Transcript A Quantum Framework for `Sour grapes` in

QUANTUM INTERACTION 2013, LEICESTER, UK
A Quantum Framework for ‘Sour grapes’ in
Cognitive dissonance.
Polina Khrennikova,
School of Management,
University of Leicester
COGNITIVE DISSONANCE BIAS
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The bias of cognitive dissonance firstly discovered by Leon Festinger in
1950th
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Is often referred to as an action- opinion theory, where actions can
influence a person’s opinions , beliefs, identity, see also Self- Perception
theory by Bem.
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The term refers to various situations ( contexts) were the individual is
faced with conflicting cognitions ( e.g a smoker, a person engaging in an
unpleasant activity to reach a goal)
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An tension in the person’s mental state occurs influenced by his/ her
beliefs, emotions, attitudes, identity, and other dissonant elements
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The choices, opinions and actions are influenced by the personal
endeavor to balance the conflicting cognitions and restore the mental
harmony
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‘The agents that are comfortable with dissonance will likely be able to
maintain attitudes that do not conform to their actions while those who prefer
a consistent cognitive state will experience a significant swing in attitude as a
result of actions that they choose to take’ (Kitto, Boschetti, Bruza, 2012, p.8)
COGNITIVE DISSONANCE AND VIOLATION OF BAYESIAN
UPDATING
•
Cognitive dissonance type of behavior is biased and inconsistent
with the postulates of rational homo economicus.
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Incorrect updating of new information and the violation of
classical probabilistic framework takes place.
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Individuals are processing information incompletely ignoring
some factors not to cause uneasiness and cognitive discomfort
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making excuses and lowering the significance of the dissonant
element
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or exaggerating the importance of some factors
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Disjunction and Conjunction errors take place
e.g the information about unhealthiness of smoking, liking of an ‘ unpleasant
boring’ task or job.
OUTLINE OF THE PAPER
To show how this type of behavior works in practice we present :
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a) a ‘gedanken experiment’ in a simplified context
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b) illustrative experiment ‘the forbidden toy paradigm’ by
Aronson and Carlsmith (1963 ) with real data.
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After presenting the experimental data we will strive for a solution
of the cognitive dissonance problem with the help of the quantum
framework
•
We use the quantum probabilistic framework to find the
interference effect and to test if Born’s rule can be applied for
decision probabilities
BAYES RULE AND THE LAW OF TOTAL PROBABILITY
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Bayesian probability widely applied in modern economics and
decision making through the 20th Century is explicating a
particular concept of rational choice.
•
The Bayes Formula for an event A ( also called prior probability)
is updated as new event B takes place (information, data).
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P (A|B) depicts the conditioning of A by the new event B, also
known as posterior probability.
• 𝑷 𝑨𝑩 =
•
𝑷 𝑩ǀ𝑨 𝑷 𝑨
𝑷 𝑩
=
𝑷
𝑩𝑨
𝑷 𝑩ǀ𝑨 𝑷 𝑨
𝑷 𝑨 +𝑷 𝑩 𝑨
−
𝑷 𝑨−
The Bayes formula can be also expressed thought the Law of Total
Probability (right hand side)
THEORETICAL GEDANKEN EXPERIMENT
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The aim is to show in a simplified context the violation of Bayesian
updating procedure occurring with Cognitive Dissonance.
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Part A: The children are asked to choose a toy which they would mostly like to
play with ( we depict G+). After the experiment takes place where each child is
left in a room with a variety of toys for 10 min and covertly observed. Two
threat contexts are introduced: one group of the children is told that they are
severely prohibited to play with the favorite toy ( S) the second group are
mildly prohibited to play with the favorite toy (M).
•
In line with the postulates of rationality and Bayes updating an “inequality of
rationality” should hold:
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𝑃 (𝐺 + |𝑀) > 𝑃 (𝐺 + |𝑆)
(1)
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Part B: The experimenter returns to the room and ‘removes’ the experimental
context, allowing each child to play with any toy including the favorite one. The
children who play with the experimental toy (G+) are asked in which groups
they were: S or M.
•
The probabilities of observed data (S|G+) and (M|G+) are obtained ( the
likelihood function)
“GEDANKEN EXPERIMENT” – OUTCOMES
Bayes formula would give :
𝑃 𝑀ǀ𝐺 + =
𝑃 𝐺+ǀ𝑀 𝑃 𝑀
𝑃 𝐺+
(2)
𝑃 𝑆ǀ𝐺 + =
𝑃 𝐺+ǀ𝑆 𝑃 𝑆
𝑃 𝐺+
(3)
P (G+) for both contexts = 1
P (S) = P(M)= ½
We can *switch* the probabilities and (1) would predict that following “contextual
behavior inequality” would hold:
(𝑴ǀ𝑮+) > 𝑷(𝑺ǀ𝑮+) (4)
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Cognitive dissonance context gives a violation of (4) and consequently
(1) involving a mismatching of the Bayesian updating procedure.
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This gives an indication of the impossibility to use the apparatus of
classical probability theory for such decision making context.
EXPERIMENT FORBIDDEN TOY PARADIGM
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Cognitive dissonance is measured indirectly via the change of a toy’s
attractiveness among the participants.
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The aim of the experiment is to show that cognitive dissonance exists
and can be enhanced or reduced in this example by the level of
prohibition.
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Two threat contexts Mild and Severe are applied. The design is similar to
our ‘Gedanken experiment’.
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Here the context is multipart and instead the desirability of the toy before
and after the experimental conditions is measured. It is established
through a ranking of the toys before and after the experiment.
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We denote it by L+/ L- (the same or increased attractiveness of the toy or
decreased attractiveness of the toy)
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Note: 𝑃 𝐺 + ǀ𝑆 = 𝑃 𝐺 + ǀ𝑀 = 0 In
plays with the toy
both threat contexts none of the children
METHOD
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22 preschool children ( 11 boys and 11 girls ranging in age from
3.8 to 4.6 years)
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all children took part in both experimental conditions (Mild and
Severe) with a time interval of 45 days
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Part A: a ranking of the toys two by two, until a choice between 10
pairs of toys is established. The experimenter takes second
ranked toy places it on a low board and leaves the room. The
child is observed for 10 min through a one way mirror .
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Part B: the experimenter comes back and gives each child a
chance to play with the toys again. After the second ranking list is
established.
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an increase of attractiveness of the toy after the threat in the
Severe condition and a decrease in the Mild condition is observed.
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the authors make an additional experiment on 11 children to
establish a baseline for the effect of ´increased desirability’ in S
context
ANALYSIS OF EXPERIMNTAL DATA
•
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The results support the hypothesis that cognitive dissonance
arises in M context ( the children reduce the dissonance by lowering
the attractiveness of the key toy)
The data from the experiment
gives following frequencies:
P(L+|M) 0.636
P(L-|M)
0.364
P(L+|S)
1
P(L-|S)
0
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We check if the Formula of total probability holds:
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(𝐿+) = (𝑆)(𝐿 + |𝑆) + 𝑃(𝑀)𝑃(𝐿 + |𝑀) = 0.818 ≠ 1
The principle of additivity of FTP is violated
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Similar violation of FTP as observed in the disjunction effect in Savage
Sure Thing Principle violation
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we cannot directly envisage whether the Bayes formula was violated
because of the altered context
MOTIVATION FOR USAGE OF QUANTUM FRAMEWORK
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The aim of applying the QL representation of observables is to predict
the prior probability 𝑃(𝐿+) with the aid of conditional probabilities 𝑃 𝐿 + 𝑆
and 𝑃 𝐿 + 𝑀 .
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From quantum physics we borrow the notion of incompatible
observational contexts that lead to complementarity of quantum
measurements
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The usage of quantum probabilistic framework provides us a numerical
measure of incompatibility of cognitions (the so-called coefficient of
interference)
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In our case: cognitive dissonance phenomenon is regarded as
interference of children’s cognitions that are incompatible.
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the emotional part also plays an important role namely the liking for the
toy. It can also be incompatible with the cognition part of the mental state
( see eg. Busemeyer and Bruza, 2012)
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Mathematically : probability amplitudes for L+ and M or S threat
conditions ( to ‘like the toy’ and ‘not to be allowed to play with it’) interfere
with each other
QUANTUM PROBABILISTIC REPRESENTATION OF DATA
we apply the quantum probability formula as an extension of the
traditional probability equation with the ‘interference term’.
𝑷 𝑩 = 𝑷(𝑩|𝑨)𝑷 (𝑨) + 𝑷(𝑩|𝑨−) 𝑷(𝑨−) + 𝟐𝒄𝒐𝒔𝜽 (𝑷(𝑨)𝑷(𝑩|𝑨)𝑷(𝑨−) 𝑷(𝑩|𝑨−)
For our problem it has the form:
𝑷 (𝑳+) = 𝑷 (𝑴) 𝑷 (𝑳 + |𝑴) + 𝑷(𝑺) 𝑷 (𝑳 + |𝑺) + 𝟐𝒄𝒐𝒔𝜽 𝑷(𝑴)(𝑷𝑳 + |𝑴)𝑷(𝑺)𝑷(𝑳 + |𝑺)
we obtain:
angle θ= 1.34 radian
cos θ= 0.228 – Positive interference of probability amplitudes for our observables
QUANTUM PROBABILISTIC REPRESENTATION OF DATA
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interference of probability amplitudes for severe punishment condition
and the liking of the toy possibly give positive interference?
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Born’s rule check : whether we can use the final probability amplitudes
to determine the prior probability ( L+ or L-)
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𝑷 𝑳+ = 𝜳
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𝜳 =
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𝜳 𝟐 = |𝟎. 𝟕𝟐𝟏 + 𝟎. 𝟔𝟖𝟗𝒊| 𝟐 = 0.52+ 0.475 = 0.995≈ 1
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𝟐
𝑷(𝑴)𝑷(𝑳 + |𝑴) + 𝒆𝒊𝜽
𝑷(𝑺) 𝑷(𝑳 + |𝑺)
Born’s Rule holds and we can proceed by using it to reconstruct the
wave function from transition probabilities and decision making context.
QUANTUM PROBABILISTIC REPRESENTATION OF DATA
•
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Transition probabilities for our experiment
𝟎. 𝟔𝟑𝟔 𝟎. 𝟑𝟔𝟒
𝟏
𝟎
we note that the matrix is not doubly stochastic as it should be
in quantum physics:
• Statistics of the experiment is neither quantum nor classical. The
‘Quantumness’ is merely present in the phenomenological
application of mathematical calculus.
• Observables (in our case the choices L+/L- and events S/M) are
not completely captured by the two dimensional Hilbert space
and a state space of higher dimension would be needed.
SUMMARIZING REMARKS
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we observed non – classicality of children's behavior, were
Kolmogorov’s probabilistic framework was violated
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an illustration of a direct violation of the Bayes formula ( in the
gedanken experiment part) was shown
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We applied to our problem the quantum probabilistic framework and
found a positive interference of transition probability amplitudes
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We checked whether Born’s rule can be applied for this context to
obtain the final probabilities
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As a next step we propose to use dynamical quantum equation for
modeling the state transition and finding final choice probabilities
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we remark that there is no claim about universality of Quantum
framework possibly alternative classical probabilistic frameworks
could be suggested
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We primarily strive to accurate and more general mathematical
framework
• thank you for your attention 