- Email: [email protected]
How constant is the constant of risk-aversion?
Anim. Behav., 1986, 34, 1659-1667
H o w constant is the constant of risk-aversion? D. W. S T E P H E N S * & S. R. P A T O N
Department of Zoology, University of British Columbia, Vancouver, British Columbia V6T 1WS, Canada
Abstract. We describe an experiment designed to distinguish between two models of risk-sensitive feeding behaviour: the variance discounting model and the z-score model. The variance discounting model assumes that mean reward levels do not affect preferences over reward variability, but the z-score model assumes that mean reward levels do affect preferences over variability. We presented two choices to feeding rufous hummingibrds, Selaphoruous rufus. One alternative had a higher mean and a higher variance than the other. After measuring preference, we increased the mean of both alternatives by adding the same amount to all possible outcomes. The variance discounting model predicts that such a general shift should not change preferences, but the z-score model predicts that preferences will change. Our results support the z-score model. The variance discounting model's assumption of constant risk-aversion fails.
A forager may prefer a constant reward of 10 calories to a gamble that yields zero calories half of the time and 20 calories half of the time, or it may prefer the gamble. If a forager prefers the constant reward, then its preference is called risk-averse, but if a forager prefers the gamble then its preference is called risk-prone (sometimes risk-loving or riskpreferring). Many studies have shown that such preferences exist (Leventhal et al. 1959; Caraco et al. 1980; Caraco 1981, 1983; Real 1981; Waddington et al. 1981; Barnard & Brown 1985; Battalio e t al. 1985; Wunderle & O'Brien 1985), and two papers have studied the more complicated case in which the alternatives vary in both mean and variance (Real et al. 1982; Caraco & Lima 1985). Risk is synonymous with variance when food rewards are normally distributed. These preferences are called risk-sensitive foraging behaviour, because they show that foragers 'care' about the risk in food reward as well as the mean food reward. Risk-sensitivity has forced a re-evaluation of foraging theory because foraging theory had assumed that mean reward was enough to predict forager preferences (see Stephens & Krebs 1986). Now that the existence of risk-sensitive preferences is well established, there are more subtle questions to investigate: how much mean will a risk-averse forager sacrifice to gain a unit reduction in variance? How do foragers trade off means and variances? Behavioural ecologists have proposed two ways to evaluate combinations of mean and * Present address: Department of Biology, University of Utah, Salt Lake City, UT 84112, U.S.A.
variance: the variance discounting and z-score models. These models make different assumptions about how mean and variance interact to influence preference. We briefly describe each model, and report the results of an experiment designed to distinguish between these two models.
THE MODELS Variance Discounting Many authors (Oster & Wilson 1978; Caraco 1980; Real 1980a, b) have argued that foraging theorists should maximize a linear combination of the mean and variance instead of maximizing the mean alone (many foraging models use mean maximization, see Stephens & Krebs 1986). Real (1980a, b) has called this the variance discounting model. This model proposes that foragers should maximize:
p-- ka2 where p is the mean food reward, a z is the variance in food reward, and k is assumed to be 'a constant associated with the degree to which variance (uncertainty) is evolutionarily or behaviorally undesirable' (Real 1981, page 20) called the 'constant of risk-aversion'. This model assumes that the fitness value of food increases (or decreases) continuously and smoothly with increasing amounts of food, specifically, the utility function is assumed to be exponential. According to variance discounting a forager should be willing to 'pay' up to k units of mean for a
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Animal Behaviour, 34, 6
Variance
Discounting C
A
/~+C" /~+C.
E ~3 O)
/~r
Variance
as
Z-score /~ D
~
~2Variance
02
Z3 Z
~
2
Z:
~ C-
1
/zt+c -
:~ R.
R-
/~1"
I
Standard deviation a
al
I
o2 Standar~ deviation
a
Figure 1. (A) The indifference relationships that variance discounting predicts: parallel lines with slope k. A variancemean pair on a higher line is preferred to a pair on a lower line. (B) The indifference relationships that the z-score predicts: lines with the same #-intercept (R). A standard deviation-mean pair on a higher line is preferred to a pair on a lower line. (C) and (D) represent hypothetical preference tests in which a forager is offered a choice between the end points of one of the broken lines. (C) According to variance discounting, preferences should not change when there is a general shift in means that does not affect the variances; * marks the preferred alternative. (D) According to the z-score, preferences should change when there is a general shift in means that does not affect the variances.
unit decrease in variance, a n d a unit decrease in variance is w o r t h k units o f m e a n regardless of the m e a n ' s size. This property is called c o n s t a n t riskaversion: m e a n a n d variance are perfect substitutes for one another, a n d k is the c o n s t a n t m a r g i n a l rate of substitution. A n analysis of indifference curves shows the
implications of c o n s t a n t risk-aversion. Variance discounting predicts t h a t the forager should be indifferent between all (~r2, #) pairs o n the line: F = # - k a 2 or # = F + k o "2 where F is a constant. F i g u r e 1A shows t h a t these indifference curves are parallel straight lines with
Stephens & Paton." Hummingbird risk-aversion slopes equal to k and #-intercepts at different F values. The forager should prefer any (crz,#) pair on a higher indifference line to any other point on a lower indifference line. Variance discounting predicts that the forager should maximize the #intercepts (F) given the fixed slope of k. For example, suppose an experimenter offers a forager two alternatives: the first alternative yields a high mean (#2) and a high variance (or22),and the second alternative yields both a lower mean (#1) and lower variance (a21; Fig. 1C). According to variance discounting, the best choice depends on the value of k: if k is less than the slope of the line segment connecting (a21, #l) and (a22, #2) (the broken line in Fig. 1C) then the high mean, high variance choice should be preferred, but if k is greater than this slope, then the low mean, low variance alternative should be preferred. The slope of the connecting line would not change if the experimenter increased the means of both alternatives by the same amount, so this general increase in mean should not change the forager's preference (Fig. 1C). This reflects the assumption of constant risk-aversion: if a relatively high mean is worth suffering a. higher variance when means are generally low, then a relatively high mean must still be worth a higher variance when means are generally high. The z-score Model Other authors have proposed that risk-sensitivity might be explained by minimizing the probability of falling short of food (Caraco 1980; Stephens 1981; McNamara & Houston 1982; Pulliam & Millikan 1982; Stephens & Charnov 1982). For example, a small non-breeding bird may need a fixed number of calories to survive a winter night. Stephens (1981) and Stephens & Charnov (1982) have shown that a 'shortfall minimizer' should care about both the mean and the variance in food reward. They formulated the z-score model: when total food gains are normally distributed, minimizing the probability of a shortfall is equivalent to mlmmlzmg R-# Z~ G
where # is the mean food reward, a is the standard deviation of food reward and R is the amount of food that the forager requires (usually for survival). Unlike variance discounting, the z-score model does not predict constant risk-aversion. According
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to the z-score, the amount of mean a forager will 'pay' to gain a unit reduction in variance depends on the present mean. Caraco & Lima (1985) have pointed out that the z-score predicts decreasing risk-aversion: as the mean increases a unit increase in variance (or2) requires a smaller increase in mean to maintain indifference. Under decreasing riskaversion, mean and variance are imperfect substitutes, and mean is substituted for variance at a decreasing marginal rate.
Comparison of the Models To compare the z-score model to variance discounting, consider the indifference relationships that the z-score predicts. A forager should be indifferent between all (a, #) points that lie on the line described by
z=R--# or # = R - z a where z is a constant. All indifference lines have an intercept at the requirement (R) on the mean (#) axis (Fig. 1B), and a forager should prefer any (a, #) point on a line of higher slope to any other point on a line of lower slope: a high slope means a low z, because the slope is - z . These indifference lines differ from those predicted by variance discounting in two respects. First, the z-score predicts a linear indifference relationship between mean and standard deviation (variance discounting predicts a linear relationship between mean and variance). Second, the z-score predicts slope maximizing, in contrast to variance discounting's intercept maximizing. Consider the same example discussed above: a forager is offered two alternatives, the first alternative yields a high mean (#2) and a high variance (cr:2), and the second alternative yields both a lower mean (#1) and lower variance (a21). According to the z-score, the forager's choice should depend on the i~ositions of the points (0"1, ill) and (az, #2) relative to R. The line between these points defines an intercept, called b, on the # axis. If the requirement (R) is greater than this intercept (b) then the z-score predicts preference for the high mean/high variance alternative, but if R is less than b, then the z-score predicts preference for the low mean/low variance alternative. Figure ID shows that if the means are low, then the forager might prefer the high mean/high variance alternative. However, if the same amount is added to both
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Animal Behaviour, 34, 6
alternatives and if this addition is large enough, then the forager should switch its preference to the low mean/low variance alternative. This reflects the z-score's prediction of decreasing risk-aversion: a relatively high mean may be worth suffering the higher variance when means are generally low, but when means are increased a lower variance becomes more important than a relatively high mean.
How Constant is k? The two models differ when faced with the general shift in means shown in Figs 1C, D: when the slope between the alternatives is maintained, the z-score predicts that general changes in mean can affect preference, but variance discounting predicts that general changes in mean cannot affect preference (compare Fig. 1C with Fig. 1D). We used this difference to distinguish between variance discounting's prediction of constant risk-aversion and the z-score's prediction of decreasing riskaversion. Put another way, our experiment asks: how constant is the constant of risk-aversion? We can ask this question without actually measuring any k values, by exploiting the different predictions that Figs 1C, D show. This question is important for two reasons, first, it is important to know whether the value of variance is really independent of mean food levels as variance discounting predicts, or whether the mean affects the value of variance as the z-score predicts. Second, variance discounting does not define k independently of observed risk-taking behaviour: an experimenter must measure the constant of risk-aversion by observing risk-taking. This means that variance discounting can only make quantitative predictions if a k measured in one situation predicts risk-taking in another (Staddon 1983). In contrast, the z-score defines its parameter (R) independently of observed risktaking: R is the food requirement. The predictive value of variance discounting hinges on whether k is constant, and on what k is a constant property of: k might be a constant that describes a species specific response to risk, or k might describe an individual's response to risk. If k were a constant property of individuals or species, then variance discounting would be a powerful predictive model. In the following experiment, we asked whether the constant of risk-aversion is constant. Foragers were offered two alternatives: one alternative had
high mean and high variance, and the other alternative had low mean and low variance. We then measured the forager's preferences. After measuring preferences, the means were increased without changing the variances by adding the same amount to each alternative; this maintained the slope between the alternatives (the slope of the broken lines in Figs 1C, D). This shift should not change preferences if the constant of risk-aversion is constant (with respect to mean). However, if the z-score model is correct, then preferences should change towards the low mean, low variance alternative after the shift up.
M A T E R I A L S AND M E T H O D S
Subjects The subjects were six wild-caught rufous hummingbirds (Selaphoruous rufus): three females and three males. The subjects lived in individual experimental chambers throughout the experiment. We took the following steps to control the birds' energy budgets. (1) Each chamber had its own house light which provided a 10: l 4 h light: dark cycle. (2) The experiments were conducted in a temperature controlled room, kept at 10~ (3) Each bird had access to an ad libitum feeder (dispensing 25% sucrose solution) when it was not taking part in an experiment. The ad libitum feeder was removed after dark on the night before an experiment, and the experiment was begun at first light on the following morning (0800 hours). (d) The experiment usually ended before 1230 hours, and ad libitum food was not returned until 1230 each day. When an experiment ran past 1230 hours, we returned the bird's food immediately after the experiment ended, We only conducted experiments 6 days a week to ensure that the birds had enough protein, and to reduce any cumulative energy deficits. We gave the birds ad libitum food (a mixture of sucrose, nutrients and water: Tooze & Gass, in press), and adult Drosophilia on the day they were not tested.
Chambers The six experimental chambers were identical, and measured 60 x 95 x 40 cm. The chambers were divided into three compartments. The home compartment (measuring 60 x 35 x 40 cm) was always illuminated during the day. At the far end of the
Stephens & Paton: Hummingbird risk-aversion two other compartments (each measuring 30 x 60 x 40 cm) we placed a red circle (a 'flower'). The birds learned to extract a solution of water and sucrose through a small hole in the centre of this red circle. The bird could enter each 'flower compartment' through small holes (13x 13 cm) in the partition that separated the home compartment from the two flower compartments. We controlled the birds' access to each flower by deciding whether to illuminate its compartment. The birds never entered a completely dark compartment.
Flowers
A spring loaded device holding a standard plastic 250 #1 microcentrifuge tube (cut to 21 mm length) was located behind the portion of the flower that the bird could see. A Brinkman adjustable microdispenser was used to fill the tubes with a 25~o solution of sucrose and distilled water (nectar). The flowers were arranged so that the bird's beak had to pass through a computer monitored photodarlington when it drank nectar (see Gass 1985). The computer flashed tell-tale lights, visible only to the experimenter, to show when a visit had occurred, and when nectar tubes should be changed.
Trials
The computer controlled the feeding opportunities by controlling the lights in the choice compartments. When a bird visited a flower the computer turned off the choice chambers' lights 1 s after the bird withdrew its beak. The computer offered another choice 1 min after the previous flower visit ended (beak withdrawal). We could not completely control the interval between choices because the birds could choose when to 'take up' an offer. A bird's daily experimental session consisted of 80 feeding opportunities which were divided into forced choice and free choice trials. In forced choice trials the computer illuminated only one 'flower compartment'. The forced choice trials ensured that the birds were equally familiar with both alternatives. The first 40 feeding opportunities each day were forced choice trials: 20 forced choice trials on each side. The computer determined the order of presentation using six random schedules. For the free choice trials both sides of the chamber were illuminated for the last 40 feeding opportunities each day.
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Treatments
We tested hummingbird preferences in two treatments as suggested by Figs 1C, D. The low-line treatment offered a 'large' alternative that gave 45 #1 or 5 #1 of food with equal probabilities of 0.5: a mean (#) of 25 /21 and a standard deviation (0) of 20. The low-line treatment also offered a 'small' alternative that gave 10 #1 or 15#1 with equal probabilities: /~= 12.5 #1, ~=2.5. Notice that the small alternative has a smaller mean and a smaller variance than the large alternative. The high-line treatment also offered both a large and small alternative, but both alternatives had larger means because we added 20 #1 to each possible outcome. Adding a constant value increases the mean of each alternative without changing the variance. Specifically, in the high-line treatment the large alternative gave 65 #1 or 25/~1: /2 = 45/~1, ~ = 20. The small alternative gave 30 #1 or 35 #1:#=32.5 #1, ~=2.5. According to variance discounting, preferences should be the same in each treatment: if the large alternative is preferred on the low-line treatment, then it must be preferred in the high-line treatment. However, the z-score model predicts that preference for the 'large' alternative should be stronger in the low-line treatment. We assigned birds to high-line (birds 1, 3 and 4) and low-line (birds 2, 5 and 6) treatments at random. Birds 1 and 3 experienced the low-line treatment after finishing the high-line treatment, and birds 2 and 5 experienced the high-line treatment after finishing the low-line treatment. Birds 4 and 6 experienced only one treatment.
Preference Scores
Measuring preferences presents two problems. First, both models assume that the forager 'knows' what the alternatives are. To allow the birds time to 'learn about' the alternatives being offered, we did not score a bird's choices until its behaviour had stabilized. We considered the behaviour to be stable when choice proportions from two successive days were within 10~o. Moreover, we required that treatments last for at least 3 days (the average treatment lasted 4.9 days). We scored preference using only the last two 'stable' days in a treatment. Second, many animals have strong side preferences in dichotomous preference tests. To control
Animal Behaviour, 34, 6
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Table I. Summary of the data Preference scores Bird (sex)
Rl0 _+sE(N) ml
1st 2nd 3rd 4th 5th 6th 7th 8th
Mean__+sE
11
10 17 16 20 11 16 10
10 16 17 18 10 16 10
12 17 15 19 11 17 11
10 17 13 18 10 17 12
10 16 17 20 12 15 11
10 17 15 19 11 16 11
10.5+__0.33 17.1__+0.72 14.9+__0'72 18.9+0.29 10.6+0.26 16.1_+0.44 10.7+0-18
6.29__+0.55(5) High-line 3 Low-line 17
8 17
5 14
3 13
3 19
5 15
7 10
6 16
5.0_+0.68 15.1+0.99
4.71 +_0.32(7) High-line Low-line 10
11
12
8
15
10
I1
10
10.9_+0.72
1 (F)
9.25.+--0-77(4) High-line I0 Low-line 15
12 14
2 (F)
6.86_+0.47(3) High-line 11 Low-line 19
15 18
3 (M)
7,2+0.48(6)
High-line 10 Low-Iine 14
10 18
4 (F)
7.67+0.66(3)
High-line 10 Low-line
5 (M) 6 (M)
for side preferences, we always switched the sides of the small and large alternatives and repeated the learning and stability trials as indicated above. The 80 choices (40 from each of the last two stable days of a treatment) were divided into eight successive blocks of 10 choices, and we measured the number of times a bird chose the large alternative. Written in symbols, this preference test yielded eight scores B1, B 2 , . . . B8 (B stands for 'before side switch'). This procedure was repeated after the sides were switched, yielding eight new scores A~, A 2 , . . 9 A 8 (A stands for 'after side switch'). The scores used to test whether the large or small alternative was preferred are the eight sums BL + A ~, B 2 + A 2 , . . . Bs+A8. This gave eight preference scores, ranging between 0 and 20, for each bird in each treatment. A score of 0 means that the bird always took the small alternative, a score of 10 indicates indifference, and a score of 20 means the bird always took the large alternative.
Requirement Workers have usually assumed that the z-score model's R (requirement) is the forager's daily food requirement, although it could be the requirement for a longer or shorter period. Following Caraco et al. (1980) we measured our birds' daily energy requirements by observing how much food they ate per day in experimental temperature and photo-
period conditions (see Table I). We took these measurements on days when no preference tests were made. Analysis of variance shows that requirements differ significantly (at the 5 ~ level) from one bird to the next. This gave us a measure of the amount a forager must eat in the 10 h of light under experimental conditions. To calculate the amount that an individual needed in each of the 40 free choice trials (Ro), we used the formula R~ = (0.45R10-- Gvor)/40. Where Rl0 is the measured 10 h requirement (Table I shows Rl0 values for each individual), which is multiplied by 0'45, because the birds had ad libitum food for 5.5 h after the experiment each day; GFor is the gain during the forced choice trials (750 ~1 in the low-line treatment, and 1550 pl in the high-line treatment).
Graded Preference Both models predict only three types of preference: preference for the large alternative, preference for the small alternative or indifference. However, we did not expect to observe 100% preference for either alternative, instead we expected to observe graded changes in preference as Stephens (1985) has argued. Equal preferences occur when mean preferences scores are equal, and not when preferences are simply in the same direction. So, we analysed our data for quantitative changes in preference.
Stephens & Paton: Hummingbird risk-aversion RESULTS Table I shows the data for each bird in each treatment. We analysed these data in three steps. First, we asked whether the same constant of riskaversion could explain all our hummingbirds' behaviour: is k a constant property of the species? Second, we asked whether it is constant between treatments for the same individual: is k a property of individuals? Last, we asked whether the measured food requirements combined with the z-score model give an a priori explanation of the preferences observed. Is k Constant for Species? To answer this question we considered only the first treatment of each of the six birds. Ideally, we would have liked to do a nested analysis of variance of these data, with individuals nested within treatments (high-line and low-line). However, we found that these data were markedly non-normal with unequal variances even when the appropriate transformations were tried (Sokal & Rohlf 1981), so analysis of variance was not appropriate. We performed, a Friedman test viewing each of the six individuals as a treatment, and each of the eight preference scores as a block (see Conover 1980). This test showed significant (the 5% level of significance is used throughout) differences between the individuals. However, we wanted to know whether the treatment or intrinsic variability of individuals caused these differences. We tested the difference between birds in the high-line and low-line treatments using a test for unplanned comparisons in the Friedman test (Conover 1980). This test showed significant differences between the birds in the high-line group and the birds in the lowline group. We conclude that the constant of riskaversion is unlikely to be a constant property of the species: the k value measured from the low-line group would not successfully predict the behaviour of the high-line group. Is k Constant for Individuals? To ask this question we considered only the four birds that experienced both high-line and low-line treatments. For a given bird the data were eight differences: we paired the first score in the high-line treatment with the first score in the low-line treatment and subtracted the high-line from the low-line score, and so on for the second, third, fourth, fifth, sixth, seventh and eighth pairs of
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scores. We found that these differences met the assumptions of analysis of variance. We allowed for the effects of individual birds and for the effects of repeated measurement by performing a repeated measures analysis of variance on these differences (score sequence, 1 through 8, was the 'treatment' in this analysis). This analysis of variance tested the hypothesis that the mean difference equalled zero (using BMDP statistical software, Dixon 1983) as the variance discounting model predicts. This test showed that the mean difference is significantly different from zero. We conclude that the constant of risk-aversion is not a constant property of individual hummingbirds, because different k values would be required to predict an individual's behaviour in the high-line and low-line treatments. Can the z-score Model Explain This? Our data supported the z-score model over the variance discounting model because groups of individuals, and individuals themselves, behaved differently in the high-line and low-line treatments, and their preference for the large alternative was stronger in the low-line treatment, as the z-score model predicts. The z-score model makes stronger predictions if an a priori estimate of the forager's requirement (R~) can be made. Since Rc is a property of individuals, the z-score model might explain the differences between individuals that we observed. Each treatment defines an intercept (b) on the # axis of Fig. 1B, D (br~gh=30"69 #1 and blow= 10-69 /A). When the requirement is greater than this intercept the z-score predicts preference for the larger alternative, but when the requirement is less than this intercept, the z-score predicts preference for the smaller alternative. We calculated R~- b for each bird in each treatment, and we expected (under the graded preference view, Stephens 1985) that preference scores (preference for the large alternative) should increase with R e - b . Figure 2 plots mean preference scores against Rc-b. Non-parametric measures of correlation (Kendall's, 0.71, and Spearman's, 0.84, rank correlation coefficients) showed significant positive correlations between preference for the large alternative and R e - b . Moreover, the z-score predicts that indifference, preference scores of 10, should occur when R~- b = 0. We found that the intercept of a regression line through these points lies between 2-94 and 16.15 with 95% confidence, and the estimated intercept is 9-32 (Fig. 2). These data suggest that the z-score model, or a model with
Animal Behaviour, 34, 6
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f
v #.
Requirement
minus intercept
Figure 2. Mean preference scores from all birds in all treatments are plotted against R~--b, the difference between the requirement per choice (Re) and the #intercept defined by the alternatives offered (b). The regression line shown has equation: mean score= 9-32+ 0-12(Ro- b). The parametric correlation coefficient is 0.78, and it is significantly different from zero.
similar properties, might account for the differences we observed.
DISCUSSION Our results contradict variance discounting's prediction of constant risk-aversion. We found that the general level of mean food reward affected the trade off between mean and variance as predicted by the z-score. The constant of risk-aversion, k, is not constant. A k value that is consistent with one observed preference is inconsistent with preferences after a general increase in means of only 20#1. Our results do not support the predictive use of variance discounting. Until now the strongest evidence supporting the z-score was the changing risk-sensitivity observed in granivorous birds and shrews (Caraco et al. 1980; Caraco 1981, 1983; Barnard & Brown 1985). These experiments showed that foragers were riskaverse when they expected to gain more than they required (#>R), and risk-prone when they expected to require more than they gained (R > #). This behaviour supports the z-score model, and contradicts variance discounting because the constant of risk-aversion measured in the risk-averse case (k>0) could not predict the behaviour observed in the risk-prone case (k < 0). However, the variance discounting's proponents could argue that, although experimentalists can show risk-prone behaviour, risk-proneness is not common in nature, and both models explain dayto-day risk-averse foraging decisions equally well.
Caraco & Lima's (1985) results support this argument. Caraco & Lima tried to determine whether measured mean-standard deviation indifference curves were linear (as predicted by the z-score) or quadratic (as predicted by variance discounting). They found that both models worked equally well. However, estimating even a single indifference point takes many hours of observation, so Caraco & Lima were in the difficult situation of trying to distinguish a line from a curve With only a handful of points. Our results show that the two models make different predictions about the interactions of mean and variance, and these differences might be important in day-to-day foraging decisions, even though both models predict that animals will usually be risk-averse. For example, our experiment shows that a hummingbird's preferences might be different in rich and poor habitats, even if means are always correlated with variances. Variance discounting might be saved by one or two types of post hoc argument. First, a variance discounting proponent might make a specific claim about when he expects k to be constant: for example, 'k will be constant when the change in means is less than X% of the energy requirement'. Second, a variance discounting proponent might argue that the model is purely descriptive: k simply measures observed risk-taking. This relegates variance discounting to a statistical technique, rather than a theoretical insight. However, variance discounting has one unassailable use. It is an extremely convenient formulation that modellers might use to study how risk-sensitivity affects mean-maximizing models. A modeller might say, for example, the k values between k~ and k2 do not change the mean-maximizing result'. Variance discounting, broadly defined, can include models of the form m a x { # - k(#)~2)} where k(#) is now some function of the mean (#, Real, personal communication). Our results and comments only deal with the version of variance discounting where k(#) is a constant for all values of #. However, a constant k value, and its consequence of constant risk-aversion, is explicitly or implicitly assumed in most uses of variance discounting (e.g. Real 1980a, b, 1981; Real et al. 1982; Lacey et al. 1983; Caraco & Lima 1985; Wunderle & O'Brien, in press). Moreover, if the coefficient of risk-sensitivity' (k(#); Real et al. 1981) is left as an
Stephens & Paton: Hummingbird risk-aversion unspecified function of mean, t h e n variance disc o u n t i n g becomes untestably vague: k(#) can be c h a n g e d to a c c o m m o d a t e any observatiOn. A critic might charge t h a t b o t h o f the models considered here are so simple t h a t an a t t e m p t to distinguish between t h e m is a battle of straw men. W e agree t h a t neither model is likely to explain everything a b o u t forager risk-taking. The models are i m p o r t a n t because they help us to fix ideas: viewing t h e m as alternatives suggested our experiments. However, our experiment distinguishes between two contrasting properties of the models: variance discounting's c o n s t a n t risk-aversion a n d the z-score's decreasing risk-aversion. These properties are more general t h a n the simple models considered here, a n d o u r results suggest t h a t the successors of these models s h o u l d n o t assume c o n s t a n t risk-aversion as variance discounting has. ACKNOWLEDGMENTS W e are grateful to C. L. Gass, T. Phillipi, A. E. Sorensen, S. T a m m a n d Z. Tooze for their help a n d advice. W e t h a n k C. P. L. Barkan, T. Caraco, E. L. C h a r n o v , M. E. Ellis, J. R. Lucas, G. H. Orians, L. A. Real, C. T. S n o w d o n , R. C. Y d e n b e r g a n d the reviewers for their c o m m e n t s o n the manuscript. This work was partially s u p p o r t e d by N S E R C g r a n t 6 7 4 8 7 6 to C. L. Gass. D W S was s u p p o r t e d by a N A T O postdoctoral fellowship, and S R P was s u p p o r t e d by a N S E R C studentship. We t h a n k N A T O a n d N S E R C for their support.
REFERENCES Barnard, C. J. & Brown, C. A. J. 1985. Risk-sensitive foraging in common shrews (Sorex araneus L.). Behav. Ecol. Sociobiol., 16, 161-164. Battalio, R. C., Kagel, J. H. & McDonald, D. N. 1985. Animals' choices over uncertain outcomes: some initial experimental results. Am. Econ. Rev., 75, 597-613. Caraco, T. 1980. On foraging time allocation in a stochastic environment. Ecology, 61, 119-128. Caraco, T. 1981. Energy budgets, risk and foraging preferences in dark-eyed juncos (Junco hymelais). Behav. Ecol. Sociobiol., 8, 820-830. Caraco, T. 1983. White crowned sparrows (Zonotrichia leucophrys): foraging preferences in a risky environment. Behav. Ecol. Sociobiol., 12, 63~59. Caraco, T. & Lima, S. L. 1985. Foraging juncos: interaction of reward mean and variability. Anim. Behav., 33, 216-224. Caraco, T., Martindale, S. & Whitham, T. S. 1980. An empirical demonstration of risk-sensitive foraging preferences. Anim. Behav., 28, 820-830.
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Conover, W. J. 1980. Practical Nonparametric Statistics. 2nd edn. New York: John Wiley. Dixon, W. J. 1983. BMDP Statistical Software. Berkeley: University of California Press. Gass, C. L. 1985. Behavioral foundations of adaptation. In: Perspectives in Ethology (Ed. by P. P. G. Bateson & P. H. Klopfer), pp. 63 107. New York: Plenum Press. Lacey, E. P., Real, L., Antonovics, J. & Heckel, D. G. 1983. Variance models in the study of life histories. Am. Nat., 122, 114~131. Leventhal, A. M., Morretl, R. F., Morgan, E. F., Jr & Perkins, C. C., Jr. 1959. The relation between mean reward and mean reinforcement. J. Exp. Psychol., 57, 284 287. McNamara, J. & Houston, A. I. 1982. Short-term behaviour and lifetime fitness. In: Functional Ontogeny (Ed. by D. J. McFarland), pp. 60-87. London: Pitman. Oster, G. F. & Wilson, E. O. 1978. Caste andEcology in the Social Insects. Princeton, NJ: Princeton University Press. Pultiam, H. R. & Millikan, G. C. 1982. Social organization in the non-reproductive season. In: Avian Biology, VoL 6 (Ed. by D. S. Farner & J. R. King). pp. 169-197. New York: Academic Press. Real, L. A. 1980a. Fitness, uncertainty, and the role of diversification in evolution and behavior. Am. Nat., 115, 623-638. Real, L. A. 1980b. On uncertainty and the law of diminishing returns in evolution and behavior. In: Limits to Action: the Allocation of Individual Behavior (Ed. by J. E. R. Staddon), pp. 37-64. New York: Academic Press. Real, L. A. 1981. Uncertainty and plant pollinator interactions: the foraging behavior of bees and wasps on.artificial flowers. Ecology, 62, 20~26. Real, L. A., Ott, J. & Silverfine, E. 1982. On the tradeoff between mean and variance in foraging: an experimental analysis with bumblebees. Ecology, 63, 1617-1623. Sokal, R. R. & Rohlf, F. J. 1981. Biometry." the Principles and Practice of Statistics in Biological Research. 2nd edn. San Francisco: W. H. Freeman. Staddon, J. E. R. 1983. Adaptive Behavior and Learning. New York: Cambridge University Press. Stephens, D. W. 1981. The logic of risk-sensitive foraging preferences. Anim. Behav., 29, 628 629. Stephens, D. W. 1985. How important are partial preferences? Anim. Behav., 33, 667-669. Stephens, D. W. & Charnov, E. L. 1982. Optimal foraging: some simple stochastic models. Behav. Ecol. Sociobiol., 10, 251 263. Stephens, D. W. & Krebs, J. R. 1986. Foraging Theory. Princeton, NJ: Princeton University Press. Tooze, Z. J. & Gass, C. L. In press. Responses of rufous hummingbirds to a midday fast. Can. J. Zool. Waddington, K. D., Allen, T. & Heinrich, B. ! 981. Floral preferences of bumblebees (Bombus edwardsii) in relation to intermittent versus continuous rewards. Anita. Behav., 29, 779-784. Wunderle, J. M. & O'Brien, T. G. 1985. Risk aversion in hand-reared bananaquits. Behav. Ecol. Sociobiol., 17, 371-380. (Reeeived 20 July 1985," revised 26 October 1985; MS. number: A4576)
H o w constant is the constant of risk-aversion? D. W. S T E P H E N S * & S. R. P A T O N
Department of Zoology, University of British Columbia, Vancouver, British Columbia V6T 1WS, Canada
Abstract. We describe an experiment designed to distinguish between two models of risk-sensitive feeding behaviour: the variance discounting model and the z-score model. The variance discounting model assumes that mean reward levels do not affect preferences over reward variability, but the z-score model assumes that mean reward levels do affect preferences over variability. We presented two choices to feeding rufous hummingibrds, Selaphoruous rufus. One alternative had a higher mean and a higher variance than the other. After measuring preference, we increased the mean of both alternatives by adding the same amount to all possible outcomes. The variance discounting model predicts that such a general shift should not change preferences, but the z-score model predicts that preferences will change. Our results support the z-score model. The variance discounting model's assumption of constant risk-aversion fails.
A forager may prefer a constant reward of 10 calories to a gamble that yields zero calories half of the time and 20 calories half of the time, or it may prefer the gamble. If a forager prefers the constant reward, then its preference is called risk-averse, but if a forager prefers the gamble then its preference is called risk-prone (sometimes risk-loving or riskpreferring). Many studies have shown that such preferences exist (Leventhal et al. 1959; Caraco et al. 1980; Caraco 1981, 1983; Real 1981; Waddington et al. 1981; Barnard & Brown 1985; Battalio e t al. 1985; Wunderle & O'Brien 1985), and two papers have studied the more complicated case in which the alternatives vary in both mean and variance (Real et al. 1982; Caraco & Lima 1985). Risk is synonymous with variance when food rewards are normally distributed. These preferences are called risk-sensitive foraging behaviour, because they show that foragers 'care' about the risk in food reward as well as the mean food reward. Risk-sensitivity has forced a re-evaluation of foraging theory because foraging theory had assumed that mean reward was enough to predict forager preferences (see Stephens & Krebs 1986). Now that the existence of risk-sensitive preferences is well established, there are more subtle questions to investigate: how much mean will a risk-averse forager sacrifice to gain a unit reduction in variance? How do foragers trade off means and variances? Behavioural ecologists have proposed two ways to evaluate combinations of mean and * Present address: Department of Biology, University of Utah, Salt Lake City, UT 84112, U.S.A.
variance: the variance discounting and z-score models. These models make different assumptions about how mean and variance interact to influence preference. We briefly describe each model, and report the results of an experiment designed to distinguish between these two models.
THE MODELS Variance Discounting Many authors (Oster & Wilson 1978; Caraco 1980; Real 1980a, b) have argued that foraging theorists should maximize a linear combination of the mean and variance instead of maximizing the mean alone (many foraging models use mean maximization, see Stephens & Krebs 1986). Real (1980a, b) has called this the variance discounting model. This model proposes that foragers should maximize:
p-- ka2 where p is the mean food reward, a z is the variance in food reward, and k is assumed to be 'a constant associated with the degree to which variance (uncertainty) is evolutionarily or behaviorally undesirable' (Real 1981, page 20) called the 'constant of risk-aversion'. This model assumes that the fitness value of food increases (or decreases) continuously and smoothly with increasing amounts of food, specifically, the utility function is assumed to be exponential. According to variance discounting a forager should be willing to 'pay' up to k units of mean for a
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Animal Behaviour, 34, 6
Variance
Discounting C
A
/~+C" /~+C.
E ~3 O)
/~r
Variance
as
Z-score /~ D
~
~2Variance
02
Z3 Z
~
2
Z:
~ C-
1
/zt+c -
:~ R.
R-
/~1"
I
Standard deviation a
al
I
o2 Standar~ deviation
a
Figure 1. (A) The indifference relationships that variance discounting predicts: parallel lines with slope k. A variancemean pair on a higher line is preferred to a pair on a lower line. (B) The indifference relationships that the z-score predicts: lines with the same #-intercept (R). A standard deviation-mean pair on a higher line is preferred to a pair on a lower line. (C) and (D) represent hypothetical preference tests in which a forager is offered a choice between the end points of one of the broken lines. (C) According to variance discounting, preferences should not change when there is a general shift in means that does not affect the variances; * marks the preferred alternative. (D) According to the z-score, preferences should change when there is a general shift in means that does not affect the variances.
unit decrease in variance, a n d a unit decrease in variance is w o r t h k units o f m e a n regardless of the m e a n ' s size. This property is called c o n s t a n t riskaversion: m e a n a n d variance are perfect substitutes for one another, a n d k is the c o n s t a n t m a r g i n a l rate of substitution. A n analysis of indifference curves shows the
implications of c o n s t a n t risk-aversion. Variance discounting predicts t h a t the forager should be indifferent between all (~r2, #) pairs o n the line: F = # - k a 2 or # = F + k o "2 where F is a constant. F i g u r e 1A shows t h a t these indifference curves are parallel straight lines with
Stephens & Paton." Hummingbird risk-aversion slopes equal to k and #-intercepts at different F values. The forager should prefer any (crz,#) pair on a higher indifference line to any other point on a lower indifference line. Variance discounting predicts that the forager should maximize the #intercepts (F) given the fixed slope of k. For example, suppose an experimenter offers a forager two alternatives: the first alternative yields a high mean (#2) and a high variance (or22),and the second alternative yields both a lower mean (#1) and lower variance (a21; Fig. 1C). According to variance discounting, the best choice depends on the value of k: if k is less than the slope of the line segment connecting (a21, #l) and (a22, #2) (the broken line in Fig. 1C) then the high mean, high variance choice should be preferred, but if k is greater than this slope, then the low mean, low variance alternative should be preferred. The slope of the connecting line would not change if the experimenter increased the means of both alternatives by the same amount, so this general increase in mean should not change the forager's preference (Fig. 1C). This reflects the assumption of constant risk-aversion: if a relatively high mean is worth suffering a. higher variance when means are generally low, then a relatively high mean must still be worth a higher variance when means are generally high. The z-score Model Other authors have proposed that risk-sensitivity might be explained by minimizing the probability of falling short of food (Caraco 1980; Stephens 1981; McNamara & Houston 1982; Pulliam & Millikan 1982; Stephens & Charnov 1982). For example, a small non-breeding bird may need a fixed number of calories to survive a winter night. Stephens (1981) and Stephens & Charnov (1982) have shown that a 'shortfall minimizer' should care about both the mean and the variance in food reward. They formulated the z-score model: when total food gains are normally distributed, minimizing the probability of a shortfall is equivalent to mlmmlzmg R-# Z~ G
where # is the mean food reward, a is the standard deviation of food reward and R is the amount of food that the forager requires (usually for survival). Unlike variance discounting, the z-score model does not predict constant risk-aversion. According
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to the z-score, the amount of mean a forager will 'pay' to gain a unit reduction in variance depends on the present mean. Caraco & Lima (1985) have pointed out that the z-score predicts decreasing risk-aversion: as the mean increases a unit increase in variance (or2) requires a smaller increase in mean to maintain indifference. Under decreasing riskaversion, mean and variance are imperfect substitutes, and mean is substituted for variance at a decreasing marginal rate.
Comparison of the Models To compare the z-score model to variance discounting, consider the indifference relationships that the z-score predicts. A forager should be indifferent between all (a, #) points that lie on the line described by
z=R--# or # = R - z a where z is a constant. All indifference lines have an intercept at the requirement (R) on the mean (#) axis (Fig. 1B), and a forager should prefer any (a, #) point on a line of higher slope to any other point on a line of lower slope: a high slope means a low z, because the slope is - z . These indifference lines differ from those predicted by variance discounting in two respects. First, the z-score predicts a linear indifference relationship between mean and standard deviation (variance discounting predicts a linear relationship between mean and variance). Second, the z-score predicts slope maximizing, in contrast to variance discounting's intercept maximizing. Consider the same example discussed above: a forager is offered two alternatives, the first alternative yields a high mean (#2) and a high variance (cr:2), and the second alternative yields both a lower mean (#1) and lower variance (a21). According to the z-score, the forager's choice should depend on the i~ositions of the points (0"1, ill) and (az, #2) relative to R. The line between these points defines an intercept, called b, on the # axis. If the requirement (R) is greater than this intercept (b) then the z-score predicts preference for the high mean/high variance alternative, but if R is less than b, then the z-score predicts preference for the low mean/low variance alternative. Figure ID shows that if the means are low, then the forager might prefer the high mean/high variance alternative. However, if the same amount is added to both
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Animal Behaviour, 34, 6
alternatives and if this addition is large enough, then the forager should switch its preference to the low mean/low variance alternative. This reflects the z-score's prediction of decreasing risk-aversion: a relatively high mean may be worth suffering the higher variance when means are generally low, but when means are increased a lower variance becomes more important than a relatively high mean.
How Constant is k? The two models differ when faced with the general shift in means shown in Figs 1C, D: when the slope between the alternatives is maintained, the z-score predicts that general changes in mean can affect preference, but variance discounting predicts that general changes in mean cannot affect preference (compare Fig. 1C with Fig. 1D). We used this difference to distinguish between variance discounting's prediction of constant risk-aversion and the z-score's prediction of decreasing riskaversion. Put another way, our experiment asks: how constant is the constant of risk-aversion? We can ask this question without actually measuring any k values, by exploiting the different predictions that Figs 1C, D show. This question is important for two reasons, first, it is important to know whether the value of variance is really independent of mean food levels as variance discounting predicts, or whether the mean affects the value of variance as the z-score predicts. Second, variance discounting does not define k independently of observed risk-taking behaviour: an experimenter must measure the constant of risk-aversion by observing risk-taking. This means that variance discounting can only make quantitative predictions if a k measured in one situation predicts risk-taking in another (Staddon 1983). In contrast, the z-score defines its parameter (R) independently of observed risktaking: R is the food requirement. The predictive value of variance discounting hinges on whether k is constant, and on what k is a constant property of: k might be a constant that describes a species specific response to risk, or k might describe an individual's response to risk. If k were a constant property of individuals or species, then variance discounting would be a powerful predictive model. In the following experiment, we asked whether the constant of risk-aversion is constant. Foragers were offered two alternatives: one alternative had
high mean and high variance, and the other alternative had low mean and low variance. We then measured the forager's preferences. After measuring preferences, the means were increased without changing the variances by adding the same amount to each alternative; this maintained the slope between the alternatives (the slope of the broken lines in Figs 1C, D). This shift should not change preferences if the constant of risk-aversion is constant (with respect to mean). However, if the z-score model is correct, then preferences should change towards the low mean, low variance alternative after the shift up.
M A T E R I A L S AND M E T H O D S
Subjects The subjects were six wild-caught rufous hummingbirds (Selaphoruous rufus): three females and three males. The subjects lived in individual experimental chambers throughout the experiment. We took the following steps to control the birds' energy budgets. (1) Each chamber had its own house light which provided a 10: l 4 h light: dark cycle. (2) The experiments were conducted in a temperature controlled room, kept at 10~ (3) Each bird had access to an ad libitum feeder (dispensing 25% sucrose solution) when it was not taking part in an experiment. The ad libitum feeder was removed after dark on the night before an experiment, and the experiment was begun at first light on the following morning (0800 hours). (d) The experiment usually ended before 1230 hours, and ad libitum food was not returned until 1230 each day. When an experiment ran past 1230 hours, we returned the bird's food immediately after the experiment ended, We only conducted experiments 6 days a week to ensure that the birds had enough protein, and to reduce any cumulative energy deficits. We gave the birds ad libitum food (a mixture of sucrose, nutrients and water: Tooze & Gass, in press), and adult Drosophilia on the day they were not tested.
Chambers The six experimental chambers were identical, and measured 60 x 95 x 40 cm. The chambers were divided into three compartments. The home compartment (measuring 60 x 35 x 40 cm) was always illuminated during the day. At the far end of the
Stephens & Paton: Hummingbird risk-aversion two other compartments (each measuring 30 x 60 x 40 cm) we placed a red circle (a 'flower'). The birds learned to extract a solution of water and sucrose through a small hole in the centre of this red circle. The bird could enter each 'flower compartment' through small holes (13x 13 cm) in the partition that separated the home compartment from the two flower compartments. We controlled the birds' access to each flower by deciding whether to illuminate its compartment. The birds never entered a completely dark compartment.
Flowers
A spring loaded device holding a standard plastic 250 #1 microcentrifuge tube (cut to 21 mm length) was located behind the portion of the flower that the bird could see. A Brinkman adjustable microdispenser was used to fill the tubes with a 25~o solution of sucrose and distilled water (nectar). The flowers were arranged so that the bird's beak had to pass through a computer monitored photodarlington when it drank nectar (see Gass 1985). The computer flashed tell-tale lights, visible only to the experimenter, to show when a visit had occurred, and when nectar tubes should be changed.
Trials
The computer controlled the feeding opportunities by controlling the lights in the choice compartments. When a bird visited a flower the computer turned off the choice chambers' lights 1 s after the bird withdrew its beak. The computer offered another choice 1 min after the previous flower visit ended (beak withdrawal). We could not completely control the interval between choices because the birds could choose when to 'take up' an offer. A bird's daily experimental session consisted of 80 feeding opportunities which were divided into forced choice and free choice trials. In forced choice trials the computer illuminated only one 'flower compartment'. The forced choice trials ensured that the birds were equally familiar with both alternatives. The first 40 feeding opportunities each day were forced choice trials: 20 forced choice trials on each side. The computer determined the order of presentation using six random schedules. For the free choice trials both sides of the chamber were illuminated for the last 40 feeding opportunities each day.
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Treatments
We tested hummingbird preferences in two treatments as suggested by Figs 1C, D. The low-line treatment offered a 'large' alternative that gave 45 #1 or 5 #1 of food with equal probabilities of 0.5: a mean (#) of 25 /21 and a standard deviation (0) of 20. The low-line treatment also offered a 'small' alternative that gave 10 #1 or 15#1 with equal probabilities: /~= 12.5 #1, ~=2.5. Notice that the small alternative has a smaller mean and a smaller variance than the large alternative. The high-line treatment also offered both a large and small alternative, but both alternatives had larger means because we added 20 #1 to each possible outcome. Adding a constant value increases the mean of each alternative without changing the variance. Specifically, in the high-line treatment the large alternative gave 65 #1 or 25/~1: /2 = 45/~1, ~ = 20. The small alternative gave 30 #1 or 35 #1:#=32.5 #1, ~=2.5. According to variance discounting, preferences should be the same in each treatment: if the large alternative is preferred on the low-line treatment, then it must be preferred in the high-line treatment. However, the z-score model predicts that preference for the 'large' alternative should be stronger in the low-line treatment. We assigned birds to high-line (birds 1, 3 and 4) and low-line (birds 2, 5 and 6) treatments at random. Birds 1 and 3 experienced the low-line treatment after finishing the high-line treatment, and birds 2 and 5 experienced the high-line treatment after finishing the low-line treatment. Birds 4 and 6 experienced only one treatment.
Preference Scores
Measuring preferences presents two problems. First, both models assume that the forager 'knows' what the alternatives are. To allow the birds time to 'learn about' the alternatives being offered, we did not score a bird's choices until its behaviour had stabilized. We considered the behaviour to be stable when choice proportions from two successive days were within 10~o. Moreover, we required that treatments last for at least 3 days (the average treatment lasted 4.9 days). We scored preference using only the last two 'stable' days in a treatment. Second, many animals have strong side preferences in dichotomous preference tests. To control
Animal Behaviour, 34, 6
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Table I. Summary of the data Preference scores Bird (sex)
Rl0 _+sE(N) ml
1st 2nd 3rd 4th 5th 6th 7th 8th
Mean__+sE
11
10 17 16 20 11 16 10
10 16 17 18 10 16 10
12 17 15 19 11 17 11
10 17 13 18 10 17 12
10 16 17 20 12 15 11
10 17 15 19 11 16 11
10.5+__0.33 17.1__+0.72 14.9+__0'72 18.9+0.29 10.6+0.26 16.1_+0.44 10.7+0-18
6.29__+0.55(5) High-line 3 Low-line 17
8 17
5 14
3 13
3 19
5 15
7 10
6 16
5.0_+0.68 15.1+0.99
4.71 +_0.32(7) High-line Low-line 10
11
12
8
15
10
I1
10
10.9_+0.72
1 (F)
9.25.+--0-77(4) High-line I0 Low-line 15
12 14
2 (F)
6.86_+0.47(3) High-line 11 Low-line 19
15 18
3 (M)
7,2+0.48(6)
High-line 10 Low-Iine 14
10 18
4 (F)
7.67+0.66(3)
High-line 10 Low-line
5 (M) 6 (M)
for side preferences, we always switched the sides of the small and large alternatives and repeated the learning and stability trials as indicated above. The 80 choices (40 from each of the last two stable days of a treatment) were divided into eight successive blocks of 10 choices, and we measured the number of times a bird chose the large alternative. Written in symbols, this preference test yielded eight scores B1, B 2 , . . . B8 (B stands for 'before side switch'). This procedure was repeated after the sides were switched, yielding eight new scores A~, A 2 , . . 9 A 8 (A stands for 'after side switch'). The scores used to test whether the large or small alternative was preferred are the eight sums BL + A ~, B 2 + A 2 , . . . Bs+A8. This gave eight preference scores, ranging between 0 and 20, for each bird in each treatment. A score of 0 means that the bird always took the small alternative, a score of 10 indicates indifference, and a score of 20 means the bird always took the large alternative.
Requirement Workers have usually assumed that the z-score model's R (requirement) is the forager's daily food requirement, although it could be the requirement for a longer or shorter period. Following Caraco et al. (1980) we measured our birds' daily energy requirements by observing how much food they ate per day in experimental temperature and photo-
period conditions (see Table I). We took these measurements on days when no preference tests were made. Analysis of variance shows that requirements differ significantly (at the 5 ~ level) from one bird to the next. This gave us a measure of the amount a forager must eat in the 10 h of light under experimental conditions. To calculate the amount that an individual needed in each of the 40 free choice trials (Ro), we used the formula R~ = (0.45R10-- Gvor)/40. Where Rl0 is the measured 10 h requirement (Table I shows Rl0 values for each individual), which is multiplied by 0'45, because the birds had ad libitum food for 5.5 h after the experiment each day; GFor is the gain during the forced choice trials (750 ~1 in the low-line treatment, and 1550 pl in the high-line treatment).
Graded Preference Both models predict only three types of preference: preference for the large alternative, preference for the small alternative or indifference. However, we did not expect to observe 100% preference for either alternative, instead we expected to observe graded changes in preference as Stephens (1985) has argued. Equal preferences occur when mean preferences scores are equal, and not when preferences are simply in the same direction. So, we analysed our data for quantitative changes in preference.
Stephens & Paton: Hummingbird risk-aversion RESULTS Table I shows the data for each bird in each treatment. We analysed these data in three steps. First, we asked whether the same constant of riskaversion could explain all our hummingbirds' behaviour: is k a constant property of the species? Second, we asked whether it is constant between treatments for the same individual: is k a property of individuals? Last, we asked whether the measured food requirements combined with the z-score model give an a priori explanation of the preferences observed. Is k Constant for Species? To answer this question we considered only the first treatment of each of the six birds. Ideally, we would have liked to do a nested analysis of variance of these data, with individuals nested within treatments (high-line and low-line). However, we found that these data were markedly non-normal with unequal variances even when the appropriate transformations were tried (Sokal & Rohlf 1981), so analysis of variance was not appropriate. We performed, a Friedman test viewing each of the six individuals as a treatment, and each of the eight preference scores as a block (see Conover 1980). This test showed significant (the 5% level of significance is used throughout) differences between the individuals. However, we wanted to know whether the treatment or intrinsic variability of individuals caused these differences. We tested the difference between birds in the high-line and low-line treatments using a test for unplanned comparisons in the Friedman test (Conover 1980). This test showed significant differences between the birds in the high-line group and the birds in the lowline group. We conclude that the constant of riskaversion is unlikely to be a constant property of the species: the k value measured from the low-line group would not successfully predict the behaviour of the high-line group. Is k Constant for Individuals? To ask this question we considered only the four birds that experienced both high-line and low-line treatments. For a given bird the data were eight differences: we paired the first score in the high-line treatment with the first score in the low-line treatment and subtracted the high-line from the low-line score, and so on for the second, third, fourth, fifth, sixth, seventh and eighth pairs of
1665
scores. We found that these differences met the assumptions of analysis of variance. We allowed for the effects of individual birds and for the effects of repeated measurement by performing a repeated measures analysis of variance on these differences (score sequence, 1 through 8, was the 'treatment' in this analysis). This analysis of variance tested the hypothesis that the mean difference equalled zero (using BMDP statistical software, Dixon 1983) as the variance discounting model predicts. This test showed that the mean difference is significantly different from zero. We conclude that the constant of risk-aversion is not a constant property of individual hummingbirds, because different k values would be required to predict an individual's behaviour in the high-line and low-line treatments. Can the z-score Model Explain This? Our data supported the z-score model over the variance discounting model because groups of individuals, and individuals themselves, behaved differently in the high-line and low-line treatments, and their preference for the large alternative was stronger in the low-line treatment, as the z-score model predicts. The z-score model makes stronger predictions if an a priori estimate of the forager's requirement (R~) can be made. Since Rc is a property of individuals, the z-score model might explain the differences between individuals that we observed. Each treatment defines an intercept (b) on the # axis of Fig. 1B, D (br~gh=30"69 #1 and blow= 10-69 /A). When the requirement is greater than this intercept the z-score predicts preference for the larger alternative, but when the requirement is less than this intercept, the z-score predicts preference for the smaller alternative. We calculated R~- b for each bird in each treatment, and we expected (under the graded preference view, Stephens 1985) that preference scores (preference for the large alternative) should increase with R e - b . Figure 2 plots mean preference scores against Rc-b. Non-parametric measures of correlation (Kendall's, 0.71, and Spearman's, 0.84, rank correlation coefficients) showed significant positive correlations between preference for the large alternative and R e - b . Moreover, the z-score predicts that indifference, preference scores of 10, should occur when R~- b = 0. We found that the intercept of a regression line through these points lies between 2-94 and 16.15 with 95% confidence, and the estimated intercept is 9-32 (Fig. 2). These data suggest that the z-score model, or a model with
Animal Behaviour, 34, 6
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f
v #.
Requirement
minus intercept
Figure 2. Mean preference scores from all birds in all treatments are plotted against R~--b, the difference between the requirement per choice (Re) and the #intercept defined by the alternatives offered (b). The regression line shown has equation: mean score= 9-32+ 0-12(Ro- b). The parametric correlation coefficient is 0.78, and it is significantly different from zero.
similar properties, might account for the differences we observed.
DISCUSSION Our results contradict variance discounting's prediction of constant risk-aversion. We found that the general level of mean food reward affected the trade off between mean and variance as predicted by the z-score. The constant of risk-aversion, k, is not constant. A k value that is consistent with one observed preference is inconsistent with preferences after a general increase in means of only 20#1. Our results do not support the predictive use of variance discounting. Until now the strongest evidence supporting the z-score was the changing risk-sensitivity observed in granivorous birds and shrews (Caraco et al. 1980; Caraco 1981, 1983; Barnard & Brown 1985). These experiments showed that foragers were riskaverse when they expected to gain more than they required (#>R), and risk-prone when they expected to require more than they gained (R > #). This behaviour supports the z-score model, and contradicts variance discounting because the constant of risk-aversion measured in the risk-averse case (k>0) could not predict the behaviour observed in the risk-prone case (k < 0). However, the variance discounting's proponents could argue that, although experimentalists can show risk-prone behaviour, risk-proneness is not common in nature, and both models explain dayto-day risk-averse foraging decisions equally well.
Caraco & Lima's (1985) results support this argument. Caraco & Lima tried to determine whether measured mean-standard deviation indifference curves were linear (as predicted by the z-score) or quadratic (as predicted by variance discounting). They found that both models worked equally well. However, estimating even a single indifference point takes many hours of observation, so Caraco & Lima were in the difficult situation of trying to distinguish a line from a curve With only a handful of points. Our results show that the two models make different predictions about the interactions of mean and variance, and these differences might be important in day-to-day foraging decisions, even though both models predict that animals will usually be risk-averse. For example, our experiment shows that a hummingbird's preferences might be different in rich and poor habitats, even if means are always correlated with variances. Variance discounting might be saved by one or two types of post hoc argument. First, a variance discounting proponent might make a specific claim about when he expects k to be constant: for example, 'k will be constant when the change in means is less than X% of the energy requirement'. Second, a variance discounting proponent might argue that the model is purely descriptive: k simply measures observed risk-taking. This relegates variance discounting to a statistical technique, rather than a theoretical insight. However, variance discounting has one unassailable use. It is an extremely convenient formulation that modellers might use to study how risk-sensitivity affects mean-maximizing models. A modeller might say, for example, the k values between k~ and k2 do not change the mean-maximizing result'. Variance discounting, broadly defined, can include models of the form m a x { # - k(#)~2)} where k(#) is now some function of the mean (#, Real, personal communication). Our results and comments only deal with the version of variance discounting where k(#) is a constant for all values of #. However, a constant k value, and its consequence of constant risk-aversion, is explicitly or implicitly assumed in most uses of variance discounting (e.g. Real 1980a, b, 1981; Real et al. 1982; Lacey et al. 1983; Caraco & Lima 1985; Wunderle & O'Brien, in press). Moreover, if the coefficient of risk-sensitivity' (k(#); Real et al. 1981) is left as an
Stephens & Paton: Hummingbird risk-aversion unspecified function of mean, t h e n variance disc o u n t i n g becomes untestably vague: k(#) can be c h a n g e d to a c c o m m o d a t e any observatiOn. A critic might charge t h a t b o t h o f the models considered here are so simple t h a t an a t t e m p t to distinguish between t h e m is a battle of straw men. W e agree t h a t neither model is likely to explain everything a b o u t forager risk-taking. The models are i m p o r t a n t because they help us to fix ideas: viewing t h e m as alternatives suggested our experiments. However, our experiment distinguishes between two contrasting properties of the models: variance discounting's c o n s t a n t risk-aversion a n d the z-score's decreasing risk-aversion. These properties are more general t h a n the simple models considered here, a n d o u r results suggest t h a t the successors of these models s h o u l d n o t assume c o n s t a n t risk-aversion as variance discounting has. ACKNOWLEDGMENTS W e are grateful to C. L. Gass, T. Phillipi, A. E. Sorensen, S. T a m m a n d Z. Tooze for their help a n d advice. W e t h a n k C. P. L. Barkan, T. Caraco, E. L. C h a r n o v , M. E. Ellis, J. R. Lucas, G. H. Orians, L. A. Real, C. T. S n o w d o n , R. C. Y d e n b e r g a n d the reviewers for their c o m m e n t s o n the manuscript. This work was partially s u p p o r t e d by N S E R C g r a n t 6 7 4 8 7 6 to C. L. Gass. D W S was s u p p o r t e d by a N A T O postdoctoral fellowship, and S R P was s u p p o r t e d by a N S E R C studentship. We t h a n k N A T O a n d N S E R C for their support.
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