Lower Cumulative Independence

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Lower Cumulative Independence

• Michael H. Birnbaum
• California State University, Fullerton

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LCI is implied by CPT

• EU satisfies LCI
• CPT, RSDU, RDU satisfy LCI.
• RAM and TAX violate LCI.
• Violations are direct internal contradiction in
  RDU, RSDU, CPT, EU.

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In this test, we increase z in both gambles and
coalesce it with y (in R), and we increase y and
coalesce it with x (in S only). So we have
improved S relative to R.
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Lower Cumulative Independence (3-2-LCI)
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LCI implied by any model that satisfies

• Consequence monotonicity
• Transitivity
• Coalescing
• Comonotonic restricted branch independence

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Example Test of 3-2-LCI
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Generic Configural Model
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CPT Cannot Handle Violation of 3-2-LCI
Suppose CPT satisfies coalescing
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2 Types of Reversals
SR This is a violation of LCI. If systematic,
it refutes CPT.
RS This reversal is perfectly consistent with
LCI. (We Improved S relative to R.)
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EU allows no violations

• EU implies LCI

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RAM Weights
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RAM Violations

• RAM violates 3-LCI. Even when the a(i,n) are all
  equal, if t(p) is negatively accelerated, RAM
  violates coalescing coalescing branches with
  better consequences makes the gamble worse and
  coalescing the branches leading to lower
  consequences makes the gamble better. Even
  though we improved S, we made R seem better.

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TAX Model
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TAX Violates LCI

• Special TAX model violates 3-LCI. Like RAM, the
  model violates coalescing.
• Predictions were calculated in advance of the
  studies, which were designed to test those
  specific predictions.

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TAX Model Violates LCI
According to parameters estimated from previous
data, TAX implies a violation of LCI in the
example used. The prediction is fairly robust
with respect to the parameters, and holds for d gt
.36 and g lt 1.06. As shown later, violations are
observed here, showing that at least some
participants have parameters in this region. By
adjusting the consequences, we can test other
combinations of parameters.
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Summary of Predictions

• EU, CPT satisfy LCI
• TAX RAM violate LCI
• Here CPT defends the null hypothesis against a
  specific prediction made by both RAM and TAX.
  Predictions were made in advance and studies
  designed to test them.

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Birnbaum (99), n 124
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Lab Studies of LCI

• Birnbaum Navarrete (1998) 27 tests n 100
  (p, q) (.25, .25), (.1, .1), (.3, .1), (.1,
  .3).
• Birnbaum, Patton, Lott (1999) n 110 (p, q)
  (.2, .2).
• Birnbaum (1999) n 124 (p, q) (.1, .1),
  (.05, .05).

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Web Studies of LCI

• Birnbaum (1999) n 1224 (p, q) (.1, .1),
  (.05, .05).
• Birnbaum (2004b) 12 studies with total of n
  3440 participants different formats for
  presenting gambles probabilities (p, q) (.1,
  .1), (.05, .05).

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Additional Replications

• A number of as yet unpublished studies have
  replicated the basic findings with a variety of
  different procedures in choice.

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Error Analysis

• We can fit true and error model to data with
  replications to separate real violations from
  those attributable to error.
• Model implies violations are real and cannot be
  attributed to error.

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Violations predicted by RAM TAX, not CPT

• EU and CPT are refuted by systematic violations
  of LCI.
• TAX RAM, as fit to previous data correctly
  predicted the modal choices. Predictions
  published in advance of the studies.
• Violations of LCI are to CPT as the Allais
  paradoxes are to EU.

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To Rescue CPT

• CPT cannot handle the results unless it becomes a
  configural model.
• For CPT to handle these data, allow different
  W(P) functions depending on the number of
  branches in the gambles. Let g lt 1 for two-branch
  gambles and g gt 1 for three-branch gambles.

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Add to the case against CPT/RDU/RSDU

• Violations of Lower Cumulative Independence are a
  strong refutation of CPT model as proposed.

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Next Program UCI

• The next programs reviews tests of Upper
  Cumulative Independence (UCI).
• Violations of 3-2-UCI contradict any form of RDU,
  CPT, including EU.
• They are consistent with (and were predicted by)
  RAM and TAX.

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For More Information
mbirnbaum_at_fullerton.edu
http//psych.fullerton.edu/mbirnbaum/
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