Test of a Subtractive Theory of "Fair" Allocations

Transcription

1 Journal of Personality and Social Psychology 1989, Vol. 56, No. 5, Copyright 1989 by the American Psychological Association, Inc /89/M0.75 Test of a Subtractive Theory of "Fair" Allocations Barbara Mellers and Elizabeth Hartka University of California, Berkeley Tested models of "fair" allocation judgments in two experiments in which Ss were given information about how hard two people worked on a job and how much one of them was paid. The task was to assign a "fair" payment to the other person. Both the present results and previous research can be described by a subtractive theory of equity. According to this theory, an equitable state is one in which differences between subjective outcomes are equivalent to differences between subjective inputs. "Fair" allocation judgments are then a monotonic function of subjective differences. When the response scale produces ceiling and floor effects, the judgment function has a sigmoidal shape. The question of what constitutes an equitable state has long been of interest to philosophers, psychologists, sociologists, and others (Aristotle, cited in Ross, 1966; also Adams, 1965; Homans, 1961; Walster, Walster, & Berscheid, 1978). Some researchers have studied "fairness" by investigating what contributes to feelings of inequity in the workplace, the marketplace, and interpersonal relationships (Austin, McGinn, & Susmilch, 1980; Messick & Sentis, 1979,1983). Others have asked what rules best describe how people, young and old, "fairly" divide resources or rewards among a group of contributing individuals (Anderson, 1976; Harris, 1976, 1980, 1983; Hook & Cook, 1979;Keil&McClintock, 1983; Mellers, 1982). The present article further examines rules for "fairly" allocating resources and rewards between two persons. Most of the two-person models proposed can be treated as special cases of the following general representation: F(0 A, 0 B ) = G(I A, (1) where F is a function that describes how subjective outcomes for Persons A and B (O A and O B ) are compared and G is a function that describes how subjective inputs (I A and I B ) are compared. It is generally assumed that F and G are the same, and considerable debate has occurred over their functional form. Two of the most popular representations are ratio models (i.e., an equitable state is one in which inputs and outcomes are, in some sense, proportional) and subtractive models (i.e., an equitable state is one in which the difference in the subjective inputs matches the difference in subjective outcomes). The question of how to distinguish between ratio and subtractive operations is not new to equity theory. Both of these representations have been proposed as theories of choice behavior (Luce, 1959; Restle, 1961; Thurstone, 1927), similarity judgment (Beals, Krantz, & Tversky, 1968; Ekman & Sjoberg, The authors would like to express appreciation to Michael Birnbaum, Jerome Busemeyer, and Richard Harris for valuable comments or discussions, to Karen Biagini for help with data collection, and to Norman Anderson for providing his data for reanalysis. Correspondence concerning this article should be addressed to Barbara Mellers, Department of Psychology, University of California, Berkeley, California ), and psychophysical judgment (Birnbaum, 1978, 1980; Stevens, 1971). Developing ordinal tests to distinguish between these operations can, in some cases, be quite difficult. The purpose of the present article is twofold. First, we argue that the present data and previous research on fair allocation judgments are inconsistent with a generalization of the ratio model referred to as the similarity prediction (Anderson, 1976). Second, we propose an alternative model that can describe both sets of results. This model asserts that when making "fair" allocations, people consider the difference between the inputs and then assign an outcome that is a monotonic function of the subjective difference. Changes in the judgment function that are due to floor and ceiling effects are also discussed. A Two-Person Allocation Task Anderson (1976) asked subjects to "fairly" assign a reward to Person B on the basis of Person A's and Person B's inputs and Person A's outcome. For example, if A's input was average, B's input was above average, and A was paid $ 1, how much should Person B receive? Anderson (1976) considered the following ratio model: where O B is the judgment of B's outcome; I B and I A are the scale values that depend on the performance of Persons A and B; and O A is the scale value of the payment for Person A. The ratio model (Equation 2) implies that if O B is plotted as a function of I A and I B for any given O A, the curves should form a divergent bilinear fan (i.e., curves which differ only in slope and intersect at a common point). Anderson pointed out that for his data the curves diverged but no spacing of I B would make them bilinear, and he rejected the ratio model. The Similarity Prediction Anderson then discussed a more general ratio model, referred to as the similarity prediction. It relaxes the assumption about how I A and I B are combined, and it can be written: (2) = fii A,I B)-0 A, (3) 691

2 692 BARBARA MELLERS AND ELIZABETH HARTKA where the symbols are as defined in Equation 2, and f is an unspecified function that compares subjective inputs. Anderson (1976, 1983) concluded his data were consistent with the similarity prediction. The similarity prediction implies that judged outcomes for Person B should be proportionately related across different levels of O A. To test the similarity prediction (Equation 3), one can divide judged outcomes for Person B by the amount that Person A receives (e.g., O B /O A ). The similarity prediction implies that these adjusted outcomes for Person B should not depend on O A. We computed the adjusted means (O B /O A ) for the data of Anderson (1976). Interactions between I A and I B were not identical across all levels of O A. The interactions systematically decrease and become more compressed as A's payment increases. For example, when A's input is very much below average and B's input is very much above average, B's adjusted outcome is 2.74, 2.14, and 1.94, when A's outcome is $1.00, $1.50, and $2.00. In other words, when B works hard and A does not, B receives proportionally less as A's outcome increases. The Linearity Prediction It is interesting to consider an even less stringent assumption of the ratio model than the similarity prediction, which we call the linearity prediction, and can be written as follows: = a A fi;i A,I B ) (4) where a A and b A are linear constants than depend on O A. According to the linearity prediction, judged outcomes for Person B should be linearly related across different values of O A. Linearity is a generalization of similarity. To test the linearity prediction, one can compare ratios of differences between curves across levels of A's outcome. Linearity implies that ratios of differences should be independent of O A. For the data of Anderson (1976), we computed ratios of differences between judged outcomes for Person B at the two most extreme levels of I B when A's input was very much below average and when it was very much above average. Ratios of differences were 2.57, 1.97, and 1.80 when A's outcome was $1.00, $1.50, and $2.00, and did not remain constant, as implied by the linearity prediction. In sum, neither the similarity prediction nor a weaker condition of linearity are consistent with the results of Anderson (1976). Subtractive Theory Mellers (1982) proposed that in two-person tasks, equity occurs when differences in subjective inputs are equal to differences in subjective outcomes. For a task in which subjects are given A's input, B's input, and A's outcome and judge Person B's outcome, the theory can be written: 0 B = J[0 A - I A (5) where J is a monotonic function. 1 One variable that presumably influences the shape of the judgment function is the response scale. When subjective differences (O A - I A + I B ) cover a wide range relative to the response scale, there may be ceiling and floor effects that produce nonlinearities in the judgment function. If the underlying process is a subtractive one (Equation 5) and the judgment function is linear, then when Os is plotted as a function of I B with a separate curve for each level of I A and a separate panel for each level of O A, the curves should appear parallel. The Shape of the Judgment Function If there were floor effects in the response scale, then many small subjective differences would be mapped into similar judged outcomes at the bottom of the scale. In that case, the judgment function would be concave upward, and the curves would diverge (differences between the upper and lower curves would increase from left to right). If there were ceiling effects, then many larger subjective differences would be mapped into similar judged outcomes at the top of the scale. The judgment function would be concave downward, and the curves would converge (differences between the upper and lower curves would decrease from left to right). If both ceiling and floor effects occur, the judgment function might have a sigmoidal shape. A sigmoidal-shaped judgment function was also postulated by Mellers (1982) to account for inequity judgments and by Birnbaum and Rose (1975) to account for nonlinearities produced by a linemark response similar to that used by Anderson (1976). Figure 1 presents an illustration of a sigmoidal-shaped judgment function. Values of O A I A + I B were generated using the numbers 1,0, and 1 for I A and I B ; and using 2, 0, and 2 for O A. The predicted response, O B, is plotted on the ordinate, and the integrated impression prior to the response, J~'(O B ) = O A I A + I B, is on the abscissa. Figure 2 shows the effects of a sigmoidal-shaped judgment function on subjective differences. Upper panels show the predicted responses for the same values of O A, I A, and I B used in Figure 1. B's outcome is plotted as a function of B's input with a separate curve for each level of A's input. The curves are divergent when A's outcome is lower (and subjective differences are small) and convergent when A's outcome is higher (and subjective differences are large). Lower panels show predictions before transformation to the response scale, J~'(O B ). The curves are parallel, consistent with the subtractive model (O A I A + I B ). In Anderson's experiment, subjects judged Person B's outcome on a 20-cm rating scale labeled $0 and $5 at the ends. Since Anderson (1976) used values of O A that ranged only from $ 1 to $2 (compared with a response scale that ranged from $0 to $5), values of O A I A + I B would have tended to be small, as illustrated by Line Segment 1 at the base of Figure 1. This bounded response scale could have produced floor effects, and ' One might argue that the theory should be stated in a more restrictive form. If subjective outcomes (O A ) are some function (h) of actual outcomes, then O A = h(5 A ), where 5 A is the actual outcome for Person A, and subjective inputs, I A = g(0 A ), where </> A is the actual input for Person A, then the model that follows from Equation 1 is as follows: Although Equation 5 is more general in that the response function (J) is not constrained to be h~', this generalization is permitted because other factors presumably influence the judged outcome for Person B, including floor and ceiling effects in the response scale.

3 "FAIR" ALLOCATIONS 693 GO o Hypothesized J Function Six anchor trials were also included with the experimental trials, after Anderson (1976). Those trials had more extreme levels of I A and I B than were used in the experimental trials. According to Anderson (1976, p. 293), "their function is to define the endpoints of the rating scale and avoid end effects by keeping the main data interior to the scale." The stimulus combinations were presented in random order using a booklet format with one page of instructions followed by two pages of test trials. Subjects completed the task individually or in small groups of no more than five. Participants -II Subjective Differences H2 Figure 1. An illustration of the proposed judgment function (J) for the subtractive theory. (This function is concave upward for small values of O A - I A + IB, which implies divergent interactions. For large values of O A I A + IB, it is concave downward, which implies convergent interactions. O = outcomes. I = inputs. A = Person A. B = Person B.) the judgment function would be concave upward. Values of O B would tend to diverge, as found by Anderson (1976). If a wider range of O A were used, as illustrated by Line Segment 2, then, according to the subtractive theory, values of O B should diverge for low values of O A and converge for high values of O A. The ratio model implies divergence for all levels of O A. The present experiments tested this implication of the subtractive model with a nonlinear response scale using an allocation task similar to that used by Anderson (1976). In Experiment 1, we use a wider range of A's outcomes, which would be expected to produce both ceiling and floor effects. If the subtractive model is appropriate and the judgment function has a sigmoidal shape, the interaction between I A and I B should change from divergent to convergent as O A increases. In Experiment 2, we examine judgments for the same task with no restrictions on the maximum value of the response scale. Twenty-one graduate and undergraduate volunteers at the University of California, Berkeley, who were naive with respect to the experimental questions, served as subjects in Experiment 1. A separate group of 20 undergraduates who received credit in a lower division psychology course served as subjects in Experiment 2. Results The upper panels of Figure 3 show mean judgments for Experiment 1, in which subjects were told to use a response scale from $0 to $5. Mean outcomes for Person B are plotted as a function of B's input with a separate curve for each level of A's input and a separate panel for each amount of A's outcome. When A's outcome is relatively small ($1), the curves diverge, and when A's outcome is relatively large ($4), the curves converge. A multivariate test of the contrast formed from the linear component of A's input crossed with that of B's input and A's outcome was performed. This test could be done as a withinsubject t test comparing the bilinear contrasts. It was significant CO O Method Subjects received information about the inputs of two people, A and B, and how much Person A was paid. In Experiment 1, they were asked to assign a "fair" payment for Person B on a scale from $0 to $5. In Experiment 2, they were simply asked to assign a fair payment with no restrictions on the response scale, except that payments are generally assumed to be nonnegative. Design and Procedure The levels of input for Person A and Person B were the same as those used by Anderson (1976, Experiment 1) and referred to how well two people did a job: very much below average (VBA), below average (BA), average (A), above average (AA), and very much above average (Y\A). The input factors for Person A and Person B were factorially combined with Person A's outcome (I A X I B X O A ). Levels of A's outcome were either $ 1 or $4, instead of $ 1.00, $ 1.50, or $2.00, as used by Anderson. CD O I -3 Figure 2. Effects of a sigmoidal-shaped judgment function (J) on responses. (Upper and lower panels show predicted responses and transformed responses for the subtractive theory plotted as function of I B with a separate curve for each level of! A and a separate panel for each level of O A. I = inputs. O = outcomes. A = Person A. B = Person B.) I B H

4 694 BARBARA MELLERS AND ELIZABETH HARTKA means are shown in the lower panels of Figure 4. The value of stress was 5.4%. The curves appear quite parallel, as predicted by the subtractive model. Mean responses for the data of Anderson (1976) were also rescaled to three-factor parallelism, and the results were generally consistent with the model. The value of stress was 8.0%, and the curves appeared to be fairly parallel. Fit of the Subtractive Theory VBA BA A AA VAAVBA BA A AA VAA IB Figure 3. Mean responses for Experiment 1, plotted as in Figure 2. (Notice that when Person A's outcome (O) is $1, the curves diverge. When Person A's outcome is $4, the curves converge, in violation of the similarity prediction. Lower panels show that, in general, the curves can be rescaled to parallelism, consistent with the subtractive model. VBA = very much below average. BA = below average. A = average. AA = above average. "VAA = very much above average.) at the.01 level, t(20) = 7.49, which indicates that the convergence and divergence in Figure 3 are different. Furthermore, 20 out of 21 subjects gave responses which were divergent when O A was $1 and convergent when O A was $4. Thus, the divergent interactions predicted by the ratio model do not occur when there are ceiling and floor effects that presumably prevent subjects from assigning allocations they think are "fair". The lower panels of Figure 3 show a test of the subtractive model and will be discussed in more detail later. The upper panels of Figure 4 show mean judgments for Experiment 2, plotted as in Figure 3. Without restrictions on the response scale, both sets of curves show divergent interactions between I A and I B. Sixteen out of 20 subjects gave responses that were divergent whether O A was $ 1 or $4. Test of the Subtractive Theory To test the subtractive model of "fair" allocations, mean responses were rescaled to three-factor parallelism as a function of O A, I A, and IB, using Kruskal and Carmone's (1969) monotone analysis of variance (MONANOVA). MONANOVA fits a reformulation of Equation 5, that is, J~'(O B ) = O A - I A + I B to the data. The lower panels of Figure 3 show rescaled means for Experiment 1. The value of stress was 8.5%. With the exception of a small crossover of the two highest curves and a slight bulge of the lowest curve when A's outcome was $4, the data appear generally consistent with the subtractive model. The same analysis was performed on the data in Experiment 2. Rescaled The subtractive model was simultaneously fit to the data of Experiments 1 and 2 with a computer program that selected parameters to minimize the sum of squared deviations between predicted and observed responses. The minimization was accomplished with the aid of Chandler's (1969) subroutine, STEPIT. The model had 13 free parameters (5 levels of input with the first level arbitrarily set to 1.0, 2 levels of outcome with the first level arbitrarily set to 1.0, and 2 response functions that were estimated with cubic polynomial functions). Data (dots) and predictions (lines) for Experiments 1 and 2 are shown in Figure 5. The subtractive model can account for the divergence and convergence of the curves in Experiment 1. Furthermore, this model can describe the divergence of the curves in Experiment 2. The proportion of variance unaccounted for by the model was 3.20% in Experiment 1 and 1.57% in Experiment 2. The square root of the average squared error was only 23<t. 2 The estimated judgment functions for Experiments 1 and 2 are shown in Figure 6 with predictions on the ordinate and subjective differences (O A - I A + IB) on the abscissa. In Experiment 1, the function is flat at both the upper and lower ends. This shape is presumably the result of floor and ceiling effects created by the bounded response scale. In Experiment 2, the function is flat at the lower end but not at the upper end. The unbounded response scale presumably removed the ceiling effects, but subjects may have felt constrained by the minimum response of $0, and the implied lower bound produced floor effects. The estimated values of input were 1.00,2.26,3.67,4.97, and 6.81 for the five levels ranging from very much below average to very much above average. The estimated values of O A were 1.00 and In Experiment 1, the coefficients in the response function were 0.86, 0.31, 0.04, and 0.00 for the additive constant, the linear term, the quadratic term, and the cubic term, respectively. In Experiment 2, the coefficients were 0.81, 0.27, 0.04 and 0.00, respectively. The shape of the judgment function is initially flat and then concave upward, as expected if there were floor effects produced by the response scale. The data of Anderson (1976) were also fit to the subtractive model in the same fashion. The model required estimation of 10 parameters (5 levels of input with the first level arbitrarily set to 1.0, 3 levels of outcome with the first level set to 1.0, and a cubic polynomial response function). The subtractive model 2 The ratio model with a linear judgment function was also fit to the data in Experiments 1 and 2 for comparative purposes. The model required 10 free parameters and left 7.64% and 3.31% of the variance unexplained in Experiment 1 and 2, respectively. This model predicts divergent interactions between I A and I B for all four sets of data in Experiments 1 and 2.

5 "FAIR" ALLOCATIONS 695 VBA BA A AA VAA VBA BA A AA VAA IB Figure 4. Mean responses for Experiment 2, plotted as in Figure 2. (When there are no restrictions on the response scale, both sets of curves show a divergent interaction. The parallelism of the curves in the lower panels is consistent with the subtractive theory. O = outcomes. I = inputs. A = Person A. B = Person B. VBA = very much below average. BA = below average. A = average. AA = above average. VAA = very much above average.) appeared consistent with the data and could account for all but 1.85% of the variance. The square root of the average squared error was only 17<t. The estimated values of input were 1.00,1.12,1.27,1.42, and Estimated values of outcomes were 1.00, 1.14, and The coefficients in the polynomial function were 1.79, -5.41, 6.03, and 1.36, respectively. Fit of the Weighted Solutions Model Harris (1983) proposed another model of "fair" allocations referred to as the weighted solutions model. According to this model, "fair" allocation judgments are a weighted average of three allocation rules: (a) equal divisions, (b) proportional divisions, and (c) equal excess divisions (Komorita & Kravitz, 1979). For the present experiment, the model can be expressed as follows: 0 B = ao A + b^oa + (1 - a - b)[o A - I A + I B ], (6) IA where a and b are the weights of the equal divisions and proportional divisions, respectively. This model was fit to the means in Experiments 1 and 2 and used 11 parameters (5 levels of input, 2 levels of A's outcome, 2 weights for Experiment 1 [the third was assumed to be one minus the values of the other two], and 2 weights for Experiment 2). Although this model could account for all but 3.39% of the variance in Experiment 2, the VBA A A AA VAA VBA BA A AA VAA Figure 5. Data (dots) and predictions (lines) of the subtractive model for Experiments 1 and 2. (I = inputs. A = Person A. B = Person B. VBA = very much below average. BA = below average. A = average. AA = above average. VAA = very much above average.) proportion of variance unaccounted for in Experiment 1 was more than twice as much (7.59%). The model predicts slight divergent interactions for both sets of data in Experiment 1 and more pronounced divergent interactions in Experiment 2. For Experiment 1, the estimated weights (constrained to fall between 0 and 1) were 0.77, 0.02, and 0.21 for equal divisions, proportional divisions, and equal excess divisions, respectively. For Experiment 2, the weights were 0.00,0.92, and 0.08, respectively. In sum, the subtractive model with a nonlinear judgment function could account for the data in Experiments 1 and 2. In Experiment 1, the J function had a sigmoidal shape. The flattening of J at both ends could explain why the curves diverge V 5 I 4 3 * 2 o e - 6 I IB Exp. I Exp. 2 Subjective Difference (Oj-I^Ig) Figure 6. Estimated judgment functions for Experiment 1 (left hand panel) and Experiment 2 (right hand panel). (Predicted outcome for Person B plotted against estimated differences, e.g., O A I A + IB- O = outputs. I = inputs.)

6 696 BARBARA MELLERS AND ELIZABETH HARTKA when O A is $ 1 and converge when O A is $4. Experiment 2 shows that when there are no restrictions on the response scale (but an assumed lower bound of $0), the judgment function is positively accelerated, and the curves diverge. Because the data from all three experiments can be described well by the subtractive model and the scale values are all positive, the results can also be rescaled to fit a ratio model. However, this ratio theory would require a nonlinear judgment function, unlike the model in Equation 2 proposed by Anderson (1976). When the judgment function for the ratio model is assumed to be nonlinear, neither the present experiment nor that of Anderson provides a diagnostic test between ratio and subtractive theories of equity, because the two theories are ordinally indistinguishable. Additional tests and assumptions must be made (see Mellers, 1982,1985). Discussion Anderson (1976) claimed that in his data, the divergent interactions between I A and I B were similar across levels of O A and argued that the response scale was linear and the similarity prediction was supported. However, both the present results and those of Anderson show systematic violations of the similarity prediction. Even if the similarity prediction had been supported, it would be incorrect to assume that the response scale was necessarily linear. If one assumes that the similarity prediction is real, then in order to argue that the response scale is linear, one would have to show that no monotonic transformation of the data existed that would both retain the bilinear interaction between O A and ((If,, IB) and leave unchanged the interaction between I A and I B (Birnbaum, 1974; Birnbaum& Veil, 1974). The Subjective Zero Point Birabaum(1978; 1980; 1982) investigated ratio and subtractive models in psychophysical judgment. He concluded that when subjects are asked to compare stimuli using either "ratio" or "difference" tasks, the data are best represented by subtraction. Ratio judgments can be made to fit a ratio model (i.e., to exhibit bilinearity) whenever the task produces an exponential judgment function. Birnbaum argued that ratio models do not describe judgments based on pairs of single stimuli because a ratio model requires a subjective zero point. However, subjects can use a ratio operation when asked to judge "ratios of differences" between pairs of stimulus pairs. In this case, the zero point for stimulus pairs (represented by subtraction) is well-defined, even if the individual stimuli are only known to an interval scale. It seems possible that for "fair" allocation judgments, subjective inputs do not have a well-defined zero point and should be thought of as locations on an interval scale. Harris (1976, 1980) has considered problems with the zero point for inputs (including negative inputs) and has also argued for a subtractive model, which requires no such zero point. Conclusion In conclusion, both the present results and those of Anderson (1976) can be accounted for by a subtractive model with a monotonic function. This theory correctly predicts that the divergent interactions between I A and I B can be reversed by using larger values of O A. It can also account for the data when there is no specified response scale. The shape of the judgment function is affected by the size of the subjective differences relative to the upper and lower bounds of the response scale. This function has a sigmoidal shape when the response scale is bounded at both ends and is positively accelerated when the response scale is unbounded at the upper end. Extensions of the subtractive theory for two persons can also describe the multiple-person data. Mellers (1982) argued that multiple-person judgments of fair allocations and inequity can be explained by a subtractive model that allows contextual effects due to variations in the stimulus distributions (Mellers & Birnbaum, 1982). According to this model, an equitable state occurs when the relative position of a person's inputs in the distribution of inputs matches the relative position of that person's outcome in the distribution of outcomes. For inequity judgments, subjects consider the difference between the two relative positions. The relative position of a stimulus in its distribution is postulated to be its range-frequency value according to Parducci's (1965, 1974, 1982) theory. In conclusion, the subtractive theory provides a coherent framework for understanding psychological comparisons in equity judgment. References Adams, S. J. (1965). Inequity in social exchange. In L. Berkowitz (Ed.), Advances in experimental social psychology (Vol. 2, pp ). New \brk: Academic Press. Anderson, N. H. (1976). Equity judgments as information integration theory. Journal oj'personality and Social Psychology, 33, Anderson, N. H. (1983). Ratio models of equity and inequity: Comment on Mellers. Journal oj'experimental Psychology: General, 112, Austin, W., McGinn, N. C, & Susmilch, C. (1980). Internal standards revisited: Effects of social comparisons and expectancies on judgments of fairness and satisfaction. Journal of Experimental Social Psychology, 16, Seals, R., Krantz, D., & Tversky, A. (1968). Foundations of multidimensional scaling. Psychological Review, 75, Birnbaum, M. H. (1974). The nonadditivity of personality impressions. [Monograph] Journal of Experimental Psychology, 102, Birnbaum, M. H. (1978). Differences and ratios in psychological measurement. In N. J. Castellan & F. Restle (Eds.), Cognitive Theory (Vol. 3, pp ). Hillsdale, NJ: Erlbaum. Birnbaum, M. H. (1980). A comparison of two theories of "ratio" and "difference" judgments. Journal of Experimental Psychology: General, 109, Birnbaum, M. H. (1982). Controversies in psychological measurement. In B. Wegner (Ed.), Social Attitudes in psychophysical measurement (pp ). Hillsdale, NJ: Erlbaum. Birnbaum, M. H., & Rose, B. A. (1975). Judgments of differences and ratios of numerals. Perception & Psychophysics, 3, Birnbaum, M. H., & Veil, C. T. (1974). Scale-free tests of an additive model for the size-weight illusion. Perception & Psychophysics, 16, Chandler, J. P. (1969). STEPIT Finds local minima of a smooth function of several variables. Behavioral Science, 14, Ekman, G., & Sjoberg, L. (1965). Scaling. Annual Review of Psychology, 76,

7 "FAIR" ALLOCATIONS 697 Harris, R. (1976). Handling negative inputs: On the plausible equity formulae. Journal of Experimental Social Psychology, 12, Harris, R. (1980). Equity judgments in hypothetical, four-person partnerships. Journal of Experimental Social Psychology, 16, Harris, R. (1983). Pinning down the equity formula. In D. M. Messick & K. S. Cook (Eds.), Equity Theory (pp ). New York: Praeger. Romans, G. C. (1961). Social behavior: Its elementary forms. New York: Harcourt, Brace & World. Hook, J., & Cook, T. (1979). Equity theory and the cognitive ability of children. Psychological Bulletin, 86, Keil, L. J., & McClintock, C. G. (1983). A developmental perspective on distributive justice. In D. M. Messick, & K. S. Cook, (Eds.), Equity theory (pp ). New York: Praeger. Komorita, S., & Kravitz, D. (1979). The effects of alternatives on bargaining. Journal of Experimental Social Psychology, 15, Kruskal, J. B., & Carmone, F. L. (1969). MONANOVA: A FORTRAN iv program for monotone analysis of variance. Behavioral Science, 14, Luce, D. (1959). Individual choice behavior: A theoretical analysis. New York: Wiley. Mellers, B. A. (1982). Equity judgment: A revision of Aristotelian views. Journal of Experimental Psychology: General, 111, Mellers, B. A. (1985). A reconsideration of two-person inequity judgments: A reply to Anderson. Journal of Experimental Psychology: General, 111, Mellers, B. A., & Birnbaum, M. H. (1982). Loci of contextual effects in judgment. Journal of Experimental Psychology: Human Perception and Performance, 8, Messick, D., & Sentis, K. (1979). Fairness and preference. Journal of Experimental Psychology, 15, Messick, D., & Sentis, K. (1983). Fairness, preference, and fairness biases. In D. M. Messick, & K. S. Cook (Eds.), Equity theory (pp ). New York: Praeger. Parducci, A. (1965). Category judgment: A range-frequency model. Psychological Review, 72, Parducci, A. (1974). Contextual effects: A range-frequency analysis. In E. C. Carterette & M. P. Friedman (Eds.), Handbook of Perception (Vol. 2). New York: Academic Press. Parducci, A. (1982). Category ratings: Still more contextual effects! In B. Wegner (Ed.), Social attitudes and psychophysical measurement. Hillsdale, NJ: Erlbaum. Restle, F. (1961). Psychology of judgment and choice. New "York: Wiley. Ross, W. D. (Ed.). (1966). The works of Aristotle (Vol. 9). London: Oxford University Press. Stevens, S. S. (1971). On the psychophysical law. Psychological Review, 78, Thurstone, L. L. (1927). A law of comparative judgment. Psychological Review, 34, Walster, E., Walster, G., & Berscheid, E. (1978). Equity: Theory and research. Boston: Allyn & Bacon. Received July 2,1985 Revision received September 13,1988 Accepted October 13,1988 Low Publication Prices for APA Members and Affiliates Keeping You Up-to-Date: All APA members (Fellows; Members; and Associates, and Student Affiliates) receive as part of their annual dues subscriptions to the American Psychologist and the APA Monitor. High School Teacher and Foreign Affiliates receive subscriptions to the APA Monitor and they can subscribe to the American Psychologist at a significantly reduced rate. In addition, all members and affiliates are eligible for savings of up to 50% on other APA journals, as well as significant discounts on subscriptions from cooperating societies and publishers (e.g., the British Psychological Society, the American Sociological Association, and Human Sciences Press). Essential Resources: APA members and affiliates receive special rates for purchases of APA books, including the Publication Manual of the APA, the Master Lectures, and APA's Guide to Research Support. Other Benefits of Membership: Membership in APA also provides eligibility for low-cost insurance plans covering life; medical and income protection; hospital indemnity; accident and travel; Keogh retirement; office overhead; and student/school, professional, and liability. For more information, write to American Psychological Association, Membership Services, 1200 Seventeenth Street NW, Washington, DC 20036, USA