Dynamic Quantum Decision Models Jennifer S. Trueblood University of California, Irvine Thursday, September 5, 13
Aug 28, 2014
Dynamic Quantum Decision Models
Jennifer S. TruebloodUniversity of California, Irvine
Thursday, September 5, 13
Outline
1. Disjunction Effect
2. Comparing Quantum and Markov Models with Prisoner’s
Dilemma Game
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Disjunction Effect
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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
• Therefore you should prefer A over B even when S is unknown
S
S̄
• People violate the Sure Thing Principle (Tversky & Shafir, 1992)
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Disjunction Effect using Tversky & Shafir (1992) Gambling Paradigm
• Chance to play the following gamble twice:
• Even chance to win $250 or lose $100
• Condition Win:
• Subjects told ‘Suppose you won the first play’
• Result: 69% choose to gamble
• Condition Lost:
• Subjects told ‘Suppose you lost the first play’
• Result: 59% choose to gamble
• Condition Unknown:
• Subjects told: ‘Don’t know if you won or lost’
• Result: 35% choose to gamble
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Failure of a 2-D Markov Model
Law of Total Probability:
p(G|U) = p(W |U)p(G|W ) + p(L|U)p(G|L)
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Failure of a 2-D Markov ModelLaw of Total Probability:
p(G|U) = p(W |U)p(G|W ) + p(L|U)p(G|L)
p(G|W ) = 0.69 > p(G|U) > p(G|L) = 0.59
But Tversky and Shafir (1992) found that p(G|U) = .35 < p(G | L) = 0.59 < p(G |W) = 0.69
violating the law of total probability
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2-D Quantum Model
Law of Total Amplitude:
p(G|U) = || < W |U >< G|W > + < L|U >< G|L > ||2
amplitude for transitioning to the “lose” state from
the “unknown” state
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Quantum Model Account Violation of Sure Thing Principle
= || < W |U > ||2|| < G|W > ||2 + || < L|U > ||2|| < G|L > ||2 + Int
p(G|U) = || < W |U >< G|W > + < L|U >< G|L > ||2
Int = 2 ·Re[< W |U >< G|W >< L|U >< G|L >]
To account for Tversky and Shafir (1992) we require Int < 0
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Tversky and Shafir’s Intuition?
• If you win on first play, you play again because you have extra “house” money
• If you lose on first play, you play again because you need to make up for your losses
• If you don’t know, these two reasons interfere and leaving you without any reason coming to mind
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Failure of 2-D Quantum Model!
• Quantum Model must satisfy Double stochasticity
• In particular
• ||<G | W>||2 + ||<G|L>||2 = 1
• But Tversky & Shafir found that
• p(G | W) = 0.69 and p(G|L) = 0.59
• Violates double stochasticity!
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2-D Transition Matrix
General 2-D transition matrix
•Columns of T must sum to1•Rows of T do not have to sum to 1
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Markov Process
•Obeys law of total probability, but allows for general transition matrix
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Quantum Process
•Obeys law of total amplitude and not law of total probability. But U must transform a unit length vector Ψ(0) into another unit length vector Ψ(t)•To preserve lengths, U must be unitary
hN |SihG|Si
�=
hN |W i hN |LihG|W i hG|Li
�·hW |SihL|Si
�
=
hN |W i · hW |Si+ hN |Li · hL|SihG|W i · hW |Si+ hG|Li · hL|Si
�
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Quantum Unitary Matrix
Unitary Matrix
Transition Matrix
•T must be Doubly stochastic: Both rows and columns of T must sum to unity
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Disjunction Effect using Prisoner Dilemma Game (Shafir & Tversky, 1992)
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• Condition 1: You know the other defected, and now you must decide whether to defect or cooperate
• Condition 2: You know the other cooperated, and you must decide whether to defect or cooperate
• Condition 3: You do not know, and you must decide whether to defect or cooperate
Disjunction Effect using Prisoner Dilemma Game (Shafir & Tversky, 1992)
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Results from 4 Experiments(Entries show % to defect)
StudyKnown to
defectKnown 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
Violates the law of total probability
Violates the law of double stochasticity
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Another Failure: Both 2-D Models fail to explain PD Game results
• The Markov model fails because the results once again violate the law of total probability
• The quantum model fails because the results once again violate the law of double stochasticity
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Compatible vs. Incompatible Measures
• The failed QP model assumes beliefs and actions are incompatible
• Previously we assumed that beliefs and actions were represented by different bases within the same 2-D vector space
• Now we need to switch to a compatible representation which requires a 4-D space.
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Inference-Action State Space
4 dimensional space
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Classic EventsSuppose:Observe start at t=0 in state I1A1Do not observe during t=1Observe end at t=2 in state I2A2
Classic Events:I1A1➝ I1A1➝ I2A2 orI1A1➝ I2A2➝ I2A2 orI1A1➝ I2A1➝ I2A2 orI1A1➝ I1A2➝ I2A2
These 4 are the only possibilities in 2 steps; We just don’t know which is true
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Quantum Events
Suppose:Observe start at t=0 in state I1A1Do not observe during t=1Observe end at t=2 in state I2A2
We cannot say there are only 4 possible ways to get there; At t=1, the state is a superposition of all four;There is deeper uncertainty
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Compare 4-D Markov and Quantum Models for PD game
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Markov Model Assumption 1Four basis states: {|DD⟩, |DC⟩, |CD⟩, |CC⟩ }e.g. |DC⟩ ➝ you infer that opponent will defect but you decide to cooperate
e.g. ΨDC = Initial probability that the Markov system starts in state |DC⟩
X i = 1
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Initial inferences affected by prior information (Markov)
Condition 1Known Defect
Condition 2Known Coop
Condition 3Unknown
U = 0.5 D + 0.5 C
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Quantum Model Assumption 1Four basis states: {|DD⟩, |DC⟩, |CD⟩, |CC⟩ }e.g. |DC⟩ ➝ you infer that opponent will defect but you decide to cooperate
e.g. ΨDC = Initial probability amplitude that the Quantum system starts in state |DC⟩
Probability = |ΨDC|2
|Ψ|2 = 1
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Initial inferences affected by prior information (Quantum)
Condition 1Known Defect
Condition 2Known Coop
Condition 3Unknown
U =p0.5 D +
p0.5 C
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Markov Model Assumption 2Strategy Selection
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Strategies affected by game payoffs and processing time
dΨ(t)/dt = K·Ψ(t) (Kolmogorov Forward Equation)
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Intensity MatrixK = KA +KB
KA =
KAd 00 KAc
�KAi =
�1 µi
1 �µi
�
KB =
2
664
�1 0 +� 00 0 0 0+1 0 �� 00 0 0 0
3
775+
2
664
0 0 0 00 �� 0 +10 0 0 00 +� 0 �1
3
775
!
µi depends on the pay-offs associated with different actions
transforms the state probabilities to favor either defection or cooperation depending on pay-offs
Cognitive dissonance - beliefs change to be consistent with actions
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Quantum Model Assumption 2
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Strategies affected by Game Payoffs and Processing Time
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The HamiltonianH = HA +HB
HA =
HAd 00 HAc
�HAi =
1p1 + µ2
i
µi 11 �µi
�
µi depends on the pay-offs associated with different actions
HB =�p2
2
664
+1 0 +1 00 0 0 0+1 0 �1 00 0 0 0
3
775+
2
664
0 0 0 00 �1 0 +10 0 0 00 +1 0 +1
3
775
!
transforms the state probabilities to favor either defection or cooperation depending on pay-offs
Cognitive dissonance - beliefs change to be consistent with actions
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Markov Model Assumption 3output vector
e.g. ϕDC = final probability that the Markov system ends in state |DC⟩.
measurement operator for decision to defect
Probability defect = L·ϕ
T · = � =
2
664
�DD
�DC
�CD
�CC
3
775
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Markov PredictionIf the opponent is known to defect:
If the opponent is known to cooperate:
Under the unknown condition:
L·ϕD = L·TΨD
L·ϕC = L·TΨC
L·ϕU = L·TΨU = L·T(p·ΨD + q·ΨC)= p·L·TΨD + q·L·TΨC
= p· L·ϕD + q· L·ϕC
Known to defect
Known to cooperate
Unknown
Busemeyer et al. (2006)
91 84 66
Markov Model 91 84 between 91 and 84
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Quantum Model Assumption 3output vector
e.g. ϕDC = final probability amplitude that the Quantum system ends in state |DC⟩.
measurement operator for decision to defect
Probability defect = |M·ϕ|2
U · = � =
2
664
�DD
�DC
�CD
�CC
3
775Probability = |ϕDC|2
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Quantum PredictionIf the opponent is known to defect:
If the opponent is known to cooperate:
Under the unknown condition:
Known to defect
Known to cooperate
Unknown
Busemeyer et al. (2006)
91 84 66
Markov Model 91 84 69
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Quantum Prediction
The probability of defection under the unknown condition minus the average for the two known conditions. (Negative values indicate an interference effect.
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Thank You
• Want to learn more...
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Bayesian Analysis of Individual Data
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Model Complexity Issue
• Perhaps quantum probability succeeds where traditional models fail because it is more complex
• Bayesian model comparison provides a coherent method for comparing models with respect to both accuracy and parsimony
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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 unknown2. 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?
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Two Stage Decision Task
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Barkan And Busemeyer (2003) Results
Risk averse after a win
Risk seeking after a loss
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Two Competing Models
1. Quantum Model
2. Markov model
• Reduction of the quantum model when one key parameter is set to zero
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Quantum Model• Four outcomes: W = win first gamble, L = lose first gamble
T = take second gamble, R = reject second gamble
• 4-D vector space corresponding to the four possible events: W ∧ T, W ∧ R, L ∧ T, L ∧ R
F
I
F = U · I
• State of the decision maker:
1.Before first gamble
2.Before second gamble
• From first gamble to second gamble
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Unitary Transformation• From first gamble to second gamble:
F = U · I
allows for changes in beliefs
using one free parameter calculates the utilities for taking the
gamble using two free parameters (loss aversion, , and risk aversion, ) �b
a
• The Markov model is a special case of the quantum model when � = 0
U = exp(�i · ⇡2· (HA +HB))
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Comparing Fits
• Fit both models to the dynamic consistency data:
1. Quantum
• Three parameters: a and b to determine the utilities and for changing beliefs to align with actions
• R2 = .82
(� = 0)
�
2. Markov
• R2 = .78
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Hierarchical Bayesian Parameter Estimation
• Used hierarchical Bayesian estimation to evaluate whether or not H0: for the quantum model
L(Di|✓i)
� = 0
q(✓i|⇡)
r(⇡)
Likelihood of data given model parms for person i
Prior probability of parms for person i dependent on hierarchical parms - binomial distribution
Prior probability over hierarchical parms - uniform distribution [0, 1]
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Distributions
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Estimates of Group Level Parameters
The risk aversion hierarchical parameter is located below 0.5 indicating somewhat strong risk aversion
The loss aversion hierarchical parameter is located above 0.5 indicating higher sensitivity to losses
Busemeyer, J. R., Wang, Z., Trueblood, J. S. (2012). Hierarchical Bayesian estimation of quantum decision model parameters. In J. R. Busemeyer et al. (Ed.), QI 2012, LNCS 7620. Berlin, Germany. Springer-Verlag.
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Estimate of the Quantum Parameter
The hierarchical distribution of the quantum parameter lies below 0.5 implying the mean value is below zero
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