Question

A television network is attempting to decide during the summer which of the following three football games to televise on the Saturday following Thanksgiving Day: Alabama vs. Auburn, Georgia vs. Georgia Tech, Miami vs. Virginia Tech. The estimated viewer ratings (millions of homes) for the games depend on the win-loss records of the six teams, as shown in the following payoff table. Determine the best game (most viewers) to televise using the following decision criteria.

		States of Nature - Number of viewers
		One team has
	Both Teams	Winning record	Both teams
	Have winning	One team has	Have losing
Games	Records	Losing record	Records
Alabama vs. Auburn	10.3	7.3	5.4
Georgia vs. Georgia Tech	9.6	8.1	4.8
Miami vs. Viriginia Tech	12.5	6.5	3.2

Determine the best decision using the following decision criteria. Calculate and show values for each alternative (except for #7). Assume that the probability of both teams having a winning record is 20%, and both teams having a losing record is 25%.

            1. Maximax

            2. Maximin

            3. Criterion of Realism (a = .4)

            4. Minimax

            5. Expected Value

            6. Expected Opportunity Loss

            7. Expected Value of Perfect Information

Answer 1

Decision: Among Alabama vs. Auburn, Georgia vs. Georgia Tech and Miami vs. Virginia Tech which game to telecast for maximum viewers. The estimated viewer ratings (millions of homes) for the games depend on the win-loss records of the six teams.

Criteria for making decisions under uncertainty;

1. Maximax(Optimistic):  Used to find the alternative that maximizes the maximum payoff – maximax criterion  

• Locate the maximum payoff for each alternative   
• Select the alternative with the maximum number

	States of Nature - Number of viewers
Games	Both Teams Have winning Records	One team has Winning record One team has Losing record	Both Teams Have losing Records	Max of each game
					Alabama vs. Auburn	10.3	7.3	5.4	10.3
					Georgia vs. Georgia Tech	9.6	8.1	4.8	9.6
Miami vs. Virginia Tech	12.5	6.5	3.2	12.5

So according to Maximax criteria Miami vs. Virginia Tech fits the criteria of the maximum of all. So it should be telecasted.

2. Maximin (Pessimistic): Used to find the alternative that maximizes the minimum payoff – maximin criterion

• Locate the minimum payoff for each alternative
• Select the alternative with the maximum number

	States of Nature - Number of viewers
Games	Both Teams Have winning Records	One team has Winning record One team has Losing record	Both Teams Have losing Records	Min of each game
					Alabama vs. Auburn	10.3	7.3	5.4	5.4
					Georgia vs. Georgia Tech	9.6	8.1	4.8	4.8
					Miami vs. Virginia Tech	12.5	6.5	3.2	3.2

So according to Maximax criteria, Alabama vs. Auburn fits the criteria of the maximum of the minimum. So it should be telecasted.

3. The criterion of Realism (alpha = 0.4): Often called weighted average. It is a compromise between optimism and pessimism. Compute the weighted averages for each alternative

Weighted average = alpha(best in a row) + (1?alpha)(worst in a row)

Select the alternative with the highest value

For the first row = 0.4*10.3+0.6*5.4 = 7.36

For the second row = 0.4*9.6+0.6*4.8 = 6.72

For the third row = 0.4*12.5+0.6*3.2 = 6.92

	States of Nature - Number of viewers
Games	Both Teams Have winning Records	One team has Winning record One team has Losing record	Both Teams Have losing Records	Criterion of realism
					Alabama vs. Auburn	10.3	7.3	5.4	7.36
					Georgia vs. Georgia Tech	9.6	8.1	4.8	6.72
					Miami vs. Virginia Tech	12.5	6.5	3.2	6.92

Since we need to select maximum of the weighted averages,  Alabama vs. Auburn should be telecast.

4. Minimax Regret: Based on opportunity loss or regret. The difference between the optimal profit and actual payoff for a decision (Opportunity loss is the amount lost by not picking the best alternative in a given state of nature)

1. Create an opportunity loss table by determining the opportunity loss from not choosing the best alternative

2. Calculate opportunity loss by subtracting each payoff in the column from the best payoff in the column

3. Find the maximum opportunity loss for each alternative and pick the alternative with the minimum number

Opportunity loss table is created by subtracting the maximum of a column with all values of the col.

So for the first col, 12.5-10.3 = 2.2, 12.5-9.6 = 2.9 and 12.5-12.5 = 0

For the second col, 8.1-7.3 = 0.8, 8.1-8.1=0 and 8.1-6.5 =1.6

For the third col, 5.4-5.4=0,5.4-4.8=0.6 and 5.4-3.2=2.2

Opportunity Loss table for the games:

Games	Records	Losing record	Records	max of each game
Alabama vs. Auburn	2.2	0.8	0	2.2
Georgia vs. Georgia Tech	2.9	0	0.6	2.9
Miami vs. Virginia Tech	0	1.6	2.2	2.2

Minimax decision using opportunity loss will be either of Alabama vs. Auburn or Miami vs. Virginia Tech as both has a minimum of the maximum of opportunity loss of each game.

5. Expected Value: When there are several possible states of nature and the probabilities associated with each possible state are known. Most popular method – choose the alternative with the highest expected value.

where Xi: is the number of records for each alternative

P(Xi): the probability of each alternative

Assume that the probability of both teams having a winning record is 20%, and both teams having a losing record is 25%.

Thus we would choose Georgia vs. Georgia Tech according to this criteria as it has maximum expected value.

6. Expected opportunity loss:

where Xi: is the opportunity loss value for each alternative

P(Xi): the probability of each alternative

Assume that the probability of both teams having a winning record is 20%, and both teams having a losing record is 25%.

Thus we would choose Georgia vs. Georgia Tech according to this criteria as it has minimum expected opportunity loss.

7. The expected value of perfect information: EVPI (expected value of perfect information) places an upper bound on what you should pay for additional information.

EVwPI (expected value with perfect information) is the long run average return if we have perfect information before a decision is made.

EVPI = EVwPI - Maximum Expected value

EVwPI = sum of (best record in the state of nature* probability of that state of nature)

= 12.5*0.2+8.1*0.55+5.4*0.25 = 2.5+4.455+1.35 = 8.305

EVPI = 8.305 - 7.575 = 0.73