Generalized Exemplar Model of Sampling

Jing QianMax Planck Institute for Human

Development, BerlinFURXII RomeJune 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?

Range Frequency Theory

(functional form)

€

RFT i = weight ×

Mi

− Mmin

Mmax

− Mmin

+ ( 1 − weight ) ×

i − 1

N − 1

RFT is a model of magnitude estimation which rely on two rules:

Relative distance to lowest and highest values

Relative rank in the context

Range Frequency Theory

(example)

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.

Stimulus Magnitude

w=1

w=0

w=0.5

Stimulus Magnitude HighLow

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. )

RFT (conclusion)

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)

RFT is sensitive to the distribution of contextual stimuli

Limitations of RFTRFT 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

Low HighTarget

1 1 1 111

Elements of Sampling 1:

Similarity-based SamplingIf a target retrieves mainly similar

exemplars, exemplars close to the target should have greater weight:

Low HighTarget

1 .2.5 1.5.2

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 SamplingFor wages, and perhaps also prices, it

is also plausible that distant items might carry more weight:

Low HighTarget

1 3 2 123

Such accounts assume that distant items (e.g. an individual earning much more) have greater effectSeveral 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

If γ < 0: similar js will have greater effect— similarity-weighted sampling

If γ > 0: distant js will have greater effect—distance-weighted sampling

γ

γ

Elements of Sampling 3: Loss

AversionLosses loom larger than gains

Low Target

2 11122

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.

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.

Integrating the three elements of

Sampling

Ui (x) =wxi −xmin

xmax −xmin

+ (1−w) 0.5 +α (xi −xj )

γ

j=1

i−1

∑ −β (xj −xi )γ

j=i+1

N

∑

2(α (xi −xj )γ

j=1

i−1

∑ + β (xj −xi )γ )

j=i+1

N

∑

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

γ αβ

GEMS

Utility of xi

Ui (x) =wxi −xmin

xmax −xmin

+ (1−w) 0.5 +α (xi −xj )

γ

j=1

i−1

∑ −β (xj −xi )γ

j=i+1

N

∑

2(α (xi −xj )γ

j=1

i−1

∑ + β (xj −xi )γ )

j=i+1

N

∑

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Range value (as in RFT, weighted by

w)

Downward Comparisons(weighted by )

Upward Comparisons(weighted by )

α

β

Sampling parameter γ

GEMS: Effect of γAssuming an evenly-spaced distribution

(flat or rectangular), let w=0, ,

and only vary γ α =β =1

Similarity

Sampling

RFT

Distance

Sampling

(γ=0)

(γ<0)

(γ>0)

GEMS: Effect of Assuming an evenly-spaced distribution, let w=0, , and only vary the ratio between and

α,β

γ = 0 α β

Downward

Comparison

RFT

Upward

Comparison

(α=β=1)

(α<β)

(α>β)

Application of GEMS: Wage

SatisfactionA test between RFT and Model of Inequality AversionAs both models are nested within GEMS as follows

Ui (x) =wxi −xmin

xmax −xmin

+ (1−w)i −1n−1RFT:

€

Ui(x)=xi−αimax(xj−xi,0)

j≠i∑n−1 −βi

max(xi−xj,0)j≠i∑n−1MIA:

γ =0, α =β =1

γ =1, plus a scaling constant

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.

α,β

α < β

Experiment: Wage Satisfaction

RatingsMethod: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 StimuliTwo different wage distributions are used (positively-skewed and negatively-skewed) Positively and Negatively skewed distributions

16.2 19.5 22.8 26.1 29.4

starting wage /1000

Result: Model ComparisonThree models (GEMS, RFT, MIA) were

fitted to the data, RFT was selected to be the best model.

Range Frequency Model Fit

1234567

17 22 27Wage

PS Mean DataNS Mean DataRFT ModelRFT Model

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.

Fehr-Schmidt Model Fit

1234567

17 22 27Wage

PS Mean DataNS Mean DataFehr-Schmidt ModelFehr-Schmidt Model

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.

α β

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).

αβ γ

General Likelihood Ratio Test for comparisons of nested models

revealed that RFT is the most efficient model that captured

the data.

Background: Probability

Weighting FunctionProspect 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).

€

w ( p ) =δ p

γ

δ pγ

+ ( 1 − p )γ

where w(p) is weighted probability; p is objective probability; parameter primarily controls elevation, and parameter primarily controls curvature of the PWF curve.

γ

δ

Effect of δ Effect of γ

Application of GEMS: Probability Weighting

FunctionWhen 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:

w(p) =αpγ

αpγ + β(1−p)γ

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

This presentation will be repeated

after

10 seconds.

This presentation will be repeated

after

9 seconds.

This presentation will be repeated

after

8 seconds.

This presentation will be repeated

after

7 seconds.

This presentation will be repeated

after

6 seconds.

This presentation will be repeated

after

5 seconds.

This presentation will be repeated

after

4 seconds.

This presentation will be repeated

after

3 seconds.

This presentation will be repeated

after

2 seconds.

This presentation will be repeated

after

1 second.