Utility Theory

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Utility Theory
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• Apply the 6 step method to the following report
  and the hypothesis that there is a correlation
  between being a man and being found guilty of
  killing your spouse in America.
• The US Justice Department's Bureau of Justice
  statistics published the following information on
  cases in which the defendant was accused of
  killing a spouse, in the wake of the OJ Simpson
  trial. Their study was based on 540 cases
  resolved in 1988.In 318 cases, the accused was
  male, and in 222 cases the accused was female.
  277 of the men and 155 of the women were declared
  guilty. Among the male defendants, 11 were not
  prosecuted, 46 pleaded guilty, 41 were
  convicted in a trial, and only 2 were acquitted.
  Among female defendants, 16 were prosecuted, 39
  pleaded guilty, 31 were convicted in a trial,
  and 14 were acquitted.

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Assignment 4

• 1. Real World. The population of interest is the
  population who kill their spouse in America (not
  necessarily Americans).
• The population sampled is the population who are
  prosecuted in America in 1988.
• 2. Data. In 318 cases, the accused was male, and
  in 222 cases the accused was female. 277 of the
  men and 155 of the women were declared guilty.

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• 3. Model. Variables Gender and verdict.
• Values Men and women found guilty and not found
  guilty.
• The hypothesis is that there is a correlation
  between being a man and being found guilty.
• 4. Random sampling. The report doesnt say. It is
  possible that the 540 cases are ALL the cases
  from 1988. In that case the sample is trivially
  representative of the population sampled.
• If the population sampled is considered to be all
  the trials over a broader period of time than
  1988, then there is not random sampling from the
  population

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No overlap Evidence of correlation Estimate of
strength 31,3
94 80
77 63
Sample of men Sample of women
87
70
Guilty
0
318
222
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• 6. Summary. The year 1988 is probably
  representative of other years.
• Study performed by a govt body.
• Good evidence that there is a weak correlation.

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Overview

• Concepts of Decision Theory
• Expected Monetary Value
• Utility
• Ordinal and Cardinal scales
• Measuring Utility
• Challenges to utility theory

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Decisions

• The entire course so far has been about what we
  should believe propositions, correlations,
  causation
• Passive.
• Beliefs should fit how the world is.
• The world should fit our desires.
• Active.

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Belief
Desire
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The Big Picture

• Decision theory aims to provide a framework in
  which we can choose the rational choice.
• Which choice you make always depends on two
  things.
• 1. What will (probably) happen?
• 2. Will whatever happens be good?

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Example

• Should you order the chicken?
• 1. What will happen if you order the trout?
• Trout? Sprout? Snouts?
• Probabilities
• 2. How good will the chicken / sprouts / snouts
  be?
• Utilities

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Basic Concepts of Decision Theory1. Options

• Options are choices, or decisions
• There must be at least two options in any
  decision model.
• Eg. 100 on red 7

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2. States of the world

• States of the world are ways the world might be.
  Possible worlds. There is only one way the world
  is. The actual world.
• We might be uncertain what the world is like.
• Eg. The ball lands on red 7

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3. Outcomes

• An outcome is the result of picking a particular
  option given some state of the world.
• Eg. Winning 3600

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Decision matrix

• The states of the world along the top combine
  with the options down the side

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Decision matrix

• The states of the world along the top combine
  with the options down the side

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Decision matrix

• The states of the world along the top combine
  with the options down the side

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Expected Monetary Value

• The expected monetary value of a decision is
  calculated by multiplying the monetary value in
  each square by the probability of that square and
  summing across the row.

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Decision matrix

• The states of the world along the top combine
  with the options down the side

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From money to utility

• Why cant we simply use EMV in decision theory?
• 1. Not everything has a price.
• 2. Utility doesnt always match EMV.

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Utility ? Money

• You have 1.50. The bus fare home is 3. You are
  offered a bet that costs 1.50 and pays 3 only
  if a fair die lands 1 or 2.
• EMV(Bet) 1
• EMV (No Bet) 1.50
• Nevertheless, you would take the bet.

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Utility ? Money

• You win your bet and have 3.
• You are now offered a bet that costs 3 and pays
  6 if a fair coin lands 1, 2, 3 or 4.
• EMV(Bet) 4
• EMV (No Bet) 3
• Nevertheless, you would not take the bet.

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Utility ? Money

• The diminishing marginal value of money
• Every new dollar earned is a bit less valuable
  than the last.
• Giving Bill Gates 1m doesnt make him as happy
  as giving Jamal Malik 1m.
• This explains insurance, sweat shops,

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Utilities

• The utility of an agent in an outcome is a
  representation of his preference for choosing
  that outcome.
• If agent S chooses outcome A over outcome B then
  S has more utility in A than B.

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Utilities and Preferences

• If the agent doesnt eat the cookie, then not
  eating the cookie has a higher utility than
  eating the cookie.
• Utilities are derived from choices (preferences)

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Ordinal vs. Interval rankings

• How precisely must we describe the agents
  utilities in order to know what their rational
  decision is?
• 1. Ordinal ranking
• 2. Cardinal / Interval ranking

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Ordinal utilities

• Ordinal rankings 1st, 2nd, 3rd.
• Ordinal rankings of utilities are not enough to
  determine the agents rational choices.

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Ordinal utilities
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Ordinal utilities

• Same ordinal rankings, different interval
  rankings.
• Preference changes.

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Interval Scales

• Moral We dont just need the order of
  preferences, we also need the interval between
  preferences.

Best Worst
A B C D
Best Worst
A B C D
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Measuring Utility

• We know how to measure ordinal utility.
• But how do we measure utility intervals?
• Von Neumann-Morgenstern utility theory
• Measure the strength of an agents preference for
  something by the risks he is willing to take to
  get it.

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• Suppose your ranking order is going to New York,
  then LA, then Washington.
• Do you prefer New York to LA more or less than LA
  to Washington?
• Suppose you had a trip to LA.
• You can sell the trip to LA for a gamble that
  wins a trip to New York and loses a trip to
  Washington.
• What probability of winning New York would you
  need?
• That probability is the utility of LA.

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• Suppose New York is by far the best. Then you
  would readily give up the trip to LA.

New York 1
LA
Washington 0
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• Suppose New York is by far the best. Then you
  would readily give up the trip to LA.

New York 1
LA P(New York) 0.2) ? 0.2
Washington 0
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• Suppose New York is nothing special. Then you
  would not readily give up the trip to LA.

New York 1
LA
Washington 0
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• Suppose New York is nothing special. Then you
  would not readily give up the trip to LA.

New York 1
LA P(New York) 0.8 ? 0.8
Washington 0
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• If youre willing to take odds of 1/3 to win 3
  and risk 1.50, then U(1.50) 1/3

3 1
1.50 P(1/3) 1/3
0 0
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Von Neumann-Morgensterns Assumptions

• 1. Ordering condition
• 2. Continuity condition
• 3. Better-prizes condition
• 4. Better-chances condition
• 5. Reduction-of-compound lotteries

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Representation Theorem

• If an agents preferences have enough structure
  (that is, they satisfy (1) (5)), then they can
  be represented by an interval scale u so that the
  utility of any gamble is its expected utility.

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Challenges to VN-Ms assumptions

• Their assumptions entail that there are odds at
  which I would risk dying on the spot to gaining
  one cent.
• But is this false?
• Wouldnt you cross a busy road to pick up a 100
  bill?

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Challenges to VN-Ms assumptions

• Mom has a treat to give to either Abby or Ben.
  She is indifferent to who gets it, but prefers
  that one of them get it than neither.
• However, she prefers that it is decided by a fair
  coin flip than by any sure method.
• Does Mom violate the assumptions of decision
  theory?

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!
Decision theory requires that an agent who is
indifferent between two options also be
indifferent to a 50/50 lottery. So Moms
preference for the third option looks irrational.
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Allaiss Paradox
Gamble A

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Gamble B
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Gamble A
Gamble B

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Risk-aversion

• Humans are risk-averse.
• According to VNM decision theory, this is
  irrational.
• Some think it is irrational.
• But risk-aversion persists for careful reasoners.
• And isnt it perfectly reasonable to avoid risk?

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St. Petersburg Paradox

• A coin will be flipped until it lands Heads. Let
  n be the number of times the coin is flipped. You
  will receive 2n units of utility.
• How much would you play to pay this game?
• The expected utility is infinite.
• But would you give up everything you have to play
  this game?