Decision Theory Final

DECISION THEORY

PROF. KAUSHIK PAUL ASSOCIATE PROFESSOR OPERATIONS AREA IILM GRADUATE SCHOOL OF MANAGEMENT GREATER NOIDA

TOPICS OF DISCUSSION 

INTRODUCTION



PAYOFF MATRIX



REGRET TABLE



DECISION MAKING UNDER UNCERTAINTY



DIFFERENT METHODS OF DECISION MAKING UNDER UNCERTAINTY



DECISION MAKING UNDER RISK

2

TOPICS OF DISCUSSION 

DIFFERENT METHODS OF DECISION MAKING UNDER RISK 

MAXIMUM LIKELIHOOD PRINCIPLE



MAXIMUM EXPECTATION PRINCIPLE



CONCEPT OF DECISION TREE



DECISION MAKING WITH PERFECT INFORMATION AND ECONOMIC VALUE OF PERFECT INFORMATION (EVPI)



DECISION MAKING WITH REVISED PROBABILITIES ( USING BAYE’S THEOREM)



EXPECTED OPPORTUNITY LOSS/REGRET PRINCIPLE 3

INTRODUCTION 

EXTERNAL ENVIRONMENT IMPOSES CERTAIN CONSTRAINTS ON US. BASED ON OUR RESPONSE TO SUCH CONSTRAINTS, WE GET DIFFERENT PAYOFFS.



WE CAN NOT CHANGE THE PAYOFF MATRIX.



AT BEST, WE CAN USE THE AVAILABLE INFORMATION JUDICIOUSLY TO ARRIVE AT THE OPTIMAL DECISION AND MAXIMIZE OUR PAYOFFS IN THE LONG RUN.

4

EXAMPLE 

A WHOLE SALER OF FRUITS BUYS STRAWBERRIES AT $ 20 A CASE AND SELLS THEM AT $ 50 A CASE. THE PRODUCT IS PERISHABLE BY NATURE AND CAN NOT BE STORED OVERNIGHT. IT HAS BE SOLD ON THE DAY OF THE PURCHASE ITSELF.



FROM EXPERIENCE, THE WHOLE SALER KNOWS THAT THE DAILY DEMAND WILL RANGE BETWEEN 10 TO 13 CASES.



EVERY CASE OF STRAWBERRIES BOUGHT AND NOT SOLD WOULD LEAD TO A MARGINAL LOSS OF $ 20, WHILE EVERY CASE THAT COULD NOT BE SOLD BECAUSE OF STOCK OUT WOULD LEAD TO AN OPPORTUNITY LOSS OF $ 30. 5

PAYOFF MATRIX IN US $ POSSIBLE DEMAND IN CASES

POSSIBLE STOCK ACTION IN CASES 10

11

12

13

10

300

280

260

240

11

300

330

310

290

12

300

330

360

340

13

300

330

360

390 6

REGRET TABLE IN US $ POSSIBLE DEMAND IN CASES

POSSIBLE STOCK ACTION IN CASES 10

11

12

13

10

0

20

40

60

11

30

0

20

40

12

60

30

0

20

13

90

60

30

0 7

DECISION MAKING UNDER UNCERTAINTY 

HERE, WE HAVE NO INFORMATION ABOUT THE LIKELIHOOD( PROBABILTY) OF ANY PARTICULAR STATE OF DEMAND.



IN SUCH A CONDITION, WE ARE MAKING A DECISION UNDER UNCERTAINTY.



FOLLOWING PRINCIPLES ARE USED TO TAKE DECISIONS UNDER UNCERTAINTY:     

LAPLACE PRINCIPLE MAXIMIN OR MINIMAX CRITERION MAXIMAX. OR MINIMIN. CRITERION HURWICZ CRITERION SALVAGE PRINCIPLE 8

LAPLACE PRINCIPLE 



ASSUMES ALL EXTERNAL CONSTRAINTS ( HERE DEMAND OF STRAWBERRIES IN CASES TO BE EQUIPROBABLE). MAXIMUM PAYOFF IS ACHIEVED BY PURSUING THE STRATEGY OF STOCKING 12 UNITS.

STOCK ACTION ( CASES)

PAYOFF (US $)

10

300

11

317.5

12

322.5

13

315

9

MAXIMIN. OR MINIMAX PRINCIPLE 



HERE, MINIMUM PAYOFFS FROM EACH STRATEGY IS CONSIDERED. THE STRATEGY WITH MAX. OF THESE MINIMUM PROFITS IS SELECTED. THIS IS KNOWN AS MAXIMIN. PRINCIPLE. FOR COSTS, THE STRATEGY WITH MIN. OF MAXIMUM COSTS IS CHOSEN. THAT IS KNOWN AS MINIMAX. PRINCIPLE. THIS IS A PESSIMISTIC DECISION MAKING. MAXIMUM PAYOFF IS ACHIEVED BY PURSUING THE STRATEGY OF STOCKING 10 UNITS.

STOCK ACTION ( CASES)

MIN. PAYOFF (US $)

10

300

11

280

12

260

13

240

10

MAXIMAX. OR MINIMIN. PRINCIPLE 



HERE, MAXIMUM PAYOFFS FROM EACH STRATEGY IS CONSIDERED. THE STRATEGY WITH MAX. OF THESE MAXIMUM PROFITS IS SELECTED. THIS IS KNOWN AS MAXIMAX. PRINCIPLE. FOR COSTS, THE STRATEGY WITH MIN. OF THE MINIMUM COSTS IS CHOSEN. THAT IS KNOWN AS MINIMIN. PRINCIPLE. THIS IS A HIGHLY OPTIMISTIC DECISION MAKING. MAXIMUM PAYOFF IS ACHIEVED BY PURSUING THE STRATEGY OF STOCKING 13 UNITS.

STOCK ACTION ( CASES)

MAX. PAYOFF (US $)

10

300

11

330

12

360

13

390

11

HURWICZ PRINCIPLE 

INDEX OF OPTIMISM= α



CRITERION VALUE = α( MAX. PROFIT) + (1- α)(MIN. PROFIT)



FOR COSTS, CRITERION VALUE = α( MIN.COST) + (1- α)(MAX.COST)



α = 0 STANDS FOR MAXIMIN. OR MINIMAX. PRINCIPLE



α = 1 STANDS FOR MAXIMAX. OR MINIMIN. PRINCIPLE



FOR OUR EXAMPLE, LET US ASSUME AN INDEX OF OPTIMISM OF 60% (α = 0.6) 12

HURWICZ PRINCIPLE CONTINUED 



THE STRATEGY WITH THE MAXIMUM HURWICZ CRITERION VALUE IS CHOSEN. HERE, THE MAXIMUM HURWICZ CRITERION VALUE IS ACHIEVED BY PURSUING THE STRATEGY OF STOCKING 13 UNITS.

STOCK ACTION ( CASES)

HURWICZ CRIETRION VALUE (US $)

10

300

11

310

12

320

13

330

13

SALVAGE PRINCIPLE 



HERE, WE SELECT THE STRATEGY THAT MINIMIZES THE MAXIMUM REGRET. THIS IS ALSO A PESSIMISTIC DECISION MAKING. THIS IS CLOSE TO THE MINIMAX. PRINCIPLE BUT RESULTS MAY VARY. HERE, THE MINIMUM OF MAX. REGRET (SALVAGE) VALUE IS ACHIEVED BY PURSUING THE STRATEGY OF STOCKING 12 UNITS.

STOCK ACTION ( CASES)

MAX. REGRET VALUE (US $)

10

90

11

60

12

40

13

60

14

DECISION MAKING UNDER RISK 

NOW, LET US ASSUME THAT BASED ON THE SALES OF PAST 100 DAYS, THE WHOLESALER HAS FOLLOWING INFORMATION ABOUT THE MARKET DEMAND.

DAILY SALES ( CASES) 10

NO. OF DAYS SOLD 15

PROB. OF DEMAND 0.15

11

20

0.20

12

40

0.40

13

25

0.25

TOTAL

100

1.00 15

DECISION MAKING UNDER RISK 

HERE, WE HAVE ADDITIONAL INFORMATION ON THE PROBABILITY OF EACH DEMAND STATE.



WHEN WE HAVE PROBABILITIES ASSOCIATED WITH EACH DEMAND STATE AVAILABLE TO US, THE DECISION MAKING TECHNIQUE IS CALLED DECISION MAKING UNDER RISK.



FOLLOWING PRINCIPLES ARE USED TO ARRIVE AT THE OPTIMAL DECISION.   

MAXIMUM LIKELIHOOD PRINCIPLE MAXIMUM EXPECTATION PRINCIPLE MINIMUM EXPEXCTED OPPORTUNITY LOSS/REGRET PRINCIPLE

16

MAXIMUM LIKELIHOOD PRINCIPLE 

HERE, WE CHOOSE TO STOCK ACCORDING TO THAT DEMAND STATE WHICH HAS MAXIMUM PROBABILITY OF OCCURRENCE. IN THIS CASE, THE WHOLE SALER SHOULD STOCK 12 CASES, IF HE ADOPTS THIS PRINCIPLE.

DAILY SALES ( CASES) 10 11 12 13 TOTAL

PROB. OF DEMAND 0.15 0.20 0.40 0.25 1.00

17

MAXIMUM EXPECTATION PRINCIPLE 

HERE, WE CHOOSE THAT STRATEGY WHICH HAS MAXIMUM EXPECTED PAYOFF. THIS IS THE MOST ACCEPTABLE PRINCIPLE SINCE, THE EXPECTED PAYOFF WILL ALWAYS COME TRUE IN THE LONG RUN.



IN OUR EXAMPLE, THE WHOLE SALER SHOULD CHOOSE THAT STRATEGY WHICH WILL MAXIMIZE HIS EXPECTED PROFIT.



NOW, WE HAVE TO EXAMINE THE EXPECTED PROFIT FOR EACH STRATEGY ( STOCK ACTION).

18

EXPECTED PROFIT FROM PURSUING A STRATEGY OF STOCKING 10 CASES DEMAND IN CASES

CONDITIONAL PROFIT

PROB. OF DEMAND

EXPECTED PROFIT (US $)

10

300

0.15

45.0

11

300

0.2

60.0

12

300

0.4

120.0

13

300

0.25

75.0

1.00

300.0

TOTAL

19

EXPECTED PROFIT FROM PURSUING A STRATEGY OF STOCKING 11 CASES DEMAND IN CASES

CONDITIONAL PROFIT

PROB. OF DEMAND

EXPECTED PROFIT (US $)

10

280

0.15

42.0

11

330

0.2

66.0

12

330

0.4

132.0

13

330

0.25

82.5

1.00

322.5

TOTAL

20

EXPECTED PROFIT FROM PURSUING A STRATEGY OF STOCKING 12 CASES DEMAND IN CASES

CONDITIONAL PROFIT

PROB. OF DEMAND

EXPECTED PROFIT (US $)

10

260

0.15

39.0

11

310

0.2

62.0

12

360

0.4

144.0

13

360

0.25

90.0

1.00

335.0

TOTAL

21

EXPECTED PROFIT FROM PURSUING A STRATEGY OF STOCKING 13 CASES DEMAND IN CASES

CONDITIONAL PROFIT

PROB. OF DEMAND

EXPECTED PROFIT (US $)

10

240

0.15

36.0

11

290

0.2

58.0

12

340

0.4

136.0

13

390

0.25

97.5

1.00

327.5

TOTAL

22

DECISION TREE 

SO, THE MAXIMUM PROFIT COMES FROM PURSUING A STRATEGY OF STOCKING 12 CASES.



NOW, HOW DO WE REPRESENT THIS LOGIC DIAGRAMATICALLY? WELL, WE USE A DIAGRAM CALLED A DECISION TREE.



IN A DECISION TREE, WE REPRESENT A DECISION ( STATEGY) WITH A TECTANGLE WHILE AN OUTCOME ( DEMAND IN THIS CASE) IS REPRESENTED BY A CIRCLE.

23

DECISION TREE FOR THE WHOLESALER PROBABILITIES

10,45.0, 0.15 11,60,0.2

$ 300.0

12,120.0,0.4

$ 335.0

$ 322.5

13,75,0.25

$ 335.0

$ 327.5 24

DECISION MAKING WITH PERFECT INFORMATION (EVPI) 

IF WE GO BACK TO THE PAYOFF MATRIX, WE HAVE MARKED OUR BEST DECISIONS FOR EACH DEMAND SCENARIO. IF WE COULD HAVE PERFECT INFORMATION ABOUT THE MARKET DEMAND, OUR EXPECTED PAYOFF TABLE WOULD HAVE LOOKED LIKE THIS.



PAYOFF MATRIX IN US $

MARKET DEMAND ( CASES)

CONDITIONAL PROFIT

PROB. OF DEMAND

EXPECTED PROFIT ( US $)

10

300

0.15

45.0

11

330

0.2

66.0

12

360

0.4

144.0

13

390

0.25

97.5

1.00

352.5

TOTAL

25

DECISION TREE FOR THE WHOLESALER WITH PERFECT INFORMATION

10, 300.0

$ 300.0

11, 280.0 12, 260.0

$ 330.0

13, 240.0

$ 352.5

$ 360.0

$ 390.0 26

ECONOMIC VALUE OF PERFECT INFORMATION ( EVPI) 

INCREASE IN EXPECTED PROFIT WITH PERFECT INFORMATION = ( 352.5- 335) $ = 17.5 $



THIS IS KNOWN AS THE ECONOMIC VALUE OF PERFECT INFORMATION ( EVPI).

27

DECISION MAKING WITH REVISED PROBABILITIES 

LET, THERE BE A MARKET RESEARCH FIRM, THAT PROVIDES ADDITIONAL INFORMATION ( FORECASTING) ABOUT THE POSSIBLE STATE OF DEMAND, AND CHARGES A FEE FOR THE SAME.



WHAT IS THE ADDITIONAL VALUE OF THIS FORECAST AND HOW MUCH CAN THE WHOLESALER PAY FOR IT?



THE AGENCY FORECASTS THE DEMAND BY RATING IT AS ABOVE NORMAL (AN) OR BELOW NORMAL (BN).



IT HAS BEEN OBSERVED IN THE PAST THAT IN 80% OF THE INSTANCES, WHEN THE DEMAND WAS 10 CASES, THE AGENCY HAD FORECASTED BELOW NORMAL (BN). IN 60% OF THE INSTANCES, WHEN THE DEMAND WAS 11 CASES, THE AGENCY HAD FORECASTED BELOW NORMAL (BN). IN 30% OF THE INSTANCES, WHEN THE DEMAND WAS 12 CASES, THE AGENCY HAD FORECASTED BELOW NORMAL (BN). IN 20% OF THE INSTANCES, WHEN THE DEMAND WAS 13 CASES, THE AGENCY HAD FORECASTED BELOW NORMAL (BN). 28

REVISED PROBABILITIES (USING BAYE’S THEOREM) FORECAST

EVENT

P( EVENT)

ABOVE NORMAL (AN)

10

0.15

0.2

0.03

0.05

11

0.2

0.4

0.08

0.14

12

0.4

0.7

0.28

0.47

13

0.25

0.8

0.2

0.34

TOTAL

1.00

P(AN) =

0.59

1.00

10

0.15

0.8

0.12

0.29

11

0.2

0.6

0.12

0.29

12

0.4

0.3

0.12

0.29

13

0.25

0.2

0.05

0.13

TOTAL

1.00

P(BN) =

0.41

1.00

BELOW NORMAL (BN)

P(FORECAST/ P(FORECAST EVENT) & EVENT)

REVISED PROB. (EVENT/FORE CAST)

29

DECISION TREE FOR THE REVISED PROBABILITIES DO NOT BUY FORECAST A $ 335.0

10 B

AN, 0.59 $ 348.14 $ 335.165

C 13

$335.165

D A’

BUY FORECAST

11 B’

$ 316.5 EVPI= $ 0.165

C’ BN, 0.41

D’

30

12

NODE ‘A’

STOCKING 10 CASES 10,300.0,0.05 $ 300.0 11,300,0.14

12,300.0, 0.47

13, 300.0, 0.34

31

NODE ‘B’

STOCKING 11 CASES 10,280.0,0.05 $ 327.46 11,330,0.14

12,330.0, 0.47

13, 330.0, 0.34

32

NODE ‘C’

STOCKING 11 CASES 10,260.0,0.05 $ 348.14 11, 310.0,0.14

12,360.0, 0.47

13, 360.0, 0.34

33

NODE ‘D’

STOCKING 11 CASES 10,240.0,0.05 $ 345.08 11, 290.0,0.14

12,340.0, 0.47

13, 390.0, 0.34

34

NODE ‘ A’ ’

STOCKING 10 CASES 10,300.0,0.29 $ 300.0 11,300,0.29

12,300.0, 0..29

13, 300.0, 0.13

35

NODE ‘ B’ ’

STOCKING 11 CASES 10,280.0,0.29 $ 315.50 11,330,0.29

12,330.0, 0..29

13, 330.0, 0.13

36

NODE ‘ C’ ’

STOCKING 12 CASES 10,260.0,0.29 $ 316.50 11,310,0.29

12,360.0, 0..29

13, 360.0, 0.13

37

NODE ‘ D’ ’

STOCKING 13 CASES 10,240.0,0.29 $ 303.00 11,290,0.29

12,340.0, 0..29

13, 390.0, 0.13

38

MINIMUM EXPECTED REGRET/OPPORTUNITY LOSS PRINCIPLE 



THIS PRINCIPLE WOULD GIVE THE SAME ANSWER AS THE PREVIOUS PRINCIPLE.

THIS IS BECAUSE, ER(j)+ EP(j)= EPPI ( EXPECTED PAYOFF UNDER PERFECTINFORMATION)



HENCE, THE STRATEGY THAT WOULD MINIMIZE ER(j) WOULD AUTOMATICALLY MAXIMIZE EP(j).



THE MIN. ER(j) IS ALSO THE EVPI IN THIS CASE.

39

EXPECTED OPPORTUNITY LOSS/REGRET FOR DIFFERENT STOCK ACTIONS STOCK ACTION ( CASES) 10

EXPECTED REGRET ( US $) 52.5

11

30.0

12

17.5

13

25.0 40

EVSI AND EFFICIENCY OF EVSI 

IN OUR EXAMPLE, ECONOMIC VALUE OF SAMPLE INFORMATION ( EVSI) = (335.165- 335) $ = 0.165 $



EFFICIENCY OF EVSI = 0.165/335 = 0.05 %

41

QUESTIONS PLEASE

THANK YOU 42