Decision Analysis Chapter 6

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Decision AnalysisChapter 6

• Xiaolong (Jonathan) Zhang
• Department of Finance and Quant. Analysis
• College of Business Administration
• Georgia Southern University
• xzhang_at_georgiasouthern.edu

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Decision Analysis as a Problem Solver

• Identify decision options and definerisks
• Formulate a decision problemas a payoff table or
  decision treemodel
• Interpret and evaluate the decisionsrecommended
  by the model
• Communicate results
• Tracking performance of the decision

Unknown decision options and environment
Decision problem with discrete options and risks
Evaluation
Payoff table or decision tree methods
Forecasting
DecisionAnalysis
Optimization
Queueing
Optimal decisions
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Outline

• Uncertainty spectrum
• Decision problem with discrete decision options
  and state variable Payoff-Table Analysis
• Decision criteria under uncertainty
• Decision criteria under risk
• Value of perfect and imperfect information

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Uncertainty Spectrum

• Decision problems involve varying degrees of
  uncertainty and our knowledge about the
  uncertainty and decision alternatives. Assume we
  know the alternatives (options), the uncertainty
  spectrum orders the level of uncertainty from
  perfectly known on one extreme to completely
  unknown on the other
• Certainty ? Risk ? Uncertainty ?
  Ignorance

No uncertainty about the states of nature
Know the states of nature and their probability
Know the states of nature but not their prob.
Dont even know the states of nature
Increasing uncertainty
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Modeling Decision Problems under Risk and
Uncertainty with Payoff Table

• When decision alternatives (options) and the
  states of nature are discrete, payoff table can
  be constructed to describe the decision problem
• Decision alternatives as rows
• States of nature as columns
• Cells of the table are payoffs associated with
  each combination of state and decision alternative

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Examples

• Example 1 Georgia Lottery Cash 3 with a draw and
  1 ticket. The states of nature are win or lose.
  Decision alternatives are to play or not to
  play. The winning payoff is 500.
• Example 2 Movie making
• Jeff Mogul is a film producer and he is
  currently reviewing a script from a new
  screenwriter. If he makes this movie, Jeff
  estimates that there is a 20 chance that it will
  be a hit and the movie will gross 100 million.
  If it is a flop, the film will only gross 10
  million. It costs the studio 20 million to make
  and market the new movie.
• Example 3 Tom Brown InvestmentPage 331

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Solutions to Examples Assuming No Knowledge of
Probability Dist.

• In-class construct the payoff table and the
  associated regret table and apply the decision
  criteria
• Fill in with class notes?
• Excel

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Decision-Making under Uncertainty

• When a decision maker does not have an idea
  about the probability distribution of random
  states of nature, he/she may resort to any of the
  following decision criteria when payoffs
  represent rewards

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Decision-Making under Risk

• When a decision maker does have an idea about
  the probability distribution associated with
  states of nature, he/she can maximize the
  expected payoff from selecting an alternative.
  The expected payoff of each decision alterative
  is first calculated, the alternative that yields
  the largest expected payoff will be chosen.

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Solutions to Examples with Knowledge of
Probability Distribution

• In-class
• Excel

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Interpretation of Expected Payoff Decision
Criterion

• If the decision problem occur repeatedly, the
  expected payoff represents the average payoff
  over a long run
• If the decision problem is one-time decision, the
  expected payoff can be interpreted as
  representing the decision-makers preference
  under risk in which the probabilities are often
  judgmental.

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Expected Value of Perfect Information under Risk

• Assume that you have elicited the help of someone
  who can tell you the future. In this case you
  have perfect information about which state is
  going to occur, what is your expected payoff?
• The difference between the expected payoff you
  got in the above scenario and when you do not
  have perfect information measures the value of
  the prescient person to you.

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Expected Value of Perfect Information for the
Three Examples

• Example 1
• Example 2
• Example 3

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Expected Value of Imperfect Information under Risk

• The expected value of perfect information
  represents the maximum we can gain from acquiring
  information to help us make better decisions.
  In reality, we only have imperfect information,
  i.e., our predictions about the states of nature
  are not always right. For example, weather
  forecasts are not 100 accurate, expert opinions
  do not always materialize, and so on.
• To quantify the value of imperfect information,
  we need to have a way of figuring out how
  information alters the probability distribution
  of the states. If we know how to do this, we can
  easily compare the expected payoffs with and
  without the information.
• Bayes rule is what the doctor ordered

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Review of Bayes Rule and Extended Example 2

• The Bayes Rule provides an intuitive way to
  adjust the probability distribution of the states
  when additional information is made available
• Jeff Mogul, the movie producer, wants to have a
  better idea of how likely the new script will be
  a hit. He wants to get advise from Ebert. The
  record shows that Ebert is right 70 of the time
  on the hit movies when he gives a thumbs-up and
  90 right on the flop movie when he gives a
  thumbs-down. How should Jeff adjust his
  probability of hit or miss after seeking Eberts
  service?

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Key Concepts

• Prior distribution of Jeff P(Hit) 0.2, P(Flop)
  0.8
• Jeffs believe prior to obtaining any advice
  from Ebert
• Conditional Likelihood Quality of Eberts advice
  for flop or hit movies
• P(Up Hit) 0.7 P(Down Hit) 0.3
• P(Up Flop) 0.1 P(Down Flop) 0.9
• Posterior probability distribution Jeffs
  updated probabilities given Eberts advice
• P(Hit Up) ? P(Hit Down) ?
• P(Flop Up) ? P(Flop Down) ?

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Bayes Rule in a Graphical View
PosteriorProbability
Thumbs-Up
EV with Up
EbertsAdvice and Likelihood
PriorProbability
PosteriorProbability
EV with Down
Thumbs-Down
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Figuring Out Posterior Probability A Tabular
Approach
Input Prior prob of hit or flop and likelihood
of Eberts advice given hit or flop row prob.
Intermediate output Joint probability
Output of Posterior probability Conditional
probability given up or downcol prob.
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Expected Value of Imperfect (Sample) Information

• Expected value with sample information (EVSI) is
  a weighted average of the expected payoff given
  Eberts advice. The weights are the probability
  distribution of Eberts advice.
• EVSI P(Up) EP with up-advice
• P(Down) EP with down-advice
• EVSI is the difference between the expected
  payoff with and without the sample information.
• Examples Movie Making and Tom Brown Investment ?
  Go to Excel and your notes

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Value of Eberts Advice
No advice sought EV 8, Make
Advice sought, Up EV 520/11, Make
Advice sought, Down EV 0, No make
EVSI (0.22 (520/11) 0.78 0) 8 10. 4
8 2.4 Million
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Expected Value of Imperfect (Sample) Information

• Expected payoff with sample information is a
  weighted average of the expected payoff given
  information about states of nature. The weights
  are the probability distribution of the
  information.
• EVSI is the difference between the expected
  payoff with and without the sample information.
• Examples Movie Making and Tom Brown Investment ?
  Go to Excel and your notes

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Summary

• Decision analysis involves
• Defined alternatives
• Knowledge about the state of nature and its
  probability distribution
• Accurately estimated payoffs under each
  combination of decision alternative and the state
  of nature
• Attitude and outlook of the decision maker
  towards risk
• Information

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Caveat

• Decision-making dilemma Best decision vs. best
  outcome
• Expected payoff optimal in the long run, but not
  in the short-run
• Maximax Thrill of risk-taking (utility), but may
  wind up with a huge loss
• Maximin Satisfaction of security, but may miss
  the opportunity for big payoff
• Minimax regret 20/20 hindsight, sensitive to
  adding non-optimal alternatives
• Limitations of decision criteria
• People do not treat wins and losses the same
• Alternatives sometimes are dependent
• People do not treat large some of money and
  pocket change the same
• Payoffs are rough estimates