Ldb Convergenze Parallele_trueblood_03

An Introduction to Quantum
Models of Cognition and
Decision-making
Jennifer S.Trueblood
University of California, Irvine
Thursday, September 5, 13

Motivation
• Judgments often deviate from classical probability theory
• Conjunction & disjunction fallacies, base rate fallacies,
subadditivity, etc.
• Heuristics and biases approach lacks a coherent
theoretical explanation for judgment effects
• Quantum probability theory has the potential to explain
judgments from a theoretical standpoint

Probability Theories
• Classic (Kolmogorov 1933) Theory
• Boolean logic: follows the extension rule A ⊆ B, p(A) < p(B)
• Quantum (von Neumann, 1932) Theory
• A generalization of classical theory
• Drops unicity (a single sample space)
• Can violate the extension rule

Six Reasons for a Quantum Approach
to Cognition and Decision
1. Judgments are based on indeﬁnite states
2. Judgments create rather than record
3. Judgments disturb each other, introducing uncertainty
4. Judgments do not always obey classic logic
5. Judgments do not obey the principle of unicity
6. Cognitive phenomena may not be decomposable

Reason 1: Judgments are based
on indefinite states
• Stochastic model: a particle producing a definite sample path through a state
space
• Quantum model: a wave moving across time over the state space
• superposition state: an indefinite state capturing ambiguity and/or
confusion

Reason 2: Judgments create
rather than record
• Classic model: answers are “read outs” from stored states
• Quantum model: answers are constructed from the interaction of an
indeﬁnite state and the question we ask
• Example: ambiguity of emotional state
• “Are you excited?”
• “Are you sad?”

Reason 3: Judgments disturb
each other
• Classical models cannot capture the effects of measurement disturbances
• Quantum theory can allow for one question to disturb the answer to
another.

Reason 4: Judgments do not
always obey classic logic
• Defendant is guilty or innocent; defendant is good or bad
• Classical logic obeys the distributive axiom:
• Guilty ⋀ (Good ⋁ Bad) = (Guilty ⋀ Good) ⋁ (Guilty ⋀ Bad)
• The law of total probability in classical probability theory is derived from the
distributive axiom
• p(Guilty) = p(Good)p(Guilty | Good) + p(Bad)p(Guilty | Bad)
• Quantum logic does not always obey the distributive axiom and thus can
violate the law of total probability

Reason 5: Judgments do not
obey the principle of unicity
• Classical probability theory assumes a single sample space - a complete and
exhaustive description of all events
• Quantum probability allows for multiple sample spaces that are pasted
together in a coherent way
Future events

Reason 6: Cognitive phenomena
may not be decomposable
• Two seemingly distinct and separated systems behave as one: quantum
correlated
• Example, words in a memory experiment
• Two theories:
1.Spreading activation model: words are discrete nodes that can be highly
connected
2.Study word and its associates behave as one: a word’s associative network
arises in synchrony with the word being studied

Examples from Cognition and
Decision
• Examples of paradoxical ﬁndings from cognition and decision that can be
modeled with quantum probability
1. Interference effects in perception
2. Interference of categorization on decision-making
3. The disjunction effect
4. Violations of dynamic consistency
5. Survey question order effects
6. Conceptual combinations

Interference Effects in
Perception
• Consider two different perceptual judgment tasks:
• Task A: binary forced choice response
• Task B: conﬁdence rating on a 7 point scale
• Participants are randomly assigned to two groups:
• Group 1:Task B
• Group 2:Task A followed by Task B

Response Probabilities

Response Probabilities
Task B
.
.
.
1
2
7
S
pB
(RB
=1)
pB
(RB
=7)
Task A Task B
.
.
.
11
2
1
1
2
7
S
pAB
(RB
=1|RA
=1)
pAB
(RB
=7|RA
=2)
pAB
(RA
=2)
pAB
(RA
=1)

Total Probability and
Interference Term
• The law of total probability gives:
TP(RB = k) =
2X
j=1
pAB(RA = j) · pAB(RB = k | RA = j)
• The interference effect for level k of the response to task B:
IntB(k) = pB(RB = k) TP(RB = k)

Conte et al. (2009)
• Experiments on perceptual judgment tasks with pairs of ambiguous ﬁgures

Conte et al. (2009) Experiment 1
• Task was to decide whether the objects (circles or horizontal lines) were
equal or not
Figure A Figure B

Experiment 1 Results
• The inference effect for level B+ (horizontal lines are equal):
IntB(+) = pB(RB = +) TP(RB = +)
= 0.6667 0.5000 = 0.1667
Total Probability for
A followed by B

Interference of Categorization
on Decision-making
• Townsend et al. (2000) Task:
• Categorize faces as ‘Good guy’ or ‘Bad guy’
• Decide to act Friendly (withdraw) or Aggressive (attack)
• Narrow faces had a 0.6 probability of being ‘Bad’ and wide faces had a 0.6
probability of being ‘Good’

Townsend et al. (2000)
• Two conditions

Busemeyer et al. (2009)
• Replication of Townsend (2000)
• Results:
• Total probability of attacking after categorization = 0.59
• Probability of attacking without categorization = 0.69
Interference Effect

Savage’s Sure Thing Principle
• Suppose
• when is the state of the world, you prefer action A over B
• when is the state of the world, you also prefer action A
over B
S
¯S
• Therefore you should prefer A over B even when S is
unknown

The Disjunction Effect
• Violations of the Sure Thing Principle (Tversky & Shaﬁr, 1992) in
the Prisoner’s Dilemma game
You Defect
You
Cooperate
Other
Defects
other: 10
you: 10
other: 25
you: 5
Other
Cooperates
other: 5
you: 25
other: 20
you: 20
Study
Known to
defect
Known to
cooperate
Unknown
Shaﬁr &
Tversky
(1992)
97 84 63
Croson
(1999)
67 32 30
Li & Taplan
(2002)
83 66 60
Busemeyer
et al. (2006)
91 84 66
Observed proportion of defections

Dynamic Consistency
• Dynamic consistency: Final decisions agree with planned decisions (Barkan
and Busemeyer, 2003)
• Two stage gamble
1. Forced to play stage one, but outcome remained unknown
2. Made a plan and ﬁnal choice about stage two
• Plan:
• If you win, do you plan to gamble on stage two?
• If you lose, do you plan to gamble on stage two?
• Final decision
• After an actual win, do you gamble on stage two?
• After an actual loss, do you now choose to gamble on stage two?

Barkan and Busemeyer (2003)
Results
Risk averse
after a win
Risk seeking
after a loss

Question Order Effects
• A Gallup Poll question in 1997 (N = 1002, split sample)
• Do you generally
think Bill Clinton is
honest and
trustworthy?
• How about Al Gore?
think Al Gore is
honest and
trustworthy?
• How about Bill
Clinton?

Question Order Effects:
Assimilation
• Proportion of “Yes” responses
think Bill Clinton is
honest and
trustworthy?
(50%)
• How about Al Gore?
(60%)
think Al Gore is
honest and
trustworthy?
(68%)
• How about Bill
Clinton?
(57%)
18%
3%

Question Order Effects:
Contrast
• Proportion of “Yes” responses
think Newt Gingrich
is honest and
trustworthy?
(41%)
• How about Bob
Dole?
(64%)
think Bob Dole is
honest and
trustworthy?
(60%)
• How about Newt
Gingrich?
(33%)
19%
31%

Conceptual Combinations
• Hampton (1988) asked subjects to rate typicality
1. rate whether an item belonged to category A
2. rate whether it belonged to category B
3. rate whether it belonged to A or B
4. rate whether it belonged to A and B

Results
• Is drinking beer a member of games (.78)
• Is drinking beer a member of hobbies (.20)
• Is drinking beer a member of games or hobbies (.58)
• Is a spider a pet (.40)
• Is a spider a farmyard animal (.33)
• Is a spider both a pet and a farmyard animal (.65)
• Is guppy a pet (.09)
• Is guppy a ﬁsh (.08)
• Is guppy a pet ﬁsh (.39)

Two Dimensional Example

Uncertainty Principle
• Given True is observed
• State changes to True (S T)
• Prob True now equals 1.0
• Prob of Good equals .50
• Certain about True implies uncertain
about Good and visa versa

Interference Principle
• If initially asked True vs
False, and True is observed,
state changes to True
(becoming uncertain about
Good)
• If next asked about Good vs
Bad and Good is observed,
state changes to Good
(becoming uncertain about
True)

Distributive Rule
• Classic Theory
G = G⋀(T⋁F)
= (G⋀T)⋁(G⋀F)
• Quantum Theory
G = G⋀(T⋁F)
≠ (G⋀T)⋁(G⋀F)
(G⋀T) doesn’t exist
(G⋀F) doesn’t exist

Unicity (closure)
• Classic Theory
If G,T are events then (G⋀T)
is an event
• Quantum Theory
If G,T are compatible, then
(G⋀T) is an event
If G,T are incompatible,
then (G⋀T) doesn’t exist

Three Dimensional Example
• Voting Event:
1. democrat (outcome D)
2. republican (outcome R)
3. independent (outcome I)
• Ideology Event:
1. liberal (outcome L)
2. conservative (outcome C)
3. moderate (outcome M)

Classic Set Representation
L C M
D D ∩ L D ∩ C D ∩ M
R R ∩ L R ∩ C R ∩ M
I I ∩ L I ∩ C I ∩ M
Vote
Ideology

Classic Probability Function
p(L) p(C) p(M)
p(D) p(D ∩ L) p(D ∩ C) p(D ∩ M)
p(R) p(R ∩ L) p(R ∩ C) p(R ∩ M)
p(I) p(I ∩ L) p(I ∩ C) p(I ∩ M)
Probabilities
sum to 1 for
nine joint
outcomes

Compatibility
• Compatible events
• Two events can be realized
simultaneously
• Order of events does not matter
• Incompatible events
• Two events cannot be realized
simultaneously
• Order of events does matter
}
}Quantum
Probability
Classic
Probability

Compatible Events in Quantum
Probability
• State vector within nine dimensional space
↵ij
same as classical
probability
q(i j) = ||Pij| i||2
= ||↵ij||2
= p(i j)
Projector (9x9 matrix with
all zeros except a 1 for i∩j)
| i =
2
6
6
6
6
6
6
6
6
6
6
6
6
4
↵DL
↵DC
↵DM
↵RL
↵RC
↵RM
↵IL
↵IC
↵IM
3
7
7
7
7
7
7
7
7
7
7
7
7
5
• Probability amplitude for voting outcome i and
ideology outcome j:
• Quantum probability for voting outcome i and
ideology outcome j

Vector Space For Incompatible
Events
• Represented by two basis for the same 3
dimensional vector spaceD
R
I
C
M
L
• Ideology Basis:
L = liberal
C = conservative
M = moderate
• Voting Basis:
D = democrat
R = republican
I = independent
• Ideology Basis is a unitary
transformation of theVoting Basis:
Id = {U|Di, U|Ri, U|Ii}
V = {|Di, |Ri, |Ii} Id = {|Li, |Ci, |Mi}

Calculating Quantum
Probabilities
• Belief State:
D
R
I
C
M
L
| i = S
| i = L|Li + C|Ci + M |Mi
= U L|Di + U C|Ri + U M |Ii
| i = ↵D|Di + ↵R|Ri + ↵I|Ii
↵ = U = U†
↵
q(D) = ||PD| i||2
= ||↵D||2
q(C) = ||PC| i||2
= || C||2
• Unitary Transformations relate the
probability amplitudes:
• Calculating probabilities

Mixed vs Superposed States
• Suppose voter is NOT independent
• Mixed State:
• 0.5 probability state = D
• 0.5 probability state = R
• Superposition:
S =
1
p
2
(|Di + |Ri)

Mixed vs Superposed States
• For both mixed and superposed states
• Equal probability of voting democrat or republican
• If democrat, probability moderate equals 0.5
• If republican, probability moderate equals 0.25
• In a classical mixed state
• Either democrat or republican exactly
• Total probability moderate = P(D)P(M|D) + P(R)P(M|R) = (.5)x(.5) +(.5)x(.25) = .375
S =
1
p
2
(|Di + |Ri)• In a quantum superposed state
• Neither a democrat or republican exactly
• The probability of being a moderate is zero because S is orthogonal to the
moderate event

ThankYou
• What’s coming next...
• Quantum models of human judgments
• Dynamic quantum decision models

Ldb Convergenze Parallele_trueblood_03

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