1. What is Payoff?
Payoff means return. Payoff occurs as a result of some
unexpected events. Matrix means rectangular array of some
numbers placed within brackets.
So, payoff is the amount of profit that a decision maker wants or
expects under different conditions of uncertainty and for taking
different strategies and when it is expressed in a systematic
array of numbers within brackets it is called payoff matrix.
2. Significance of payoff matrix in business:
A businessman conducts his business under different
conditions of uncertain environment. The economic
conditions are also different in different environments such
as boom, depression, recession, slump etc.
A businessman takes different strategies to earn profit
under different economic conditions. Thus a businessman
can be aware of the outcomes of different strategies taken
under various economic conditions through payoff matrix.
3. For conceptualizing about decision making process
payoff matrix converts the decision making process into
formal:
1) Description of objectives.
2) Selection of payoffs.
3) Evaluation of alternative
payoffs , and
4) Selection from among the
alternative strategies.
In addition to this, payoff matrix plays a great role
in game theory. A player knows not only his own
list of possible courses of action but also of his
opponents. We can also consider the payoffs
under different stages of the product like cycle.
4. Procedure of decision making:
Decisions are taken under three conditions. The strategy
to be taken by a decision maker depends on the conditions
of (a) certainty (b) risk, and (c) uncertainty.
The payoff matrix of different strategies taken under
different economic conditions is discussed below:
States of nature
Alternative
Strategies
Boom(N1) Recovery(N2) Recession(N3) Depresion(N4)
S1 6 6 6 4
S2 25 7 7 -15
S3 20 20 7 -1
S4 19 16 9 -2
S5 20 15 15 -3
5. Decision Making Under Certainty:
Under conditions of certainty a decision maker is fully aware
of what will happen in future. Consequently, the matrix
becomes a one column matrix and the decision maker takes
that strategy that yields the highest payoff. In our above
example, if the decision maker becomes certain that boom
economic condition (N1) will prevail then he will take S2
strategy. If he is assured that the economy is recovering then
he will take S3 strategy. If the economic condition is in the
Recession (N3) stage then he will take S5 strategy and in
case of depression he will take S1 strategy.
6. Decision Making Under Risk:
In this case, each alternative strategy yields a definite outcome
and the probability of each outcome is known to the decision
maker i.e., the probability of outcome can be guessed
beforehand and the decision maker can prepare a distribution of
different strategy’s payoffs.
Criterion for Decision Making Under Risk:
Under this method the payoff is multiplied by the expected
probability and the strategy with the highest expected value is
taken. For example, if the decision maker thinks that the
probability of happening Boom is 20% Recovery-65%,
Recession-10%, and Depression is 5% then the strategy to be
taken is determined by the following way:
7. Here the expected value of S4 is calculated as follows:
E(S(4) = 0.20*19+0.65*16+0.10*9+0.05*(-2) =15.00 and the
expected value of each strategy is done the same way. Here S3
strategy shows the highest expected payoff. So, S3 strategy is to
be taken. But if any two strategies show the same expected
payoff then the standard deviation of them is calculated. Then
the strategy is taken on the basis of desire of risk taking.
8. Decision Making Under Uncertainty:
There are 4 criteria for decision making under
uncertainty which are as follows:
•The Wald Decision Criterion
•The Hurwicz Alpha Decision Criterion
•The Savage Decision Decision Criterion, also called the
minimax regret.
•The Laplace Decision Criterion.
Wald Decision Criterion :
This decision making process is called the process of the
pessimists. This is also called the most conservative method.
Murphy’s Law is fully effective here. Under this method the
worst outcome of each strategy is chosen first. Then the
highest of them is taken as the selected strategy.
9. Example of Wald Decision Criterion
Alternative
Strategies
States of nature Criterion
N1 N2 N3 N4 Maximin Minimax
S1 6 6 6 4 4 6
S2 25 7 7 -15 -15 25
S3 20 20 7 -1 -1 20
S4 19 16 9 -2 -2 19
S5 20 15 15 -3 -3 20
Here S1 strategy is acceptable because among the maximin,
S1 strategy yields the highest payoff.
10. The Hurwicz Alpha Decision Criterion:
Under this method the maximum value of weighted estimated
payoff is taken. Then element of weight is the coefficient of
optimism which is expressed by alpha (α). The highest payoff
is multiplied by alpha. Then the lowest payoff is multiplied
by (1- α). These two products are added. The strategy with
the highest added value is selected.
The following formula is used:
di = αMi + (1- α) m
11. Hurwicz Alpha Decision Criterion
Alternative
Strategies
Mi α αMi m 1- α (1- α)m d
S1 6 .7 4.2 4 0.3 1.2 5.4
S2 25 .7 17.5 -15 0.3 -4.5 13.0
S3 20 .7 14.0 -1 0.3 -0.3 13.7*
S4 19 .7 13.3 -2 0.3 -0.6 12.7
S5 20 .7 14.0 -3 0.3 -0.9 13.1
Here S3 strategy should be accepted because it
yields the highest payoff.
12. The Savage Decision Criterion:
This method is also called the maximum regret criterion. Under
this method steps are taken to reduce the regret i.e. steps are
taken to reduce the opportunity cost of wrong decision. At first a
regret matrix is constructed from the payoff matrix. Each cell
value of column is deducted from the highest value of a column
to get the regret. The difference must be the absolute value. The
regret matrix is calculated in the following manner:
13. Alternati
ve
Strategy
Payoff Matrix Regret Matrix Maximum
Regret
N1 N2 N3 N4 N1 N2 N3 N4
S1 6 6 6 4 19 14 9 0 19
S2 25 7 7 -15 0 13 8 19 19
S3 20 20 7 -1 5 0 8 5 8
S4 19 16 9 -2 6 4 6 6 6
S5 20 15 15 -3 5 5 0 7 7
Here S4 strategy should be accepted because it reduces to the
minimum the maximum punishment of wrong assumption about
the state of nature.
The Savage Decision Criterion:
14. The Laplace Decision Criterion:
This method is also called the Bayes Criterion. According
to this method if the probability of occurring an event is
unknown we should think that the probability of
occurring any event is equal. The strategy with the
highest expected payoff is accepted.
In our previous example, the expected value of S1, S2,
S3, S4, and S5 are 22/4, 24/4, 46/4, 42/4 and 47/4. So, S5
strategy should be accepted.
