Presentation on theme: "Introduction to Decision Analysis"— Presentation transcript:
1 Introduction to Decision Analysis The field of decision analysis provides framework for making important decisions.Decision analysis allows us to select a decision from a set of possible decision alternatives when uncertainties regarding the future exist.The goal is to optimized the resulting payoff in terms of a decision criterion.
2 Components of a Decision Model A set of possible decisionsdiscrete, continuousfinite, infiniteA set of possible events that could occur (states of nature)Payoffs (returns or costs) associated with each decision/state of nature pair
3 Payoff Table Analysis Payoff Tables Payoff Table analysis can be applied when -There is a finite set of discrete decision alternatives.The outcome of a decision is a function of a single future event.In a Payoff Table -The rows correspond to the possible decision alternatives.The columns correspond to the possible future events.Events (States of Nature) are mutually exclusive and collectively exhaustive.The body of the table contains the payoffs.
4 EXAMPLE TOM BROWN INVESTMENT DECISION Tom Brown has inherited $1000.He has decided to invest the money for one year.A broker has suggested five potential investments.Gold.(Junk) Bond.Growth Stock.Certificate of Deposit.Stock Option Hedge.Tom has to decide how much to invest in each investment.
5 Constructing a Payoff Table Determine the set of possible decision alternatives.for Tom this is the set of five investment opportunities.Defined the states of nature.Tom considers several stock market states (expressed by changes in the DJA)State of Nature DJA CorrespondenceS.1: A large rise in the stock market Increase over 1000 pointsS.2: A small rise in the stock market Increase between 300 and 1000S.3: No change in the stock market Change between -300 and +300S.4: A small fall in stock market Decrease between 300 and 800S5: A large fall in the stock market Decrease of more than 800The States of Nature are Mutually Exclusiveand Collective Exhaustive
6 Determining Payoffs Our broker reasons: Stocks and bonds generally move in the same direction of the marketGold is an investment hedge that seems to move opposite to the market directionA C/D account pays the same interest irrespective of market conditionsSpecifically our broker’s analysis leads to the following payoff table:
8 Dominated Decision Alternatives The Stock Option hedge is dominated by the Bond Alternative, because the payoff for each state of nature for the stock option the payoff for the bond option. Thus the stock option hedge can be eliminated from any consideration.
9 Decision Making Criteria Classifying Decision Making CriteriaDecision making under certaintyThe future state of nature is assumed knownDecision making under riskThere is some knowledge of the probability of the states of nature occurring.Decision making under uncertainty.There is no knowledge about the probability of the states of nature occurring.
10 Decision Making Under Certainty Given the “certain” state of nature, select the best possible decision for that state of nature
11 Decisions for Decision Making Under Certainty If we knew Large Small No Small Largethis ===> Rise Rise Change Fall FallGoldBondStockC/DChoose => Stock Stock Gold Gold C/D
12 Decision Making Under Uncertainty The decision criteria are based on the decision maker’sattitude toward life.These include an individual being pessimistic or optimistic, conservative or aggressive.CriteriaMaximin Criterion - pessimistic or conservative approach.Minimax Regret Criterion - pessimistic or conservative approach.Maximax criterion - optimistic or aggressive approach.Principle of Insufficient Reasoning.
13 Decision Making Under Uncertainty The decision criteria are based on the decision maker’s attitude toward life.These include an individual being pessimistic or optimistic, conservative or aggressive.CriteriaMaximax (Minimin), Maximin (Minimax), Minimax Regret, Principle of Insufficient Reasoning
14 The Maximax CriterionThis criterion is based on the best possible scenario.It fits both an optimistic and an aggressive decision maker.An optimistic decision maker believes that the best possible outcome will always take place regardless of the decision made.An aggressive decision maker looks for the decision with the highest payoff (when payoff is profit)
15 To find an optimal decision. Find the maximum payoff for each decision alternative.Select the decision alternative that has the maximum of the “maximum” payoff.TOM BROWN - continuedThe Optimal decision
16 The Maximin CriterionThis criterion is based on the worst-case scenario.It fits both a pessimistic and a conservative decision maker.A pessimistic decision maker believes that the worst possible result will always occur.A conservative decision maker wishes to ensure a guaranteed minimum possible payoff.
17 To find an optimal decision Record the minimum payoff across all states of nature for each decision.Identify the decision with the maximum “minimum payoff”.TOM BROWN - ContinuedThe Optimal decision
18 The Minimax Regret Criterion (“I shoulda”) This criterion fits both a pessimistic and a conservative decision maker.The payoff table is based on “lost opportunity,” or “regret”.The decision maker incurs regret by failing to choose the “best” decision.
19 To find an optimal decision For each state of nature.Determine the best payoff over all decisions.Calculate the regret for each decision alternative as the difference between its payoff value and this best payoff value.For each decision find the maximum regret over all states of nature.Select the decision alternative that has the minimum of these “maximum regrets”.
20 TOM BROWN - continued 500 - (-100) = 600 500 -100 500 500 -100 -100 = 600500TOM BROWN - continued-100500500-100-100500-100-100Investing in Gold incurs a regretwhen the market exhibitsa large rise500500The Optimal decisionLet us build the Regret Table
21 The Principle of Insufficient Reason This criterion might appeal to a decision maker who is neither pessimistic nor optimistic.It assumes all the states of nature are equally likely to occur.The procedure to find an optimal decision.For each decision add all the payoffs.Select the decision with the largest sum (for profits).
22 TOM BROWN - continued Sum of Payoffs Gold $600Bond $350Stock $50C./D $300Based on this criterion the optimal decision alternative is to invest in gold.
24 Decision Making Under Risk The Expected Value CriterionIf a probability estimate for the occurrence of each state of nature is available, payoff expected value can be calculated.For each decision calculate its expected payoff byExpected Payoff = SBased on past market performance the analyst predicts:P(large rise) = .2, P(small rise) = .3, P(No change) = .3,P(small fall) = .1, P(large fall) = .1(Probability)(Payoff)Over States of Nature
25 TOM BROWN - continued The Optimal decision (0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130
26 When to Use the Expected Value Approach The Expected Value Criterion is useful in cases where long run planning is appropriate, and decision situations repeat themselves.One problem with this criterion is that it does not consider attitude toward possible losses.
27 Expected Value of Perfect Information (EVPI) The Gain in Expected Return obtained from knowing with certainty the future state of nature is called:Expected Value of Perfect Information (EVPI)It is also the Smallest Expect Regret of any decision alternative.Therefore, the EVPI is the expected regretcorresponding to the decision selectedusing the expected value criterion
28 TOM BROWN - continued Stock If it were known with certainty that there will be a “Large Rise” in the marketLarge rise-10025050060Stock... the optimal decision would be to invest in...Similarly,Expected Return with Perfect information =0.2(500)+0.3(250)+0.3(200)+0.1(300)+0.1(60) = $271EVPI = ERPI - EV = $271 - $130 = $141
29 Bayesian Analysis - Decision Making with Imperfect Information Bayesian Statistic play a role in assessing additional information obtained from various sources.This additional information may assist in refining original probability estimates, and help improve decision making.
30 Should Tom purchase the Forecast ? TOM BROWN - continuedTom can purchase econometric forecast results for $50.The forecast predicts “negative” or “positive” econometric growth.Statistics regarding the forecast.Should Tom purchase the Forecast ?When the stock market showed a large rise theforecast was “positive growth” 80% of the time.
31 SOLUTIONTom should determine his optimal decisions when the forecast is “positive” and “negative”.If his decisions change because of the forecast, he should compare the expected payoff with and without the forecast.If the expected gain resulting from the decisions made with the forecast exceeds $50, he should purchase the forecast.
32 Tom needs to know the following probabilities P(Large rise | The forecast predicted “Positive”)P(Small rise | The forecast predicted “Positive”)P(No change | The forecast predicted “Positive ”)P(Small fall | The forecast predicted “Positive”)P(Large Fall | The forecast predicted “Positive”)P(Large rise | The forecast predicted “Negative ”)P(Small rise | The forecast predicted “Negative”)P(No change | The forecast predicted “Negative”)P(Small fall | The forecast predicted “Negative”)P(Large Fall) | The forecast predicted “Negative”)
33 [ P(B | A 1)P(A 1)+ P(B | A 2)P(A 2)+…+ P(B | A n)P(A n) ] Bayes’ Theorem provides a procedure to calculate these probabilitiesP(B |A i)P(A i)[ P(B | A 1)P(A 1)+ P(B | A 2)P(A 2)+…+ P(B | A n)P(A n) ]P(A i | B) =Posterior Probabilities for “positive” economic forecast0.160.56X=0.2860.3750.2680.0710.0000.20.30.1The probability that the stock marketshows “Large Rise” given thatthe forecast predicted “Positive”The Probability that the forecast is “positive” and the stock market shows “Large Rise”.Observe the revision inthe prior probabilities
34 Bayesian Prob. for “Negative” Forecast s P(s) P(Neg|s) P(Neg and s) P(s|Neg)Large Rise /.44= .091Small Rise /.44= .205No Change /.44= .341Small Fall /.44= .136Large Fall /.44= .227.44
35 Expected Value of Sample Information The expected gain from making decisions based on Sample Information.With the forecast available, the Expected Value of Return is revised.Calculate Revised Expected Values for a given forecastas follows.EV(Invest in……. |“Positive” forecast) ==.286( )+.375( )+.268( )+.071( )+0( ) =EV(Invest in ……. | “Negative” forecast) ==.091( )+.205( )+.341( )+.136( )+.227( ) =GoldBond-100250-100100200100150200200-100300300-150$84$180-100100200300-100Bond100200300250-100100200150200-100300-150$ 65$120
36 Should Tom purchase the Forecast ? The rest of the revised EV s are calculated in asimilar manner.EREV = Expected Value Without Sampling Information = 130Expected Value of Sample Information - ExcelSo,Should Tom purchase the Forecast ?Invest in Stock when the Forecast is “Positive”ERSI = Expected Return with sample Information =(0.56)(250) + (0.44)(120) = $193Invest in Gold when the forecast is “Negative”
37 EVSI = Expected Value of Sampling Information = ERSI - EREV = = $63.Yes, Tom should purchase the Forecast.His expected return is greater than the Forecast cost.Efficiency = EVSI / EVPI = 63 / 141 = 0.45
38 Decision TreesThe Payoff Table approach is useful for a single decision situation.Many real-world decision problems consists of a sequence of dependent decisions.Decision Trees are useful in analyzing multi-stage decision processes.
39 Characteristics of the Decision Tree A Decision Tree is a chronological representation of the decision processThere are two types of nodesDecision nodes (represented by squares)State of nature nodes (representing by circles).The root of the tree corresponds to the present time.The tree is constructed outward into the future with branches emanating from the nodes.A branch emanating from a decision node corresponds to a decision alternative. It includes a cost or benefit value.A branch emanating from a state of nature node corresponds to a particular state of nature, and includes the probability of this state of nature.
40 BILL GALLEN DEVELOPMENT COMPANY B. G. D. plans to do a commercial development on a property.Relevant dataAsking price for the property is $300,000Construction cost is $500,000Selling price is approximated at $950,000Variance application costs $30,000 in fees and expensesThere is only 40% chance that the variance will be approved.If B. G. D. purchases the property and the variance is denied, the property can be sold for a net return of $260,000A three month option on the property costs $20,000 which will allow B.G.D. to apply for the variance.A consultant can be hired for $5000P (Consultant predicts approval | approval granted) = 0.70P (Consultant predicts denial | approval denied) = 0.90
41 What should BGD do? Construction of the Decision Tree Initially the company faces a decision about hiring the consultant.After this decision is made more decisions follow regardingApplication for the variance.Purchasing the option.Purchasing the property.
42 Let us consider the decision to not hire a consultant 11 3Do nothing2Buy land-300,0004Apply for variance-30,000Do not hire consultantHire consultant-50001Purchase option-20,000Let us consider the decision tonot hire a consultant11Apply for variance-30,000
43 Apply for varianceBuy land and-70,00010120,000867BuildSell950,000-500,000ApprovedDenied0.40.65260,000Sell9-300,000-500,000950,000131415Buy landBuildSell17-50,000100,00016ApprovedDenied0.40.612Purchase option andApply for variance
44 Let us consider the decision to hire a consultant This is where we are at this stage
45 Let us consider the decision to 2-5000212844373620Apply for variance-30,000Do not hire consultantLet us consider the decision tohire a consultant1935Do NothingBuy land-300,000Purchase option-20,0001Hire consultant-500018PredictApprovalDenial0.40.6
46 The consultant serves as a source for additional information -75,00027115,000252324BuildSell950,000-500,00022ApprovedDenied0.17650.8235?260,000Sell26Posterior Probability of approval | consultant predicts approval) =Posterior Probability of denial | consultant predicts approval) =Consultant predicts approvalThe consultant serves as a source for additional informationabout denial or approval of the variance.Therefore, at this point we need to calculate theposterior probabilities for the approval and denialof the variance application
47 Determining the Optimal Strategy The rest of the Decision Tree is built in a similar manner.Now,Work backward from the end of each branch.At a state of nature node, calculate the expected value of the node.At a decision node, the branch that has the highest ending node value is the optimal decision.The highest ending node value is the value for the decision node.Let us illustrate by evaluating one branch --Hire consultant and consultant predicts approval
48 Thus if consultant recommends approval -- BUY LAND 80500-22500115,000-75,000115,000-75,000115,000-75,000(115,000)(.8235)=(-75,000)(.1765)=115,000-75,000115,000-75,000-75,000115,000-2250080500-75,000115,000BuildSell80500-2250023242580500--500,000950,000Predict approvalApproved?0.823522Denied81483Sell0.17652627260,000$81,483 is the expected value if consultant predicts approval and land is boughtIn a similar manner, the expected value if consultant predicts approval and the option is purchased is $68,525The expected value if the consultant predicts approval and BGD does nothing is -$5,000 (the consultant’s fee).Thus if consultant recommends approval -- BUY LAND
49 Evaluation of Other Branches DO NOT HIRE CONSULTANTDo nothing EV = $0Buy land EV = $6,000Purchase option EV = $10,000 <==BESTHIRE CONSULTANT -- PREDICTS DENIALDo nothing EV = -$5,000 <===BESTBuy land EV = -$40,458Purchase option EV = -$27,730
50 BEST COURSE OF ACTION OPTIMAL STRATEGY -- EV(DO NOT HIRE CONSULTANT) = $10,000EV(HIRE CONSULTANT) =.4(81,483) + .6(-5000) = $29,593.20OPTIMAL STRATEGY --HIRE CONSULTANTIf consultant predicts approval -- buy the landIf consultant predicts denial -- do nothing
51 Game TheoryGame theory can be used to determine optimal decision in face of other decision making players.All the players are seeking to maximize their return.The payoff is based on the actions taken by all the decision making players.
52 Classification of Games Number of PlayersTwo players - ChessMultiplayer - More than two competitors (Poker)Total returnZero Sum - The amount won and amount lost by all competitors are equal (Poker among friends)Nonzero Sum -The amount won and the amount lost by all competitors are not equal (Poker In A Casino)Sequence of MovesSequential - Each player gets a play in a given sequence.Simultaneous - All players play simultaneously.
53 IGA SUPERMARKETThe town of Gold Beach is served by two supermarkets: IGA and Sentry.Market share can be influenced by their advertising policies.The manager of each supermarket must decide weekly which area of operations to discount and emphasize in the store’s newspaper flyer.
54 DataThe weekly percentage gain in market share for IGA, as a function of advertising emphasis.A gain in market share to IGA results in equivalent loss for Sentry, and vice versa (i.e. a zero sum game)
55 IGA needs to determine an advertising emphasis that will maximize its expected change in market shareregardless of Sentry’s action.
56 IGA’s Perspective - A Linear Programming model SOLUTIONIGA’s Perspective - A Linear Programming modelDecision variablesX1 = the probability IGA’s advertising focus is on meat.X 2 = the probability IGA’s advertising focus is on produce.X 3 = the probability IGA’s advertising focus is on groceries.Objective Function For IGAMaximize expected market change (in its own favor) regardless of Sentry’s advertising policy.Define the actual change in market share as V.
57 ConstraintsIGA’s market share increase for any given advertising focus selected by Sentry, must be at least V.The Model
58 Sentry’s Perspective - A Linear Programming model Decision variablesY1 = the probability that Sentry’s advertising focus is on meat.Y2 = the probability that Sentry’s advertising focus is on produce.Y3 = the probability that Sentry’s advertising focus is on groceries.Y4 = the probability that Sentry’s advertising focus is on bakery.Objective functionMinimize changes in market share in favor of IGAConstraintsSentry’s market share decrease for any given advertising focus selected by IGA, must not exceed V.