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Decision Models


Decision Models Making Decisions Under Risk Decision Making Under Risk When doing decision making under uncertainty, we assumed we had no idea about which state ... – PowerPoint PPT presentation

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Title: Decision Models

Decision Models
  • Making Decisions
  • Under Risk

Decision Making Under Risk
  • When doing decision making under uncertainty, we
    assumed we had no idea about which state of
    nature would occur.
  • In decision making under risk, we assume we have
    some idea (by experience, gut feel, experiments,
    etc.) about the likelihood of each state of
    nature occurring.

The Expected Value Approach
  • Given a set of probabilities for the states of
    nature, p1, p2 etc., for each decision an
    expected payoff can be calculated by
  • ?pi(payoffi)
  • If this is a decision that will be repeated over
    and over again, the decision with the highest
    expected payoff should be the one selected to
    maximize total expected payoff.
  • But if this is a one-time decision, perhaps the
    risk of losing much money may be too great --
    thus the expected payoff is just another piece of
    information to be considered by the decision

Expected Value Decision
  • Suppose the broker has offered his own
    projections for the probabilities of the states
    of nature
  • P(S1) .2, P(S2) .3, P(S3) .3, P(S4) .1,
    P(S5) .1

Expected Value
.2(250).3(200).3(150) .1(-100).1(-150)
.2(500).3(250).3(100) .1(-200).1(-600)
.2(60).3(60).3(60) .1(60).1(60)
Perfect Information
  • Although the states of nature are assumed to
    occur with the previous probabilities, suppose
    you knew, each time which state of nature would
    occur -- i.e. you had perfect information
  • Then when you knew S1 was going to occur, you
    would make the best decision for S1 (Stock
    500). This would happen p1 .2 of the time.
  • When you knew S2 was going to occur, you would
    make the best decision for S2 (Stock 250).
    This would happen p2 .3 of the time.
  • And so forth

Expected Value of Perfect Information (EVPI)
  • The expected value of perfect information (EVPI)
    is the gain in value from knowing for sure which
    state of nature will occur when, versus only
    knowing the probabilities.
  • It is the upper bound on the value of any
    additional information.

Calculating the EVPI

Expected Return With Perfect Information (ERPI)
.2(500) .3(250) .3(200) .1(300) .1(60)
Expected Return With No Additional Information
EV(Bond) 130
Expected Value Of Perfect Information (EVPI)
ERPI - EV(Bond) 271 - 130 141
Using the Decision Template
Sample Information
  • One never really has perfect information, but can
    gather additional information, get expert advice,
    etc. that can indicate which state of nature is
    likely to occur each time.
  • The states of nature still occur, in the long run
    with P(S1) .2, P(S2) .3, P(S3) .3, P(S4)
    .1, P(S5) .1.
  • We need a strategy of what to do given each
    possibility of the indicator information
  • We want to know the value of this sample
    information (EVSI).

Sample Information Approach
  • Given the outcome of the sample information, we
    revise the probabilities of the states of nature
    occurring (using Bayesian analysis).
  • Then we repeat the expected value approach (using
    these revised probabilities) to see which
    decision is optimal given each possible value of
    the sample information.

Example -- Samuelman Forecast
  • Noted economist Milton Samuelman gives an
    economic forecast indicating either Positive or
    Negative economic growth in the coming year.
  • Using a relative frequency approach based on past
    data it has been observed
  • P(Positivelarge rise) .8
    P(Negativelarge rise) .2
  • P(Positivesmall rise) .7 P(Negativesmall
    rise) .3
  • P(Positiveno change) .5 P(Negativeno change)
  • P(Positivesmall fall) .4 P(Negativesmall
    fall) .6
  • P(Positivelarge fall) 0 P(Negativelarge
    fall) 1

Bayesian ProbabilitiesGiven a Positive Forecast

Prob(Positive) P(PositiveLarge Rise)P(Large
Rise) P(PositiveSmall Rise) P(Small Rise)
P(PositiveNo Change)P(No Change)
P(PositiveSmall Fall) P(Small Fall)
P(PositiveLarge Fall) P(Large Fall)
Prob(Positive) P(Positive and Large Rise)
P(Positive and Small Rise) P(Positive
and No Change) P(Positive and Small Fall)
P(Positive and Large Fall)
P(Large RisePos) P(PosLg. Rise)P(Lg.
Rise)/P(Pos) P(Small RisePos) P(PosSm.
Rise)P(Sm. Rise)/P(Pos) P(No ChangePos)
P(PosNo Chg.)P(No Chg.)/P(Pos) P(Small FallPos)
P(PosSm. Fall)P(Sm. Fall)/P(Pos) P(Large
FallPos) P(PosLg. Fall)P(Lg. Fall)/P(Pos)
(.80) (.20) /.56 .286
(.70) (.30) /.56 .375
(.50) (.30) /.56 .268
(.40) (.10) /.56 .071
(0) (.10) /.56 0
Best Decision With Positive Forecast
Expected Value
When Samuelman predicts positive -- Choose the
Bayesian ProbabilitiesGiven a Negative Forecast

Prob(Negative) P(NegativeLarge Rise)P(Large
Rise) P(NegativeSmall Rise) P(Small Rise)
P(NegativeNo Change)P(No Change)
P(NegativeSmall Fall) P(Small Fall)
P(NegativeLarge Fall) P(Large Fall)
Prob(Negative) P(Negative and Large Rise)
P(Negative and Small Rise) P(Negative
and No Change) P(Negative and Small Fall)
P(Negative and Large Fall)
P(Large RiseNeg) P(NegLg. Rise)P(Lg.
Rise)/P(Neg) P(Small RiseNeg) P(NegSm.
Rise)P(Sm. Rise)/P(Neg) P(No ChangeNeg)
P(NegNo Chg.)P(No Chg.)/P(Neg) P(Small FallNeg)
P(NegSm. Fall)P(Sm. Fall)/P(Neg) P(Large
FallNeg) P(NegLg. Fall)P(Lg. Fall)/P(Neg)
(.20) (.20) /.44 .091
(.30) (.30) /.44 .205
(.50) (.30) /.44 .341
(.60) (.10) /.44 .136
(1) (.10) /.44 .227
Best Decision With Negative Forecast
Expected Value
When Samuelman predicts negative -- Choose Gold!
Strategy With Sample Information
  • If the Samuelman Report is Positive --
  • Choose the stock!
  • If the Samuelman Report is Negative --
  • Choose the gold!

Expected Value of Sample Information (EVSI)
  • Recall, P(Positive) .56 P(Negative) .44
  • When positive -- choose Stock with EV 249
  • When negative -- choose Gold with EV 120

Expected Return With Sample Information (ERSI)
.56 (249) .44 (120) 192.50
Expected Return With No Additional Information
EV(Bond) 130
Expected Value Of Sample Information (EVSI)
ERSI - EV(Bond) 192.50 - 130 62.50
  • Efficiency is a measure of the value of the
    sample information as compared to the theoretical
    perfect information.
  • It is a number between 0 and 1 given by
  • Efficiency EVSI/EVPI
  • For the Jones Investment Model
  • Efficiency 62.50/141 .44

Using the Decision Template
Output -- Posterior Analysis
Indicator ProbabilitiesRevised
Probabilities Optimal Strategy EVSI, EVPI,
  • Expected Value Approach to Decision Making Under
  • EVPI
  • Sample Information
  • Bayesian Revision of Probabilities
  • P(Indicator Information)
  • Strategy
  • EVSI
  • Efficiency
  • Use of Decision Template