| 3.
Decision Rules:
A tool commonly used to display information needed
for the decision process is a payoff matrix or decision table. The
table shown below is an example of a payoff matrix. The A's stand
for the alternative actions available to the decision maker. These
actions represent the controllable variables in the system. The uncertain
events or states of nature are represented by the S's. Each S has
an associated probability of its occurance, denoted P. (However,
the only decsion rule that makes use of the probabilities is the Laplace
criterion.) The payoff is the numerical value associated with an
action and a particular state of nature. This numerical value can
represent monetary value, utility, or both. This type of table will
be used to illustrate each type of decision rule.
| Actions\States |
S1 (P=.25) |
S2 (P=.25) |
S3 (P=.25) |
S4 (P=.25) |
| A1 |
20 |
60 |
-60 |
20 |
| A2 |
0 |
20 |
-20 |
20 |
| A3 |
50 |
-20 |
-80 |
20 |
Table 1: General Payoff
Matrix style from Chankong [4]. This generic/hypothetical example
illustrates 3 different actions that can be taken, and 4 different possible,
uncertain states of nature with their respective payoffs.
i. Hurwicz criterion.
This approach attempts to strike a balance between
the maximax and maximin criteria. It suggests that the minimum and
maximum of each strategy should be averaged using a and 1 -
a
as weights. a represents the index of pessimism and the alternative
with the highest average is selected. The index a reflects
the decision makerТs attitude towards risk taking. A cautious decision
maker will set a = 1 which reduces the Hurwicz criterion to the
maximin criterion. An adventurous decision maker will set
a
= 0 which reduces the Hurwicz criterion to the maximax criterion. [1]
A decision table illustrating the application of this criterion (with a
= .5) to a decision situation is shown below.
| Actions\States |
S1 |
S2 |
S3 |
S4 |
a = .5 |
| A1 |
20 |
60 |
-60 |
20 |
0 |
| A2 |
0 |
20 |
-20 |
20 |
0 |
| A3 |
50 |
-20 |
-80 |
20 |
-15 |
Table 2: Hurwicz criterion
illustration (a = .5); Here the probability of each state is not considered;
results in a tie between the first two alternatives.
ii. Laplace insufficient reason criterion.
The Laplace insufficient reason criterion postulates
that if no information is available about the probabilities of the various
outcomes, it is reasonable to assume that they are equally likely.
Therefore, if there are n outcomes, the probability of each is 1/n.
This approach also suggests that the decision maker calculate the expected
payoff for each alternative and select the alternative with the largest
value. The use of expected values distinguishes this approach from
the criteria that use only extreme payoffs. This characteristic makes
the approach similar to decision making under risk. A table illustrates
this criterion below. [1]
| Actions\States |
S1 (P=.25) |
S2 (P=.25) |
S3 (P=.25) |
S4 (P=.25) |
Expected Payoff: |
| A1 |
20 |
60 |
-60 |
20 |
0 |
| A2 |
0 |
20 |
-20 |
20 |
5 |
| A3 |
50 |
-20 |
-80 |
20 |
-7.5 |
Table 3: Laplace insufficiency
illustration; Second alternative wins when expected payoff is calculated
between equiprobable states.
iii. Maximax criterion.
The maximax criterion is an optimistic approach.
It suggests that the decision maker examine the maximum payoffs of alternatives
and choose the alternative whose outcome is the best. This criterion
appeals to the adventurous decision maker who is attracted by high payoffs.
This approach may also appeal to a decision maker who likes to gamble and
who is in the position to withstand any losses without substantial inconvenience.
See the table below for an illustration of this criterion. [1]
| Actions\States |
S1 |
S2 |
S3 |
S4 |
Max Payoff |
| A1 |
20 |
60 |
-60 |
20 |
60 |
| A2 |
0 |
20 |
-20 |
20 |
20 |
| A3 |
50 |
-20 |
-80 |
20 |
50 |
Table 4: Maximax illustration;
First alternative wins.
iv. Maximin criterion.
The maximin criterion is a pessimistic approach.
It suggests that the decision maker examine only the minimum payoffs of
alternatives and choose the alternative whose outcome is the least bad.
This criterion appeals to the cautious decision maker who seeks to ensure
that in the event of an unfavorable outcome, there is at least a known
minimum payoff. This approach may be justified because the minimum
payoffs may have a higher probability of occurrence or the lowest payoff
may lead to an extremely unfavorable outcome. This criterion is illustrated
in the table below. [1]
| Actions\States |
S1 |
S2 |
S3 |
S4 |
Min payoff |
| A1 |
20 |
60 |
-60 |
20 |
-60 |
| A2 |
0 |
20 |
-20 |
20 |
-20 |
| A3 |
50 |
-20 |
-80 |
20 |
-80 |
Table 5: Maximin illustration.
Second alternative wins.
v. Savage minimax regret criterion.
The Savage minimax regret criterion examines the
regret, opportunity cost or loss resulting when a particular situation
occurs and the payoff of the selected alternative is smaller than the payoff
that could have been attained with that particular situation. The
regret corresponding to a particular payoff Xij is defined as Rij = Xj(max)
Ц Xij where Xj(max) is the maximum payoff attainable under the situation
Sj. This definition of regret allows the decision maker to transform
the payoff matrix into a regret matrix. The minimax criterion suggests
that the decision maker look at the maximum regret of each strategy and
select the one with the smallest value. This approach appeals to
cautious decision makers who want to ensure that the selected alternative
does well when compared to other alternatives regardless of what situation
arises. It is particularly attractive to a decision maker who knows
that several competitors face identical or similar circumstances and who
is aware that the decision makerТs performance will be evaluated in relation
to the competitors. This criterion is applied to the same decision situation
and transforms the payoff matrix into a regret matrix. This is shown
below. [1]
| Actions\States |
R1 |
R2 |
R3 |
R4 |
Max Regret |
| A1 |
30 |
0 |
40 |
0 |
40 |
| A2 |
50 |
40 |
0 |
0 |
50 |
| A3 |
0 |
80 |
60 |
0 |
80 |
Table 5: Minimax illustration.
First alternative wins.
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