironmental variance, as we shall see be-low.

Empirical evidence consistent with thisaccount comes from tasks in which the sub-ject reproduces the value of a foraging pa-rameter from its memory, or is required tomake a decision based on its memory for aforaging parameter. Because of their rele-vance for risk-sensitivity we describe twoexamples, one related to delays to food andthe other to amounts of food. In the first(Kacelnik et al., 1990; Brunner et al.,

RISKY THEORIES 427

1992), starlings were given a task simulat-ing foraging in patches with fixed inter-preyinterval and sudden depletion. Within eachpatch, an unpredictable number of preycould be obtained until the patch was ex-hausted, and the interval between succes-sive prey was constant. The birds couldonly detect patch exhaustion by the intervalsince the last food delivery: if the typicalinter-prey interval was exceeded, then thepatch would yield no more prey and theonly option was to give up and travel to anew patch. The birds learned this task andshowed giving up times that were propor-tional to the inter-prey intervals, and thestandard deviations of their giving up timeswere proportional to the mean intervals, asrequired if Weber's law applies to time de-lays.

In a second experiment (Bateson and Ka-celnik, 1995&), starlings faced a choice be-tween two sources of reward. One wasfixed (the standard) and the other gave anamount of food that increased when thestandard was preferred and diminishedwhen the opposite happened. The oscilla-tion of the amount in the adjustable alter-native gives a measure of the accuracy ofdiscrimination. Further, because the dis-crimination is not between two amounts offood placed in front of the animal, but be-tween two stimuli (coloured pecking keys)associated with these amounts, the task de-pended on memory. By changing the valueof the standard, it was possible to examinethe accuracy of this discrimination. As ex-pected, the spread of the oscillations of theadjustable alternative increased when thevalue of the standard was increased as re-quired if Weber's law also applies toamount of food.

Scalar expectancy theoryThe most immediate implication of We-

ber's law to the problem of variability inforaging is to give a new explanation forwhy food sources with variability in delayare preferred to alternatives with equalmean delay and no variance, and converselywhy food sources offering fixed amount offood are generally preferred to alternativesoffering variable amounts with the samemean (Reboreda and Kacelnik, 1991; Bate-

son and Kacelnik, 1995a; see also Perezand Waddington, 1996). Reboreda and Ka-celnik have developed a model based on theobservation that Weber's law affects mem-ory for both amount and delay. In theirmodel, which is based on Scalar Expectan-cy Theory (Gibbon, 1977; Gibbon et al,1984; Gibbon et al., 1988), the memoryformed for time intervals or amounts offood looks like a normal distribution withmean equal to the real value of the stimulusand standard deviation proportional to themean. The constant of proportionality re-lating the mean to the standard deviation{i.e., the coefficient of variation) is assumedto be fixed for a given subject in a givenexperiment.

In the analyses of variability discussedabove, variable food sources offer equalfrequency of two outcomes. According toReboreda and Kacelnik's version of SET, acoloured cup signalling a food source of-fering 1 or 7 seeds with equal probabilitywould be remembered as the joint memo-ries of 1 and 7 seeds. Similarly, a stimulusfollowed by food after either 5 or 15 s willbe remembered as the joint memories ofthese two intervals. However, because ofWeber's Law, a uniformly distributed mix-ture of intervals or amounts (as in these ex-periments) is represented in memory as apositively skewed distribution. This skeweddistribution is generated by combining thenormal distributions that represent each ofthe constituent elements of the mixture(Fig. 4).

Subjects are assumed to choose betweentwo options by retrieving a sample frommemory for each option, comparing thesesamples, and preferring the option offeringthe better sample (bigger reward size orshorter delay). Because the memory repre-sentations of variable options are skewed tothe right, whereas memory representationsof fixed options with the same mean aresymmetric around the true mean, in morethan half of comparisons the sample fromthe representation of the variable optionwill be smaller than the sample from therepresentation of the fixed option. For op-tions with equal long-term rates of food in-take the model predicts that an option of-fering variability in delay to food should be


FIXED OPTION VARIABLE OPTIONProbability ofexperience1.0 1

0.5 -

Probability densityin memory0.4 1

0.3-

0.2-

0.1 -

8

Probability ofexperience1.0

0.5 •

Probability densityin memory

0.4 1

Joint distribution formed byadding the two constituentdistributions.

8

FIG. 4. The upper two panels represent the experienced distribution of outcomes (amounts or delays) in a fixed(left) and variable (right) option. The lower tow panels represent the distributions that are assumed to be formedin memory as a result of the above experiences. Note the skew in the distribution of the memory for the variableoption that results from the constant accuracy with which the constituent stimuli are represented (this Fig. isreproduced from Bateson and Kacelnik, 1995a).

chosen more often than a fixed alternativeand that a source offering variability inamount of food should be chosen less oftenthan a fixed alternative.

Although the assumptions of this accountare supported by the data showing that We-ber's law applies to memory for bothamount and delays and its predictions forpreference for variability in delay (see alsoGibbon et al., 1988) and aversion to vari-ability in amount have been confirmed, themodel fails in some respects, which we listbelow, (i) Although the model predicts thatanimals should choose a fixed amount offood more often than a variable amount inour studies the risk-aversion for amountwas not found in the frequency of choicesbut on other measures of preference such asthe latency to choose and the number ofpecks made. It is likely that these results

depend on the particular value of amountvariance used in the tests and thus furtherexperiments with larger variance are re-quired (Bateson and Kacelnik, 1995a). (ii)Regarding delays, the qualitative agreementwith the model was much stronger, butthere is a quantitative discrepancy. Themodel predicts that animals should be in-different between a fixed and variable op-tion when the fixed option is equal to thegeometric mean of the two outcomes in thevariable option (Bateson and Kacelnik,1995a), whereas subsequent research hasdemonstrated that indifference occurs at ap-proximately the harmonic mean (equivalentto Equation 2b or EoR) (Mazur, 1984; Ma-zur, 1986; Bateson and Kacelnik, 1996).(iii) The model cannot accommodate theobserved effects of energy budget on pref-erence. As discussed above, these results

RISKY THEORIES 429

are not as well documented as static pref-erence for delay variability, but they do ex-ist. While the model is unable to produce aswitch in preference from risk aversion torisk proneness or vice versa, it could ac-commodate shifts in the extent of prefer-ence (such as those reported by Reboredaand Kacelnik, 1991) if it is assumed thatenergy budget could affect the degree ofperceptual error, (iv) In its present form themodel does not predict what animals shouldchoose if the alternatives differ in the vari-ance of both amount and delay simulta-neously, since it provides no way of com-bining predictions relating amount and de-lay. A more recent version of SET (B runneret al., 1994) addresses some of the aboveproblems, including that of the indifferencepoint and the integration of amount andtime, however it also brings with it newlimitations that we do not have space to dis-cuss here.

DISCUSSION AND CONCLUSIONS

Our aims in this paper have been two-fold: first to review the current state of ourknowledge of how animals respond to en-vironmental variability and second to de-scribe and compare a number of competingexplanations for this behavior, that althoughclearly related have seldom been broughttogether. One outcome of the literature re-view is the identification of methodologicalproblems that need to be considered wheninterpreting or planning experiments onrisk-sensitivity. We found that both thepresence and the direction of risk-sensitivepreferences can be influenced by whatcould appear to be trivial differences in thechoice procedure, the schedule of reinforce-ment, the means used to deliver rewards orthe index used to measure the subjects'preferences. However, in spite of these dif-ficulties we did find general patterns in risk-sensitive preferences that candidate theoret-ical explanations need to address. In brief,when variability is in the amount of reward:

• Animals are usually risk-averse, some-times indifferent to risk and rarely risk-prone.

• There is some evidence that energy bud-get affects the direction of preference.

And, when variability is in the delayto reward:

• Animals are always risk-prone.• There is currently little evidence to sug-

gest that energy budget affects the degreeof risk proneness.

• The time between a subject focusing ona stimulus or making a choice and re-ceiving an associated reward has a largerimpact on the value of a foraging optionthan other times in the foraging cycle.

• Animals are indifferent between a fixedand a variable delay when the expectedratio of amount over the stimulus-rewardinterval (EoR) is equal in the two op-tions.

Our review highlights substantial gaps inthe existing literature. There are few studiesthat compare variability of both amount anddelay in the same species and under thesame budget conditions. Of particular inter-est, studies investigating the effects of vari-ability in amount have used smaller speciesthan those investigating variability in delay.Therefore there has not yet been an ade-quate test of whether energy budget can in-fluence the direction of risk-sensitive pref-erences when variability is in delay to re-ward. To date no studies have investigatedthe dynamics of risk-sensitive preferences.This could be interesting since risk-sensi-tive foraging theory and our mechanisticmodel based on associative learning makeopposite predictions here. Risk-sensitiveforaging theory predicts that after a run ofgood luck on a variable option animals willhave improved their state more than after arun of bad luck, will be more likely to beon a positive budget, and thus more likelyto be risk-averse. In contrast, the learningmodel predicts that animals should have astronger preference for options that have re-cently rewarded them well.

The theories that we have considered toexplain the above phenomena rely on twomain types of explanation: functional (ornormative) arguments that consider the cir-cumstances under which risk-sensitive be-havior is adaptive, and causal argumentsthat consider what is known of the generalmechanisms used by animals to perceive,learn and remember information about the


environment. Risk-sensitive foraging theo-ry and the interruption-based rationale fortime discounting are strictly functional,whereas the other models combine causaland functional reasoning. In this latter cat-egory, empirically observed risk sensitivityis treated as a side effect of behaviouralmechanisms that are not direct or exclusiveadaptations to cope with risk. Favouringone of these models does not imply takingan anti-optimality stance, because themechanisms responsible for the observedsensitivity to variance are interpreted asproducts of evolutionary trade-offs due toselection at some broader level. For in-stance, our associative learning accountproduces risk proneness for variability indelay by assuming that the non-linear dropin the efficiency with which the associationbetween an arbitrary stimulus and a mean-ingful event is learnt, has evolved becauseof its suitability for detecting causal rela-tions in the environment. This non-linearityis present whether or not there is variability,but in the presence of variability strongerassociations are formed as a by-product. Ina different example, our discussion of ratealgorithms assumes that the selective forcehas been for rate maximization but that be-cause animals use a particular algorithm(EoR) to compute their average rate of in-take they show sensitivity to variability indelay. Similarly, the perceptual errors pos-tulated in the models of Yoccoz et al.(1993) and of Reboreda and Kacelnik(1991) can also be seen as the result of anevolutionary trade-off. It is likely that theerrors could be reduced if more time orneural processing capacity were devoted tothe representation of the relevant parame-ters, but that it is not adaptive to allocateresources in such a way. These mechanistichypotheses are not ad hoc additions solelyfor the purposes of explaining observed re-sults. The hypotheses are often based onknown features of animal cognition, andwill often lead to the generation of new pre-dictions that can be tested empirically jus-tifying this approach to optimality as a pro-gressive program of research (Kacelnik andCuthill, 1987; Mitchell and Valone, 1990).

Our stress on causal accounts of risk sen-sitivity over the purely functional models

may be seen as neither mainstream behav-ioural ecology nor particularly desirable,especially if it led to relinquishing thesearch for sophisticated and detailed behav-ioural adaptations. In this case we advocatethe approach because when judged overall,a consideration of the role of mechanismhas contributed more to behavioural re-search on risk than unfettered adaptationistthinking. Part of the reason for this is thatrisk-sensitive foraging theory is not a singlemodel but a variegated set of models per-tinent to different biological scenarios.These models make different predictionsabout how an animal should respond to riskdepending on a number of variables (Mc-Namara and Houston, 1992). The existenceof these alternative models makes risk-sen-sitive foraging theory both sophisticatedand powerful, but also very hard to testsince failures can often be attributed to amismatch between the experimental situa-tion and the scenario in which the animal'sbehavior has evolved. It is virtually impos-sible to have accurate knowledge of theecological parameters required to select theappropriate model to test, let alone makequantitative or even qualitative predictionsfor a given species. This has the conse-quence that negative results cannot be usedto reject risk-sensitive foraging theory. Apotential response to this problem is to findand test predictions that are general to allrisk-sensitive foraging models. One suchprediction is that risk-sensitive preferencesare due to uncertainty (true risk) and notjust to variance which can in some circum-stances be predictable.

A second reason why risk-sensitive mod-els may have been hard to test lies in thefact that adaptive behavior must be imple-mented by behavioural mechanisms thatwill themselves have evolved in specific en-vironments. Unless the conditions of a risk-sensitive foraging experiment are close tothose under which an adaptive response torisk evolved the mechanisms may fail toproduce an adaptive outcome. This problemcould be relevant for the assessment of vari-ables such as energy budget, variance inrate of gain or probability of interruption.These considerations are bound to be sig-nificant given that the vast majority of risk-

RISKY THEORIES 431

sensitive foraging experiments are per-formed in the artificial conditions of labo-ratories.

Mechanistic models have the obvious ad-vantage that they can be falsified under lab-oratory conditions. Our claims about therate-maximising algorithm under constraint(the use of EoR) would be abandoned if ourbirds maximised the arithmetic rather thanthe harmonic mean in a variable delay ex-periment. No speculation about the irrele-vance of the lab can save a failed mecha-nistic model. However, causal explanationscan be seen as unsatisfactory on their ownsince they shift the explanatory burdenelsewhere: showing that risk proneness fordelay may be a by-product of the way an-imals learn begs the question of why ani-mals learn this way.

To date none of the available theories ad-equately accounts for all the crucial behav-ioural observations in studies of the effectsof variability. Risk-sensitive foraging theo-ry does not provide any insight into whyrisk-aversion should be more commonwhen variability is in amount and riskproneness when it is in delay. The fact thatshifts in preference have sometimes beenobserved when budgets are manipulated re-mains the single empirical observation thatprevents us from abandoning the theory en-tirely. It is therefore crucial that these keyexperiments are replicated and extended.The interruption model, EoR and the asso-ciative learning model can all account forrisk proneness when variability is in delay,but they are unable to account for the gen-eral observation of risk aversion when vari-ability is in amount. The associative learn-ing model explains why the time delay be-tween focusing on a stimulus and receivingreward should be more important than othertimes in the foraging cycle. Although EoRmaximising can explain risk aversion tovariability in amount if there is a positivecorrelation between amount and handlingtime and if handling time is a large propor-tion of the foraging cycle. The SET modelpredicts risk aversion when variability is inamount and risk proneness when variabilityis in delay, but fails to predict indifferenceat the harmonic mean, and the shift in pref-erence with energy budget. Of course it is

possible that the phenomena described inthis review, although superficially related,may be the products of very different un-derlying mechanisms, or have evolved un-der different selective pressures, in whichcase they may never yield to one unifyingexplanation.

We close by asking the question ofwhether risk-sensitive foraging theory isstill a useful theoretical tool to guide be-havioural research. Our tentative reply isthat the jury is still out. We can concludethat in this area of research, in commonwith most others, the achievements of pure-ly evolutionary or purely mechanistic theo-rising are severely limited. A persistencewith Tinbergen's (1963) call for simulta-neous attention to both the functions andmechanisms of behavior is not optional butimperative; we can only make progress byfollowing his lead.

ACKNOWLEDGMENTS

Financial support was provided by theWellcome Trust (Research Grant 046101 toAlex Kacelnik and Advanced Training Fel-lowship to Melissa Bateson). We thank RalfCartar, David Krakauer and especially PeterSmallwood for helpful discussion and com-ments on the manuscript.

REFERENCES

Ahearn, W. and P. N. Hineline. 1992. Relative pref-erences for various bivalued ratio schedules.Anim. Learn. Behav. 20:407-415.

Banschbach, V. S. and K. D. Waddington. 1994. Risk-sensitivity in honey-bees—no consensus amongindividuals and no effect of colony honey stores.Anim. Behav. 47:933-941.

Barkan, C. P. L. 1990. A field test of risk-sensitiveforaging in black-capped chickadees (Pans atri-capillus). Ecology 71:391-400.

Barnard, C. J. and C. A. J. Brown. 1985. Risk sen-sitive foraging in common shrews (Sorex araneusL.). Behav. Ecol. Sociobiol. 16:161-164.

Barnard, C. J., C. A. J. Brown, A. Houston, and J. M.McNamara. 1985. Risk-sensitive foraging incommon shrews: An interruption model and theeffects of mean and variance in reward rate. Be-hav. Ecol. Sociobiol. 18:139-146.

Bateson, M. and A. Kacelnik. 1995a. Preferences forfixed and variable food sources: Variability inamount and delay. J. Exp. Anal. Behav. 63:313-329.

Bateson, M. and A. Kacelnik. 1995fc. Accuracy ofmemory for amount in the foraging starling (Stur-nus vulgaris). Anim. Behav. 50:431-443.


Bateson, M. and A. Kacelnik. Starlings' preferencesfor predictable and unpredictable delays to food:Risk-sensitivity or time discounting? Anim. Be-hav. (In press).

Bateson, M. and A. Kacelnik. 1996. Rate currenciesand the foraging starling: The fallacy of the av-erages revisited. Behav. Ecol. (In press).

Bateson, M. and S. C. Whitehead. 1996. The ener-getic costs of alternative rate currencies in the for-aging starling. Ecology 77:1303-1307.

Battalio, R. C , J. H. Kagel, and D. N. McDonald.1985. Animals choices over uncertain outcomes:Some initial experimental results. Amer. Econ.Rev. 75:596-613.

Bednekoff, P. A. 1996. Risk-sensitive foraging, fit-ness, and life histories: Where does reproductionfit into the big picture? Amer. Zool. 36:471-483.

Behar, I. 1961. Learned avoidance of non reward.Psych. Rep. 9:43-52.

Brunner, D., J. Gibbon, and S. Fairhurst. 1994. Choicebetween fixed and variable delays with differentreward amounts. J. Exp. Psych.: Anim. Behav.Processes 20:331-346.

Brunner, D., A. Kacelnik, and J. Gibbon. 1992. Op-timal foraging and timing processes in the starling,Sturnus vulgaris: Effect of inter-capture interval.Anim. Behav. 44:597-613.

Caraco, T. 1981. Energy budgets, risk and foragingpreferences in dark-eyed juncos (Junco hyemalis).Behav. Ecol. Sociobiol. 8:213-217.

Caraco, T. 1982. Aspects of risk-aversion in foragingwhite crowned sparrows. Anim. Behav. 30:719-727.

Caraco, T. 1983. White crowned sparrows (Zonotrich-ia leucophrys) foraging preferences in a risky en-vironment. Behav. Ecol. Sociobiol. 12:63-69.

Caraco, T. and S. L. Lima. 1985. Foraging juncos:Interaction of reward mean and variability. Anim.Behav. 33:216-224.

Caraco, T., W. U. Blanckenhorn, G. M. Gregory, J. A.Newman, G. M. Recer, and S. M. Zwicker. 1990.Risk-sensitivity: Ambient temperature affects for-aging choice. Anim. Behav. 39:338-345.

Caraco, T, S. Martindale, and T. S. Whittam. 1980.An empirical demonstration of risk sensitive for-aging preferences. Anim. Behav. 28:820-830.

Caraco, T., A. Kacelnik, N. Mesnick, and M. Smule-witz. 1992. Short-term rate maximization whenrewards and delays covary. Anim. Behav. 44:441-447.

Cartar, R. V. 1991. A test of risk sensitive foraging inbumble bees. Ecology 72:888-895.

Cartar, R. V. and L. M. Dill, 1990. Why are bumblebees risk sensitive foragers? Behav. Ecol. Socio-biol. 26:121-127.

Case, D. A., P. Nichols, and E. Fantino. 1995. Pi-geon's preference for variable-interval water re-inforcement under widely varied water budgets. J.Exp. Anal. Behav. 64:299-311.

Charnov, E. L. 1976a. Optimal foraging: Attack strat-egy of a mantid. Am. Nat. 110:141-151.

Charnov, E. L. 1976ft. Optimal foraging: The margin-al value theorem. Theor. Popul. Biol. 9:129-136.

Cicerone, R. A. 1976. Preference for mixed versus

constant delay of reinforcement. J. Exp. Anal. Be-hav. 25:257-261.

Clements, K. C. 1990. Risk-aversion in the foragingblue jay, Cyanocitta cristala. Anim. Behav. 40:182-195.

Croy, M. I. and R. N. Hughes. 1991. Effects of foodsupply, hunger, danger and competition on choiceof foraging location by the fifteen-spined stickle-back, Spinachia spinachia L. Anim. Behav. 42:131-139.

Davison, M. C. 1969. Preference for mixed-intervalversus fixed-interval schedules. J. Exp. Anal. Be-hav. 12:247-252.

Davison, M. C. 1972. Preference for mixed-intervalversus fixed-interval schedules: Number of com-ponent intervals. J. Exp. Anal. Behav. 17:169-176.

Dickinson, A. 1980. Contemporary Animal LearningTheory. Cambridge University Press, Cambridge.

Dunning, J. B. 1993. CRC handbook of avion bodymasses. Boca Raton, Florida.

Epstein, R. 1981. Amount consumed as a function ofmagazine-cycle duration. Behav. Anal. Lett. 1:63-66.

Essock, S. M. and E. P. Rees. 1974. Preference forand effects of variable as opposed to fixed rein-forcer duration. J. Exp. Anal. Behav. 21:89-97.

Fantino, E. 1967. Preference for mixed versus fixed-ratio schedules. J. Exp. Anal. Behav. 10:35—43.

Gibbon, J. 1977. Scalar expectancy theory and We-ber's law in animal timing. Psych. Rev. 84:279-325.

Gibbon, J., M. D. Baldock, C. Locurto, L. Gold, andH. S. Terrace. 1977. Trial and intertrial durationsand autoshaping. J. Exp. Psych.: Anim. Behav.Processes 3:264-284.

Gibbon, J., R. M. Church, and W. H. Meek. 1984.Scalar timing in memory. In J. Gibbon and L. Al-lan (eds.), Timing and Time Perception, pp. 52-77. New York Academy of Sciences, New York.

Gibbon, J., R. M. Church, S. Fairhurst, and A. Kacel-nik. 1988. Scalar Expectancy Theory and choicebetween delayed rewards. Psychol. Rev. 95:102-114.

Gilliam, J. F., R. F Green, and N. E. Pearson. 1982.The fallacy of the traffic policeman: A responseto Templeton and Lawlor. Amer. Nat. 119:875-878.

Green, L. and J. Myerson. 1996. Exponential versushyperbolic discounting of delayed outcomes: Riskand waiting time. Amer. Zool. 36:496-505.

Ha, J. C. 1991. Risk-sensitive foraging: The role ofambient temperature and foraging time. Anim.Behav. 41:528-529.

Ha, J. C, P. N. Lehner, and S. D. Farley. 1990. Risk-prone foraging behaviour in captive grey jays Per-isoreus canadensis. Anim. Behav. 39:91—96.

Hamm, S. L. and S. J. Shettleworth. 1987. Risk aver-sion in pigeons. J. Exp. Psych.: Anim. Behav. Pro-cesses 13:376-383.

Harder, L. and L. A. Real. 1987. Why are bumblebees risk-averse? Ecology 68:1104-1108.

Hastjarjo, X, A. Silcrberg, and S. R. Hursh. 1990.

RISKY THEORIES 433

Risky choice as a function of amount and variancein food supply. J. Exp. Anal. Behav. 53:155-161.

Herrnstein, R. J. 1964. Aperiodicity as a factor inchoice. J. Exp. Anal. Behav. 7:179-182.

Hodges, C. M. and Wolf, L. L. 1981. Optimal for-aging in bumblebees: Why is nectar left behind inflowers? Behav. Ecol. Sociobiol. 9:41-44.

Houston, A. and J. McNamara. 1985. The choice oftwo prey types that minimises the probability ofstarvation. Behav. Ecol. Sociobiol. 17:135-141.

Houston, A. I. 1991. Risk-sensitive foraging theoryand operant psychology. J. Exp. Anal, of Behav.56:585-589.

Houston, A. I. and J. M. McNamara. 1982. A se-quential approach to risk-taking. Anim. Behav.30:1260-1261.

Houston, A. I. and J. M. McNamara. 1990. Risk-sen-sitive foraging and temperature. Trends Ecol.Evol. 5:131-132.

Kacelnik, A. and I. C. Cuthill. 1987. Starlings andoptimal foraging theory: Modelling in a fractalworld. In A. C. Kamil, J. R. Krebs, and H. R.Pulliam (eds.). Foraging theory. Plenum Press,New York.

Kacelnik, A., D. Brunner, and J. Gibbon. 1990. Tim-ing mechanisms in optimal foraging: Some appli-cations of scalar expectancy theory. In R. N.Hughes (ed.), NATO AS1 Series, Series G: Eco-logical Sciences, Vol. G 20, pp. 63—81. Springer-Verlag, Berlin.

Kagel, J. H., L. Green, and T. Caraco. 1986a. Whenforagers discount the future: Constraint or adap-tation? Anim. Behav. 34:271-283.

Kagel, J. H., D. N. MacDonald, R. C. Battalio, S.White, and L. Green. 1986b. Risk-aversion in rats(Rattus norvegicus) under varying levels of re-source availability. J. Comp. Psychol. 100:95—100.

Kahneman, D. and A. Tversky. 1984. Choices, valuesand frames. Am. Psych. 39:341-350.

Krebs, J. R., D. W. Stephens, and W. J. Sutherland.1983. Perspectives in optimal foraging. In A. H.Brush and G. A. Clark (eds.), Perspectives in or-nithology, pp. 165-216. Cambridge UniversityPress, New York.

Lattal, K. A. 1991. Scheduling positive reinforcers. InI. H. Iversen and K. A. Lattal (eds.) Experimentalanalysis of behavior, part 1, pp. 87-134. Elsevier,Amsterdam.

Lawes, M. J. and M. R. Perrin. 1995. Risk-sensitiveforaging behaviour of the round-eared elephantshrew {Macroscelides proboscideus). Behav. Ecol.Sociobiol. 37:31-37.

Leventhal, A. M., R. E Morell, E. J. Morgan, and C.C. Perkins. 1959. The relation between mean re-ward and mean reinforcement. J. Exp. Psychol.59:284-287.

Logan, F. A. 1965. Decision making by rats: Uncer-tain outcome choices. J. Comp. Physiol. Psychol.59:246-251.

Mazur, J. E. 1984. Tests of an equivalence rule forfixed and variable reinforcer delays. J. Exp. Psy-chol.: Anim. Behav. Proc. 10:426-436.

Mazur, J. E. 1985. Probability and delay of reinforce-

ment as factors in discrete-trial choice. J. Exp.Anal. Behav. 43:341-351.

Mazur, J. E. 1986. Fixed and variable ratios and de-lays: Further tests of an equivalence rule. J. Exp.Psychol.: Anim. Behav. Proc. 12:116-124.

Mazur, J. E. 1987. An adjusting procedure for study-ing delayed reinforcement. In M. L. Commons, J.E. Mazur, J. A. Nevin and H. Rachlin (eds.),Quantitative Analyses of Behaviour: The effect ofdelay and of intervening events on reinforcementvalue, pp. 55—73. Erlbaum, Hilsdale, New Jersey.

Mazur, J. E. 1989. Theories of probabilistic reinforce-ment. J. Exp. Anal. Behav. 51:87-99.

McNamara, J. 1983. Optimal control of the diffusioncoefficient of a simple diffusion process. Math.Oper. Res. 8:373-380.

McNamara, J. M. 1984. Control of a diffusion byswitching between two drift-diffusion coefficientpairs. SIAM J. Cont. 22:87-94.

McNamara, J. M. 1996. Risk-prone behaviour underrules which have evolved in a changing environ-ment. Amer. Zool. 36:484-495.

McNamara, J. M. and A. I. Houston. 1987. A generalframework for understanding the effects of vari-ability and interruptions on foraging behaviour.Acta Biotheoretica 36:3-22.

McNamara, J. M. and A. I. Houston. 1990. Starvationand predation in a patchy environment. In B.Shorrocks and I. R. Swingland (eds.), Living in apatchy environment, pp. 23-43. Oxford Univer-sity Press, Oxford.

McNamara, J. M. and A. I. Houston. 1992. Risk-sen-sitive foraging: A review of the theory. Bull.Math. Biol. 54:355-378.

McNamara, J. M., S. Merad, and A. I. Houston, 1991.A model of risk-sensitive foraging for a reproduc-ing animal. Anim. Behav. 41:787-792.

Menlove, R. L., H. M. Inden, and E. G. Madden.1979. Preference for fixed over variable access tofood. Anim. Learn. Behav. 7:499-503.

Mitchell, W. A. and T. J. Valone. 1990. The optimi-zation research program: Studying adaptations andtheir function. Quart. Rev. Biol. 65:43-52.

Montague, P. R., P. Dayan, C. Person, and T. J. Se-jnowski. 1995. Bee foraging in uncertain envi-ronments using predictive hebbian learning. Na-ture 377:725-728.

Moore, F. R. and P. A. Simm. 1986. Risk-sensitiveforaging by a migratory bird {Dendroica corona-ta). Experientia 42:1054-1056.

Morris, C. J. 1986. The effects of occasional short(FR1) reinforcement ratios on choice behaviour.Psych. Rec. 36:63-68.

Myerson, J. and L. Green. 1995. Discounting of de-layed rewards: Models of individual choice. J.Exp. Anal. Behav. 64:263-276.

Perez, S. M. and Waddington, K. D. 1996. Carpenterbee (Xylocopa micans) risk-indifference and a re-view of nectarivore risk-sensitivity studies. Amer.Zool. 36:435-446.

Possingham, H. P., A. I. Houston, and J. M. McNa-mara. 1990. Risk-averse foraging in bees: A com-ment on the model of Harder and Real. Ecology71:1622-1624.


Pubols, B. H. 1962. Constant versus variable delay toreinforcement. J. Comp. Physiol. Psychol. 55:52-56.

Real, L. A. 1981. Uncertainty and pollinator-plant in-teractions: The foraging behavior of bees andwasps on artificial flowers. Ecology 62:20-26.

Real, L. A., J. Ott, and E. Silverfine. 1982. On thetrade-off between mean and variance in foraging:An experimental analysis with bumblebees. Ecol-ogy 63:1617-1623.

Reboreda, J. C. and A. Kacelnik. 1991. Risk sensitiv-ity in starlings: Variability in food amount andfood delay. Behav. Ecol. 2:301-308.

Rider, D. P. 1983a. Preference for mixed versus con-stant delays of reinforcement: Effect of probabilityof the short, mixed delay. J. Exp. Anal. Behav.39:257-266.

Rider, D. P. 1983b. Choice for aperiodic versus peri-odic ratio schedules: A comparison of concurrentand concurrent-chains procedures. J. Exp. Anal.Behav. 40:225-237.

Rodriguez, M. L. and A. W. Logue. 1988. Adjustingdelay to reinforcement: Comparing choice in pi-geons and humans. J. Exp. Psych.: Anim. Behav.Processes 14:105-117.

Samuelson, P. A. 1937. A note on the measurementof utility. Rev. Econ. Stud. 4:155-161.

Smallwood, P. D. 1996. An introduction to risk-sen-sitivity: Using Jensen's inequality to clarify evo-lutionary arguments of adaptation and constraint.Amer. Zool. 36:392-401.

Snyderman, M. 1987. Prey selection and self-control.In M. L. Commons, J. E. Mazur, J. A. Nevin andH. Rachlin (eds.), Quantitative Analyses of Be-haviour, pp. 283-308. Lawrence Erlbaum Asso-ciates, Hillsdale, New Jersey.

Staddon, J. E. R. and N. K. Innis. 1966. Preferencefor fixed vs. variable amounts of reward. Psychon.Sci. 4:193-194.

Stephens, D. W. 1981. The logic of risk-sensitive for-aging preferences. Anim. Behav. 29:628-629.

Stephens, D. W. and J. R. Krebs. 1986. Foraging the-ory. Princeton University Press, Princeton.

Stephens, D. W. and R. C. Ydenberg. 1982. Risk aver-sion in the great tit. In D. W. Stephens Stochas-ticity in foraging theory: Risk and information,Unpublished DPhil Thesis, Oxford University.

Templeton, A. R. and L. R. Lawlor. 1981. The fallacyof the averages in ecological optimisation theory.Am. Nat. 117:390-393.

Tinbergen, N. 1963. On aims and methods of ethol-ogy. Z. Tierpsychol. 20:410-433.

Turelli, M., J. H. Gillespie, and T. W. Shoener. 1982.The fallacy of the fallacy of the averages in eco-logical optimization theory. Amer. Nat. 119:879-884.

Tuttle, E. M., L. Wulfson, and T. Caraco. 1990. Risk-aversion, relative abundance of resources and for-

aging preferences. Behav. Ecol. Sociobiol. 26:165-171.

Waddington, K. D. 1995. Bumblebees do not respondto variance in nectar concentration. Ethology 101:33-38.

Waddington, K. D., T. Allen, and B. Heinrich. 1981.Floral preferences of bumblebees (Bombus ed-wardsii) in relation to intermittent versus contin-uous rewards. Anim. Behav. 29:779-784.

Wagenaar, W. A. 1988. Paradoxes of gambling be-haviour. Erlbaum, Hove.

Wunderle, J. M. and Z. Cotto-Navarro. 1988. Con-stant vs. variable risk-aversion in foraging bana-naquits. Ecology 69:1434-1438.

Wunderle, J. M. and T. G. O'Brien. 1985. Risk-aver-sion in hand reared bananaquits. Behav. Ecol. So-ciobiol. 17:371-380.

Wunderle, J. M., M. Santa-Castro, and N. Fletcher.1987. Risk-averse foraging by bananaquits onnegative energy budgets. Behav. Ecol. Sociobiol.21:249-255.

Yoccoz, N. G., S. Engen, and N. C. Stenseth. 1993.Optimal foraging: The importance of environmen-tal stochasticity and accuracy in parameter esti-mation. Am. Nat. 141:139-157.

Young, J. S. 1981. Discrete-trial choice in pigeons:Effects of reinforcer magnitude. J. Exp. Anal. Be-hav. 35:23-29.

Young, R. J., H. Clayton, and C. J. Barnard. 1990.Risk sensitive foraging in bitterlings, Rhodeus ser-icus: Effects of food requirement and breedingsite quality. Anim. Behav. 40:288-297.

Zabludoff, S. D., J. Wecker, and T. Caraco. 1988. For-aging choice in laboratory rats: Constant vs. vari-able delay. Behav. Processes 16:95-110.

APPENDIX

Definitions of different types of schedulesInterval schedule (I): As soon as the programmed

time interval has elapsed the first response made isreinforced.

Time schedule (T): reinforcement occurs after a pro-grammed time interval, independent of the behavior ofthe subject.

Ratio schedule (R): reinforcement occurs as soon asthe subject has completed the programmed number ofresponses.

All of these schedule types can be either fixed (F),variable (V) or mixed (M), where a fixed schedule isalways the same programmed value, a variable sched-ule is any of a range of values usually described bytheir mean, and a mixed schedule is one of two valuesgiven probabilistically. For example, FI30 specifiesthat the first response after 30 s has elapsed is rein-forced, VR20 that reinforcement is possible after anunknown number of responses with a mean of 20 havebeen completed, and MT5/10 that reinforcement isequally likely to occur after either 5 s or 10 s. Formore details, a reference work such as Lattal (1991)should be consulted.

risky theories—the effect of variancs e on foraging … · lumba livid) are given a choice ......

Documents