Generalized Exemplar Model of Sampling

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Generalized Exemplar Model of Sampling

• Jing Qian
• Max Planck Institute for Human Development,
  Berlin
• FURXII Rome
• June 2006

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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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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.
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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.
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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?
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Range Frequency Theory(functional form)
RFT is a model of magnitude estimation which rely
on two rules

• Relative distance to lowest and highest values
• Relative rank in the context

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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 w0, 0.5, 1. )
Stimulus Magnitude
High
Low
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.
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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)

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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
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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).
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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)
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A sampling parameter ?
Effect of item j (magnitude Mi ) on perception of
i Make effect depend on Mi-Mj If ? 0 each
j will have equal effect ( Mi-Mj
1 ) as in RFT If ? lt 0 similar js will
have greater effect similarity-weighted
sampling If ? gt 0 distant js will have greater
effect distance-weighted sampling
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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.
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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.
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GEMS
Sampling parameter
Utility of xi
Downward Comparisons (weighted by )
Range value (as in RFT, weighted by w)
Upward Comparisons (weighted by )
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GEMS Effect of ?
Assuming an evenly-spaced distribution (flat or
rectangular), let w0, , and only vary ?
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GEMS Effect of
Assuming an evenly-spaced distribution, let w0,
, and only vary the ratio between and
? 0
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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.
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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)
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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.
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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.
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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 (p0
and p1), and let w0 (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.
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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_at_mpib-berlin.mpg.de

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