TESTING THEORIES TESTING ERRORS

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TESTING THEORIES TESTING ERRORS
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THE COMMON RATIO-EFFECT
Choice 1 A 30, 1 B 40, 0.80 0, 0.20
Choice 2 C 30, 0.25 0, 0.75 D 40, 0.20 0,
0.80
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• THE ASYMMETRIC PATTERN

• Assume
• 80 truly prefer A over B and C over D.
• Error in choice 1 5
• Error in choice 2 30

IT IS IMPORTANT TO KNOW MORE ABOUT ERRORS
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THE FECHNER MODEL
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THE FECHNER MODEL

• The above pattern could be due to next model
• Consider two lotteries, F and G perceived as
  SVF e and SVg e
• Probability of judging G preferable to H
• Pr ((SVG SVH) (eG eH)) gt 0
• If true difference bigger, less chance of error
• This is the essence of any Fechner model true
  (core) error
• Is this model credible? If so, how should we
  model e?

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OUR EXPERIMENT

• ME of each lottery (6 times)
• error estimated as SD
• A total of 274 respondents.
• On-line in 3 spanish universities (Olavide,
  Murcia, Vigo)

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A (84, 0.25 0, 0.75) EV 21.00 D
(60, 0.25 8, 0.75) B (36, 0.55 0, 0.45)
EV 19.80 E (36, 0.40 9, 0.60) C (22,
0.80 0, 0.20) EV 17.60 F (20, 0.80
8, 0.20)
The sdevMEs fall (significantly) and in line with
the ME values This COULD be consistent with
Fechner with the variance of e changing with the
magnitude of the subjective value
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• Lot of overlap between A and C. More room for
  errors than between C and F.

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PROBABILITY EQUIVALENTS

• Assume we elicit the SV of lotteries using
  Probability Equivalents.
• Yardstick lottery (q, 1200)
• Again 6 times for each lottery

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A (84, 0.25 0, 0.75) EV 21.00 D
(60, 0.25 8, 0.75) B (36, 0.55 0, 0.45)
EV 19.80 E (36, 0.40 9, 0.60) C (22,
0.80 0, 0.20) EV 17.60 F (20, 0.80
8, 0.20)
In fact, sdevPEs vary significantly (except D v
E) in the OPPOSITE direction This is contrary to
Fechner Under Fechner, variances should show a
similar pattern with both MEs and PEs because it
is a property of the lottery.
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CHOICES

• A further test of Fechner considers the
  relationship between MEs/PEs and Choices.
• Fechner suggests choice is much like comparing a
  randomly picked ME value from the MEA
  distribution with an independently-picked value
  from the MEB distribution
• Or equally, picking a PEA and, independently, a
  PEB and choosing whichever turns out higher.

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• Assume we have one subject that gives next
  responses to the six CE questions
• For lottery A 20, 30, 45, 55, 65, 75 euro.
• For lottery B 40, 50, 60, 70, 80, 90 euro.
• If we assume that this reflects some kind of
  distribution and that from this distribution the
  subjects also decides choices, that would predict
  something like

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• B chosen 26 times

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9 choices A, BB, CA, CD, EE, FD, FA,
DB, EC, F 6 times
C (22, 0.8 0, 0.2) F (20, 0.8 8, 0.2)
From ME C lt F 47.9
Remember the shape of functions (large overlap)
From PE C lt F 44.9
From actual pairwise choice (each person repeats
6 times)
F chosen on 97.67 of occasions
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152
1
0
0
10
1
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QUESTIONS

• How can we explain that
• Errors follow such different pattern between ME
  and PE.
• The distribution of choices.
• The Fechner (Trueerror) does not seem to work

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A DIFFERENT THEORY

• THE RANDOM PREFERENCE MODEL

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From moment to moment, state of mind may change
(slightly) At any moment / in any particular
state, individual behaves according to some
core theory but the parameters of the
preference function(s) may vary from one occasion
/ state of mind to another For example, under
EU, degree of risk aversion might vary from one
occasion to another as if, for any particular
decision, the individual picks a ut(.) at random
and applies to that decision picks afresh for
next decision Need not just be EU if RDEU,
also allow distribution over probability
transformation function parameter(s) What can
we expect from RP?
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A (84, 0.25 0, 0.75) EV 21.00 D
(60, 0.25 8, 0.75) B (36, 0.55 0, 0.45)
EV 19.80 E (36, 0.40 9, 0.60) C (22,
0.80 0, 0.20) EV 17.60 F (20, 0.80
8, 0.20)

• Assume an EU maximizer with utility function Xr.
  For this subject there is an r such that
  U(A)U(C) in our case r0.87.

• Larger variance for lottery A under CE and for C
  in PE. Variance increases with SV in CE and the
  opposite in PE.

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..NOW TO CHOICES.
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• Consider two lotteries G and H where both have
  the same EV but where the spread of G is
  slightly, but unambiguously, greater than the
  spread of H.
• Consider an individual whose ut(.) are all
  concave but the degree of risk aversion varies
  from one state of mind to another.
• The ME of both lotteries will be always lower
  than EV but we would expect a lot of overlap in
  ME.
• Pairwise choices will display a very different
  pattern if every ut(.) is concave, so she will
  exhibit some degree of risk aversion on every
  occasion of choice and always prefer H to G (as
  in each occasion H and G are evaluated using the
  same r)

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IN SUMMARY

• If we assume
• A transitive core theory (EU, RDU)
• Errors according to RP (imprecision)
• We can explain results that cannot be explain
  with Fechner (Trueerror)

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BUT RP IS NOT ENOUGH
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A (84, 0.25 0, 0.75) EV 21.00 D
(60, 0.25 8, 0.75) B (36, 0.55 0, 0.45)
EV 19.80 E (36, 0.40 9, 0.60) C (22,
0.80 0, 0.20) EV 17.60 F (20, 0.80
8, 0.20)
Our story would predict that people choose S
or R in the same proportion in each caseThis
does not happen. ---BUT MORE IMPORTANT..--------
--
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A (84, 0.25 0, 0.75) EV 21.00 D
(60, 0.25 8, 0.75) B (36, 0.55 0, 0.45)
EV 19.80 E (36, 0.40 9, 0.60) C (22,
0.80 0, 0.20) EV 17.60 F (20, 0.80
8, 0.20)
Mean and median PEs move in the opposite
direction to MEs
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PREDICTIONS AT THE INDIVIDUAL LEVEL
For these values of r, A is better than C.
This distribution is producing Choices, PE and
ME. Then we should expect 1. C chosen more than
50 of cases (median C gt median A). 2. Median
ME(C)gtmedian ME(A) 3. Median PE (C)gtmedian
PE(A) This is the prediction of RP transitive
core ut(.) functions can be ordered
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CONCLUSION

• It seems difficult to reconcile these data with
  any descriptive theory that assumes transitive
  preferences.
• Potential applications of these findings
  welfare economics (behavioral welfare economics),
  equilibrium and all that..., understanting
  consumer behavior.

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AT THE INDIVIDUAL LEVEL

• Consider first the case of an EU (RDU) maximiser
  whose ut(.) functions can be ordered by some risk
  aversion (Prob. Transf.) parameter.
• Suppose that he is asked to undertake choice and
  valuation tasks involving two binary lotteries, G
  and H, where both the variance and expected value
  of G is greater.
• For some ut(.) at the risk-seeking/risk-neutral/le
  ss risk-averse end of the distribution, the
  higher EV of G is sufficient for GgtH.
• Let the proportion of ut(.) that entail GgtH be
  denoted by a then a is the probability of
  observing GgtH on any occasion when he is asked to
  make a Choice.

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THE END
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• If a gt 0.5, the median ut(.) would entail Ggt H,
  and we should expect
• The median MEGgtMEH
• The median PEGgtPEH
• G is chosen more than 50 of the time.