CSCI 6962 Lecture 2

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CSCI 6962 -- Lecture 2

• Volkan Isler

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Utility

• You receive 2 if a fair coin toss yields heads
  and lose 1 otherwise
• You have a fortune of 10 million. You receive
  20M if heads and lose 10M otherwise.
• You want to go out and drink beer tonight. You
  have 30 which buys you a good deal of beer. You
  receive 30 extra if heads and lose 30
  otherwise.
• There is a concert you are desperate to see. The
  tickets are 50 but you have only 30. You
  receive 20 if heads and lose 30 otherwise.

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Utility

• Is a measure of value and assigns prospects P to
  numbers u(P)
• What are desired properties of a utility function?

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Utility Property 1

• u(P1) gt u(P2) if and only if the individual
  prefers P1 to P2

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Utility Property 2

• If P is the prospect where with probability p,
  the individual faces P1 and with probability
  (1-p) the individual faces P2, then
• u(P) p u(P1) (1-p) u(P2)

What can you say about the example where you had
10 Million?
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Von Neumann and Morgenstern Theorem

• If
• A person is able to express preferences between
  every possible pair of gambles, where the gambles
  are taken over some basic set of alternatives
• Then,
• One can introduce utility associations to the
  basic alternatives, such that if the person is
  guided solely by the utility expected value, he
  is acting according to his true taste.
• ... provided only that there is an element of
  consistency in his tastes.

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Formalizing the statement of the theorem

• The element of consistency is formalized by a
  list of assumptions.

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Assumption 1

• The individual, faced with P1 and P2, can decide
  if prefers P1 or P2 or if he is indifferent.

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Assumption 2

• If
• P1 is regarded at least as well as P2 and
• P2 is regarded at least as well as P3,
• Then,
• P1 is regarded at least as well as P3

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Assumption 3

• If P1 is preferred to P2 which is preferred to
  P3, then
• there is a mixture of P1 and P3 which is
  preferred to P2 and
• there is a mixture of P1 and P3 over which P2 is
  preferred.

(For those who believe in hell) Say, P1 is
living a happy life, P2 is living a miserable
life and P3 is going to hellhow would you
interpret this assumption?
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Assumption 4

• Suppose the individual prefers P1 to P2 and let
  P3 be another prospect. Then the individual will
  prefer
• a mixture of P1 to P3 to the same mixture of P2
  to P3.

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Back to the theorem

• Under these four assumptions, it was shown that
  there is a utility function that is consistent
  with the individuals tastes.

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An application of utility St. Peters paradox

• Lets play the following game.
• Toss a fair coin until you receive heads.
• Let n be the number of coin tosses, you earn X
  2n.
• What is EX? In other words, how much would you
  be willing to pay for the privilidge of playing
  this game.

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EX

In other words, the expectation is infinite, so
you should be willing to bet any finite amount of
money to play this game. What are the issues
with this argument?
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Probability Review

• On your own, please make sure that you are
  comfortable with
• Mean, variance, standard deviation
• Expectation
• Tail inequalities (deviation from the mean)
• Markov, Chebyshev and Chernoff bounds
• These tools provide means for reasoning about
  uncertainty due to randomness.
• Lets go back to uncertainty due to ignorance.

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Uncertainty due to ignorance about the state of
Nature

• Lets start with the 2D case, i.e. there are two
  possible states of Nature

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Parameters

• States of nature
• ?1 today will be sunny (not rainy)
• ?2 today will be rainy
• Actions
• a1 wear fair-weather outfit
• a2 wear a raincoat
• a3 wear a raincoat boots, hat, umbrella

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Loss of Utility table
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Observations weather indicator
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Strategy observations to actions

• s1 (a1, a1,a1)
• s2 (a1, a1,a2)
• s3 (a1, a1,a3)
• s4 (a1, a2,a1)
• s5 (a1, a2,a2)
• s6 (a1, a2,a2)

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Expected loss of utility L(?,s)

• s5 (a1, a2,a2)
• L(?1, s5) 0.600.2510.151 0.40
• L(?2, s5) 0.250.330.53 3.40

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Admissable, inadmissable (dominated)strategies?
Is this the set of all strategies?
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Randomized strategies. This set is always
convex. Why?
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Utilizing prior information about Nature

• Let w be the probability that the true state is
  ?1.
• The strategy s that minimizes the costw L(?1,
  s) (1-w) L(?2, s)
• is called the Bayes strategy

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A geometric interpretation

• For notation simplicity, letx L(?1, s) and y
  L(?2, s)
• Let w be the probability that the true state is
  ?1.
• Consider the functionf(x,y) wx (1-w)y
• Note the implicit dependence on s
• For f(x,y) constant, we obtain a ?
• What we are looking for is a strategy s, that
  minimizes f(x,y)

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w 1/3
A supporting line for our convex set at s18 (a2
a3 a3)
1/3x 2/3y 1
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Supporting lines

• A line is said to be a supporting line for a set
  S at the boundary point u if
• The line passes through u
• Interior of S completely lies on one side of the
  line

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A property of convex sets

• For any given boundary point u of a convex set S,
  there is a supporting line of S at u.
• We will prove this theorem later

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A second property

• If two planar convex sets U and V are disjoint, a
  line can be drawn such that U is on one side and
  V is on the other side of the line.

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Implication

• Every admissible strategy is an optimal strategy
  for some prior w.

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How about the converse?

• That is, if (1-w) and w are positive priors, are
  the corresponding Bayes strategies admissible?
• What if w 1?

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Minimax Strategies
Max(x,y) 30/14
max(x,y) 1
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Notes

• Note that the minimax strategy is
  randomized/mixed
• Further, it is a boundary point. Hence it is the
  bayesian strategy for some prior w.

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Questions?

• We will stop here..