Lottery Contract

Document Sample
Lottery Contract Powered By Docstoc
					                    Chapter 6
     It is a truth very certain that when it is not
     in our power to determine what is true we
     ought to follow what is probable.—
     Descartes


    Decision Making and Risk:
      Certainty Equivalents
1
            and Utility
          Decision Making Using
           Certainty Equivalents
    The certainty equivalent (CE) is the payoff
     amount we would accept in lieu of under-
     going the uncertain situation.

      Shirley Smart would pay $25 to insure her 1983
       Toyota against total theft loss. CE = – $25.

      For $1,000, Willy B. Rich would sell his Far-
       Fetched Lottery rights. CE = $1,000.

      Lucky Chance would pay somebody $100 to fill
2
       her Far-Fetched Lottery contract. CE = -$100.
                Risk Premiums
     A situation’s risk premium (RP) is the
      difference its between expected payoff (EP)
      and certainty equivalent (CE):
                      RP = EP - CE
     Shirley Smart’s car is worth $1,000 and
      there is a 1% chance of its being stolen.
      Thus, going without insurance has
          EP = (– $1,000)(.01) + ($0)(.99) = – $10
          RP = – $10 – (– $25) = $15
3
                 Risk Premiums
     Playing the Far-Fetched Lottery has EP =
      $2,500. Thus,
       For Willy B. Rich,
         RP = EP – CE = $2,5000 – ($1,000) = $1,500
       For Lucky Chance,
         RP = EP – CE = $2,5000 – (– $100) = $2,600
     Different people will have different CEs
      and RPs for the same circumstance.
       They have different attitudes toward risk.
4
          Attitude Toward Risk
  People with positive RPs are risk averters.
     Lucky Chance has greater risk aversion than
      Willy B. Rich, as reflected by her greater RP.
     We cannot compare Shirley’s risk aversion to
      the others’ because circumstances differ.
  Risk averse persons have RPs that increase:
     When the downside amounts become greater.
     Or when the chance of downside increases.
   A risk seeker will have negative RP.
5  A risk neutral person has zero RP.
    Maximizing Certainty Equivalent
     A plausible axiom:
      Decision makers will prefer the act yielding
      greatest certainty equivalent.
     A logical conclusion:
      The ideal decision criterion is to maximize
      certainty equivalent.
       Doing so guarantees taking the preferred action.
     But CEs are difficult to determine. One
      approach is to discount the EPs.
       RP = EP – CE implies that CE = EP – RP.
6
       Using Risk Premiums to Get
          Certainty Equivalents
     Ponderosa Records president has the following
      risk premiums, keyed to the downside.




     These were found by extrapolating from three
      equivalencies (white boxes).
        Exact amounts are unknowable, but these values seem
7        to fit his risk profile.
    Decision Tree Analysis with CEs
     (Discounted Expected Payoffs)




8
     How Good is the Analysis?
  This result is different from that of ordinary
   back folding (Bayes decision rule).
     It specifically reflects underlying risk aversion.
     The result must be correct if CEs are right.
  The major weakness is the ad hoc manner
   for getting the RPs, and hence the CEs.
     Many assumptions are made in extrapolating to
      get the table of RPs.
   There is a cleaner way to achieve the same
9
    thing using utilities.
                   Utility Theory
      Consider a set of outcomes, O1, O2, ..., On.
       The following assumptions are made:
        Preference ranking can be done.
        Transitivity of preference: A is preferred to B
         and B to C, then A must be preferred to C.
        Continuity: Consider Obetween. Take a gamble
         between two more extreme outcomes; winning
         yields Obest and losing Oworst. There is a win
         probability q making you indifferent between
         getting Obetween and gambling. Such a gamble is
10       called a reference lottery.
             Utility Assumptions
      Continuity (continued):
         e.g., +$1,000 v. Far-Fetched Lottery, you pick q.
         For Willy B. Rich, q = .5. (His CE was = +$1,000.)
         For Lucky Chance, q = .9.
         If the win probability were .99, would you risk
          +$1,000 to gamble? What is your q?
      Substitutability: In a decision structure, you
       would willingly substitute for any outcome a
       gamble equally preferred.
         One outcome on Lucky Chance’s tree is +$1,000;
          she would accept substituting for it the Far-Fetched
11
          Lottery gamble with .9 win probability.
     Utility Assumptions and Values
       Increasing preference: Raising q makes any
        reference lottery more preferred.
          Anybody would prefer the revised Far-Fetched
           Lottery when two coins are tossed and just one head
           will win the $10,000. (The win probability goes
           from .5 to .75.) You still might not like that gamble!
       Outcomes can be assigned utility values
        arbitrarily, so that the more preferred always
        gets the greater value:
         u(Obest) = 10     u(Oworst)=0      u(Obetween)=50
          Willy has u(+$10,000) = 500, u(-$5,000) = 0 and
12         u(+$1,000) = 250. These are his values only.
                  Utility Values
         Lucky has different values: u(+$10,000) = 50,
          u(-$5,000) = -99, and u(+$1,000) = 35.1.
      Like temperature, where 0o and 100o are
       different states on Celsius and Fahrenheit
       scales, so utility scales may differ.
      The freezing point for water is 0o C and the
       boiling point 100o C. In-between states will
       have values in that range, and hotter days will
       have greater temperature values than cooler.
      So, too, with utility values. They will fall into
       the range defined by the extreme outcomes,
       Oworst and Obest. More preferred outcomes will
13     have greater utilities
                     Utility Values
      A reference lottery can be used to find the
       utility for an outcome Obetween by:
        First, establish an indifference win probability
         qbetween making it equally preferred to the
         gamble:
           Obest with probability qbetween and
            Oworst with probability 1 - qbetween
        Second, compute the lottery’s expected utility:
           u(Obest)(qbetween) + u(Oworst)(1 - qbetween)
           The above is u(Obetween).
14
                   Utility Values
      Using the Far-Fetched Lottery as reference:
           Lottery           Willy       Lucky
         Outcomes       Prob. Utility Prob. Utility
      Obest (+$10,000) q=.5 500 q=.9          50
      Oworst (-$5,000) 1 -.5       0  1 -.9 -99
              Expected Utility: 250          35.1
           Obetween (+$1,000):   250           35.1

        The indifference q plays a role analogous to the
         thermometer, reading attitude towards the
15       outcome similarly to measuring temperature.
              The Utility Function
      Utility values assigned to monetary
       outcomes constitute a utility function.
      From a few points we may graph the utility
       function and apply it over a monetary range.
      Those points may be obtained from an
       interview posing hypothetical gambles.
        Using u(+$10,000)=100 and u(-$5,000)=0
         Shirley Smart gave the following equivalencies:
           A: +$10,000 @ qA v -$5,000 ≡ +$1,000 if qA =.70
           B: +$10,000 @ qB v +$1,000 ≡ +$5,000 if qB =.75
           C1: +$1,000 @ qC1 v -$5,000 ≡ -$500 if qC1 =.70
16
           C2: +$1,000 @ qC2 v -$5,000 ≡ -$2,000 if qC2 =.30
        Shirley’s Utility Function
      Shirley’s utilities for the equivalent amounts are
       equal to the respective expected utilities:
         u(+$1,000) = u(+$10,000)(.70) + u(-$1,000)(1-.70)
                     = 100(.70) + 0(1-.70) = 70
         u(+$5,000) = u(+$10,000)(.75) + u(+$1,000)(1-.75)
                     = 100(.75) + 70(1 - .75) = 92.5
         u(-$500) = u(+$1,000)(.70) + u(-$5,000)(1-.70)
                   = 70(.70) + 0(1 - .70) = 49
         u(-$2,000) = u(+$1,000)(.30) + u(-$5,000)(1-.30)
                     = 70(.30) + 0(1 - .30) = 21
      Altogether, Shirley gave 6 points, plotted on the
       following graph. The smoothed curve fitting
17     through them defines her utility function.
     Shirley’s Utility Function




18
          Using the Utility Function
      This utility function applies to the
       Ponderosa decision.




19
       Using the Utility Function
  Read the utility payoffs corresponding to
   the net monetary payoffs.
  Apply the Bayes decision rule, with either:
     A utility payoff table, computing the expected
      payoff each act.
     Or a decision tree, folding it back.
    The certainty equivalent amount for any act
     or node may be found from the expected
     utility by reading the curve in reverse.
    The following Ponderosa Records decision
20
     tree was folded back using utility payoffs.
     Decision Tree Analysis
         with Utilities




21
         Shape of Utility Curve and
           Attitude Toward Risk
      The following shapes generally apply.




        The risk averter has decreasing marginal util-
         ility for money. He will buy casualty insurance
         and losses weigh more heavily than like gains.
        Risk seekers like some unfavorable gambles.
22      Risk neutrality values money at its face amount.
      Important Utility Ramifications
        Hybrid shapes (like Shirley’s) imply shifting
         attitudes as monetary ranges change.
      Regardless of shape, maximizing expected
       utility also maximizes certainty equivalent.
        Therefore, applying Bayes decision rule with
         utility payoffs discloses the preferred action.
      Primary impediments to implementation:
        Clumsiness of the interview process.
        Multiple decision makers.
        Attitudes change with circumstances and time.
23
             Ratification of
           Bayes Decision Rule
  Over narrow monetary ranges, utility curves
   resemble straight lines.
  For a straight line, expected utility equals
   the utility of the expected monetary payoff.
  Maximizing expected monetary payoff then
   also maximizes expected utility. Thus:
     The Bayes decision rule discloses the preferred
      action as long as the outcomes are not extreme.
    Managers can then delegate decision
     making without having to find utilities.
24
     Preferred actions will be found by the staff.
     Using Utility Functions with
           PrecisionTree
     PrecisionTree can be used to evaluate
     decision trees with with exponential and
     logarithmic utility functions.
     To get started, click on the name box of a
     decision tree and the Tree Setting dialog box
     appears, as shown next.

                        A              B
       1      tree #1                  1
       2                               0
25
              Tree Settings Dialog Box   2. Select the
                                         type of utility
                       (Figure 6-14)     function in the
                                         Function line.
                        1. Check the
                                         Here exponential
                        Use Utility
                                         is chosen.
                        Function box.


3. Select the
risk coefficient,
R, in the R value
line. Here
10,000 is used.
4. Select
Expected Utility
in the Display
line. Other
options are
Certainty
Equivalent and
Expected Value.

5. Click OK.
         26
                     Decision Tree with Exponential Utility
                  A
                       Function for R = 10,000 (Figure 6-15)
                                           B                              C                             D                        E               F
    1                                                                                                                       Success
                                                                                                                                     80.0%      0.0
The2 optimal strategy is:                                                                                                           $90,000      1
    3                                                                                        Market nationally
                                                                                                                   FALSE
    4
1. Not test market and to                                                                                        -$50,000       -48
    5                                                                                                                       Failure
                                                                                                                                       20.0%     0.0
abort.
    6                                                                                                                                      $0   -244
    7                                                         Favorable
                                                                                   50.0%
2. The corresponding
    8                                                                             $10,000               -1
    9                                                                                                              TRUE           0
expected utility is 0.
   10
                                                                                             Abort
                                                                                                                     $0          -1
   11                              Test market
                                                   FALSE
   12                                            -$15,000                 -2
   13                                                                                                                       Success
                                                                                                                                       20.0%    0.0
   14                                                                                                                                 $90,000    1
   15                                                                                        Market nationally
                                                                                                                   FALSE
   16                                                                                                            -$50,000      -531
   17                                                                                                                       Failure
                                                                                                                                       80.0%     0.0
   18                                                                                                                                      $0   -664
   19                                                         Unfavorable
                                                                                    50.0%
   20                                                                                   $0              -3
   21                                                                                        Abort
                                                                                                                   TRUE           0
   22                                                                                                                $0          -3
   23   Ponderosa Record Company
   24                                       0
   25                                                                                        Success
                                                                                                                   50.0%         0
   26                                                                                                            $100,000        1
   27                                                         Market nationally
                                                                                    FALSE
   28                                                                             -$60,000             -201
   29                                                                                        Failure
                                                                                                                   50.0%         0
   30                                                                                                                  $0      -402
   31                              Don't test market
                                                       TRUE
   32                                                    $0               0
   33         27                                              Abort
                                                                                    TRUE                 1
   34                                                                                 $0                 0

				
DOCUMENT INFO
Description: Lottery Contract document sample