Problems With Decision Criteria

1
Problems With Decision Criteria

• Transparencies for chapter 2

2
The Payoff Matrix

• The Simplest structure for a decision model
  consist of a set of possible course of actions, a
  list of possible outcomes that could occur and
  straight forward evaluation of each decision
  outcome pairs. Formulated as follows

3
Payoff matrix

• aj Course of action j
• ?i Outcome variable
• yij The value for the decision maker if taking
  action aj and ?i occurs
• a1 a2 aj .. am
• ?1
• ?2
• ?i
• ?n

yij
4
Preptown Book Store

• The manger of the bookstore at the Preptown
  college must decide how many copies to order of
  the book thought thinking
• to order for the course Creative Thinking.
  The maximum enrollment is 70. So far 50 are
  enrolled and this could go up or down. The book
  will make 15 on every sold book. The course
  will not repeat and any book unsold book will be
  disposed at 5 loss. The manger must decide how
  much to order.

5
Payoff Matrix

• Considering only orders in units of 10
• 0 10 20 30 40 50 60
  70
• 0 0 -50 -100 -150 200 250 300
  -350
• 10 0 150 100 50 0 -50
  -100 -150
• 20 0 150 300 250 200 150
  100 50
• 30 0 150 300 450 400 350
  300 250
• 40 0 150 300 450 600 550
  500 450
• 50 0 150 300 450 600 750
  700 650
• 60 0 150 300 450 600 750
  900 850
• 70 0 150 300 450 600 750
  900 950

6
Non-stochastic Criteria

• Outcome dominance
• Option aj dominates ak if and only if
• yij ? yik for all I
• and yij gt yik for at least one ?i
• This criterion is useful for eliminating options
  that are inferior. It reduces number of options
  and problem complexity.

7
Example for Outcome Dominance

• Consider the following payoff matrix
• a1 a2 a3
• ?1 6 3 8
• ?2 5 4 2
• ?3 7 6 3
• a1 dominates a2

8
Maximin Criterion

• Action aq is optimal in maximin criterion if and
  only if for each aj there exist a yij which is
  the minimum yij over all and yiq is the maximum
  of all yij

9
Example for Maximin Criterion

• Examine the payoff matrix for the bookstore
  problem.
• The minimums for the course of actions for a1
  through a8 are
• 0 -50 -100 -150 -200 -250 -300 -350
• The maximum of these minimums is 0 and therefore
  the optimal course of action is a1.

10
Problem with Maxmin

• Looking at the worst scenarios can be misleading
  as in the following matrix (Conservative
  criterion)
• a1 a2
• ?1 31 32
• ?2 10,000 33
• The optimal under maxmin is a2 which is misleading

11
Maxmax Criterion

• Action aq is optimal under maxmax criterion if
  and only if there exist ?p such that
  
• ypq ? yij for all i and j.

12
Example for Maxmax Criterion

• Consider the preptown store . The maximax
  criterion selects a8.
• Maximax criterion is risky and can result in
  huge losses.

13
Example for Maxima

• This example demonstrates that maximax is risky.
• a1 a2
  
  ?1 9
  10
• ?2 8 -50,000
• Maximax will select a2

14
Minimax Regret Criterion

• This criterion advocated by Savage
• Regret rij Max (yij) yij
• j
• a1 a2
  Regret matrix
• ?1 8 9
  1 0
• ?2 12 10 0
  2

15
Minimax Regret

• aq is optimal in minimax regret sense iff
  
  
• rq ? rj for all j
• where rj max rij

16
Minmax Regret

• To apply minimax regret perform the following
  steps
• Compute the regret matrix
• For each aj compute the maximum regret
• Select aj with the minimum maximum regret.

17
Minimax regret

• Minimax regret violates the coherence principle.
  The following example demonstrate that.
• a1 a2
  Regret matrix
• ?1 8 2
  0 6
• ?2 0 4 4
  0
• a1 is the optimal using minmax regret

18
Minmax and Coherence ( cont.)

• Payoff Matrix Regret Matrix
• a1 a2 a3 a1 a2 a3
• ?1 8 2 1 0 6
  7
• ?2 0 4 7 7
  3 0
• a2 is optimal when a3 is added. This is rank
  reversal. Any criterion that reverses the rank
  of alternatives is incoherent.

19
Stochastic Criteria

• The previous criteria do not take into account
  the relative chances of the occurrence of the
  outcomes
• Use the concept of probability. How to get the
  probability?

20
Probability Distributions

• Review Probability
• Expected Value
• Mode
• Variance
• Possible application

21
Model Outcome

• aq is optimal in modal outcome sense iff exist
  ?p such that
• P(?p) ? P(?i) for all i
• and ypq ? ypj for all j

22
Modal Outcome

• This criterion look at the most likely outcome
  and then select the course of action that has the
  maximum payoff with respect to this outcome.
• ? p(? ) a1 a2 a3

?1
18 15
19 20
22 19
40 30
20
0,2
?2
0.7
?3
0.3
23
Modal Outcome

• The following counter example shows that the
  model criterion could be misleading
• ? p(? ) a1 a2 a3

?1
0,24
0 99
98 0
99
99 21 20
20
?2
0.25
?3
0.51
24
Modal Outcome

• Linley presented an example to show modal
  outcome is incoherent.
• ? p(? ) a1 a2

?1
5 3
5
3
8 9

2/9
?2
3/9
?3
4/9
25
Modal Outcome

• Linley presented an example to show modal
  outcome is incoherent. Justify?
• ? p(? ) a1 a2

?1
?2
5 3
8
9

5/9
?3
4/9
26
Expected Value

• EV(aj) ?yijP(?i )
• Action aq is optimal in expected value sense iff
  
• E(aq) E(aj) for all j

i
27
Expected Value Example

• ? p(? ) a1 a2 a3

?1
0.2
18 15
19 20
22 19
40 30
20
?2
0.7
?3
0.1
EV(Aj)
21.6
21.4
19.1
a1 is the optimal in expected value sense
28
Expected Regret

• Compute the regret matrix
• Compute expected regret
• Select the option that minimizes expected
  regret.
• The solution that maximizes expected value is
  the same as the one that minimizes expected
  regret. Proof on page 29 text.

29
Expected Regret Example

• ? p(? ) a1 a2 a3

?1
0.2
1 4
0 2
0 3 0
10
20
?2
0.7
?3
0.1
ER(aj)
1.6
1.8
4.1
a1 is the optimal in minimizing expected regret
30
Modal Outcome

• This criterion look at the most likely outcome
  and the select the course of action that has the
  maximum payoff with respect to this outcome.
• ? p(? ) a1 a2 a3

?1
18 15
19 20
22 19
40 30
20
0,2
?2
0.7
?3
0.3
31
Payoff Distribution Analysis

• Payoff matrix versus payoff distribution
• For each action there is a payoff for each state
  of nature (outcome) ?. Let us denote the payoff
  vector for aj by Yj and its probability
  distribution by pj(? ). The payoff vector with
  its probability distribution is called the payoff
  distribution

32
Payoff Formulation of DP

• The DP problem can be formulated as follows
• Given a set of payoff distributions Yj
  (Options), select the best payoff
  distribution among them. Best, in what sense?
• Expected value sense, medial payoff sense, modal
  payoff sense, etc.

33
Modal Payoff Versus Modal Outcome

• ? p(? ) a1 a2

?1
0.3
10 20
10
20 15
10
?2
0.3
?3
0.4
This example shows modal payoff not the same as
modal outcome. How see page 31 Bunn book
Applied Decision Analysis
34
Risk Analysis

• Risk is the Likelihood of greater losses. It is
  the probability of undesirable event occurring.
• In financial analysis risk is taken to be the
  dispersion of payoff distribution. (variance).
• In insurance industry risk refers to the maximum
  amount of money which can lost under a particular
  policy. It is referred to as the expected value
  of a detritus proposal.

35
Measures of Risk

• Variance focuses on dispersion
• Semi-variance Focuses on the largest possible
  values greater than a certain value c.

36
Measures of Risk

• Critical probability Same as semi-variance but
  risk is measured in terms of probability.
• P( y c ) F(c)
• Why we need all these measures?

37
Generalization of Risk Measures

• Fishburn generalizes the last two measures of
  risks by making the power ?. If ? 2 it becomes
  the semi-variance. If ? 0 it becomes the
  critical value and so on. The use of critical
  value is more applicable in practice.

38
Mean -Variance Dominance

• Option aj dominates option ai iff
• E(aj) ? E(ai) and V(aj) V(ai) with one of
  them an inequality.
• Option j E(aj) V(aj)
• 1 7 1
• 2 8 2
• 3 9 2
• 4 7 1.5
• 5 10 3
• Option 3 dominates options 2, option 1
  dominates option 4. The efficient set
• ES Options 1, 3 and 5