Lecture 3: quantum probabilities and bounded rationality

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Lecture 3quantum probabilities and bounded
rationality
reinhard blutner http//www.blutner.de blutner_at_uva
.nl
Institute for Logic,Language and Computation
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Outlook

• Quantum probabilities and Gleasons theorem
• Bounded rationality
• The conjunction puzzle
• The disjunction puzzle
• Other puzzles

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1 Quantum probabilities and Gleasons theorem
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Classical and non-classical observables

• Classical observables are defined by commuting
  operators, non-classical ones by non-commuting
• Typical but unacceptable claim The macro-world
  can always be described by classical observables
  (and the micro-world by non-classical, quantum
  observables)
• The emergence of quantal macrostates does not
  necessarily require the reference to
  corresponding quantal microstates (Aerts)
• Complementary observables can arise in classical
  dynamic systems with incompatible partitions of
  the phase space (b. Graben Atmanspacher)

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Qantum probabilities

• Let H be a Hilbert space. Define an additive
  probability measure on a projection lattice of H,
  i.e.
• ?(ab) ?(a) ?(b) if a and b are orthogonal
  projections
• ?(1) 1 (1 projects the whole Hilbert space H)

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Gleasons theorem

• Let H have dimension gt 2. Then every countably
  additive probability measure on a projection
  lattice of H has a unique linear extension
  defined for all (bounded) operators on H.

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Consequence of Gleasons theorem

• Each additive probability measure ? on a
  projection lattice of H (dimension gt 2) can be
  represented by a density operator
• ? ?j m(j) j??j
• on H in the following way
• ?(a)  Tr(?a) 1
• supposed the vectors j? form an orthonormal
  system of eigenvectors of a, and ?j m(j) 1
• 1 Tr(X) ?i ?i X i? (for an orthonormal
  system of eigenstates i of X)

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Proof

• ?(a)  ?(?ij i??iaj??j)
• ?(?i i??i) ?iai? (since aj? ?j j?)
• ?i ?(i??i) ?iai?
• ?i m(i) ?iai?
• ?ij m(i) ?ij??jai?
• ?i ?i ?j(m(j)j??j) a i?
• Tr(?a)

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Conditioned probabilities1

• For defining conditioned probabilities we
  stipulate the following condition for all
  projections a, b in H
• ?(ba) ?(b)/?(a) if b ? a (i.e. ba b), ?(a)
  ? o
• Fact ?(ba) ?(abaa) ?(aba)/?(a)
• Proof b aba a'ba aba' a'ba'
• ?(ba) ?(abaa) ?(a'baa) ?(aba'a)
  ?(a'ba'a)
• ?(a'baa) ?(aba'a) ?(a'ba'a) 0
• (e.g. ?(a'baa) ? ?(a'a) 0, etc. )
• Thus, ?(ba) ?(abaa) ?(aba)/?(a)

1 I follow the very elegant formulation given by
Gerd Nistegge (2008) in annals of physics
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Asymmetric conjunction

• Use the notion (a b) for a sequence of two
  projection operators
• The definition for the probability for sequences
  is as follows ?(a b) ? ?(a) ?(ba) ?(aba)
• If a and b commute we get ?(a b) ?(b a),
  i.e. ?(a) ?(ba) ?(b) ?(ab) (Bayesian
  formula)
• If a and b dont commute the Bayesian formula can
  be violated!

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Conditioning

• Theorem Let a and b be projection operators,
  then 1
• ?(b) ?(a b) ?(a' b) ?(aba' a'ba)
• Proof a a' b b' 1 aa' bb' 0
• b aba a'ba aba' a'ba'
• Using Gleasons theorem yields the result!
• Remark The term ?(aba' a'ba) describes an
  interference effect which vanishes in the
  classical case (where all operators are
  commuting).
• 1 a' means the same as a?

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Explicit calculation of the interference term

• Assume that a and a' represent pure states a
  a??a, a' a'??a', aa' 0, aa' 1.
• Fact ?(aba' a'ba) 2 ?1/2(a b) ?1/2 (a' b)
  cos(?)
• Proof ?(aba' a'ba)
• ?ab??a'b? a??a' ?ab??a'b? a'??a
• use ?ab??a'b? ?ab??a'b? exp(i?)
• 2?ab??a'b??((a??a' exp(i?) a'??a
  exp(-i?))
• 2 ?1/2(a b) ?1/2 (a' b) cos(?)

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2 Bounded rationality
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Herbert Simon 1955

• Boundedly rational agents experience limits in
  formulating and solving complex problems and in
  processing (receiving, storing, retrieving,
  transmitting) information.
• There is a number of dimensions along which
  "classical" models of rationality can be made
  more realistic without giving up rigorous
  formalization.
• limiting what sorts of utility functions there
  might be
• recognizing the costs of gathering and processing
  information
• the possibility of having a "multi-valued"
  utility function.

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Hard problems for bounded rationality

• Disjunction effects
• Conjunction effects
• Ellsberg paradox
• Allais paradox
• Framing effects

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2.1 Disjunction Effects
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Savages sure-thing principle

• A businessman contemplates buying a certain
  piece of property. He considers the outcome of
  the next presidential election relevant to the
  attractiveness of the purchase. So, to clarify
  the matter for himself, he asks whether he would
  buy if he knew that the Republican candidate were
  going to win, and decides that he would do so.
  Similarly, he considers whether he would buy if
  he knew that the Democratic candidate were going
  to win, and again finds that he would do so.
  Seeing that he would buy in either event, he
  decides that he should buy, even though he does
  not know which event obtains. (Savage, 1954, p.
  21)

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Tversky and Shafir

• B A p
• B ?A q
• B (A ? ?A) between p q
• Tversky and Shafir (1992) show that
  significantly more students report they would
  purchase a nonrefundable Hawaiian vacation if
  they were to know that they have passed or failed
  an important exam than report they would purchase
  if they were not to know the outcome of the exam.
• Disjunction effect ?(B) ? ?(BA) ?(A)
  ?(B?A) ? (?A)

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Khrennikov
Test A Test B
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Disjunction effect

• Subjects have to decide whether the two objects
  in Test A (Test B) are of the same size or not
• Test A and Test B are realized separately and in
  turn (first A then B after 2 seconds)
• ?(B) ? ?(BA) ?(A) ?(B?A) ? (?A)

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Results

• ?(B) 0.45 ?(? B) 0.55
• ?(A) 0.7 ?(? A) 0.3
• ?(B/A) 0.43 ?(? B/A) 0.57
• ?(B/?A) 1 ?(?B/?A) 0
• ?(B) ? ?(BA) ?(A) ?(B?A) ? (?A) 0.15
  (sign.)

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Disjunction effect interference effect

• ?(b) ?(ab) ?(a'b) ?
• ? 2 ?1/2(ab) ?1/2 (a'b) cos(?)
• Note The phase factors ? correspond to the
  expressions ?ab??a'b? ?ab??a'b? ?
  exp(i ?)
• ?(b) ? ?(a b) ?(a b') ?
• 2?0.3? cos(?) 0.15
• ? cos(?) 0.25 ? ? 75,5?

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2.2 Conjunction Effects
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Tversky Kahnemann (1983)

• Linda is 31 years old, single, outspoken and
  very bright. She majored in philosophy. As a
  student, she was deeply concerned with issues of
  discrimination and social justice, and also
  participated in anti-nuclear demonstrations.
• Linda is a teacher in elementary school.
• Linda works in a bookstore and takes Yoga
  classes.
• Linda is active in the feminist movement.
  (F) (6,1)
• Linda is a psychiatric social worker.
• Linda is a member of the League of Women Voters.
• Linda is a bank teller. (T) (3,8)
• Linda is an insurance salesperson.
• Linda is a bank teller and is active in the
  feminist movement. (TF) (5,1)

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Different experimental procedures

• Probability judgments
• ranking probabilities of the different evens
• probability assessment on a 9 point scale
• Frequency judgments
• Judging representativeness
• degree to which an event is representative of an
  appropriate mental model
• degree of correspondence between an instance and
  a category, an outcome and a model
• Degree of prototypicality

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Asymmetric conjunction resolves the conjunction
puzzle in the probabilistic case

• ?(b) ?(ab) ?(a'b) ?
• ? 2 ?1/2(ab) ?1/2 (a'b) cos(?)
• Conjunction effect ?(ab) ? ?(b) ??(a'b) ? 2
  ?1/2(ab) ?1/2 (a'b) cos(?)
• Example for ?? ?(ab) ? ?(b) ?1/2 (a'b)
  ?1/2 (a'b) 2 ?1/2(ab)

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Example

• ?(b) 0.38 (Linda is a bank teller)
• ?(a) 0.61 (Linda is a feminist)
• ?(a b) 0.51 (Linda is a feminist bank teller)
• Conjunction effect 0.13
• ?(ba) 0.84, ?(a' b) 0.71
• 0.13 ?1.2 cos(?) ? 0.71,
• i.e. cos(?) -0.7, ? 2.35 ? 270?

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2.3 Ellsberg Paradox
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Ellsberg Paradox

• Urn containing 30 red balls 60 balls you
  cannot really see, black or yellow proportion
  unknown.

• Game A you receive 100 if you draw a red ball
• Game B you receive 100 if you draw a black ball
  
• Most people prefer A ? B
• Game C you receive 100 if you draw a red or
  yellow ball
• Game D you receive 100 if you draw a black or
  yellow ball
• Most people prefer D ? C

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Expected utility theory

• Preferring A over B means that the expected
  utilty for A is higher than the expected utility
  for B
• A ? B
• ?(R)U(100)(1- ?(R))U(0) gt ?(B)U(100)(1-
  ?(B))U(0)
• i.e. ?(R) gt ?(B) assuming U(100) gt U(0)
• D ? C
• ?(B)U(100)?(Y)U(100)?(R)U(0) gt
  ?(R)U(100)?(Y)U(100)?(B)U(0)
• i.e. ?(B) gt ?(R) assuming U(100) gt U(0)
• This contradiction indicates that the preferences
  are inconsistent with expected-utility theory

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Sure thing principle
In formal terms, the principle states that the
choice between two actions is unaffected by the
pay-offs in a constant column
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2.4 Allais paradox
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A choice problem

• The choice problem designed by Maurice Allais
  shows an inconsistency of actual observed choices
  with the predictions of expected utility theory
• The problem arises when comparing participants'
  choices in two different experiments, each of
  which consists of a choice between two gambles, A
  and B.
• The particular payoffs and chances are essential

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Example
1 U(1M) gt 0.89 U(1M) o.o1 U(0) 0.1
U(5M)
0.89 U(0) 0.11 U(1M) lt 0.9 U(0) 0.1
U(5M) i.e. 0.11 U(1M)lt o.o1 U(0) 0.1
U(5M)