Ldb Convergenze Parallele_trueblood_01

Dynamic Quantum
Decision Models
Jennifer S.Trueblood
University of California, Irvine
Thursday, September 5, 13

Outline
1. Disjunction Effect
2. Comparing Quantum and Markov Models with Prisoner’s
Dilemma Game

Disjunction 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
• Therefore you should prefer A over B even when S is
unknown
S
¯S
• People violate the Sure Thing Principle (Tversky & Shaﬁr, 1992)

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’
• Condition Unknown:
• Subjects told:‘Don’t know if you won or lost’

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)

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)
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

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

Quantum Model AccountViolation 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

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

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!

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

Markov Process
•Obeys law of total probability, but allows for general
transition matrix

Quantum Unitary Matrix
Unitary Matrix
Transition Matrix
•T must be Doubly stochastic: Both rows and
columns of T must sum to unity

Disjunction Effect using Prisoner Dilemma
Game (Shaﬁr & Tversky, 1992)

• 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)

Results from 4 Experiments
(Entries show % to defect)
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
Violates the law of total
probability
Violates the law of double
stochasticity

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

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.

Inference-Action State Space
4 dimensional space

Classic Events
Suppose:
Observe start at t=0 in state I1A1
Do not observe during t=1
Observe end at t=2 in state I2A2
Classic Events:
I1A1➝ I1A1➝ I2A2 or
I1A1➝ I1A2➝ I2A2
These 4 are the only possibilities in 2 steps; We just
don’t know which is true

Quantum Events
Suppose:
Observe start at t=0 in state I1A1
Do not observe during t=1
Observe 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

Compare 4-D Markov and
Quantum Models for PD game

Markov Model Assumption 1
Four 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

Initial inferences affected by
prior information (Markov)
Condition 1
Known Defect
Condition 2
Known Coop
Condition 3
Unknown
U = 0.5 D + 0.5 C

Quantum Model Assumption 1
Four 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

Initial inferences affected by
prior information (Quantum)
Condition 1
Known Defect
Condition 2
Known Coop
Condition 3
Unknown
U =
p
0.5 D +
p
0.5 C

Strategy Selection

Strategies affected by game payoffs
and processing time
dΨ(t)/dt = K·Ψ(t) (Kolmogorov Forward Equation)

Intensity Matrix
K = KA + KB
KA =

KAd 0
0 KAc
KAi =

1 µi
1 µi
KB =
2
6
6
4
1 0 + 0
0 0 0 0
+1 0 0
0 0 0 0
3
7
7
5 +
2
6
6
4
0 0 0 0
0 0 +1
0 0 0 0
0 + 0 1
3
7
7
5
!
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

Strategies affected by Game
Payoffs and Processing Time

The Hamiltonian
H = HA + HB
HA =

HAd 0
0 HAc
HAi =
1
p
1 + µ2
i

µi 1
1 µi
µi depends on the
pay-offs associated
with different actions
HB = p
2
2
6
6
4
+1 0 +1 0
0 0 0 0
+1 0 1 0
0 0 0 0
3
7
7
5 +
2
6
6
4
0 0 0 0
0 1 0 +1
0 0 0 0
0 +1 0 +1
3
7
7
5
!
transforms the state
probabilities to favor
either defection or
cooperation depending
on pay-offs
Cognitive dissonance - beliefs change to be
consistent with actions

output 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
6
6
4
DD
DC
CD
CC
3
7
7
5

Markov Prediction
If 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

output 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
6
6
4
DD
DC
CD
CC
3
7
7
5
Probability = |ϕDC|2

Quantum Prediction
If 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

Quantum Prediction
The probability of defection under the
unknown condition minus the average for
the two known conditions. (Negative
values indicate an interference effect.

ThankYou
• Want to learn more...

Bayesian Analysis of Individual
Data

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

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?

Two Stage Decision Task

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

Two Competing Models
1. Quantum Model
2. Markov model
• Reduction of the quantum model when one key
parameter is set to zero

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

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))

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

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]

Distributions

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.

Estimate of the Quantum
Parameter
The hierarchical distribution of the
quantum parameter lies below 0.5
implying the mean value is below
zero

Ldb Convergenze Parallele_trueblood_01

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