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Generalized Exemplar Model of Sampling. Jing Qian Max Planck Institute for Human Development, Berlin FURXII Rome June 2006. Content of presentation. This presentation contains one model (GEMS) and its application in two areas: wage satisfaction and probability weighting function.
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Generalized Exemplar Model of Sampling • Jing Qian • Max Planck Institute for Human Development, Berlin • FURXII Rome • June 2006
Content of presentation This presentation contains one model (GEMS) and its application in two areas: wage satisfaction and probability weighting function
Abstract 1/2 An integrative model (GEMS: Generalized Exemplar Model of Sampling) is proposed for measuring three factors that influence preference formation: Loss aversion, sensitivity to distances to other exemplars in the context, and sensitivity to relative rank information.
Abstract 2/2 The GEMS model incorporate two models, namely Range Frequency Model (Parducci, 1965; 1995) from psychophysics, and Model of inequality aversion (Fehr & Schmidt, 1999) from Economics as nested special cases.
What is Range Frequency Theory? Range Frequency Theory is a simple mathematical model of subjective magnitude judgement of stimuli in a context. Originally developed in psychophysics (Parducci; 1960s) “How loud is this sound?” (1 - 7 scale) “How heavy is this weight?” Now being applied to judgement of other quantities: How satisfied are you with your wage? How expensive is this price? How attractive is this face?
Relative distance to lowest and highest values Relative rank in the context Range Frequency Theory(functional form) RFT is a model of magnitude estimation which rely on two rules:
Range Frequency Theory(example) The figure below shows the subjective rating of the stimuli in the two distributions according to Range Frequency Theory (different weighting were applied: w=0, 0.5, 1. ) Stimulus Magnitude Low High Above are stimuli drawn from two distributions: Unimodal (red) and Bimodal (black) Note the two stimuli pointed by the two arrows. They are of the same magnitude in the two distributions, but their relative ranking were different.
RFT (conclusion) RFT is sensitive to the distribution of contextual stimuli • As shown in the example, the stimuli of same magnitude were rated as subjectively higher in a bimodal distribution than in a unimodal distribution due to their difference in relative ranking within each perspective distributions. • RFT effect is very robust, and can be found in a wide array of applications, such as Judgement of equity or fairness (Mellers, 1982, 1990), Price perception (Neidrich, Sharma & Wedell, 2001; Qian & Brown, 2005), Wage satisfaction (Brown, Gardner, Oswald & Qian, 2004). • Explaining the shape of probability weighting function (Brown & Qian, 2005)
Limitations of RFT RFT is an elegant model which works well in pre-defined context, but… The issue of sampling is left unspecified: RFT assumes that all contextual information from the environment are sampled and remembered RFT assumes all contextual information is weighted equally
Elements of Sampling 1: Similarity-based Sampling If a target retrieves mainly similar exemplars, exemplars close to the target should have greater weight: For example, if a target price of £200 is being evaluated, a similar price of £220 should exert more influence than a distant price of £280. Indeed, experimental evidence showed £220 is weighted twice as great as £280 when judging £200. (Qian & Brown, 2005).
Elements of Sampling 2: Distance-based Sampling For wages, and perhaps also prices, it is also plausible that distant items might carry more weight: Such accounts assume that distant items (e.g. an individual earning much more) have greater effect Several models within economics work like this: Models of inequality aversion (Fehr & Schmidt, 1979) Models of relative deprivation (Deaton, 2001)
A sampling parameter γ Effect of item j (magnitude Mi ) on perception of i: Make effect depend on |Mi-Mj| If γ = 0: each j will have equal effect ( |Mi-Mj|= 1 ) — as in RFT Ifγ< 0: similar js will have greater effect — similarity-weighted sampling If γ > 0: distant js will have greater effect —distance-weighted sampling
Elements of Sampling 3: Loss Aversion Losses loom larger than gains Usher and McClelland (2004) showed loss aversion in price perception: Lower prices carry more weights than higher prices. In Fehr and Schmidt (1999)’s model of inequality aversion, income below or above the reference point were weighted differently.
Integrating the three elements of Sampling where Ui is utility of a contextual stimulus xi , other contextual stimuli are denoted as xj within a range of [xmin , xmax]. w is the weighting assigned to the range-based component, and (1-w) to the exemplar based component. is the sampling parameter, and are weights assigned to downward comparison and upward comparisons respectively.
GEMS Sampling parameter Utility of xi Downward Comparisons (weighted by ) Range value (as in RFT, weighted by w) Upward Comparisons (weighted by )
GEMS: Effect of γ Assuming an evenly-spaced distribution (flat or rectangular), let w=0, , and only varyγ
GEMS: Effect of Assuming an evenly-spaced distribution, let w=0, , and only vary the ratio between and γ = 0
Application of GEMS: Wage Satisfaction A test between RFT and Model of Inequality Aversion As both models are nested within GEMS as follows RFT: MIA: Where represent the weightings given to downward and upward comparisons respectively. The inequality aversion model assumes , i.e., higher wages will have greater impact than lower wages.
Experiment: Wage Satisfaction Ratings Method: Participants (24 psychology students) were presented simultaneously with 11 possible starting salaries offered to themselves and their peers for a similar first job after graduation. They were asked to rate (on a 1-7 scale) how satisfied one would be with each starting salary knowing exactly what other people would get at the same time. Design of Stimuli Two different wage distributions are used (positively-skewed and negatively-skewed)
Result: Model Comparison Three models (GEMS, RFT, MIA) were fitted to the data, RFT was selected to be the best model. When all four parameters are allowed to freely vary, The General Model obtained a fit of R2=.963; when w= .32; =1.01; and =.98; =.008. The plot of model fit looks very similar to Fig. 1 (therefore not shown here). The MIA Model obtained a fit of R2=.896 (when w=.43, and = 1.10, =1.03) for data from all participants, although the points in the figure shows only the mean data. General Likelihood Ratio Test for comparisons of nested models revealed that RFT is the most efficient model that captured the data. The RFT Model obtained a fit of R2=.952 (when w=.38) for data from all participants, although the points in the figure shows only the mean data.
Background: Probability Weighting Function Prospect Theory (Kahneman & Tversky, 1979) assumes that, when evaluating risky outcomes, small probabilities are over-weighted, and large probabilities are under-weighted. A Probability Weighting Function (PWF) is used to transform objective probabilities into subjective probabilities. PWF can be captured in a psychologically meaningful way as the functional form below. (Gonzalez & Wu, 1999). Effect of Effect of where w(p) is weighted probability; p is objective probability; parameter primarily controls elevation, and parameter primarily controls curvature of the PWF curve.
Application of GEMS: Probability Weighting Function When a probability is evaluated, against a contextual set of other probabilities, the subjective magnitude of the target probability is a function of its distances with other exemplars. In the extreme case, if a target probability is evaluated against only the two default state (p=0 and p=1), and let w=0 (because all probabilities are scaled between [0, 1] naturally, then GEMS can be simplified as: The above formulation is equivalent to that of Gonzalez & Wu(1999), which means that an exemplar-based interpretation is suitable to explain the origin of the probability weighting function.
Conclusions • GEMS can be used to model similarity/distance based sampling and loss aversion in judgement of contextual magnitudes such as wages, or probability of winning a gamble. • GEMS is also used in modelling ratings of price attractiveness (Qian & Brown, 2005), and evidence for similarity sampling and loss aversion were obtained. • GEMS provides a method to look into two influences of contextual influences: the distribution of contextual stimuli; and the different weighting given to similar vs. dissimilar stimuli and stimuli that lie below or above the reference point of judgement. • Further details and papers with regard to GEMS and experimental results can be obtained upon request: Qian@mpib-berlin.mpg.de
